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Spike phase synchronization in multiplex cortical neural networks
Physica A xx (xxxx) xxx–xxx
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Spike phase synchronization in multiplex cortical neural networks Mahdi Jalili Department of Electrical and Computer Engineering, School of Engineering, RMIT University, 3001 Melbourne, VIC, Australia
highlights • Chemical synapses and gap junctions in C. elegance networks have different synchronizability. • The cortical networks have better synchronizability than the random networks. • Modularity of the networks is one of the main driving effects for their synchronizability.
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Article history: Received 5 February 2016 Received in revised form 2 August 2016 Available online xxxx Keywords: Dynamical networks Phase synchronization Transmission delay Coupled oscillators Hindmarsh–Rose neuron model Spiking neurons
abstract In this paper we study synchronizability of two multiplex cortical networks: wholecortex of hermaphrodite C. elegans and posterior cortex in male C. elegans. These networks are composed of two connection layers: network of chemical synapses and the one formed by gap junctions. This work studies the contribution of each layer on the phase synchronization of non-identical spiking Hindmarsh–Rose neurons. The network of male C. elegans shows higher phase synchronization than its randomized version, while it is not the case for hermaphrodite type. The random networks in each layer are constructed such that the nodes have the same degree as the original network, thus providing an unbiased comparison. In male C. elegans, although the gap junction network is sparser than the chemical network, it shows higher contribution in the synchronization phenomenon. This is not the case in hermaphrodite type, which is mainly due to significant less density of gap junction layer (0.013) as compared to chemical layer (0.028). Also, the gap junction network in this type has stronger community structure than the chemical network, and this is another driving factor for its weaker synchronizability. © 2016 Elsevier B.V. All rights reserved.
1. Introduction Many real systems can be modeled as networks; a collection of individual nodes interacting through (directed or undirected, weighted or unweighted) edges [1–3]. The last two decades have witnessed tremendous progress on understanding statistical and dynamical properties of complex networked systems. It has been shown that many real networks share some common properties such as small-worldness [4], scale-free degree distribution [5] and community structure [6]. Such properties significantly influence how dynamical processes evolve on networks and the way collective actions emerge [7]. Synchronization is the most widely studied collective behavior in networked systems [8]. It happens when two (or more) dynamical systems meet and interact; if the interactions between the individual dynamical units are strong enough, their behavior shows a time-correlated activity, i.e., they get into synchrony. There are different types of synchronization phenomenon such as complete, bubbling and lag synchronization. The synchronization type observed in
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.physa.2016.09.030 0378-4371/© 2016 Elsevier B.V. All rights reserved.
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many real systems is phase synchronization that is due to (often) weak coupling between the dynamical units [9]. Two (or more) dynamical systems are phase synchronized if their phases get into coherency. Recently, there has been much effort in the community of network science on moving from single-layer modeling of network systems to multi-layer modeling, which is mainly due to enhanced resolution of network datasets [10]. In singlelayer modeling of a networked system, all node-to-node interactions are treated the same and the only difference between them is characterized by their weight (in some applications the links can have positive or negative weights). However, in a multi-layer framework, the connection links are organized in different layers, which allows considering temporal- or context-related properties of the interactions. Indeed, multi-layer networks include a set of nodes and several layers of connections, accurately describing the node-to-node interactions, and/or the whole system’s parallel functioning. Examples of such multi-layer networks include road and rail traffic networks [11], air transportation networks [12], online social networks with several types of relations such as friendship, vicinity, membership and partnership [13], and international trade networks [14]. Although there are many research studies addressing the problem of synchronization (or consensus) in single-layer networked systems, there are few works investigating the problem in multi-layer networks [15,16]. In this work we study the role of layers on the synchronization phenomenon in two real multi-layer cortical neural networks. Temporal synchronization of neuronal activities plays an important role in neural binding and information processing mechanisms [17,18]. Various brain disorders such as schizophrenia and Alzheimer’s disease are linked to abnormality in the synchronization level of the brain [19–22]. Often, a specific mathematical neuron model is employed, and real or synthetic networks are used to study the synchronization phenomenon in neural networks [23–26]. There are two types of connections in neuronal networks: uni-directional chemical synapses and bi-directional electrical couplings through gap junctions. Various studies reported that these two modalities of synaptic transmission closely interact in brain’s functioning, see a review in Ref. [27]. For example, studying the role of these two types of synaptic connections on the central respiratory rhythm-generating system showed that the chemical couplings are mainly responsible for the production of respiratory cycle timing, while both electrical and chemical connections are involved in short-time-scale synchronization [28]. Both chemical synapses and gap junctions have been shown to be important in synchronizing the neural activity [28–31]. It has been shown that combined electrical and chemical couplings entrain synchronized gamma oscillations, which is required to many cognitive functions of the brain [32,33]. These two types of connections orchestrate action potential timings in oscillatory interneuronal networks. Electrical coupling through gap junctions have been frequently reported to enhance synchronization in the gamma frequencies [29]. A computational study suggests that electrical coupling have the main role in providing synchrony among neuronal networks, while chemical connections have the complementary role [25]. Jhou et al. introduced multistate synchronization in combined chemically and electrically coupled neural networks [34]. They identified the regions for coupling strength to achieve the synchronization. Baptista et al. studied the combined action of chemical and gap junction connections in model small-world networks [35]. They provided numerical simulations on Hindmarsh–Rose neurons coupled through excitatory/inhibitory chemical synapses and gap junctions. Previous works studied the role of chemical and electrical coupling on model networks (e.g., random, small-world and scale-free network topologies). In this work, we consider two real multiplex networks: whole-cortex connectivity network in hermaphrodite C. elegans [36] and that of posterior cortex in male C. elegans [37], and study the phase synchronization of spike trains. Each of these networks has two distinct connection layers, chemical synapses and gap junctions, each with its own functionality in the system. We study the role of each layer in the synchronization. We also compare the synchronizability of each layer with corresponding randomized networks, allowing to study the role of network synchronization in its evolution process.
