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Partial synchronization patterns in brain networks

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Schoell, E. (2021). Partial synchronization patterns in brain networks. In EPL (Europhysics Letters). IOP

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epl draft

Partial synchronization patterns in brain networks

Eckehard Sch¨

oll1,2,3

1Institut f¨ur Theoretische Physik, Technische Universit¨at Berlin, Hardenbergstraße 36, 10623 Berlin, Germany

2Potsdam Institute for Climate Impact Research, Telegrafenberg A 31, 14473 Potsdam, Germany

3Bernstein Center for Computational Neuroscience Berlin, Humboldt-Universit¨at, 10115 Berlin, Germany

PACS 05.45.Xt – Synchronization; coupled oscillators

PACS 87.19.lj – Neuronal network dynamics

Abstract –Partial synchronization patterns play an important role in the functioning of neuronal

networks, both in pathological and in healthy states. They include chimera states, which consist of

spatially coexisting domains of coherent (synchronized) and incoherent (desynchronized) dynam-

ics, and other complex patterns. In this perspective article we show that partial synchronization

scenarios are governed by a delicate interplay of local dynamics and network topology. Our focus

is in particular on applications of brain dynamics like unihemispheric sleep and epileptic seizure.

Introduction. – Synchronization is a widespread

natural phenomenon occurring in dynamical networks of

nonlinear oscillators [1, 2]. Probably the first example

was given by Christiaan Huygens (1629 - 1695), who ob-

served that while two individual pendulum clocks show

slightly deviating times, they spontaneously synchronize

at exactly the same frequency if they are weakly cou-

pled via a wooden beam. In the human brain, synchro-

nization of neurons is essential for normal physiological

functioning [3], for instance in the context of cognition

and learning, but it is also strongly related to patholog-

ical conditions such as Parkinson’s disease [4], or epilep-

tic seizures, which are the cardinal symptom of epilepsy

[5–7]. This neurological disease is currently understood

as a network disease [8], and a better understanding of

the role of the epileptic network’s topology in seizure gen-

eration and termination is highly desirable. Sleep is as-

sociated with specific synchronized oscillations, i.e., sleep

spindles and slow oscillations in the thalamocortical sys-

tem [9]. While the synchronization processes can differ

between adults and children [10], transitions from wakeful-

ness to sleep are widely accompanied by synchronization

phenomena [11]. A particularly intriguing phenomenon

in nature is unihemispheric slow-wave sleep, exhibited by

aquatic mammals including whales, dolphins and seals,

and multiple bird species. Unihemispheric sleep, as the

name suggests, is the remarkable ability to engage in deep

(slow-wave) sleep with a single hemisphere of the brain

while the other hemisphere remains awake [12–14]. In-

terestingly, sleep and wakefulness are characterized by a

high and low degree of synchronization, respectively [12].

In the human brain the first-night effect, which describes

troubled sleep in a novel environment, has been related

to asymmetric dynamics recently, i.e., a manifestation of

one hemisphere of the brain being more vigilant than the

other [15].

Using complex networks of coupled oscillators, one can

simulate synchronization phenomena observed in the hu-

man brain. Often coupled oscillators of FitzHugh-Nagumo

type are employed since these are a paradigmatic model

for neural dynamics [16]. The model describes the non-

linear dynamics of individual neurons or whole brain ar-

eas by a fast excitatory and a slow inhibitory variable.

The coupling between different neurons or different ar-

eas of the brain is mediated by a coupling matrix, which

may be mathematically constructed using standard pro-

cedures from network science, or taken from empirical

structural brain connectivities of human subjects mea-

sured, e.g, by diffusion-weighted magnetic resonance imag-

ing (MRI) [17]. The role of distant-dependent transmis-

sion time delays in large-scale brain synchronization has

been stressed in [18].

In this perspective article we focus on partial syn-

chronization patterns related to unihemispheric sleep and

epileptic seizures. We review computer simulations of net-

works based upon empirical connectivities and FitzHugh-

Nagumo dynamics on the nodes of the network, which

sheds light on dynamical phenomena in the brain. In our

first application, we find dynamical symmetry-breaking

between the two hemispheres, and in a minimum model

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E. Sch¨oll

discuss the modalities of unihemispheric sleep in human

brain, where one hemisphere sleeps while the other re-

mains awake [19]. In the second application, we observe

spontaneously occurring episodes of strong synchroniza-

tion, which resemble the ones seen during epileptic seizures

recorded by electroencephalography (EEG) [20, 21]. For

a better insight into the network properties giving rise

to such pathological events, we simulate the dynamics

on various artificial network topologies: we randomly

rewire links in a small-world fashion, consider fractal con-

nectivities, and exchange equal weights with empirical

weights from diffusion-weighted magnetic resonance imag-

ing. Moreover, we explore how global aspects of the net-

works – as assessed with the average clustering coeffi-

cient and the mean shortest path length – impact on the

dynamics of the epileptic-seizure-related synchronization

phenomena.

