# Diversity of neuronal activity is provided by hybrid synapses

Kesheng Xu  Faculty of Science, Jiangsu University, Zhenjiang, Jiangsu 212013,
PR China<EMAIL_ADDRESS>Centro Interdisciplinario de Neurociencia de
Valparaíso, Universidad de Valparaíso, Valparaíso, 2360102, Chile  Jean Paul
Maidana  Patricio Orio  Centro Interdisciplinario de Neurociencia de
Valparaíso, Universidad de Valparaíso, Valparaíso, 2360102, Chile  Instituto
de Neurociencia, Facultad de Ciencias, Universidad de Valparaíso, Valparaíso,
2360102, Chile<EMAIL_ADDRESS>
(Received: date / Accepted: date)

###### Abstract

Many experiments have evidenced that electrical and chemical synapses – hybrid
synapses – coexist in most organisms and brain structures. The role of
electrical and chemical synapse connection in diversity of neural activity
generation has been investigated separately in networks of varying
complexities. Nevertheless, theoretical understanding of hybrid synapses in
diverse dynamical states of neural networks for self-organization and
robustness still has not been fully studied. Here, we present a model of
neural network built with hybrid synapses to investigate the emergence of
global and collective dynamics states. This neural networks consists of
excitatory and inhibitory population interacting together. The excitatory
population is connected by excitatory synapses in small world topology and its
adjacent neurons are also connected by gap junctions. The inhibitory
population is only connected by chemical inhibitory synapses with all-to-all
interaction. Our numerical simulations show that in the balanced networks with
absence of electrical coupling, the synchrony states generated by this
architecture are mainly controlled by heterogeneity among neurons and the
balance of its excitatory and inhibitory inputs. In balanced networks with
strong electrical coupling, several dynamical states arise from different
combinations of excitatory and inhibitory weights. More importantly, we find
that these states, such as synchronous firing, cluster synchrony, and various
ripples events, emerge by slight modification of chemical coupling weights.
For large enough electrical synapse coupling, the whole neural networks become
synchronized. Our results pave a way in the study of the dynamical mechanisms
and computational significance of the contribution of mixed synapse in the
neural functions.

Hybrid synapses, Balanced netoworks , Dynamical coexistence state

##  I Introduction

Over the past decades, spatiotemporal patterns has been observed in
experimental recordings of neural activity, such as synchronous states[1, 2],
complex spatiotemporal patterns[3, 4, 5, 6, 7, 8, 9] and chimera states[10,
11], has long been of interest, but their underlying mechanism of the
generation of these patterns has not been fully understood yet.

It is generally accepted that there are two main modalities of synaptic
transmission – chemical and electrical[12, 13]. At chemical synapses,
information is transferred through the release of a neurotransmitter from
presynaptic neuron and detection of the neurotransmitter by an adjacent
postsynaptic cell[14], whereas in electrical synapses, the cytoplasm of
adjacent cells is directly connected by clusters of intercellular channels
called gap junctions[15]. From this different characteristics and
functionality, chemical synaptic communication is typically unidirectional and
episodic. By contrast, electrical synaptic transmission through gap junctions
underlies direct, continuous, reliable and rapid neuronal communication in the
CNS[15]. In addition, electrical synapses are bidirectional and local
connections between neurons which are spatially very close. Theoretical work
has shown that diversity of firing patterns and cortical functions is provided
by balance between coactivated excitatory and inhibitory synaptic inputs[16,
17, 18, 19] (for a review see ref.[20]). For example, when externally driven,
circuits of recurrently connected by excitatory and inhibitory neurons with
strong synapses settle rapidly into a state in which population activity
levels ensure a balance of excitatory and inhibitory currents[21, 22, 23].
Over all, excitatory neurons are thought to do only one thing: excite each
other, whereas inhibitory neurons can generate nonlinear effects[24], such as
modulating stimulus response gain, sharpening tuning to stimuli, and pacing
cortical oscillations[20, 24]. For electrical coupling, it is responsible for
a variety of network effects, particularly in networks that generate rhythmic
activity, some of which are well established, such as regulation of phase
relationships, synchrony, and pattern formation[25, 26, 27, 28, 29], and some
are novel, such as a direct role in rhythmogenesis (also see review[2]).

The relationship between spatiotemporal patterns of neuronal activity and the
types of synaptic transmission can be investigated in modeling studies[30, 31,
32, 33]. Computational models have shown that the synchronous occurrence of
action potentials in many neurons is mainly due to connections of chemical
inhibitory synapses[34, 35, 36, 37, 38] or electrical synapses between
neurons[1, 39, 40, 41, 42, 2]. The results obtained by Börgers et al. [43]
also suggest that synchronous oscillations often comes from interplay of
chemical synapses connections between E-cells (excitatory neurons) and I-cells
(inhibitory neurons) but under ideal conditions – homogeneity in relevant
network parameters and all-to-all-connections. Synchronous states have
traditionally been thought to involve precise zero-lag synchrony[44, 45].
Instead of precise zero-lag synchrony, a range of flexible phase offsets –
phase differences between two (or more) oscillations– is possible. These
flexible phase relationships can, in their simplest form, be traveling waves
of various shapes (including plane, radial and spiral waves[46]). The
combination of multiple traveling waves can form complex spatiotemporal
patterns [47, 4]. It has been shown, in spiking neural networks[46, 48, 49,
50, 51]that transmission delays[41], the spatial reach of connections[7] and
the strength of inhibition of the excitatory-inhibitory networks influence the
emergence of spatiotemporal patterns, such as asynchronous and irregular
activity, or propagating waves. Most previous modeling studies have shown that
the formation of spatiotemporal patterns such as traveling waves in neural
networks has mainly been investigated by means of phenomenological neural-
field models. These models uncover the existence and uniqueness of diverse
patterns which are stationary or nonstationary in space and time, such as
waves, wave fronts, bumps, pulses, and periodic patterns[32, 52, 53].
Recently, much attention has been paid to the chimera state[54, 55, 56, 57].
In chimera state, populations of coupled oscillators may exhibit two
coexisting subpopulations, one with synchronized oscillations and the other
with unsynchronized oscillations, even though all of the oscillators are
coupled to each other in an equivalent manner[58, 59]. This symmetry-breaking
behavior has been studied in a variety of complex dynamical systems and has
attracted growing attention in neural system studies from both theoretical and
experimental perspectives[60, 61, 62].

The role of purely electrical synapse or chemical synapse connections in
generation of the diversity of neural activity has been investigated deeply in
networks of varying complexities as discussed above. Many experiments have
evidenced that diverse neural circuits use a combination of electrical and
chemical synapses – hybrid synapses – coexisting in most organisms and brain
structures[63, 64, 65, 66], to convey signals between neurons (reviewed in[67,
12] ). At first, the synchronous dynamical behavior of the neuronal systems
with two types of synapses have attracted more and more attention. Kopell et
al.[68] found that both electrical and chemical synapses play the different
but complementary roles in the synchronization of neuronal networks. Baptista
et al.[69] further combined two types of synapses to study the process of
synchronization and the information production. Liu et al.[70] studied the
combined effects of the information transmission delay and the ratio of the
electrical and chemical synapses on the synchronization transitions. Secondly,
the concept in which chimera-like states emerge spontaneously with suitable
tunning of electrical and chemical coupling strengths, surfaced in the
pioneering theoretical and computational work of Hizanidis et al.[60]. In
another work, Majhi et al.[61] analyzed chimera states in a two-layer neural
network connected with hybrid synapses. The results obtained by them showed
that the emergence of chimera states depends significantly on chemical
synapses, not electrical ones. More recently, Calim et al.[62] highlighted the
importance of the excitability level and nonlocal network features on
spatiotemporal patterns in networks of neurons coupled only via chemical
synapses. Furthermore, they mapped the transitions between incoherent states,
traveling waves, chimeras, coherent states, and global amplitude death in the
parameter space. More importantly, Pernelle et al.[71] showed that gap
junction plasticity acts as a mechanism to regulate oscillations in the
networks connected by hybrid synapses.

All these approaches mentioned above only identify the essential mechanisms
for generating various complex spatiotemporal patterns (such as synchrony
state, chimeras state, and etc) in neural networks coupled via chemical
synapse or hybrid synapses. Nevertheless, they give little insight on how the
electrical and chemical synapses are related in the emergence of a diversity
of dynamical states in neural networks for self-organization and robustness.
In other words, the relationship between spatiotemporal patterns of neuronal
activity and the types of synaptic transmission together with corresponding
synaptic weights has not been systematically investigated yet. For example,
whether the excitatory-inhibitory balance or gap junction make significant
different contributions to generate firing patterns , and whether these
different states can coexist in the same parameter space or the same time bins
of interest still have not been fully studied yet.

