Document [original]

Computational modeling of

glutamate-induced calcium signal generation

and propagation in astrocytes

vorgelegt von

Master of Science

Franziska Oschmann

geb. in Aachen

von der Fakultät IV – Elektrotechnik und Informatik

der Technischen Universität Berlin

zur Erlangung des akademischen Grades

Doktor der Naturwissenschaften

- Dr. rer. nat. -

genehmigte Dissertation

Promotionsausschuss:

Vorsitzender: Prof. Dr. Henning Sprekeler

Gutachter: Prof. Dr. Klaus Obermayer

Gutachter: Prof. Dr. Christine Rose

Gutachter: Prof. Dr. Susanne Schreiber

Gutachter: Dr. Hugues Berry

Tag der wissenschaftlichen Aussprache: 26. Oktober 2018

Berlin 2018

Für meinen lieben Thomas, meine Eltern und ihre Ehepartner . . .

Acknowledgements

First of all I would like to thank Prof. Dr. Klaus Obermayer for his support and

supervision of my doctoral thesis. I am especially grateful that he gave me the

possibility to write a master’s thesis and later a doctoral thesis in a computational

field, although I had a mainly experimental background. I found this time very

instructive and highly enjoyed the work in his interdisciplinary Neural Information

Processing Group at Technische Universität Berlin.

Next, I would like to thank the other members of my committee: Prof. Dr. Chris-

tine Rose, Prof. Dr. Susanne Schreiber, Dr. Hugues Berry and Prof. Dr. Henning

Sprekeler, for the examination and evaluation of my doctoral thesis. Furthermore, I

would like to thank Prof. Dr. Christine Rose and Dr. Hugues Berry for their scientific

cooperation over the past years and for the discussions and suggestions concerning

my PhD project that were quite important to me.

I thank the Bernstein Center Berlin for their scientific and financial support, which

allowed me to invite speakers from the field and to attend international conferences.

Especially, I would like to thank Margret Franke and Robert Martin, who were sup-

portive throughout my time in Berlin.

I thank the whole NI-group very much, for just being the best group one could

imagine working in. I highly enjoyed being a part of the group and will miss our

daily lunch and coffee breaks, the summer and Christmas parties or just the small

chats on the hallway. Also I would like to thank Camilla for making everything run

smoothly in the group.

I would also like to thank my family, who supported me mentally during my studies

and my doctoral thesis. In particular, I would like to thank Thomas for bringing

me down to earth in stressful situations and for his support and encouragement

throughout the whole time.

Abstract

Since the 1990s researchers have shown that astrocytes generate calcium oscilla-

tions in response to neuronal activity and propagate them as intercellular calcium

waves over long distances. Moreover, astrocytes release transmitters in a calcium-

dependent manner and by that signal to neurons. These discoveries have made

astrocytes and especially calcium signal generation and propagation in astrocytes

an important research area in the neuroscience field. However, although the impact

of astrocytes at single synapses is well understood, the functional role of astrocytes

in neuronal networks is not captured yet. Therefore, it is of high importance to fully

understand the generation and propagation of calcium signals, in order to predict

the behavior of neuron-astrocyte networks. Coupled with that, the development of

computational models has become an important method in the analysis and pre-

diction of astrocytic calcium dynamics.

In the first part of my thesis, I develop a computational model reproducing the

calcium signal generation at different positions along a subcellular compartment of

the astrocyte, the astrocytic process. The novelty of my approach is the considera-

tion of two interacting mechanisms for the generation of astrocytic calcium signals,

namely the calcium entry from the extracellular space and the calcium release from

internal stores. In addition, I apply parameters defining the astrocyte morphology

in order to predict the calcium signal generation at different positions across the

astrocyte. With this model I show that 1) seemingly there is a spatial separation of

these two calcium signal generation mechanisms across the astrocyte, and 2) a high

activity of both mechanisms evokes a depletion of the internal calcium store and

the suppression of intracellular calcium oscillations.

In the second part of my thesis, I develop a reduced model for calcium signal gen-

eration in astrocytes and perform a stability analysis of this reduced model. The

model reduction is based on the separation of time-scales of the dynamical variables

and the subsequent derivation and application of the time-independent solutions

of the fast-reacting variables. The stability analysis of the reduced system revealed

that 1) the fixed-points of all dynamical variables are independent of those two pa-

rameters determining the impact of either the calcium release from internal stores

or the calcium entry from the extracellular space and are solely determined by the

extracellular stimulation, 2) the stabilities of all fixed points, however, are deter-

mined by these two parameters, and 3) the eigenvalues of the fixed points predict

that the in part 1 observed depletion of internal calcium stores can be prevented by

an increased transport of calcium into internal stores.

In the third part of my thesis, I study the propagation of calcium signals along

astrocytic outgrowths, astrocytic processes, with the help of a multi-compartment

model. I derive the multi-compartment model by the diffusive coupling of single

point-models of astrocytes which I introduce in the first part of the thesis. With the

help of this spatial model I discover, that there is a strong interaction between the

sodium and the calcium signal propagation, and that the sodium signal carries the

calcium signal in astrocytic regions devoid of the internal calcium store.

In summary, this thesis demonstrates the high benefit of computational modeling

in the investigation of calcium dynamics in astrocytes and contributes to a better

understanding of calcium signal generation and propagation in astrocytes.

Zusammenfassung

Seit den 1990er Jahren haben Forscher gezeigt, dass Astrozyten als Reaktion auf

neuronale Aktivität Calciumschwingungen erzeugen und diese als interzelluläre

Calciumwellen über weite Strecken weiterleiten. Darüber hinaus schütten As-

trozyten Transmitter in Abhängigkeit von den generierten Calciumsignalen aus

und beeinflussen damit Neuronen. Diese Entdeckungen haben Astrozyten und

insbesondere die Erzeugung und Vermehrung von Calciumsignalen in Astrozyten

zu einem wichtigen Forschungsgebiet der Neurowissenschaften gemacht. Obwohl

der Einfluss von Astrozyten auf einzelne Synapsen gut erforscht ist, ist die funk-

tionelle Rolle von Astrozyten in neuronalen Netzwerken noch unklar. Daher ist

es von großer Bedeutung, die Erzeugung und Ausbreitung von Calciumsignalen

vollständig zu verstehen, um das Verhalten von Neuronen-Astrozyten-Netzwerken

vorherzusagen. Damit verbunden ist die Entwicklung von mathematischen Mod-

ellen zu einer wichtigen Methode zur Analyse und Vorhersage der Calciumdynamik

in Astrozyten geworden.

Im erstenTeil meiner Doktorarbeit entwickle ich ein mathematisches Modell, das die

Calciumsignalerzeugung an verschiedenen Positionen entlang eines subzellulären

Kompartiments des Astrozyten, dem astrozytischen Prozess, reproduziert. Die

Neuheit meines Ansatzes ist die Berücksichtigung zweier interagierender Mecha-

nismen zur Erzeugung der Calciumsignale in Astrozyten, nämlich der Calciumein-

strom aus dem extrazellulären Raum und die Calciumausschüttung aus den inneren

Calciumspeichern. Zusätzlich wende ich Parameter an, die die Astrozytenmor-

phologie definieren, wie der Volumenanteil des internen Calciumspeichers oder das

Oberflächen-Volumen-Verhältnis des Astrozyten. Mit diesem Modell zeige ich, dass

1) scheinbar eine räumliche Trennung dieser beiden Erzeugungsmechanismen für

das Calciumsignal entlang des astrozytischen Prozesses stattfindet, und 2) ein ho-

her Calciumeinstrom aus dem extrazellulären Raum eine Erschöpfung des inneren

Calciumspeichers und die Unterdrückung intrazellulärer Calciumschwingungen

hervorruft.

Im zweiten Teil meiner Doktorarbeit entwickle ich ein reduziertes Modell für die

Generierung von Calciumsignalen in Astrozyten und führe eine Stabilitätsanalyse

dieses reduzierten Modells durch. Die Modellreduktion basiert auf der Trennung

der Zeitskalen der dynamischen Variablen und der anschließenden Berechnung

und Anwendung der zeitunabhängigen Lösungen der schnell reagierenden Vari-

ablen. Die Stabilitätsanalyse des reduzierten Systems ergab, dass 1) die Fixpunkte

aller dynamischen Variablen unabhängig von den beiden Parametern sind, die den

Einfluss entweder der Calciumausschüttung aus internen Speichern oder des Cal-

ciumeintrags aus dem extrazellulären Raum bestimmen und ausschließlich durch

die extrazelluläre Stimulation bestimmt werden, 2) die Stabilitäten aller Fixpunkte

jedoch durch diese beiden Parameter bestimmt werden, und 3) die Eigenwerte der

Fixpunkte vorhersagen, dass die in Teil 1 beobachtete Erschöpfung der internen

Calciumspeicher durch einen erhöhten Rücktransport von Calcium in interne Spe-

icher verhindert werden kann.

Im dritten Teil meiner Doktorarbeit untersuche ich die Ausbreitung von Calcium-

signalen in astrozytären Prozessen mit Hilfe eines Mehrkammermodells. Ich leite

das Mehrkammermodell durch die diffusive Kopplung der Punktmodelle der As-

trozyten ab, die ich im ersten Teil der Arbeit vorgestellt habe. Mit Hilfe dieses

räumlichen Modells entdeckte ich, dass 1) die Wechselwirkung zwischen den bei-

den Erzeugungsmechanismen für Calciumsignale auch die Calciumausbreitung in

Abhängigkeit von der Stärke beider Mechanismen begünstigt oder behindert, und

2) das ausbreitende Natriumsignal in der Lage ist, das Calciumsignal in astrozytis-

chen Teilen ohne interne Calciumspeicher zu transportieren.

Zusammenfassend zeigt diese Arbeit den hohen Nutzen der mathematischen Mod-

ellierung bei der Untersuchung der Calciumdynamik in Astrozyten und trägt zu

einem besseren Verständnis der Erzeugung und Ausbreitung von Calciumsignalen

bei.

Contents

Introduction 1

1 Discovery of astrocytes . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Morphology of astrocytes . . . . . . . . . . . . . . . . . . . . . . . . . 2

3 Physiology of astrocytes . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3.1 Intracellular calcium dynamics . . . . . . . . . . . . . . . . . . 3

3.2 Intracellular sodium dynamics . . . . . . . . . . . . . . . . . . 5

3.3 Intracellular potassium dynamics . . . . . . . . . . . . . . . . 6

4 Computational models for calcium signal generation in astrocytes . . 7

5 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1 Computational modeling of astrocytic calcium dynamics

within a single compartment 9

Introduction 11

Model 13

1 Computational model for Ca2+ release from internal stores . . . . . . 13

1.1 Computational model for Ca2+ release from internal stores by

DePitta et al., 2009 . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.2 Changes of the model by De Pittà et al. (2009) . . . . . . . . . 16

2 Extension of the model of De Pittà et al. (2009) and introduction of

Na+, K+and V dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.1 Dynamics of the ion concentrations and the membrane voltage 18

3 Neuronal stimulation of the astrocyte compartment . . . . . . . . . . 22

4 Model parameter values . . . . . . . . . . . . . . . . . . . . . . . . . . 23

5 Computational methods . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Results 27

1 Ca2+ release from the internal Ca2+ store along the astrocytic process 27

2 Ca2+ entry from the extracellular space . . . . . . . . . . . . . . . . . . 28

2.1 Na+entry into the astrocyte . . . . . . . . . . . . . . . . . . . . 28

2.2 Ca2+ transport through the plasma membrane . . . . . . . . . 30

2.3 Impact of the glutamate transporter activity on the Ca2+ re-

sponse under synaptic stimulation . . . . . . . . . . . . . . . . 33

2.4 Interactionofthe mGluR-dependent andGluT-dependentpath-

way . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

Discussion 37

2 Model reduction 39

Introduction 41

Model 43

1 Model reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

1.1 Reduced model . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

1.2 Model behavior in response to parameter variations and sta-

bility analysis of the reduced model . . . . . . . . . . . . . . . 50

1.3 Initialization of the dynamical system . . . . . . . . . . . . . . 53

1.4 Computational methods . . . . . . . . . . . . . . . . . . . . . . 54

Results 55

1 Comparison of the full and the reduced model . . . . . . . . . . . . . 55

1.1 Analysis of the reduced model . . . . . . . . . . . . . . . . . . 56

1.2 Prediction of the model behavior based on the eigenvalues of

the Jacobian matrix . . . . . . . . . . . . . . . . . . . . . . . . . 59

Discussion 61

3 Multi-compartment model for the signal propagation 65

Introduction 67

Model 69

1 Astrocyte morphology . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

2 Multi-compartment model . . . . . . . . . . . . . . . . . . . . . . . . . 71

2.1 General form of the diffusion equation . . . . . . . . . . . . . . 71

2.2 Time- and space-dependent changes of Ca2+, Na+, K+and IP372

2.3 Time- and space-dependent changes of the membrane potential 73

3 Morphology of the multi-compartment model . . . . . . . . . . . . . 75

3.1 Connectivity matrix for open and sealed end condition . . . . 75

3.2 Connectivity matrix for a branching process . . . . . . . . . . 76

4 Model parameter values . . . . . . . . . . . . . . . . . . . . . . . . . . 77

Results 79

1 Na+diffusion in astrocytic processes . . . . . . . . . . . . . . . . . . . 79

2 Ca2+ signal propagation within an astrocytic process . . . . . . . . . . 86

3 Model morphologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

3.1 Branching of the astrocytic process . . . . . . . . . . . . . . . . 91

3.2 Open and sealed end of the astrocytic process . . . . . . . . . 92

4 Modeling of the perisynaptic astrocytic process . . . . . . . . . . . . . 94

Discussion 97

1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

List of Figures

0.1 Drawing of astrocytes by Cajal. . . . . . . . . . . . . . . . . . . . . . . 2

1.1 Geometry of a single astrocytic compartment. . . . . . . . . . . . . . . 16

1.2 The volume ratio of the endoplasmatic reticulum (ER) as a function

of the surface volume ratio (SVR). . . . . . . . . . . . . . . . . . . . . . 17

1.3 Ca2+ signal generation in astrocytes. . . . . . . . . . . . . . . . . . . . 19

1.4 Ca2+ concentration dynamics in the intracellular compartment dur-

ing synaptic activation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

1.5 Intracellular Na+concentration dynamics during a constant extracel-

lular glutamate concentration. . . . . . . . . . . . . . . . . . . . . . . . 29

1.6 Intracellular Ca2+ concentration dynamics for different values of the

maximal pump current of the Na+-Ca2+ exchanger. . . . . . . . . . . . 31

1.7 Ca2+ oscillation frequency and amplitude for different parameter

combinations of the volume fraction of the internal store and the

maximal pump currents of the membrane transporters. . . . . . . . . 32

1.8 Intracellular Ca2+ concentration dynamics under synaptic stimula-

tion for a blocked glutamate transporter (GluT) in comparison to the

control condition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

1.9 Increase of the intracellular Na+concentration during synaptic stim-

ulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

1.10 Behavior of the dynamical variables as a function of the maximal

pump current of the Na+-Ca2+ exchanger and the volume fraction of

the internal Ca2+ store. . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.1 The time-dependent behavior of the dynamical variables of the full

model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.2 The current strengths of the membrane transporters. . . . . . . . . . . 44

2.3 The Na+i, K+iand V dynamics as a function of the glutamate concen-

tration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.4 Calculated steady state values of K+(K+ss) and V (Vss) given the

computed steady state values of Na+(Na+ss). . . . . . . . . . . . . . . 46

2.5 The values of Na+ss and glutamate determine the values of K+ss and

Vss. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.6 The simulated and fitted steady state values Na+ss, K+ss and Vss as a

function of the applied glutamate concentration. . . . . . . . . . . . . 48

2.7 Comparison of a dynamic and a constant extracellular Ca2+ concen-

tration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

2.8 The oscillatory behaviors of the full and the reduced model. . . . . . 55

2.9 Stability behavior of the reduced model. . . . . . . . . . . . . . . . . . 57

2.10 The real parts of the eigenvalues mainly predict the transition be-

tween oscillatory and non-oscillatory model behavior. . . . . . . . . . 58

2.11 The eigenvalues of the Jacobian matrix predict the boundary between

the oscillatory and the non-oscillatory range of the reduced model. . 60

3.1 Scheme of the astrocyte morphology. . . . . . . . . . . . . . . . . . . . 69

3.2 Morphological parameters of the astrocytic process. . . . . . . . . . . 70

3.3 Scheme of a branching process. . . . . . . . . . . . . . . . . . . . . . . 76

3.4 Amplitude and pace of the Na+signal propagation within an astro-

cytic process. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

3.5 Amplitude and pace of the Na+signal propagation within an astro-

cytic process assuming the same tortuosity in the intra- and extracel-

lular space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

3.6 Time-dependent Na+signal propagation along the astrocytic process. 82

3.7 Space-dependent Na+signal propagation along the astrocytic process. 83

3.8 Effect of the tortuosity on the Na+diffusion in the intra- and extra-

cellular space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

3.9 Ca2+ signal propagation as a function of the IP3and Ca2+ diffusion

coefficients. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

3.10 Ca2+ signal propagation during a low activity of the Na+-Ca2+ ex-

changer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

3.11 Ca2+ signal propagation during a moderate activity of the Na+-Ca2+

exchanger. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

3.12 Ca2+ signal propagation during a high activity of the Na+-Ca2+ ex-

changer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

3.13 Na+signal propagation in a branching astrocytic process. . . . . . . . 91

3.14 Ca2+ and Na+signal propagation along an astrocytic process with

either a sealed or an open end. . . . . . . . . . . . . . . . . . . . . . . 93

3.15 Intracellular Na+signals drive the Ca2+ signal propagation at perisy-

naptic astrocytic processes. . . . . . . . . . . . . . . . . . . . . . . . . . 95

List of Tables

1.1 Initial values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1.2 Model parameters for the dynamics of h. . . . . . . . . . . . . . . . . 23

1.3 Model parameters for the Ca2+ currents at the endoplasmatic reticulum. 24

1.4 Model parameters for the production and degradation of IP3. . . . . 24

1.5 Model parameters for the membrane currents. . . . . . . . . . . . . . 25

1.6 Parameters for the Tsodyks and Markram model. . . . . . . . . . . . . 25

1.7 Physical constants used in the model. . . . . . . . . . . . . . . . . . . 25

2.1 Model parameters for the steady-state values of Na+i, K+iand V. . . . 47

3.1 Morphological parameters of astrocyte. . . . . . . . . . . . . . . . . . 71

3.2 Parameters for the propagation of Ca2+, Na+, IP3and K+. . . . . . . . 77

3.3 Morphological parameters of the multi-compartment model. . . . . . 77

Introduction

1 Discovery of astrocytes

The brain consists of two main cell types: neurons and glial cells. While the impor-

tant function of neurons was recognized early and intensively studied, the relevance

of glial cells was unknown for a long time. However, already in the 1850s Rudolf

Virchow discovered glial cells and described them as “substance. . . which lies be-

tween the proper nervous parts, holds them together and gives the whole its form in

a greater or lesser degree” (Virchow, 1858; Kettenmann and Verkhratsky, 2008). His

finding was followed by numerous reports about different types of glial cells dis-

covered in the brain like Müller cells or Bergmann glia. The black staining reaction

developed by Golgi allowed the detailed study of the glial cell morphology. Based

on his drawings Golgi suggested that glial cells form the metabolic link between

blood vessels and the brain parenchyma and by that take on a nutritive role. Finally,

Ramòn y Cajal developed the first specific staining method for astrocytes and by

that allowed the specific investigation of astroglia (Kettenmann and Verkhratsky,

2008).

Unlike neurons, however, astrocytes are electrically non-excitable and only display

small changes in their membrane potential in response to stimulations like current

injection (Kang et al., 1998; Nedergaard et al., 2003). This is why astrocytes were ini-

tially not considered as active partners in information transmission. Instead, it was

thought that astrocytes function as metabolic supporters of neurons and mediate

the glutamate synthesis for neurons (Norenberg and Martinez-Hernandez, 1979).

Only when the first researchers detected calcium signals in astrocytes in response to

neuronal activity, the view on astrocytes changed (Cornell-Bell et al., 1990; Charles

et al., 1991). Since then more and more evidence was collected supporting the ac-

tive role in information transmission of astrocytes like for example at the tripartite

synapse (Perea et al., 2009). The main assumption is that calcium is the key com-

ponent of the astrocyte physiology and plays an active role in the detection and

control of neuronal network activity.

In the following, I will give a short overview on the morphology and physiology

of astrocytes as well as on the mathematical models describing the calcium signal

generation in astrocytes.

Figure 0.1: Drawing of astrocytes by Cajal. a Drawing of astrocytes in the pyramidal layer of the

human hippocampus. Sublimated gold chloride method. bDifferent astrocytes surrounding neuronal

somas in the pyramidal layer of the human hippocampus. Sublimated gold chloride method. Figure

and caption adapted from (García-Marín et al., 2007)

2 Morphology of astrocytes

Basically, the appearance of astrocytes can be described as star-shaped. Astrocytes

have a soma and several outgrowths, called bigger branches (see Figure 0.1). These

branches split up into smaller and smaller branches called processes. The ends of

these processes are spatially adjacent to the neuronal synapses and are therefore

called perisynaptic astrocytic processes. Since these ends of the processes wrap

themselves around the synaptic cleft and are thus in direct contact with the neurons,

the synapse consists of three parts: the pre- and postsynaptic neuron as well as

the astrocyte. A further astrocytic subcellular compartment, the astrocytic endfeet,

ensheaths blood-vessels (Mathiisenet al., 2010; McCaslin etal., 2011)and contributes

to the generation and maintenance of the blood-brain barrier (Abbott et al., 2006).

The cellular morphology and anatomical location of the astrocytes can be divided

into two main groups: protoplasmic and fibrous astrocytes.

Protoplasmic astrocytes are found in gray matter. Characteristic for the morphology

of protoplasmic astrocytes is that they occupy a spherical volume. The soma gives

rise to several radially spreading stem branches which split into fine processes

(Kettenmann and Ransom, 2013, p. 38). These cell extensions occupy 50% of the

volume and even 80% of the surface of the astrocyte. This results in a high surface-

to-volume ratio which allows the astrocytes to contact as much neurons as possible

(Kettenmann and Ransom, 2013, p. 38).

Fibrous astrocytes are found in white matter. The somas of fibrous astrocytes are

organized in rows between the axon bundles and give rise to many long fiber-like

processes which are oriented in parallel to the axons. The processes of fibrous

astrocytes are longer than those of protoplasmic astrocytes. In mice, for example,

the length of fibrous astrocytes is 300 µm, while processes of protoplasmic astrocytes

are less than 50 µm long (Kettenmann and Ransom, 2013, p. 39).

Introduction 3

3 Physiology of astrocytes

Unlike neurons astrocytes do not generate action potentials, but exhibit changes

in their intracellular calcium concentration in response to transmitter release by

neurons (Cornell-Bell et al., 1990; Charles et al., 1991). This form of astrocyte ex-

citability not only reflects the integration of neuronal activity by astrocytes, but also

allows astrocytes to affect neuronal activity by the release of transmitters into the

synaptic cleft (Pasti et al., 2001; Henneberger et al., 2010; Sahlender et al., 2014).

Besides elevations in the calcium concentration, also sodium and potassium signals

have been found to dependent on ion and transmitter changes in the extracellular

space evoked by neuronal activity (Kirischuk et al., 2007; Langer and Rose, 2009).

However, sodium and potassium signals differ substantially in their propagation

radius and their propagation mechanism to calcium signals, so that unlike calcium

they are not attributed to the integration of neural network activity.

Astrocytes express a wide variety of ion channels and transporters as well as trans-

mitter receptors, which allows them to sense neuronal activity (for reviews see

Verkhratsky et al. (1998) and Verkhratsky and Steinhäuser (2000)). In general, these

channels, transporters and receptors can be divided into two subgroups: those ac-

tivated by changes of the extracellular calcium, sodium or potassium concentration

on the one side, and those activated by changes of the extracellular neurotransmit-

ter concentration on the other side. In the following the dynamics of intracellular

calcium, sodium and potassium and the contributing channels, transporters and

receptors are described.

3.1 Intracellular calcium dynamics

Calcium signals in astrocytes are generated either by calcium release from the

internal calcium store (endoplasmatic reticulum) or by calcium entry from the ex-

tracellular space. Both the membrane of the internal calcium store and the plasma

membrane contain several families of calcium channels, which allow a flow of

calcium between the internal calcium store and the intracellular space as well as

between the intracellular space and the extracellular space.

The expression of metabotropic receptors, which are activated by the binding of

neurotransmitters, enables astrocytes to sense neuronal activity. Upon binding of

neurotransmitters to the metabotropic receptors, the second-messenger inositol-

triphosphate (IP3) is produced, which activates the calcium release from internal

stores (Verkhratsky et al., 1998; Agulhon et al., 2008).

Channels responsible for the release and uptake of calcium at the internal store

are for example the IP3-gated calcium channels/receptors (IP3R) (Streb et al., 1983;

Spät et al., 1986) or the SERCA pump. While the IP3-gated calcium channels are

mainly responsible for the calcium release from the internal store and thus for the

generation of cytoplasmic calcium signals, the SERCA pumps mediate the energy-

dependent calcium uptake into the internal store.

A link between the plasma membrane and the internal calcium store allows a store-

operated calcium entry into the astrocyte (Putney, 1986). A depletion of the internal

calcium store from releasable calcium activates the store-operated calcium entry into

the astrocyte through specific channels and the replenishment of the internal cal-

cium store (Parekh and Lewis, 2005).

