# Chimera-like states in modular neural networks

Johanne Hizanidis  Crete Center for Quantum Complexity and Nanotechnology,
Physics Department, University of Crete, 71003 Heraklion, Greece  National
Center for Scientific Research “Demokritos”, 15310 Athens, Greece  Nikos E.
Kouvaris  Department of Physics, University of Barcelona, Martí i
Franqu$\grave{e}$s 1,E-08028, Barcelona, Spain  Gorka Zamora-López  Center for
Brain and Cognition, Universitat Pompeu Fabra, Barcelona, Spain  Department of
Information and Communication Technologies, Universitat Pompeu Fabra,
Barcelona, Spain  Albert Díaz-Guilera  Department of Physics, University of
Barcelona, Martí i Franqu$\grave{e}$s 1,E-08028, Barcelona, Spain  Chris G.
Antonopoulos  Department of Mathematical Sciences, University of Essex,
Wivenhoe Park, CO4 3SQ Colchester, UK

###### Abstract

Chimera states, namely the coexistence of coherent and incoherent behavior,
were previously analyzed in complex networks. However, they have not been
extensively studied in modular networks. Here, we consider a neural network
inspired by the connectome of the C.elegans soil worm, organized into six
interconnected communities, where neurons obey chaotic bursting dynamics.
Neurons are assumed to be connected with electrical synapses within their
communities and with chemical synapses across them. As our numerical
simulations reveal, the coaction of these two types of coupling can shape the
dynamics in such a way that chimera-like states can happen. They consist of a
fraction of synchronized neurons which belong to the larger communities, and a
fraction of desynchronized neurons which are part of smaller communities. In
addition to the Kuramoto order parameter $\rho$, we also employ other measures
of coherence, such as the chimera-like $\chi$ and metastability $\lambda$
indices, which quantify the degree of synchronization among communities and
along time, respectively. We perform the same analysis for networks that share
common features with the C.elegans neural network. Similar results suggest
that under certain assumptions, chimera-like states are prominent phenomena in
modular networks, and might provide insight for the behavior of more complex
modular networks.

## Introduction

One of the most challenging complex system is the human brain in which neurons
and their interconnections through synapses form a very complicated structure,
the cortical network. The complexity of the circuitry of the nervous system of
the human brain is still a big challenge to be resolved as it contains about
86 billion neurons and thousands times more synapses Azevedoetal2009 . Neurons
are linked together to perform certain tasks and cognitive functions, such as
pattern recognition, function approximation, data processing, etc. It has been
revealed that the cortical network is hierarchical and clustered with a
complex connectivity Hilgetag_2004 , known as the modular organization of the
brain. This demands an inherent parallel nature of brain computations
Meunieretal2010 . Modular processors have to be sufficiently isolated and
dynamically differentiated to achieve independent computations, but at the
same time also globally connected to be integrated in coherent functions
Zamora2010 ; Meunieretal2010 . A possible network description for this modular
organization is that brain networks may be small-world structured Heetal2007
with properties similar to many other complex networks Stam2004 . This
viewpoint has been driven by the systematic finding of small-world topology in
a wide range of human brain networks derived from structural Heetal2007  and
functional Eguiluzetal2005  studies. Small-world topology has also been
identified in functional cortical neural networks in mammals Yuetal2008  and
also in the nervous system of the nematode Caenorhabditis elegans (C.elegans)
soil worm Wattsetal1998 ; Antonopoulosetal2015 . This topology seems to be
relevant for brain function, as it is affected by diseases Yuetal2007 , normal
ageing, and by pharmacological blockade of dopamine neurotransmission
Achardetal2007 .

In recent years, enormous research has been devoted on the C.elegans which has
revealed its ability to learn about mechano-, chemo- and thermo-sensory
stimuli Bargmann2006 ; Kapla1993 . It was also shown that its neural system
has the ability to distinguish between tastes, odours or any indication
related to the presence or absence of food. Moreover, it shows different kinds
of learning behavior, including associative learning such as classical
conditioning and differential classical conditioning, and non-associative
forms of learning such as habituation and dishabituation Gallyetal2003 . These
properties, though quite “simple”, are reminiscent of the human brain ability
to adapt to different stimuli and environments. Moreover, many
neurotransmitters such as Glutamate, GABA, Acetylcholine and Dopamine are
common in the human brain and the C.elegans neural network. The genome of the
C.elegans is almost 30 times smaller than that of humans but still, encodes
almost 22000 proteins and it is almost 35% similar to that of humans
Blumenthaletal1996 . Therefore, the study of the C.elegans neural network may
give an insight of the possible behavior of more complex systems, such as the
human brain.