Our first finding is that, in the E/I balanced networks of excitatory
population in absence of gap junction, the excitatory population has a
tendency to cluster synchronization as the weight of inhibitory synaptic
currents into the excitatory part is increase. In other words, the larger the
chemical inhibitory synapse strength used for excitatory part the higher
degree of synchronization to achieve. However, introducing the gap junction to
the excitatory population, it usually can go through several phases of network
state, that are asynchronous activity, synchronous firing, various ripples
events (such as traveling wave) and lastly subthreshold oscillations without
spikes. For example, we find that for stronger inhibitory synaptic strengths
larger than a certain amount leads to suppression of the spikes oscillation,
and for moderate inhibitory synaptic strengths one should expect to find
chimera-likeness behaviors, which is also metastable state in some time
windows. More importantly, it first time to show the coexistence of these
dynamics states, such as coexistence of coherent state, traveling wave and
oscillation without spikes, depending on the inhibition level . Finally, our
results suggest that if excitatory populations are connected with hybrid
synapses, the excitation and inhibition balance can make a great contribution
to generate various spatiotemporal patterns.

##  II Models and methods

###  II.1 Neuronal dynamics

Wang-Buzsáki model. The Wang-Buzsáki model we used has been presented detailly
in Ref[72, 73], here we give a brief overview. This conductance-based model
has three states variables for each neurons: membrane potential of each cells
$V$, the Sodium inactivation variable $h$ and Potassium activation variable
$n$ corresponding to the spikes-generating $Na^{+}$ and $K^{+}$ voltage-
dependent ion currents ($I_{Na}$ and $I_{K}$), respectively. Sodium activation
variable $m$ is considered to be instantaneous. The neuronal dynamics can be
described as :

$\displaystyle C_{m}\frac{dV}{dt}$
$\displaystyle=-I_{Na}-I_{K}-I_{L}-I_{Syn}+I_{app}$
$\displaystyle\frac{dh}{dt}$ $\displaystyle=\phi(\alpha_{h}(1-h)-\beta_{h}h)$
$\displaystyle\frac{dn}{dt}$ $\displaystyle=\phi(\alpha_{n}(1-n)-\beta_{n}n)$
(1)

Where $I_{L}=g_{L}(V-E_{L})$, $I_{Na}=g_{Na}m^{3}_{\infty}h(V-E_{Na})$ and
$I_{K}=g_{K}n^{4}(V-E_{K})$ represent the leak currents, transient sodium
currents and the delayed rectifier currents, respectively. $I_{syn}$ stands
for the synaptic currents and $I_{app}$ is the injected currents (in $\mu
A/cm^{2}$). The parameters $g_{L}$, $g_{Na}$, $g_{K}$ are the maximal
conductance density, $E_{L}$, $E_{Na}$, $E_{K}$ are the reversal potential and
function $m_{\infty}$ is the steady-state activation variables $m$ of the
Hodgkin-Huxley type[74].

Figure 1:  Network architecture and parameters. A. The balanced neural
networks containing excitatory (red) and inhibitory populations (green). The
synaptic input (excitatory synaptic input or inhibitory synaptic input)
between the populations are generated by all-to-all connections. Their
coupling strength between the populations is denoted with constants $J_{IE}$
and $J_{EI}$. Self-inhibitory connections (of the weights $J_{II}$) is all to
all connections. Excitatory (inhibitory) connections are represented in red
(black). $\nu_{ext}$ is the external drive to each neuron in the excitatory
population by external excitatory inputs. B. Diagram of excitatory populations
connected by electrical (linear diffusive coupling) and chemical (nonlinear
coupling) synapses. The excitatory population is connected by excitatory
synaptic (red solid lines) strength $J_{EE}$ in a small world topology and
adjacent neurons are also connected by electrical synaptic strength $J_{gap}$
(black dashed lines ). Without loss of generality, we fix $J_{II}=0.04$,
$J_{IE}=0.01$.

In the simulation, unless stated otherwise, the parameters and functions used
are given in Table 1.

Table 1:  Parameters and functions of Wang-Buzsáki model(1996)[72]

$g_{L}=0.1$ |  $g_{Na}=35$ |  $g_{K}=9$ ($mS/cm^{2}$)
---|---|---
$E_{L}=-65$ |  $E_{Na}=55$ |  $V_{K}=-90$ (mV)
$C_{m}=1$ ($\mu F)$ |  $\phi=5$ |  $I_{app}=0$ ($\mu A/cm^{2}$)

$\alpha_{m}(V)=-0.1(V+35)/[exp(-0.1(V+28))+1]$ |  $\beta_{m}(V)=4exp[-(V+60)/18]$
---|---
$\alpha_{h}(V)=0.07exp[-(V+58)/20]$ |  $\beta_{h}(V)=1/exp[-0.1(V+28)]+1$
$\alpha_{n}(V)=-0.01(V+34)/[exp(-0.1(V+34))-1]$ |  $\beta_{n}(V)=0.125exp[-(V+44)/80]$
$m_{\infty}=\alpha_{m}/(\alpha_{m}+\beta_{m})$ |

Synaptic dynamics. By applying the Wang-Buzsáki model to the nodes of the
network that incorporates both types of synapses, neurons in the networks are
connected by electrical (linear diffusive coupling) and chemical (nonlinear
coupling) synapses. The total synaptic input currents into neuron $i$ within
excitatory population, is given by:

$\displaystyle I_{SynE,i}$ $\displaystyle=J_{gap}\sum_{j=1,j\neq
i}^{N_{E}}C_{gap}^{ij}(V_{i}-V_{j})+J_{EE}\sum_{j=1,j\neq
i}^{N_{E}}L_{EE}^{ij}g_{syn,i}(V_{i}-E_{synE})$
$\displaystyle+J_{EI}\sum_{j=1,j\neq
i}^{N_{I}}L_{EI}^{ij}g_{syn,i}(V_{i}-E_{synI})+\sum_{i=1}^{N_{E}}g_{syn,i}(V_{i}-E_{synE})$
(2)

The four terms on right-hand side of Eq.(II.1) corresponds to electrical
synaptic inputs, self-excitatory synaptic inputs, inhibitory synaptic inputs
(from inhibitory population) and external excitatory synaptic input to each
neuron in the excitatory population, respectively. The connection matrices
$C_{gap}^{ij},L_{EE}^{ij},L_{EI}^{ij}$ are described in the network
connectivity section. Lastly purely chemical synaptic input currents for
inhibitory population is as follows:

$\displaystyle I_{SynI,i}=J_{II}\sum_{j=1,j\neq
i}^{N_{I}}L_{II}^{ij}g_{syn,i}(V_{i}-E_{synI})+J_{IE}\sum_{j=1,j\neq
i}^{N_{E}}L_{IE}^{ij}g_{syn,i}(V_{i}-E_{synE})$ (3)

The two terms on right-hand side of Eq.(3) represent self-inhibitory synaptic
input and excitatory synaptic inputs (from excitatory population).

We use a second-order differential equation to model chemical synaptic
conductances $g_{syn}$[75, 76], provided by:

$\displaystyle\frac{d^{2}g_{syn}}{dt^{2}}={\bar{g}}_{s(E,I,\nu)}f\delta\left(t_{0}+t_{d}-t\right)-\left(\frac{1}{\tau_{1}}+\frac{1}{\tau_{2}}\right)\frac{dg_{syn}}{dt}-\frac{g_{syn}}{\tau_{1}\tau_{2}}$
(4)

The delta function $\delta\left(t_{0}+t_{d}-t\right)$ represents the spike
signal from presynatpic cell. The time $t_{0}$ is the time of the presynaptic
spike and $t_{d}$ is synaptic delays from the presynatpic to postsynaptic
cell. The conductance peaks occurs at time
$t_{peak}=t_{0}+t_{d}+\frac{\tau_{1}\tau_{2}}{\tau_{1}-\tau_{2}}ln(\tau_{1}/\tau_{2})$.
The normalization factor $f$, ensures that the amplitude equals
${\bar{g}}_{s(E,I,\nu)}$, follows:

$\displaystyle
f=\dfrac{1}{e^{-\left(t_{peak}-t_{0}-t_{d}\right)/\tau_{1}}-e^{-\left(t_{peak}-t_{0}-t_{d}\right)/\tau_{2}}}$
(5)

In simulations and for practical reasons, second order differential equations
are solved as follows:

$\displaystyle\frac{dy}{dt}$
$\displaystyle={\bar{g}}_{s(E,I,\nu)}f\delta\left(t_{0}+t_{d}-t\right)-\left(\frac{1}{\tau_{1}}+\frac{1}{\tau_{2}}\right)y-\frac{g_{syn}}{\tau_{1}\tau_{2}}$
$\displaystyle\frac{dg_{syn}}{dt}$ $\displaystyle=y$ (6)

In simulations, both peak synaptic conductances ${\bar{g}}_{s(E,I,\nu)}$ and
synaptic delays $t_{d}$ are Gaussian distributed random variables with
prescribed means $g_{s(E,I)}$ and $\overline{D}$, and standard deviation
$\sigma_{g_{s(E,I)}}$ and $\sigma_{\overline{D}}$. The background drive
$v_{ext}$ to each neuron in the excitatory population was provided by external
excitatory inputs, modeled as independent and identically distributed Poisson
processes with rate $\nu$ for different neurons. Peak conductances of the
external drive were also heterogeneous and assumed values sampled from a
truncated Gaussian distribution with mean $g_{s\nu}$ and standard deviation
$\sigma_{g_{s\nu}}$. These rules are the same as the previous study[73].
Parameters of synaptic model are summarized in Table 2.