The calcium entry from the extracellular space is mediated by several calcium chan-

nels located in the plasma membrane. Among others the calcium entry pathway

including the sodium-calcium exchanger is of particular interest since it can be

activated by an elevation of the extracellular glutamate concentration (Kirischuk

et al., 2007). The sodium-calcium exchanger can operate either in the forward or in

the reverse mode by exchanging intracellular calcium with extracellular sodium or

the other way round. The stoichiometry of that exchange is 3 sodium : 1 calcium.

The switch between both modes is regulated by the ion gradients of sodium and

calcium as well by the membrane potential (Goldman et al., 1994). For example,

an activation of astrocytic glutamate transporters or NMDA receptors induce the

accumulation of sodium within the astrocyte (Rose and Karus, 2013). Upon stim-

ulation with glutamate the glutamate transporter mediates the cotransport of one

glutamate molecule together with three sodium ions and the countertransport of

one potassium ion as well as one proton (Tzingounis and Wadiche, 2007). The gen-

erated sodium accumulation in turn activates the sodium-calcium exchanger and

gives rise to the transport of calcium into the cell (Rojas et al., 2007).

Recent experimental results suggest that the generation of intracellular calcium sig-

nals varies between subcellular compartments. With the discovery that the shape

and frequency of calcium signals in the astrocytic processes are significantly dif-

ferent from those in soma (Kanemaru et al., 2014), experimentalists studied the

location-dependent generation of calcium signals by knocking out or blocking re-

ceptors and transporters (Srinivasan et al., 2015). While a knockout of the IP3

receptor 2 (IP3R2) at the internal calcium store abolished the majority of somatic

calcium transients, it had a lower effect on calcium signals in the astrocytic processes

(Srinivasan et al., 2015; Stobart et al., 2016). Moreover, while the removal of extra-

cellular calcium had only a minor effect on calcium signals in the soma, it clearly

reduced the frequency of calcium transients in the astrocytic processes (Srinivasan

et al., 2015). From that it was concluded, that the main source for calcium signal

generation in the soma is calcium release from internal stores, while in the astro-

cytic processes calcium signals are generated by calcium entry from the extracellular

space (Bazargani and Attwell, 2016). This finding is also supported by the fact that

astrocytic regions close to the synapse, perisynaptic astrocytic processes, are devoid

of internal calcium stores. Thus, these regions do not allow calcium release from

internal stores Patrushev et al. (2013).

Astrocytic calcium signals are not a local event, but spread throughout the whole

astrocyte and even between neighboring astrocytes. The propagation of these cal-

cium waves, however, differ between subcellular compartments of the astrocyte.

On the one side, in astrocytic compartments which contain internal calcium stores,

the propagation of calcium waves is driven by the diffusion of calcium and of the

second messenger IP3, which then evokes an intracellular amplification of the cal-

cium signal by calcium release from internal stores (Golovina and Blaustein, 2000;

Scemes, 2000; Sheppard et al., 1997). On the other side, in thin astrocytic processes

the generation of strong calcium signals by transmembrane calcium transport and

Introduction 5

their propagation via diffusion is favored (Rusakov et al., 2011). Single-cell calcium

signals do not stop at the astrocyte cell border, but propagate into neighboring as-

trocytes and by that create an intercellular calcium wave. This intercellular spread

of calcium waves through gap junctions travels over long distances ( 300-400 µm)

and is able to excite up to hundreds of cells (Giaume and Venance, 1998; Scemes and

Giaume, 2006). The generation and maintenance of calcium waves is not achieved

by the diffusion of calcium itself but by the diffusion of IP3through gap junctions

and the subsequent release of calcium from internal stores (Scemes and Giaume,

2006).

3.2 Intracellular sodium dynamics

The role of sodium in astrocyte physiology consists of ion regulation and homeosta-

sis on the one side, and the generation of sodium signals in response to neuronal

activity on the other side (Rose and Karus, 2013). Although much evidence for both

aspects has been found in experiments, these two functions are nevertheless con-

tradictory. Sodium homeostasis provides a low intracellular sodium concentration

to maintain the ion gradient, and in contrast, the generation of sodium signals is

characterized by a strong increase of the intracellular sodium concentration (for a

review see: (Rose and Karus, 2013)).

The sodium homeostasis in astrocytes consists of maintaining a low intracellular

sodium concentration and a large inwardly directed concentration gradient. The

intracellular sodium concentration is around 15 mM and is therefore significantly

lower than the extracellular sodium concentration, which is 145 mM (Kirischuk

et al., 2012). Therefore, the transport of sodium against its concentration gradient

and thus out of the cell requires energy. This transport is largely handled by the

sodium-potassium pump, which exchanges three sodium ions with two potassium

ions (Kaplan, 2002). This strong inwardly directed sodium gradient generated by

the sodium-potassium pump supports the uptake of ions and transmitters by astro-

cytes. For example, the uptake of glutamate mediated by the glutamate transporter

is accompanied by the transport of three sodium ions into the astrocyte and one

potassium ion out of the astrocyte Danbolt (2001). Thus, the uptake of glutamate

produces an accumulation of sodium in the intracellular space (Chatton et al., 2000).

The close spatial association of the glutamate transporter and the sodium-potassium

pump, however, favors the glutamate uptake by astrocytes as the sodium-potassium

pump generates a strong inwardly directed sodium gradient and outwardly directed

potassium gradient (Rose et al., 2009).

In contrast to the energy-dependent maintenance of a low intracellular sodium

concentration is the generation of activity dependent sodium signals. Intracellular

sodium signals are induced by an enhanced neuronal activity and subsequent ac-

cumulation of neurotransmitters in the extracellular space (Rose, 2002). The uptake

of glutamate or GABA from the extracellular space by astrocytes is coupled to the

co-transport of sodium (Chatton et al., 2000, 2003).

Moreover, the neurotransmitter-dependent accumulation of sodium in the intracel-

lular space mediates calcium entry into the astrocyte. The rise of the intracellular

sodium concentration evoked by the uptake of, for example, glutamate or GABA

cause a switch of the sodium-calcium exchanger into the reverse mode such that

calcium is transported into the astrocyte and sodium is transported out of the as-

trocyte (Rojas et al., 2007; Doengi et al., 2009). This sodium-dependent pathway

is of particular interest since it also allows calcium signal generation in response

to neuronal transmitter release in astrocytic subcellular compartments which are

devoid of internal calcium stores (Patrushev et al., 2013).

Unlike the propagation of calcium signals, the propagation of sodium signals is

not affected by intracellular amplification mechanisms and is only determined by

sodium diffusion (Rose and Karus, 2013). The diffusion speed of sodium is about

60 µm/s in somatic regions (Langer et al., 2012), and probably even higher in astro-

cytic processes due to their elaborate morphology (Nedergaard et al., 2003). Within

astrocytic endfeet sodium signals spread with a maximum velocity of 120 µm/s

(Langer et al., 2016).

3.3 Intracellular potassium dynamics

Astrocytes express various types of potassium channels and transporters. These

channels and transporters contribute significantly to the membrane potential of

astrocytes, since the membrane potential is largely determined by the potassium

gradient (Kuffler et al., 1966). Thus, small changes of the extracellular potassium

concentration lead to an activation of the potassium transporters and channels as

well as a potassium uptake. Therefore, astrocytes play a major role in the buffering

of excess extracellular potassium, which is released during neuronal activity. At

rest, the extracellular potassium concentration is 3 mM. During intense neuronal ac-

tivity or epileptiform bursting the extracellular potassium concentration can rise up

to 5-15 mM (Somjen, 1979). According to the spatial buffering hypothesis, astrocytes

take up this excess potassium from the extracellular space, distribute it within their

astrocyte network and release the potassium at sites of lower extracellular potas-

sium concentration (Orkand et al., 1966). Channels and transporters attributed to

the uptake of excess extracellular potassium are the sodium-potassium-pump (Walz

and Hinks, 1986) or the inward rectifying potassium (Kir) channel (Orkand et al.,

1966).

The sodium-potassium-pump is responsible for the maintenance of the sodium

and potassium ion gradients across the plasma membrane. These ion gradients

are maintained by an exchange of intracellular sodium for extracellular potassium

with a stoichiometry 3:2. This resulting inwardly directed sodium gradient is used

by numerous channels and transporters to generate gradients for other ions such

as calcium or potassium (Kettenmann and Ransom, 2013, p. 190). The gluta-

mate uptake mediated by the glutamate transporter, for example, is mainly driven

by the inwardly directed sodium and the outwardly directed potassium gradient.

Moreover, the glutamate transporters are directly coupled to the sodium-potassium

pumps such that the sodium-potassium pumps can control the glutamate uptake

(Rose et al., 2009).

Introduction 7

4 Computational models for calcium signal generation in

astrocytes

The discovery of the generation of neuronal-activity dependent calcium signals in

astrocytes was accompanied with the development of computational models ana-

lyzing and predicting the signal generation in astrocytes (for a review see Oschmann

et al. (2017a) and Manninen et al. (2018)).

In the original, two-dimensional model calcium signals are evoked by only one

mechanism: the calcium- and IP3-dependent release of calcium from the internal

calcium store (Li and Rinzel, 1994). Here, solely the calcium dynamics in the inter-

nal calcium store and in the intracellular space are considered. This original model

constitutes the core mechanism and has been adopted in most subsequent models

for calcium signal generation in astrocytes (Höfer et al., 2002; Goto et al., 2004; De

Pittà et al., 2009).

The mathematical description of the IP3-dynamics in astrocytes is more heteroge-

neous. In the early models IP3is assumed to be a constant parameter (Dupont

and Goldbeter, 1993; Li and Rinzel, 1994). These models predict the generation

of calcium oscillations also for the application of constant IP3levels, despite the

fact the time-dependent dynamics of IP3is essential for the generation of calcium

oscillations. Later models apply a more complete description of the IP3dynamics

consisting of the agonist-, calcium- and IP3-dependent synthesis and degradation

of IP3(Goto et al., 2004; Nadkarni and Jung, 2007; De Pittà et al., 2009).

Some computational models also consider the calcium entry from the extracellular

space and by that assume a combination of calcium generation mechanisms. Ap-

plied mechanisms for the calcium entry are voltage-gated calcium channels (Postnov

et al., 2008) or store-operated calcium channels (Handy et al., 2017). Most of these

mechanisms are not solely driven by calcium, but also by other ions or the membrane

potential. Thus, a consideration of these mechanisms results in the consideration

of additional model variables.

The spread of calcium waves across astrocyte-networks was investigated by numer-

ous computational models. In general, the coupling of several astrocytes via gap

junctions allows the spread of calcium and IP3to neighboring astrocytes and by that

facilitates intercellular communication. For example, Ullah et al. (2006) showed that

the synchronization of intracellular calcium signals across several cells depends on

the coupling strength between the astrocytes. Here, the astrocyte-network only con-

sists of two cells. Subsequent studies, however, increased the number of considered

astrocytes (Kang and Othmer, 2009) or also included neurons (Amiri et al., 2013).

5 Motivation

Although already numerous models addressed the generation of astrocytic calcium

signals by calcium release from internal stores, most of them lack the additional

contribution of calcium entry from the extracellular space driven by neurotransmit-

ters like glutamate.

Schummers et al. (2008) showed the high dependence of the calcium signal in astro-

cytes on the activity of the glutamate transporter. They observed a strong attenua-

tion of the calcium signal during a block of the glutamate transporter. The glutamate

transporter couples the transport of glutamate and sodium and by that induces an

accumulation of sodium in the intracellular space. This accumulated sodium is able

to activate the sodium-calcium exchanger such that calcium is transported into the

astrocyte. By that the transport of glutamate is linked to the calcium transport into

the astrocyte. Thus, the glutamate-driven calcium entry via the sodium-calcium

exchanger could explain the neurotransmitter-dependent generation of calcium sig-

nals in astrocytic regions devoid of internal calcium stores. Moreover, this second

pathway also allows a strong interaction of propagating calcium and sodium sig-

nals. Thus, based on this mechanism the propagating sodium signal could drive

the calcium signal in astrocytic regions devoid of the internal calcium store.

In the first part of my thesis I focus on the development of a computational model

for calcium signal generation in astrocytes, which accounts for calcium release from

internal stores and calcium entry form the extracellular space both driven by an

elevated glutamate level in the extracellular space. Moreover, I include the volume

of the internal calcium store as a morphological parameter into the model in order

to scale the calcium release from internal stores. With this model I test whether the

sodium-calcium exchanger serves as a source for calcium signals in astrocytes and

how this second mechanisms affects the calcium release from internal stores.

In the second part of my thesis, I develop a reduced version of the model presented

in the first part, which allows an analytic analysis. Starting from that, l first investi-

gate, if the reduced model quantitatively and qualitatively reproduces the behavior

of the full model. In a second step I determine, if the analytic determined fixed

points and their stabilities are sufficient in order to predict the calcium dynamics

without performing a numerical integration.

In the third part of my thesis, I develop a multi-compartment model in order to

investigate the calcium signal propagation within the astrocyte. With the help of

that model I study how the propagating calcium and sodium signals interact with

each other.

Part 1

Computational modeling of

astrocytic calcium dynamics

within a single compartment

Introduction

The discovery that astrocytes integrate and process synaptic activity by the gener-

ation of intracellular calcium signals (Perea et al., 2009) was accompanied with the

development of numerous computational models reproducing and investigating

the calcium dynamics of astrocytes (for a review see Oschmann et al. (2017a) or

Manninen et al. (2018)). One of the first models designed for this purpose was

published by Li and Rinzel (1994) and accounts for calcium oscillations mediated

by the second-messanger inositol trisphosphate (IP3). Although this model was not

explicitly developed for astrocytes, it constitutes the core mechanisms for numerous

models describing astrocytic calcium dynamics (Postnov et al., 2008; Nadkarni and

Jung, 2007; De Pittà et al., 2009). The model published by De Pittà et al. (2009) stands

out from the other ones since it includes an explicit computation of the IP3concen-

tration and by that allows a link between the stimulus, the extracellular glutamate

concentration, and the intracellular oscillations of calcium as well as IP3.

Although experimental studies have demonstrated that calcium signals are shaped

by calcium release from internal stores as well as calcium entry from the extra-

cellular space, the majority of the computational models focuses on the former

mechanism. The former mechanism consists of the glutamate dependent produc-

tion of the second messanger IP3by metabotropic glutamate receptors (mGluRs)

and the subsequent IP3- and calcium-dependent exchange of calcium between the

intracellular space and the internal calcium store. The latter mechanisms describes

the calcium entry into the astrocyte via for example the sodium-calcium exchanger,

which is activated by the glutamate-driven sodium accumulation within the intra-

cellular space of astrocytes.

However, astrocytic calcium signals are not only shaped by different mechanisms,

but also by the weighting of these mechanisms, which varies between different sub-

cellular compartments of the astrocyte. These different subcellular compartments

are the soma and the astrocytic processes. While calcium release from internal stores

primary generates calcium signals in the soma, calcium entry from the extracellular

space forms the majority of calcium signals in astrocytic processes (Srinivasan et al.,

2015; Stobart et al., 2016; Bindocci et al., 2017). These findings are supported by the

fact that the volume fraction of internal calcium stores decreases along the astrocytic

process from the soma towards the synapse and perisynaptic astrocytic processes

are devoid of internal calcium stores (Patrushev et al., 2013).

Based on the above cited results I hypothesize that calcium signals in astrocytes are

generated by two mechanisms whose impact changes along the astrocytic process

Computational modeling of astrocytic calcium dynamics

within a single compartment

and in dependence on the volume fraction of the internal calcium store. Here, I pro-

pose a mathematical model which accounts for glutamate driven calcium signals in

astrocytes shaped by both calcium release from internal stores as well as calcium

entry from the extracellular space. Key element of the model is the consideration of

the volume fraction of the internal calcium store, which allows a model parametriza-

tion for different positions along the astrocytic process between the soma and the

synapse. With the help of the model I investigate the impact of the volume fraction

of the internal calcium store on the calcium signal generation mediated by both

mechanisms for different positions along the astrocytic process.

The results of this paper have been previously published (Oschmann et al., 2017b):

1 2.

1The authors contribution to the original article are (FO: Franziska Oschmann, KM: Konstantin

Mergenthaler, EJ: Evelyn Jungnickel, KO: Klaus Obermayer): Conceptualization: FO KM EJ KO.

Funding acquisition: FO KM KO. Investigation: FO KM EJ. Methodology: FO KM EJ KO. Software:

FO KM EJ. Writing – original draft: FO KO.

2The article is published under the Creative Commons Attribution License (CC BY 4.0).

Link: https://doi.org/10.1371/journal.pcbi.1005377

Model

The objective of this first part of my thesis is the development of a computational

model which accounts for (1)the generation of calcium (Ca2+) signals by Ca2+ release

from internal stores as well as Ca2+ entry from the extracellular space together with

(2) the varying weighting of these mechanisms in the different subcellular compart-

ments of the astrocyte like the soma or an perisynaptic astrocytic process. For the

purpose of addressing the former task, I extend a consisting model for Ca2+ release

from internal stores (De Pittà et al., 2009) with Ca2+ entry from the extracellular

space. I choose the model of De Pittà et al. (2009) as the basis of the model extension

since it allows a direct stimulation of the Ca2+ dynamics via glutamate. In order

to approach the latter problem, I include geometrical parameters into the model,

which allow a model parametrization for different subcellular compartments.

In the following, first, I introduce the computational model for Ca2+ release from

internal stores which serves as a basis for the model extension and, second, I extend

this model with the mechanism for Ca2+ entry from the extracellular space as well

as with the geometrical parameters which allow the investigation of Ca2+ signals in

different subcellular compartments.

1 Computational model for Ca2+ release from internal stores

1.1 Computational model for Ca2+ release from internal stores by DePitta

et al., 2009

De Pittà et al. (2009) developed a computational model for Ca2+ signal generation

in astrocytes, which accounts for Ca2+ release from internal stores evoked by the

stimulation with extracellular glutamate. The computational model describes the

temporal changes of the intracellular concentrations of Ca2+ (Ca2+i) and IP3(IP3i)

as well as the opening probability of receptor channels at the internal Ca2+ store

(h). Ca2+ currents crossing the outer membrane are neglected, such that the model

builds a closed system with a constant overall Ca2+ concentration in the intracellular

space and the internal Ca2+ store.

Three Ca2+ currents determine the exchange of Ca2+ between the internal Ca2+ store

and the intracellular space and by that shape the intracellular Ca2+ concentration.

These Ca2+ currents crossing the membrane of the internal Ca2+ store are: the

IP3receptor current, the SERCA pump and a calcium leak current. While the IP3

receptor current is sensitive for the concentrations of Ca2+ and IP3, the SERCA pump

Computational modeling of astrocytic calcium dynamics

within a single compartment

and the Ca2+ leak current solely depend on the concentrations of Ca2+ in the internal

store and in the intracellular space.

1.1.1 Dynamical variables

Intracellular Ca2+ concentration The intracellular Ca2+ concentration is defined

by the sum of all Ca2+ currents, which contribute to a change of the intracellular

Ca2+ concentration: dCa2+

dt IIP3R−ISerca +ICER leak.(1.1)

These Ca2+ currents flowing between the internal Ca2+ store and the intracellular

space are the SERCA pump (ISerca), a leak current (ICERleak) and the IP3receptor

current (IIP3R). Detailed mathematical descriptions of the named currents can be

found in Section 1.1.2.

Intracellular IP3concentration The concentration change of the second messen-

ger IP3is determined by the production and degradation of IP3. The production

is mediated by the phosphoinositide-specific phospholipase C β(PLCβ) and the

phosphoinositide-specific phospholipase C δ(PLCδ). The degradation is mediated

by the IP33-kinase (IP3-3K) and the inositol polyphosphate 5-phosphatase (IP-5P).

dIP3i

dt prodPLCβ+prodPLCδ−de grIP3-3K−degrIP-5P

prodPLCβvβ·g0.7

g0.7+(KR+Kp·Ca2+i

Ca2+i+Kπ)0.7

prodPLCδvδ

1+IP3i

κδ

·Ca2+2

Ca2+2

i+K2

PLCδ

degrIP3-3Kv3K·Ca2+4

Ca2+4

i+K4

D·IP3i

IP3i+K3

degrIP-5Pr5p·IP3i

(1.2)

The production of IP3by the phosphoinositide-specific phospholipase C (PLC) β

is linked to the level of the extracellular glutamate concentration g. The maximal

rate of IP3production by PLCβis described by vβand the glutamate affinity of

the receptor is set by KR. Kpis the Ca2+-/PLC-dependent inhibition factor and Kπ

determines the Ca2+ affinity of PLC.

The maximal rate of IP3production by PLCδis described by vδ. The activity of

PLCδis inhibited according to the inhibition constant kδ. The Ca2+ affinity of PLCδ

is set by KPLCδ.

The maximal degradation rate of IP3by IP3-3K is determined by v3K. KDis the Ca2+

affinity of IP3-3K and K3is the IP3affinity of IP3-3K.

The degradation of IP3through dephosphorylation by the inositol polyphosphate

5-phosphatase (IP-5P) depends on the maximal rate, r5P, of degradation by IP-5P.

Values of the model parameters can be found in Table 1.4.

Model 15

Activation of the IP3receptor channel The activation of the IP3receptor channel

is defined by h:

dt h∞−h

τh

,(1.3)

with: h∞Q2

Q2+Ca2+

,τh1

a2(Q2−Ca2+

i)and Q2d2

IP3i+d1

IP3i+d3.

Here, a2determines the IP3receptor binding rate for Ca2+ inhibition. The inactiva-

tion dissociation constants of Ca2+ and IP3are d2and d3, respectively.

Ca2+ concentration in the internal calcium store The Ca2+ concentration within

the internal Ca2+ store is defined by the change of the intracellular Ca2+ concen-

tration. Since De Pittà et al. (2009) neglected the flow of Ca2+ through the outer

membrane, the total free Ca2+ (Ca2+free) concentration remains constant and the

Ca2+ concentration in the internal Ca2+ store is defined by:

Ca2+

ER (Ca2+

f ree −Ca2+

i)/ratioER.(1.4)

Here, ratioER is the ratio between the volumes of the internal Ca2+ store and the

intracellular space.

1.1.2 Ca2+ currents at the internal calcium store

Calcium current through IP3receptor channels IP3receptor channels mediate

the Ca2+- and IP3-dependent Ca2+ transport from the internal Ca2+ store into the

intracellular space. The channels are build up by four subunits. Each subunit

consists of three binding sites: two Ca2+ binding sites and one IP3binding site. The

binding of Ca2+ and IP3to these binding sites determines the activity of the receptor

channel. While the binding of one Ca2+ ion and one IP3molecule opens the channel,

the binding of a second Ca2+ ion to the third binding site closes the channel:

IIP3RrC·(IP3i

IP3i+d1

)3·(Ca2+i

Ca2+i+d5

)3·h3·(Ca2+

ER −Ca2+

i).(1.5)

The IP3and Ca2+ binding affinities of the channels’ subunits are determined by d1

and d5, respectively. The maximal channel permeability is rC.

SERCA pump The SERCA pump mediates the Ca2+ transport from the intracel-

lular space into the internal Ca2+ store:

ISerca υer ·Ca2+2

Ca2+2

i+K2

.(1.6)

Here, the maximal rate of Ca2+ uptake by the SERCA pump is vER and KER deter-

mines the Ca2+ affinity of the SERCA pump.

Computational modeling of astrocytic calcium dynamics

within a single compartment

Figure 1.1: Geometry of a

single astrocytic compart-

ment. The geometry of a

single astrocytic compart-

ments is described by the

diameter of the intracellu-

lar space (dICS), the diam-

eter of the internal Ca2+

store (dER) and the length

of a single compartment

(h).

dER

dICS

Ca2+ leak from the ER The unspecific Ca2+ leak current follows the Ca2+ concen-

tration gradient between the internal Ca2+ store and the intracellular space multi-

plied with the maximal Ca2+ leakage from the internal Ca2+ store, rL:

ICER leak rL·(Ca2+

ER −Ca2+

i).(1.7)

1.2 Changes of the model by De Pittà et al. (2009)

With the intention to allow a model parametrization for different astrocytic subcel-

lular compartments, I change the model developed by De Pittà et al. (2009) regarding

the geometry of the considered astrocyte element as well as the computation of the

Ca2+ concentration within the intracellular space and the internal Ca2+ store.

Geometry A single astrocyte consists of numerous subcellular compartments: one

soma and several big branches which originate at the soma and split up into smaller

branches (processes). These subcellular compartments differ in their surface area,

their volume and in the volume fraction of the internal Ca2+ store. Thus, in order to

apply these geometric parameters to the model, I first define the surface area, the

volume and the volume fraction of the internal Ca2+ store for the different subcel-

lular compartments.

The outer shell of a single compartment of an extended astrocyte can be approxi-

mated by a cylinder. A second cylinder of same length and smaller diameter lying

within this compartment then defines the shape and the position of the internal

Ca2+ store (see Figure 1.1). Since the diffusion to neighboring compartments is

neglected the surface area of the compartment is defined by the shell surface of the

cylinder. The surface area (A) and the volume (Vol) of the astrocytic compartment

are described by:

AdICS ·π·hICS,

Vol (dICS

2)2·π·hICS.

Here, dICS and hICS are the diameter and the length of the intracellular space of the

astrocytic compartment, respectively.

Model 17

0 20 40

SVR [μm−1]

0μ00

0μ05

0μ10

0μ15

ratioER

Figure 1.2: The volume ratio of the endoplas-

matic reticulum (ER) as a function of the sur-

face volume ratio (SVR). The data has been

adapted from Patrushev et al. (2013). Figure

and caption adapted from (Oschmann et al.,

2017b).