Neural networks, among other complex systems, self-organize in ways that
synchronous spatiotemporal patterns can emerge. Insightful findings regarding
synchronization in complex networks have been reviewed in Ref. Arenasetal2008
and, recently, synchronization in complex modular or clustered networks has
been investigated in Ref. Gardenesetal2011 . It appears as the interplay
between the intrinsic dynamics associated to the nodes of the network and, its
topology and connecting functions Nicosia2013 . Synchronization in neural
networks is important for normal and various cognitive functions Mathias2011 ,
but may also reflect pathological brain states Traub_1982 . There is also
increasing evidence that various types of brain diseases such as Alzheimer’s
disease, schizophrenia and brain tumors may be associated with deviations of
the network topology from the optimal small-world pattern Stametal2007 . It
has been found that burst synchronization of neural systems may be strongly
influenced by many factors, such as coupling strengths and types Belykh_2005 ,
noise Buric_2007 , and the existence of clusters in neural networks.

Over a decade ago, a very peculiar phenomenon of synchronization was reported
in coupled oscillators, where a hybrid state combining both coherent and
incoherent parts can spontaneously emerge PhysRevA.39.4835 ; KUR02a , which
was later termed chimera state ABR04 . Surprisingly, these states were first
found in systems of identical oscillators coupled with a symmetric interaction
function. Since then, there has been increasing interest in chimera states
LAI09 ; MOT10 ; MAR10 ; OME10a ; BOU14 ; LAZ15  and it has been shown that
they are not limited to phase oscillators, but can also appear in a large
variety of different systems including neural systems OME13 ; HIZ13 ; OME15
which is the focus of the present work. Apart from the classical chimera
states, which consist of a coherent and incoherent domain, chimeras with
multiple incoherent regions SET08 ; OME13 ; VUE14a , as well as amplitude
chimera states SET14  and chimera death states ZAK14  have also been recently
reported. The study of chimera states goes beyond numerical observations.
Experimental verification was first demonstrated in chemical TIN12  and
optical HAG12  systems. Further experiments were realized in mechanical MAR13
, electronic LAR13 , and electrochemical SCH14a ; WIC13  oscillator systems.
Until recently, it was widely assumed that identical elements and symmetric
coupling topology were prerequisites for chimera states. These limitations,
however, can be overcome and chimera-like states can also be found in systems
with non-identical elements, or with non-regular or even global topologies
(see PAN15  and references within). Potential applications of chimera states
in nature include bump states in neural systems LAI01 ; SAK06a  and the
phenomenon of unihemispheric sleep in birds and dolphins RAT00 , which sleep
with one eye open, meaning that half of the brain is synchronized with the
other half being desynchronized. This is relevant for studies of
synchronization in community-based networks where only a few works have
focused on this interesting phenomenon SHA10 ; WIL12d .

Our aim is to contribute in this direction by considering a topology based on
the C.elegans cmtkdataset  network, whose neurons are found, by employing a
community detection algorithm Ponsetal2005 , to be organized in six
communities (see Fig. 1). We assume that neurons are connected using two types
of synapses: electrical and chemical footnote1 , which connect the neurons
within and across the communities, respectively Baptistaetal2010 ;
Antonopoulosetal2015 .

We model the neuron dynamics in terms of the Hindmarsh-Rose system. Our
primary focus lies on the conditions for the existence of chimera-like states
in this modular neural network, and at the same time on the relationship
between the properties of these states with the topological characteristics of
the considered network.

## Results

We first study the level of synchronization within each community and of the
entire network as a function of the electrical and chemical couplings. To
measure synchronization, we use the Kuramoto order parameter $\rho$ (Eq. (5)).
The parameter $\rho$ is bounded in the interval $[0,1]$ and is equal to 1 when
neurons in the considered population are completely synchronized and 0 when
they are totally desynchronized. Note that, one actually averages $\rho(t)$
over time to obtain the value to which the order parameter converges in a
sufficiently long time interval.

Figures 2(a-g) show the order parameter for each community and for the entire
network, in the $(g_{ch},g_{el})$ parameter space. All six communities have a
region of high synchronization for low chemical and high electrical coupling,
which is also reflected in the parameter space of the entire network (Fig.
2(g)). This is reasonable since in this case, the neurons are strongly
connected within each community and at the same time the coupling between
communities is weak, meaning that the different populations do not affect each
other significantly. In particular, for communities 2 and 4, this region of
synchronization extends also to smaller electrical couplings. This is because
these communities have the largest number of nodes and electrical links within
them (see Fig. 1(a) and 1(b)). Small communities are not easily synchronizable
due to the strong influence from the large ones and to the sparsity of their
electrical synapses.