A: Global simulation paramters
---
Description |  Symbol |  Value
Simulation duration |  $T_{sim}$ |  2000 ms
Start-up transient |  $T_{trans}$ |  400 ms
Time step |  dt |  0.02 ms
B: Populations and external input
Description |  Symbol |  Value
Population size of excitatory neurons |  $N_{E}$ |  1000
Population size of inhibitory neurons |  $N_{I}$ |  250
Possion input rate (external excitatory inputs) |  $\nu$ |  6000 Hz
C: Connection parameters
Description |  Symbol |  Value
Weight of gap junction |  $J_{gap}$ |  0.1
Weight of self-excitatory connection |  $J_{EE}$ |  0.5
Weight of self-inhibitory connection |  $J_{II}$ |  0.04
Weight of inhibitory connections for excitatory populations |  $J_{EI}$ |  0.03
Weight of excitatory connections for inhibitory populations |  $J_{IE}$ |  0.01
Probability of local inhibitory connections |  $P_{I}$ |  0.2
Synaptic dynamics (Difference of Two Exponentials)
Reversal potential for excitatory synapses |  $E_{synE}$ |  0 mV
Reversal potential for inhibitory synapses |  $E_{synI}$ |  -80 mV
Excitatory synaptic decay time |  $\tau_{1E}$ |  3 ms
Ihibitory synaptic decay time |  $\tau_{1I}$ |  4 ms
Excitatory synaptic time |  $\tau_{2E}$ |  1 ms
Inhibitory synaptic time |  $\tau_{2I}$ |  1 ms
Mean synaptic delay |  $\overline{D}$ |  1.5 ms
Mean synaptic excitatory conductance |  $g_{sE}$ |  5 nS
Mean synaptic inhibitory conductance |  $g_{sI}$ |  200 nS
Mean synaptic input conductance |  $g_{s\nu}$ |  3 nS
Standard deviation delay |  $\sigma_{\overline{D}}$ |  0.1 ms
Standard synaptic excitatory conductance |  $\sigma_{g_{sE}}$ |  1 nS
Standard inhibitory conductance |  $\sigma_{g_{sI}}$ |  10 nS
Standard input conductance |  $\sigma_{g_{s\nu}}$ |  1 nS
Small-world networks (SW)
Probability of replacing new links |  $P_{sw}$ |  0.01
Degree of nearest neighbours connections |  K |  10
Table 2:  List of simulation and default network parameters in the E/I
balanced networks.

###  II.2 Network connectivity

To build the connectivity matrix, we consider a balanced neural network,
consisting of $N_{E}$ excitatory and $N_{I}$ inhibitory neurons shown in
Fig.1. We will use the subscript $E$ to denote the excitatory populations and
$I$ for the inhibitory ones. The connections between the $i$th neurons of the
$kth$ population and the $jth$ neurons of the $lth$ population, denoted by
$L_{kl}^{ij}$. Here $k,l=E,I$. The neurons in excitatory population are
connected by chemical synapses in a small world topology and adjacent neurons
are connected by gap junctions. The connection matrices of chemical and
electrical synapses in the excitatory populations are defined as $L_{EE}^{ij}$
and $C_{gap}^{ij}$, while the connection matrices of chemical synapses inside
the inhibitory one are given by $L_{II}^{ij}$. The weights of self-excitatory
connections or self-inhibitory ones is symbolized by $J_{EE}$ or $J_{II}$. The
coupling strengths of adjacent electrical connections are denoted with
$J_{gap}$. The link matrices between populations with all-to-all connections
are $L_{IE}^{ij}$ and $L_{EI}^{ij}$. Their coupling strength between the
populations is denoted with constants $J_{IE}$ and $J_{EI}$. The connectivity
matrix $L_{kl}^{ij}=1$ ($L_{kl}^{ij}=0$) corresponds to connections between
neuron $i$ and $j$ (or not). The default network parameters used are shown in
Table 2.

The small-world topology mentioned above is implemented as two basic steps of
the standard algorithm[77]: 1. Construct a regular ring lattice with $N$ nodes
of mean degree $2K$. Each node is connected to its $K$ nearest neighbors on
either side. 2. For every node $i=0,...,N-1$ take every edge connecting $i$ to
its $K$ rightmost neighbors – that is every edge ($i,j$ mod $N$) with $i<j\leq
i+K$ , and rewire it with probability $P_{sw}$. Rewiring is done by replacing
($i,j$ mod $N$) with ($i,k$) where $k$ is chosen uniformly at random from all
possible nodes . Self-inhibitory connections is all-to-all connections. The
rewiring edge cannot be a duplicate or self-loop.

###  II.3 Network Activity Characterization

In this paper we first use the synchronization index $\chi(N)$, previously
introduced by Ref [78, 79] , to account for the synchronization level of the
neural activity of the considered networks, where:

$\displaystyle\chi^{2}=\dfrac{N\sigma_{V(t)}^{2}}{\sum_{i=1}^{N}\sigma_{V_{i}(t)}^{2}}$
(7)

Here $V(t)=\frac{1}{N}\sum_{i=1}^{N}V_{i}(t)$ is populations average of the
membrane potential $V_{i}(t)$. The variables $\sigma_{V(t)}$ and
$\sigma_{V_{i}(t)}$ denote the standard deviation of $V(t)$ over time, or
respectively, of the membrane potential traces $V_{i}(t)$ of the each isolated
neuron $i$.

To further investigate the global dynamical behaviour of the neural networks,
we then introduce the order parameter and its the variance in time,
metastability. The order parameter $R$, describes the global level of phase
synchrony in a system of $N_{E}$ oscillators[80, 81], given by:

$R=\left\langle\left|\frac{1}{N_{E}}\sum_{k=1}^{N_{E}}e^{2\pi
i[(t-t_{k}^{n})/(t_{k}^{n+1}-t_{k}^{n})]}\right|\right\rangle$ (8)

Where the vertical bars ($=\phi_{c}(t)$) denotes the modulus of the complex
number and the angle brackets is the temporal average value of that quantity,
while the exponent, $2\pi[(t-t_{k}^{n})/(t_{k}^{n+1}-t_{k}^{n})]$, is the
phase we assign to each neurons in the excitatory populations. The closer to 0
(1) $R$ becomes, the more asynchronous (synchronous) the dynamics is. $N_{E}$
is the total number of excitatory populations, $t_{k}^{n}$ is the time of the
$n$th spikes of neurons $k$ and $t_{k}^{n+1}$ is the time of the following
$(n+1)$th spikes.

In order to quantify the metastability and chimera-likeness of the observed
dynamics, the global metastability $Met$[82] , is the variance in time of
order parameter $R$, by:

$Met=\frac{1}{\Delta t}\sum\limits_{t\leq\Delta
t}\left(\phi_{c}(t)-R\right)^{2}$ (9)

where $\Delta t$ in Eq.(9) is the time windows to quantify the metastability
of the excitatory populations. Metastability is $0$ if the system is either
completely synchronized or completely desynchronized – a high value is present
only when periods of coherence alternate with periods of incoherence.

##  III Results

### Influence of the electrical coupling in the generation of several firing
patterns

In what follows, our main goal is to investigate how the electrical and
chemical coupling can affect the firing patterns diversity within the
excitatory population of the E/I balanced networks. To do this, we simulated
balanced networks containing excitatory and inhibitory populations. Neurons in
the excitatory population are connected by both electrical and chemical
synapses, while neurons in inhibitory one are only coupled by chemical
synapses (See II.2 section for detail). In this work, we consider the network
composed of Wang-Buzsáki neurons. To measure the level of synchronization, we
use the synchronization index $\chi(N)$ (Eq.(7)) and Kuramoto order parameter
$R$ (Eq.(8)). We also introduce the metastability $Met$ Eq.(9) to quantify the
metastability and chimera-likeness of the observed dynamics within excitatory
populations.