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2. Dynamical equations
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In this work we consider multiplex networks as the connection structure. In these networks, the connections exist in different layers, and the nodes are identical across the layers. We study phase synchronization among N neurons with the same dynamics. On each node of the connection graph a dynamical system sits and the equations of the motion of the dynamical network read as x˙ i (t ) = F (xi (t )) +
M
σl
l=1 47 48 49 50 51 52
53
N
al,ij Hl xj (t ) , xi (t ) ;
i = 1, 2, . . . , N ,
(1)
j =1
where xi ∈ Rd are the state vectors and F : Rd → Rd defines the individual system’s dynamical equation. M is the number of layers, and the individual dynamical systems are coupled via a unified coupling strength σl and coupling matrix Al = (al,ij ) in each layer. Here we consider binary connections that is al,ij = 1, if there is a link from node i to j in layer l, and zero otherwise. There are no self-loops that means the diagonal entries of Al equal to zero. Hl (.) is a projection function showing the coupling function between the individual units in layer l. Considering linear coupling between the dynamical systems in all layers, Eq. (1) can be rewritten as x˙ i (t ) = F (xi (t )) +
M l=1
σl
N j =1
al,ij H xj (t ) − xi (t ) ;
i = 1, 2, . . . , N .
(2)
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As individual dynamical systems (represented by F (.) in Eqs. (1) and (2)), we consider Hindmarsh–Rose neuron model [38], which is capable of producing many of patterns observed in real neuronal systems such as spiking, bursting, adaptation and chaotic motion [39]. It has three first-order differential equations as follows
x˙ (t ) = y (t ) + ax2 (t ) − x3 (t ) − z (t ) + I y˙ (t ) = 1 − dx2 (t ) − y (t ) z˙ (t ) = µ (b (x (t ) − x0 ) − z (t )) ,
(3)
where x represents the membrane potential (dimensionless), y and z are virtual states representing the fast and slow current dynamics, respectively. I is the external input current injected to the neuron and a governs the qualitative behavior. µ and b control spiking frequency and adaptation behavior of the model [39,40]. µ controls the variation speed of z (t ) – the slow variable – which corresponds to the efficiency of slow channels in exchanging ions. When the neuron is spiking, this parameter governs the spiking frequency, and in the case of bursting behavior, it affects the number of spikes per burst. b governs the adaptation behavior of the model; values around b = 4 result in strong adaptation and subthreshold overshoot, whereas a unitary value for this parameter leads to spiking behavior without adaptation. x0 sets the resting potential of the system and d is a positive value. Coupling in real neural networks can in general be from two types: electrical and slow coupling through chemical synapses. The above couplings (Eq. (2)) represent the first type. In order to perform numerical simulations under the second type, Fast Threshold Modulation (FTM) model is used [41], in which the synapses are considered to be fast enough compared to the dynamics of the model. In FTM model the influence of neuron i to neuron j is modeled as a current injected from i to j, which is a nonlinear function of the membrane potential xi of i and a linear function of the membrane potential xj of j, as follows [41]
Iij = σ Vs − xj (t ) Θ (xi (t )) ,
(4)
where σ is the strength of coupling and Vs is the synaptic reversal potential. If Vs > xj , the current injected to the cell depolarizes it, and thus the coupling is excitatory, whereas for Vs < xj the negative current injected to the cell hyperpolarizes it, and thus introducing inhibitory coupling. Activation function Θ (xi ) is a sigmoid function that takes the form as
Θ (xi (t )) =
1 1 + exp {−λ (xi (t ) − θs )}
,
(5)
where θs is the threshold and λ is a positive constant. In the limit λ → ∞, a hard threshold is obtained and the above sigmoid function reduces to a Heaviside step function. Having this type of coupling between the individual neurons (in both gap junction and chemical networks), the equations of motion read M
x˙ i (t ) = F (xi (t )) −
N
σl (Vs − xi (t ))
l =1
j=1
al,ij
1
; 1 + exp −λ xj (t ) − θs
i = 1, 2, . . . , N .