Apart from this region which is common in all communities, there are other
“islands” of synchronization where one or more communities achieve high values
of the order parameter. Prominent examples are Fig. 2(c) and especially Fig.
2(f), where communities 3 and 6 show high synchronization levels for high
chemical couplings with the rest of them and the network globally, being
incoherent. For community 6, in particular, this region is located at high
values of the chemical coupling and gradually shrinks as the electrical
coupling increases. This means that community 6 is dominated by the connected
communities, a fact which is also clear from the size of the arrowheads
directing to it in Fig. 1(b). This effect, although weaker, is also
responsible for the additional high synchronization area in the parameter
space of community 3, located at large values of both chemical and electrical
couplings.

Another measure related to synchronization, which is very relevant in modular
networks, is the metastability index $\lambda$, which measures the coherence
among the communities along time. Figure 2(h) shows the parameter space for
$\lambda$ (Eq. (6)). It is clear that the region in which $\lambda$ attains
values close to zero coincides with the region where all communities are in a
synchronized state. In other parts of the parameter space $\lambda$ achieves
higher values, indicating that the system often switches between synchronous
and asynchronous states.

Figure 2(i) corresponds to the parameter space for the chimera-like index (Eq.
(8)). It is evident that $\chi$ achieves its highest values in the two
synchronization “islands” of communities 3 and 6, as well as on the border of
the common synchronization region separating coherent from incoherent
behavior. All of the aforementioned parameter spaces allow us to gain a
complete picture of the collective behavior in our system, both in the
community and global scale.

In order to highlight some characteristic patterns that emerge in our system,
we select 3 points of interest on the $(g_{ch},g_{el})$ parameter space,
marked by letters A, B and C. They are chosen so that the following three
cases are covered, A: both $\lambda$ and $\chi$ are low-valued, B:
metastability prevails, i.e. $\lambda\gg\chi$ (when normalized), and C:
“chimera-like” states prevail over metastability i.e. $\chi\gg\lambda$ (when
normalized). Figure 3 shows the space-time plots of the membrane potential $p$
(see Eqs. (The Hindmarsh-Rose System.)) for the three points, one typical
snapshot in time where each community is coloured in accordance to Fig. 1, and
the time series of node 100, which belongs to community 3. The nodes in the
communities are relabelled so that each community is placed next to each
other.

Point A corresponds to low metastability and low chimera-like index (see Fig.
2(h) and 2(i)). This means that the network as a whole does not switch in time
to different synchronization patterns frequently, and simultaneously, the 6
communities are, to a large extent, in synchrony with each other (see also
Supplementary Movie S1). This is expected for the combination of high
electrical and low chemical coupling and, is in agreement with the high value
of the global order parameter shown in Fig. 2(g). The corresponding time
series exhibits spiking behavior with short quiescent periods between spike
appearance (Fig. 3(a)).

The metastability effect for low chimera-like index is shown for point B. In
the space-time plot of Fig. 3(b) this is illustrated by the rather regular
spatial pattern (due to low $\chi$), which alternates in time between slow
quiescent periods (yellow-red) and fast spiking intervals (blue-green) that
correspond to synchronous and incoherent behaviors, respectively (see also
Supplementary Movie S2). From the time series, it is evident that, for these
parameters, the system is in the bursting regime.

Point C corresponds to a chimera-like state, in which the metastability of the
system attains low values. Communities 2 and 4 are (on average) the most
synchronized ones, illustrated in Fig. 3(c) and also verified by the
corresponding high values of the order parameter (Fig. 2(b), (d)). This is
reasonable, since these two communities are the largest ones in the network
(see Fig. 1(a), (b)). The remaining communities alternate more frequently
between spiking and quiescent behavior (see also Supplementary Movie S3). They
are also perturbed by the inputs from the large communities making it harder
for them to synchronize, as is also captured in the snapshot of Fig. 3(c).

The above analysis has been carried out for other modular networks (see
Supplementary Information) and the results are in qualitative agreement with
those presented in this section for the C.elegans -based neural network. This
agreement shows that our results do not depend on the community detection
algorithm but, rather, on the interaction of the dynamics with the topology of
the considered network.