Figure 2:  Examples of spatiotemporal firing patterns (A) and their
corresponding action potentials (B) in each dynamical regimes against several
level of electrical strengths $J_{gap}$ in ($J_{EE},J_{EI}$) parameter space.
The 1$th$ row and 2-4$th$ rows in (A) correspond the excitatory population
built with or without gap junction, respectively. Raster plots of
spatiotemporal firing patterns within 5 different dynamical regimes (a-e) that
are shown in Fig.3. Example of chimera-likeness and incoherent state
respectively at time $t_{1}$ and $t_{2}$ in point $d$ with $J_{gap}=0.1$.
Metastable state is that synchronous part of chimera-likeness state alternates
between synchronous and incoherent behaviors with different times, eg. $t_{1}$
and $t_{2}$. Weights of gap junction $J_{gap}$ from top to bottom panels are
$0.0$, $0.1$, $0.5$, $1.0$. The red Ex (green In) of $y$ labels in (A)
indicates the neurons index of excitatory population (inhibitory population).
In all traces the scale bar in (B) is 10 ms.

Fig.2 shows the typical examples of spatiotemporal firing patterns (A) and
corresponding action potentials (B) from each dynamical regime (corresponding
marked by letters $a,b,c,d,e$ in Fig.3), and their changes with different
degree of electrical synapse connections. If excitatory population of networks
are only built with the chemical synaptic connections, raster plots (Row 1 in
Fig.2A) show that the larger the inhibitory strengths used for excitatory
population the higher level of synchronization to achieve, and denote three
different dynamical regimes – incoherent state, metastable state and
generalized synchronous state (or cluster synchronous states), respectively.
Incoherent state (Columns $a$ to $c$ in Row 1 of Fig.2) is clear in the
network of excitatory population and note that, firing patterns of panels
$b,c$ easily tend to clustering. An interesting behavior is metastable state
(Column $d$ of Fig.2), but it is difficult to show in this case. The panel $e$
displays generalized synchronous state, this is because of stronger
inhibition. The action potentials corresponding to these raster plots in
excitatory population are plotted in the first panels of Fig.2B.

In the case of networks built with hybrid synapses, it is clear, from the
raster plot of column $a$ with $J_{gap}=0.1$ (Row 2 of Fig.2A), to find that
there is an emergence of traveling wave (wave propagation), which can not be
observed in the same parameter space without gap junction (see Row 1 of Column
$a$). On the other sides, traveling waves observed in excitatory populaton has
a tendency to synchrony with increasing the weights of gap junction (see Rows
2,3,4 of Column $a$ with $J_{gap}=0.1,0.5,1$,respectively). Columns $b,c,e$
(illustrated in Row 2 of Fig.2A with $J_{gap}=0.1$) correspond to dynamical
regimes with almost completely synchrony state, and the corresponding time
series exhibits spiking behavior shown in Fig.2B. Note that, we have shown
that raster plots and action potentials provide totally discrimination in the
observed synchrony regimes. The type of oscillatory synchronization occurs in
columns $b,c$ (Row 2) with suitable inhibition corresponding to the
oscillations with period 2 and 1, respectively, while column $e$ is related to
subthreshold oscillatory synchronization without spikes under stronger
inhibition. The result in column $e$ shows that the cooperation of electrical
synapse connections and stronger inhibition leads to suppression of the spikes
oscillation, namely subthreshold synchronized oscillations. More importantly,
increasing the value of electrical synapse connections weights causes no
obvious change in the modes of synchronized oscillation (see Columns $b,c,e$
with $J_{gap}=0.5,1$ in Fig.2A). An interesting highly metastable behavior is
found in the regime with higher inhibitory level, where $d$ is located in
Fig.3. Raster plot of this point (in Row 2 of Column d with $J_{gap}=0.1$)
displays chimera-likeness state, which coexists two subpopulations of
synchronous states and incoherent state with various ripples events (such as
traveling waves) in the same certain time, e.g. at time $t_{1}$. More
importantly, chimera-likeness state sometimes is also a metastable state 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– such as
dynamics switching between time $t_{1}$ and $t_{2}$ shown in Fig.2A (see also
Fig.5). The firing patterns located in column $d$ have a slightly tendencty to
synchrony as electrical connections are increase. From the corresponding
actions potentials shown in Fig.2B, it is evident that the system is in the
bursting regime. These results imply that the level and balance of excitation
and inhibition, when introducing the gap junction to the excitatory
population, leads to various firing patterns, such as wave propagation,
synchronized oscillations, chimera-likeness and metastable state.

Figure 3:  Dynamical regimes of spatiotemporal firing patterns on
($J_{EE},J_{EI}$) parameter space with different weights $J_{gap}$ of
electrical synaptic connections. Density plots show synchronization index
$\chi(N)$ (Upper panels) as well as order parameter $R$ (Middle panels) and
metastability $Met$ (Bottom panels) of excitatory populations in the same
plane. The marked points $a,b,c,d,e$ denote different dynamical regimes.
A.Collective dynamics behavior of excitatory populations with purely chemical
synaptic connections ($J_{gap}=0$). (a-c) Incoherent state corresponds to low
values of both, $\chi$ and $Met$; (d)Metastable state where $\chi$ is low and
$Met$ is high; (e) Generalized synchronous state (not fully synchronous state)
where $\chi$ is much higher but less than 1 and $Met$ is low. Subplots B-D
illustrate different dynamics regimes in network of excitatory population
built with hybrid synapses: (a) Traveling wave (wave propagation) where both
$\chi$ and $Met$ are low values; (b) and (c) Synchronous states; (d)
Metastable state or chimera-likeness where $\chi$ is low and $Met$ is high;
This indicates a highly metastable behavior and desynchronous dynamics; (e)
Subthreshold synchronous states (but neurons within excitatory populations
oscillate without spikes). The electrical weights for each subplot are
$J_{gap}=0.1$, $J_{gap}=0.5$, $J_{gap}=1$, respectively.

The dynamics of spatiotemporal firing patterns, in the excitatory populations
with 4 different level of electrical coupling weights, are characterized in
Fig.3 of the ($J_{EE},J_{EI}$) parameter spaces. These dynamical regimes are
quantified via synchronization index as well as order parameter and
metastability. In order to highlight some characteristic patterns that emerge
in our system, we select 5 dynamical regimes of interest, marked by letters a,
b, c, d and e in parameter spaces. Excitatory population network built with
hybrid synapses, are chosen so that the following cases are covered (see
Fig.3B-D). a: both $\chi$ (or $R$ ) and $Met$ are low-valued; b,c and e:
$\chi$ is high-value but $Met$ is low-value; d: $\chi$ is middle-value and
$Met$ is high-value. To further understand the role of electrical coupling on
emergence of the diversity firing patterns, the corresponding 5 points also
have been selected in the same parameter space of excitatory population but
without gap junction shown in Fig.3A .

We first investigate collective dynamical behavior of excitatory population
with purely chemical synaptic connections ($J_{gap}=0$) in Fig.3A. The results
obtained in this figure display a region $e$ of lower level synchronization
for high inhibitory chemical coupling with the rest of dynamical regimes being
incoherent. This is because the level of recurrent inhibition controls the
overall level of synchrony of the excitatory population, which is the first
known reason to induce networks synchronization. However, when networks of
excitatory population built with hybrid synapses, with weaker electrical
coupling shown in Fig.3B, it exhibits higher level synchronization ($\chi$ or
$R$ is close to 1) than those of networks without gap junction in the same
region $e$ for stronger inhibition. In addition, there are other two emergence
regions ($b,c$) of high-level synchronzaiton where their synchronization index
(or order parameter) are close to 1 for weaker inhibition. These two regions
can not be observed clearly in networks without electrical synapse connections
(see corresponding regions $b,c$ in Fig.3A). Apart from that, the contribution
of electrical coupling on synchronization is also reflected in the same
parameter space, but with slight stronger weights of gap junction when
$J_{gap}=0.5,1$ (Fig.3C-D). Note that, increasing electrical synapse weights
of excitatory population in network can enlarge regimes $b,c,d,e$ and shrink
regime $a$, comparing Fig.3B,C and D. This is the second mainly mechanism to
increase the level of synchronization in a way that enhancing the electrical
synapse weights. Besides, it is also easy to see that the boundaries between
each regime are obvious.