(6)
It is worth mentioning that in real neural network, the gap junction neurons are coupled through fast electrical synapses, while the neurons in chemical layers are connected through chemical synapses (that can be modeled using FTM). Since we are interested in comparing the synchronization of gap junction and chemical networks (i.e., the effect of their topology in the synchronization phenomenon), the coupling is considered to be either electrical (Eq. (2)) or FTM (Eq. (6)) for both of them. This allows factoring out the influence of coupling type and providing a fair comparison between the topologies of the layers. 3. Spike phase synchronization
ϕj (t ) = 2π
t− j Ti+1
− Tij
,
j
j
Ti ≤ t ≤ Ti+1 , j = 1, . . . , N ,
(7)
with N as the number of neurons. As the spike trains are extracted from the time series of action potentials, an index indicating the amount of phase synchronization among them is calculated. Two oscillators with phases ϕ1 and ϕ2 are called to be phase synchronized when [9,44]
|ϕ1 − ϕ2 | < constant.
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As numerical simulations run for a number of time steps, the time series of the action potentials (i.e., x-components of the individual dynamical units) are obtained. Often, it is assumed that the shape of the action potential and background activity do not have much information, and only spike trains (the time histories of spike emissions) are processed [42]. Spike trains are believed to carry the major information of neuronal codes. Here we use thresholding method in order to decide whether to count a spike that is if the action potential exceeds a certain threshold value, the spike is counted. Let us suppose that Ti (i = 1, . . . , M ) are the spike times where M is the total number of spikes. The next step is to obtain phase signal from the spike trains, and then, calculate the synchronization among these phases. Having the spike train of neuron j, its phase can be calculated as [23,43] j Ti
1
(8)
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Table 1 Topological properties of C. elegans cortical networks used in this study. Network type
Size Number of links
Global efficiency
Clustering coefficient
Modularity
Assortativity
Chemical in C. elegans hermaphrodite [36] Gap junction in C. elegans hermaphrodite Chemical in male C. elegans [37] Gap junction in male C. elegans
279 279 170 170
0.29 0.21 0.38 0.41
0.21 0.18 0.29 0.26
0.20 0.46 0.21 0.24
−0.04 −0.12
2194 (uni-directional) 514 (bi-directional) 2184 (uni-directional) 918 (bi-directional)
0.09 −0.01
Fig. 1. Degree distribution (out-degree and in-degree is for the chemical networks, and degree for the gap junction networks) of the networks.
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When there are more than two oscillators, a possible approach to quantify phase synchrony is to compute the order parameter as [45,46]
3
S=
N 1 iϕj (t ) e N j =1
,
(9)
t
6
where ϕj (t ) represents the instantaneous phase of the jth oscillator, and ⟨. . .⟩t makes time averaging. i is the imaginary unit. This index scales as 0 ≤ S ≤ 1, where one has S ∼ 0 for completely independent motion (uncoupled oscillators), and the case S ∼ 1 indicates that the dynamical systems are phase synchronized [47].
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4. Connectivity data
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As real multiplex networks, we consider cortical neural networks of two types of C. elegans: whole-cortex of hermaphrodite type [36] and posterior cortex of male type [37]. In each type, there are two layers of connections between the nodes: chemical synapses and gap junctions. Table 1 shows statistical information of these networks and their degree distributions (in- and out-degree for chemical networks) are shown in Fig. 1. In both types, the chemical layer is denser (i.e., higher average degree) than the gap junctions layer. The gap junction network in the hermaphrodite type shows significantly higher community structure (characterized by the modularity index) than other networks. This network also shows moderate levels of disassortativity.