## Discussion

In this work we have quantified and compared certain measures of dynamical
complexity, such as synchronization, which may underpin the “differentiation”
in sub-domains and the “integration” as the system exhibits coherent behavior
as a whole Srinivasanetal1999  and, the metastability $\lambda$ and chimera-
like $\chi$ indices that allow for the quantification of the degree of
metastability and chimera-like behavior exhibited by the system and its
communities. In Antonopoulosetal2015 , various statistical quantities
associated with the C.elegans neural network, such as the global clustering
coefficient, the average of local clustering coefficients, the mean shortest
path, the degree probability distribution function of the network and the
small-worldness measure have been computed. The latter property is
characterized by a relatively short minimum path length on average between all
pairs of nodes, together with a high clustering coefficient. Even though
small-worldness captures important aspects of complex networks at the local
and global scale of the structure, it does not provide information about the
intermediate scale. These properties can be better described by the community
structure or modularity of the network. Since nodes within the same module are
densely intra-connected, the number of triangles in a modular network is
larger than in a random graph of the same size and degree distribution, while
the existence of a few links between nodes in different modules plays the role
of topological shortcuts in the small-world topology. Networks characterized
by this property tend to be small-world, with a high clustering coefficient
and short path length with respect to random networks.

Here, we modeled a neural network based on the C.elegans soil worm connectome
in terms of Hindmarsh-Rose dynamics and divided its network into six
communities employing the walktrap method. Based on the numerical simulations
of this system, we analyzed conditions under which coherent and incoherent
neural phase synchronization emerges simultaneously among communities for
certain values of the chemical and electrical couplings. We related this
phenomenon to structural network characteristics, such as the number of nodes
of the communities, the ratio of the number of chemical synapses connecting
their nodes, the ratio of the number of chemical synapses divided by the mean
degree of the target community, the absolute participation and contribution of
nodes to the modular structure and finally, to participation and global
hubness. We found out that none of the nodes with low participation are hubs
of the network, which is characteristic of networks with well-defined
communities, even though we found a significant number of nodes with
intermediate participation, showing that most of the neurons share connections
across several communities. We also found that the hubs of the network are
among the nodes with the largest participation, extending their connections
among most of the communities. All these findings clearly delineate the
influence of one community on the others, and especially that of the two
largest ones.

The phenomenon of chimera-like states we identified here is reminiscent of
classical chimeras observed in the Kuramoto and other models. In our study, we
showed that chimera-like states are spontaneously formed at chemical and
electrical coupling values for which the chimera-like index is much higher
than the metastability index and that they are driven by the largest
communities, which were found to be the most influential ones. This remarkable
behavior was found to be prominent in the two synchronization “islands” of two
communities, as well as on the border of the common synchronization region
separating coherent from incoherent behavior. In the Supplementary
Information, we compared our results with those for modular networks sharing
common features with that of the C.elegans-based network, grouped into six
Erdős-Rényi and small-world communities, respectively. We concluded that
chimera-like states can also be found in the modular networks, suggesting that
they appear to be prominent phenomena which might provide insights for the
functioning of the human brain.

## Methods

### The Hindmarsh-Rose System.

In the current study we aim to analyze how neural dynamics can be collectively
shaped by the coaction of two distinct types of synapses, namely of electrical
and chemical. For this purpose, we consider a C.elegans-based neural network
Varshneyetal2011 , where we employ a community detection method that finds six
communities. A schematic representation of this network is shown in Fig. 1(a).
Based on the detected communities, we assume that neurons within the same
community are connected with electrical synapses (black links in Fig. 1(a))
and neurons across communities with chemical synapses (gray links in Fig.
1(a)). Finally, we endow each neuron with Hindmarsh-Rose dynamics.

Before we incorporate in the dynamics the coupling terms arising from the
electrical and chemical synapses of the individual neurons, we briefly discuss
the different characteristics and functionality of these types of synapses. A
synapse is a junction between two neurons and serves as the means by which
neurons communicate with each other. In particular, an electrical synapse
(electrical link) is a physical connection between two neurons that allows
electrons to pass through neurons by a very small gap between them. Electrical
synapses are bidirectional and of a local character, happening between neurons
which are spatially very close. Mutual coupling through these synapses
promotes phase synchronization and coherence, resulting into groups of
synchronized neurons. On the other hand, chemical synapses (chemical links)
are typically unidirectional and the pre-synaptic signals are transmitted via
release of neurotransmitters from the pre-synaptic neuron, which attaches to
receptors at the post-synaptic neuron. Depending on the neurotransmitter, a
chemical synapse can either be excitatory or inhibitory. Since the empirical
methods used to identify the neuronal wiring in the C.elegans cannot
distinguish between the two types of synapses, here we consider only
excitatory chemical ones Varshneyetal2011 .