Another usefully measurement related to network activity is the metastability,
which measures the chimera-likeness/metastable state in the excitatory
population along time. More surperisely, we found that introducing electrical
connections, even with weaker coupling weights, to excitatory population leads
to chimera-likeness/metastable in region $d$ of the observed dynamics, as
captured by metastability (bottom row of Fig.3B-D). More importantly,
enhancing weights of the electrical connections can expand the metastability
region $d$. It is clear that the region in which $Met$ attains values close to
0 coincides with all the neurons where excitatory population are in a
synchronized state (region $b,c,e$) or incoherent state (region $a$). In other
parts of the parameter space (region d) where $Met$ achieves higher values,
indicating that the system is in chimera-likeness state, coexisting two
subpopulations of synchronous states and incoherent state, or metastable
state, often switching between synchronous and asynchronous states. The
results shown above imply that in excitatory networks build with hybrid
synapses, diverse dynamical regimes arises from different combinations of
excitatory and inhibitory weights, and their boundary between each dynamical
regime are clear, which can not display in the network coupled by purely
chemical synapses (bottom panel of Fig.3A).

Next we focus on the influence of excitation and inhibition together with
their balance to network firing patterns, in the networks of excitatory
population built with hybrid synapses.

### Influence of excitation and inhibition on excitatory population behavior

We investigate now the emergence of various firing patterns with variation of
inhibitory level accessible by the weights parameter $J_{EI}$. To do this, we
separately choose excitatory level $J_{EE}=0.5$ and $2.5$, in the networks of
excitatory population built with ($J_{gap}=0.1$) or without ($J_{gap}=0$)
electrical synapse connections.

Figure 4:  Influence of the inhibitory level $J_{EI}$ on the emergence of
spatiotemporal dynamical states shown in A ( $J_{gap}=0$) and B (
$J_{gap}=0.1$) where $J_{EE}=0.5$. The emergence of collective dynamics in the
excitatory population quantified by the order parameters (blue circle lines)
and metastability (red star lines). Inset plots show a detail of the typical
spatiotemporal firing patterns in each dynamics regimes of sweeping parameter
$J_{EI}$.

To clearly distinguish these different types of population behaviors and
characterize the effect of inhibition on their firing patterns, we compute
both order parameter $R$ and metastability $Met$ as a function of inhibitory
level $J_{EI}$ in the networks of excitatory population, which built with
(without) gap junction when $J_{EE}=0.5$. The obtained results are presented
in Fig.4 where each subplot illustrates measures of synchronization and
metastability as well as examples of several firing patterns in each dynamics
regime, indicated in the inset plots. As seen in the Fig.4A, in the network of
excitatory population in absence of gap junction, variation of $J_{EI}$ only
induces incoherent state, metastable state (it is still irregular but with
slight clustering firing pattern) and generalized synchronous state when both
$R$ and $Met$ satisfy the following conditions: $R\approx 0,Met\approx 0$;
$0.4<R<0.8,0<Met$; $0.7<R<0.8,Met\approx 0$, respectively. It is confirmed
again that increasing $J_{EI}$ has a tendency to induce the lower level of
synchronization, namely generalized synchronous state. However, it is seen
that the excitatory population built with gap junction, in Fig.4B, exhibits
characteristically distinct types of dynamical behaviors as $J_{EI}$ varies.
Note that their respective dynamics were previously illustrated in Fig.3 and
Fig.2. More precisely, we observe traveling wave patterns with lower level of
inhibition for $J_{EI}\leqslant 0.1$ when $R\approx 0.2,Met<0.01$. Excitatory
population starts to exhibit synchrony patterns with period 2 as shown inset
plot with a slight increase in $J_{EI}$ which is in range of
$0.1<J_{EI}\leqslant 0.25$ when $R>0.91,Met<0.01$. Then, a further increase in
inhibitory level to the range of $0.25<J_{EI}\leqslant 0.3$ when $R$ (around
0.65) is middle-value and $Met$ (greater than 0.07) is high-value, we observe
the metastable state in which the dynamics in excitatory population switches
between synchronous and asynchronous state (traveling wave patterns or
subthreshold oscillations state). If the inhibitory level is chosen in the
range of $0.3<J_{EI}\leqslant 0.45$ when $R\approx 1,Met\approx 0$, we find
that it gives rise to another synchrony neural activity with period 1.
Subsequently, when $J_{EI}$ lies in the range of $0.45<J_{EI}\leqslant 0.675$
with middle-value of both $R$ and $Met$, we see a peculiar chimeraness
behavior in which the excitatory population splits into three domains:
coherent state, various ripples events (such as traveling waves) and
oscillation without spikes. Coexistence of these three dynamics states
depending on the inhibition level is the first time to be observed.
Eventually, the excitatory populations exhibits subthreshold synchronous
states – oscillation without spikes– due to stronger inhibitory level in the
range of $0.675<J_{EI}$. It is evident that inhibitory level $J_{EI}$ is a
significant system parameter which plays a major role in determining the
emergence of the above mentioned firing patterns, including traveling wave,
synchrony state with period 2 (or 1), a peculiar chimera-likeness behavior and
subthreshold synchronous states. More importantly, there are transitions
between dynamical states depending on inhibitory level among individual
neurons.

Figure 5:  Distinct types of dynamical behavior as $J_{EI}$ varies as
illustrated in A ($J_{gap}=0$ ) and B ($J_{gap}=0.1$) but with high excitatory
level where $J_{EE}=2.5$. Inset plots present typical firing patterns in each
dynamics regimes.

We have so far focused on changes in the inhibitory level to study the
emergence of several firing patterns with fixed weaker excitability level. In
the following, we continue investigating the population dynamics as a
functions of inhibitory level $J_{EI}$ but with stronger excitability level.
As an extension of the findings about the effect of inhibitory level in the
emergence of various firing patterns, we still consider excitatory population
built with (or without) gap junction that are shown in Fig.5. For each case,
we calculate the order parameter and metastability of various firing patterns
by varying the inhibitory level $J_{EI}$ for a fixed value of stronger
excitatory strength $J_{EE}=2.5$. The typical firing patterns of excitatory
population for each dynamics regimes are shown in inset plots of Fig.5. As
seen in the Fig.5A, variation of inhibitory level of the excitatory part built
without gap junction causes the similar modes of firing pattern, which has
been shown in Fig.4A. In network built with hybrid synapses, it is seen that
there are three different types of neuronal activity in the excitatory
population obtained as shown in Fig.5B. There exists a much more regular
population activity, however it is still an incoherent state, for weak
inhibitory synaptic connections. It is because the range of $J_{EI}$ becomes
narrower for traveling waves and wider for chimera-likeness state with
increasing excitability levels that are shown in Fig.3B. This indicates that
traveling waves vanish at very high excitability levels (see Fig.3B-D and
Fig.5B) , which is similar with previous study[62]. If the inhibitory coupling
strength is sufficiently increased among range $0.25<J_{EI}\leqslant 0.7$
where both $R$ and $Met$ are high-value, we observe chimera-likeness behavior
in which the excitatory population coexists the synchrony state and traveling
waves at the same time. In addition, we also see metastable state that
alternates between synchronous and incoherent behaviors with different time
windows. To check the persistence of these two special behavior, we further
inscreased $J_{EI}$ and observed that chimera-likeness and metastable state
does no longer exist, instead a subthreshold oscillations (without spikes)
emerges for $0.7<J_{EI}$. To sum up, we show that balanced and hybrid synaptic
connected network of Wang-Buzsáki neurons can displays various types of
different levels of synchronous activity that imply vastly different
computational properties. For weak excitatory coupling, the network of
excitatory population displays rich collective dynamics – traveling waves,
synchrony patterns with period $2$, metastable state, synchrony patterns with
period $1$, chimera-likeness/metastable state and lastly oscillation without
spikes – as inhibitory level $J_{EI}$ varies. However, for strong excitatory
couplings, we find that the increasing inhibitory level $J_{EI}$ in the
networks of excitatory population only leads to three fundamentally different
types of neuronal activity, namely, less regular population activity, chimera-
likeness behavior/metastable state and subthreshold synchronous states
(oscillation without spikes).

Figure 6:  Influence of probability $P_{sw}$ for small-world topology on
different levels of synchronization and its robustness on ($J_{EE},J_{EI}$)
plane, unfolding a clear comparison with Fig.3.

### Influence of probability $P_{sw}$ on excitatory population behavior

To present a broader perspective on different levels of synchronization and to
confirm its robustness, we choose a bigger probability $P_{sw}$ ($=0.15$) for
small-world topology of excitatory synapse connections and show excitatory
population behavior on ($J_{EE},J_{EI}$) plane as depicted in Fig.6. It is
clear, in the network built with hybrid synapses, that the bigger probability
for small-world topology still supports rich excitatory population behavior
emerging as five different dynamical regimes marked by latters $a,b,c,d,e$ –
that is traveling waves, two types of synchronous states, chimera-likeness or
metastable state with various ripples events, and subthreshold synchronous
states. Our results reveal that the role of gap junction and excitatory-
inhibitory balance to network states are robust with network densities.