The volume of the cylinder describing the internal Ca2+ store (VolER) is defined by

the multiplication of the volume of the astrocytic compartment with the volume

fraction of the internal Ca2+ store (ratioER):

VolER Vol ·ratioER.(1.8)

The volume of the intracellular space (VolICS) results from the difference between

the volumes of the astrocytic compartment and the internal Ca2+ store:

VolICS Vol −Vol ·ratioER Vol ·(1−ratioER).(1.9)

Since the cylinder describing the internal Ca2+ store only differs in its radius com-

pared to the cylinder describing the whole astrocytic compartment, the surface area

(AER) of the internal Ca2+ store is defined as follows:

AER A·pratioER.(1.10)

Along the astrocytic process variations of the surface volume ratio (SVR A

Vol )

and also the volume fraction of the internal Ca2+ store (ratioER) were observed

(Patrushev et al., 2013) (see Figure 1.2). Moreover, a dependency between both

parameters has been reported, which can be quantified by:

ratioER 0.15 ·e−(0.002µm·SVR)2.32 .(1.11)

In order to relate the dynamics of the intracellular Ca2+ concentration to the surface

area and the volume of an astrocytic compartment as well as the volume fraction

of internal Ca2+ store, I define the intracellular Ca2+ concentration by the sum of all

currents contributing to a change of the intracellular Ca2+ concentration multiplied

with the area of the internal Ca2+ store (AER) and divided by the volume of the

intracellular space (VolICS) and the Faraday constant (F):

dCa2+i

dt AER

F·VolICS ·(IIP3R−ISerca +ICER leak).(1.12)

Computational modeling of astrocytic calcium dynamics

within a single compartment

For the purpose of maintaining the unit of the differential equation of the intracel-

lular Ca2+ concentration, I adapt the unit of the Ca2+ currents. De Pittà et al. (2009)

defined the unit of the Ca2+ currents to be µM

sec . The changed differential equation

requires currents of the unit A

m2. Therefore, I multiply the rates of IIP3R, ISerca and

ICERleak with F·Vol

AER .

Ca2+ concentration in the internal Ca2+ store. In order to relate the Ca2+ concen-

tration in the internal Ca2+ store to the surface area and the volume of the internal

Ca2+ store, the change of this concentration was defined by the sum of all Ca2+ cur-

rents crossing the membrane of the internal Ca2+ store and divided by the volume

of the internal Ca2+ store (VolER) multiplied with the Faraday constant (F):

dCa2+

dt AER

F·VolER ·(−IIP3R+ISerca −ICER leak).(1.13)

2 Extension of the model of De Pittà et al. (2009) and intro-

duction of Na+, K+and V dynamics

In order to account for a mechanism for Ca2+ entry from the extracellular space,

I extend the model for Ca2+ release from internal stores by De Pittà et al. (2009)

with a mechanisms for Ca2+ entry from the extracellular space. The mechanism

for Ca2+ entry is performed by the interaction of the glutamate transporter (GluT),

the sodium-potassium pump (NKA), the sodium-calcium exchanger (NCX) and

a sodium (Na+) as well as a potassium (K+) leak current (see Figure 1.3). The

glutamate transporter mediates the transport of glutamate from the extracellular

space into the intracellular space. Together with each glutamate molecule three

Na+ions are transported into the astrocyte and one K+ion out of the astrocyte. The

elevated Na+concentration within the astrocyte activates the Na+-Ca2+ exchanger

in the reverse mode and gives rise to Ca2+ entry from the extracellular space. The

Na+-K+pump promotes the transport of three Na+ions out of the astrocyte and two

K+ions into the astrocyte. By that the pump produces the concentration gradient

which is necessary for the glutamate uptake by astrocytes mediated by the glutamate

transporter. Unspecific flows of Na+and K+through the membrane are covered by

a Na+and a K+leak current.

2.1 Dynamics of the ion concentrations and the membrane voltage

Dynamics of ion concentrations In general the change of the ion concentration is

given by the following equation:

dion

dt A

F·Vol ·XIion.(1.14)

It depends on the sum of all ionic currents carrying the respective ion (PIion) with

respect to the number of carried ions, multiplied with the area (A), the ionic cur-

rents are flowing through, and divided by the volume (Vol) of the space the ions

Model 19

Endoplasmatic reticulum

IIP3R

ISerca

ICERleak

mGluR

IGluT INCX INKA

IKleak

INaleak

[Ca2+]i

[IP3]i

[Na+]i

[K+]i

glu [Ca2+]o[Na+]o[K+]o

[Ca2+]ER

Ca2+ release from

internal store Ca2+ entry from extracellular space

Intracellular space

Extracellular space

Figure 1.3: Ca2+ signal generation in astrocytes. Astrocytic compartments consist of three parts:

the intracellular space, the internal Ca2+ store (endoplasmatic reticulum) and the extracellular space.

Ca2+ signals in the intracellular space are generated by two different pathways: Ca2+ release from

internal stores and Ca2+ entry form the extracellular space. The former mechanisms is driven by

the glutamate dependent production of IP3, which then evokes IP3and Ca2+ dependent exchange of

Ca2+ between the intracellular space and the internal Ca2+ store. The other mechanisms describes the

glutamate transporter driven transport of Ca2+ between the extracellular and the intracellular space.

Figure and caption adapted from (Oschmann et al., 2017b).

are located in as well as by the Faraday constant.

The change of the intracellular Ca2+ concentration is determined by currents cross-

ing either the membrane of the internal Ca2+ store (IIP3R,ISerca,ICER leak) or of the

outer cell membrane (INCX). Consequently, I change the differential equation of the

intracellular Ca2+ concentration to:

dCa2+i

dt A

F·VolICS ·INCX +AER

F·VolICS ·(IIP3R−ISerca +ICER leak).(1.15)

Aand AER denote the area of the outer cell membrane and of the internal Ca2+ store,

respectively. The volume of the intracellular space is defined by VolICS.

The change of the Ca2+ concentration in the internal Ca2+ store is solely determined

by currents crossing the membrane of the internal Ca2+ store:

dCa2+

dt AER

F·VolER ·(−IIP3R+ISerca −ICER leak),

here AER and VolER describe the area and the volume of the internal Ca2+ store,

respectively.

Computational modeling of astrocytic calcium dynamics

within a single compartment

The change of the intracellular Na+concentration is determined by all Na+currents

crossing the outer membrane:

dNa+

dt A

F·VolICS ·(3IGluT −3INKA −3INCX −INaleak ).(1.16)

The change of the intracellular K+concentrations is described in the same manner:

dK+

dt A

F·VolICS ·(−IGluT +2INKA −IKleak ).(1.17)

Dynamics of the membrane voltage The change of the membrane voltage V is

determined by:

dt −1

(−2IIP3R+2ISerca −2ICER leak +INCX−

2IGluT +INKA +INaleak +IKleak ).

(1.18)

The right hand side of the equation consists of the sum of all ionic currents crossing

either the membrane of the internal Ca2+ store or the outer cell membrane with the

consideration of carried charges per ion (see Figure 1.3). Here, it is important to

realize that the transport of Na+and K+mediated by the glutamate transporter lead

to a net transfer of two positive charges per cycle across the membrane. Cmdenotes

the membrane capacitance.

Extracellular ion concentrations In order to set the overall concentrations of Ca2+,

Na+and K+to constant levels, I define the extracellular concentrations by:

Ca2+

o−Ca2+

orest Ca2+

irest +Ca2+

ERrest −Ca2+

i−Ca2+

ER,(1.19)

Na+

o−Na+

orest Na+

irest −Na+

i,(1.20)

K+

o−K+

orest K+

irest −K+

i.(1.21)

The extracellular concentrations were defined as a function of the intracellular

concentrations. This definition of the extracellular concentration was based on the

assumption that the volume of the intracellular and the extracellular space of an

astrocytic compartment are the same as well as that the overall concentration in the

intracellular and the extracellular space of an astrocytic compartment stays constant.

Values of the model parameters can be found in Table 3.3.

Transmembrane Transporters

Glutamate Transporter The transport of glutamate mediated by the glutamate

transporter (GluT) is determined by:

IGluT IGluTmax ·K+

K+

i+KGluTmK ·Na+o3

Na+o3+KGluTmN

3·g

g+KGluTmg

.(1.22)

Model 21

Here, IGluTmax is the maximal transport current of the glutamate transporter. The

half saturation constants of Na+, K+and glutamate are given by KGluTmN, KGluTmK

and KGluTmg, respectively. The half saturation constant of K+is not known from

experimental results. Since the half saturation constant of Na+is close to its intra-

cellular resting concentration, I define the half saturation constant of K+in the same

manner.

The transport of glutamate is coupled to the cotransport of three Na+ions and

one H+ion and the countertransport of one K+ion (Tzingounis and Wadiche,

2007; Kanner and Bendahan, 1982). Additionally, glutamate transporters enable the

movement of anions across the membrane, which is not coupled to the transported

of glutamate (Wadiche et al., 1995). However, since the uncoupled anion current

has only a little effect on the transport of glutamate, it is neglected.

The intracellular glutamate concentration does not affect the glutamate uptake by

the glutamate transporter. Therefore, I do not consider the intracellular glutamate

concentration in the model.

Values of the model parameters can be found in Table 1.5.

Na+-K+-ATPase The transport of Na+and K+against its concentration gradient is

performed by the Na+-K+-ATPase (NKA). Here, I apply the mathematical expression

of Luo and Rudy (1994) in a simplified form:

INKA INKAmax ·Na+i1.5

Na+i1.5+K1.5

NKAmN ·K+

K+

o+KNKAmK

.(1.23)

INKAmax defines the maximal pumping activity of the Na+-K+pump. KNKAmN and

KNKAmK determine the half saturation constants of Na+and K+, respectively.

The Na+-K+-ATPase transports three Na+ions out of the cell and two K+ions into

the cell. Its pumping activity depends on the intracellular Na+concentration and

the extracellular K+concentration.

Values of the model parameters can be found in Table 1.5.

Na+-Ca2+ exchanger The Na+-Ca2+ exchanger (NCX) mediates the exchange of

three Na+ions with one Ca2+ ion. Here, I apply the mathematical description of the

Na+-Ca2+ exchanger developed by Luo and Rudy (1994):

INCX INCXmax ·Na+o3

KNCXmN3+Na+o3·Ca2+o

KNCXmC +Ca2+o·

Na+i3

Na+o3·exp(η·V·F

R·T)−Ca2+

Ca2+

o·exp(η−1·V·F

R·T)

1+ksat ·exp(η−1·V·F

R·T).

(1.24)

INCXmax is the maximal pump current of the exchanger. The half saturation constants

for Na+and Ca2+ are given by KNCXmN and KNCXmC, respectively. The position of

the energy barrier ηcontrols the voltage dependence. ksat is a saturation factor

ensuring saturation at large negative potentials.

Computational modeling of astrocytic calcium dynamics

within a single compartment

The exchanger works either in the forward or in the reverse mode. In the forward

mode Ca2+ is transported out of the astrocyte and Na+is transported into the

astrocyte. The reverse mode works the other way round. A switch into the reverse

mode is induced by an increased intracellular Na+concentration (Blaustein and

Santiago, 1977). The current strength of the Na+-Ca2+ exchanger depends on the

intra- and extracellular Na+and Ca2+ concentrations.

Values of the model parameters can be found in Table 1.5.

Leak currents The unspecific flow of Na+and K+ions across the membrane is

given by the leak currents:

INaleak gNaleak ·(V−ENa),(1.25)

IKleak gKleak ·(V−EK).(1.26)

Here, gNaleak and gKleak are the corresponding conductances of the Na+and K+

currents. The Nernst potentials of Na+and K+are given by ENa R·T

Flog(Na+

Na+

))

and EKR·T

Flog(K+

K+

)). Here, Ris the gas constant, Tis the temperature and Fis

the Faraday constant.

Values of the model parameters can be found in Table 1.5.

3 Neuronal stimulation of the astrocyte compartment

The release of neurotransmitters from an activated nearby synapse is calculated

using the Tsodykis and Makram model (Tsodyks and Markram, 1997; Fuhrmann

et al., 2002) in its for glutamate release adapted form published by Wallach et al.

(2014):

r(t)x(t)·y(t),

dt (1−x(t))

τrec −x(t)·y(t)·s(t),

dt −y(t)

τfacil

+U0(1−y(t)) ·s(t),

dt −g

τclear

+ρCGT·r(t).

The ratio of glutamate released during each spike is given by the product r(t)of

the fraction of recovered resources (x) and active resources (y). During each spike

a fraction of active synaptic resources (y) is released into the synaptic cleft and

increases with a step increase determined by U0. Between spikes the fraction of

these active synaptic resources (y) decays back to a baseline level with time constant

τfacil. At the same time the fraction of recovered resources (x) recovers to 1 with the

time constant τrec. The change of the glutamate concentration in the synaptic cleft

Model 23

is determined by the product of the total glutamate content of readily releasable

vesicles (GT) and of the volume ratio between the synaptic vesicles and the synaptic

cleft (ρC). Glutamate is removed from the synaptic cleft with the time constant

τclear.

Values of the model parameters can be found in Table 1.6.

4 Model parameter values

I determine the initial values of the intracellular IP3concentration, the fraction h

of active IP3receptor channels, and the Ca2+ in the internal store as well as the

model parameters gNaleak and gKleak in the same manner. For example, since the

model parameters for the production and degradation of IP3and the intracellular

resting concentration of Ca2+ are known from literature, the zero of dIP3i

dt reveals the

initial concentration of IP3. In the same way I calculate the initial ratio of activated

IP3receptor channels, h, and the initial concentration of the Ca2+ concentration in

the internal Ca2+ store. In this way a stable resting state is ensured. The model

parameter gNaleak is calculated by setting dNa+

dt equal to zero and solving the equation

for gNaleak. The model parameter gKleak is calculated the same way by setting dK+

equal to zero.

Table 1.1: Initial values.

Initial values of the

ion concentrations, the

membrane voltage, IP3

and the fraction of the

activated IP3receptor

channels, h. The calcu-

lation of Ca2+ER, IP3i and

h is explained above.

Parameter Value Source

Ca2+

i0.073 µM Reyes et al. (2012)

Ca2+

ER 19 µM see text

Ca2+

o1800 µM Luo and Rudy (1994)

Na+

i15 mM Østby et al. (2009)

Na+

o145 mM Østby et al. (2009)

K+

i100 mM Østby et al. (2009)

K+

o3 mM Østby et al. (2009)

V -85 mV McKhann et al. (1997)

IP3i0.15659 µM see text

h 0.7892 see text

Table 1.2: Model parameters

for the dynamics of h.

Parameter Value Source

a20.2 1

sDe Pittà et al. (2009)

d21.049 µM De Pittà et al. (2009)

d30.9434 µM De Pittà et al. (2009)

Computational modeling of astrocytic calcium dynamics

within a single compartment

Table 1.3: Model parameters

for the Ca2+ currents at the

endoplasmatic reticulum.

Parameter Value Source

IP3receptor channel

rC61

sDe Pittà et al. (2009)

d10.13 µM De Pittà et al. (2009)

d50.08234 µM De Pittà et al. (2009)

SERCA pump

vER 4µM

(Falcke et al., 1999; Ul-

lah et al., 2006)

KER 0.1 µM De Pittà et al. (2009)

Ca2+ leak

rL0.11 1

sDe Pittà et al. (2009)

Table 1.4: Model parameters

for the production and degra-

dation of IP3.

IP3production is medi-

ated by PLCβand PLCδ

and IP3degradation is

mediated by IP3- 3K and

IP - 5P.

Parameter Value Source

IP3production by PLCβ

vβ0.05 µM

sDe Pittà et al. (2009)

KR1.3 µM De Pittà et al. (2009)

Kp10 µM De Pittà et al. (2009)

Kπ0.6 µM De Pittà et al. (2009)

IP3production by PLCδ

vδ0.02 µM

sDe Pittà et al. (2009)

kδ1.5 µM De Pittà et al. (2009)

KPLCδ0.1 µM De Pittà et al. (2009)

IP3degradation by IP3- 3K

v3K 2µM

sDe Pittà et al. (2009)

KD0.7 µM De Pittà et al. (2009)

K31µM De Pittà et al. (2009)

IP3degradation by IP - 5P

r5P 0.04 1

sDe Pittà et al. (2009)

Model 25

Table 1.5: Model parameters

for the membrane currents.

The Glutamate Trans-

porter, the Na+/K+AT-

Pase, the Na+/Ca2+ ex-

changer and the leak

currents for Na+and

K+. The determination

of the model parameters

IGluTmax and INKAmax can

be found in the Results

Section. The definition

of the model parameter

KGluTmK can be found in

the Model Section. The

calculation of gNaleak and

gKleak is explained above.

Parameter Value Source

Glutamate Transporter

IGluTmax 0.75 pA

µm2see text

KGluTmN 15 mM Horak et al. (1990)

KGluTmK 5 mM see text

KGluTmg 34 µM Horak et al. (1990)

Na+/K+ATPase

INKAmax 1.52 pA

µm2see text

KNKAmN 10 mM Luo and Rudy (1994)

KNKAmK 1.5 mM Luo and Rudy (1994)

Na+/Ca2+ exchanger

INCXmax 0.1 pA

µm2see text

KNCXmN 87500 µM Luo and Rudy (1994)

KNCXmC 1380 µM Luo and Rudy (1994)

ksat 0.1 Luo and Rudy (1994)

η0.35 Luo and Rudy (1994)

Leak Currents

gNaleak 13.34 S

m2see text

gKleak 162.46 S

m2see text

Table 1.6: Parameters for

the Tsodyks and Markram

model.

Parameter Value Source

τfacil 2s−1Wallach et al. (2014)

τrec 1s−1Wallach et al. (2014)

τclear 60 s−1Wallach et al. (2014)

U00.25 Wallach et al. (2014)

ρC6.5 ·10−4Wallach et al. (2014)

Table 1.7: Physical constants

used in the model.

Parameter Value Description

F96 500 C

mol Faraday constant

R8.314 J

mol·KGas constant

T311 K Temperature

5 Computational methods

All simulations were performed with Python 2.7 using the packages Brian (Good-

man and Brette, 2008), NumPy and Matplotlib. The Brian Simulator used the Euler

integration as numerical integration method for the non-linear differential equa-

tions with time step dt=1ms.

Computational modeling of astrocytic calcium dynamics

within a single compartment

Results

1 Ca2+ release from the internal Ca2+ store along the astro-

cytic process

As a first step I analyze the Ca2+ release from the internal Ca2+ store and its depen-

dence on the considered position along the astrocytic process. A parametrization

for different positions along the astrocytic process is achieved by a variation of the

volume fraction of the internal Ca2+ store (ratioER) together with the surface volume

ratio (SVR) (see Model Section 1). The object of this first experiment is to investigate

solely the Ca2+signal generation evoked by Ca2+ release (see Figure 1.4). For this

purpose, I set all currents related to the Ca2+ entry from the extracellular space to

zero.

The decrease of the volume fraction of the internal Ca2+ store (ratioER) causes a

bifurcation of the dynamical system and thus a transition from oscillatory to non-

oscillation behavior (see Figure 1.4 b). Astrocytic compartments with a high volume

fraction of the internal Ca2+ store (ratioER >= 0.6) show long-lasting Ca2+ oscillations.

In this parameter range a reduction of ratioER causes a decrease of the oscillation

amplitude and an increase of the oscillation frequency (see Figure 1.4 c). A further

decrease of the volume fraction of the internal Ca2+ store (ratioER < 0.6) results in a

bifurcation. Volume fractions below 0.6 do not allow the generation of intracellular

Ca2+ oscillations. Instead the Ca+concentration increases to a steady level. How-

ever, if the astrocytic compartment is devoid of the internal Ca2+ store (ratioER = 0),

the Ca2+ concentration remains unchanged.

In summary, the consideration of Ca2+ release from internal stores in isolation

produces Ca2+ oscillations in astrocytic regions with a large volume fraction of

the internal Ca2+ store. Ca2+ elevations in astrocytic compartments devoid of the

internal Ca2+ store, however, can not be explained by this mechanisms.

Computational modeling of astrocytic calcium dynamics

within a single compartment

a b c

Figure 1.4: Dynamics of the Ca2+ concentration in the intracellular compartment during synaptic

activation. a Sample stimulus (spikes). The astrocytic compartment is stimulated for 200 seconds

with a Poisson spike train of 100 Hz. The corresponding glutamate concentration in the extracellular

compartment as a function of time is calculated using the Tsodyks-Markram model. bCa2+ifor

different values of the volume ratio (ratioER) between the internal Ca2+ store and the intracellular

compartments. The upper and lower lines for ratioER>0.06 denote the average height of peaks and

troughs of the emerging Ca2+ oscillations. For ratioER≤0.06 no Ca2+ oscillations are present and

the line denotes the average concentration of Ca2+ over the stimulation period. cFrequency of Ca2+

oscillations as a function of ratioER. Figure and caption adapted from (Oschmann et al., 2017b).

2 Ca2+ entry from the extracellular space

As a next step, I analyze the Ca2+ entry from the extracellular space and its effect on

the Ca2+ release from internal stores. The Ca2+ entry from the extracellular space is

mediated by the Na+-Ca2+ exchanger (NCX). The NCX can operate in the forward

as well as in the reverse mode. While in the forward mode Ca2+ is transported

out of the astrocyte and Na+into the astrocyte, the reverse mode operates the

other way round. A switch between these two modes is evoked by changes in the

concentrations of Ca2+ and Na+. For example an accumulation of intracellular Na+

leads to a switch into the reverse mode and thus to Ca2+ entry into the astrocyte.

Therefore, the intracellular Na+dynamic is crucial for the transport of Ca2+ into the

astrocyte. Theintracellular Na+dynamicis determined by theglutamate transporter

(GluT), the Na+-K+pump (NKA) as well as the Na+-Ca2+ exchanger (NCX). Thus,

as a first experiment I analyze the glutamate dependent increase of the intracellular

Na+concentration in dependence on the above named currents and compare the

results to experimental data. Once I find a weighting of the Na+currents reflecting

the experimental data, I investigate the Ca2+ entry into the astrocyte itself and its

effect on Ca2+ release from internal stores.

2.1 Na+entry into the astrocyte

In order to study the glutamate dependent increase of the intracellular Na+concen-

tration, I vary the maximal pump currents of the considered Na+currents (IGluTmax ,

INKAmax , INCXmax ) and analyze the effect of those variations on the intracellular Na+

concentration.

While the maximal pump currents of the glutamate transporter and the Na+-K+

pump have a strong effect on the intracellular Na+concentration, the effect of the

Na+-Ca2+ exchanger is negligible (see Figure 1.5). The maximal intracellular Na+

concentration (∆Na+) increases with an increase of the current strength of the glu-

Results 29

tamate transporter and a decrease of the current strength of the Na+-K+pump (see

Figure 1.5 a). Thus, the glutamate transporter is mainly responsible for the trans-

port of Na+into the astrocyte. The Na+-K+pump in turn counteracts the function

of the glutamate transporter, so that Na+is pumped out of the astrocyte and the

intracellular Na+concentration saturates at lower concentration levels.

Figure 1.5: Increase of the Na+concentration in the intracellular compartment, [Na+]i, during a

constant extracellular glutamate concentration for different values of the maximal pump currents

of the Na+-Ca2+ exchanger (INCXmax ), the glutamate transporter (IGluTmax ), the Na+/K+-ATPase

(INKAmax ). The astrocytic compartment is stimulated for 200 seconds with a constant extracellular

glutamate concentration of 100 µM. The surface volume ratio (SVR) is set equal to 1-1 µm, which

corresponds to astrocytic compartments close to the soma. a[Na+]iafter 200 seconds with respect

to its resting concentration ([Na+]rest = 15 mM, ∆Na+= [Na+]End —[Na+]rest) for a maximal pump

current of the Na+-Ca2+ exchanger (INCXmax ) equal to 0 A

m2(left) or equal to 1 A

m2(right) and different

values of the maximal pump current of the glutamate transporter (IGluTmax ) and the Na+/K+-ATPase

(INKAmax ). bTime until saturation is reached for a maximal pump current of the Na+-Ca2+ exchanger

(INCXmax ) equal to 0 A

m2(left) or equal to 1 A

m2(right) and different values of the maximal pump current

of the glutamate transporter (IGluTmax ) and the Na+/K+-ATPase (INKAmax ). The time to saturation is

defined as the time required for the intracellular Na+concentration to reach a constant concentration.

Figure and caption adapted from (Oschmann et al., 2017b).

The investigated parameters also determine the time until the Na+concentration

saturates (see Figure 1.5 b). Since a low activity of the glutamate transporter results

in a low increase of the intracellular Na+concentration, the saturation of the Na+

Computational modeling of astrocytic calcium dynamics

within a single compartment

concentration is reached comparably fast. At the same time a high activity of the

Na+-K+pump intensifies this effect, as the reached maximal Na+concentration is

even lower.

In experimental studies an increase of the intracellular Na+concentration between

10 mM and 20 mM in response to external glutamate stimulation was observed.

The maximal Na+concentration saturated with an increase of the applied gluta-

mate concentrations and reached its steady state level in under 60 seconds (Rose

and Karus, 2013). Based on the comparison of these experimental results and the

performed parameter exploration, I choose the following values for the maximal

pump currents of the glutamate transporter and the Na+-K+pump for the upcom-

ing simulations: IGluTmax = 0.75 A

m2and INKAmax = 1.52 A

m2.

In summary, the glutamate transporter promotes a high increase of the Na+con-

centration. The Na+-K+pump counteracts this effect and the Na+concentration

saturates at lower levels for a higher activity of the Na+-K+pump. The time until

the Na+increase saturates is in general favored by a high activity of the glutamate

transporter and a low activity of the Na+-K+pump.

2.2 Ca2+ transport through the plasma membrane

As a next step, I include Ca2+ entry from the extracellular space into the model

and study its effect on the Ca2+ release from the internal store. For this purpose

I vary the maximal pump currents of both the Na+-Ca2+ exchanger (INCXmax ) and

the glutamate transporter (IGluTmax ), as these currents modulate the Ca2+ entry from

the extracellular space. Moreover, in order to parameterize the model for different

positions along the astrocytic process, I study the impact of the Ca2+ entry on

the Ca2+ release for different values of the volume fraction of internal Ca2+ stores

(ratioER).

