Investigations of the neural firing threshold
Abstract
The variability of the neuronal spike initiation points (SIP) contradicts the classical concept of a simple voltage threshold for spike initiation. Mathematically such a phenomenon is an indicator for the state space missing (one or more) state variables. Because neurons can be, however, depolarized clearly above their firing threshold if this is done slowly enough, the first time derivative of the membrane voltage, dU/dt, is introduced as the second state variable. Within this 2D state space the spike generator can be interpreted as a dynamical system with two attractors (resting potential and spike maximum). They are separated by a separatrix that is defined by the SIPs of all spikes. Within the 2D state space the SIPs are phenomenologically characterized by a sharp kink of the trajectories where the spike nearly vertically emerges from the sub-threshold activity. An algorithm has been developed that is able to detect the SIPs reliably, precise and robustly. This procedure is verified on Hodgkin-Huxley (HH) models whose real SIPs were additionally determined by successively shortening the stimulus ramps. In this context we could show that most HH models show an unphysiologically shallow spike onset. For models with a fast spike onset the found SIP was in good accordance with the real SIPs. However, the shallower the model's spike onset, the later the found SIPs appeared. As all measured cells showed a very abrupt spike onset it can be assumed that the algorithm is able to detect the SIPs correctly within neurons. Both neurons and models were now tested with ramps of different slopes in order to test the dU/dt dependence of the threshold. For the first spikes induced by these ramps the SIP finder algorithm was used to determine the SIPs. To these SIPs a function was fitted within the U-dU/dt state space resembling the cells' or models' separatrix. It could be shown that all cells exhibit a separatrix. All possible orientations have been observed: vertical (voltage threshold), horizontal (dU/dt threshold) and their combinations (slash-type and backslash-type). HH models only showed slash-type separatrices. Pre-defined separatrices that were built into spiking leaky integrate-and-fire models could be precisely reproduced by SIP finder algorithm. Neural separatrices showed a high sensitivity for small changes of cellular state parameters. Pharmacological experiments did not show a systematic effect within this experimental setup. Already the application of a synaptic block cocktail necessary for the prevention of epileptiform activity changed the separatrix clearly before the actual potassium channel block was applied.
Full text
In es iga ions o he Neu al Fi ing Th eshold
Ca s en E dmann
Sep embe 2011
Im Fachbe eich Biologie, Chemie, Pha mazie de F eien Uni e si ä Be lin
einge eich e Disse a ion
1. Gu ach e : P o . R. Menzel, F eie Uni e si ä Be lin
2. Gu ach e : P o . U. Heinemann, Cha i é Be lin
Tag de P ü ung: 17.4.2012
Hie mi e siche e ich, die o liegende Disse a ion selbs s ändig ange e ig
und keine auÿe den angegebenen Hil smi eln e wende zu haben.
O e sbe g, 30.9.2011 .................................................
(Ca s en E dmann)
Auch die längs e Reise beginn mi einem Sch i .
Chinese saying
Papa, das geh doch nich , dass Du jedes Wochenende das ganze Wochenende
nu an Deine blöden A bei sch eibs !
Lena E dmann
Con en s
1 In oduc ion 6
1.1 App oach ............................. 7
1.2 Concep s.............................. 8
1.2.1 2-Dimensional S a e Space . . . . . . . . . . . . . . . . 8
1.2.2 The Spike Ini ia ion Poin (SIP) . . . . . . . . . . . . . 8
1.2.3 The Th eshold Sepa a ix . . . . . . . . . . . . . . . . 10
1.3 Ques ions o he Sys em . . . . . . . . . . . . . . . . . . . . . 12
2 P ecise SIP De ec ion 13
2.1 Ma e ials and Me hods . . . . . . . . . . . . . . . . . . . . . . 13
2.1.1 Finding he SIP . . . . . . . . . . . . . . . . . . . . . . 13
2.1.2 Checking he SIP nde . . . . . . . . . . . . . . . . . 18
2.2 Resul s............................... 18
2.2.1 SIPFinde ......................... 18
2.2.2 La e Spike Phenomenon . . . . . . . . . . . . . . . . . 21
2.3 Discussion............................. 23
2.3.1 Poin o Spike Ini ia ion . . . . . . . . . . . . . . . . . 23
2.3.2 Speed o Spike Dynamics . . . . . . . . . . . . . . . . . 24
2.3.3 La e Spike Phenomenon . . . . . . . . . . . . . . . . . 24
3 Neu ons ha e a 2D Fi ing Th eshold 26
3.1 Ma e ials and Me hods . . . . . . . . . . . . . . . . . . . . . . 26
3.1.1 Objec i e ......................... 26
3.1.2 Cells............................ 26
3.1.2.1 Animals..................... 26
3.1.2.2 Slice P epa a ion . . . . . . . . . . . . . . . . 27
3.1.2.3 Reco ding Elec odes . . . . . . . . . . . . . . 28
3.1.2.4 The Expe imen al Se up . . . . . . . . . . . . 28
3.1.2.5 Cell Sea ch and S imulus Pa ame iza ion . . 29
3.1.2.6 S imula ion . . . . . . . . . . . . . . . . . . . 30
3.1.2.7
In Vi o
Expe imen s . . . . . . . . . . . . . 31
1
CONTENTS
2
3.1.3 Models........................... 32
3.1.3.1 Modied Leaky In eg a e-And-Fi e Model . . 32
3.1.3.2 Hodgkin-Huxley-Model . . . . . . . . . . . . . 35
3.1.3.3 Wang-Buzsáki-Model . . . . . . . . . . . . . . 35
3.1.3.4 Awiszus-Model . . . . . . . . . . . . . . . . . 35
3.1.3.5 Reduced Awiszus-Model . . . . . . . . . . . . 36
3.1.3.6 Modied Awiszus-Models . . . . . . . . . . . 36
3.1.3.7
In Silico
Expe imen s . . . . . . . . . . . . . 36
3.1.4 Da a Analysis . . . . . . . . . . . . . . . . . . . . . . . 37
3.1.4.1 Cell Quali y C i e ia . . . . . . . . . . . . . . 37
3.1.4.2 Da a Pos p ocessing . . . . . . . . . . . . . . 37
3.1.4.3 Da a Analysis . . . . . . . . . . . . . . . . . . 37
3.2 Resul s............................... 38
3.2.1 Cells............................ 38
3.2.1.1 Sepa a ices . . . . . . . . . . . . . . . . . . . 38
3.2.1.2 Ose ...................... 41
3.2.1.3 S abili y and Sensi i i y . . . . . . . . . . . . 41
3.2.1.4 Pha macology . . . . . . . . . . . . . . . . . . 44
3.2.2 Models........................... 44
3.2.2.1 Co ec ness o Sepa a ix Cons uc ion . . . . 44
3.2.2.2 Sepa a ices . . . . . . . . . . . . . . . . . . . 44
3.2.2.3 Real s. Found Sepa a ices . . . . . . . . . . 46
3.2.2.4 Ose ...................... 46
3.2.2.5 Blocking he A- ype Po assium Channel . . . 51
3.3 Discussion............................. 53
3.3.1 Model Sepa a ices . . . . . . . . . . . . . . . . . . . . 54
3.3.2 Sepa a ices in Gene al . . . . . . . . . . . . . . . . . . 54
3.3.3 Sensi i i y......................... 56
3.3.4 Pha macology . . . . . . . . . . . . . . . . . . . . . . . 57
3.3.5 Conclusion......................... 58
3.3.6 Ou look .......................... 59
Bibliog aphy 60
A Supplemen s 64
A.1 Abs ac .............................. 64
A.2 Zusammen assung......................... 66
A.3 Acknowledgemen s . . . . . . . . . . . . . . . . . . . . . . . . 68
Lis o Figu es
1.1 A spike in he 2D s a e space . . . . . . . . . . . . . . . . . . 9
1.2 The SIP and sepa a ix concep s . . . . . . . . . . . . . . . . 11
2.1 SIP Finde : s snapsho . . . . . . . . . . . . . . . . . . . . 15
2.2 SIP Finde : second snapsho . . . . . . . . . . . . . . . . . . . 16
2.3 SIP Finde : nal snapsho . . . . . . . . . . . . . . . . . . . . 17
2.4 Found SIPs a e plausible . . . . . . . . . . . . . . . . . . . . . 19
2.5 Die ences be ween SIPs and SIPs in models . . . . . . . . . 20
2.6 Spike dynamics in neu ons and models . . . . . . . . . . . . . 21
2.7 The la e spike phenomenon . . . . . . . . . . . . . . . . . . . 22
3.1 S imulusscheme.......................... 32
3.2 Beha io o he modied leaky in eg a e-and- e model . . . . 34
3.3 The 4 ypes o sepa a ices . . . . . . . . . . . . . . . . . . . . 39
3.4 All measu ed sepa a ices . . . . . . . . . . . . . . . . . . . . 40
3.5 Neu on sepa a ices a e ose independen . . . . . . . . . . . 42
3.6 Sepa a ix empo al s abili y . . . . . . . . . . . . . . . . . . . 43
3.7 The sepa a ix is a sensible indica o . . . . . . . . . . . . . . 45
3.8 Algo i hm can ep oduce eal sepa a ices . . . . . . . . . . . 46
3.9 Ose dependence o model sepa a ices: LIF models . . . . . 47
3.10 Ose dependence o model sepa a ices: Wang-Buzsáki . . . . 48
3.11 Ose dependence o model sepa a ices: Awiszus . . . . . . . 49
3.12 Ose dependence o model sepa a ices: Hodgkin-Huxley . . . 50
3.13 Modeled A- ype channel block . . . . . . . . . . . . . . . . . . 52
3
Nomencla u e
2D Two-dimensional
3D Th ee-dimensional
4-AP 4-Aminopy idine
ACSF A icial ce eospinal uid
AMPA Alpha-amino-3-hyd oxy-5-me hyl-4-isoxazolep opionic acid
AP Ac ion Po en ial
APV (2R)-amino-5-phosphono ale ic acid, a selec i e NMDA ecep o an-
agonis
CA co nu ammonis, a cu ed s uc u e o densly packed neu onal soma a
wi hin he hippocampal o ma ion
CGP-55845A [(2S)-3-[[(1S)-1-(3,4-dichlo ophenyl)e hyl]amino]-2-hyd oxyp opyl]
(phenylme hyl) phosphinic acid, a GABA
B
an agonis
CNQX 6-cyano-7-ni oquinoxaline-2,3-dione, a compe i i e AMPA/Kaina e
ecep o an agonis
DG den a e gy us, a V- o med s uc u e o neu onal soma a wi hin he
hippocampus
SIP ound SIP: SIP ound by using he o wa d/backwa d eg ession Me hod
GABA Gamma-aminobu y ic acid
LIF leaky in eg a e-and- e
NMDA N-Me hyl-D-Aspa a
PCI Pe iphe al Componen In e connec , a compu e bus o a aching
ha dwa e de ices o a compu e
4
LIST OF FIGURES
5
SIP eal SIP: SIP de e mined by sho ening he s imulus un il no spike
appea ed.