A better knowledge of the interplay between dynamics

and network properties leading to complex synchroniza-

tion phenomena is essential for understanding unihemi-

spheric sleep and epileptic seizures, as well as other syn-

chronization phenomena in the brain like full or partial

relay synchronization between distant areas of the brain.

The paradigm is to use simple nonlinear oscillator mod-

els like single-variable phase oscillators or two-variable

activator-inhibitor models in combination with complex

network structures obtained empirically or from artificially

constructed topologies. This opens up promising perspec-

tives for future research on partial synchronization pat-

terns in the brain.

Partial synchronization patterns. – There ex-

ist different forms of synchronization, i.e., complete or

isochronous (zero-lag) synchronization, generalized syn-

chronization (where the oscillations of the individual el-

ements of the network are not identical, but functionally

related), phase synchronization (where only the phases but

not the amplitudes of the oscillations are synchronized),

frequency synchronization (where only the frequencies but

not the phases are the same), cluster or group synchro-

nization (where within each cluster all elements are com-

pletely synchronized, but between the clusters there is a

phase lag), and many other forms. Some progress has

been made in generalizing synchronization, for instance,

towards adaptive networks [22, 23] (where the strength

of the links is adapted dynamically), inhomogeneous lo-

cal dynamics [24] and heterogeneous delay times [18], dis-

tributed, state-dependent, or time-varying delays.

Recent research interest has focussed on more com-

plex partial synchronization patterns, where the whole

system is not completely in synchrony, but only parts

of it have the same phase and frequency. An intriguing

example of partial synchronization patterns, which has

recently gained much attention, are chimera states, i.e.,

symmetry-breaking states of partially coherent and par-

tially incoherent behavior, for recent reviews see [25–27].

Chimera states in dynamical networks consist of spa-

tially separated, coexisting domains of synchronized (spa-

tially coherent) and desynchronized (spatially incoher-

ent) dynamics. They are a manifestation of spontaneous

symmetry-breaking in systems of identical oscillators, and

occur in a variety of physical, chemical, biological, neu-

ronal, ecological, technological, or socio-economic systems.

Other examples of partial synchronization include solitary

states [28], or hierarchical multifrequency clusters [23].

In the following we model each node of the network,

corresponding to a brain region, by the FitzHugh-Nagumo

(FHN) model, a paradigmatic model for neuronal spiking

[29, 30]. Note that while the FHN model is a simplified

model of a single neuron, it is also often used as a generic

model for excitable media on a coarse-grained level [31,32].

Thus the dynamics of the network reads:

ε˙uk=uk−u3

3−vk

+σX

j∈NH

Akj [Buu(uj−uk) + Buv(vj−vk)] (1a)

+ςX

j /∈NH

Akj [Buu(uj−uk) + Buv(vj−vk)] ,

˙vk=vk+a

+σX

j∈NH

Akj [Bvu(uj−uk) + Bvv(vj−vk)] (1b)

+ςX

j /∈NH

Akj [Bvu(uj−uk) + Bvv(vj−vk)] ,

with k∈NHwhere NHdenotes either the set of nodes k

belonging to the left (NL) or the right (NR) hemisphere,

and ε= 0.05 describes the timescale separation between

fast activator variable or neuron membrane potential u

and the slow inhibitor or recovery variable v[29]. Depend-

ing on the threshold parameter a, the FHN model may ex-

hibit excitable behavior (|a|>1) or self-sustained oscilla-

tions (|a|<1). Here we use the FHN model in the oscilla-

tory regime and fix the threshold parameter at a= 0.5 suf-

ficiently far from the Hopf bifurcation point. The coupling

within the hemispheres is given by the intra-hemispheric

coupling strength σwhile the coupling between the hemi-

spheres is given by the inter-hemispheric coupling strength

ς. The interaction scheme between nodes is characterized

by a rotational coupling matrix B. Employing a rotational

matrix Bis a simple way to parameterize the possibility of

either diagonal coupling (Buu, Bvv) or activator-inhibitor

cross-coupling (Buv, Bvu) by a single parameter ϕ:

B=Buu Buv

Bvu Bvv =cosϕsinϕ

−sinϕcosϕ.(2)

In the following we choose ϕ=π

2−0.1, causing dom-

inant activator-inhibitor cross-coupling [33], which is a

commonly employed mechanism in biology. In the neuro-

sciences, the microscopic coupling schemes are very com-

plex [34], but in our coarse-grained macroscopic descrip-

tion of a whole brain area by a pair of activator and in-

hibitor variables, activator-inhibitor coupling is a natural

p-2

Title

(a) (b)