First, I investigate the impact of the Ca2+ entry from the extracellular space mediated

by the Na+-Ca2+ exchanger on the intracellular Ca2+ signal along the astrocytic

process (see Figure 1.6). During a blocked Ca2+ entry from the extracellular space

(INCXmax = 0 pA

µm2), Ca2+ oscillations solely occur for a high volume fraction of the

internal Ca2+ store (ratioER > 0.06) (see Figure 1.6 a, f and g). During an increase

of the maximal pump current of the Na+-Ca2+ exchanger this critical point for

the onset of Ca2+ oscillations shifts to higher values of the volume fraction of the

internal Ca2+ store (see Figure 1.6 a) and culminates in a complete suppression of

the Ca2+ oscillations (see Figure 1.6 e). Moreover, astrocytic compartments lacking

the internal Ca2+ store (ratioER = 0) show an increase of the intracellular Ca2+

concentration during an active Ca2+ transport (INCXmax > 0 pA

µm2) (see Figure 1.6 c and

b).

Second, I analyze the impact of the Na+transport into the astrocyte mediated by

the glutamate transporter on the intracellular Ca2+ signal. For this purpose, I vary

the maximal pump current of the glutamate transporter (IGluTmax) (see Figure 1.7).

The influence of the glutamate transporter activity on the Ca2+ signal is mainly

determined by the maximal pump current of the Na+-Ca2+ exchanger (INCXmax) and

the volume fraction of the internal Ca2+ store (ratioER). For example in astrocytic

Results 31

compartments close to the soma, an increase of the glutamate transporter activity

enhances the Ca2+ oscillation frequency up to a maximal value (see Figure 1.7 a and

b). A further increase of the glutamate transporter activity beyond this point leads to

a decrease of the oscillation frequency. An increase of the activity of the Na+-Ca2+

exchanger, however, lowers the maximal value of the oscillation frequency. The

cause of the suppressing effect of the Na+-Ca2+ exchanger on the Ca2+ oscillations

is investigated in the Results Section 2.4.

PAPs

ratioER = 0

soma

ratioER = 0.15

Figure 1.6: Dynamics of the Ca2+ concentration in the intracellular compartment during synaptic

activation for different values of the maximal pump current of the Na+-Ca2+ exchanger (INCXmax).

The astrocytic compartment is stimulated for 200 seconds with a Poisson spike train of 100 Hz.

The corresponding glutamate concentration in the extracellular compartment as a function of time is

calculated using the Tsodyks and Markram model. aCa2+ias a function of the volume ratio (ratioER) of

internal Ca2+ stores and the maximal pump current of the Na+-Ca2+ exchanger (INCXmax). The upper

and lower symbols denote the average height of peaks and troughs of the emerging Ca2+ oscillations

(in [µM]). In case no oscillations are present, the symbols denote the average concentration of Ca2+

over the stimulation period. b-d Time course of the Ca2+ concentration for ratioER = 0 and INCXmax

equal to 0 pA

µm2(blue), 0.01 pA

µm2(gray) and 1 pA

µm2(red). e-g Time course of the Ca2+ concentration for

ratioER = 0.15 and INCXmax equal to 0 pA

µm2(blue), 0.01 pA

µm2(gray) and 1 pA

µm2(red). Figure and caption

adapted from (Oschmann et al., 2017b).

Computational modeling of astrocytic calcium dynamics

within a single compartment

Figure 1.7: Ca2+ oscillation frequency and amplitude for different values of the volume ratio

between the internal Ca2+ store and the intracellular space (ratioER), as well as the maximal

pump currents of the Na+-Ca2+ exchanger (INCXmax ) and the glutamate transporter (IGluTmax ). The

astrocytic compartment is stimulated for 200 seconds with a Poisson spike train of 100 Hz. aCa2+

oscillation frequency for three different values of ratioER (0.12, 0.13 and 0.15), as a function of IGluTmax

and INCXmax . The colored lines correspond to INCXmax equal to 0.00001 pA

µm2(blue), 0.0001 pA

µm2(yellow),

0.001 pA

µm2(gray) and 0.01 pA

µm2(green). The dashed line corresponds to IGluTmax equal to 0.75 pA

µm2.b

Ca2+ oscillation frequencies for three different values of ratioER (0.12 0.13 and 0.15), as a function of

IGluTmax and INCXmax . The colored symbols denote the values of INCXmax shown in a.cCa2+ oscillation

amplitudes for three different values of ratioER (0.12, 0.13 and 0.15), and as a function of IGluTmax and

INCXmax . Figure and caption adapted from (Oschmann et al., 2017b).

Results 33

Moreover, the volume of the internal Ca2+ store determines the Ca2+ oscillation

amplitude (see Figure 1.7 c). An increase of the volume fraction of the internal Ca2+

store is accompanied with a decrease of the surface volume ratio and thus also with

the increase of the volumes of both the internal Ca2+ store and the intracellular

space. The increased volume of the internal Ca2+ stores allows an enhanced release

of Ca2+ from internal stores and thus also an increased oscillation amplitude.

In summary, taking the Ca2+ entry from the extracellular space into account shows

the expected Ca2+ fluctuations in astrocytic compartments devoid of the internal

Ca2+ store. However, with increasing activity of the Na+-Ca2+ exchanger Ca2+ oscil-

lations are suppressed, especially in astrocytic compartments with a high volume

fraction of the internal Ca2+ store. Moreover, an increase of the glutamate trans-

porter activity prevents a suppression of the Ca2+ oscillations.

2.3 Impact of the glutamate transporter activity on the Ca2+ response

under synaptic stimulation

During visual experiments in awake ferrets a clear attenuation of the astrocytic Ca2+

signal was observed when the glutamate transporter was blocked (Schummers et al.,

2008). Based on these experimental results I investigate the Ca2+ signal generation

during a blocked glutamate transporter.

In order to determine the influence of the glutamate transporter on the Ca2+ signal,

I stimulate the astrocyte model for an active and a blocked glutamate transporter

and I compute the difference between both curves. I perform this experiment for

various values of the activity of the Na+-Ca2+ exchanger activity(INCXmax ) as well as

the volume fraction of the internal Ca2+ store (ratioER). Figure 1.8 summarizes the

results of this experiment.

The impact of the glutamate transporter on the Ca2+ signal is highest for a high

activity of the Na+-Ca2+ exchanger (INCXmax > 0.1 pA

µm2) and a low volume fraction of

the internal Ca2+ store (ratio < 0.1) (see Figure 1.8 b). This impact of the glutamate

transporter on the Ca2+ signal decreases with a decrease of the Na+-Ca2+ exchanger

activity and an increase of the volume fraction of the internal Ca2+ store.

The impact of the glutamate transporter is highest in compartments with a low

volume fraction of the internal Ca2+ store (see Figure 1.8 b and e), because in

these compartments Ca2+ signals are mainly generated by Ca2+ entry from the

extracellular space (see Figure 1.4). A blocked glutamate transporter prevents the

accumulation of Na+in the intracellular space (see Figure 1.9) and a switch of the

Na+-Ca2+ exchanger into the reverse mode. Therefore, a block of the glutamate

transporter inhibits the Ca2+ entry mediated by the the Na+-Ca2+ exchanger into

the astrocyte and evokes an attenuation of the Ca2+ signals in astrocytic regions

with a small volume fraction of the internal Ca2+ store. An increase of the volume

fraction of the internal Ca2+ store allows Ca2+ release from internal stores and thus

also the impact of the glutamate transporter on the Ca2+ signal decreases (see Figure

1.8 b, c and d).

In summary, these simulation results suggest that a clear attenuation of the Ca2+

signal during a block of the glutamate transporter can primarily be observed in

Computational modeling of astrocytic calcium dynamics

within a single compartment

astrocytic compartments with a low volume fraction of the internal Ca2+ store.

> 80%

40 -

80%

< 40%

96%

60%

23%

PAPs soma

Figure 1.8: Dynamics of the Ca2+ concentration in the intracellular compartment under synaptic

stimulation for a blocked glutamate transporter (GluT) in comparison to the control condition. a

The astrocytic compartment is stimulated for 10 seconds with a Poisson spike train of 10 Hz. The

corresponding glutamate concentration in the extracellular compartment as a function of time is

calculated using the Tsodyks and Markram model. bReduction of the Ca2+ response under block

of the GluT as a function of the maximal pump current of the Na+-Ca2+ exchanger (INCXmax) and

the volume ratio (ratioER) between the internal Ca2+ store and the intracellular compartment. The

reduction was quantified by the difference of the average Ca2+ concentration under control condition

and block normalized by the difference between the Ca2+ concentration in the control condition and

the Ca2+ concentration without stimulation. Solid lines separate the parameter space concerning the

reduction: larger than 80%, between 40% and 80% and under 40%. c-e Ca2+ response as a function of

time for different values of INCXmax and ratioER. These traces correspond to a reduction of the Ca2+

signal of 29 %, 67 % and 97 %. Solid and dashed lines correspond to control condition and block. The

block was simulated by setting IGluTmax equal to 0 pA

µm2. Figure and caption adapted from (Oschmann

et al., 2017b).

Results 35

a b c d

Figure 1.9: Increase of the intracellular Na+concentration during synaptic stimulation. Dynamics

of the Na+concentration in the intracellular compartment under synaptic stimulation for a blocked

glutamate transporter in comparison to the control condition. The astrocytic compartment is stimu-

lated for 10 seconds with a Poisson spike train of 10 Hz (see Figure 1.8). aTime course of the glutamate

concentration in the extracellular compartment calculated with the Tsodyks and Markram model. b-d

Time course of the intracellular Na+concentration for three different parameter combinations of the

volume ratio between the internal Ca2+ store and the intracellular space (ratioER) and the maximal

pump current of the Na+-Ca2+ exchanger (INCXmax ). The intracellular Na+concentration is shown for

the same parameter combinations of INCXmax and ratioER as Ca2+ in Figure 1.8 c, d and e (b: ratioER

= 0.14 and INCXmax = 0.1 pA

µm2,c: ratioER = 0.12 and INCXmax = 0.4 pA

µm2,d: ratioER = 0.03 and INCXmax =

0.5 pA

µm2). Solid and dashed lines corresponds to the control condition and a block of the glutamate

transporter, respectively. Figure and caption adapted from (Oschmann et al., 2017b).

2.4 Interaction of the mGluR-dependent and GluT-dependent pathway

Finally, I study the mechanisms underlying the interaction of Ca2+ release from

internal stores and Ca2+ entry from the extracellular space (see Figure 1.10). For

that purpose I change the impact of both pathways by varying the volume ratio

of internal Ca2+ stores (ratioER) and the maximal pump current of the Na+-Ca2+

exchanger (INCXmax ) and study the Ca2+ level in all three spaces.

First, I analyze the impact of Ca2+ entry on the intracellular Ca2+ concentration

in isolation and neglect Ca2+ release from internal Ca2+ stores (see Figure 1.10 a).

During external stimulation with glutamate the Na+-Ca2+ exchanger produces Ca2+

entry into the cell. In this case the steady state of the intracellular Ca2+ concentration

is independent of the maximal pump current of the Na+-Ca2+ exchanger.

During active Ca2+ release from internal stores and Ca2+ entry from the extracellular

space a high maximal pump current of the Na+-Ca2+ exchanger causes a suppression

of Ca2+ oscillations in all three spaces (see Figure 1.10 d, e and f). The same

oscillatory behavior is also reflected in the dynamics of IP3and h. When comparing

the Ca2+ levels in all three spaces during a high activity of the Na+-Ca2+ exchanger

(INCXmax = 1 A

m2) to the resting concentrations, I observe a strong deviation of these

levels from the resting concentrations. At rest the concentrations of Ca2+ in the

three spaces are: Ca2+i= 0.073 µM, Ca2+o= 1800 µM and Ca2+ER = 19 µM. A

strong maximal pump current of the Na+-Ca2+ exchanger decreases the the Ca2+

concentration in the internal Ca2+ store and increases the Ca2+ concentration in the

intracellular and the extracellular space compared to their resting concentrations

(see Table 3.3). Thus, a strong maximal pump current leads to a depletion of the

internal Ca2+ store and by that prevents the generation of Ca2+ oscillations.

Computational modeling of astrocytic calcium dynamics

within a single compartment

PAPs soma PAPs soma

INCXmax = 0 A

m2INCXmax = 0.001 A

m2INCXmax = 1 A

Figure 1.10: Ca2+ concentrations in the intracellular space, the internal Ca2+ store and the extracel-

lular space, the IP3concentration in the intracellular space and the ratio h of active IP3receptor

channels as a function of the maximal pump current of the Na+-Ca2+ exchanger (INCXmax) and the

volume ratio between the internal Ca2+ store and the intracellular space (ratioER). The astrocytic

compartment is stimulated for 200 seconds with a Poisson spike train of 100 Hz. The corresponding

glutamate concentration in the extracellular compartment as a function of time is calculated using

the Tsodyks and Markram model. Blue, gray and red lines denote the dynamics of [IP3]i, h [Ca2+]i,

[Ca2+]ER, [Ca2+]ofor values of INCXmax equal to 0 pA

µm2, 0.001 pA

µm2and 1 pA

µm2.aAnalysis of the Ca2+

entry from the extracellular space in isolation. Ca2+iis shown as a function of ratioER and for different

values of INCXmax.b-f Analysis of both pathways. IP3i, h, Ca2+i, Ca2+ER and Ca2+oare shown as a

function of ratioER and for different values of INCXmax. Figure and caption adapted from (Oschmann

et al., 2017b).

Discussion

Within this first part of my thesis I developed and studied a computational model for

calcium signal generation along an astrocytic process. The novelty of this compu-

tational model is, the consideration of two different pathways for the generation of

calcium signals in astrocytes, and the parametrization for different positions along

an astrocytic process by a change of the volume fraction of the internal calcium

store.

The analysis of the computational model revealed four main findings: 1) during

the isolated consideration of the pathway for calcium release from internal stores

calcium oscillations were solely observed for model parametrizations with a high

volume fraction of the internal calcium store, 2) during the additional consideration

of the pathway for calcium entry from the extracellular space I observed that a

high maximal pump current of the sodium-calcium exchanger suppresses calcium

oscillations and that the sodium-calcium exchanger generates calcium fluctuations

in astrocytic compartments devoid of the internal calcium store, 3) the impact of

the calcium entry from the extracellular space on the calcium signal generation was

highest for a high activity of the sodium-calcium exchanger and a low volume frac-

tion of the internal calcium store, 4) the suppression of calcium oscillations during a

high activity of the sodium-calcium exchanger was caused by an increased efflux of

calcium from the internal calcium store into the intracellular as well as extracellular

space.

In their study Srinivasan et al. (2015) proposed a spatial separation of calcium sig-

nal generation pathways along the astrocytic process. During their experiments

they observed that in astrocytic compartments close to the soma calcium signals

are mainly induced by calcium release from internal stores through the IP3recep-

tor channel. However, a knock-out of this receptor channel did not lead to a clear

reduction of calcium signals in astrocytic compartments close the synapse. These

experimental results are in great agreement with my simulation results. The con-

sideration of the pathway for calcium release from internal stores in isolation caused

long-lasting oscillations for model parametrizations accounting for a high volume

fraction of the internal calcium store. For for model parametrizations with an absent

internal calcium store the intracellular calcium concentration remained unchanged.

Moreover, Srinivasan et al. (2015) showed that in astrocytic compartments close to

the synapse a significant proportion of calcium signals is evoked by transmembrane

calcium fluxes. Also these experimental results can be reproduced with my model,

as the consideration of a transmembrane calcium current allows the generation of

Computational modeling of astrocytic calcium dynamics

within a single compartment

calcium signals in astrocytic compartments devoid of an internal calcium store.

However, the presented simulation results also showed that a high activity of the

sodium-calcium exchanger suppressed the calcium oscillations generated by cal-

cium release from internal stores. In contrast, a high activity of the sodium-calcium

exchanger favored the impact of calcium entry from the extracellular space on

the calcium signal generation in astrocytic compartments with a low volume of

the internal calcium store. Based on these simulation results I propose that the

transporter activity of the sodium-calcium exchanger varies along the astrocytic

process with a low activity in astrocytic compartments close to the soma and a high

activity in astrocytic compartments close to the synapse. This hypothesis is also sup-

ported by experimental results, which report a concentration and colocalization of

sodium-calcium exchangers, sodium-potassium pumps and glutamate transporters

in perisynaptic astrocytic processes (Minelli et al., 2007; Danbolt, 2001).

The above named findings also hint to different functions of astrocytic processes

and the soma regarding the sense and integration of neuronal activity. For example,

the high surface volume ratio of perisynaptic astrocytic processes support localized

accumulations of sodium and calcium within the intracellular space (Rusakov et al.,

2011), which enables these astrocytic compartments to sense individual synaptic

events. Store-operated calcium signals, however, respond to larger elevations of

calcium or IP3within the intracellular space, which suggests that these astrocytic

compartments act as integrators of local network activity (Patrushev et al., 2013).

Within this first part of the thesis I developed and studied a computational model

for calcium signal generation in astrocytes. Moreover, I discussed the roles of dif-

ferent calcium signal generation pathways and subcellular compartments in the

detection of synaptic events. However, as this model only describes the calcium

signal generation at one point within an extended astrocyte, the propagation of sig-

nals along the astrocytic process is not considered. For the purpose of investigating

calcium signal propagation, I also developed a multi-compartment model, which is

described and analyzed in part 3.

Part 2

Model reduction

Introduction

As already discussed within the previous part of my thesis, astrocytic calcium sig-

nals are produced by two mechanisms: calcium release from internal stores and

calcium entry from the extracellular space. While the calcium release from internal

stores is usually mediated by the calcium efflux through calcium- and IP3-sensitive

receptor channels at the internal calcium store, the calcium entry form the extra-

cellular space can be mediated by different pathways. These mechanisms either

include calcium transporters or calcium-permeable ion channels (Bazargani and

Attwell, 2016). Both calcium transporters and calcium-permeable ion channels are

indirectly or directly activated by neurotransmitters in the extracellular space. For

example glutamate release by neurons activates glutamate transporters and NMDA

receptors and promotes sodium elevations in the intracellular space of astrocytes

(Rose and Karus, 2013). The increased sodium concentration activates sodium-

calcium exchangers within the astrocytic membrane and gives rise to calcium entry

into astrocytes (Rojas et al., 2007). Calcium-permeable ion channels, however, are

directly activated by the binding of neurotransmitters. These ion channels are for

example AMPA, NMDA or P2X receptors, or TRPA1 channels (Hamilton et al.,

2008; Shigetomi et al., 2012). Thus, due to the high variety of mechanisms for either

calcium release from internal stores or calcium entry from the extracellular space,

calcium signal generation in astrocytes is a highly versatile process.

Due to the variety of calcium signal generation mechanisms in astrocytes, mathe-

matical models for calcium signal generation differ very much in their composition

of considered ion channels and receptors and are thus difficult to compare. Most

mathematical models for calcium signal generation in astrocytes focus on calcium

release from internal stores (Nadkarni and Jung, 2007; De Pittà et al., 2009) and

neglect the impact of calcium entry from the extracellular space. Recently, vari-

ous models extended existing models for calcium release from internal stores with

calcium entry from the extracellular space (Postnov et al., 2008; Oschmann et al.,

2017b; Handy et al., 2017). Thereby, different mechanisms for calcium entry are

applied like store operated calcium channels (Handy et al., 2017), voltage-gated

calcium channels (Postnov et al., 2008) or sodium-calcium exchangers (Oschmann

et al., 2017b). However, the above named models not only differ in their mecha-

nisms for calcium entry from the extracellular space, but also in their considered

mathematical description of calcium release from internal stores. In addition, most

mathematical models considering both mechanisms are rather complex and do not

allow a profound analytic analysis, which also hinders the handling of these mod-

42 Model reduction

els. Thus, a comparison of the various models and the model parametrization is

hardly possible.

On the basis of the above named limitations of current mathematical models con-

cerning their handling, their comparability and their informative value my aim is

to develop a reduced model for calcium signal generation in astrocytes. For that

purpose I reduce the former published model for calcium signal generation evoked

by both calcium release and calcium entry (see part 1 and Oschmann et al. (2017b)).

The derivation of the reduced model is based on a separation of time-scales of the

dynamical variables. The behavior of the derived reduced model and of the full

model coincide quantitatively and qualitatively. Moreover, the stability analysis

reveals important insights about the steady state behavior of the reduced model in

response to parameter variations.

Model

The subject of the following section is the derivation and analysis of a reduced

model for Ca2+ signal generation in astrocytes. For this purpose I reduce the full,

seven-dimensional model, which was introduced in the previous part (see part 1), to

a four-dimensional one. The model reduction is based on a separation of time-scales

of the dynamical variables. Finally, I outline how I perform the stability analysis of

this four-dimensional model.

1 Model reduction

Considering the simulation results of the full model it can easily be seen that the

model variables show oscillations with either a high or a very low amplitude (see

Figure 2.1). Whereas the model variables Ca2+i, Ca2+ER, IP3and h show long lasting

oscillations with an amplitude many times higher than their resting concentration,

the model variables Na+i, K+iand V show small oscillations in the nM and nV range.

0 100

Time [sec]

0.25

0.50

0.75

Ca2 +

i[μM]

0 100

Time [sec]

0.2

0.4

0.6

IPμ[μM]

0 100

Time [sec]

0.6

0.7

0.8

0 100

Time [sec]

Ca2 +

ER [μM]

0 100

Time [sec]

Na+

i[mM]

60 120

0.004

0.007 +2.121e1

0 100

Time [sec]

100

K+

i[mM]

60 120

9μ.786

9μ.790

0 100

Time [sec]

−80

−70

−60

V[mV]

60 120

−5μ.25

−5μ.2μ

Figure 2.1: The time-dependent behavior of the dynamical variables of the full model. The full

model is stimulated with a constant glutamate concentration of 0.1 mM. The oscillations of Na+i, K+i

and V are displayed in the image magnification due to their small amplitude. The volume fraction of

the internal Ca2+ store (ratioER) is set to 0.15.

The mechanisms at the internal calcium store (including IIP3R, ISerca, ICERleak)

and at the plasma membrane (including IGluT, INKA, INCX, leak currents) act almost

44 Model reduction

independent of each other. The only similarity is their dependence on Ca2+i. While

changes of Ca2+iseemingly have a huge impact on Ca2+ER, IP3and h, Ca2+ihas a

rather small impact on Na+i, K+iand V. This point is particular obvious considering

the strengths of the currents crossing the cell membrane (see Figure 2.2). While

the Ca2+-dependent Na+-Ca2+ exchanger has a current strength between -0.5 and

0.5 µA

m2, the Ca2+-independent glutamate transporter and the Na+-K+pump operate

with a current strength between 0.01 and 1 A

m2. Thus, Na+i, K+iand V are mostly

determined by the glutamate transporter and the Na+-K+pump. For that reason,

I assume that the impact of Ca2+ion Na+i, K+iand V is negligible and that the

dynamics of Na+i, K+iand V can be analyzed separately from those of Ca2+i, Ca2+ER,

IP3and h.

0 20 40

Time [sec]

0.08

0.10

0.12

IGluT [A

m2]

glut = 0.005 mM

glut = 0.01 mM

glut = 0.051 mM

glut = 0.101 mM

glut = 0.535 mM

glut = 1.0 mM

0 20 40

Time [sec]

0.7

0.8

0.9

1.0

1.1

INKA [A

m2]

0 20 40

Time [sec]

0.0000005

0.0000000

0.0000005

0.0000010

INCX [A

m2]

Figure 2.2: The current strengths of the glutamate transporter (IGluT) and the Na+-K+pump (INKA)

exceed the one of the Na+-Ca2+ exchanger (INCX) by several orders of magnitude. The current

strengths of the glutamate transporter, the Na+-K+pump and the Na+-Ca2+ exchanger are simulated

for different glutamate concentrations between 0.005 mM and 1 mM. The colored lines represent the

time-dependent behavior of the three membrane currents IGluT, INKA and INCX for these applied

glutamate concentrations. The volume fraction of the internal Ca2+ store (ratioER) is set to 0.15.

The simulation results of Na+i, K+iand V suggest that the steady state values

rather than the time-dependent behavior of these variables have the main impact

on the model dynamics. During a constant application of glutamate the model

variables Na+i, K+iand V show a rapid increase to a steady state level, which

depends on the applied glutamate concentration (see Figure 2.3). For that reason,

I aim to derive the steady state solutions of Na+i, K+iand V in dependence of

glutamate and include these time-independent solutions into the model.

Model 45

0 20 40

Time [sec]

Na+

i[mM]

glut = 0.0mM

glut = 0.2mM

glut = 0.4mM

glut = 0.6mM

glut = 0.8mM

glut = 1.0mM

0 20 40

Time [sec]

100

K+

i[mM]

0 20 40

Time [sec]

−85

−65

−45

V[mV]

Figure 2.3: The dynamical variables Na+i, K+iand V converge to their glutamate-dependent

steady-state values within a few seconds. The three-dimensional system describing Na+i, K+iand

V is simulated for different constant glutamate concentrations (0 - 1 mM). The colored lines represent

the time-dependent behavior of the dynamical variables Na+i, K+iand V for the different applied

constant glutamate concentrations.

1.0.1 Steady state values of Na+i, K+iand V

The dynamics of Na+i, K+iand V are defined by the following three-dimensional

dynamical system:

dNa+

dt A

F·Vol ·(1−ratioER)·(3IGluT −3INKA −INaleak ),(2.1)

dK+

dt A

F·Vol ·(1−ratioER)·(−IGluT +2INKA −IKleak ),(2.2)

dt 1

(2IGluT −INKA −INaleak −IKleak ).(2.3)

Here, the Na+-Ca2+ exchanger is neglected due to its low impact on Na+and V.