SEM S anda d e o o mean
SIP spike ini ia ion poin , he poin o no e u n wi hin he cell's ol age
ace. I he signal exceeds his poin a spike is una oidable.
U
Vol age, he e he memb ane ol age
˙
U
s ime de i a i e o he ol age
Chap e 1
In oduc ion
Conce ning he b ain's in o ma ion p ocessing capabili ies, neu ons a e he
cen al elemen s o b ain unc ion. They communica e ia ac ion po en ials
(APs, also called spikes) sen o each o he along he axons and ansmi ed
h ough synapses.
Despi e he ac ha mo e and mo e de ailed knowledge e ode many
neu obiological dogma a i seems s ill gene ally co ec o s ick o he iews
ha dend i e and soma a e he inpu and p ocessing egions o incoming
signals, ha he ac ion po en ials a e gene a ed om he memb ane po en ial
uc ua ions somewhe e a ound he axon hillock, ha spikes a e he bina y
elemen s o he neu al language, and ha hey a e ansmi ed along he
axon.
Looking a he spikes as he key elemen o neu al language, and ocusing
a he widely eco ded in acellula memb ane po en ial, his wo k will ocus
on he cen al ques ion: wha a e he ol age condi ions wi hin a neu al
soma ha elici a spike? I is gene ally ag eed ha spikes a e gene a ed by
a ol age h eshold wi hin he neu on, i.e. he e is one xed h eshold le el
he exceedance o which elici s a spike. Howe e , i is also long known ha
he ol age h eshold may a y in cen al neu ons (e.g. see [44, 9, 5] and also
[36]) as well as in dis an elemen s such as he neu omuscula junc ion (e.g.
see [38]).
When looking a he spike gene a o om he iewpoin o dynamical
sys ems ma hema ics, he a iable
U
o he ol age would hus be i s s a e
a iable, and he associa ed s a e space would hen be one-dimensional. Bu
he ac ha he h eshold a ies wi hin he neu on immedia ely indica es
ha he s a e space is oo low-dimensional and demands he in oduc ion o
u he s a e a iables.
6
Chap e 2
A new Algo i hm o P ecise
De ec ion o he Poin o Spike
Ini ia ion
2.1 Ma e ials and Me hods
The analysis p ocedu es desc ibed he e we e pe o med on ol age da a o
ac ion po en ials, hei ime de i a i e as well as he app op ia e s imulus
signals. All signals we e sampled wi h 35 kHz. They we e eco ded
in i o
and simula ed
in silico
acco ding o he me hods desc ibed in sec ion 3.1.
Please e e o ha sec ion o de ails on ma e ials, eco ding, modeling o
he like.
2.1.1 Finding he SIP
As has been s a ed abo e (see 1.2.1), he SIP denes he ansi ion om
he passi e p e-spike (i.e. sub- h eshold) dynamic o he ac i e (i.e. supe -
h eshold) spike dynamic. The algo i hm p esen ed he e will make use o he
dis inc ea u es o he wo ypes o dynamic in he
U
-
˙
U
-plane.
To nd he ansi ion poin , we will y o es ima e i om wi hin he
p e-spike egime as well as om wi hin he spike egime (see g. 2.1). Wi hin
he p e-spike dynamic, we ake a 2.8 ms (100 da a poin s a 35 kHz) long da a
window which is nea he spike, bu s ill clea ly inside he passi e egime.
This is assu ed by dening a xed dis ance o 0.6 ms (20 da a poin s a
35 kHz) om he spike's ol age maximum which is easy o de ec . We
calcula e a linea eg ession om hese da a, hus p ojec ing i s ace beyond
he SIP. This is jus iable as he p e-spike dynamics is slow in ela ion o he
13
CHAPTER 2. PRECISE SIP DETECTION
14
Algo i hm 2.1
Algo i hm o he SIP nde
START
# inding he SIPs om a ol age ace
Find spike maxima
Con e ol age da a o 2-D U-dU da a
De ine da a window ela i e o max(U) di ec ly be o e, bu clea ly no inside spike
De e mine sui able ex apola ion unc ion h ough hese da a
De ine da a window le o max(dU) spanning abou 1/8 o he spike ci cle
De e mine linea eg ession unc ion h ough hese da a
Fo (each spike max)
de e mine in e sec ion be ween linea eg ession unc ions
While (SIP no ound)
mo e spike window one da a poin backwa d
de e mine ac ual in e sec ion be ween linea eg ession unc ions
I (ac ual in e sec ion is igh o las in e sec ion)
las in e sec ion = ac ual in e sec ion
Else
iden i y da a poin nea es o las in e sec ion poin
SIP = iden i ied nea es da a poin
Endi
Endwhile
Append SIP o SIP-Lis
End o
Re u n SIP-Lis
STOP
sho da a window as well as in compa ison o he spike dynamic, especially
o he simple amp s imuli used in his s udy.
Wi hin he spike egime, we s a a
max( ˙
U)
and dene a simila window
le om he e wi h a leng h o 0.14 ms (4 da a poin s a 35 kHz). We calcu-
la e he linea eg ession om his window as well and use i as a backwa d
es ima e o he spike dynamic. As he spike ace is nea ly linea a spike
s a , his u ns ou o be a sui able assump ion
1
. As an es ima e o he
SIP, we hen calcula e he in e sec ion poin o bo h eg ession lines which
will lie a le a his s s ep. In each subsequen i e a ion, he in-spike e-
g ession window is shi ed le (i.e. backwa d in ime) one da a poin a each
s ep while he window size is kep cons an . As a consequence, he in-spike
eg ession line will become s eepe and s eepe , and he in e sec ion poin
will shi igh on he p e-spike eg ession line wi h each i e a ion. Wi h he
in-spike da a window mo ing backwa d in ime, i will e en ually en e he
p e-spike egime, hus esul ing in a shallowe eg ession line and hus in he
in e sec ion poin shi ing le again. The algo i hm s ops he e, e u ning he
igh mos in e sec ion poin as he bes es ima e o he SIP. This es ima e
is hen mapped o he eal da a by choosing he da a poin wi h he smalles
euclidean dis ance o he SIP es ima e.
1
Mo e sophis ica ed s ha e also been es ed, bu hey u ned ou o be oo p one o
sligh a ia ions in spike onse dynamics.
CHAPTER 2. PRECISE SIP DETECTION
15
-50 -40 -30 -20 -10
0
50
100
150
200
-41.5-41-40.5-40-39.5-39-38.5-38
-4
-2
0
2
4
6
8
10
Figu e 2.1: SIP Finde : s snapsho .
This gu e illus a es an ea ly s age o he SIP nde p ocess (see gu es 2.2
and 2.3 o subsequen snapsho s). Wi hin all 3 gu es he le plo displays
he spike onse in he s a e space, wi h he dense do s esembling he passi e
p e-spike dynamics ollowed by he s a o he (ac i e) spike ajec o y.
The igh plo in each gu e shows a magnied iew o he ansi ion poin .
The g ay poin s a e he da a clea ly wi hin he p e-spike dynamic which
a e used o calcula e he linea p e-spike eg ession (blue line, blue do s a e
he calcula ed suppo ing poin s). The magen a-colo ed poin s a e clea ly
wi hin he spike dynamic, hei linea eg ession is he g een line (g een do s
a e he calcula ed suppo ing poin s). The ed do is he cu en in e sec ion
o bo h eg ession lines, he yellow do s esemble he p e ious in e sec ion
poin s. The ed ci cle deno es he p e ious in e sec ion poin ela i e o he
cu en one.
U
[mV] is on ho izon al axis,
˙
U
[mV/ms] a e ical axis.
CHAPTER 2. PRECISE SIP DETECTION
16
-50 -40 -30 -20 -10
0
50
100
150
200
-41.5-41-40.5-40-39.5-39-38.5-38
-4
-2
0
2
4
6
8
10
Figu e 2.2: SIP Finde : second snapsho .
This gu e shows he nex s ep a e gu e 2.1. The in-spike eg ession
window has been shi ed backwa d in ime by one da a poin . The esul ing
g een eg ession line in e sec s wi h he blue p e-spike line mo e igh han
be o e (see ed do ). The p e ious in e sec ion poin ( ed do in le gu e2.1)
is now yellow, ma ked wi h a ed ci cle. Wi h each s ep he in e sec ion poin
shi s mo e and mo e igh on he blue p e-spike eg ession line. See gu e
2.1 o gu e legend.
CHAPTER 2. PRECISE SIP DETECTION
17
-50 -40 -30 -20 -10
0
50
100
150
200
-41.5-41-40.5-40-39.5-39-38.5-38
-4
-2
0
2
4
6
8
10
Figu e 2.3: SIP Finde : nal snapsho .
This gu e illus a es he nal s age o he SIP nde sequence. The igh
gu e is he las s ep in he p ocess: now he le bounda y o he in-spike
eg ession window has eached he p e spike dynamic, esul ing in a shallowe
g een eg ession line. This causes he eg ession in e sec ion o be mo e le
on he blue eg ession line (compa e ed do wi h ed ci cle deno ing he
las in e sec ion poin . This causes he p ocess o s op, and he igh mos
in e sec ion poin is aken o be he s es ima e o he SIP. See gu e 2.1
o gu e legend.