##  IV Discussion

We briefly discuss the different characteristics and functionality of these
two types of chemical and electrical synapses (For reviews, see[12, 13]). The
most common mechanism for transmission between neurons is chemical synapses,
which is typically unidirectional and relies on the intermittent presence of
coming spikes from presynaptic terminal to generate transient changes. From
this point, chemical synaptic communication is episodic. In addition,
neurotransmitter release is probabilistic, and failures in transmission will
occasionally occur despite the presence of coming spikes from presynaptic
terminal. Depending on the neurotransmitter, a chemical synapse can either be
excitatory or inhibitory. By contrast, electrical synaptic transmission
through gap junctions underlies direct, continuous, reliable and rapid
neuronal communication in the CNS. Electrical synapses are bidirectional and
of local connections between neurons which are spatially very close. The gap
junctional conductance is remarkably and characteristically constant for
sustained periods of time. However, recent studies had evidenced that both
chemical and electrical modalities of synaptic transmission are plastic and
capable of modifying their strength in a bidirectional fasion. For example,
activity-dependent changes in the strength of chemical synapses are thought to
underlie mechanisms of learning and memory (for a review, see ref [83]).
Regarding to the functional role of gap junction plasticity, such as changes
in light conditions[84], has been experimentally demonstrated [85, 86, 87, 88,
71] (also reviewed in refs [12, 89]).

The distribution and role of electrical synapses have been investigated in the
previous studies[90, 91]. A series of reports had demonstrated that electrical
synapses occur between excitatory projection neurons of the inferior olivary
nucleus and between interneurons in several regions of the adult brain,
including the cerebellum, neocortex, hippocampus and so on (see review[90]
detailly). Computer simulations had greatly contributed to revealing the
impact and functional role of gap junctions between inhibitory interneurons on
network activity. Firstly, gap junctions between inhibitory interneurons have
been shown to enhance synchronized neuronal firing[91, 92], which is one of
the most distinct and recognizable properties of the electrical synapses.
Secondly, it has been suggested that the presence of electrical synapses could
mediate close synchronization of subthreshold and spiking activity among
clusters of neurons[13, 93]. In all, the role of electrical synapses for
network activity has been the topic of various review articles (see Ref[90,
94]) and therefore will not be extensively discussed here. The influence of
electical coupling for balanced networks on generation of distinct dynamical
regimes has been investigated in this paper. Introducing the gap junction,
even with weaker coupling weights, to excitatory population leads to various
firing patterns – such as traveling wave, synchronous states etc. However, gap
junctions are often observed between inhibitory neurons. The role and function
of gap junctions between inhibitory neurons, in the generation of
spatiotemporal firing patterns, will be the subject for the future work.

Coactivated inhibitory and excitatory synaptic inputs plays a role in
diversity of cortical functions and general information processing[24, 95]. It
is generally accepted that activity of excitatory networks, if only excitatory
cells were present in networks, is simple and predictable – excitation just
leads to futher excitation. Inhibitory networks are totally different[20],
namely, it can generate nonlinear effects. In the absence of inhibition, any
type of external input, would generate more or less the same one-way patterns.
Theoretical work has shown that networks built from both excitatory and
inhibitory elements can self-oranize and generate complex properties[16, 17,
18, 19]. For example, study of Rubin et al.[95] shown that the ratio of
excitation-inhibition balance is essential for robust neuronal selectivity and
crucial for stability in associative memory networks in the presence of noise.

In this work, we investigate the emergence of various firing patterns in
excitation and inhibition balance networks of itself excitatory cells coupled
via hybrid synapses. Our findings in this class of network illustrate how
level of inhibition can synchronize or desynchronize, detect coincidences,
inhibit on population behavior. More detail, with weaker excitability level,
variation of inhibitory level for excitatory population leads to rich
collective dynamics, such as traveling waves, synchrony and desynchrony state
(chimera-likeness or metastable state) and oscillation without spikes due to
stronger inhibition. With stronger excitability level, increasing inhibitory
level for the same networks causes three fundamentally different types of
neuronal activity – less order population activity, chimera-likeness behavior
(or metastable state at some times) and subthreshold synchronous states.

Brain dynamics is often inherently variable and unstable, consisting of
sequences of transient spatiotemporal patterns[96, 97]. These squences of
transients are a hallmark of metastable dynamics that are neither entrely
stable nor completely unstable. Our results pay a possible way to uncover the
underlying mechanisms of generating metastable dynamics, such as chimera-
likeness, that may mediate perception and cognition.

###### Acknowledgements.

We thank the funding from Fondecyt Project Nos.3170342 (K.X.) and 1181076
(P.O.) from CONICYT, Chile. P.O. is partially funded by the Advanced Center
for Electrical and Electronic Engineering (FB0008 Conicyt, Chile). The Centro
Interdisciplinario de Neurociencia de Valparaíso (CINV) is a Millennium
Institute supported by the Millennium Scientific Initiative of the Ministerio
de Economía (ICM-MINECON, Proyecto P09-022-F,CINV, Chile).

## Conflict of interest

The authors declare that they have no conflict of interest.