The nonlinear system shown above is under-determined and thus it is not possible

to derive a unique solution of the steady state. This statement applies to glutamate

concentrations equal to and greater than zero. However, in case the fixed point

solution of one of the three variables is known, there is a unique solution for the

fixed points of the remaining two variables. Here, I assume the fixed point solution

of Na+ias given and solve the remaining system for the fixed points of K+iand V.

The resulting fixed point solution of V reads as follows:

V1

gNaleak ·(3IGluT −3INKA)+ENa.

Based on the assumption that the Nernst potential of K+depends on the intra-

and extracellular concentrations of K+, the fixed point solution of K+can not be

determined analytically. For this reason, I apply the bisection method in order to

determine the zeros of the following equation:

−IGluT ·(3gKleak +gNaleak )+INKA ·(3gKleak +2gNaleak )−(gNaleak ·gKleak )·(ENa −EK)0

46 Model reduction

0.0 0.5 1.0

Glutamate [mM]

Na+

ss [mM]

0.0 0.5 1.0

Glutamate [mM]

100

K+

ss [mM]

0.0 0.5 1.0

Glutamate [mM]

Vss [mV]

simulation

calculated

Figure 2.4: The steady states of the dynamic variables K+iand V in dependence on the applied

glutamate concentration can be calculated given that the fixed points of the Na+iconcentration

(Na+ss) are known. First, the three-dimensional system describing Na+i, K+iand V is simulated

for different constant glutamate concentrations (0 - 1 mM). The volume fraction of the internal Ca2+

(ratioER) is equal to 0.15. The blue lines denote the simulated fixed point curves of all dynamical

variables. Second, the fixed points of K+iand V (K+ss and Vss) are determined analytically based on

the simulated fixed point curve of Na+iand the applied glutamate concentration. The dashed red

lines denote the calculated fixed point curves of K+iand V.

The calculated fixed points of K+iand V exactly match the simulated fixed points

of these variable (see Figure 2.4).

Moreover, the application of different values of assumed constant glutamate

levels and steady state values of Na+(Na+ss) to these equations produces different

values of computed steady state levels of K+and V (see Figure 2.5).

0.0 0.5 1.0

Glutamate [mM]

Na+

ss [mM]

100

K+

ss [mM]

0.0 0.5 1.0

Glutamate [mM]

Na+

ss [mM]

−100

−80

−60

−40

−20

Vss [mV]

Figure 2.5: The applied values of the steady state values of Na+(Na+ss) and the applied glutamate

concentration determine the values of the steady state values of K+(K+ss) and V (Vss). The steady

state values of K+and V are computed assuming different combinations of Na+ss (15 mM - 30 mM)

and glutamate (0 mM - 1 mM) by applying the equations obtained in Model Section 1.0.1. The white

line denotes the steady state curve of Na+in dependence on the glutamate concentration.

Given the fact, that the system is under-determined, the system produces an

infinite amount of solutions depending on the initial conditions. Moreover, the

stability analysis of these steady state solutions reveals that the system is marginally

unstable. One of the eigenvalues is close to zero and either positive or negative

depending on the chosen initial condition (λ2∈[−10−13,10−13]). The other two

eigenvalues are negative (λ1−1700 and λ3−5). As λ2is comparable small, I

assume the influence of that eigenvalue to be insignificant. Since I am only interested

Model 47

in the achieved levels of Na+, K+and V during constant glutamate applications and

an already well-established parametrization, I continue to use the computed steady-

state curves and proceed with the model simplification.

In order to obtain simple equations for the fixed points of all three dynamical

variables, I choose to fit nonlinear functions to the simulated steady state curves

of Na+i, K+iand V (see Figure 2.6). These desired functions should only depend

on the input, the applied glutamate concentration, as all other parameters of the

system are well established. Since the dependence of the fixed point curves of Na+i,

K+iand V on glutamate shows a saturating curve progression, I describe the fitted

curves by a logistic function.

The glutamate-dependent functions for the fixed points of Na+i, K+iand V are as

follows:

Na+

i,ss aNa ·glut

glut +bNa

+cNa

K+

i,ss aK·glut

glut +bK

+cK

Vss aV·glut

glut +bV

+cV.

Here, aNa, aKand aVdefine the slope of the glutamate-dependent function. The pa-

rameters bNa, bKand bVdetermine the saturation by glutamate. And cNa, cKand cV

correspond to the maximal values Na+, K+and V can reach during glutamate appli-

cation. These parameters are fitted using the python package scipy.optimize.curve_

fit, which applies non-linear least squares fit. All parameters of these fitted curves

can be found in Table 2.1 .

The resulting fit of the fixed point curves of Na+, K+and V is in accordance with the

simulated fixed point curves for physiological values of Na+, K+and V (see Figure

2.6 b and c).

Table 2.1: Model parameters for the steady-state values of Na+i, K+iand V (see Section 1.0.1).

Parameter Value Description

aNa -6.72 mM Scaling of Na+iincrease

bNa -8.24 mM Half-saturation constant of glutamate

cNa 23.14 mM Maximal Na+iconcentration

aK6.72 mM Scaling of K+iincrease

bK8.24 mM Half-saturation constant of glutamate

cK91.86 mM Maximal K+iconcentration

aV-28 mV Scaling of the membrane voltage

bV11.54 mM Half-saturation constant of glutamate

cV48 mV Maximal membrane voltage

48 Model reduction

0.0 0.5 1.0

Glutamate [mM]

Na [mM]

0.0 0.5 1.0

Glutamate [mM]

100

K [mM]

0.0 0.5 1.0

Glutamate [mM]



V [mV]

simulation



0.2 1.6 3.0

I [ ]

NKA m

max 2

0.1

0.7

1.3

I [ ]

GluT m

max 2



fit

100

101

102

MSE

0.2 1.6 3.0

I [ ]

NKA m

max 2

0.1

0.7

1.3



fit

100

101

102

MSE

0.2 1.6 3.0

I [ ]

NKA m

max 2

0.1

0.7

1.3



fit

100

101

102

MSE

0.2 1.6 3.0

I[ ]

NKA m

max 2

0.1

0.7

1.3

I [ ]

GluT m

max 2

Na+

0.2 1.6 3.0

I[ ]

NKA m

max 2

0.1

0.7

1.3

K [mM]

0.2 1.6 3.0

I[ ]

NKA m

max 2

0.1

0.7

1.3



100

V [mV]

Figure 2.6: The simulated and analytically calculated steady state values Na+ss, K+ss and Vss match

in their dependence on the applied glutamate concentration. a The simulated results of Na+ss, K+ss

and Vss are obtained by simulating the three-dimensional dynamical system describing Na+i, K+i

and V for constant glutamate concentrations between 0 and 1 mM and for a duration of 200 seconds.

The volume fraction of the internal Ca2+ (ratioER) is equal to 0.15. The values of Na+i, K+iand V at

the end of the simulation time are defined as the simulated steady state values Na+ss, K+ss and Vss.

The analytic solution is derived as described in Section 1.0.1. The simulation results are shown in blue

and the fit is shown in red. bThe fitted fixed point curves of Na+, K+and V are in accordance with the

simulation results for large parts of the parameter space of the maximal pump currents of the Na+-K+

pump (INKAmax ) and the glutamate transporter (IGluTmax ). The fixed points curves of Na+, K+and V are

simulated and subsequently fitted for different values of INKAmax and IGluTmax . The difference between

the simulated and the fitted fixed point curves is computed with the mean-squared error. The figure

shows the mean-squared error of the fitted fixed point curves in comparison to the simulated ones of

Na+ss, K+ss and Vss. The white lines denote the boundary between physiological and unphysiological

values of Na+i, K+iand V. The white cross corresponds to the chosen default parameter combination

of INKAmax and IGluTmax .cLow values of the maximal pump current of the Na+-K+pump (INKAmax )

and high values of the the maximal pump current of the glutamate transporter (IGluTmax ) produce

unphysiological values of Na+ss, K+ss and Vss. The fixed points of Na+, K+and V are simulated for

different values of INKAmax and IGluTmax . The figure shows the values of the simulated fixed points of

Na+, K+and V. The following values are classified as unphysiological values: Na+ss >50 mM, K+ss <

50 mM and Vss >0 mV. The white cross corresponds to the chosen default parameter combination of

INKAmax and IGluTmax .

Model 49

1.0.2 Extracellular Ca2+ concentration (Ca2+o)

Due to the large differences of the Ca2+ concentration in the intracellular space,

on the one side, and in the extracellular space, on the other side, changes of the

Ca2+ concentration in the extracellular space have only a minor effect on the con-

centration change in the intracellular space (see Figure 2.7). Therefore, I assume the

extracellular Ca2+ concentration to be constant and set it to its resting concentration.

0.0 0.5 1.0 1.5

Ca2 +

i[μM]

Ca2 +

ER [μM]

Ca2 +

o+KNCXmC

0.998

0.999

1.000

1.001

1.002

ratio

0.0 0.5 1.0 1.5

Ca2 +

i[μM]

Ca2 +

ER [μM]

Ca2 +

0.996

0.998

1.000

1.002

1.00μ

1.006

ratio

Figure 2.7: A dynamic extracellular Ca2+ concentration (Ca2+o) has only little impact on the Ca2+o-

dependent terms of the Na+-Ca2+ exchanger. The two Ca2+o-dependent terms of the Na+-Ca2+

exchanger were calculated for either a dynamic or a constant extracellular Ca2+ concentration and for

value ranges of the Ca2+ concentration in the internal Ca2+ store (Ca2+ER) and the intracellular space

(Ca2+i). The results for the dynamic and the constant Ca2+owere divided by each other in order to

obtain a ratio. The two figures show the ratios calculated for the two Ca2+o-dependent terms. The

two different Ca2+o-dependent terms are specified in the title of each figure.

1.1 Reduced model

The reduced model consists of the four-dimensional system of differential equations

describing the dynamics of the intracellular Ca2+ concentration, the Ca2+ concen-

tration in the internal Ca2+ store, the intracellular IP3concentration and the fraction

of activated IP3receptor channels h within a single astrocytic compartment of an

astrocytic process.:

dCa2+i

dt A

F·Vol ·(1−ratioER)·INCX −dCa2+

dt ·ratioER (2.4)

dCa2+ER

dt A·√ratioER

F·Vol ·ratioER ·(−IIP3R+ISerca −ICER leak)(2.5)

dIP3

dt prodPLCβ+prodPLCδ−de grIP3−3K−degrIP−5P(2.6)

dt h∞−h

τh

.(2.7)

50 Model reduction

The definitions of the current equations INCX, IIP3R, ISerca and ICERleak, of the dynam-

ical equation for the intracellular IP3concentration and h as well as all parameter

values are mentioned in the Model Section of part 1.

The intracellular concentrations of Na+and K+as well as the membrane voltage V

are calculated by the fitted steady state curves derived in Section 1.0.1:

Na+

i,ss aNa ·glut

glut +bNa

+cNa (2.8)

K+

i,ss aK·glut

glut +bK

+cK(2.9)

Vss aV·glut

glut +bV

+cV(2.10)

(2.11)

The extracellular concentrations of Na+and K+depend on their respective intracel-

lular concentration changes and are described by:

Na+

o,ss −Na+

o,rest Na+

i,rest −Na+

i,ss (2.12)

K+

o,ss −K+

o,rest K+

i,rest −K+

i,ss .(2.13)

The extracellular Ca2+ concentration is determined by its resting concentration:

Ca2+

oCa2+

orest.(2.14)

1.2 Model behavior in response to parameter variations and stability

analysis of the reduced model

Parameter variations of the maximal pump current of the Na+-Ca2+ exchanger

(INCXmax ) andthe volumefractionof the internalCa2+ store (ratioER) allow a parametriza-

tion of the dynamical system for any position along the astrocytic process between

the soma and the perisynaptic astrocytic processes. Moreover, by setting either

INCXmax or ratioER equals to zero the astrocyte model is reduced by one Ca2+ pool

and the Ca2+ signal generation mechanisms can be analyzed separately. Thus, pa-

rameter variations of both INCXmax and ratioER have a huge impact on the dynamics

of the system and its stability. For that reason, I analyze the impact of these pa-

rameters on the dynamical system, its fixed points and the stability of the fixed

points.

1.2.1 INCXmax >0 and ratioER >0

Fixed points For a model parametrization with INCXmax and ratioER larger than

zero the astrocyte model consists of all three Ca2+ pools: the internal Ca2+ store,

the intracellular space and the extracellular space. Thus, both the Ca2+ release from

internal stores and the Ca2+ entry from the extracellular space affect the intracellular

Ca2+ concentration.

This model parametrization considering three Ca2+ pools allows the calculation

Model 51

of the fixed points of the dynamical system by forward substitution. The steady

state solution of a dynamical system does not change with time and all differential

equations are equals to zero. As a result the fixed point of the intracellular Ca2+

concentration loses its dependence on dCa2+

dt and solely depends on the Ca2+ entry

from the extracellular space. On that condition the analytic solution of the fixed

point of Ca2+iis:

Ca2+

i,f p 

Na+

i,ss

3·Ca2+

o,0

Na+

o,ss3·exp(Vss ·F

R·T).(2.15)

Since the differential equation for IP3only depends on Ca2+iand IP3, I determine

the fixed point of IP3(IP3, fp) by setting Ca2+i, fp into the differential equation and

calculating the root of this differential equation. Due to the structure of that differ-

ential equation the solution can only be derived numerically with the help of a root

finding method.

In the next step I determine the fixed point of the fraction of activated IP3receptor

channels, h, by again setting the fixed points of Ca2+iand IP3into the differential

equation of h and determining the zero of this differential equation. The fixed point

of h (hfp) is defined by:

hf p 

d2·IP3,f p+d1

IP3,f p+d3

d2·IP3,f p+d1

IP3,f p+d3

+Ca2+

i,f p

.(2.16)

In the last step I determine the fixed point of the Ca2+ concentration in the internal

Ca2+ store by again applying the above mentioned methods. The fixed point of

Ca2+ER is defined by:

Ca2+

ER,f p ratioER ·(ISerca

rL+rC·IP3,f p

IP3,1+d1

Ca2+

i,f p

Ca2+

i,f p+d5

·h3

f p

)+Ca2+

i,f p (2.17)

Consequently, none of the fixed points depend on INCXmax or ratioER. Thus, changes

of both parameters do not affect the fixed points of the four-dimensional dynamical

system.

Stability of fixed points The stability of the fixed points is determined by the

eigenvalues of the Jacobian matrix. The Jacobian matrix (J) consists of the first-

order partial derivatives of the dynamical system:

J







∂f1

∂Ca2+

∂f1

∂Ca2+

∂f1

∂IP3

∂f1

∂h

∂f2

∂Ca2+

∂f2

∂Ca2+

∂f2

∂IP3

∂f2

∂h

∂f3

∂Ca2+

∂f3

∂Ca2+

∂f3

∂IP3

∂f3

∂h

∂f4

∂Ca2+

∂f4

∂Ca2+

∂f4

∂IP3

∂f4

∂h







,(2.18)

52 Model reduction

with

f1

dCa2+

dt ,f2

dCa2+

dt ,f3dIP3

dt ,f4dh

dt .(2.19)

In consideration of the partial derivatives the Jacobian matrix can be simplified to:

J







∂f1

∂Ca2+

∂f1

∂Ca2+

∂f1

∂IP3

∂f1

∂h

∂f2

∂Ca2+

∂f2

∂Ca2+

∂f2

∂IP3

∂f2

∂h

0∂f3

∂Ca2+

∂f3

∂IP30

0∂f4

∂Ca2+

∂f4

∂IP3

∂f4

∂h







.(2.20)

The fixed points are then classified according to the eigenvalues of the Jacobian

matrix.

1.2.2 INCXmax = 0

A maximal pump current of the Na+-Ca2+ exchanger equals to zero (INCXmax =

m2) reduces the astrocyte model by one Ca2+ pool and changes the Ca2+ signal

generation in the intracellular space. With INCXmax equals to zero the Ca2+ entry from

the extracellular space is nonexistent. Thus, this model parametrization reduces

the model by one Ca2+ pool: the extracellular space. Moreover, in this case the

intracellular Ca2+ concentration solely depends on the Ca2+ release from internal

stores. Therefore, the differential equation for the intracellular Ca2+ concentration

changes to:

dCa2+

dt −ratioER ·dCa2+

dt ⇔dCa2+

dt −1

ratioER ·dCa2+

dt .(2.21)

Thus, with INCXmax equals to zero dCa2+

dt is solvable and the solution of this differential

equation is:

Ca2+

ER (t)(Ca2+

irest −Ca2+

i(t)) ·1

ratioER

+Ca2+

ERrest .(2.22)

The remaining differential equations are not affected by a change of INCXmax and

read as follows:

dCa2+i

dt A·√ratioER

F·Vol ·(IIP3R−ISerca +ICER leak)(2.23)

dIP3

dt prodPLCβ+prodPLCδ−de grIP3−3K−degrIP−5P(2.24)

dt h∞−h

τh

.(2.25)

These modified model conditions do not allow a straightforward solution of Ca2+i, fp.

This also means, that all other fixed points can not be solved by forward substitution.

Instead, I determine the solution of all fixed points numerically.

Model 53

1.2.3 ratioER = 0

In case the volume fraction of the internal Ca2+ store is equals to zero (ratioER = 0)

both the Ca2+ concentration in the internal Ca2+ store (Ca2+ER) as well as the fraction

of activated IP3receptor channels (h) loose their meaning and their differential

equations can be neglected. This parametrization with ratioER equals to zero sets

the volume of the internal Ca2+ store equals to zero and the astrocyte model consists

of only two Ca2+ pools. Thus, solely the Ca2+ entry from the extracellular space

determines the intracellular Ca2+ concentration. Therefore, the dynamical system

reduces to two dynamical variables: the intracellular concentrations of Ca2+ and IP3.

At the same time the differential equation for the intracellular Ca2+ concentration

changes compared to the four-dimensional model:

dCa2+i

dt A

F·Vol ·INCX a·(b−d·Ca2+

i),(2.26)

a

A·INCXmax ·Na+o3

KNCX mN 3+Na+o3·Ca2+o

KNCX mC+Ca2+o

F·Vol ·(1+ksat ·exp(η−1·V·F

R·T)) ,(2.27)

bNa+i3

Na+o3·exp(η·V·F

R·T),(2.28)

d1

Ca2+

o·exp(η−1·V·F

R·T).(2.29)

The solution of this changed differential equations is:

Ca2+

i(t)b

d+(Ca2+

irest −b

d)·e−adt (2.30)

The differential equation for the intracellular IP3concentration remains the same

compared to the four-dimensional model.

The fixed point of Ca2+iis determined by taking the limit of time to infinity. Thus

the fraction b

ddefines the fixed point of Ca2+i. Applying the definitions of b and d

reveals that the fixed point of Ca2+ido not change compared to the four-dimensional

model. Thus, also the fixed point of IP3remains the same.

Due to the altered model structure the model does not produce unstable behavior

like oscillations any more. For the volume fraction of the internal Ca2+ (ratioER)

equals to zero the dynamical system reduces to only two dimensions. Moreover,

one of the two differential equations solely depends on itself and can be solved.

A dynamical system can only produce oscillations if both differential equations

depend on each other. Since this is not the case here, the model solely produces

stable behavior.

1.3 Initialization of the dynamical system

The initial values of the intracellular and extracellular concentrations of Ca2+, Na+

and K+are known from literature and can be found in Table 3.3 (part 1). The initial

values of IP3and h are determined by deriving the zero of dIP3

dt and dh

dt , respectively.

54 Model reduction

The zero of the current equation INCX reveals the initial value of the membrane

voltage V, since an inactive current ensures a stable steady state at time point zero.

The parameter values gNaleak and gKleak were calculated by solving dNa+

dt and dK+

for the respective parameters.

1.4 Computational methods

All simulations were performed with Python 2.7 using the packages NumPy, SciPy

and Matplotlib. The function odeint of the SciPy package was used for the numerical

integration of the non-linear differential equations.

Results

1 Comparison of the full and the reduced model

Full

model

Red.

model

Figure 2.8: The oscillatory behaviors of the full and the reduced model coincide with each other.

a - b The full model (in a) and the reduced model (in b) are simulated for constant glutamate

concentrations of 6µM, 13µM and 55µM as well as for different values of the maximal pump current

of the Na+-Ca2+ exchanger (INCXmax ) and the volume fraction of the internal calcium store (ratioER).

The two different model variations were simulated for a duration of 200 seconds each. The color

plots present the amplitudes of the full model (in a) and the reduced model (in b) for the different

parameter combinations of the applied glutamate concentration, INCXmax and ratioER. The white areas

correspond to the non-oscillatory ranges. cThe figure shows the bifurcation curves of the intracellular

Ca2+ concentration for three different applied glutamate concentrations (6µM, 13µM and 55µM), for

one specific value of the maximal pump current of the Na+-Ca2+ exchanger (INCXmax =0.01 A

m2) and as

a function of the volume fraction of the internal Ca2+ store (ratioER). The solid and the dashed lines

correspond to the bifurcation curves of the full and the reduced model, respectively.

A comparison of the full and the reduced model reveals that both model varia-

tions coincide very well in their steady-state behaviors and that the reduced model

is sufficient for the analysis of the Ca2+ signal generation in astrocytes (see Figure

2.8). In order to compare the oscillatory behaviors of the full and the reduced model

I simulate both model variations for a variety of parameter combinations and com-

pare the results regarding the Ca2+ oscillation amplitudes, the upper and lower

points of the Ca2+ oscillations as well as the bifurcation points (see Figure 2.8 a and

b). The upper and lower points of the Ca2+ oscillations differ on average around

0.007µM between the two model types. This difference corresponds to a change of

56 Model reduction

0.01 % as measured by an oscillation amplitude of 0.6µM. Moreover, the bifurcation

points of the full and the reduced model coincide for all parameter combinations

(see Figure 2.8 c). Thus, the steady-state behavior of the reduced model is in accor-

dance with that one of the full model. Therefore, I conclude that the reduced model

is sufficient for the analysis of Ca2+ signal generation in astrocytes without loosing

qualitative or quantitative information generated by the full model.

1.1 Analysis of the reduced model

The analysis of the reduced model regarding its fixed points and its fixed point

stabilities allows a detailed investigation of the Ca2+ signal generation in astrocytes.

The fixed points of the intracellular Ca2+ concentration during an inactive (INCXmax

= 0 A

m2) or an active Ca2+ influx (INCXmax >0A

m2) from the extracellular space are

either determined by Ca2+ release from internal stores or by Ca2+ influx from the

extracellular space, respectively (see Figure 2.9a-b). During an inactive Ca2+ influx

from the extracellular space the fixed point of the intracellular Ca2+ concentration

decreases with the volume of the internal Ca2+ store (decrease of ratioER) and in-

creases with the applied glutamate concentration (see Figure 2.9a). During an active

Ca2+ influx from the extracellular space, however, the fixed point of the intracellu-

lar Ca2+ concentration solely depends on the applied glutamate concentration and

is independent of the volume of the internal Ca2+ store (see Model Section 1.2.1

and Figure 2.9b). From these semi-analytic results I conclude that an inactive Ca2+

influx from the extracellular space promotes the Ca2+ release from internal stores

whereas an active Ca2+ influx from the extracellular space primarily favors the Ca2+

transport into the astrocyte mediated by the Na+-Ca2+ exchanger. Moreover, during

a finite Ca2+ influx from the extracellular space the fixed point of the intracellular

Ca2+ concentration reaches higher concentration values (1.6µM) compared to the

inactive Ca2+ transport (0.4µM).

Results 57

a b c

osc no osc

Figure 2.9: The applied glutamate concentration, the maximal pump current of the Na+-Ca2+

exchanger (INCXmax ) and the volume fraction of the internal Ca2+ store (ratioER) determine the

stability behavior of the reduced model. a,b The fixed points of the intracellular Ca2+ (Ca2+ifp )

concentration for either an inactive (INCXmax =0 A

m2) or an active Ca2+ influx (INCXmax >0A

m2) from the

extracellular space. For INCXmax =0 A

m2the colored area denotes the values of Ca2+ifp and the white area

corresponds to the oscillatory range of the reduced model. For INCXmax >0A

m2(here: INCXmax =1 A

m2)

Ca2+ifp is shown as a function of the applied glutamate concentration. cThe stability behavior of the

reduced model as a function of the applied glutamate concentration, of INCXmax and of ratioER. The

colored lines denote the boundary between the oscillatory and the non-oscillatory range for three

different values of glutamate (6µM, 13µM and 55µM), and for parameter combinations of INCXmax as

well as of ratioER.dThe nullclines of the Ca2+ concentration and h, the intersection of the nullclines

and the vector field as a function of INCXmax . The three figures denote the nullclines with their

corresponding vector fields and trajectories for INCXmax equals to 0 A

m2(orange), 10-5 A

m2(yellow) and

m2(blue), respectively. Here, the applied glutamate concentration is equal to 100µM and ratioER

is equal to 0.05. The two other dynamical variables (IP3and Ca2+ER) are hold fixed at their fixed

points. eThe time course of the intracellular Ca2+ concentration for different values of INCXmax and

three different time durations (100 seconds 30 000 seconds, and 300 000 seconds). The colored lines

present the time course of Ca2+ifor INCXmax equals to 0 A

m2(orange), 10-6 A

m2(grey), 10-5 A

m2(yellow)

and 1 A

m2(blue). Here, the applied glutamate concentration is equal to 100µM and ratioER is equal to

0.05.