By applying the Hindmarsh-Rose dynamics to the nodes of the network that
incorporates both types of synapses, we create an undirected neural network,
in which neurons are connected by electrical (linear diffusive coupling) and
chemical (nonlinear coupling) synapses, described by the equations,

$\displaystyle\dot{p}_{i}$ $\displaystyle=$ $\displaystyle
q_{i}-ap_{i}^{3}+bp_{i}^{2}-n_{i}+I_{\text{ext}}$ $\displaystyle+$
$\displaystyle
g_{el}\sum_{j=1}^{N}L_{ij}H(p_{j})-g_{ch}(p_{i}-V_{\text{syn}})\sum_{j=1}^{N}T_{ij}S(p_{j}),$
$\displaystyle\dot{q}_{i}$ $\displaystyle=$ $\displaystyle
c-dp_{i}^{2}-q_{i},$ $\displaystyle\dot{n}_{i}$ $\displaystyle=$
$\displaystyle r[s(p_{i}-p_{0})-n_{i}],$ (1)

where $i=1,\ldots,N$ is the neuron index, $p_{i}$ is the membrane potential of
the $i$-th neuron, $q_{i}$ is associated with the fast current, either
$Na^{+}$ or $K^{+}$, and $n_{i}$ with the slow current, for example $Ca^{2+}$.
The parameters of Eqs. (The Hindmarsh-Rose System.) are chosen such that
$a=1$, $b=3$, $c=1$, $d=5$, $s=4$, $p_{0}=-1.6$ and $I_{\text{ext}}=3.25$, for
which the system exhibits a multi-scale chaotic behavior characterized as
spike bursting. $r$ modulates the slow dynamics of the system and is set to
$0.005$ so that each neuron lies in the chaotic regime. For these parameters,
the Hindmarsh-Rose model enables the spiking-bursting behavior of the membrane
potential observed in experiments made with single neurons in vitro. It is a
relatively simple model that provides a good qualitative description of many
different patterns empirically observed in neural activity. The connectivity
structure of the electrical synapses is described in terms of the Laplacian
matrix $\mathbf{L}$, whose elements are defined as
$L_{ij}=E_{ij}-\delta_{ij}k_{i}$, where $\delta_{ij}=1$ if $i=j$, and
$\delta_{ij}=0$ otherwise. $\mathbf{E}$ is an adjacency matrix whose elements
are $E_{ij}=1$ if there is an electrical synapse connecting the neurons $i$
and $j$, and $E_{ij}=0$ otherwise. The strength of the electrical coupling is
given by the parameter $g_{el}$ and its functionality is governed by the
linear function $H(p)=p$.

The connectivity structure of the chemical synapses is described in terms of
the adjacency matrix $\mathbf{T}$, whose elements are $T_{ij}=1$ if there is a
chemical synapse between neurons $i$ and $j$, and $T_{ij}=0$ otherwise. The
chemical coupling is nonlinear and its functionality is described by the
sigmoidal function
$S(p)=\\{1+\exp[-\lambda(p-\theta_{\text{syn}})]\\}^{-1}\,$, which acts as a
continuous mechanism for the activation and deactivation of the chemical
synapses. The coupling strength associated to this type of synapses is
$g_{ch}$. For the chosen set of parameters, $|p_{i}|<2$ and thus
$(p_{i}-V_{\text{syn}})$ is always negative, meaning that the chemical
coupling is excitatory if $V_{\text{syn}}=2$. The other parameters are
$\theta_{\text{syn}}=-0.25$ and $\lambda=10$ following Refs. Baptistaetal2010
; Antonopoulosetal2015 . For simplicity, we assume that both types of synapses
are bidirectional.

Additionally to Eqs. (The Hindmarsh-Rose System.), we introduce the
instantaneous angular frequency of the $i$-th neuron as Pereiraetal2007 ,

$\dot{\phi}_{i}=\frac{\dot{q}_{i}p_{i}-\dot{p}_{i}q_{i}}{p_{i}^{2}+q_{i}^{2}},$
(2)

where $\phi_{i}$ is the phase defined by the fast variables, $p_{i}$ and
$q_{i}$ of the $i$-th neuron; $i=1,\ldots,N$.

In the following, we analyze the dynamical behavior of the system of Eqs. (The
Hindmarsh-Rose System.) by means of numerical simulations for a network of six
small-world communities, detected in the C.elegans neural network. We also
link the emergent dynamics to the coaction of the two types of synapses. The
large-scale interaction between communities is illustrated in Fig. 1(b), where
the communities are represented by circles of sizes proportional to their
number of nodes. The width of each arrow is proportional to the number of
chemical synapses between two communities and the size of the arrowhead is
proportional to the relative influence of one community to the other, that is,
the number of the chemical synapses divided by the mean degree of the
electrical synapses of the target community. A more detailed analysis reveals
that only $20\%$ of the neurons are exclusively connected within their
community (Fig. 1(c)). The rest of neurons are linked with neurons in other
communities through chemical synapses. As in other brain networks Zamora2011 ;
Klimm2014 , a trend is appreciated for which, sparsely connected neurons have
low participation while the hubs span their links among many communities (Fig.
1(d)). In this case, all hubs of the network are condensed into community 4
and spread links among five or six communities. In conclusion, community 4 is
not only found to be the largest but also the most influential one in terms of
its inter-modular connectivity.