Fig. 1: Model for the hemispheric brain structure: (a)

Weighted coupling matrix Akj of the averaged empirical

structural brain network derived from twenty healthy hu-

man subjects. The 90 brain regions k, j are taken from

the Automated Anatomic Labeling atlas [39], labeled such

that k= 1, ..., 45 and k= 46, ..., 90 correspond to the

left and right hemisphere, respectively. (b) Schematic

representation of the graph of the brain structure with

highlighted left (dark blue) and right (light orange) hemi-

sphere. After [19].

extension of pure activator-activator coupling. Mathemat-

ically, this means that signals of other neuronal areas are

coupled via a coupling phase, which introduces a phase lag

or time delay. The subtle interplay of excitatory and in-

hibitory interaction enables intermittent periods of either

high or low synchronization. This is typical of the critical

state at the edge of different dynamical regimes in which

the brain operates [35, 36]. The coupling phase ϕis sim-

ilar to the phase-lag parameter of the paradigmatic Ku-

ramoto phase oscillator model [37], which is widely used to

describe synchronization phenomena in coupled oscillator

networks. The coupling phase has been shown to be cru-

cial for the modeling of chimera patterns in the Kuramoto

model [38] and in the FHN model [33].

First, we consider an empirical structural brain network,

obtained from diffusion-weighted MRI data measured in

healthy human subjects. For details regarding the exper-

imental setup, data acquisition and processing, see [40].

Obtaining such connectivity information using diffusion

tractography is known to face a range of challenges [41].

The 90 brain areas of the Automated Anatomical Label-

ing (AAL) atlas [39] correspond to the 90 nodes of our

network, and the connecting white-matter fibers between

the areas correspond to the links. To eliminate individ-

ual variation, the matrices of 20 subjects were averaged

over the coupling between two brain regions kand j, giv-

ing rise to the topology of Fig. 1(a), (b). Brain areas

k∈NL={1,2,...,45}and k∈NR={46,...,90}corre-

spond to the left and right hemisphere, respectively, as in

[19,21]. The structure of the brain hemispheres can be eas-

ily distinguished: In the adjacency matrix in Fig. 1(a), the

connections within one hemisphere are much stronger than

the connections between both hemispheres. In Fig. 1(b)

the hemispheric brain structure is schematically shown.

Note that there is a very slight structural asymmetry of

the two brain hemispheres, related to the known asymme-

tries in localization of psychological functions, such as the

prevalence of language functions in the left brain hemi-

sphere in humans.

These empirical structural connectivities have been used

in computer simulations of unihemispheric sleep [19], of

epileptic seizure [20,21], and of the influence of sound on

brain networks, i.e., synchronization patterns induced by

the frequency of an external sound source [42].

Methods. – The dynamical behavior can be char-

acterized by the mean phase velocity ωk= 2πMk/∆Tfor

each node k, where Mcomplete rotations are realized dur-

ing ∆T. Further, we use the global Kuramoto order pa-

rameter rto measure the degree of synchronization of a

network:

r(t) = 1

N

k=1

exp[iφk(t)]

,(3)

utilizing an abstract dynamical phase φkobtained from

the standard geometric phase ˜

φk(t) = arctan(vk/uk) by

a transformation which yields constant phase velocity ˙

φk.

For an uncoupled FHN oscillator, the function t(˜

φk) is cal-

culated numerically, assigning a value of time 0 < t(˜

φk)<

Tfor every value of the geometric phase, where Tis the

oscillation period. The dynamical phase is then defined

as φk= 2πt(˜

φk)/T, which yields ˙

φk= const. Only by

using the dynamical phase φk, rather than the geometri-

cal phase ˜

φk(t), strong temporal fluctuations of r(t) due to

the slow-fast time scales of inhibitor and activator are sup-

pressed, and a change in rindeed reflects a change in the

degree of synchronization. The Kuramoto order parame-

ter may vary between 0 and 1, where r= 1 corresponds to

complete phase synchronization, small values characterize

desynchronized states, and intermediate values correspond

to partial synchronization.

One may introduce the hemispheric Kuramoto order pa-

rameters RL(t) and RR(t) characterizing the left and the

right hemisphere, respectively, by restricting the summa-

tion in Eq. (3) to the respective hemisphere.

Unihemispheric sleep. – In this section we apply

the FitzHugh-Nagumo model with the empirical structural

connectivity introduced in the previous section to study

the phenomenon of unihemispheric sleep [19]. We show

that the dynamical asymmetry of the two brain hemi-

spheres, induced by the slight natural structural asymme-

try, can be enhanced by introducing the inter-hemispheric

coupling strength as a control parameter for partial syn-

chronization patterns. It has been speculated that uni-

hemispheric sleep is related to the spontaneous symmetry-

breaking phenomenon of chimera states in oscillator net-

works [43,44].