58 Model reduction

Figure 2.10: The real parts of the eigenvalues mainly predict the transition between os-

cillatory and non-oscillatory model behavior. The eigenvalues of the four-dimensional

system were computed as a function of the applied glutamate concentration, the maximal

pump current of the Na+-Ca2+ exchanger (INCXmax ) and the volume fraction of the internal

Ca2+ store (ratioER. The real parts of the eigenvalues where divided into two groups rep-

resenting either the appearance of at least one positive real eigenvalue and the appearance

of no positive real eigenvalue (blue areas). The eigenvalues were also divided according to

the fact if the eigenvalue with the largest real part is also complex.

Although neither the maximal pump current of the Na+-Ca2+ exchanger nor the

volume of the internal Ca2+ store affect the fixed points of the intracellular Ca2+

concentration during an active Ca2+ influx from the extracellular space, they affect

the oscillation behavior of the model and thus the stability of the fixed points (see

Figure 2.9c and Fig 2.10). Figure 2.9 c shows the border between parameter com-

binations, for which oscillations occur, and combinations, which do not produce

oscillations. This model behavior in response to parameter variations is also re-

flected by the real and imaginary parts of the eigenvalues (see Figure 2.10). The real

parts of the eigenvalues show a clear transition from at least one positive eigenvalue

to solely negative eigenvalues and by that build the boundary between unstable

to stable model behavior (see Figure 2.10). The oscillatory range of the reduced

and the full model shrink for an increase of the Ca2+ influx from the extracellular

space and a decrease of the volume of the internal Ca2+ store (see Figure 2.8 and

Figure 2.9c). Moreover, the applied glutamate concentration increases the oscilla-

tory range for values up to 13 µM and decreases the oscillatory range for larger

values (see Figure 2.9 b). The applied glutamate concentration determines the con-

centrations of Ca2+ and IP3in the intracellular space (see Figure 2.9a,c) and thus

also the opening probability of IP3receptor channels at the internal Ca2+ store (Li

and Rinzel, 1994). The opening probability of these channels in dependence on the

intracellular Ca2+ concentration has a bell-shaped course and reaches its maximal

value for intracellular Ca2+ concentrations around 0.2 µM. Thus, by enhancing the

intracellular Ca2+ concentration due to an increase of the glutamate concentration

the opening probability exceeds its maximal value and less Ca2+ is released from

internal Ca2+ stores. Therefore, glutamate concentrations larger than 13 µMevoke

less Ca2+ release from internal stores. For that reason, Ca2+ oscillations vanish and

the size of the oscillatory range decreases. Consequently, for the generation of Ca2+

oscillations within the intracellular space the cooperation of Ca2+ release from in-

ternal stores and Ca2+ influx from the extracellular space is essential.

Results 59

An inactive or active Ca2+ influx from the extracellular space does not only de-

termine the fixed points of the intracellular Ca2+ concentration, but also the time

scale of the Ca2+ dynamics (see Figure 2.9d-e). During an inactive Ca2+ influx from

the extracellular space the intracellular Ca2+ concentration reaches its steady state

within several seconds. In this case, the vector field is comparable small. The fixed

point of Ca2+ is close to its resting value and only small changes are needed to reach

the fixed point. The onset of the Ca2+ entry from the extracellular space changes

the Ca2+ nullcline, the intersection of the nullclines and also the vector field (see

Fig 2.9d). Moreover, the strength of the Ca2+ influx from the extracellular space

(IINCXmax ) determines the time-dependent dynamics of the Ca2+ concentration (see

Fig 2.9e). A low Ca2+ influx from the extracellular space (IINCXmax =10−6A

m2) results in

a small vector field. This means that the Ca2+ concentration moves slowly from the

resting value to its fixed point. Thus, a low IINCXmax decelerates the model dynamics

drastically and it takes up to 300 000 seconds until the Ca2+ concentration reaches

its steady state. An increase of the Ca2+ influx from the extracellular space increases

the vector field and the Ca2+ concentration moves fast from the resting value to its

fixed point. Thus a high IINCXmax accelerates the Ca2+ dynamics again and the Ca2+

concentration reaches its fixed point within several seconds (see Figure 2.9 d-e).

1.2 Prediction of the model behavior based on the eigenvalues of the

Jacobian matrix

The eigenvalues of the Jacobian matrix of the reduced model predict the model

behavior. The eigenvalues of the Jacobian matrix determine the stability of the

fixed points. A transition from positive to negative real parts of an eigenvalue

corresponds to the change from unstable to stable behavior of the model. Thus,

the real parts of the eigenvalues predict the boundary between unstable and stable

model behavior (see Figure 2.10 and Figure 2.11 a). For the reduced model two out

of four eigenvalues show such a transition from positive to negative values and thus

predict a transition from unstable to stable model behavior. The real parts of other

two eigenvalues are solely negative.

The boundary between positive and negative real parts of the eigenvalues also

predicts the model behavior in response to parameter variations (see Figure 2.11

b). The previous analysis of the reduced model revealed that during an increase

of the Ca2+ influx the oscillatory range shrinks due to increased Ca2+ release from

internal stores. This raises the question whether an altered Ca2+ release and uptake

at internal Ca2+ stores could reverse the effect of a high Ca2+ influx. For that

reason I compute the eigenvalues of the Jacobian matrix for variations of those

parameters which determine the Ca2+ release and uptake at the internal Ca2+ store.

The computed eigenvalues predict that the oscillatory range increases for a decrease

of Ca2+ release and an increase of Ca2+ uptake.

60 Model reduction

0.15 0.05 0.0

ratioER

1e-07

0.001

10.0

I [ ]

NCX m

max 2



0.15 0.05 0.0

ratioER





103



101



0.15 0.05 0.0

ratioER

1e-07

0.001

10.0

I [ ]

NCX m

max 2

v =

4.5







0.15 0.05 0.0

ratioER

v =

9.0







0.15 0.05 0.0

ratioER

v =

22.5







Figure 2.11: The eigenvalues of the Jacobian matrix predict the boundary between the oscillatory

and the non-oscillatory range of the reduced model. a The figure shows the real part of two out of

four eigenvalues (λ1and λ2) of the Jacobian matrix for different values of the maximal pump current

of the Na+-Ca2+ exchanger (INCXmax ) and the volume fraction of the internal Ca2+ store (ratioER). The

white lines denote the boundary between positive and negative values and hence also between the

oscillatory and non-oscillatory range. The real parts of the other two eigenvalues are negative for all

parameter combinations. bThe figure shows the boundary between positive and negative real parts

of the eigenvalues for different values of the rates for Ca2+ uptake (vER) and Ca2+ release (rc) at the

internal Ca2+ store as well as for parameter combinations of INCXmax and ratioER. The colored lines

denote the boundaries between the positive and the negative eigenvalues for rcequals to 3 1

sec (grey),

sec (green) and 15 1

sec (yellow). The solid and dashed lines correspond to the boundaries between

positive and negative values of λ1and λ2, respectively.

Discussion

Within this second part of my thesis I showed that first a reduced model for calcium

signal generation in astrocytes is in high accordance with the considered full model,

and that second the reduced model allows a profound analytic analysis and gives

in-depth insights into the model behavior.

The comparison of the full and the reduced model for calcium signal generation

in astrocytes revealed that the reduced model reproduces the model behavior of

the full model quite well. Both the comparisons of oscillation amplitudes and

bifurcation points showed that the reduced and the full model coincide. This

result illustrates that the time-dependent behavior of sodium, potassium and the

membrane voltage is irrelevant for the calcium entry from the extracellular space.

The calcium entry is only determined by the steady-state solutions of these variables.

This dependency also emphasizes the difference of the fast time-scale of sodium,

potassium and the membrane voltage and the slow time-scale of the other dynamical

variables.

The mathematical structure of the reduced model allowed me to compute the fixed

points and their stabilities. The reduced model consists of four differential equations

describing the intracellular concentrations of calcium and IP3, the concentration

of calcium in the internal calcium store and the opening probability of calcium

channels at the internal calcium store. The dependencies of the single differential

equations allowed the computation of the steady-state solutions of all dynamical

variables by forward-substitution. The stabilities of these fixed points were then

determined by the eigenvalues of the Jacobian matrix. Thus, the reduced model

allowed the derivation of the equilibrium solution of the dynamical system as well

as the investigation of its steady-state behavior. Consequently, the analysis of the

reduced model is much more simplified compared to the full model since it does

not require the numerical simulation of the system.

Moreover, based on the reduced model I could study the two borderline cases

for either a nonexistent internal calcium store or an inactive calcium entry from

the extracellular space (see part 2, Model Section 1.2). In both cases the amount of

considered calcium pools reduced from three to two. This reduction is accompanied

by a simplification of the model. Considering the volume of the internal calcium

store equals to zero also means that the intracellular calcium concentration solely

depends on the calcium entry from the extracellular space. In the case of an inactive

calcium entry from the extracellular space the calcium signal is only determined by

calcium release from internal stores. In addition, these two mentioned borderline

62 Model reduction

cases can directly be associated with different subcellular compartments of the

astrocyte. Astrocytic endfeet and perisynaptic astrocytic processes are devoid of

internal calcium stores (Patrushev et al., 2013). Thus, by setting the volume of

the internal calcium store equal to zero the calcium signal generation in these

subcellular compartments can be studied. In the astrocytic soma the calcium release

from internal stores is the predominant mechanisms for calcium signal generation

(Srinivasan et al., 2015; Stobart et al., 2016; Bazargani and Attwell, 2016; Bindocci

et al., 2017). This is also demonstrated by the model results showing that calcium

oscillations are only observed for a high volume fraction of the internal calcium

store and a low maximal pump current of the sodium-calcium exchanger. Both

the high volume fraction of the internal calcium store and the low maximal pump

current of the sodium-calcium exchanger are characteristic for the soma.

Based on the fixed points and their stability I could predict the model behavior

in response to parameter variations. Here, I have addressed the question whether

altered calcium release and uptake mechanisms at the internal calcium store can

reverse the suppression of calcium oscillations during an increased calcium entry

from the extracellular space (Oschmann et al., 2017b). The computed eigenvalues of

the Jacobian matrix predicted that an an increase of calcium transport back into the

internal calcium store as well as a decrease of calcium release from internal stores led

to an increase of the oscillatory range. These simulation results support the impact of

store-operated calcium channels on calcium signal generation in astrocytes (Pizzo

et al., 2001; Sergeeva et al., 2003). Store-operated calcium channels are in close

proximity to the internal calcium store (Golovina, 2005). Depletion of internal

stores activates calcium influx mediated by store-operated calcium channels and

refilling of internal calcium stores (Boulay et al., 1999). In experiments it has been

shown that a block of store-operated calcium entry prevents calcium oscillations in

astrocytes (Pizzo et al., 2001; Sergeeva et al., 2003). Thus, although my model does

not contain store-operated calcium channels I could show that a refilling of internal

calcium stores is essential to sustain calcium oscillations in astrocytes.

Thereduced model alsoserves as ageneral framework forthe studyof calciumsignal

generation in astrocytes. Besides the sodium driven sodium-calcium exchanger

also other transporters can account for calcium entry from the extracellular space.

Since most of these transporters solely depend on calcium itself (like store-operated

calcium channels) or on one of the reduced dynamical variables sodium, potassium

or the membrane voltage (like NMDA receptors or voltage-gated ion channels)

these mechanisms can easily be applied to the reduced model without increasing its

complexity. Moreover, the extension of the reduced model with these transporters

still allows a profound analytic analysis. As long as the applied mathematical

descriptions of the calcium transporters solely depend on calcium, the fixed point

and the fixed point stability of the intracellular calcium concentration and also of

all other dynamical variables can easily be computed.

Insummary, I developed areduced modelfor calciumsignal generationin astrocytes

which could serve as a general framework to investigate the interaction of calcium

signaling mechanisms in astrocytes as well as the different mechanisms for calcium

entry from the extracellular space. The advantages of such a reduced model are

Discussion 63

that it allows the computation of its equilibrium solution and thus supersede a

numerical integration of the dynamical system, and that based on the steady-state

solution of the system predictions about the model behavior as well as about the

biological system in response to parameter variations can be made.

64 Model reduction

Part 3

Multi-compartment model for the

signal propagation

Introduction

Calcium, sodium and potassium signals in astrocytes are not local events, but

propagate through the whole astrocyte and even to neighboring astrocytes via gap

junctions. The propagation of calcium, sodium and potassium signals differ in their

propagation mechanisms and propagation radius. On the one side, the calcium sig-

nal propagates by both the renewal through the all-or-non-like release of calcium

from internal stores as well as by calcium diffusion. On the other side, the prop-

agation of sodium and potassium is solely driven by diffusion. This circumstance

also allows calcium signals to travel larger distances than sodium and potassium

signals.

The propagation of calcium signals also differs between subcellular compartments.

In those subcellular compartments containing internal calcium stores, the calcium

signal propagation requires the diffusion of calcium and inositol-triphosphate (IP3).

The diffusion of both components then evokes an intracellular amplification of the

calcium signal by calcium release from internal stores (Golovina and Blaustein, 2000;

Scemes, 2000; Sheppard et al., 1997). However, perisynaptic astrocytic processes are

devoid of internal calcium stores (Patrushev et al., 2013), which makes it impossible

to amplify the calcium signal by intracellular mechanisms. Therefore, the propaga-

tion of calcium signals in astrocytic processes is driven by diffusion (Rusakov et al.,

2011).

Moreover, calcium signals do not only spread throughout single astrocytes, but also

propagate into neighboring astrocytes and by that create an intercellular calcium

wave. These intercellular waves of calcium signals propagating via gap junctions

spread over long distances (300 - 400 µm) with a speed ranging between 15-20 µm/s

and are able to excite up to hundreds of cells (Giaume and Venance, 1998; Scemes

and Giaume, 2006).

In contrast to calcium signals, sodium and potassium signals are not affected by

intracellular amplification and are solely driven by diffusion. Since the sodium

and potassium signals spread exclusively by diffusion they propagate much faster

than calcium signals. Moreover, the different subcellular compartments also play a

considerable role in the propagation speed of sodium and potassium signals. For

example, the diffusion speed of sodium is about 60 µm/s in somatic regions (Langer

et al., 2012), and probably even higher in astrocytic processes due to their elaborate

morphology (Nedergaard et al., 2003). In astrocytic endfeet the diffusion speed is

even higher and the maximum velocity is 120 µm/s (Langer et al., 2016).

Although the propagation of the different ion signals within individual astrocytes

68 Multi-compartment model for the signal propagation

and astrocyte networks has already been studied, only little is known about the

interaction of the individual propagating signals. For example, an interaction of

sodium and calcium signals in astrocytic processes could be of great importance.

Since the sodium and the calcium signals are directly linked via the sodium-calcium

exchanger (Goldman et al., 1994; Kirischuk and Ketfenmann, 1997; Reyes et al., 2012),

the calcium signal propagation in astrocytic processes could also be driven by the

fast sodium signal propagation and its effect on the sodium-calcium exchanger. In

order to study the interaction of traveling ion signals in astrocytes I develop a multi-

compartment model which allows to investigate the propagation of calcium, sodium

and potassium signals in different subcellular compartments of the astrocyte.

Model

The subject of this last part of my thesis is the development of a multi-compartment

model for the computation of signal propagation in astrocytes. This model accounts

for the movement of ions (Ca2+, Na+, K+) and molecules (IP3) in the astrocytic

process.

1 Astrocyte morphology

soma

process

Figure 3.1: Scheme of the astrocyte morphology. Scheme depicting the division of astrocytes into

subcellular compartments (soma and processes).

Each astrocyte consists of several subcellular compartments: one soma and several

processes originating at the soma and splitting up into smaller processes (see Figure

3.1). These single subcellular compartments can be represented by basic geometric

bodies. Thus, for example, a sphere represents the soma and cylinders of different

lengths and diameters represent processes of different sizes.

70 Multi-compartment model for the signal propagation

Figure 3.2: Morphological parameters of the astrocytic process. Mean values and the standard

deviation for the length and diameter of astrocytic processes based on experimental data (Diniz et al.,

2016; Canchi et al., 2017; Zhang et al., 2016).

The geometrical parameters like the length and the diameter of these subcellular

compartments applied in the model are based on experimental data. Due to the

fact, that more and more experimentalists pay more attention to the generation and

propagation of Ca2+ signals in different subcellular compartments, the published

results also contain detailed information about the geometry of the considered

astrocytes. The geometric parameters of the processes are based on experimental

data published on the open-source platform NeuroMorpho.org (Diniz et al., 2016;

Canchi et al., 2017; Zhang et al., 2016). This platform contains publicly accessible

three-dimensional reconstructions of astrocytes. Here, the morphology of each cell

is given by points within a three-dimensional cartesian coordinate system. Thus,

for example the length of a single astrocytic process can be obtained by summing

up the euclidean distances between all points of an astrocytic process (see Figure

3.2 and Table 3.1). Noticeable is the big difference in the diameter of the processes

between the individual studies. This difference could be explained by the use of

different species and brain regions in the experiments. While Diniz et al. (2016)

and Canchi et al. (2017) investigate astrocytes in the hippocampus of rats and mice,

Zhang et al. (2016) use the corpus callosum of rabbits for their experiments. Based

on the average diameter across all studies, I assume the diameter of the processes as

1µm. I assume the length of the process to be 40 µmin order to be able to investigate

the signal propagation along process unhindered of the ends of the process.

Model 71

Table 3.1: Morphological

parameters of astrocyte.

Morphological parameters of

an astrocytic process applied

in the multi-compartment

model.

Parameter Value

length 40 µm

diameter 1 µm

length of single compartment 0.5 µm

In order to obtain a multi-compartment model, each subcellular compartment

has to be divided into several parts. In the case of the processes, implementation is

relatively simple, as the cylinder only needs to be dismantled into smaller cylinders

with a shorter length and the same diameter.

Since the movement of ions and molecules in the extracellular space is also to

be calculated, the extracellular space is divided into individual compartments in

the same way as the intracellular space. The considered volume of the extracellular

space makes up only a fraction of the volume of the intracellular space. The assumed

ration between the volumes of the intra- and the extracellular space is 0.5 and is

adopted from (Halnes et al., 2013).

2 Multi-compartment model

2.1 General form of the diffusion equation

The diffusion of ions and molecules within a subcellular compartment like the

soma or the processes is approximated by a multi-compartment model. For this

purpose the subcellular compartments are divided into a number of segments. The

movement of ions and molecules within the whole subcellular compartment is then

defined by the diffusion of ions and molecules between the single segments.

The movement of ions between neighboring segments is defined by the diffusion

equation which consists of the product of the diffusion coefficient (Dion) and the

partial derivative of the respective ion concentration:

di f f Dion

λ2·∂2[ion](x,t)

∂x2.(3.1)

Here, the diffusion coefficient is scaled by the parameter λ, which is a measure for

the tortuosity. The diffusion coefficient is calculated relative to the cross-section

areas between the individual segments. This reduces the diffusion coefficient for

a smaller cross-section area and increases the coefficient for a larger cross-section

area. As in the following experiments cylinders with a constant diameter are used,

the diffusion constant is the same for all segments of a cylinder.

The concentration change within a single segment is determined by the sum of all

ion currents carrying the respective ion (PIion) multiplied with the area (A) all

currents are going through:

Icomp A·XIion(x,t).(3.2)

Consequently, the change of the ion concentrations in time and space consists of the

sum of the changes within a single segment as well as the ion movement between

72 Multi-compartment model for the signal propagation

the segments:

∂[ion](x,t)

∂tA

F·Vol XIion(x,t)+Dion

λ2·∂2[ion](x,t)

∂x2.(3.3)

Here, Icomp is divided by the product of the Faraday constant and the volume of the

segment (F·Vol) in order to convert the current into a concentration change.

2.2 Time- and space-dependent changes of Ca2+, Na+, K+and IP3

Intracellular concentrations of Ca2+, Na+, K+and IP3For the purpose of com-

puting the time- and space-dependent changes of the intracellular concentrations

of Ca2+, Na+, K+and IP3, I extend the differential equations for the time-dependent

change within a single compartment (see Equations 1.15, 1.16, 1.17 and 1.2 in the

Model Section of part 1) by the diffusion equation. The time-dependent concen-

tration change within a single compartment is defined by the sum of those ion

currents contributing to a change of the respective ion (PIion). The intracellular

Ca2+ concentration forms a special case, since it is determined by Ca2+ currents

at the outer membrane (ICa2+

M) as well as at the internal Ca2+ store (ICa2+

ER ). The

intracellular concentrations of Na+and K+are solely defined by the sum of the

respective ion currents at the plasma membrane, PINa+Mand PIK+M. Only the

intracellular concentration of IP3is not determined by membrane currents, but by

the production (prodIP3) and degradation (de grIP3) of this molecule. Thus, the time-

and space-dependent change of the intracellular IP3concentration is defined by the

IP3-production and degradation as well as the diffusion of IP3between neighboring

compartments:

d[Ca2+]i(x,t)

dt A

F·VolICS ·ICa2+

+AER

F·VolICS ·XICa2+

+DCa2+

λ2·∂2[Ca2+]i(x,t)

∂x2,

(3.4)

d[Na+]i(x,t)

dt A

F·VolICS ·XINa+

+DNa+

λ2·∂2[Na+]i(x,t)

∂x2,(3.5)

d[K+]i(x,t)

dt A

F·VolICS ·XIK+

+DK+

λ2·∂2[K+]i(x,t)

∂x2,(3.6)

d[IP3]i(x,t)

dt prodIP3−de grIP3+DIP3

λ2·∂2[IP3]i(x,t)

∂x2.(3.7)

The subscript irefers to the considered volume of the concentration change, the

intracellular space.

Ca2+ concentration in the internal Ca2+ store The time- and space-dependent

change of the Ca2+ concentration in the internal Ca2+ store is defined in the same

manner. The Ca2+ currents at the internal Ca2+ store (ICa2+ER ) are scaled by the area

(AER) and the volume (VolER) of the internal Ca2+ store (see Equation 1.13 in the

Model Section of part 1). The diffusion of Ca2+ between neighboring compartments

Model 73

determines the propagation of Ca2+ events in the internal Ca2+ store:

d[Ca2+]ER (x,t)

dt AER

F·VolER ·XICa2+

+DCa2+

λ2·∂2[Ca2+]ER (x,t)

∂x2.(3.8)

Extracellular concentration In order to allow ion diffusion in the extracellular

space, the extracellular ion concentrations are computed explicitly. In the first part

of this thesis I determined the extracellular ion concentrations implicitly by linking

them to the change of the respective intracellular ion concentrations (see Model

Section part 1). Since this implicit computation is not sufficient for the analysis of

ion diffusion in the extracellular space, an explicit computation of the extracellular

ion concentration is required. I define the extracellular ion concentrations in the

same way as the intracellular ion concentrations with the exceptions that the sign

of the ion currents is reversed and the volume of the extracellular space (VolECS)

differs from the volume of the intracellular space (VolICS):

d[Ca2+]o(x,t)

dt A

F·VolECS ·(−ICa2+

M)+DCa2+

λ2·∂2[Ca2+]o(x,t)

∂x2,(3.9)

d[Na+]o(x,t)

dt A

F·VolECS ·(−XINa+)+DNa+

λ2·∂2[Na+]o(x,t)

∂x2,(3.10)

d[K+]o(x,t)

dt A

F·VolECS ·(−XIK+)+DK+

λ2·∂2[K+]o(x,t)

∂x2.(3.11)

The subscript orefers to the considered volume of the concentration change, the

extracellular space.

2.3 Time- and space-dependent changes of the membrane potential

The spatial dependency of the membrane voltage is usually defined by the cable

equation (Koch, 1999, p. 25):

∂V

∂t1

2arL

(a2·∂2V

∂x2)−Im+Ie.(3.12)

The cable equation is composed of one term describing the longitudinal flow of

currents ( 1

2arL(a2·∂2V

∂x2)), one term describing those membrane currents crossing

one piece of the membrane (Im) and one term specifying the external input (Ie).

This first term of the cable equation consist of the second-order partial derivative

of the membrane potential multiplied with the radius aand the inverse of the

intracellular resistivity rL. The currents Imand Ieare expressed as currents per

unit area. The multiplication of both currents with the surface area of the segment

is canceled during the derivation of the cable equation. The parameter Cmis the

specific membrane capacitance.

One of the assumptions the cable equation is based on, states that diffusive ion

currents are neglected. Thus, this standard cable equation is not applicable for a

model which considers diffusive currents in the intra- and extracellular space.

74 Multi-compartment model for the signal propagation

Instead, the time- and space-dependent change of the membrane potential can

be derived from the charge density. The following derivation of the space- and

time-dependent change of the membrane potential is taken from the publication of

Halnes et al. (2013). The membrane of a cell functions like a parallel-plate capacitor.

Proceeding from this assumption, a capacitor with capacitance ∂Cseparates two

opposite charges (∂Qand −∂Q) and by that generates a voltage difference:

V∂Q

∂C.(3.13)

The charge within a piece of membrane of length ∂xand with volume Vol a2π·∂x

is then defined by:

∂Qρ·Vol.(3.14)

The charge can be calculated for both the intracellular space (∂Qi) as well as for the

extracellular space (∂Qo). In this case, either the intracellular charge density (ρi) or

the extracellular (ρo) charge density is then taken into account. For simplicity, the

index nwill from now on stand for both the intracellular (i) and the extracellular

case (o).