CHAPTER 2. PRECISE SIP DETECTION
18
2.1.2 Checking he SIP nde
In o de o e i y he p ecise unc ion o he SIP nde we need an al e na i e
way o de e mine he ansi ion poin om p e-spike o spike dynamic. As
his poin is dened as he poin -o -no- e u n we can nd i by making he
s imulus amp sho e and sho e un il no spike is elici ed any mo e. This
would hus mean o swi ch o he s imulus exac ly a he SIP. Despi e all
eo s o a clean adjus men o elec ode and eco ding pa ame e s a i ac s
in he eco ded ol age ace (due o he sha p edge a s imulus swi ch-o)
could no be a oided. As he delica e analysis o he SIPs would ha e been
ho oughly aec ed by hese a i ac s his p ocedu e was ca ied ou solely
on models.
In his app oach he SIP is sea ched o using he bina y sea ch algo i hm.
The i e a i e p ocedu e s a s wi h a amp s imulus o an ini ial leng h
L0
.
In he nex s eps he amp is ei he elonga ed (i he las i e a ion did no
elici a spike) o
Ln+1 =Ln+Ln
2
o sho ened (i he las i e a ion did elici
a spike) o
Ln+1 =Ln−Ln
2
o i e a ion
n
wi h
n= 0
a he ini ial s ep.
This is epea ed un il a empo al esolu ion o
∆ 50.001
ms is eached.
These eal SIPs ( SIPs) can hen be compa ed o he nde 's SIPs ( SIPs)
by calcula ing he empo al die ence be ween bo h poin s. Using he sep-
a a ix concep p esen ed in chap e 3 we can also analyze he empo al
dis ance be ween comple e sepa a ices.
2.2 Resul s
2.2.1 SIP Finde
Fo all na u al spikes analyzed we can p ecisely and obus ly de e mine he
phenomenological SIP wi h he algo i hm desc ibed he e. P ecisely means
ha he SIPs - he only ones a ailable o na u al spikes - a e bo h plausible
in
U
and
˙
U
as well as e y close o he p e-spike sub h eshold dynamic (see
g. 2.4). Robus ly means ha he SIP nde is able o nd he SIP in all
cases o heal hy spikes.
Wi h he s imulus sho ening p ocedu e using he simula ed models we
ha e an al e na i e ool a hand (see 2.1.2) o de e mine he eal SIPs. Com-
pa ing hese wi h he ound SIPs, we encoun e la ge die ences be ween
SIPs and SIPs (up o 1.3 ms) wi h mos Hodgkin-Huxley ype models used
he e. This means ha eaching he SIP does induce he i e e sible dynamic
ha will lead o a spike, bu ob iously in he beginning his dynamic is much
oo weak o esul in any de ec able eec s in he memb ane ol age.
CHAPTER 2. PRECISE SIP DETECTION
19
-4 -3 -2 -1 0
-60
-40
-20
0
20
-60 -40 -20 0 20
0
100
200
300
Figu e 2.4: Found SIPs a e plausible.
SIP loca ion is plausible in
U
(le plo ),
˙
U
as well as in he
U
/
˙
U
s a e
space ( igh plo ). Colo s code om ed ia o ange, yellow, g een o blue
o s eepness o s imula ion amp: ed means shallow amps, blue a e s eep
amps. Black do s indica e posi ion o de ec ed SIPs.
Le plo shows ol age [mV] agains ime [ms], spikes a e aligned wi h hei
ol age maximum a =0. Same g oup o spikes is shown in igh plo in s a e
space wi h
U
[mV] on he ho izon al axis and
˙
U
[mV/ms] on he e ical axis.
CHAPTER 2. PRECISE SIP DETECTION
20
-4 -3 -2 -1 0
-60
-40
-20
0
20
40
SIPs
SIPs
-60 -40 -20 0 20 40
-200
0
200
400
600
SIPs
SIPs
Figu e 2.5: Die ences o SIPs o SIPs in models.
This Plo shows he Wang/Buzsáki a ia ion o he Hodgkin/Huxley ype
model, answe ing o amp s imula ion. Colo s code om ed ia o ange,
yellow, g een o blue o s eepness o s imula ion amp: ed means shallow
amps, blue a e s eep amps. Black do s indica e posi ion o de ec ed SIPs:
while he igh -hand cloud o SIPs indica e he SIPs in bo h plo s, he le -
hand cloud in he igh plo as well as co espondingly he le -hand s ipes
o do s in he le plo show he SIPs.
Le plo shows ol age [mV] o e ime [ms], spikes a e aligned wi h hei
ol age maximum a =0. Same g oup o spikes is shown in igh plo in
s a e space.
U
[mV] is on ho izon al axis,
˙
U
[mV/ms] a e ical axis.
CHAPTER 2. PRECISE SIP DETECTION
21
-60 -40 -20 0 20 40
-50
0
50
100
150
200
250
300
-60 -40 -20 0 20
-400
-200
0
200
400
600
800
-60 -40 -20 0 20
0
100
200
300
Figu e 2.6: Spike dynamics in neu ons and models.
Le : T ajec o ies o he o iginal Hodgkin-Huxley model. Spike onse is
e y shallow. Middle: T ajec o ies o he modied Awiszus model. No e
ha spike onse dynamic is much as e he e. Righ : T ajec o ies o a eal
neu on. Spikes a e s ill as e a spike onse , he ajec o ies eme ge nea ly
e ical om he p e-spike dynamic.
See gu e 2.5 o he eec s o kinkiness on eal and ound SIPs.
Luckily, we see an in e es ing eec among he models. Con a y o he
o he Hodgkin-Huxley models es ed, he modied Awiszus model does de ec
he SIPs much close o he SIPs (see g. 3.10). This model phenomenolog-
ically die s om he o he ones in so a as i s spike dynamic shows a much
mo e ab up onse .
Close inspec ion o he ajec o ies o models and neu ons e eal signi-
can die ences conce ning he speed o he dynamic a spike onse (see g.
2.6). We can see ha he as e he spike dynamic a spike onse , he close
he SIPs a e o he SIPs. As he modied Awiszus model has a much as e
dynamic han he o iginal Hodgkin-Huxley model, i allows he de ec ion o
he SIPs much close o he SIPs.
Compa ing he ajec o ies o a eal neu on o hose o he models, we
see ha i is s ill much as e a spike onse han any o he Hodgkin-Huxley
ype models. This jus ies he in e p e a ion ha he SIPs ound on he
neu on's ajec o ies a e simila o he SIPs we canno p ecisely de ec .
2.2.2 La e Spike Phenomenon
Du ing he sho ening o he s imulus amps we encoun e ed an in e es ing
phenomenon in all Hodgkin-Huxley ype models. When he amp was sho -
ened o de e mine he p ecise SIP, i occu ed o ce ain slope/du a ion
combina ions ha a spike occu ed a (up o 60 ms) a e he s imulus had
been swi ched o.
CHAPTER 2. PRECISE SIP DETECTION
22
0
200
400
600
800
1000
-20
0
20
40
60
80
100
0
200
400
600
800
1000
-20
0
20
40
60
80
100
0
500
1000
1500
2000
2500
3000
3500
-80
-60
-40
-20
0
20
0
500
1000
1500
2000
2500
3000
3500
-80
-60
-40
-20
0
20
Figu e 2.7: La e spike phenomenon.
Top le : a amp o leng h 5.23438 ms and slope 758 mV/ms in he classic
Hodgkin-Huxley model induces a spike which occu s a a e he amp has
been swi ched o. This illus a es he concep o a poin o no e u n as
a deni ion o he SIP. Top igh : a sligh ly sho e Ramp (5.2334 ms) does
no p oduce a spike bu only a bump ins ead.
The la e spike phenomenon depends on he pa ame e se o he espec-
i e model. The lowe wo gu es show a much longe ime delay o he
Wang/Buzsáki a ia ion o he Hodgkin/Huxley ype model. Ramp pa-
ame e s a e: Leng h 4.29688 ms and slope 758 mV/ms (lowe le ), leng h
4.2959 ms wi h he same slope (lowe igh ). All gu es: ed cu e shows
s imulus wi hou scale o iming in o ma ion only. G ey ec angle maps
beginning and end o s imulus o he ol age ace.
CHAPTER 3. NEURONS HAVE A 2D FIRING THRESHOLD
29
he uppe end h ough which ca bogena ed ACSF was supplied, and a sink
on he opposi e side. The ACSF eeding pipes lead h ough he wa e be-
low, hus wa ming up he ACSF. Each expe imen ing chambe was equipped
wi h wo AgCl pelle s as g ounding elec odes. The in e ace chambe was
xa ed by pe manen magne s on he hea y s eel op o a ac i e pneuma ic
expe imen ing able (Science P oduc s GmbH, Ho heim, Ge many) which
isola ed he se up om ib a ions. The slice was illumina ed by a cold ligh
sou ce (Olympus, Tokyo, Japan) ia be op ics. A s e eo mic oscope (Leica
Mic osys ems GmbH, We zla , Ge many) p o ided op ical con ol o slice
quali y and coa se elec ode posi ion.
Be o e s a ing he expe imen , he oo o he slice chambe was co e ed
wi h one laye o lens cleaning pape (62647-B, Kodak, Roches e , New Yo k,
USA). Fo d ainage, he inne chambe walls we e aligned wi h 2 mm wide
s ipes o nylon s ocking ab ic which also led in o he chambe 's sink. Li le
shee s o Kodak lens cleaning pape o an app op ia e layou ensu ed ha
he e e ence elec ode pelle s we e in good elec ical con ac o he ACSF.
The chambe was cons an ly own h ough by ca bogena ed and wa med
ACSF, supplied by a pe is al ic pump (Minipuls 3, Gilson Inc., Middle on,
UK) ia he wo nozzles a a a e o abou 1.6 ml/min. Fo expe imen s,
slices we e ans e ed in o he expe imen ing chambe on hei li le shee s
o lens cleaning pape using a pai o weeze s and placed nea o he ACSF
supplying nozzle. Fo pha macological expe imen s, he ACSF hose was
emo ed om he s anda d ACSF ese oi and placed inside a second ACSF
ese oi wi h added d ugs. I ook abou 3 minu es o he new solu ion
o each he chambe 's nozzle. Once a week, all componen s o he se up
ha come in o con ac wi h ACSF (hoses, s o age chambe , expe imen ing
chambe ) we e cleaned using a 0.3 M solu ion o H
2
O
2
.
3.1.2.5 Cell Sea ch and S imulus Pa ame iza ion
The elec ode was posi ioned o e he CA1 egion whe e all cells we e eco ded.
I was hen lowe ed un il elec ical con ac wi h he issue was es ablished.