It is presumed that a certain degree of structural in-

terhemispheric separation is a necessary condition for

this pattern to persist [12]. Therefore we propose to

model unihemispheric sleep by a two-community network

p-3

E. Sch¨oll

(a) (b)

(d)(c)

Fig. 2: (color online) Partial synchronization pattern for

σ= 0.70, ς= 0.15 with low and high degree of synchro-

nization in the left (a, c) and right (b, d) hemisphere,

respectively. (a),(b) Mean phase velocity profiles ωk, av-

eraged over ∆T= 5 000. (c),(d) inner panels: space-time

plots of node-wise phase velocity ω1

kaveraged over a sin-

gle oscillation, outer panels: hemispheric Kuramoto or-

der parameter RL,R as a function of time t. Parameters:

ε= 0.05, a= 0.5, ϕ=π

2−0.1. After [19].

of the two hemispheres where the inter-hemispheric cou-

pling strength is smaller than the intra-hemispheric cou-

pling. Similar results are expected if the longer propaga-

tion time delays between the two hemispheres are taken

into account [18]. We consider the empirical structural

brain network shown in Fig. 1, where each region of in-

terest is modeled by a single FitzHugh-Nagumo oscillator

Eq. (1). To achieve partial synchronization patterns we

consider the inter-hemispheric coupling strength ςas an

independent parameter that allows us to reduce the cou-

pling between the hemispheres. In a certain intermediate

interval of inter-hemispheric coupling strength ς < σ we

find the partial synchronization pattern shown in Fig. 2

where the left hemisphere is incoherent while the right

is frequency-synchronized, except for three small brain re-

gions (hippocampus, gyrus parahippocampalis, and amyg-

dala). Interestingly, these three special regions occasion-

ally perform an extra oscillation thus leading occasionally

to a higher instantaneous frequency, and within the net-

work they have a pacemaker role. The partial synchroniza-

tion shows up in the space-time plot, in the mean phase

velocity profile, and in the hemispheric Kuramoto order

parameter (although there is no perfect phase synchro-

nization, and hence RR<1). Note that the incoherent,

left hemisphere occasionally exhibits a high degree of syn-

chronization that, in contrast to the right hemisphere, is

unstable and vanishes after a short while.

In conclusion, we have obtained a symmetry-broken

state, where the (right) frequency-synchronized hemi-

Fig. 3: Epileptic-seizure-like synchronization phenomena

in a FitzHugh-Nagumo network with empirical connec-

tivity. (a) Weighted adjacency matrix. (b) Schematic

network structure, where the left (right) semicircle corre-

sponds to the left (right) hemisphere (nodes are numbered

clockwise sequentially 1,...,90 starting from the bottom

of the circle). (c) Global Kuramoto order parameter rvs

time for a time interval of 1 hour. (d) rvs time for a time

interval of 30 seconds. The horizontal dashed grey line de-

notes the time average hriover 3 hours. The horizontal full

red line marks the threshold of r= 0.8. If r > 0.8 for more

than 8 seconds, we define this as a seizure (pink shaded

region). (e) Space-time plot of the dynamical phases cor-

responding to panel (d). The left (right) hemisphere is

shown in the lower (upper) half. Simulation parameters:

ε= 0.05, a= 0.5, ϕ=π

2−0.1, N= 90, σ=ς= 0.6.

After [21].

sphere is reminiscent of the sleep state, and the (left)

desynchronized hemisphere ressembles an awake state.

Epileptic-seizure-related synchronization phe-

nomena. – Epilepsy is a neurological disorder that af-

fects almost 70 million people worldwide. In epileptol-

ogy, the development of the concept of an epileptic net-

work [45, 46] received a strong impetus from network-

theoretical concepts. An epileptic network comprises

anatomically, and more importantly, functionally con-

nected cortical and subcortical brain structures and re-

gions. Seizures may emerge from network constituents

that generate and sustain normal, physiological brain dy-

namics during the seizure-free interval [45].

For networks of neurons, modeled with the FitzHugh-

Nagumo neuronal dynamics, epileptic-seizure-like dynam-

ics has previously been investigated in an empirical struc-

tural brain connectivity and a mathematically constructed

network with modular fractal connectivity [20]. Further,

p-4

Title

the role of partial synchronization phenomena for mech-

anisms of seizure initiation [47] and termination [48] has

been explored. Here we present the results of simulations

for different network topologies that shed light on the role

of the coupling structure for spontaneous synchronization

[21].