The charge density (ρn) of either the intra- or the extracellular space is composed

of the sum of all ion concentrations multiplied with their valence zion:

ρn(x,t)FX

ion

zion[ion]n(x,t).(3.15)

The capacitance of a piece of membrane of length ∂xand with area A2πa·∂xis

defined by the product of the membrane capacitance per membrane area (Cm) and

the membrane area:

∂CCm·A.(3.16)

If one now inserts the definitions of the charge (Equation 3.14) and the capacitance

(Equation 3.16) into the aforementioned equation of the membrane potential (Equa-

tion 3.13), one obtains an equation for the membrane potential based on the charge

density. Vcan be expressed in terms of the ion concentrations in the intra- or

extracellular space:

V∂Qi

∂Cρi·VolICS

Cm·Aρi

Cm·SVRICS

,(3.17)

and

V−∂Qo

∂Cρo·VolECS

Cm·Aρo

Cm·SVRECS

.(3.18)

Here, the negative sign follows from the convention, that Vis positive when the

intracellular space has a positive charge. Moreover, by demanding consistency

between Equations 3.17 and 3.18, Halnes et al. (2013) derived the charge symmetry

condition. This condition states that the charge on the inside of a piece of membrane

has the same magnitude but opposite sign compared to the outside of that piece

of membrane. Since the ion concentrations considered in Equations 3.17 and 3.18

are both time- and space-dependent, the resulting definition of the membrane

potential has the same properties. Moreover, the diffusive currents in the intra- or

extracellular space are taken into account.

Model 75

3 Morphology of the multi-compartment model

3.1 Connectivity matrix for open and sealed end condition

The partial derivatives of the diffusion term are approximated by finite differences.

The partial derivative defining the concentration change between neighboring com-

partments is defined by:

∂2[ion](x,t)

∂x2[ion](x+h,t)−2·[ion](x,t)+[ion](x−h,t)

h2.(3.19)

Thus, the concentration of an ion at a specific position and time ([ion](x,t)) is deter-

mined by those particles entering the compartment ([ion](x+h,t)and [ion](x−h,t))

as well as those particles leaving the compartment (2·[ion](x,t)). All compartments

have the same length, h.

For the purpose of obtaining the space-dependent partial derivative of the ion con-

centrations (Equation 3.19), a matrix defining the finite differences (D) is multiplied

with a vector containing the concentration of the respective ion for a specific time

point and all compartments:

∂[ion](x,t)

∂xDion

λ2·−−−→

[ion](t)·D,(3.20)

with

D







−1 1 0 0 . . . 0 0 0

1−2 1 0 . . . 0 0 0

0 1 −2 1 . . . 0 0 0

.....

0 0 0 0 . . . 1−2 1

0 0 0 0 . . . 0 1 −1







.(3.21)

Thus, the finite difference matrix specifies the connectivity of single compartments

within an extended cell. Moreover, it also defines whether an subcellular compart-

ment has an open or a sealed end. In case of a sealed end condition (Equation 3.21)

the ends of the subcellular compartment are caped and no currents flow across the

ends of the subcellular compartment. The open end condition can be thought of a

cut cable without any seal of the ends. In this condition currents can flow freely

out of the ends of the subcellular compartment and the extracellular solution is in

direct contact with the intracellular space.

These two types of termination also result in different finite difference matrices. For

the sealed end condition (Equation 3.21) the end compartments solely receive input

from one neighboring compartment. This fact is represented by the number -1

instead of -2 as the first and last entries of the main diagonal. Moreover, the ending

compartments give input to only one neighboring compartment, which is why the

first and last entries of the main diagonal have only one adjoining field. The finite

difference matrix for the open end condition (Equation 3.22) has one additional

column compared to the one of the sealed end condition. This additional column

76 Multi-compartment model for the signal propagation

allows the consideration of currents crossing the ends of the subcellular compart-

ment. Moreover, each compartment not only receives input from two neighboring

compartments but also gives input to two neighboring compartments.

D







1−2 1 0 0 . . . 0 0 0 0

0 1 −2 1 0 . . . 0 0 0 0

0 0 1 −2 1 . . . 0 0 0 0

.....

0 0 0 0 0 . . . 1−2 1 0

0 0 0 0 0 . . . 0 1 −2 1







.(3.22)

3.2 Connectivity matrix for a branching process

With the intention of creating structures which represent the astrocyte geometry,

single subcellular compartments are combined. As a basic structure I choose the

branching of one process into two smaller processes. The connections between the

individualsubcellular compartments aredefined usinga Hines matrix (Hines, 1984).

Prerequisite for the creation of a Hines matrix is that the individual compartments

of a branch are numbered continuously (see Figure 3.3). The structure of the

matrix itself is similar to the one of the connectivity matrix: each entry of the

matrix corresponds to a connection between two compartments. The continuous

numbering of all compartments now allows a branching or connection of subcellular

compartments like two processes. The following figure is intended to illustrate the

structure of the Hines matrix for a simplified branching. Here, each process solely

consists of three compartments.

1 2 3

45 6

7 8 9

Figure 3.3: Scheme of a branching process. Scheme depicting the division of a branching astrocytic

process into single compartments of the multi-compartment model.

Model 77

D







D1,1D1,20000000

D2,1D2,2D2,30 0 0 0 0 0

0D3,2D3,3D3,40 0 D3,70 0

0 0 D4,3D4,4D4,50 0 0 0

0 0 0 D5,4D5,5D5,60 0 0

0 0 0 0 D6,5D6,60 0 0

0 0 D7,30 0 0 D7,7D7,80

0 0 0 0 0 0 D8,7D8,8D8,9

0000000D9,8D9,9







.(3.23)

4 Model parameter values

The following table shows those parameters, which were introduced within this

third part. All other parameter values are taken from the first part of the thesis.

Some of these parameters are obtained during parameter explorations. The corre-

sponding figures can be found in the Results Section.

Unless otherwise stated, the default parameter values mentioned here are used in

all subsequent simulations.

Table 3.2: Parame-

ters for the propa-

gation of Ca2+, Na+,

IP3and K+.

ParameterValue Source

DCa2+5·10−11 m2

sec see Results Section

DIP35·10−12 m2

sec see Results Section

DNa+1.33 ·10−9m2

sec Qian and Sejnowski (1989)

DK+1.96 ·10−9m2

sec Qian and Sejnowski (1989)

λi3.2 Halnes et al. (2013)

λo1.6 Halnes et al. (2013)

IGluTmax 6.8 A

m2see Results Section

INKAmax 5.8 µA

m2see Results Section

Table 3.3: Morpho-

logical parameters

of the multi-

compartment

model.

ParameterValue

SVRICS 41

µm

SVRECS 81

µm

SVRER 10.32 1

µm

ratioER 0.15

78 Multi-compartment model for the signal propagation

Results

Subject of the results section is first the parametrization of the multi-compartment

model, followed by the analysis of the multi-compartment model. As it has been

shown in the first and second part of this thesis the Ca2+ and Na+dynamics act

independently of each other. Therefore, I first investigate those model parameters

shaping the propagation of the Na+signal. Then, I perform the model parametriza-

tion for the Ca2+ propagation. Subsequently, I study the propagation and interaction

of the Ca2+ and Na+dynamics in single astrocytic processes with either a sealed or

an open end, in branching astrocytic processes and at the perisynaptic astrocytic

process.

1 Na+diffusion in astrocytic processes

First, I parameterize the multi-compartment model in order to fit the propagation

of the Na+signals to experimental data. As it has already been shown in the first

and second part of the thesis, the Na+dynamics act almost independently of the

Ca2+ dynamics. Therefore, I determine the parameter set for the propagation of the

Na+signals considering solely the dynamics of Na+, K+and the membrane voltage

(V) at the outer membrane. The dynamics of Na+, K+and V are determined by

the glutamate transporter (GluT), the Na+-K+pump as well as the Na+and K+leak

currents. The Ca2+ dynamics at the outer membrane or the internal Ca2+ store are

neglected. Moreover, as the Na+diffusion coefficient is known from experiments

(Qian and Sejnowski, 1989), I solely investigate the effect of the maximal pump

currents of the glutamate transporter (IGluTmax ) and the Na+-K+pump (INKAmax) on

the Na+propagation by a parameter exploration. For these parameter explorations, I

study the Na+signal propagation along a cylinder with a length of 40 µm, a diameter

of 1 µm and with sealed ends. As a stimulus for the parameter exploration, I choose

to stimulate the middle compartment of the astrocytic process with a constant

glutamate concentration of 1 mM for a duration of 0.5 second.

80 Multi-compartment model for the signal propagation



erent tortuosity

a b

Figure 3.4: Amplitude and pace of the Na+signal propagation within an astrocytic process for

different values of the maximal pump currents of the glutamate transporter (IGluTmax ) and the

Na+-K+pump (INKAmax ). An astrocytic process with the length of 40 µm, diameter of 1 µm and

sealed ends is stimulated in the middle compartment (length: 0.5 µm) with a constant glutamate

concentration of 1 mM for a duration of 0.5 seconds. The tortuosities of the intra- and extracellular

space are set to their default values. aThe maximal amplitude of the Na+signal at the stimulation site

for parameter combinations of the maximal pump currents of the glutamate transporter (IGluTmax ) and

the Na+-K+pump (INKAmax ). bThe pace of the Na+signal propagation for parameter combinations

of maximal pump currents of the glutamate transporter (IGluTmax ) and the Na+-K+pump (INKAmax ).

The pace is determined by the amplitude and the traveled distance of the Na+signal within the first

100 milliseconds after the begin of the stimulation.

First, I study the Na+signal propagation assuming the default values for the

tortuosity of the intra- and extracellular space (λi= 3.2, λo= 1.6). In this case,

both a high activity of the glutamate transporter and a low activity of the Na+-K+

pump enhance the amplitude of the Na+signal (see Figure 3.4 a). The glutamate

transporter mediates the cotransport of glutamate together with three Na+ions.

Thus, the higher the maximal pump activity of the glutamate transporter (IGluTmax )

is, the more Na+is transported into the cell. The Na+-K+pump counteracts the

glutamate transporter by transporting Na+out of the cell. Therefore, a low activity

of the Na+-K+pump favors the accumulation of Na+within the astrocyte.

The propagation speed of the Na+signal is mainly determined by the activity of

the Na+-K+pump and decreases with an increase of the maximal pump current

of the Na+-K+pump (see Figure 3.4 b). Since the subcellular compartment is only

stimulated at one point in the middle, the glutamate transporter, which transports

Na+into the cell, has only a small effect on the propagation speed. The strength

of the Na+-K+pump, however, does not only determine the amplitude of the Na+

signal, but also the propagation speed. A low maximal pump current of the Na+-K+

pump favors a high amplitude and a high propagation speed, as less Na+is pumped

out of the cell.

According to experimental results, the amplitude of the Na+signal at the stimulation

site is between 25 and 30 mM and the propagation speed within an astrocytic process

is around 60 µm

sec (Langer et al., 2012). In comparison to these experimental results I

Results 81

choose the values for the maximal pump currents to be IGluTmax = 6.8 A

m2and INKAmax

= 5.8 ·10-6 A

m2.

Interestingly, assuming the same tortuosity in the intra- and extracellular (λi=λo=

3.2) space changes the speed of the Na+signal propagation, but not the amplitude

of the signal. The tortuosity scales the diffusion coefficient. Thus, by assuming

different values for the tortuosity in the intra- and extracellular space, the diffusion

coefficients of the intra- and extracellular space differ as well. In contrast, assuming

the same tortuosity in the intra- and extracellular space also results in the same

diffusion coefficients. In this case the maximal pump activities of the glutamate

transporter and the Na+-K+pump have no influence on the propagation speed,

the propagation speed is the same for all parameter combinations (see Figure 3.5

b). The amplitude of the Na+signal, however, is not affected by variations of the

tortuosity. The impact of the membrane transporters during conditions assuming

either the same or different tortuosities will be explained in detail in a following

section.

Same tortuosity

a b

Figure 3.5: Amplitude and pace of the Na+signal propagation within an astrocytic process assuming

the same tortuosity (λi=λo= 3.2) in the intra- and extracellular space for different values of the

maximal pump currents of the glutamate transporter (IGluTmax ) and the Na+-K+pump (INKAmax ).

An astrocytic process with the length of 40 µm, diameter of 1 µm and sealed ends is stimulated in the

middle compartment (length: 0.5 µm) with a constant glutamate concentration of 1 mM for a duration

of 0.5 seconds. aThe maximal amplitude at the stimulation site for parameter combinations of the

maximal pump currents of the glutamate transporter (IGluTmax ) and the Na+-K+pump (INKAmax ). b

The pace of the Na+signal propagation for parameter combinations of maximal pump currents of

the glutamate transporter (IGluTmax ) and the Na+-K+pump (INKAmax ). The pace is determined by the

amplitude and the traveled distance of the Na+signal within the first 100 milliseconds after the begin

of the stimulation.

Moreover, the consideration of either different or the same tortuosities in the

intra- and extracellular space determines whether the Na+concentration under-

shoots its resting concentration. Assuming different tortuosities in the intra- and

extracellular space, the Na+signal undershoots the Na+resting concentration at a

distance of about 10 µm from the stimulation site (see Figure 3.6 a and Figure 3.7

a). In case of the same tortuosity in the intra- and extracellular space, however, I

82 Multi-compartment model for the signal propagation

input

x [μm]

a b

Figure 3.6: The propagation of Na+along the astrocytic process for the chosen parameter combi-

nation of the maximal pump currents of the glutamate transporter and the Na+-K+pump. The

stimulated astrocytic process has a length of 40 µm, a diameter of 1 µm and sealed ends. The process

is stimulated in the middle compartment (length: 0.5 µm) with a constant glutamate concentration

of 1 mM for a duration of 0.5 seconds. aThe time course of the Na+concentration shown for six

different distances from the stimulation site along the astrocytic process. In this case, the tortuosities

in the intra- and extracellular space are set to their default values. bThe time course of the Na+

concentration shown for six different distances from the stimulation site along the astrocytic process.

In this case, the tortuosities in the intra- and extracellular space are the same (λi=λo= 3.2). cScheme

depicting the stimulation and recording sites along the astrocytic process.

do not observe this phenomenon and the Na+signal never undershoots the Na+

resting concentration (see Figure 3.6 b and Figure 3.7 b). This observed effect can

be explained by comparing the transmembrane and diffusive fluxes for these two

conditions (see Figure 3.8). When comparing the difference of the diffusive fluxes

for either assuming a different or a same tortuosity (JDNa di f f −JDNa same) huge differ-

ences between the strength of the diffusive fluxes around the stimulation site can be

observed (see Figure 3.8 b). In case of different tortuosities, the diffusive flux in the

intracellular space is weaker around the stimulation site compared to the diffusive

fluxes for same tortuosities (see Figure 3.8 b (left)). Moreover, the diffusive flux in

the extracellular space for different tortuosities is higher around the stimulation site

compared to the diffusive flux for same tortuosities (see Figure 3.8 b (right)). These

results suggest that in the intracellular space in case of different tortuosities less

Na+diffuses to neighboring compartments. This also results in a higher amplitude

of the Na+signal (see Figure 3.8 a (left)). At the same time in the extracellular space

the diffusive flux around the stimulation site is way higher for different tortuosities

compared to same tortuosities (see Figure 3.8 b (right)). Thus, in case of different

Results 83

Figure 3.7: The propagation of the Na+signal along the astrocytic process for the chosen parameter

combination of the maximal pump currents of the glutamate transporter and the Na+-K+pump.

The stimulated astrocytic process has a length of 40 µm, a diameter of 1 µm and sealed ends.

The process is stimulated in the middle compartment (length: 0.5 µm) with a constant glutamate

concentration of 1 mM for a duration of 0.5 seconds. aThe spatial profile of the Na+concentration

along the astrocytic process shown for five different time points after the start of the stimulation. In

this case, the tortuosities in the intra- and extracellular space are set to their default values. bThe

time course of the Na+concentration along the astrocytic process shown for five different time points

after the start of the stimulation. In this case, the tortuosity in the intra- and extracellular space are

the same (λi=λo= 3.2).

tortuosities this imbalance between diffusive fluxes in the intra- and extracellular

space leads to a high efflux of Na+from the intracellular space (see Figure 3.8 c).

This high Na+efflux is also illustrated by the time derivatives of the Na+concentra-

tion in the intra- and extracellular space (∂Na+

∂tand ∂Na+

∂t) (see Figure 3.8 d). With

additional consideration of the equation for the time derivative of the Na+concen-

tration in the intra- and extracellular space, it becomes clear that whenever the time

derivative is negative the transmembrane flux is stronger than the diffusive flux and

Na+is transported out of the cell. Moreover, the weaker diffusive flux of Na+for

different tortuosities within the extracellular space is also illustrated very well by

the time derivatives (see Figure 3.8 d (right)). In summary, for different tortuosities

the Na+diffusion in the extracellular space is too strong compared to the one of

the intracellular space and pulls Na+out of the cell. In case of the same tortuosity

in the intra- and extracellular space, the Na+diffusion in both spaces is the same

and no imbalance occurs. Moreover, as during this condition no Na+crosses the

membrane, except at the stimulation site, the membrane transporters do not affect

the Na+signal propagation (see Figure 3.5).

In summary, the Na+signal amplitude increases with an increasing maximal pump

current of the glutamate transporter and a decreasing maximal pump current of the

Na+-K+pump (see Figure 3.4). Moreover, the maximal pump current of the Na+-K+

pump mainly determines the propagation speed of the Na+signal as it transports

Na+out of the cell along the astrocytic process. Also the tortuosity of the intra-

and extracellular space affect the Na+signal propagation (see Figure 3.5). In case of

assuming the same tortuosity in the intra- and extracellular space, the Na+signal

84 Multi-compartment model for the signal propagation

propagates with a speed of 200 µM

sec independent of the maximal pump currents of

the glutamate transporter or the Na+-K+pump. In addition, the tortuosity deter-

mines the occurrence of an undershoot of the Na+concentration due to changes in

the Na+diffusion in the intra- and extracellular space (see Figure 3.8).

Figure 3.8 (facing page):Effect of the tortuosity on the Na+diffusion in the intra- and extracellular

space. Spatial profiles of the Na+concentration, the diffusion fluxes, the transmembrane fluxes and

the time derivative of the Na+concentration for either assuming different (blue lines) or same (dashed

orange lines, here: λi=λo= 3.2) tortuosities in the intra- and extracellular space. The stimulated

astrocytic process has a length of 40 µm, a diameter of 1 µm, sealed ends and is stimulated in the

middle compartment (length: 0.5 µm) for a duration of 0.5 seconds. All curves are shown for the time

of stimulus offset (5.5 seconds). This corresponds to the same time points for stimulus onset and offset

as in Figure 3.6. aTraces depicting the spatial profile of the Na+concentration in the intracellular (left)

and in the extracellular space (right). bSpatial profile of the difference between diffusive fluxes for

either assuming different or same tortuosities (JDNa+di f f −JDNa+same) in the interval between stimulus

onset (5 seconds) and 0.5 seconds after stimulus offset (6 seconds). The differences are shown for the

intracellular (left) and the extracellular space (right). cSpatial profile of the transmembrane Na+flux

in the intracellular (left) and in the extracellular space (right). dSpatial profile of the time derivative

of the Na+concentration in the intracellular (left) and in the extracellular space (right). eScheme

illustrating the transmembrane as well as the diffusive Na+flux either assuming a different or the

same tortuosity.

Results 85

Intra

Extra

Intra

Extra

different tortuosity same tortuosity

input

zone

input

zone

transmembrane flux

diffusive flux

86 Multi-compartment model for the signal propagation

2 Ca2+ signal propagation within an astrocytic process

For the purpose of investigating the Ca2+ signal propagation along the astrocytic

process, I first study the Ca2+ dynamics in isolation for different combinations of

the diffusion coefficients of Ca2+ and IP3. An exclusive study of the Ca2+ dynamics

includes solely the release of Ca2+ from internal stores. The Ca2+ entry from the

extracellular space as well as the Na+and K+dynamics are neglected. As a next

step, I investigate the interaction between the Na+and the Ca2+ signal propagation.

Thereby I mainly focus on how the maximal pump currents of the membrane

transporters influence the Ca2+ signal propagation.

For these parameter explorations, I study the Ca2+ signal propagation along a

cylinder with a length of 40 µm, a diameter of 1 µm and with sealed ends. As a

stimulus for the parameter exploration, I choose to stimulate the astrocytic process

with a constant glutamate concentration of 1 mM for a duration of 100 seconds.

Here, I assume the default values for the tortuosity of the intra- and extracellular

space.

First, I study the impact of the diffusion coefficients of Ca2+ and IP3on the Ca2+ signal

propagation (see Figure 3.9). Since IP3is known to drive the Ca2+ signal propagation

and favors the renewal of the signal at the internal Ca2+ store, the diffusion coefficient

of IP3plays a central role in the Ca2+ signal propagation and is therefore subject of

the parameters exploration. The Ca2+ signal propagation is examined by means of

the maximal reached Ca2+ oscillation frequency, the oscillation propagation radius

and the oscillation amplitude (see Figure 3.9).

a b c

Figure 3.9: Ca2+ signal propagation as a function of the IP3and Ca2+ diffusion coefficients. Maximal

frequency, propagation radius and amplitude of the Ca2+ oscillations for different combinations of the

IP3and Ca2+ diffusion coefficients. The Ca2+ oscillation propagation is studied along an astrocytic

process with the length of 40 µm, diameter of 1 µm and sealed ends. The middle compartment

(compartment length = 0.5 µm) is stimulated for a duration of 100 seconds with a constant glutamate

concentration of 1 mM glutamate. The tortuosities of the intra- and extracellular space are set to their

default values. aThe maximal observed Ca2+ oscillation frequency for parameter combinations of

the diffusion coefficients of IP3and Ca2+.bPropagation radius of the Ca2+ oscillations for parameter

combinations of the diffusion coefficients of IP3and Ca2+.cAmplitude of the Ca2+ oscillations for

parameter combinations of the diffusion coefficients of IP3and Ca2+.

The parameter exploration reveals that moderate values of both the Ca2+ diffu-

sion coefficient (DCa2+ : 10-15 m2

sec - 10-10 m2

sec ) and the IP3diffusion coefficient (DIP3:

Results 87

10-14 m2

sec - 10-14 m2

sec ) allow the generation and propagation of Ca2+ oscillations. The

maximal oscillation frequency and the oscillation amplitude decrease with increas-

ing values of both diffusion coefficients (see Figure 3.9 a and b). As the figure

shows, primarily high values of the DIP3seem to suppress the generation of Ca2+

oscillations. Since IP3has no blocking effect on the receptor channels at the internal

Ca2+ store, I assume that the fast diffusion of IP3leads to a Ca2+ release at several

locations simultaneously along the internal store. As a result, the internal Ca2+ store

is depleted and the oscillations are suppressed. Thus, the higher DIP3is, the more

receptors along the internal Ca2+ store are simultaneously activated, which leads

to a lower amplitude of the oscillations (see Figure 3.9 c). In contrast, low values

of DIP3lead to a too slow propagation of IP3signals, so that less IP3receptors are

activated along the astrocytic process and the Ca2+ release from internal stores is

too low to generate Ca2+ oscillations.

As a next step, I investigate the impact of the membrane transporters (glutamate

transporter, Na+-K+pump and Na+-Ca2+ exchanger) on the Ca2+ signal propagation

(see Figures 3.10, 3.11 and 3.12). Here, each figure shows the model behavior in

response to parameter variations of the maximal pump currents of the glutamate

transporter (IGluTmax = [4.8 A

m2, 6.8 A

m2, 8.8 A

m2]) and the Na+-K+pump ((INKAmax ) =

[4.8 ·10-6 A

m2, 5.8 ·10-6 A

m2, 6.8 ·10-6 A

m2]). The three different figures (3.10, 3.11 and

3.12) illustrate the model behavior for three different values of the maximal pump

current of the Na+-Ca2+ exchanger (INCXmax = [0.0001 A

m2, 0.001 A

m2, 0.01 A

m2]). In

general, I observe that an increase of the glutamate transporter activity (IGluTmax )

enhances both the Ca2+ oscillation frequency and the Ca2+ oscillation propagation

radius (see Figure 3.10). Moreover, too high values of the maximal pump current

of the glutamate transporter (IGluTmax ) lead to a suppression of the Ca2+ oscillations

and solely an increase of the intracellular Ca2+ concentration can be observed (see

Figures 3.10 or 3.11). The amplitude of this intracellular Ca2+ raise increases with

increasing maximal pump current of the glutamate transporter (IGluTmax ) due to

the enhanced Na+accumulation within the intracellular space. While the maximal

pump activity of the Na+-K+pump (INKAmax ) has a relatively small effect on the

frequency and the propagation radius of the Ca2+ oscillations (see Figures 3.10 and

3.11), it is decisive for the amplitude of the intracellular Ca2+ raise. The activity of

the Na+-Ca2+ exchanger (INCXmax ), however, strongly affects the Ca2+ signal prop-

agation and suppresses the Ca2+ oscillation generation and propagation for high

values of its maximal pump activity (INCXmax >0.001 A

m2) (see Figures 3.11 or 3.12).

In summary, the isolated study of the Ca2+ signal propagation reveals that mod-

erate values of the diffusion coefficients of Ca2+ and IP3allow the generation and

propagation of Ca2+ oscillations (see Figure 3.9). The analysis of the Ca2+ signal

propagation in combination with the Na+signal propagation reveals that primar-

ily the glutamate transporter and the Na+-Ca2+ exchanger affect the Ca2+ signal

propagation (see Figures 3.10, 3.11 or 3.12). Too high values of the maximal pump

currents of both transporters suppress the Ca2+ signal generation and propagation.

88 Multi-compartment model for the signal propagation

INKAmax

IGluTmax

distance = 0 μm

distance = 20 μm

distance = 40 μm

input

x [μm]

Figure 3.10: Ca2+ signal propagation as a function of the maximal pump activities of the glutamate

transporter (IGluTmax ), the Na+-K+pump (INKAmax ) and the Na+-Ca2+ exchanger (INCXmax ). The

considered astrocytic process has a length of 40 µm, a diameter of 1 µm and sealed ends. The

astrocytic process is stimulated at the first compartment with a constant glutamate concentration

of 1 mM for a duration of 100 seconds (see colored gray area). The tortuosities of the intra- and

extracellular space are set to their default values. aScheme illustrating the stimulation and recording

sites along an astrocytic process of length 40 µm.bThe propagation of the Ca2+ signals is depicted

for parameter combinations of IGluTmax (4.8 A

m2, 6.8 A

m2, 8.8 A

m2) as well as of INKAmax (5.3 ·10-6 A

m2,

5.8 ·10-6 A

m2, 6.3 ·10-6 A

m2) and INCXmax equal to 0.0001 A

m2. Each figure shows the intracellular Ca2+

concentration at three different positions (0 µm, 20 µm, 40 µm) along an astrocytic process.