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5. Results and discussion
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In this section we give the results obtained through numerical simulations in MatLab. The parameters are set as d = 5, b = 4, µ = 0.03, I = 3.6, x0 = −1.6, λ = 100, θs = −0.25, Vs = 2, and a = 3.8 ± 0.38 (10% of variation in this parameter), which leads to spiking pattern for the neurons [39,40]. The neurons will all have spiking behavior but with
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Fig. 2. Spike phase order parameter (S) as a function of coupling strength in gap junction network (δe ) and that of chemical network (δc ) in (A) male C. elegans and (B) its randomized version. The randomized networks are such that the nodes have the same degree as the original network. Graphs show mean values over 50 realizations.
different frequency of spikes. Identical neurons will have high levels of phase synchronization, although for very small coupling strength. The settings used in this work (i.e., different values for parameter a) make the individual neurons to be non-identical for which the synchronization is a more challenging task. Let us denote the unified coupling strength between the neurons in the chemical layer by δc and in the gap junction layer by δe . We use a threshold of th = 1.5 for deciding the spikes, i.e., if the action potential gets a value higher than this threshold, the spike counts. The differential equations are iterated, starting from random initial conditions, using the Heun algorithm [48]. We consider the simulation time as Tf = 1000 time units and the integration time-step as dT = 0.05. Neglecting a number of first iterations in order to eliminate transients, the time series of the action potentials, and hence, the spike trains are obtained. Then, the phase signals are extracted using Eq. (7), and finally, the phase order parameter, as expressed by Eq. (9), is obtained. The simulations are performed 50 times with random initial conditions and the mean values are shown in the graphs. Fig. 2(A) shows the spike phase order parameter S for male C. elegans. We also perform the numerical simulations on properly randomized versions of the original networks. To study whether the synchronization phenomenon was important in the evolution of the networks, we compare the synchronizability of the original networks with those of randomized versions. For each network, the randomized version is constructed such that the nodes degrees (in- and out-degrees for directed networks) are preserved. To this end, two random links without any common source/sink nodes are first chosen. Then, the source or sink nodes are switched for the links. This process is repeated 20,000 times for each network. Degree distribution has a major role in determining networks’ dynamical properties, and this randomization strategy provides an unbiased comparison (through keeping the degrees unchanged). Such a randomization strategy has been previously used for discovering significant motifs in complex networks [49]. Fig. 2(B) shows S as a function of δe and δc in the randomized networks of male C. elegans. In order to better illustrate the interplay between chemical and gap junction layers and their influence on the spike phase synchronization, we show 2D plots of S as a function of δe or δc when one of them is fixed (Fig. 3). The graphs show S as a function of chemical coupling strength δc (gap junction coupling strength δe ) when δe (δc ) is fixed for the corresponding layer. These results show that the original male C. elegans has better phase synchronizability than the randomized networks. When comparing the influence of different layers, the behavior is not the same for different coupling strengths. When δc is fixed at some small values (panels A and B), the gap junction layer has higher contribution on the synchronization (i.e., higher S) than the chemical layer (with the same fixed δe ). This is interesting since the gap junction layer is about 19% sparser (i.e., smaller average degree) than the chemical layer. Often, when average degree increases, the synchronizability of the network also increases [50]. This indicates that in this multiplex neural network, the gap junction network has more significant role than the chemical network in the synchronization phenomenon. Indeed, the structure of the gap junction network has specific properties that make it more synchronizable than the chemical one. This can be linked to its higher global efficiency (Table 1). This is in line with previous results suggesting that gap junctions have the major role in the synchronization process and chemical connections have a complementary role [25,32]. The results for hermaphrodite C. elegans are shown in Figs. 4 and 5. This multiplex network shows somehow different behavior as compared to the male type. In hermaphrodite C. elegans, often the randomized networks have better phase synchronizability than the original ones. Furthermore, for small and high fixed δc , the gap junction layer has less contribution in the synchronization than the chemical layer, while for some medium δc (panel D), the gap junction network contributes more. Note that in hermaphrodite C. elegans, the density of gap junction layer is almost half of the chemical layer, which is one of main factors for its poor synchronizability. Another driving effect on this behavior can be modular structure of the networks. Gap junction network of this type has much higher modularity than the chemical network, and it is well known that modular networks have weak synchronization properties [51,52].
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Fig. 3. Spike phase order parameter (S) as a function of δe (δc ) in male C. elegans. In order to plot S as a function of δe (δc ), the value of δc (δe ) is fixed at (A) 0.03, (B) 0.06, (C) 0.105, (D) 0.21, (E) 0.3, and (F) 0.45. With a fixed coupling strength in gap junction (chemical) network, S is shown as a function of δc (δe ). The graphs show the mean values (with bars corresponding to the standard error in randomized networks) over 50 realizations.
Fig. 4. Spike phase order parameter (S) as a function of δe and δc in (A) hermaphrodite C. elegans and (B) its randomized version. Other designations are as Fig. 2.