First we use the FHN model Eq.(1) with the empirical

structural brain network of Fig. 1. The coupling strength

σ=ςis chosen such that it is as high as possible while

still avoiding full synchronization for long simulations (≈

10000 time units), i.e., σ= 0.6. In order to compare

our simulations with real data (EEG recordings of absence

seizures), we transform the dimensionless time units of the

FHN oscillator model to real time units by comparing the

oscillation period of a single FHN oscillator T= 2.56 to

the dominant frequency of an absence seizure at about

f= 3Hz [49]. Therefore, the simulation time is converted

to real time by 1s= 2.56 ·3 = 7.68 simulation time units.

The results of the simulation are shown in Fig. 3. In

panels (c) and (d), we show the global Kuramoto order

parameter r(t), which measures the degree of synchroniza-

tion. Panels (c) and (d) also reveal periods of very high

and of very low synchronization of the system as a function

of time, varying in a range from 0 to almost 1 (panel (c)).

The temporal average of the order parameter hri(hori-

zontal dashed grey line in (d)) and its standard deviation

δare given by hri ± δ= 0.59 ±0.21 for the full simulation

of 164 minutes. We define a threshold of high synchrony

as rth =hri+δ= 0.8 (horizontal red line in (d)). In

the simulation presented in Fig. 3, the order parameter is

found to be in high synchrony with r > 0.8 during 17% of

the simulation time. Only if the synchronization remains

above the threshold for at least 8 seconds, we define this

time interval as a seizure.

In Fig. 3(d), the order parameter is shown versus time

for one exemplary seizure. Approximately 6 seconds prior

to the start of the seizure, the order parameter drops to a

low value of r≈0.2. Such an apparent desynchronization

can often be observed prior to the onset of focal epileptic

seizures [47, 50–52]. The order parameter then increases

above r > 0.8 (onset of seizure) and remains in high syn-

chrony for almost 10 seconds. The seizure interval is shown

as a pink shaded region. In the full simulation of 164

minutes, 11 seizures were detected, giving an average of

4 seizures per hour and an average duration of 11 sec-

onds. In Fig. 3(e), the dynamic phases of the oscillators

are shown as space-time plot for the same time interval as

in (d) exhibiting strong synchronization during seizure.

We have compared the simulations with EEG recordings

from a 12 years old subject who suffered from seizures, and

found good agreement of the simulated and the recorded

seizures (see Supplemental Material).

Empirical vs artificially constructed networks. –

In order to gain deeper insight into the interplay of dynam-

ics and network topology, especially regarding the occur-

rence of seizures, different artificially constructed networks

have been considered [21]. For details see the Supplemen-

tal Material.

First, by randomly rewiring its links, the highly orga-

nized structure of the empirical connectivity matrix can

be artificially destroyed, while keeping the weight distri-

bution and average node strength. On average the system

is less synchronized, and no seizure is found in the simula-

tion. Next, a quasi-fractal connectivity on a ring network

is considered. It synchronizes completely at a relatively

small coupling strength σ, and no clearly defined seizures

are found. To achieve a more realistic weight distribution,

all non-zero links of the fractal ring can be replaced by ran-

domly chosen weights of the empirical connectivity matrix

in Fig. 1(a). However, despite rich, brain-like modulated

synchronization behavior with a few very short events of

high synchrony r > 0.8, not a single well-defined seizure

is detected.

Finally, we consider small-world-like networks, which

can be constructed according to the Watts-Strogatz al-

gorithm [53] by starting from a nonlocally coupled ring

and randomly rewiring links with a probability p. The

impact of the average clustering coefficient and the av-

erage shortest path length on the number of observed

epileptic-seizure-related synchronization episodes is eval-

uated. Intuitively, the clustering coefficient of the net-

work measures the probability of “cliques” in the net-

work: using the language of social networks, a “clique”

is a group of people who are all connected with each other

within this clique, i.e., if A has two friends, then these

two are also friends with each other. Analogously, a net-

work has a large clustering coefficient if whenever an el-

ement is connected to two other elements (open triplet),

then these two are also likely to be connected directly

(closed triplet). We have examined both network mea-

sures, clustering coefficient and shortest path length, by

employing the Watts-Strogatz small-world algorithm with

various rewiring probabilities p. Among the artificial net-

works, a small-world network with intermediate rewiring

probability p≈0.232 results in the best agreement with

the simulations for empirical structural connectivity. For

the other network topologies, either no spontaneously oc-

curring epileptic-seizure-related synchronization phenom-

ena are found in the simulated dynamics, or the overall de-

gree of synchronization remains high throughout the sim-

ulation. This indicates that a topology with some balance

of regularity and randomness favors the self-initiation and

self-termination of episodes of high, seizure-like synchro-

nization. In particular, the value of the clustering coeffi-

cient should not be too high (as for regular ring networks,

p= 0) and not too low (as for pure random networks,

p= 1), and thus the rewiring probability should assume

intermediate values between 0 and 1. Random network

structures increase brain synchronization compared to re-

alistic brain networks. There is a subtle interplay of reg-

ularity and randomness.