The b idge balance was adjus ed a he amplie o co ec o he elec ode's
esis ance. The capaci y compensa ion was adjus ed o co ec o he elec-
ode's capaci y. Du ing cell sea ch, he elec ode was lowe ed in s eps o
abou 2 nm, using a mechanical 3D posi ioning de ice (Leica Mic osys ems
GmbH, We zla , Ge many). To clean he elec ode ip as well as o make
cell pene a ion easie , he so-called buzz
1
was used equen ly. This cell
1
Buzz means a sho inc ease o he capaci y compensa ion, yielding o an oscilla ion
o he compensa ion ci cui . The p ocesses leading o bo h an easie cell pene a ion as
well as cleaning he elec ode ip is no known, bu i is a success ul s anda d p ocedu e
CHAPTER 3. NEURONS HAVE A 2D FIRING THRESHOLD
30
sea ch p ocedu e was done wi hou mic oscopic con ol, i.e. blind. B idge
balance and capaci y compensa ion we e pe manen ly checked and adjus ed
i necessa y.
While in he ex acellula space, a epe i i e hype pola izing pulse (am-
pli ude 0.1 nA, du a ion 100 ms) was used wi h b idge balance sligh ly ou o
balance. P oximi y o a cell was suspec ed when he appa en elec ode e-
sis ance showed a sudden inc ease o when ex acellula AP appea ed. A e
pene a ion o a cell i was hype pola ized in o de o help he cell o eco e
om pene a ion. Unde pe manen con ol o he cellula heal h s a e he
hype pola izing cu en was slowly educed o 0. Cells we e hen gi en abou
10-15 minu es ime o accommoda ion and egene a ion be o e s a ing any
s imula ion.
A e his pause, hey we e oughly checked o ing h eshold, esis-
ance, ime cons an and spike ampli ude. I hese pa ame e s seemed heal hy
(see 3.1.4.1 o heal hy pa ame e s), he cells we e used o expe imen s.
Fi s , he ampli udes o he depola izing and hype pola izing ose cu -
en s we e iden ied. The depola izing ose ampli ude was chosen o be
sligh ly sub- h eshold. The hype pola izing ose was chosen o be oughly
5-10 mV below he es ing po en ial.
Now he slopes o he s imulus cu en s we e dened om he es ing
po en ial. As he p ecise o m o he memb ane po en ial du ing a amp
could no be con olled, he memb ane po en ial be o e he s imulus and
he po en ial a spike ini ia ion we e used as as xed poin s and a linea
es ima e was made. S imulus slopes we e now chosen such ha he esul ing
es ima ed linea memb ane po en ial slopes we e oughly 0.1, 0.2, 0.4, 1.0,
2.0 mV/ms. This p ocedu e was epea ed o bo h ose s po en ials in o de
o in es iga e he inuence o he ose on he h eshold.
3.1.2.6 S imula ion
S imulus gene a ion and da a eco ding was con olled by an indi idually
de eloped LabView p og am. I p o ided a g aphical use in e ace o as ,
easy and exible access o all necessa y pa ame e s and con ols. I also
p o ided au oma ic mechanisms o assu e ha all ele an eco ding and
s imula ion pa ame e s we e sa ed in he da a le heade .
The compu e -gene a ed s imulus signals we e digi ized a 35 kHz in o de
o ensu e high empo al p ecision e en wi h as signals. They we e ou pu
ia a da a in e ace ca d (Type PCI MIO 16 E 4, Na ional Ins umen s Co p.,
Aus in, Texas, USA) and hen amplied o comply wi h he needs o he
o sha p elec ode eco dings.
CHAPTER 3. NEURONS HAVE A 2D FIRING THRESHOLD
31
s imulus channel o he amplie (Model IR-183, Neu o Da a Ins umen s
Co p., New Yo k, USA), ed in o i and applied o he cell h ough he
eco ding elec ode. Addi ionally, his s imulus ou pu o he Neu o Da a
amplie was eco ded by he compu e again ( ia he same ca d) and sa ed
along wi h he cellula signal (one indi idual le pe s imulus). Thus, he
eal s imulus is accessible a any pa o he in acellula signal. Bo h he
s imulus and he cellula signal we e also displayed on an oscilloscope o
online con ol.
The eco ded in acellula memb ane po en ial was ed in o he Neu o
Da a amplie ia i s high- esis ance (abou
1013 Ω
) head s age p e-amplie ,
low-pass l e ed a 3 kHz, digi ized a 35 kHz ia he PCI ca d and s o ed on
he compu e 's ha d disk (Pen ium III 700 MHz P ocesso , Windows NT).
This high sampling a e was chosen in o de o be able o p ecisely loca e
he s a o he spike. As he spike s a con ains ela i ely low equencies
as compa ed o he l e 's cu -o equency, his does no in e e e wi h he
low-pass l e ing.
3.1.2.7
In Vi o
Expe imen s
A e iden ica ion o all s imulus pa ame e s o he indi idual cell he ex-
pe imen s could be s a ed. A s he cell was s imula ed by a 500 ms long
posi i e squa e pulse, adjus ed o an ampli ude ha ini ia ed a sequence
o spikes. In da a analysis la e on his se ed as an indica o o he cell
ype. Second, a sequence o 9 die en 200 ms long squa e pulses wi h die -
en sub- h eshold posi i e and nega i e ampli udes was gi en, hus allowing
o de e mine ol age/cu en ela ionship, he cell esis ance and he ime
cons an .
Fo each expe imen al un he e we e 5 die en slopes o be es ed a 3
die en ose s (posi i e ose , es ing po en ial and nega i e ose ), i.e. 15
pa ame e combina ions. Each pa ame e combina ion was gi en 10 imes o
enhance eliabili y, and he sequence o pa ame e se s was pseudo- andomly
selec ed. The e was a pause o a leas 2 s be ween each pa ame e se o
a oid any a e eec s o he p e ious s imula ion (see g. 3.1).
In case o s abili y analysis, his p ocedu e was epea ed a e a pause
in o de o in es iga e empo al s abili y o he esul s. In case o pha -
macological expe imen s, D ugs we e added o he ACSF and pe usion was
s a ed a leas 20 min be o e he ac ual measu emen since diusion in o he
slice is a he slow (see [35]). A e ha he p ocedu e was epea ed wi h he
same s imulus pa ame e s. A e a pha macological expe imen , he slice was
disca ded and he chambe ho oughly washed wi h s anda d ACSF be o e
using a new slice.
CHAPTER 3. NEURONS HAVE A 2D FIRING THRESHOLD
32
10 ms
20 mV
1 nA
Figu e 3.1: Scheme o a ypical s imula ion.
Lowe ace shows he s imulus, s a ing (in his case) wi h an nega i e
ose (le ) o -100 pA. Uppe ace shows he cell's answe . One s imulus
un had he cons an leng h o 1 second, which consis ed o a 300 ms delay
a e he onse o an e en ual ose , he amp i sel , and a ailing pause
a e which he ose was swi ched o ( igh side o gu e). Be ween wo
s imuli he e was a 2 second pause. Speci ying an op ional bounda y allowed
o s a eco ding be o e and a e he s imulus, i. e. be o e and a e he
ose . Typical s imulus slopes used we e be ween 1.5 and 250 nA/s. S imulus
ampli ude was 744 pA in his case.
Fo pha macological expe imen s, he idea was o es he impac o he
A- ype po assium cu en on he ing h eshold. In o de o es his hy-
po hesis, hese channels we e blocked using 50
µ
M 4-Aminopy idine (4-AP).
Applica ion o 4-AP wi h 50
µ
M aec s (among o he s) K cu en s media ed
by K 1 and K 3 channel membe s and can induce seizu e like e en s. To p e-
en gene a ion o epilep i o m discha ges synap ic ansmission was blocked
using a cock ail o glu ama e and GABA ecep o an agonis s (30
µ
M CNQX
(6-cyano-7-ni oquinoxaline-2,3-dione, a compe i i e AMPA/Kaina e ecep-
o an agonis ), 60
µ
M APV ((2R)-amino-5-phosphono ale ic acid, a selec i e
NMDA ecep o an agonis ), 5
µ
M Bicucullin (a compe i i e GABA
A
an ag-
onis ) and 1
µ
M CGP-55845A ([(2S)-3-[[(1S)-1-(3,4-dichlo ophenyl)e hyl]a-
mino]-2-hyd oxyp opyl](phenylme hyl) phosphinic acid, a GABA
B
an ago-
nis )).
3.1.3 Models
3.1.3.1 Modied Leaky In eg a e-And-Fi e Model
The leaky in eg a e-and- e (LIF) model was in oduced by Lapicque in 1907
[29]. Con a y o he models o he Hodgkin-Huxley ype, he LIF model is
a simple one. I inco po a es a cons an capaci y, a cons an esis ance and
esembles a low-pass l e :
CHAPTER 3. NEURONS HAVE A 2D FIRING THRESHOLD
33
C˙
U+1
R(U−U0) = I
This e m compu es he passi e uc ua ions o he memb ane po en ial.
In o de o use i as a neu on model a h eshold c i e ion is in oduced:
U=U0
i
U≥Uθ
Typically, i is equipped wi h a xed ol age h eshold, and i s mem-
b ane po en ial is simply ese o a gi en alue
U0
when he h eshold is
eached, i.e. no spike is p oduced. This is a p oblem o he ques ions o
be in es iga ed he e as hey explici ly need o look a he ol age ace ex-
ac ly a he ansi ion om he sub- h eshold beha io o he spike beha io .
Especially he SIP nding algo i hm needs he spike ups oke o de e mine
he SIP. In o de o cope o his, an a icial analy ical spike is a ached o
he model's ol age ace exac ly a
U( ) = Uθ
(see le plo in g. 3.2). O
cou se explici ca e is aken o assu e ha he e is a smoo h ansi ion om
he model ace o he spike ace as his will be he poin o be analyzed
la e . To do so, we de e mine he slope (
dU
d
) o he ol age ace a
U( ) = Uθ
and nd he a achmen poin wi h exac ly he same slope wi hin he onse
o he a icial spike.
The a icial spike is an analy ical unc ion ha has been p ecisely ed
o he onse and ups oke o a eal spike acco ding o he p ocedu e desc ibed
in [18]. I mimics he spike onse and ups oke in g ea de ail (see igh plo
in g. 3.2). In o de o adop i o he spike ea u es (
U
and
˙
U
ampli udes)
o he da a a hand i was comp essed in he ime domain by ac o 2. This
modica ion was used solely o make esul s compa able o he cellula da a.