In the Supplementary Information we present the same analysis using modular
networks with 277 neurons grouped into six communities, which share some
common features with those of the C.elegans neural network.

###  Modular Structure of the C.elegans Neural Network.

We identified the communities of the C.elegans neural network employing the
walktrap method Ponsetal2005  of the igraph software, using six steps,
following Ref. Antonopoulosetal2015 . The algorithm detects communities
through a series of short random walks, based on the idea that the nodes
encountered on any given random walk are more likely to be within a community.
The algorithm initially treats all nodes as communities of their own, then
merges them into larger communities, and these into even larger ones, and so
on. Essentially, it tries to find densely connected subgraphs (i.e.
communities) in a graph via random walks. The idea is that short random walks
tend to stay in the same community. Following this approach, we have been able
to identify six communities in the C.elegans neural network.

### Contribution of Nodes on the Mesoscale.

Given a modular network, its nodes can play different roles according to their
inter- and intra-modular connections Guimera2005a ; Guimera2005b . For
example, some nodes may only connect to nodes in a particular community. Such
nodes, often referred to as _peripheral_ , form the core of the community.
Other nodes might be well-connected within a module but extend connections to
other modules. These allow for the communities to link with each other. In the
other extreme case we might find nodes which are equally connected across all
modules and therefore, are difficult to be classified within modules. Such
nodes, often called _kinless nodes_ , may however allow for the integration of
information, since they have access to the information in all modules. Here,
we characterize the roles of the nodes based on a recently introduced
appropriate framework Klimm2014 .

_Hubness:_ In order to characterize the hubness of a node so that its hubness
value is comparable across networks of different sizes and densities, the
degrees of a node has to be evaluated under a common reference or statistical
frame. We define the hubness index $h_{i}$ of a node $i$ in a network of size
$N$ and density $d$ as the relative difference of the node’s degree $k_{i}$
with respect to the degree distribution of an equivalent random graph of the
same size and number of nodes,

$h_{i}=\frac{k_{i}-\left<k\right>_{R}}{\sigma_{R}}=\frac{k_{i}-(N-1)d}{\sqrt{(N-1)\,d\,(1-d)}},$
(3)

where $\left<k\right>_{R}$ is the mean degree of the equivalent random graph,
and $\sigma_{R}$ the standard deviation of its degree distribution. The
hubness is negative for nodes with degree $k_{i}<\left<k\right>_{R}$, allowing
also for the identification of outliers that are significantly less connected
than expected from randomness.

_Participation Index:_ Given a partition $\mathcal{C}$ with $M$ disjoint
communities, the aim is to quantify how distributed a node is among them. The
contribution of a node to each community depends on the community size and
therefore, its participation is characterized in terms of the likelihood the
node has to belong to a given community. A _participation vector_ $F_{i}$ with
elements $F_{im}$ represent the probability that node $i$ belongs to community
$C_{m}$, where $m=1,2,\ldots,M$. This probability is given by
$F_{im}=\frac{k_{im}}{N_{m}}$, where $N_{m}$ is the size of the community.
Since we are only interested in the relative differences, the participation
vectors are normalized such that $\sum_{m=1}^{M}F_{im}=1$. The vector of a
node devoting all its links to the second community of a network with $M=6$
communities is $F_{i}$ $=(0,1,0,0,0,0)$ and, for a node whose links are all
equally likely distributed among the six communities, $F_{i}$
$=(1/6,1/6,1/6,1/6,1/6,1/6)$. When reducing this information into an index
that quantifies how distributed the links of a node are among all communities,
for consistency with previous definitions Guimera2005a , we set $f_{i}=0$ if
the node devotes all its links to a single community and $f_{i}=1$ if its
links are equally likely distributed among _all_ modules. Therefore, we
evaluate the standard deviation $\sigma(\mathbf{F_{i}})$ of the elements of
the participation vector $F_{i}$ and, define the participation index as

$f_{i}\>=\>1-\frac{\sigma(\mathbf{F_{i}})}{\sigma_{max}(M)}\>=\>1-\frac{M}{\sqrt{M-1}}\;\sigma(\mathbf{F_{i}}).$
(4)

The normalization factor accounts for the fact that the standard deviation of
an $M$-dimensional vector with all elements equal to zero but one is
$\sigma_{max}(M)=\sqrt{M-1}\,/\,M$.