In conclusion, our simulations indicate that the human

brain seems to effectively function in a specific window

p-5

E. Sch¨oll

Fig. 4: Comparison of (a) empirical brain network, (b)

artificial brain network (small world with rewiring prob-

ability p= 0.232, clustering coefficient C=0.25). Each

black dot represents one of 90 brain areas; the left (right)

semicircle corresponds to the left (right) hemisphere.

of medium clustering. If the clustering is too large, the

neural synchronization is approximately constant in time.

The brain, however, shows both low and high synchroniza-

tion values on the EEG during different tasks and mental

states such as sleep. Moreover, epileptic brains, which

function normally most of the time, appear to synchro-

nize during seizures fully. This shows that the brain is

capable of sustaining both very coherent and very incoher-

ent oscillatory states, which is not possible for too large

clustering coefficients. On the other hand, if the cluster-

ing coefficient is too small, the synchronization fluctuates

rapidly in time and does not resemble the dynamics of

simulations with an empirical brain network. One might

speculate, based on these simulations, that the difference

between healthy and epileptic brains might show up in the

network’s slightly altered clustering coefficient [54,55].

Conclusions and future challenges. – Partial syn-

chronization patterns play an essential role in collective

brain dynamics. Computer simulations of dynamical net-

works of nonlinear oscillators can help to understand the

functioning of the brain, both in pathological and in

healthy states. It is known that the brain is operat-

ing in a critical state at the edge of different dynamical

regimes [35, 36]. We have shown that simple oscillator

models in combination with complex network structures

can explain and elucidate a plethora of observed partial

synchronization scenarios, which are governed by a del-

icate interplay of local dynamics and network topology.

Using empirical structural network connectivities obtained

from diffusion-weighted magnetic resonance imaging of

humans together with paradigmatic dynamics, e.g. the

FitzHugh-Nagumo model, on the nodes of the network,

yields realistic scenarios of partial synchronization. They

include chimera states, which consist of spatially coex-

isting domains of coherent (synchronized) and incoherent

(desynchronized) dynamics, and other complex patterns.

First, we have focussed on unihemispheric sleep, where

one hemisphere of the brain sleeps while the other remains

awake. By tuning the coupling between the hemispheres

we have shown that at intermediate inter-hemispheric cou-

pling one hemisphere becomes incoherent, giving rise to a

chimera-like partial synchronization pattern.

Second, we have shown that FitzHugh-Nagumo oscilla-

tors, coupled via empirical structural connectivities mea-

sured in human subjects, exhibit episodes of high syn-

chronization that resemble the ones seen during epileptic

seizures. Comparing our long-term simulations to EEG-

recorded epileptic seizures, the simulations show striking

similarities to the real data.

In order to gain more insight into the nature of the

empirical structural connectivities of the brain, and into

the interplay of dynamics and network topology, we have

also studied different artificially constructed complex net-

work structures, ranging from random networks, regular

nonlocally coupled ring networks, ring networks with frac-

tal connectivities, and small-world networks with various

rewiring probabilities. Although at first glance the net-

work structure of the empirical connectivities (Fig. 4a)

and the small world network with rewiring probability

p= 0.232 (Fig. 4b) do not at all look similar, the partial

synchronization scenarios are found to be very similar, giv-

ing the best match among all artificial structures studied.

This has been attributed to the intermediate value of the

average clustering coefficient, which results in a sophis-

ticated balance of synchrony and asynchrony with wide

temporal variability.

We have discussed in detail unihemispheric sleep and

epileptic seizures as examples of brain dynamics, but a

great wealth of other collective dynamical behavior in the

brain is amenable to this approach. For instance, in neuro-

science various scenarios have been uncovered where spe-

cific brain areas, e.g., thalamus or hippocampus, act as

a functional relay between other brain regions, having a

strong influence on signal propagation, brain functionality,

and dysfunctions [56–58]. Relay synchronization scenar-

ios between remote layers of a network have been studied

for three-layer networks with regular nonlocally coupled

ring topology [26], randomly diluted small-world topolo-

gies, and configurations where the relay is a single node,

i.e., a hub. Future promising perspectives of the research

on relay functions in the brain should also use empirical

connectivities.