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Fig. 6 shows the synchronizability of the multiplex male and hermaphrodite C. elegans when the coupling type is FTM (Eq. (6)), mimicking connections made by excitatory chemical synapses in neural networks. As it is seen, in male C. elegans, the gap junction network (although being sparser) has better synchronizability than the chemical network under this coupling fashion, which is similar to the observation obtained under linear coupling (Fig. 3). Also similar to the previous case, in hermaphrodite C. elegans, the chemical network has better synchronizability. Note that the coupling through chemical synapses (Eq. (6)) has been considered to be instantaneous in this work; however, transmission time-delay is often incorporated in real neural networks. Synchronization has a complex dependency on the time delay [23,53] and it has been shown that the delay can support transition between synchronization and desynchronization regimes [54]. This simplification is made to allow relating the synchronization to solely on the topology of the layers. In summary, these results show that layers’ contribution on the synchronization of multiplex networks is complex that highly depends on the specific network type. To have a further understanding of the synchronization patterns in
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Fig. 5. Spike phase order parameter (S) as a function of δe (δc ) in hermaphrodite C. elegans. Other designations are as Fig. 3.
Fig. 6. S as a function of δe (δc ) in (A) male and (B) hermaphrodite C. elegans. In order to plot S as a function of δe (δc ), the value of δc (δe ) is fixed at 0.05. With a fixed coupling strength in gap junction (chemical) network, S is shown as a function of δc (δe ). The graphs show the mean values (with bars corresponding to the standard error in randomized networks) over 50 realizations. The neurons are coupled through fast threshold modulation, a nonlinear coupling frequently used to model chemical synapses, as expressed by Eq. (6).
multiplex networks, the experiments should be repeated in more real examples of multiplex networks. In order to have better understanding on the role of electrical (i.e., gap junction) and chemical connections in the synchronization process, similar experiments should be repeated in vitro and in vivo conditions.
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6. Conclusions
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Synchronization phenomenon plays an important role in the process of information binding in neuronal systems. Neurons often get into synchrony when they are involved in some kinds of information processing. Here we considered two real multiplex networks (i.e., networks for which the connections exist in distinct layers between a group of nodes): connectome of whole-cortex in hermaphrodite C. elegans and that of posterior cortex in male C. elegans. In these multiplex networks the connections are formed in two layers: chemical synapses and gap junctions. Considering Hindmarsh–Rose neuron model on the nodes, we studied spike phase synchronization in the networks. First, the networked differential equations were numerically solved and time series of the action potentials were extracted. Then, the spike trains were obtained, and finally phase order parameter was computed quantifying spike phase synchrony in the networks. The simulation results showed that the way the layers (chemical and gap junction) contribute in the phase synchronization process is different. Having lower edge density, the gap junction layer showed higher or equal contribution in the synchronization than the chemical layer in male C. elegans. Whereas in hermaphrodite type, the chemical layer often showed higher contribution, which is due to its much higher density (edge density in the chemical layer is 2.14 times more as compared to the gap junction layer) and significantly weaker community structure. Furthermore, male cortical network showed higher synchronization level than the randomized networks, while this was not the case in hermaphrodite type.
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Acknowledgments
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This research was supported by Australian Research Council through project No DE140100620. The author would like to thank Dr. Mark Damian McDonnell for sharing the connectivity matrices of the cortical networks.
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[33] J. Szabadics, A. Lorincz, G. Tamás, β and γ frequency synchronization by dendritic GABAergic synapses and gap junctions in a network of cortical interneurons, J. Neurosci. 21 (2001) 5824–5831. [34] F.J. Jhou, J. Juang, Y.H. Liang, Multistate and multistage synchronization of Hindmarsh-Rose neurons with excitatory chemical and electrical synapses, IEEE Trans. Circuits Syst. I 59 (2012) 1335–1347. [35] M.S. Baptista, F.M. Moukam Kakmeni, C. Grebogi, Combined effect of chemical and electrical synapses in Hindmarsh-Rose neural networks on synchronization and the rate of information, Phys. Rev. E 82 (2010) 036203. [36] L.R. Varshney, B.L. Chen, E. Paniagua, D.H. Hall, D.B. Chklovskii, Structural properties of the Caenorhabditis elegans neuronal network, PLoS Comput. Biol. 7 (2011) e1001066. [37] T.A. Jarrell, Y. Wang, A.E. Bloniarz, C.A. Brittin, M. Xu, J.N. Thomson, D.G. Albertson, D.H. Hall, S.W. Emmons, The connectome of a decision-making neural network, Science 337 (2012) 437–444. [38] J.L. Hindmarsh, R.M. Rose, A model of neuronal bursting using three coupled first order differential equations, Proc. Roy. Soc. London Ser. B-Biol. Sci. 221 (1984) 87–102. [39] M. Storace, D. Linaro, E. de Lange, The Hindmarsh-Rose neuron model: Bifurcation analysis and piecewise-linear approximations, Chaos 18 (2008) 033128. [40] E. de Lange, Neuron Models of the Generic Bifurcation Type: Network Analysis and Data Modeling, Ecole Polytechnique Federal de Lausanne, Lausanne, 2006. [41] D. Somers, N. Kopell, Rapid synchronization through fast threshold modulation, Biol. Cybernet. 