Wi hin his simple model we a e able o p edene a bi a y sepa a ices.
We will hus be able o es he analy ical algo i hms used u he on o
check whe he hey a e able o ep oduce hese buil -in sepa a ices.
Th ee die en h eshold c i e ia we e used o he leaky in eg a e-and-
e models:
1. Type A (exponen ial):
UΘ=(1000 ˙
U≤0
exp 3−0.2˙
U
o he wise
2. Type B ( oo ):
UΘ=
1000 ˙
U≤0
20 ˙
U+ 1
o he wise
3. Type C (quasi linea ):
UΘ=(1000 ˙
U≤0
5+0.5˙
U
o he wise
CHAPTER 3. NEURONS HAVE A 2D FIRING THRESHOLD
34
0 1 2 3 4 5 6
0
10
20
30
40
50
60
70
0 1 2 3 4 5 6
0
10
20
30
40
50
60
Figu e 3.2: Beha io o he modied leaky in eg a e-and- e model.
Le : when s imula ed, he memb ane po en ial o he model ises o he
h eshold alue
Uθ
. While he o iginal model would now simply ese he
ol age o
U0
, an analy ically dened spike is a ached o he ol age ace.
I is aken ca e o assu e ha
dU
d
is exac ly iden ical a he ansi ion om
model ace o spike ace. As we a e only in e es ed in he spike ini ia ion
we do no need o ake ca e o he pos -spike beha io o he model.
Righ : The analy ical spike ha is a ached o he model. The slope o he
poin o a achmen on he spike is chosen o be exac ly he same as on he
model's ol age ace a
U( ) = Uθ
.
Bo h gu es show ol age [mV] agains ime [ms].
CHAPTER 3. NEURONS HAVE A 2D FIRING THRESHOLD
35
Fo easie compa ison wi h cells and he o he models he calcula ed ol age
da a o he leaky in eg a e-and- e models we e shi ed o he mean mem-
b ane po en ial o all measu ed cells be o e analysis.
3.1.3.2 Hodgkin-Huxley-Model
The o iginal Hodgkin-Huxley model ([24], used in he e sion om [28])
esul ed om measu emen s ha Hodgkin and Huxley had ca ied ou on he
gian axon o he squid. This and he ac ha he measu emen s we e done
a low Ca
+
and a a empe a u e o 10
°
C indica e ha he e is a big die ence
in model beha io compa ed o a mammalian neu on a body empe a u e.
Ne e heless, he Hodgkin-Huxley model is modeled because o i s p inciple
impo ance as a e e ence.
The cen al equa ion o he Hodgkin-Huxley model is he cu en balance
equa ion
CmdV
d =INa+IK+Ileak +Iinj,
ha is i basically depends on he sum
o all cu en s owing in and ou o he memb ane, acco ding o Ki chho's
law. The e is he sodium cu en
INa
and he po assium cu en
IK
, bo h
esembling he cu en s ha ow h ough he co esponding ion channels, he
leak cu en
Ileak
, indica ing ha he memb ane is no he me ically sealed,
and o cou se all cu en s injec ed in o he cell ia a s imula ion elec ode,
Iinj
.
Fo be e compa ison wi h cells and he o he models he calcula ed
ol age da a o he Hodgkin-Huxley model was shi ed o he mean memb ane
po en ial o all measu ed cells be o e analysis.
3.1.3.3 Wang-Buzsáki-Model
In 1996, Wang and Buzsáki [45] in oduced a Hodgkin-Huxley ype model
o hippocampal in e neu ons. Thei mo i a ion was o in es iga e gamma
oscilla ion in a hippocampal ne wo k model, so hei cu en balance equa ion
is
CmdV
d =I
Na
+I
K
+I
syn
+I
leak
+I
inj
wi h he synap ic inpu cu en
I
syn
.
The modica ions o he o iginal Hodgkin-Huxley model we e made in o de
o phenomenologically adjus he model beha io o he beha io o eal
in e neu ons in espec o an a e hype pola iza ion and high ing a es.
The e is no need, howe e , o conside any synap ic inpu o his wo k, so
he model alls back o he cu en balance equa ion known om he s anda d
Hodgkin-Huxley model. The pa ame iza ion is, o cou se, die en .
3.1.3.4 Awiszus-Model
In 1992, Awiszus [1] p esen ed a modied Hodgkin-Huxley model. His in-
en ion was o adop i o he beha io o a mammalian neu on a body
CHAPTER 3. NEURONS HAVE A 2D FIRING THRESHOLD
36
empe a u e. Basis o his modica ion was ol age-clamp da a om small
a neu ons in he sup a-op ic nucleus a ea [15, 14]. In his modica ion, an
A- ype po assium channel is added o he s anda d Hodgkin-Huxley model,
so ha he basic equa ion is
CmdV
d =I
Na
+I
K
+I
A
+I
leak
+I
inj
. This amily
o models will allow o es he impac o he A- ype po assium cu en .
3.1.3.5 Reduced Awiszus-Model
In his pape [1] Awiszus educes he model om 6 o 5 dimensions by u i-
lizing he close ela ionship o he a iables
a
and
h
. In his 5-dimensional
model, he co esponding equa ions a e combined in o one, hus educing he
dimensionali y. All o he equa ions a e iden ical wi h he ull model.
3.1.3.6 Modied Awiszus-Models
Wi h he ull Awiszus model modeling he A- ype po assium channel, we
ha e he abili y o in es iga e he inuence o his ype o channel on he
sepa a ix. We ha e hus modied he model by blocking he A- ype po as-
sium cu en . This is done by se ing he maximum conduc ance o he
A- ype po assium channel o
G
A
= 0
mS/cm
2
. As his modica ion esul ed
in a pe manen ly spiking beha io a es ing po en ial, we adjus ed he leak
cu en pa ame e s o
G
leak
= 0.1768
mS/cm
2
and
E
leak
=−76.95
mV such
ha he model did no spike a es . This model will be e e ed o as NoA.
In o de o p o ide a usable compa ison o his modica ion we modied
he ull model o he same alues o
G
leak
and
E
leak
.
3.1.3.7
In Silico
Expe imen s
The models we e s imula ed in he same way as he biological cells. Calcu-
la ions we e ca ied ou on x86 compu e a chi ec u e unning Linux ope a -
ing sys ems (RedHa Ve sion 7.1 and 7.3, Suse Ve sion 9.0, 9.1, 9.2, Ubun u
Ve sion 8.9 and 9.1), using he Ma hema ica so wa e package (Wol am Re-
sea ch, Inc.) in e sions 4 and 5. Models we e nume ically sol ed using he
buil -in nume ical sol e o die en ial equa ions wi h a maximal s ep size
o 0.1 s.
P elimina y es s we e conduc ed o check he inuence o ex e nal noise
on he sepa a ix. Fo hese we eco ded in acellula noise wi h he ex-
pe imen al se up and shi ed i s mean o 0. This noise was hen used o
noise he models by simply adding i on o he smoo h model esul s. Fu -
he compa ison be ween pu e and noisy models e ealed iden ical SIPs (and
hus iden ical sepa a ices), so all u he analysis was done wi h pu e (i.e.
noise- ee) models.
CHAPTER 3. NEURONS HAVE A 2D FIRING THRESHOLD
37
3.1.4 Da a Analysis
3.1.4.1 Cell Quali y C i e ia
Cells we e only used o u he analysis i hey me he ollowing quali y
c i e ia:
spike ampli ude mo e han 60 mV
ac ion po en ial o e shoo ing
memb ane esis ance mo e han 25 M
Ω
memb ane ime cons an mo e han 2.5 ms
3.1.4.2 Da a Pos p ocessing
The pos p ocessing p ocedu e consis ed o se e al sel -de eloped Pe l and
Ma hema ica ools o au oma ized da a p epa a ion, a angemen and ea-
u e ex ac ion. The aim o his p ocess was o p epa e and a ange he
c ucial da a o easy, s anda dized and widely au oma ized access by he
analysis p og ams. The ollowing s eps we e applied:
1. unzip da a le
2. ead pa ame e heade
3. nd maximum o s spike du ing s imulus amp
4. w i e ele an in o ma ion in o log le
5. e-zip da a le
Using a sel -de eloped Ma hema ica ool se , his ex ac ed in o ma ion was
used o ex ac he ele an da a (spikes as well as s imulus) om he aw
da a les and sa e hem in a bina y Ma hema ica o ma o easy and s an-
da dized access o all u he analysis.
3.1.4.3 Da a Analysis
All da a ha e been p ocessed in he same way, independen o hei o i-
gin (cells o models). Fi s o all, only he s spike on each amp was
used o u he analysis in o de o elimina e any spike a e eec s a he
SIP. This spike was ex ac ed om he o iginal da a le oge he wi h he
co esponding s imulus ace. F om his o iginal ol age da a he ol age
de i a i e was calcula ed using he mean o h ee subsequen da a poin s:
CHAPTER 3. NEURONS HAVE A 2D FIRING THRESHOLD
38
˙
Ui=
1000 Ui+1−Ui−1
2
wi h
˙
Ui
, he de i a i e o he
i
h
da a alue,
Ui
, he
ol age alue o he
i
h
da a alue,
, he sampling equency, and
i
, he
index. Wi h he ol age gi en in mV his calcula ion yields he de i a i e in
uni s o mV/ms. I is a simple algo i hm o calcula ing he de i a i e o
e enly spaced da a, and i only makes a mino e o wi hin he 3- alue da a
window. We also ha e es ed mo e complex de i a i e l e s like e.g. he
Sa i zky-Golay l e class, bu he e was no signican die ence wi hin he
esul s.
Ha ing
U
and
˙
U
, we can now use he algo i hm desc ibed in 2 o p ecisely
de e mine he SIPs o he spikes. All 10 SIPs gene a ed wi h he same
pa ame e combina ion we e g ouped oge he , and he cen e o g a i y
(2-dimensional mean) o each o hese clouds was calcula ed oge he wi h
he 2-dimensional s anda d de ia ion and s anda d e o o mean (SEM).