### Estimating Synchronization, Metastability and Chimera-like States.

In this paper we use the order parameter $\rho$ to account for the
synchronization level of the neural activity of the considered networks and of
their communities Gardnesetal2010 . It is originated from the theory of
measures of dynamical coherence of a population of $N$ Kuramoto phase
oscillators KUR02a  and can be computed by a complex number $z(t)$ defined as

$z(t)=\rho(t)e^{\mathrm{i}\Phi(t)}=\frac{1}{N}\sum_{j=1}^{N}e^{\mathrm{i}\phi_{j}(t)}.$
(5)

By taking the modulus $\rho(t)$ of $z(t)$, one can measure the phase coherence
of the $N$ neurons of the network, and by $\Phi(t)$ their average phase. In
this context, $\phi_{i}$ is the phase variable of the $i$-th neuron given by
Eq. (2). Actually, one averages $\rho(t)$ in time to obtain the order
parameter $\rho=\langle\rho(t)\rangle_{t}$, namely the tendency of $\rho(t)$
in time. A value of $\rho=1$ corresponds to complete synchronous activity,
whereas $\rho=0$ to complete desynchronization. We use Eq. (5), adapted
accordingly, wherever we need to compute the synchronization level of neural
networks or communities. In particular, in the case of the whole neural
network, $N$ is the total number of neurons, whereas in the case of
communities, $N$ represents the number of neurons of the particular community.

The previous paragraph focused on the synchronization of each community as
described by the time average of the order parameter. However, if one wants to
look at the instantaneous behavior of each community and of the whole network,
one will see that there are alternating time intervals of synchronization and
incoherence for each community, also seen in the incoherent regions of Fig. 2.
This is due to the chaotic bursting behavior of the Hindmarsh-Rose model which
involves quiescent periods, where neurons easily synchronize, followed by fast
spiking intervals where neurons tend to desynchronize because of the
corresponding faster time scales.

Complex dynamical systems do not necessarily settle into stationary
synchronized states. Instead, they can be metastable in time, meaning that
they can stay in the vicinity of one stable state for a certain time interval
and then, spontaneously move towards another. An even more interesting feature
of many complex systems is undoubtedly the coexistence of different, often
contradictory states. In terms of synchronization, this is illustrated through
the so-called chimera states KUR02a ; ABR04 , where one population of
oscillators synchronizes whereas other populations of identical oscillators
are desynchronized.

In order to quantify the metastability and chimera-likeness of the observed
dynamics, we employ two measures first introduced by M. Shanahan in Ref. SHA10
. In particular, the level of metastability can be calculated from the so-
called metastability index $\lambda$, given by the expression

$\lambda={\langle\sigma_{\textbf{met}}\rangle}_{C_{m}},$ (6)

where,

$\sigma_{\textbf{met}}(m)=\frac{1}{T-1}\sum_{t=1}^{T}(\rho_{m}(t)-{\langle\rho_{m}\rangle}_{T})^{2}.$
(7)

In Eq. (6), $C_{m}$ is the set of all $M$ communities and $m=1,2,\ldots,M$.
The order parameter $\rho_{m}(t)$ of each community $m$ is sampled at discrete
times $t\in{1,\ldots,T}$. For a given community, the variance
$\sigma_{\textbf{met}}(m)$ of $\rho_{m}(t)$ over all time steps, gives an
indication of how much the synchrony in this community fluctuates in time.
Averaging over all $M$ communities, one obtains an index of the metastability
present in the entire network.

Similarly, the so-called chimera-like index $\chi$ SHA10  is given by

$\chi={\langle\sigma_{\textbf{chi}}\rangle}_{T},$ (8)

where

$\sigma_{\textbf{chi}}(t)=\frac{1}{M-1}\sum_{m=1}^{M}(\rho_{m}(t)-{\langle\rho(t)\rangle}_{M})^{2}.$
(9)

In the above expression, $\sigma_{\textbf{chi}}(t)$ is an instantaneous
quantity that gives the variance of $p_{m}(t)$ over all $M$ communities at a
given time $t$. The time average of this quantity indicates how chimera-like a
certain state is.

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## Author contributions statement

J. H., N. E. K. and C. G. A. conceived and designed the study and performed
the numerical simulations. G. Z.-L. and A. D. -G. analyzed the roles of the
nodes. J. H., N. E. K., C. G. A., G. Z.-L. and A. D. -G. carried out the
analysis and wrote the article.