∗∗∗

This work was supported by the Deutsche Forschungs-

gemeinschaft (DFG, German Research Foundation) -

Project Nos. 163436311 - SFB 910, 429685422, 440145547,

and 308748074. I am grateful to R. Andrzejak, R. Berner,

T. Chouzouris, J.C. Claussen, J. Czech, F. Drauschke,

J. Hlinka, P. Jiruska, J. Koulen, K. Lehnertz, M. Myki-

etyshyn, I Omelchenko, L. Ramlow, A. Provata, G.

Ruzzene, J. Sawicki, A. ˇ

Skoch, G. Strelkova, S. Yanchuk,

and A. Zakharova for stimulating collaborations.

p-6

Title

REFERENCES

[1] A. Pikovsky, M. Rosenblum, and J. Kurths, Synchroniza-

tion: a universal concept in nonlinear sciences, 1st ed.

(Cambridge University Press, Cambridge, 2001).

[2] S. Boccaletti, A. N. Pisarchik, C. I. del Genio, and A.

Amann, Synchronization: From Coupled Systems to Com-

plex Networks (Cambridge University Press, 2018).

[3] W. Singer, Eur. J. Neurosci. 48, 2389 (2018).

[4] P. A. Tass, M. Rosenblum, J. Weule, J. Kurths, A.

Pikovsky, J. Volkmann, A. Schnitzler, and H. J. Freund,

Phys. Rev. Lett. 81, 3291 (1998).

[5] K. Lehnertz, S. Bialonski, M.-T. Horstmann, D. Krug,

A. Rothkegel, M. Staniek, and T. Wagner, J. Neurosci.

Methods 183, 42 (2009).

[6] P. Jiruska, M. de Curtis, J. G. R. Jefferys, C. A. Schevon,

S. J. Schiff, and K. Schindler, J. Physiol. 591.4, 787

(2013).

[7] V. K. Jirsa, W. C. Stacey, P. P. Quilichini, A. I. Ivanov,

and C. Bernard, Brain 137, 2210 (2014).

[8] K. Lehnertz, G. Ansmann, S. Bialonski, H. Dickten, C.

Geier, and S. Porz, Physica D 267, 7 (2014).

[9] M. Steriade, D. A. McCormick, and T. J. Sejnowski, Sci-

ence 262, 679 (1993).

[10] M. Spiess, G. Bernardi, S. Kurth, M. Ringli, F. M. Wehrle,

O. G. Jenni, R. Huber, and F. Siclari, Neuroimage 178,

23 (2018).

[11] F. Moroni, L. Nobili, F. De Carli, M. Massimini, S. Fran-

cione, C. Marzano, P. Proserpio, C. Cipolli, L. De Gen-

naro, and M. Ferrara, Neuroimage 60, 497 (2012).

[12] N. C. Rattenborg, C. J. Amlaner, and S. L. Lima, Neu-

rosci. Biobehav. Rev. 24, 817 (2000).

[13] N. C. Rattenborg, B. Voirin, S. M. Cruz, R. Tisdale, G.

Dell’Omo, H. P. Lipp, M. Wikelski, and A. L. Vyssotski,

Nat. Commun. 7, 12468 (2016).

[14] G. G. Mascetti, Nat Sci Sleep 8, 221 (2016).

[15] M. Tamaki, J. W. Bang, T. Watanabe, and Y. Sasaki,

Curr. Biol. 26, 1190 (2016).

[16] D. S. Bassett, P. Zurn, and J. I. Gold, Nat. Rev. Neurosci.

19, 566 (2018).

[17] P. Sanz-Leon, S. A. Knock, A. Spiegler, and V. K. Jirsa,

Neuroimage 111, 385 (2015).

[18] S. Petkoski and V. K. Jirsa, Phil. Trans. R. Soc. A 377,

20180132 (2019).

[19] L. Ramlow, J. Sawicki, A. Zakharova, J. Hlinka, J. C.

Claussen, and E. Sch¨oll, EPL 126, 50007 (2019).

[20] T. Chouzouris, I. Omelchenko, A. Zakharova, J. Hlinka,

P. Jiruska, and E. Sch¨oll, Chaos 28, 045112 (2018).

[21] M. Gerster, R. Berner, J. Sawicki, A. Zakharova, A.

Skoch, J. Hlinka, K. Lehnertz, and E. Sch¨oll, Chaos 30,

123130 (2020).

[22] D. V. Kasatkin, S. Yanchuk, E. Sch¨oll, and V. I. Nekorkin,

Phys. Rev. E 96, 062211 (2017).

[23] R. Berner, Patterns of Synchrony in Complex Networks of

Adaptively Coupled Oscillators (Springer, Cham, 2021).

[24] F. Sorrentino and L. M. Pecora, Chaos 26, 094823 (2016).

[25] E. Sch¨oll, Nova Acta Leopoldina 425, 67 (2020).

[26] J. Sawicki, Delay controlled partial synchronization in

complex networks (Springer, Heidelberg, 2019).