Models o cou se we e calcula ed only once pe pa ame e combina ion as
hey a e comple ely de e minis ic. We ed a unc ion o he o m
a0+
a1x+a2log(x)
wi h
an∈R
o each s a e a iable sepa a ely by minimizing
he 2-dimensional dis ance o he unc ion o he means. These ed 2-
dimensional unc ion now is he sepa a ix o he cell.
3.2 Resul s
3.2.1 Cells
3.2.1.1 Sepa a ices
We ha e been able o eco d om 22 cells ha me he quali y c i e ia (see
3.1.4.1). Among hese cells we e 10 egula spiking, 5 oscilla o y, and 6 as
spiking. One cell could no be clea ly alloca ed o a special cell ype by he
eco ded da a.
The cells showed e y die en ypes o sepa a ices. They can be g ouped
in o 4 g oups:
e ical (n=6):
The e ical sepa a ix is he ype o h eshold one
would expec om a cell wi h a pu e ol age h eshold. I a ies in he
˙
U
domain, bu all spikes a e elici ed a he same ol age
U
.
ho izon al (n=4):
This is he opposi e o he e ical sepa a ix. All
spikes s a a he same
˙
U
alue, so eec i ely we ha e a
˙
U
h eshold.
As a consequence, he SIPs span up o 20 mV, hus again s ongly
ques ioning he 1-dimensional h eshold concep .
CHAPTER 3. NEURONS HAVE A 2D FIRING THRESHOLD
45
-28 -26 -24 -22 -20 -18 -16
2.5
5
7.5
10
12.5
15
17.5
20
-52.5-50-47.5-45-42.5-40-37.5-35
2.5
5
7.5
10
12.5
15
17.5
20
-60 -55 -50 -45 -40 -35 -30
2.5
5
7.5
10
12.5
15
17.5
20
Figu e 3.7: The sepa a ix is a sensible indica o o he cell's s a e.
Le plo : The sepa a ix seems o be a e y sensible indica o o he cell's
physiological s a e. This as spiking neu on ini ially showed he ed sepa a-
ix wi hou any pha macology. Second, he g een sepa a ix was measu ed
30 min a e applica ion o he synap ic block cock ail wi hou 4-AP. This
al eady shi ed he sepa a ix 3 mV o he le . Thi d, he addi ional ap-
plica ion o 4-AP did no change i om i s shi ed s a e. Addi ionally, he
ollowing cell pa ame e s changed be ween uns: memb ane esis ance +5 /
+1 M
Ω
, spike ampli ude -4 /
±
0 mV. Middle plo : A egula spiking neu on.
Red: The ini ial sepa a ix. G een: A e 20 min unde block+4-AP. Blue:
A e 50 min unde block+4-AP. Addi ionally, he ollowing cell pa ame e s
changed be ween uns: memb ane ime cons an -1 / -2 ms, memb ane e-
sis ance -4 / -12 M
Ω
, spike ampli ude +4 /
±
0 mV. C: Fas spiking neu on
wi h i s ini ial ed sepa a ix, he g een a e 35 min unde block+4-AP, he
u quoise a e 70 min unde block+4-AP, he blue a e 35 min wash. Ad-
di ionally, he ollowing cell pa ame e s changed be ween uns: memb ane
ime cons an -2 /
±
0 / -2 ms, memb ane esis ance +1 / +1 / +4 M
Ω
, spike
ampli ude
±
0 / +1 / -11 mV.
C osshai s show he SEM o he means, p ojec ed on o he sepa a ices, he
means o he SIPs a e omi ed o be e isibili y. C osshai s o le and
igh plo s ha e been omi ed o he same eason. Colo s o c osshai s code
o he slope o he s imulus om shallow ( ed) o s eep (blue). Colo s o
sepa a ices in all 3 plo s a e solely o be e dis inguishabili y and do no
code o any pa ame e . Axes show
˙
U
[mV/ms] agains
U
[mV].
CHAPTER 3. NEURONS HAVE A 2D FIRING THRESHOLD
46
-60-57.5-55-52.5-50-47.5-45
2.5
5
7.5
10
12.5
15
17.5
20
-60-57.5-55-52.5-50-47.5-45
2.5
5
7.5
10
12.5
15
17.5
20
-60-57.5-55-52.5-50-47.5-45
2.5
5
7.5
10
12.5
15
17.5
20
Figu e 3.8: Algo i hm can ep oduce eal sepa a ices.
Real s. econs uc ed sepa a ices o he 3 die en p edened sepa a ix
ypes in he LIF models (see 3.1.3.1). Le : ype A, middle: ype B, igh :
ype C. The small de ia ions o he econs uc ed sepa a ices a e due o
he limi ed empo al esolu ion because o he sampled model signal as well
as o a sys ema ic eec o he SIP nde (see 2). The le mos sepa a ix
is he eal sepa a ix, while he ound one is sligh ly mo e igh . Howe e ,
hey a e nea enough o show ha he algo i hm p esen ed he e is able o
phenomenologically ep oduce a buil -in sepa a ix.
de ec ion o he SIPs and hus he sepa a ices. Howe e , as he s a e space
is spanned by
U
and
˙
U
, any dis o ion o he ol age ace be o e he SIP
will simply shi he SIP along he sepa a ix wi hou al e ing i s o m.
3.2.2.3 Real s. Found Sepa a ices
Using he models, we a e able o de e mine he eal SIPs (and hus he
eal sepa a ices) ia a second me hod (see chap e 2). Depending on he
ab up ness o he spike onse we see a signican di e gence be ween eal
and ound sepa a ices. Please e e o 2.1.1 o he esul s conce ning eal
s. ound SIPs.
3.2.2.4 Ose
The sepa a ices o he modied Leaky In eg a e-and-Fi e models a e - by
deni ion - independen o he ose . O cou se die en ose s lead o die -
en SIPs, especially because iden ical s imulus slopes lead o die en ol age
slopes a he SIPs. Howe e , as he p edened h eshold is independen o
any o he pa ame e s bu
U
and
˙
U
, hese eec s me ely shi he SIPs along
he sepa a ix (see he SEM ellipses elonga ed along he sepa a ix in g.
3.10).
CHAPTER 3. NEURONS HAVE A 2D FIRING THRESHOLD
47
Figu e 3.9: Ose dependence in LIF models.
The sepa a ices o he leaky in eg a e-and- e models a e
pe deni ionem
independen o he ose . This Figu e shows he eal sepa a ix o he ype
A model (see 3.1.3.1), wi h all ose s g ouped oge he . Shi s occu a he
s eepe pa s o he sepa a ix, bu he o ien a ion o he s anda d de ia ion
ellipses show ha hey ollow he sepa a ix i sel , hus e aining i s o m.
Do s deno e he SIPs o he die en s imulus amp slopes, wi h he colo
shi ing om ed (shallow amps) o blue (s eep amps), ellipsoids show he
SEM. Axes a e
˙
U
[mV/ms] agains
U
[mV].
CHAPTER 3. NEURONS HAVE A 2D FIRING THRESHOLD
48
Figu e 3.10: Ose dependence o he Wang-Buzsáki model sepa a ices.
The Wang-Buzsáki modica ion o he Hodgkin-Huxley model as an example
o he ose dependence o he sepa a ix. The h ee dis inc lines in he
bo om le co ne show he eal sepa a ices, ound ia he amp sho ening
pa adigm (see 2.1.2). The uppe igh co ne shows he h ee co esponding
ound sepa a ices, de e mined by he SIP de ec ion algo i hm p esen ed in
2. The inc eased a iabili y o he SIPs a he ound sepa a ices is due o
he much as e dynamic a his poin .
Do s deno e he SIPs o he die en s imulus amp slopes, wi h he colo
shi ing om ed (shallow amps) o blue (s eep amps). Axes a e
˙
U
[mV/ms]
agains
U
[mV].
CHAPTER 3. NEURONS HAVE A 2D FIRING THRESHOLD
49
Figu e 3.11: Ose dependence o modied Awiszus model sepa a ices.
The modied Awiszus model in he same layou . No e ha he h ee de ec ed
sepa a ices a e close o he eal sepa a ices. Also no e ha he o ien a ion
and o e all o m o he ound sepa a ices a e iden ical o he eal sepa a i-
ces, al hough hey a e shi ed in s a e space.
Do s deno e he SIPs o he die en s imulus amp slopes, wi h he colo
shi ing om ed (shallow amps) o blue (s eep amps). Axes a e
˙
U
[mV/ms]
agains
U
[mV].
CHAPTER 3. NEURONS HAVE A 2D FIRING THRESHOLD
50
Figu e 3.12: Ose dependence o Hodgkin-Huxley model sepa a ices.
The o iginal Hodgkin-Huxley model shows a simila , bu sligh ly die en
beha io . Ins ead o he h ee ose sepa a ices being shi ed along he
ajec o ies hey a e shi ed pe pendicula . Howe e , we again nd he ound
sepa a ices o be shi ed along he ajec o ies, he eby ep oducing he eal
ones.
Do s deno e he SIPs o he die en s imulus amp slopes, wi h he colo
shi ing om ed (shallow amps) o blue (s eep amps). Axes a e
˙
U
[mV/ms]
agains
U
[mV].
CHAPTER 3. NEURONS HAVE A 2D FIRING THRESHOLD
51
The Hodgkin-Huxley models a e clea ly sensi i e o ose s, independen
o hei a o . Wi hin all o hese models we see he h ee eal sepa a i-
ces o posi i e, ze o and nega i e ose s clea ly sepa a ed. Fo he o iginal
Hodgkin-Huxley model he die en ose s shi he sepa a ices
pe pendic-
ula
o he cou se o he ajec o ies (see g. 3.10 C), wi h he nega i e
ose s inducing he uppe and posi i e ones inducing he lowe sepa a ix.
This again also means ha in his case he ose shi s he
cou se
o he a-
jec o ies. Fo all o he Hodgkin-Huxley a o s es ed he e, he sepa a ices
a e shi ed
along
he ajec o ies. Con a y o he o iginal model his indi-
ca es ha he ose does
no
change he cou se o he ajec o ies. In hese
models he igh mos sepa a ix is elici ed om a hype pola ized model, he
middle one om a model a es ing po en ial, he le mos om a depola ized
model. All models howe e show ha he o e all o m o he eal sepa a ix
is e ained in any case, independen o he ose .