68 (1993) 393–407. [42] P. Dayan, L.F. Abbott, Theoretical Neuroscience: Computational and Mathematical Modeling of Neural Systems, first ed., MIT Press, 2001. [43] X. Liang, M. Tang, M. Dhamala, Z. Liu, Phase synchronization of inhibitory bursting neurons induced by distributed time delays in chemical coupling, Phys. Rev. E 80 (2009) 066202. [44] M.G. Rosenblum, A.S. Pikovsky, J. Kurths, Phase synchronization of chaotic oscillators, Phys. Rev. Lett. 76 (1996) 1804–1807. [45] S. Boccaletti, J. Kurths, G. Osipov, D.L. Valladares, C.S. Zhou, The synchronization of chaotic systems, Phys. Rep. 366 (2002) 1–101. [46] A. Pikovsky, M. Rosenblum, J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences, first ed., Cambridge University Press, 2003. [47] M. Chavez, D.-U. Hwang, A. Amann, H.G.E. Hentschel, S. Boccaletti, Synchronization is enhanced in weighted complex networks, Phys. Rev. Lett. 94 (2005) 218701. [48] A. Quarteroni, R. Sacco, F. Saleri, Numerical Mathematics, Springer-Verlag, Berlin, Germany, 2000. [49] R. Milo, S. Shen-Orr, S. Itzkovitz, N. Kashtan, D. Chklovskii, U. Alon, Network motifs: simple building blocks of complex networks, Science 298 (2002) 824–827. [50] M. Jalili, A. Ajdari Rad, Synchronizability of dynamical networks: different measures and coincidence, Int. J. Nonlinear Sci. Numer. Simul. 10 (2009) 1481–1489. [51] K. Park, Y.C. Lai, S. Gupte, J.W. Kim, Synchronization in complex networks with a modular structure, Chaos 16 (2006) 015105. [52] M. Jalili, Synchronizability of complex networks with community structure, Internat. J. Modern Phys. C 23 (2012) 1250029. [53] M. Jalili, Phase synchronizing in hindmarsh-rose neural networks with delayed chemical coupling, Neurocomputing 74 (2011) 1551–1556. [54] Z.G. Esfahani, L.L. Gollo, A. Valizadeh, Stimulus-dependent synchronization in delayed-coupled neuronal networks, Sci. Rep. 6 (2016) 23471.
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Physica A journal homepage: www.elsevier.com/locate/physa
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Spike phase synchronization in multiplex cortical neural networks Mahdi Jalili Department of Electrical and Computer Engineering, School of Engineering, RMIT University, 3001 Melbourne, VIC, Australia
highlights • Chemical synapses and gap junctions in C. elegance networks have different synchronizability. • The cortical networks have better synchronizability than the random networks. • Modularity of the networks is one of the main driving effects for their synchronizability.
article
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Article history: Received 5 February 2016 Received in revised form 2 August 2016 Available online xxxx Keywords: Dynamical networks Phase synchronization Transmission delay Coupled oscillators Hindmarsh–Rose neuron model Spiking neurons
abstract In this paper we study synchronizability of two multiplex cortical networks: wholecortex of hermaphrodite C. elegans and posterior cortex in male C. elegans. These networks are composed of two connection layers: network of chemical synapses and the one formed by gap junctions. This work studies the contribution of each layer on the phase synchronization of non-identical spiking Hindmarsh–Rose neurons. The network of male C. elegans shows higher phase synchronization than its randomized version, while it is not the case for hermaphrodite type. The random networks in each layer are constructed such that the nodes have the same degree as the original network, thus providing an unbiased comparison. In male C. elegans, although the gap junction network is sparser than the chemical network, it shows higher contribution in the synchronization phenomenon. This is not the case in hermaphrodite type, which is mainly due to significant less density of gap junction layer (0.013) as compared to chemical layer (0.028). Also, the gap junction network in this type has stronger community structure than the chemical network, and this is another driving factor for its weaker synchronizability. © 2016 Elsevier B.V. All rights reserved.
1. Introduction Many real systems can be modeled as networks; a collection of individual nodes interacting through (directed or undirected, weighted or unweighted) edges [1–3]. The last two decades have witnessed tremendous progress on understanding statistical and dynamical properties of complex networked systems. It has been shown that many real networks share some common properties such as small-worldness [4], scale-free degree distribution [5] and community structure [6]. Such properties significantly influence how dynamical processes evolve on networks and the way collective actions emerge [7]. Synchronization is the most widely studied collective behavior in networked systems [8]. It happens when two (or more) dynamical systems meet and interact; if the interactions between the individual dynamical units are strong enough, their behavior shows a time-correlated activity, i.e., they get into synchrony. There are different types of synchronization phenomenon such as complete, bubbling and lag synchronization. The synchronization type observed in
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.physa.2016.09.030 0378-4371/© 2016 Elsevier B.V. All rights reserved.