## Acknowledgments

This study was supported by the J. S. Latsis Public Benefit Foundation in
Greece. J. H. acknowledges support by the EU/FP7-REGPOT-2012-2013-1 under
grant agreement n316165. N. E. K. and A. D. -G. acknowledges support by the
LASAGNE (Contract No.318132) and MULTIPLEX (Contract No.317532) EU projects as
well as by the Generalitat de Catalunya (2014SGR608) and Spanish MINECO
(FIS2012-38266). G. Z. -L. received support from the European Union Seventh
Framework Programme FP7/2007-2013 under grant agreement number PIEF-
GA-2012-331800.

## Additional information

The authors declare no competing financial interest. Correspondence or request
for material should be addressed to J. H., N. E. K., C. G. A, G. Z. -L. or A.
D. -G..

## Figures Legends

Figure 1: Modular organization of the C.elegans neural network and
contribution of neurons to the modular structure. (a) Employing the walktrap
method Ponsetal2005 , six communities have been detected in the neural network
of C.elegans. Different colors are assigned to neurons of different
communities, 1: brown (19 neurons), 2: blue (69 neurons), 3: red (18 neurons),
4: yellow (108 neurons), 5: purple (20 neurons), 6: green (43 neurons), and
are used throughout all figures. Different type of synapses are depicted by
links of different color; black links denote electrical coupling between
neurons within the same community, whereas gray links represent chemical
coupling between neurons across different communities. (b) The communities are
represented by circles of different size which is proportional to the number
of their nodes. Directed links connecting the communities clearly delineate
the influence of one community to the other. Their width is proportional to
the number of chemical synapses connecting their nodes and the size of their
arrows is proportional to the number of those chemical synapses divided by the
mean degree of the target community. (c) Absolute participation and
contribution of nodes to the modular structure. (d) Relation between
participation and global hubness. None of the nodes with low participation is
a hub of the network, what is characteristic of networks with well-defined
communities. However, we find a significant number of nodes with intermediate
participation, showing that most neurons share connections across several
communities. The hubs of the network are among the nodes with the largest
participation, extending their connections among most of the communities.
Figure 2: Parameter spaces. Density plots show the order parameters as well as
the metastability and the chimera-like index in the same $(g_{ch},g_{el})$
plain. The order parameter of each community $\rho_{1,\ldots,6}$ is shown in
(a)-(f), and of the entire network in (g). The metastability index is shown in
(h) and the chimera-like index in (i) (note the different range of values in
the color bar in (h) and (i)). The marked points A ($g_{ch}=0.015$,
$g_{el}=1.7$), B ($g_{ch}=0.18$, $g_{el}=0.7$) and C ($g_{ch}=0.015$,
$g_{el}=0.5$) denote three different dynamical regimes. Synchronization is
clear within each community in the regime where point A is chosen and
corresponds to low values of both, $\lambda$ and $\chi$. Point B characterizes
a regime where the synchronization index of each community and of the entire
network is very low, $\chi$ is also low and $\lambda$ is high. This indicates
a highly metastable behavior and desynchronous dynamics. An interesting
behavior is found on the edge that separates those two regimes, where point C
is located. Some communities (obviously 2 and 4, less clearly 6) are
synchronous and others (1, 3 and 5) are desynchronous. Furthermore, $\lambda$
is low, whereas $\chi$ is high, meaning that this state is not metastable and
chimera-like dynamics can be observed. The dynamical behavior in these points
is illustrated clearly in Fig. 3. Figure 3: Synchronous, metastable and
chimera-like states. (a) The spatiotemporal evolution of $p_{i}$ (upper left),
with a time series of the neuron with index 100 of community 3 (upper right)
and a snapshot of the system state (bottom) are shown for $g_{ch}=0.015$ and
$g_{el}=1.7$, which correspond to point A of Fig. 2. Both, $\lambda$ and
$\chi$ are very low, thus synchronization is clearly shown. (b) The same plot
for point B ($g_{ch}=0.18$, $g_{el}=0.7$). Here, the metastable index
$\lambda$ is high, while the chimera-like index $\chi$ is low, thus a
desynchronized state is depicted. (c) The same plot for point C
($g_{ch}=0.015$, $g_{el}=0.5$), where the metastable index $\lambda$ is low
and the chimera-like index $\chi$ is high. A chimera-like state is illustrated
here, where neurons in communities 1 (brown), 3 (red), 5 (purple) and 6
(green) are desynchronized, whereas neurons in communities 2 (blue) and 4
(yellow) are synchronized. Neurons are ordered according to their community.
Same color code as in Fig. 1 is used in the snapshots.