The sepa a ices ound by he SIP de ec o (see 2) ypically lie a mo e
igh (i.e. shi ed o wa d in ime along he ajec o ies) han he eal ones,
meaning ha hey a e de ec ed signican ly a e hei eal occu ence. Only
o he modied Awiszus model we nd he de ec ed sepa a ix wi hin he
ange o he eal sepa a ices. The empo al die ence be ween eal and
ound sepa a ices is di ec ly linked o he ab up ness o he spike eme ging
om he p e-spike dynamics: he smoo he (i.e. less kinky) he spikes eme ge
(see g. 2.5 o a compa ison), he mo e dis an he ound sepa a ices a e
om he eal ones, and ice e sa.
I is wo h no ing again ha he phenomenological sepa a ices ound by
he algo i hms and p ocedu es p esen ed in his wo k a e able o ep oduce
he key ea u es o he sepa a ices in any case, be i angle, ex ension and
cu a u e. As he amoun o di e gence o eal o ound sepa a ices is clea ly
linked o he sha pness o he ajec o y's kink a spike onse , his again
nou ishes he iew ha ou algo i hm can be able o p ecisely iden i y he
sepa a ices o eal neu ons as well.
3.2.2.5 Blocking he A- ype Po assium Channel
Wi h he A- ype cu en being explici ly a ailable in he Awiszus model, we
can es he model's esponse o comple ely blocking he A- ype cu en .
To do so we ha e o compa e he modied Awiszus model (NoA) wi h he
modied Awiszus model wi h implemen ed A- ype cu en as hese wo die
only in hei A- ype channel conduc ance.
While he ound and eal sepa a ices a e a he close o he ull model
hey a e again clea ly sepa a ed o he NoA model. As ano he suppo o
ou nding ha he p oximi y o eal and ound sepa a ices is linked o he
CHAPTER 3. NEURONS HAVE A 2D FIRING THRESHOLD
52
-66
-64
-62
-60
-58
0
5
10
15
20
eal w/o a
eal wi h a
ound w/o a
ound wi h a
Figu e 3.13: Modeled A- ype channel block.
The ed and g een sepa a ices a e he ound and eal ones o he con ol
model wi h wo king A- ype channel, he u quoise and blue sepa a ices be-
long o he model wi h blocked A- ype channel. The compa ison o he eal
sepa a ices o he wo models (g een s. blue) shows a signican le -and-
down shi (-5 mV, -4 mV/ms) oge he wi h a s aigh ening. The ound
sepa a ices a e in close p oximi y and show jus a small le -and-up shi
(-1.5 mV, +1.5 mV/ms). No e ha he o e all angle o il is p ese ed o
all 4 sepa a ices.
The g ay a eas indica e he app oxima e co ido o he ajec o ies bundle.
The wa ed a ea in e connec s he wo sepa a ices o he model wi hou he
A- ype cu en , and he checke ed one in e connec s he wo sepa a ices o
he model wi h he A- ype cu en . The checke ed co ido is shi ed a li le
o he lowe igh side, hus indica ing ha he model wi h he A- ype cu -
en shows a sligh ly slowe dynamic.
Do s show he means o he SIPs elici ed by he same s imulus slope. Colo s
o do s code o he slope o he s imulus om shallow ( ed) o s eep (blue).
Colo s o sepa a ices a e solely o be e dis inguishabili y and do no code
o any pa ame e . Axes a e
˙
U
[mV/ms] agains
U
[mV].
CHAPTER 3. NEURONS HAVE A 2D FIRING THRESHOLD
53
ab up ness o he spike onse , he ull model u ns in deed ou o ha e a
as e spike onse han he NoA e sion.
Looking a he ound sepa a ices, we nd a sligh igh -shi as well as
a sligh down-shi o he NoA Ve sion as compa ed o he ull model. This
esul is simila o he esul s ound o he neu ons (see sec ion 3.2.1.4).
Howe e , he neu ons showed his eec al eady when he synap ic block
cock ail was added, he e was no sys ema ic 4-AP- ela ed eec isible.
A mo e eliable esul o he models is o cou se he eal sepa a ix.
He e we see a clea shi o abou 5 mV o he igh and abou 3 mV/ms
upwa d. The inc eased a iabili y o he SIPs (see g een sepa a ix in g.
3.13) indica es ha he spike dynamic becomes i e e sible la e , close o he
SIP sepa a ix. The eal sepa a ix becomes cu ed, bu main ains i s o e all
o ien a ion. Looking a he co ido s o he ajec o ies bundle, i seems ha
he sepa a ix is no de ec ed ea lie , bu he models spike dynamic becomes
la e i e e sible.
Howe e , due o obscu e pha macological esul s wi h neu ons we a e
unable o link hese ndings o biology.
3.3 Discussion
The esul s p esen ed in his wo k suppo he hypo hesis (see 1.1) o
˙
U
as
a second s a e a iable, esul ing in a 2-dimensional ing h eshold. These
h esholds can be seen as sepa a ices in an
U
-
˙
U
s a espace p ojec ion which
sepa a e he passi e p e-spike egime om he ac i e spike dynamic. These
sepa a ices a e able o explain he ( ol age) h eshold a iabili y ound in
ypical in acellula spike eco dings, hus making he new h eshold concep
much mo e adequa e as compa ed o he 1-dimensional one.
We could also show ha he algo i hms p esen ed he e a e able o p e-
cisely ep oduce buil -in sepa a ices om modied leaky in eg a e-and- e
models. Applied o neu al in acellula eco dings, we addi ionally could
show ha neu ons ha e a ious ypes o sepa a ices: e ical, ho izon al,
slash- ype, backslash- ype. Howe e , hese ypes a e no dis inc ly seg e-
ga ed, bu hey seem o o m a kind o sepa a ix-con inuum.
Fu he mo e sepa a ices seem o be a highly sensible indica o no only
o he cellula heal h s a e, bu also o small changes in spike- ele an cell
pa ame e s. In his con ex we could no e i y any eec o an A- ype
po assium channel block.
App oaches o a ibu e ce ain cell ypes o ce ain sepa a ix ypes did
no succeed. We ha e no been able o map he sepa a ix ype o he cell
ype.
CHAPTER 3. NEURONS HAVE A 2D FIRING THRESHOLD
54
3.3.1 Model Sepa a ices
The sepa a ices o he leaky In eg a e-and-Fi e models p ecisely ep oduce
he buil -in sepa a ices. This was o be expec ed as bo h he de ec ing al-
go i hm as well as he p edened h eshold unc ion wo k solely in he phe-
nomenological domain. I is, howe e , a p oo o he abili y o he algo i hm
o p ecisely de ec he phenomenological SIPs.
The Hodgkin-Huxley sepa a ices show a mo e complex si ua ion. He e
we see a clea dependence on he s a ing memb ane po en ial (depola ized,
hype pola ized, es ). This ea u e is p esen h oughou all Hodgkin-Huxley
models es ed, bu we nd one die ence: while he o iginal Hodgkin-Huxley
model has he h ee ose - ela ed sepa a ices shi ed along hei longi udinal
axis, all o he a o s o he Hodgkin-Huxley model ha e hei ose - ela ed
sepa a ices shi ed along he spike ajec o y (see g. 3.10).
Addi ionally, he ound sepa a ices a e unable o ep oduce any ose -
induced shi along he spike ajec o ies ha occu s in he espec i e models.
In he con a y, hey a e all g ouped mo e o less oge he . The excep ion is
he o iginal Hodgkin-Huxley model: He e we see he same shi in he ound
sepa a ices ha is p esen in he eal sepa a ices. Howe e , his was o be
expec ed as all sepa a ices a e
pe deni ionem
loca ed on he ajec o y
bundle. I a die en ose shi s he ajec o y bundle pe pendicula o he
ajec o y di ec ion, he sepa a ices - eal as well as ound ones - will be
shi ed as well.
I is an in e es ing nding ha Hodgkin-Huxley models seem o exhibi
mainly he backslash- ype sepa a ices (al hough some a e admi edly nea ly
e ical). Al hough he models es ed he e we e designed wi h die en in en-
ions he e is none among hem ha would show a die en sepa a ix ype.
E en u he pa ame e a ia ion did no esul in any o he sepa a ix o m.
I would hus su ely be an in e es ing app oach o analyze sys ema ically i
and unde which condi ions Hodgkin-Huxley models a e able o swi ch hei
sepa a ix in o ano he ype.
3.3.2 Sepa a ices in Gene al
The idea o analyzing neu al ac i i y using phase plane p ojec ions is no
new. I has been used in a ious ma hema ical pape s (see e.g. [19] and he
de ailed wo ks o Izhike ich, e.g. [25, 26]) in o de o isualize neu al beha io
in he ol age as well as in he equency domain using he concep s o limi
cycles, a ac o s, bi u ca ions and hus e en sepa a ices. Howe e , his
app oach esul s om he heo e ical
analysis
o he neu on as a dynamical
sys em. Thus, e en i he ol age
U
is used as one s a e a iable, he o he
BIBLIOGRAPHY
61
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mogeneous synap ic inpu .
Biol Cybe n
, 95(1):119, 2006.
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Biol Cybe n
,
95(2):97112, 2006.
[13] B. Ca ling. A low-dimensional, ime- esol ed and adap ing model neu-
on.
In J Neu al Sys
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Appendix A
Supplemen s
A.1 Abs ac
I is gene ally accep ed ha neu al spikes a e ini ia ed solely by a ol -
age h eshold mechanism. Howe e , when looking a in acellula ol age
eco dings, he ol age
U
exhibi s a signican ange o ol age alues, hus
con adic ing he h eshold concep as such.
F om a dynamical sys ems pe spec i e one clea ly would conclude he e-
om ha he sys em is insucien ly desc ibed using
U
alone as a s a e
a iable. Inspi ed by quali a i e epo s o neu ons ha can be d i en a
beyond hei ( ol age) h eshold when s imula ing hem slowly enough, we
in oduced he s ime de i a i e o he ol age,
˙
U
, as a second s a e a i-
able. Plo ing spikes wi hin his wo-dimensional s a e space sugges s ha
hey can be locally in e p e ed as a dynamical sys em wi h wo a ac o s:
one being he es ing po en ial, he o he one being he spike ol age max-
imum (due o he he Na equilib ium po en ial). These wo a ac o s can
be imagined o be sepa a ed by a bounda y line, a so-called sepa a ix. This
sepa a ix would hus be composed o poin s a which a spike has become
una oidably.