Results 89

INKAmax

IGluTmax

distance = 0 μm

distance = 20 μm

distance = 40 μm

input

x [μm]

Figure 3.11: Ca2+ signal propagation as a function of the maximal pump activities of the glutamate

transporter (IGluTmax ), the Na+-K+pump (INKAmax ) and the Na+-Ca2+ exchanger (INCXmax ). The

considered astrocytic process has a length of 40 µm, a diameter of 1 µm and sealed ends. The

astrocytic process is stimulated at the first compartment with a constant glutamate concentration

of 1 mM for a duration of 100 seconds (see colored gray area). The tortuosities of the intra- and

extracellular space are set to their default values. aScheme illustrating the stimulation and recording

sites along an astrocytic process of length 40 µm.bThe propagation of the Ca2+ signals is depicted

for parameter combinations of IGluTmax (4.8 A

m2, 6.8 A

m2, 8.8 A

m2) as well as of INKAmax (5.3 ·10-6 A

m2,

5.8 ·10-6 A

m2, 6.3 ·10-6 A

m2) and INCXmax equal to 0.001 A

m2. Each figure shows the intracellular Ca2+

concentration at three different positions (0 µm, 20 µm, 40 µm) along an astrocytic process.

90 Multi-compartment model for the signal propagation

INKAmax

IGluTmax

distance = 0 μm

distance = 20 μm

distance = 40 μm

input

x [μm]

Figure 3.12: Ca2+ signal propagation as a function of the maximal pump activities of the glutamate

transporter (IGluTmax ), the Na+-K+pump (INKAmax ) and the Na+-Ca2+ exchanger (INCXmax ). The

considered astrocytic process has a length of 40 µm, a diameter of 1 µm and sealed ends. The

astrocytic process is stimulated at the first compartment with a constant glutamate concentration

of 1 mM for a duration of 100 seconds (see colored gray area). The tortuosities of the intra- and

extracellular space are set to their default values. aScheme illustrating the stimulation and recording

sites along an astrocytic process of length 40 µm.bThe propagation of the Ca2+ signals is depicted

for parameter combinations of IGluTmax (4.8 A

m2, 6.8 A

m2, 8.8 A

m2) as well as of INKAmax (5.3 ·10-6 A

m2,

5.8 ·10-6 A

m2, 6.3 ·10-6 A

m2) and INCXmax equal to 0.01 A

m2. Each figure shows the intracellular Ca2+

concentration at three different positions (0 µm, 20 µm, 40 µm) along an astrocytic process.

Results 91

3 Model morphologies

This part of the results section aims to provide examples for different morphologies,

which can be applied with the developed computational model. The model allows

besides the application of a single astrocytic process either the investigation of a

branching process or a process with an open end.

3.1 Branching of the astrocytic process

input

a b

input

Figure 3.13: Na+signal propagation in a branching astrocytic process. The branching process is

stimulated either at the bigger or the smaller branch with a constant glutamate concentration of 1

mM for a duration of 500 ms. The bigger branch has a diameter of 1 µm, the smaller branches have

a diameter of 0.5 µm. All branches have the same length of 40 µm. The outer ends of the branching

process are sealed. The tortuosities of the intra- and extracellular space are set to their default values.

aSpatial profile of the Na+concentration along the bigger and one smaller branch of the astrocytic

process at the time point of the stimulation offset. Here, the bigger branch of the astrocytic process

is stimulated. The scheme below depicts the stimulation site in relation to the branching node. b

Spatial profile of the Na+concentration along the bigger and one smaller branch of the astrocytic

process at the time point of the stimulation offset. Here, the smaller branch of the astrocytic process

is stimulated. The scheme below depicts the stimulation site in relation to the branching node.

Here, I study the Na+signal propagation in a branching astrocytic process (see

Figure 3.13, bottom). The bigger branch of the astrocytic process has a diameter of

1µm, the smaller branches have a diameter of 0.5 µm. All branches have the same

length of 40 µmand the outer ends of the branching process are sealed. Moreover,

I assume the default values for the tortuosities of the intra- and extracellular space.

For the purpose of studying the signal propagation at a branching node, I stimulate

the process right before the branching site at either the branch with the larger or

the smaller diameter and record the Na+concentration.

The main difference between the stimulation of either the bigger or the smaller

92 Multi-compartment model for the signal propagation

branch is the amplitude of the generated Na+signal (see Figure 3.13). While the

stimulation of the branch with the larger diameter produces a Na+signal with a

lower amplitude (see Figure 3.13 a), the stimulation of the branch with the smaller

diameter produces a signal with a larger amplitude (see Figure 3.13 b). Since the

stimulated compartment of the bigger branch has a higher volume compared to the

one of the smaller branch, the temporal change of the Na+concentration within

the bigger branch is smaller and the amplitude is lower. The Na+traces in the two

smaller branches are the same.

3.2 Open and sealed end of the astrocytic process

As a further variation of the astrocyte morphology I assume the ends of the astro-

cytic process to be open and investigate the influence of this change on the signal

propagation. For this study, I apply a cylinder with a length of 40 µm, a diameter

of 1 µm and with either open or sealed ends. Here, I assume the default values

for the tortuosity of the intra- and extracellular space. In order to study the signal

propagation under these conditions, I stimulate the astrocytic process at its middle

compartment and record the Na+and Ca2+ concentrations at three different posi-

tions along the astrocytic process (see Figure 3.14).

The open end condition leads to a higher increase of the Na+concentration at the

stimulation site. Moreover, it narrows the signal around the stimulation site and

prevents the large undershoot of the Na+concentration (see Figure 3.14 a and b).

Due to the open end condition an infinite bath of ions it attached to both ends of

the astrocytic process. This also allows an infinite influx of ions through the ends

into the astrocytic process. For this reason, not only the diffusion of Na+ions from

the stimulation site towards the ends of the astrocytic process is weaker and causes

a narrower spatial profile of the Na+concentration, but also prevents the Na+un-

dershoot.

Moreover, the open end condition reduces the propagation radius of the Ca2+ oscil-

lations (see Figure 3.14 c). While the sealed end condition favors the propagation

of long-lasting Ca2+ oscillations within the whole astrocytic process, the open end

condition only allows the propagation of the Ca2+ oscillation within 10 µmaway

from the stimulation site. The attached infinite bath functions like a leak current

and forces the outer compartments of the astrocytic process, those adjacent to the

ends of the process, to keep their Ca2+ resting concentration. The Ca2+ resting con-

centrations is too low for the generation of Ca2+ oscillations and thus the open end

condition lets oscillations vanish in compartments close to the ends of the astrocytic

process.

In summary, the open end condition functions like an infinite bath of ions attached

to the ends of the process. This condition prevents the undershoot of the Na+

concentration, but at the same time decreases the Ca2+ signal propagation radius.

Results 93

Figure 3.14: Ca2+ and Na+signal propagation along an astrocytic process with either a sealed or an

open end. The astrocytic process has either a sealed or an open end, a length of 40 µmand a diameter

of 1 µm. The astrocytic process is stimulated at the middle compartment (compartment length = 0.5

µm) for a duration of 100 seconds with a constant glutamate concentration of 1 mM. The tortuosities

of the intra- and extracellular space are set to their default values. aNa+signal propagation along

an astrocytic processes with either a sealed end (left) or an open end(right). The Na+signal is shown

for three different distances from the stimulation site (at the stimulation site (distance = 0 µm), 10

µmaway from the stimulation site and 20 µmaway from the stimulation site.). bSpatial profile

of the Na+concentration along the astrocytic process with either a sealed end (left) or an open end

(right). The spatial profile is shown for three different time points (200 seconds, 201 seconds and 202

seconds). The astrocytic process is stimulated at time point 100 seconds and the stimulation lasts 100

seconds. cCa2+ signal propagation along an astrocytic processes with either a sealed end (left) or an

open end(right). The Ca2+ signal is shown for three different distances from the stimulation site (at

the stimulation site (distance = 0 µm), 10 µmaway from the stimulation site and 20 µmaway from

the stimulation site.).

94 Multi-compartment model for the signal propagation

4 Modeling of the perisynaptic astrocytic process

This part of the results section aims to give an example for experiments which

can be conducted with the multi-compartment model. In this case I model and

analyze the Ca2+ signal propagation at the perisynaptic astrocytic process. Ca2+

signals in astrocytes can be evoked on several pathways. In general, these pathways

can be divided into either a Ca2+ release from internal stores or a Ca2+ entry from

the extracellular space. Experimental evidence also hints to a spatial separation

of these pathways with a primary Ca2+ entry from the extracellular space at the

perisynaptic astrocytic processes and a strong Ca2+ release from internal stores in

the soma (Srinivasan et al., 2015). Moreover, perisynaptic astrocytic processes are

assumed to be devoid of internal stores, which does not allow a Ca2+ release in these

subcellular compartments. Based on these experimental results the question arises

whether Ca2+ elevations induced at perisynaptic astrocytic processes can travel

along the astrocytic process all the way to subcellular compartments containing

internal Ca2+ stores and induce Ca2+ release from internal stores and thus also

Ca2+ oscillations. The activation of Ca2+ release from internal stores in subcellular

compartments far away from the synapse is of special interest, since the activation of

Ca2+ release would ensure long-lasting and self-sustained Ca2+ oscillations which

are able to travel all the way to the soma. This form of Ca2+ propagation would play

an important role in the integration of synaptic information within the astrocyte.

For the purpose of modeling the ion dynamics at the perisynaptic astrocytic process,

I apply a cylinder with a length of 40 µm, a diameter of 1 µm and with sealed ends.

The first 5 µmof the astrocytic process are assumed to be devoid of the internal Ca2+

store (ratioER = 0). This part of the astrocytic process is stimulated and the Ca2+

and Na+signals are recorded at four different sites along the astrocytic process.

The Ca2+ signal measured at the end of the process is assumed to be equivalent

to a Ca2+ signal, which reaches the soma. If the Ca2+ signal reaches this point of

the process, it is assumed that it would also further propagate to a soma directly

following the process. Here, I assume the default values for the tortuosity of the

intra- and extracellular space.

I study the ion dynamics at the perisynaptic astrocytic process for different values

of the maximal pump current of the Na+-Ca2+ exchanger. The diffusion of Ca2+ is

very slow compared to the one of Na+. Therefore, it is assumed, that mainly the

propagating Na+signal and the associated switch of the Na+-Ca2+ exchanger drives

the Ca2+ signal along the astrocytic process to subcellular compartments containing

the internal Ca2+ store. Therefore, I study the effect of the maximal pumping

strength of the Na+-Ca2+ exchanger on the Ca2+ propagation at the perisynaptic

astrocytic process.

Results 95

stimulation site

beginning of internal store

middle of internal store

end of process

intracellular space

internal Ca2+ store

INCXmax = 0.0001 A

m2INCXmax = 0.001 A

INCXmax = 0.001 A

m2INCXmax = 0.01 A

INCXmax = 0.01 A

cinput

x [



Figure 3.15: Intracellular Na+signals drive the Ca2+ signal propagation at perisynaptic astrocytic

processes. The time course of the intracellular Na+and Ca2+ concentration for three different positions

along the astrocytic process. The time courses of Na+and Ca2+ are shown for three different values for

the maximal pump current of the Na+-Ca2+ exchanger (INCXmax = [0.0001 A

m2, 0.001 A

m2, 0.01 A

m2]). The

astrocytic process has a length of 40 µm, a diameter of 1 µm and lacks the internal Ca2+ store within

the first 5 µmof the process. The ends of the process are sealed. The Na+and Ca2+ concentrations

are recorded at four different positions along the astrocytic process: at the stimulation site, at the

position where the internal Ca2+ store begins, at the middle of the process and at the end of the

process. The tortuosities of the intra- and extracellular space are set to their default values. aTime

course of the intracellular Na+concentration for three different values of the maximal pump current

of the Na+-Ca2+ exchanger and four different positions along the astrocytic process. bTime course

of the intracellular Ca2+ concentration for three different values of the maximal pump current of the

Na+-Ca2+ exchanger and four different positions along the astrocytic process. cScheme depicting the

spatial arrangement of the internal Ca2+ store within the astrocytic process and the recording sites.

96 Multi-compartment model for the signal propagation

The maximal pump current of the Na+-Ca2+ exchanger scales the increase of

the intracellular Ca2+ concentration in compartments devoid of the internal Ca2+

store. It also determines whether Ca2+ oscillations are generated and propagate to

the end of the process. Different strengths of the Na+-Ca2+ exchanger do not affect

the amplitude nor the time course of the intracellular Na+concentration (see Figure

3.15 a). However, the maximal pump activity of the Na+-Ca2+ exchanger strongly

affects the Ca2+ signal generation and propagation along the astrocytic process (see

Figure 3.15 b). A low maximal pump current of the Na+-Ca2+ exchanger (INCXmax

= 0.0001 A

m2) merely leads to an increase of the intracellular Ca2+ concentration, but

not to Ca2+ oscillations. A moderate maximal pump current (INCXmax = 0.001) allows

the generation and propagation of Ca2+ oscillations up to the end of the astrocytic

process. A high maximal pump current in turn suppresses the generation and

propagation of Ca2+ oscillations.

In summary, based on these simulation results I assume, that the high increase of the

Na+concentration and the resulting increase of the intracellular Ca2+ concentration

leads to a Na+-driven propagation of Ca2+ in astrocytic regions devoid if internal

stores.

Discussion

Within this last part of my thesis I developed and studied a multi-compartment

model for the propagation of ion signals in astrocytic subsellular compartments.

In order to develop the multi-compartment model, I assumed the astrocyte point-

model, presented in part 1 of this thesis, to describe the signal generation at one

point of an extended astrocyte and coupled the single point-models by diffusion.

By that I was able to study signal propagation within an extended astrocytic sub-

cellular compartment, like the astrocytic process. The novelty of this model is the

consideration of the propagation of calcium, on the one side, and the propagation

of sodium and potassium, on the other side.

The key finding of this last part is the strong impact of the sodium on the calcium

signal propagation. Since the propagation of sodium and calcium is coupled via the

sodium-calcium exchanger, the maximal pump current of the sodium-calcium ex-

changer has a huge impact on the calcium signal generation. It determines whether

the calcium entry from the extracellular space promotes or suppresses the calcium

signal generation and propagation. In addition, those parameters shaping the

sodium signal propagation, the glutamate transporter and the sodium-potassium

pump, also determine the calcium signal propagation. In general, an increase of

the glutamate transporter activity and a decrease of the sodium-potassium pump

activity led to an increase in both the frequency and the propagation radius of the

calcium oscillations. Moreover, a too high calcium influx into the astrocyte, favored

by a high activity of the glutamate transporter and the sodium-calcium exchanger,

led to a suppression of the calcium oscillations. This observed effect of the glu-

tamate transporter and the sodium-calcium exchanger in the multi-compartment

model was similar to the one observed in the single-compartment model (see part

1). Also in the single-compartment model a high activity of these transporters

led to the suppression of the calcium oscillations. The reason for this effect was

an increased efflux of calcium from the internal calcium store. Thus, the calcium

concentration in the internal store decreased to a concentration level, which did not

allow the generation of calcium oscillations by calcium release from internal stores

any more.

Moreover, I used the developed multi-compartment model to investigate questions

that were difficult to address with biological experiments. One example was the

signal propagation in small perisynaptic processes as imaging methods did not

allow the exact measurement of signals in these small processes. Therefore, I used

the multi-compartment model in order to study the calcium and sodium signal

98 Multi-compartment model for the signal propagation

propagation in perisynaptic astrocytic processes. These subcellular regions were

devoid of internal calcium stores. The question to investigate was whether the

sodium signal propagation is sufficient to transport calcium signals up to a point

of the astrocytic process containing an internal calcium store, where they induce

calcium release from internal stores. This experiment revealed that especially the

strength of the sodium-calcium exchanger affected the calcium signal propagation

along the astrocytic process. While the the sodium signal was not affected by the

strength of the sodium-calcium exchanger, the exchanger primarily determined the

amount and time scale of transported calcium into the cell. For all considered max-

imal pump currents of the sodium-calcium exchanger calcium is transported into

the cell, but not in all cases the time-scale of the calcium transport allowed the gen-

eration of calcium oscillations. While during a too low transporter activity calcium

was transported not fast enough into the cell to activate the IP3receptor channels,

a too high transporter activity led to a depletion of the calcium store and a sup-

pression of the calcium oscillations. The medium strength of the sodium-calcium

exchanger enabled the generation of calcium oscillations. These simulation results

showed that especially the sodium signal propagation contributed to the calcium

signal propagation at perisynaptic astrocytic processes.

It was noticeable that the calcium signal propagation observed in the simulations

had a much smaller propagation radius than it was observed in experiments. Intra-

and intercellular calcium waves propagate via gap junctions and spread over long

distances (300 - 400 µm) (Giaume and Venance, 1998; Scemes and Giaume, 2006).

When I developed the multi-compartment model I did not adjust the maximal

pump currents of the calcium currents at the internal store. As the generation of

long-ranging calcium waves requires the release of a lot of calcium from internal

stores, it is quite likely that the maximal pump currents change to other values in

the multi-compartment model. During the additional consideration of the sodium

dynamics, however, the propagation range increased to longer distances. This could

imply that the sodium signal propagation has a promoting effect on the propagation

of calcium waves. However, the strength of those Ca2+ currents at the internal store

should be investigated in future studies.

Moreover, I could show that especially the sodium signal propagation was affected

by the assumed tortuosities in the intra- and extracellular space and by considera-

tion of either an open or a sealed end of the astrocytic process. By assuming different

tortuosities in the intra- and extracellular space, different diffusion constants were

obtained in these spaces (see Results Section 3). The application of a higher tortuos-

ity in the intracellular space compared to the extracellular space resulted in a lower

diffusion constant in the intracellular space. During the application of this imbal-

ance in the diffusion coefficients I could observe an increased flux of sodium out

of the cell and the resulting undershoot of the intracellular sodium concentration.

However, assuming the same tortuosity and thus the same diffusion coefficient in

the intra- and extracellular space prevented such an undershoot. Interestingly, also

the consideration of open ends of the astrocytic process prevented an undershoot of

the sodium concentration. Since an undershoot of the sodium concentration during

the sodium signal propagation has not been reported in experimental studies (Rose

Discussion 99

and Karus, 2013), the question arouse what the origin of the undershoot is. Various

models analyzed diffusive currents in the intra- and extracellular space of glial cells

and came to the conclusion that the hindrance of cell structures in the extracellular

space is less (Halnes et al., 2013). For this reason, I assume that the same tortuosity

in the intra- and extracellular space prevented an undershoot, but the values of the

tortuosity were not the main reason for triggering an undershoot. Rather, the closed

ends of the process could play a major role, since this condition limited the maxi-

mum number of ions in the system. Under physiological conditions the astrocytic

process would not be isolated, but coupled to other subcellular compartments like

processes and the soma. Thus, in an astrocyte sodium can flow from adjacent sub-

cellular compartments into the astrocytic process and these adjacent compartments

would then function as a sodium bath. Therefore, physiological realistic structures,

such as the combination of several subcellular compartments, might show a realistic

image of signal propagation in astrocytes.

The multi-compartment model presented in this last part of my thesis provided

important insights into the interaction between the propagation of sodium and cal-

cium signals. However, there are only few experimental results, which could be

used to parameterize and test the model. Although, experimental results on the

propagation of calcium and sodium signals exist, the impact of transporter activi-

ties on the signal propagation was usually not investigated. Moreover, experimental

studies so far focused on the propagation of either calcium or sodium signals and

did not investigate the interaction of both ions. This made the parametrization and

thus also the interpretation of the simulation results difficult. Since this is only

one of the first multi-compartment models for signal propagation in astrocytes, the

close interaction between experimentalists and theoreticians is indispensable for

the development of further multi-compartment models.

100 Multi-compartment model for the signal propagation

Discussion and Outlook

1 Discussion

Within my thesis I developed a computational model describing the calcium signal

generation within one compartment of an extended astrocyte. I further derived a

reduced version of this model in order to analyze the steady-state behavior of the

model and also combined the point-models (full model) to a multi-compartment

model in order to study the signal propagation within the astrocyte.

The novelty of my developed computational model for calcium signal generation

in astrocytes (see part 1) is the consideration of two pathways contributing to the

signal generation. These pathways are the calcium release from internal stores and

the calcium entry from the extracellular space. Whereas the mathematical descrip-

tion of calcium release from internal stores has already been used in numerous

models (Li and Rinzel, 1994; Nadkarni and Jung, 2007; De Pittà et al., 2009), the

consideration of calcium entry via the sodium-calcium exchanger is new. Pub-

lished computational models considering the calcium entry from the extracellular

space rather focus on voltage-gated calcium channels (Postnov et al., 2008; Zeng

et al., 2009; Li et al., 2012). Moreover, the model can be parameterized for different

positions along the astrocytic process by varying the surface volume ratio of the

astrocytic compartment together with the volume fraction of internal calcium store.

Thus, my point-model of an astrocyte allows to study the generation of calcium

signals under different morphological conditions.

Within the second part of my thesis I derived a reduced version of the model pre-

sented in part 1 by a separation of time scales of the model variables. On the one

hand, this reduced model has very well mapped the quantitative and qualitative

behavior of the full model, and, on the other hand, the reduced model enabled

a detailed analytic model investigation in order to predict its steady-state behav-

ior. This approach of the separate computation of the steady-state values of the

fast model variables and the subsequent analysis of the reduced model behavior

differs significantly from other computational studies investigating calcium signal

generation in astrocytes. Most other computational models either focus on only one

mechanisms for the generation of calcium signals in astrocytes and are thus easy

to analyze, or consider more than one mechanisms and are thus high-dimensional

and complicated to analyze. My approach takes different mechanisms for the gen-

eration of calcium signals into account and still allows a detailed model analysis.

Thus, my approach gives important insights into the interaction of the mechanisms,

101

102

which would not be offered by numerical simulations.

Object of the third part of this thesis is the development of a multi-compartment

model for the signal propagation in astrocytic subcellular compartments. In order

to do so I diffusively coupled the single-point models presented in the first part of

the thesis. The obtained multi-compartment model allowed me to study the prop-

agation of calcium and sodium signals along a single astrocytic process or along

a branching process. To my best knowledge the developed multi-compartment

model is the first, which combines calcium and sodium dynamics in astrocytes.

Other computational models rather focus on the propagation of calcium waves

within astrocyte networks (Ullah et al., 2006; Kang and Othmer, 2009) than on the

propagation along one astrocytic process. Thus, the multi-compartment model

allows to study complex propagation patterns like for example the interaction of

calcium and sodium signal propagation.

In summary, my developed models allow to study the generation and propagation

of calcium signals in astrocytes and by that contribute to a better understanding

of the astrocyte function. However, the investigation of astrocytes has only begun

many decades after that of neurons. Thus, computational models of astrocytes

are not yet as sophisticated in terms of the investigation of the signal propagation

or the astrocyte morphology as neuron models. Furthermore, existing astrocyte

models function either as a reproduction of experimental data or as an extension

of experiments and by that investigate problems which are not accessible with cur-

rent experimental methods. For the purpose of developing and applying astrocyte

models more specifically, a close cooperation between experimentalists and theo-

reticians is essential. In this way a close link between model predictions, which are

tested in experiments, and model fine tuning, which is based on experimental data,

could be implemented. Based on this, important aspects regarding the functional

role of astrocytes in neural information processing could be targeted.

2 Outlook

Based on the results presented in this thesis, several model extensions and future

studies are possible. This possible future studies can be divided into extensions of

the single-compartment model and the multi-compartment model.

The single-compartment model could be extended with further mechanisms me-

diating a calcium entry from the extracellular space. Possible mechanisms are

for example voltage-gated calcium channels or store-operated calcium channels.

These mechanisms could be applied to both the full and the reduced model in

order to investigate the model behavior in response to time-varying and constant

stimuli, respectively. Moreover, further channels mediating a sodium influx like

the NMDA-receptors could be added to the model. By considering additional

NMDA-receptors, the model could account for astrocytes in different brain regions

as NMDA receptors are mainly expressed in cortical astrocytes (Conti et al., 1997;

Verkhratsky et al., 1998; Conti et al., 1999). Moreover, the single-compartment model

could be extended with calcium buffering as calcium-binding proteins determine

the amount of free calcium in a cell. Experimental studies could prove the effect

Discussion 103

of calcium-binding proteins on the intra- and intercellular propagation of calcium

signals (Wang et al., 1997; Gerlai et al., 1995). In order to develop and analyze a

computational model for calcium buffering in astrocytes, I would suggest to first ex-

tend the single-compartment model with calcium buffering. Subsequently, I would

introduce this mechanisms to the multi-compartment model in order to study the

effect of calcium buffering on intracellular calcium signal propagation.

The multi-compartment model could be extended with more complex and realistic

morphologies. For example, it would be conceivable to link several processes that

form a primary branch. Furthermore, so far the signal propagation within the soma

has not been considered. Especially in the soma the signal propagation in three

directions is of great importance. Therefore, a more detailed model, which takes

the propagation of the signals into three directions into account, should be used

for the modeling of the soma. Once a detailed model of the soma and primary

astrocytic branches is developed, a combination of these structures could be used

to investigate the integration of neuronal signals within the astrocyte.

104

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