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many real systems is phase synchronization that is due to (often) weak coupling between the dynamical units [9]. Two (or more) dynamical systems are phase synchronized if their phases get into coherency. Recently, there has been much effort in the community of network science on moving from single-layer modeling of network systems to multi-layer modeling, which is mainly due to enhanced resolution of network datasets [10]. In singlelayer modeling of a networked system, all node-to-node interactions are treated the same and the only difference between them is characterized by their weight (in some applications the links can have positive or negative weights). However, in a multi-layer framework, the connection links are organized in different layers, which allows considering temporal- or context-related properties of the interactions. Indeed, multi-layer networks include a set of nodes and several layers of connections, accurately describing the node-to-node interactions, and/or the whole system’s parallel functioning. Examples of such multi-layer networks include road and rail traffic networks [11], air transportation networks [12], online social networks with several types of relations such as friendship, vicinity, membership and partnership [13], and international trade networks [14]. Although there are many research studies addressing the problem of synchronization (or consensus) in single-layer networked systems, there are few works investigating the problem in multi-layer networks [15,16]. In this work we study the role of layers on the synchronization phenomenon in two real multi-layer cortical neural networks. Temporal synchronization of neuronal activities plays an important role in neural binding and information processing mechanisms [17,18]. Various brain disorders such as schizophrenia and Alzheimer’s disease are linked to abnormality in the synchronization level of the brain [19–22]. Often, a specific mathematical neuron model is employed, and real or synthetic networks are used to study the synchronization phenomenon in neural networks [23–26]. There are two types of connections in neuronal networks: uni-directional chemical synapses and bi-directional electrical couplings through gap junctions. Various studies reported that these two modalities of synaptic transmission closely interact in brain’s functioning, see a review in Ref. [27]. For example, studying the role of these two types of synaptic connections on the central respiratory rhythm-generating system showed that the chemical couplings are mainly responsible for the production of respiratory cycle timing, while both electrical and chemical connections are involved in short-time-scale synchronization [28]. Both chemical synapses and gap junctions have been shown to be important in synchronizing the neural activity [28–31]. It has been shown that combined electrical and chemical couplings entrain synchronized gamma oscillations, which is required to many cognitive functions of the brain [32,33]. These two types of connections orchestrate action potential timings in oscillatory interneuronal networks. Electrical coupling through gap junctions have been frequently reported to enhance synchronization in the gamma frequencies [29]. A computational study suggests that electrical coupling have the main role in providing synchrony among neuronal networks, while chemical connections have the complementary role [25]. Jhou et al. introduced multistate synchronization in combined chemically and electrically coupled neural networks [34]. They identified the regions for coupling strength to achieve the synchronization. Baptista et al. studied the combined action of chemical and gap junction connections in model small-world networks [35]. They provided numerical simulations on Hindmarsh–Rose neurons coupled through excitatory/inhibitory chemical synapses and gap junctions. Previous works studied the role of chemical and electrical coupling on model networks (e.g., random, small-world and scale-free network topologies). In this work, we consider two real multiplex networks: whole-cortex connectivity network in hermaphrodite C. elegans [36] and that of posterior cortex in male C. elegans [37], and study the phase synchronization of spike trains. Each of these networks has two distinct connection layers, chemical synapses and gap junctions, each with its own functionality in the system. We study the role of each layer in the synchronization. We also compare the synchronizability of each layer with corresponding randomized networks, allowing to study the role of network synchronization in its evolution process.
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2. Dynamical equations
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In this work we consider multiplex networks as the connection structure. In these networks, the connections exist in different layers, and the nodes are identical across the layers. We study phase synchronization among N neurons with the same dynamics. On each node of the connection graph a dynamical system sits and the equations of the motion of the dynamical network read as x˙ i (t ) = F (xi (t )) +
M
σl
l=1 47 48 49 50 51 52
53
N
al,ij Hl xj (t ) , xi (t ) ;
i = 1, 2, . . . , N ,
(1)
j =1
where xi ∈ Rd are the state vectors and F : Rd → Rd defines the individual system’s dynamical equation. M is the number of layers, and the individual dynamical systems are coupled via a unified coupling strength σl and coupling matrix Al = (al,ij ) in each layer. Here we consider binary connections that is al,ij = 1, if there is a link from node i to j in layer l, and zero otherwise. There are no self-loops that means the diagonal entries of Al equal to zero. Hl (.) is a projection function showing the coupling function between the individual units in layer l. Considering linear coupling between the dynamical systems in all layers, Eq. (1) can be rewritten as x˙ i (t ) = F (xi (t )) +
M l=1
σl
N j =1
al,ij H xj (t ) − xi (t ) ;
i = 1, 2, . . . , N .