[27] A. Zakharova, Chimera Patterns in Networks: Interplay

between Dynamics, Structure, Noise, and Delay,Under-

standing Complex Systems (Springer, Cham, 2020).

[28] Y. Maistrenko, B. Penkovsky, and M. Rosenblum, Phys.

Rev. E 89, 060901 (2014).

[29] R. FitzHugh, Biophys. J. 1, 445 (1961).

[30] J. Nagumo, S. Arimoto, and S. Yoshizawa., Proc. IRE 50,

2061 (1962).

[31] A. Chernihovskyi, F. Mormann, M. M¨uller, C. E. Elger,

G. Baier, and K. Lehnertz, J. Clin. Neurophysiol. 22, 314

(2005).

[32] A. Chernihovskyi and K. Lehnertz, Int. J. Bifurc. Chaos

17, 3425 (2007).

[33] I. Omelchenko, O. E. Omel’chenko, P. H¨ovel, and E.

Sch¨oll, Phys. Rev. Lett. 110, 224101 (2013).

[34] A. E. Pereda, Nat. Rev. Neurosci. 15, 250 (2014).

[35] P. Massobrio, L. de Arcangelis, V. Pasquale, H. J. Jensen,

and D. Plenz, Front. Syst. Neurosci. 9, 22 (2015).

[36] J. Shi, K. Kirihara, M. Tada, M. Fujioka, K. Usui, D.

Koshiyama, T. Araki, L. Chen, K. Kasai, and K. Aihara,

Front. Netw. Physiol. (2021), to be published.

[37] Y. Kuramoto, Chemical Oscillations, Waves and Turbu-

lence (Springer-Verlag, Berlin, 1984).

[38] O. E. Omel’chenko, M. Wolfrum, and Y. Maistrenko,

Phys. Rev. E 81, 065201(R) (2010).

[39] N. Tzourio-Mazoyer, B. Landeau, D. Papathanassiou, F.

Crivello, O. Etard, N. Delcroix, B. Mazoyer, and M. Jo-

liot, Neuroimage 15, 273 (2002).

[40] T. Melicher, J. Horacek, J. Hlinka, F. Spaniel, J. Tintera,

I. Ibrahim, P. Mikolas, T. Novak, P. Mohr, and C. Hoschl,

Schizophr. Res. 162, 22 (2015).

[41] J. Hlinka and S. Coombes, European Journal of Neuro-

science 36, 2137 (2012).

[42] J. Sawicki and E. Sch¨oll, Front. Appl. Math. Stat. 7,

662221 (2021).

[43] D. M. Abrams, R. E. Mirollo, S. H. Strogatz, and D. A.

Wiley, Phys. Rev. Lett. 101, 084103 (2008).

[44] A. E. Motter, Nat. Phys. 6, 164 (2010).

[45] S. S. Spencer, Epilepsia 43, 219 (2002).

[46] L. Kuhlmann, K. Lehnertz, M. P. Richardson, B. Schelter,

and H. P. Zaveri, Nat. Rev. Neurol. 14, 618 (2018).

[47] R. G. Andrzejak, C. Rummel, F. Mormann, and K.

Schindler, Sci. Rep. 6, 23000 (2016).

[48] A. Rothkegel and K. Lehnertz, New J. Phys. 16, 055006

(2014).

[49] L. Sadleir, K. Farrell, S. Smith, M. Connolly, and I. Schef-

fer, Neurology 67, 413 (2006).

[50] F. Mormann, K. Lehnertz, P. David, and C. E. Elger,

Physica D 144, 358 (2000).

[51] F. Mormann, T. Kreuz, R. G. Andrzejak, P. David, K.

Lehnertz, and C. E. Elger, Epilepsy Res. 53, 173 (2003).

[52] S. Feldt, H. Osterhage, F. Mormann, K. Lehnertz, and

M. Zochowski, Phys. Rev. E 76, 021920 (2007).

[53] D. J. Watts and S. H. Strogatz, Nature 393, 440 (1998).

[54] M.-T. Horstmann, S. Bialonski, N. Noennig, H. Mai, J.

Prusseit, J. Wellmer, H. Hinrichs, and K. Lehnertz, Clin.

Neurophysiol. 121, 172 (2010).

[55] G. Ansmann and K. Lehnertz, J. Neurosci. Methods 208,

165 (2012).

[56] P. Roelfsema, A. Engel, P. K¨onig, and W. Singer, Nature

385, 157 (1997).

[57] S. M. Sherman, Nat. Neurosci. 19, 533 (2016).

[58] L. L. Gollo, C. R. Mirasso, M. Atienza, M. Crespo-Garcia,

and J. L. Cantero, PLoS ONE 6, e17756 (2011).

p-7