## References

  * [1] Robert C Elson, Allen I Selverston, Ramon Huerta, Nikolai F Rulkov, Mikhail I Rabinovich, and Henry DI Abarbanel.  Synchronous behavior of two coupled biological neurons.  Physical Review Letters, 81(25):5692, 1998.
  * [2] Jian Jing.  Network Functions and Plasticity: Perspectives from Studying Neuronal Electrical Coupling in Microcircuits.  Academic Press, 2017.
  * [3] Stephen Coombes.  Large-scale neural dynamics: simple and complex.  NeuroImage, 52(3):731–739, 2010.
  * [4] Lyle Muller, Frédéric Chavane, John Reynolds, and Terrence J Sejnowski.  Cortical travelling waves: mechanisms and computational principles.  Nature Reviews Neuroscience, 19(5):255, 2018.
  * [5] Jun Ma, Xinlin Song, Jun Tang, and Chunni Wang.  Wave emitting and propagation induced by autapse in a forward feedback neuronal network.  Neurocomputing, 167:378–389, 2015.
  * [6] Qing Yun Wang, Qi Shao Lu, and Guan Rong Chen.  Ordered bursting synchronization and complex wave propagation in a ring neuronal network.  Physica A: Statistical Mechanics and its Applications, 374(2):869–878, 2007.
  * [7] Kesheng Xu, Wenwen Huang, Baowen Li, Mukesh Dhamala, and Zonghua Liu.  Controlling self-sustained spiking activity by adding or removing one network link.  EPL (Europhysics Letters), 102(5):50002, jun 2013.
  * [8] Kesheng Xu, Xiyun Zhang, Chaoqing Wang, and Zonghua Liu.  A simplified memory network model based on pattern formations.  Scientific reports, 4:7568, 2014.
  * [9] Chris I De Zeeuw, Freek E Hoebeek, Laurens WJ Bosman, Martijn Schonewille, Laurens Witter, and Sebastiaan K Koekkoek.  Spatiotemporal firing patterns in the cerebellum.  Nature Reviews Neuroscience, 12(6):327, 2011.
  * [10] Chang-Hai Tian, Xi-Yun Zhang, Zhen-Hua Wang, and Zong-Hua Liu.  Diversity of chimera-like patterns from a model of 2d arrays of neurons with nonlocal coupling.  Frontiers of Physics, 12(3):128904, 2017.
  * [11] Siyu Huo, Changhai Tian, Ling Kang, and Zonghua Liu.  Chimera states of neuron networks with adaptive coupling.  Nonlinear Dynamics, pages 1–12, 2019.
  * [12] Alberto E Pereda.  Electrical synapses and their functional interactions with chemical synapses.  Nature Reviews Neuroscience, 15(4):250, 2014.
  * [13] Pepe Alcami and Alberto E Pereda.  Beyond plasticity: the dynamic impact of electrical synapses on neural circuits.  Nature reviews. Neuroscience, 2019.
  * [14] Morgan Sheng, Bernardo Sabatini, and Thomas.(eds) Sudhof.  The Synapse.  Cold Spring Harbor Laboratory Press, 2012.
  * [15] Michael VL Bennett and R Suzanne Zukin.  Electrical coupling and neuronal synchronization in the mammalian brain.  Neuron, 41(4):495–511, 2004.
  * [16] Jeffrey S Anderson, Matteo Carandini, and David Ferster.  Orientation tuning of input conductance, excitation, and inhibition in cat primary visual cortex.  Journal of neurophysiology, 84(2):909–926, 2000.
  * [17] Bassam V. Atallah and Massimo Scanziani.  Instantaneous modulation of gamma oscillation frequency by balancing excitation with inhibition.  Neuron, 62(4):566 – 577, 2009.
  * [18] Michael Wehr and Anthony M. Zador.  Balanced inhibition underlies tuning and sharpens spike timing in auditory cortex.  Nature, 426(6965):442–446, 2003.
  * [19] Cindy Poo and Jeffry S. Isaacson.  Odor representations in olfactory cortex: “sparse” coding, global inhibition, and oscillations.  Neuron, 62(6):850 – 861, 2009.
  * [20] Jeffry S. Isaacson and Massimo Scanziani.  How inhibition shapes cortical activity.  Neuron, 72(2):231 – 243, 2011.
  * [21] C. van Vreeswijk and H. Sompolinsky.  Chaos in neuronal networks with balanced excitatory and inhibitory activity.  Science, 274(5293):1724–1726, 1996.
  * [22] Sean R. Bittner, Ryan C. Williamson, Adam C. Snyder, Ashok Litwin-Kumar, Brent Doiron, Steven M. Chase, Matthew A. Smith, and Byron M. Yu.  Population activity structure of excitatory and inhibitory neurons.  PLOS ONE, 12(8):1–27, 08 2017.
  * [23] Ran Rubin, L. F. Abbott, and Haim Sompolinsky.  Balanced excitation and inhibition are required for high-capacity, noise-robust neuronal selectivity.  Proceedings of the National Academy of Sciences, 114(44):E9366–E9375, 2017.
  * [24] Gyorgy Buzsaki.  Rhythms of the Brain.  Oxford University Press, 2006.
  * [25] Michael Beierlein, Jay R Gibson, and Barry W Connors.  A network of electrically coupled interneurons drives synchronized inhibition in neocortex.  Nature neuroscience, 3(9):904, 2000.
  * [26] Dirk Bucher, Astrid A Prinz, and Eve Marder.  Animal-to-animal variability in motor pattern production in adults and during growth.  Journal of Neuroscience, 25(7):1611–1619, 2005.
  * [27] Eve Marder and Dirk Bucher.  Understanding circuit dynamics using the stomatogastric nervous system of lobsters and crabs.  Annual Review of Physiology, 69(1):291–316, 2007.
  * [28] Romuald Nargeot, Morgane Le Bon-Jego, and John Simmers.  Cellular and network mechanisms of operant learning-induced compulsive behavior in aplysia.  Current Biology, 19(12):975 – 984, 2009.
  * [29] Angela Wenning, Brian J. Norris, Anca Doloc-Mihu, and Ronald L. Calabrese.  Bringing up the rear: new premotor interneurons add regional complexity to a segmentally distributed motor pattern.  Journal of Neurophysiology, 106(5):2201–2215, 2011.  PMID: 21775711.
  * [30] Marlene Bartos, Imre Vida, and Peter Jonas.  Synaptic mechanisms of synchronized gamma oscillations in inhibitory interneuron networks.  Nature reviews neuroscience, 8(1):45, 2007.
  * [31] Alex Roxin, Nicolas Brunel, and David Hansel.  Role of delays in shaping spatiotemporal dynamics of neuronal activity in large networks.  Physical review letters, 94(23):238103, 2005.
  * [32] Bard Ermentrout.  Neural networks as spatio-temporal pattern-forming systems.  Reports on Progress in Physics, 61(4):353–430, apr 1998.
  * [33] Rory G Townsend and Pulin Gong.  Detection and analysis of spatiotemporal patterns in brain activity.  PLoS computational biology, 14(12):e1006643, 2018.
  * [34] Roger D Traub, John GR Jefferys, and Miles A Whittington.  Fast oscillations in cortical circuits.  MIT press Cambridge, MA, 1999.
  * [35] György Buzsáki and Xiao-Jing Wang.  Mechanisms of gamma oscillations.  Annual review of neuroscience, 35:203–225, 2012.
  * [36] Andre Fisahn, Fenella G Pike, Eberhard H Buhl, and Ole Paulsen.  Cholinergic induction of network oscillations at 40 hz in the hippocampus in vitro.  Nature, 394(6689):186, 1998.
  * [37] György Buzsáki and James J Chrobak.  Temporal structure in spatially organized neuronal ensembles: a role for interneuronal networks.  Current opinion in neurobiology, 5(4):504–510, 1995.
  * [38] Larry S Benardo.  Recruitment of gabaergic inhibition and synchronization of inhibitory interneurons in rat neocortex.  Journal of Neurophysiology, 77(6):3134–3144, 1997.
  * [39] FK Skinner, L Zhang, JL Perez Velazquez, and PL Carlen.  Bursting in inhibitory interneuronal networks: a role for gap-junctional coupling.  Journal of neurophysiology, 81(3):1274–1283, 1999.
  * [40] Goran Söhl, Stephan Maxeiner, and Klaus Willecke.  Expression and functions of neuronal gap junctions.  Nature reviews neuroscience, 6(3):191, 2005.
  * [41] Guoning Tang, Kesheng Xu, and Luoluo Jiang.  Synchronization in a chaotic neural network with time delay depending on the spatial distance between neurons.  Physical Review E, 84(4):046207, 2011.
  * [42] Kesheng Xu, Jean Paul Maidana, Samy Castro, and Patricio Orio.  Synchronization transition in neuronal networks composed of chaotic or non-chaotic oscillators.  Scientific reports, 8(1):8370, 2018.
  * [43] Christoph Börgers and Nancy Kopell.  Synchronization in networks of excitatory and inhibitory neurons with sparse, random connectivity.  Neural computation, 15(3):509–538, 2003.
  * [44] Charles M Gray, Peter König, Andreas K Engel, and Wolf Singer.  Oscillatory responses in cat visual cortex exhibit inter-columnar synchronization which reflects global stimulus properties.  Nature, 338(6213):334, 1989.
  * [45] Peter Uhlhaas, Gordon Pipa, Bruss Lima, Lucia Melloni, Sergio Neuenschwander, Danko Nikolić, and Wolf Singer.  Neural synchrony in cortical networks: history, concept and current status.  Frontiers in integrative neuroscience, 3:17, 2009.
  * [46] Adam Keane and Pulin Gong.  Propagating waves can explain irregular neural dynamics.  Journal of Neuroscience, 35(4):1591–1605, 2015.
  * [47] Lyle Muller and Alain Destexhe.  Propagating waves in thalamus, cortex and the thalamocortical system: experiments and models.  Journal of Physiology-Paris, 106(5-6):222–238, 2012.
  * [48] Carsten Mehring, Ulrich Hehl, Masayoshi Kubo, Markus Diesmann, and Ad Aertsen.  Activity dynamics and propagation of synchronous spiking in locally connected random networks.  Biological cybernetics, 88(5):395–408, 2003.
  * [49] Pierre Yger, Sami El Boustani, Alain Destexhe, and Yves Frégnac.  Topologically invariant macroscopic statistics in balanced networks of conductance-based integrate-and-fire neurons.  Journal of computational neuroscience, 31(2):229–245, 2011.