In o de o e i y he sepa a ix as a 2D ing h eshold in
U
and
˙
U
hese spike ini ia ion poin s ha e o be iden ied as p ecise as possible. In
physiological da a he spike is cha ac e ized by a sha p kink a spike onse
whe e i ab up ly and e ically eme ges om he sub- h eshold ac i i y. An
algo i hm has been de eloped ha - bo h s a ing om he sub- h eshold
as well as om he supe - h eshold domain - is able o iden i y hese poin s
eliably, p ecise and obus ly.
This p ocedu e was e ied using Hodgkin-Huxley ype and leaky in eg a e-
and- e models. He e he eal spike ini ia ion poin s could be de e mined by
64
APPENDIX A. SUPPLEMENTS
65
successi ely sho ening s imulus leng h. Du ing analysis i u ned ou ha
mos Hodgkin-Huxley ype models show a e y shallow spike onse which in
in clea con as o he sha p onse in cells. We could show ha he ound
spike ini ia ion poin s a e he mo e dis an om he eal ones he shallowe
he spike onse was. On he o he hand, i he model showed an ab up
and e ical spike onse eal and ound spike ini ia ion poin s we e e y close
oge he . Because eal neu ons show an ab up and e ical spike onse we
can hus expec he de ec ed spike ini ia ion poin s o be iden ical o he eal
ones.
Bo h cells and models we e hen s imula ed wi h amps o die en slopes
in o de o es he h eshold dependence on
˙
U
. The new algo i hm was hen
used o de ec he spike ini ia ion poin s o he s spike o each amp. A
cu e was ed o hese poin s wi hin he
U
-
˙
U
s a e space, hus esembling
he sepa a ix o he cell.
We could show ha all cells exhibi a sepa a ix. All 4 possible o ien a ion
classes we e ound: e ical (i.e. a pu e ol age h eshold), ho izon al (i.e.
a pu e
˙
U
h eshold), slash- ype (i.e. om lowe le o uppe igh ) and
backslash- ype (i.e. om lowe igh o uppe le ). Hodgkin-Huxley ype
models showed mainly slash- ype sepa a ices and a e ical one in one case.
Leaky in eg a e-and- e models wi h p edened sepa a ices we e used o
e ica ion, and he algo i hm was able o ep oduce all o hese p edened
sepa a ices.
The neu al sepa a ices showed empo al s abili y (when cellula s a e
pa ame e s we e cons an ) and independence o posi i e o nega i e ose .
On he o he hand, small changes in cell pa ame e s (memb ane ime con-
s an , esis ance, es ing po en ial) showed clea changes and shi s in he
sepa a ices, hus indica ing a hem o be a highly sensible indica o o cell
s a e. Pha macological expe imen s wi h A- ype channel blocke s did no
exhibi a sys ema ical eec as - due o he sensi i i y add essed abo e - al-
eady he necessa y block cock ail changes cell pa ame e s, hus shi ing he
sepa a ix.
APPENDIX A. SUPPLEMENTS
66
A.2 Zusammen assung
Es is ein allgemein ane kann es Kozep , dass neu onale Ak ionspo enziale
mi els eines ein achen Schwellenmechanismus de Memb anspannung
U
in-
duzie we den. Schau man sich jedoch die Ini ia ionspunk e on Spikes
inne halb eine Zelle genaue an, so wi d eine beme kenswe e Va iabili ä
de Spannungswe e zu Spikebeginn deu lich, was dem Modell eine Span-
nungsschwelle jedoch wide sp ich .
Aus Sich de ma hema ischen Konzep e de dynamischen Sys eme is
ein solches Phänomen ein Zeichen ü einen un e dimensionie en Zus and-
s aum, in dem (mindes ens) eine Zus ands a iable ehl . Inspi ie du ch
die imme wiede kolpo ie en Phänomene, dass Zellen ohne Spike deu -
lich übe ih e Spannungsschwelle depola isie we den können, wenn dies
nu langsam genug geschieh , wu de die e s e Ablei ung de Spannung nach
de Zei ,
˙
U
, als zwei e Zus ands a iable einge üh . Die Da s ellung on
Spikes in diesem Zus ands aum leg nahe, dass man sie o übe gehend als
dynamisches Sys em auassen kann, in dem sich zwei A ak o en - das Ruhe-
po en ial au de einen Sei e, das Spikemaximum (beding du ch das Na-
Gleichgewich spo enzial) au de ande en Sei e - benden, die du ch eine Sep-
a a ix, eine G enzlinie oneinande ge enn sind. Diese Sepa a ix müss e
demnach aus den Punk en bes ehen, an denen im neu onalen Zus ands aum
ein Spike unwide uich ini iie wu de.
Fü die Ve ika ion de Sepa a ix als eine 2-dimensionalen Feue schwelle
aus
U
und
˙
U
müssen diese Punk e de Spikeen s ehung möglichs p äzise
bes imm we den. Phänomenologisch sind sie in den Da en du ch einen
scha en Knick gekennzeichne , in dem de Spikeansa z sich ap up nahezu
senk ech aus de un e schwelligen Ak i i ä e heb . Es wu de ein Algo i h-
mus en wickel , de sowohl ausgehend on de un e schweligen als auch on
de übe schwelligen Domäne diesen Spikeini ia ionspunk zu e lässig, p äzise
und obus nden kann.
Dieses Ve ah en wu de an Modellen e izie , in denen zusä zlich die
ech en Spikeini ia ionspunk e du ch sukzessi e S imulus e kü zung bes imm
wu den. Dabei wu de deu lich, das die meis en Hodgkin-Huxley-Modelle
im Gegensa z zu Zellen einen unphysiologisch achen Spikebeginn besi zen.
Wäh end bei Modellen mi schnellem Spikebeginn de ge undene Spikeini ia-
ionspunk gu mi dem ech en übe eins imm , wi d e bei den ande en um so
spä e de ek ie , je ache de Spike beginn . Da die gemessenen Zellen alle
übe einen ap up en Spikebeginn e ügen, kann da on ausgegangen we den,
dass de Algo i hmus die Spikeini ia ionspunk e p äzise de ek ie .
Zellen und Modelle wu den nun mi e schieden s eilen Rampen s im-
ulie , um die Abhängigkei de Schwelle on
˙
U
zu es en. Fü die so
APPENDIX A. SUPPLEMENTS
67
induzie en Spikes wu den nun mi els des besch iebenen Algo i hmus die
Spikeini ia ionspunk e bes imm . An die so ge undenen Punk e wu de im
U
-
˙
U
-Zus ands aum eine Funk ion ge e , die die Sepa a ix ü die jeweilige
Zelle bzw. das jeweilige Modell da s ell .
Es konn e gezeig we den, dass alle Zellen eine Sepa a ix besi zen. Es
sind alle wu den alle O ien ie ungen beobach e : senk ech (en sp echend
eine einen Spannungsschwelle), waage ech (en sp echend eine einen
˙
U
-
Schwelle) sowie ih e Kombina ionen (sch äg on links un en nach ech s
oben (slash-Typ) als auch on links oben nach ech s un en (backslash-
Typ)). Hodgkin-Huxley-Modelle zeig en nu eine slash-Typ-Sepa a ix sowie
in einem Fall eine senk ech e. Fü Tes zwecke wu den spikende Leaky In eg a e-
and-Fi e-Modelle mi o denie en Schwellsepa a ices eingese z , die on
den o ges ell en Algo i hmen p äzise ep oduzie we den konn en.
Die Sepa a ices bei den Zellen zei lich s abil (bei gleichbleibenden Zell-
pa ame e n) und unabhängig on einem posi i en ode nega i en Ose . An-
de e sei s zeig en sich be ei s bei kleinen Ände ungen de Zellpa ame e deu -
liche Ände ungen in de Sepa a ix, was au eine hohe Sensibili ä de Sep-
a a ices ü den Zellzus and hinweis . Pha makologische Expe imen e mi
A-Kanal-Blocke n zeig en keinen sys ema ischen Eek , da be ei s die Gabe
des zu Ve meidung epilep i o me Ak i i ä en no wendige Block-Cock ail
die Sepa a ix e ände e, be o de A-Kanal-Block zugegeben wu de.
APPENDIX A. SUPPLEMENTS
68
A.3 Acknowledgemen s
Many people ha e con ibu ed o his wo k in a a ie y o ways. I would
like o hank hem he e, hose named below, bu also hose who I o go o
men ion.
And eas He z ga e me he oppo uni y o pu sue hese ideas in his Lab.
Jan Benda, Daniel C eme s and Ma in S emmle we e always a ailable and
willing o discuss and c i icize app oaches and concep s. Lau enz Wisko
helped me o ocus hough s. Pe a P inz and Daniela Ee also discussed
ideas and concep s, p o ided da a and especially helped in going h ough
mo i a ional lows.
Uwe Heineman suppo ed he expe imen s bo h p ac ically as well as
heo e ically. His expe ience was ex emely aluable o o e come he pi s o
p ac ical elec ophysiology and in e p e cellula esponses. He also ne e
s opped o suppo he nishing o his disse a ion. And eas D aguhn and
Die ma Schmi z suppo ed he wo k wi h discussions and hin s. The ex-
pe imen s we e done oge he wi h Gun e K eck wi h whom I ook he s
elec ophysiological s eps oge he .
Jan Benda, Daniel C eme s and And eas D aguhn kindly oe ed hei
ime o ho oughly discuss he s e sions o his disse a ion. They made
aluable ema ks and suppo ed me in nishing he wo k.
Randol Menzel was immedia ely suppo i e when in need o a supe iso .
He was in e es ed in he wo k and suppo ed my a emp o nish i . I hank
him e y much o being a ailable any ime and o his p agma ic calmness.
Ma in Naw o was so kind o supe ise he las s eps o he wo k. He ga e
aluable hin s on a gumen a ion and s uc u e.
Mos o all, howe e , I would like o hank my iend, pa ne and wi e
Silke E dmann o he pa ience as well as he impa ience. She suppo ed me
du ing he lows and co ec ed me du ing phases o igno ance. I owe he a
lo . And las bu no leas I would like o hank my daugh e Lena E dmann
o he pa ience and o (mos ly) silen ly accep ing ha he a he could no
go swimming because o his disse a ion.
APPENDIX A. SUPPLEMENTS
69
Con ac
Fo u he ques ions you may con ac he au ho a
ca s en
do
e dmann
a
email
do
de
.