  * [50] Nicole Voges and Laurent U Perrinet.  Complex dynamics in recurrent cortical networks based on spatially realistic connectivities.  Frontiers in computational neuroscience, 6:41, 2012.
  * [51] Johanna Senk, Karolína Korvasová, Jannis Schuecker, Espen Hagen, Tom Tetzlaff, Markus Diesmann, and Moritz Helias.  Conditions for traveling waves in spiking neural networks.  arXiv preprint arXiv:1801.06046, 2018.
  * [52] S. Coombes.  Waves, bumps, and patterns in neural field theories.  Biological Cybernetics, 93(2):91–108, Aug 2005.
  * [53] Paul C. Bressloff.  Waves in Neural Media.  Springer, 2014.
  * [54] Erik A Martens.  Bistable chimera attractors on a triangular network of oscillator populations.  Physical Review E, 82(1):016216, 2010.
  * [55] Iryna Omelchenko, Yuri Maistrenko, Philipp Hövel, and Eckehard Schöll.  Loss of coherence in dynamical networks: spatial chaos and chimera states.  Physical review letters, 106(23):234102, 2011.
  * [56] Changhai Tian, Liang Cao, Hongjie Bi, Kesheng Xu, and Zonghua Liu.  Chimera states in neuronal networks with time delay and electromagnetic induction.  Nonlinear Dynamics, 93(3):1695–1704, 2018.
  * [57] Erik A Martens, Carlo R Laing, and Steven H Strogatz.  Solvable model of spiral wave chimeras.  Physical review letters, 104(4):044101, 2010.
  * [58] E Omel’chenko, Yuri L Maistrenko, and Peter A Tass.  Chimera states: The natural link between coherence and incoherence.  Physical review letters, 100(4):044105, 2008.
  * [59] Mark R Tinsley, Simbarashe Nkomo, and Kenneth Showalter.  Chimera and phase-cluster states in populations of coupled chemical oscillators.  Nature Physics, 8(9):662, 2012.
  * [60] Johanne Hizanidis, Nikos E Kouvaris, Gorka Zamora-López, Albert Díaz-Guilera, and Chris G Antonopoulos.  Chimera-like states in modular neural networks.  Scientific reports, 6:19845, 2016.
  * [61] Soumen Majhi, Matjaž Perc, and Dibakar Ghosh.  Chimera states in a multilayer network of coupled and uncoupled neurons.  Chaos: An Interdisciplinary Journal of Nonlinear Science, 27(7):073109, 2017.
  * [62] Ali Calim, Philipp Hövel, Mahmut Ozer, and Muhammet Uzuntarla.  Chimera states in networks of type-i morris-lecar neurons.  Physical Review E, 98(6):062217, 2018.
  * [63] John E Rash, Kimberly GV Davidson, Naomi Kamasawa, Thomas Yasumura, Masami Kamasawa, Chunbo Zhang, Robin Michaels, Diego Restrepo, Ole P Ottersen, Carl O Olson, et al.  Ultrastructural localization of connexins (cx36, cx43, cx45), glutamate receptors and aquaporin-4 in rodent olfactory mucosa, olfactory nerve and olfactory bulb.  Journal of neurocytology, 34(3-5):307–341, 2005.
  * [64] Carmen Vivar, Roger D Traub, and Rafael Gutiérrez.  Mixed electrical–chemical transmission between hippocampal mossy fibers and pyramidal cells.  European Journal of Neuroscience, 35(1):76–82, 2012.
  * [65] Farid Hamzei-Sichani, Kimberly GV Davidson, Thomas Yasumura, William GM Janssen, Susan L Wearne, Patrick R Hof, Roger D Traub, Rafael Gutiérrez, Ole Petter Ottersen, and John E Rash.  Mixed electrical–chemical synapses in adult rat hippocampus are primarily glutamatergic and coupled by connexin-36.  Frontiers in neuroanatomy, 6:13, 2012.
  * [66] Sidney P Kuo, Gregory W Schwartz, and Fred Rieke.  Nonlinear spatiotemporal integration by electrical and chemical synapses in the retina.  Neuron, 90(2):320–332, 2016.
  * [67] James I Nagy, Alberto E Pereda, and John E Rash.  Electrical synapses in mammalian cns: past eras, present focus and future directions.  Biochimica et Biophysica Acta (BBA)-Biomembranes, 1860(1):102–123, 2018.
  * [68] Nancy Kopell and Bard Ermentrout.  Chemical and electrical synapses perform complementary roles in the synchronization of interneuronal networks.  Proceedings of the National Academy of Sciences, 101(43):15482–15487, 2004.
  * [69] MS Baptista, FM Moukam Kakmeni, and Celso Grebogi.  Combined effect of chemical and electrical synapses in hindmarsh-rose neural networks on synchronization and the rate of information.  Physical Review E, 82(3):036203, 2010.
  * [70] Chen Liu, Jiang Wang, Haitao Yu, Bin Deng, Xile Wei, Kaiming Tsang, and Wailok Chan.  Impact of delays on the synchronization transitions of modular neuronal networks with hybrid synapses.  Chaos: An Interdisciplinary Journal of Nonlinear Science, 23(3):033121, 2013.
  * [71] Guillaume Pernelle, Wilten Nicola, and Claudia Clopath.  Gap junction plasticity as a mechanism to regulate network-wide oscillations.  PLoS computational biology, 14(3):e1006025, 2018.
  * [72] Xiao-Jing Wang and György Buzsáki.  Gamma oscillation by synaptic inhibition in a hippocampal interneuronal network model.  Journal of neuroscience, 16(20):6402–6413, 1996.
  * [73] Agostina Palmigiano, Theo Geisel, Fred Wolf, and Demian Battaglia.  Flexible information routing by transient synchrony.  Nature neuroscience, 20(7):1014, 2017.
  * [74] Alan L Hodgkin and Andrew F Huxley.  A quantitative description of membrane current and its application to conduction and excitation in nerve.  The Journal of physiology, 117(4):500–544, 1952.
  * [75] Christof Koch and Idan Segev.  Methods in neuronal modeling: from ions to networks.  MIT press, 1998.
  * [76] David Sterratt, Bruce Graham, Andrew Gillies, and David Willshaw.  Principles of computational modelling in neuroscience.  Cambridge University Press, 2011.
  * [77] Duncan J. Watts and Steven H. Strogatz.  Collective dynamics of ’small-world’ networks.  Nature, 393(6684):440–442, 1998.
  * [78] David Golomb and John Rinzel.  Dynamics of globally coupled inhibitory neurons with heterogeneity.  Physical review E, 48(6):4810, 1993.
  * [79] David Golomb and John Rinzel.  Clustering in globally coupled inhibitory neurons.  Physica D: Nonlinear Phenomena, 72(3):259–282, 1994.
  * [80] Yoshiki Kuramoto.  Chemical oscillations, waves, and turbulence.  Courier Corporation, 2003.
  * [81] Elena Bertolotti, Raffaella Burioni, Matteo di Volo, and Alessandro Vezzani.  Synchronization and long-time memory in neural networks with inhibitory hubs and synaptic plasticity.  Physical Review E, 95(1):012308, 2017.
  * [82] Murray Shanahan.  Metastable chimera states in community-structured oscillator networks.  Chaos: An Interdisciplinary Journal of Nonlinear Science, 20(1):013108, 2010.
  * [83] S. J. Martin, P. D. Grimwood, and R. G. M. Morris.  Synaptic plasticity and memory: An evaluation of the hypothesis.  Annual Review of Neuroscience, 23(1):649–711, 2000.  PMID: 10845078.
  * [84] Stewart A Bloomfield and Béla Völgyi.  The diverse functional roles and regulation of neuronal gap junctions in the retina.  Nature Reviews Neuroscience, 10(7):495, 2009.
  * [85] Jennifer Claire Robinson, C. Andrew Chapman, and Richard Courtemanche.  Gap junction modulation of low-frequency oscillations in the cerebellar granule cell layer.  The Cerebellum, 16(4):802–811, Aug 2017.
  * [86] Zemin Wang, Ryan Neely, and Carole E. Landisman.  Activation of group i and group ii metabotropic glutamate receptors causes ltd and ltp of electrical synapses in the rat thalamic reticular nucleus.  Journal of Neuroscience, 35(19):7616–7625, 2015.
  * [87] Josef Turecek, Genevieve S. Yuen, Victor Z. Han, Xiao-Hui Zeng, K. Ulrich Bayer, and John P. Welsh.  Nmda receptor activation strengthens weak electrical coupling in mammalian brain.  Neuron, 81(6):1375 – 1388, 2014.
  * [88] Philippe Coulon and Carole E. Landisman.  The potential role of gap junctional plasticity in the regulation of state.  Neuron, 93(6):1275 – 1295, 2017.
  * [89] Julie S. Haas, Corey M. Greenwald, and Alberto E. Pereda.  Activity-dependent plasticity of electrical synapses: increasing evidence for its presence and functional roles in the mammalian brain.  BMC Cell Biology, 17(1):14, May 2016.
  * [90] Mario Galarreta and Shaul Hestrin.  Electrical synapses between gaba-releasing interneurons.  Nature Reviews Neuroscience, 2(6):425–433, 2001.
  * [91] Barry W. Connors and Michael A. Long.  Electrical synapses in the mammalian brain.  Annual Review of Neuroscience, 27(1):393–418, 2004.  PMID: 15217338.
  * [92] Guillaume Pernelle, Wilten Nicola, and Claudia Clopath.  Gap junction plasticity as a mechanism to regulate network-wide oscillations.  PLOS Computational Biology, 14(3):1–29, 03 2018.
  * [93] Dimitris G. Placantonakis, Anatoly A. Bukovsky, Sue A. Aicher, Hans-Peter Kiem, and John P. Welsh.  Continuous electrical oscillations emerge from a coupled network: A study of the inferior olive using lentiviral knockdown of connexin36.  Journal of Neuroscience, 26(19):5008–5016, 2006.
  * [94] Barry W. Connors.  Synchrony and so much more: Diverse roles for electrical synapses in neural circuits.  Developmental Neurobiology, 77(5):610–624, 2017.
  * [95] Ran Rubin, L. F. Abbott, and Haim Sompolinsky.  Balanced excitation and inhibition are required for high-capacity, noise-robust neuronal selectivity.  Proceedings of the National Academy of Sciences, 114(44):E9366–E9375, 2017.
  * [96] Olaf Sporns.  Networks of the Brain.  MIT press, 2010.
  * [97] Mikhail I. Rabinovich, Ramón Huerta, Pablo Varona, and Valentin S. Afraimovich.  Transient cognitive dynamics, metastability, and decision making.  PLOS Computational Biology, 4(5):1–9, 05 2008.