(2)
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As individual dynamical systems (represented by F (.) in Eqs. (1) and (2)), we consider Hindmarsh–Rose neuron model [38], which is capable of producing many of patterns observed in real neuronal systems such as spiking, bursting, adaptation and chaotic motion [39]. It has three first-order differential equations as follows
x˙ (t ) = y (t ) + ax2 (t ) − x3 (t ) − z (t ) + I y˙ (t ) = 1 − dx2 (t ) − y (t ) z˙ (t ) = µ (b (x (t ) − x0 ) − z (t )) ,
(3)
where x represents the membrane potential (dimensionless), y and z are virtual states representing the fast and slow current dynamics, respectively. I is the external input current injected to the neuron and a governs the qualitative behavior. µ and b control spiking frequency and adaptation behavior of the model [39,40]. µ controls the variation speed of z (t ) – the slow variable – which corresponds to the efficiency of slow channels in exchanging ions. When the neuron is spiking, this parameter governs the spiking frequency, and in the case of bursting behavior, it affects the number of spikes per burst. b governs the adaptation behavior of the model; values around b = 4 result in strong adaptation and subthreshold overshoot, whereas a unitary value for this parameter leads to spiking behavior without adaptation. x0 sets the resting potential of the system and d is a positive value. Coupling in real neural networks can in general be from two types: electrical and slow coupling through chemical synapses. The above couplings (Eq. (2)) represent the first type. In order to perform numerical simulations under the second type, Fast Threshold Modulation (FTM) model is used [41], in which the synapses are considered to be fast enough compared to the dynamics of the model. In FTM model the influence of neuron i to neuron j is modeled as a current injected from i to j, which is a nonlinear function of the membrane potential xi of i and a linear function of the membrane potential xj of j, as follows [41]
Iij = σ Vs − xj (t ) Θ (xi (t )) ,
(4)
where σ is the strength of coupling and Vs is the synaptic reversal potential. If Vs > xj , the current injected to the cell depolarizes it, and thus the coupling is excitatory, whereas for Vs < xj the negative current injected to the cell hyperpolarizes it, and thus introducing inhibitory coupling. Activation function Θ (xi ) is a sigmoid function that takes the form as
Θ (xi (t )) =
1 1 + exp {−λ (xi (t ) − θs )}
,
(5)
where θs is the threshold and λ is a positive constant. In the limit λ → ∞, a hard threshold is obtained and the above sigmoid function reduces to a Heaviside step function. Having this type of coupling between the individual neurons (in both gap junction and chemical networks), the equations of motion read M
x˙ i (t ) = F (xi (t )) −
N
σl (Vs − xi (t ))
l =1
j=1
al,ij
1
; 1 + exp −λ xj (t ) − θs
i = 1, 2, . . . , N .
(6)
It is worth mentioning that in real neural network, the gap junction neurons are coupled through fast electrical synapses, while the neurons in chemical layers are connected through chemical synapses (that can be modeled using FTM). Since we are interested in comparing the synchronization of gap junction and chemical networks (i.e., the effect of their topology in the synchronization phenomenon), the coupling is considered to be either electrical (Eq. (2)) or FTM (Eq. (6)) for both of them. This allows factoring out the influence of coupling type and providing a fair comparison between the topologies of the layers. 3. Spike phase synchronization
ϕj (t ) = 2π
t− j Ti+1
− Tij
,
j
j
Ti ≤ t ≤ Ti+1 , j = 1, . . . , N ,
(7)
with N as the number of neurons. As the spike trains are extracted from the time series of action potentials, an index indicating the amount of phase synchronization among them is calculated. Two oscillators with phases ϕ1 and ϕ2 are called to be phase synchronized when [9,44]
|ϕ1 − ϕ2 | < constant.
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As numerical simulations run for a number of time steps, the time series of the action potentials (i.e., x-components of the individual dynamical units) are obtained. Often, it is assumed that the shape of the action potential and background activity do not have much information, and only spike trains (the time histories of spike emissions) are processed [42]. Spike trains are believed to carry the major information of neuronal codes. Here we use thresholding method in order to decide whether to count a spike that is if the action potential exceeds a certain threshold value, the spike is counted. Let us suppose that Ti (i = 1, . . . , M ) are the spike times where M is the total number of spikes. The next step is to obtain phase signal from the spike trains, and then, calculate the synchronization among these phases. Having the spike train of neuron j, its phase can be calculated as [23,43] j Ti
1
(8)
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Table 1 Topological properties of C. elegans cortical networks used in this study. Network type
Size Number of links
Global efficiency
Clustering coefficient
Modularity
Assortativity
Chemical in C. elegans hermaphrodite [36] Gap junction in C. elegans hermaphrodite Chemical in male C. elegans [37] Gap junction in male C. elegans
279 279 170 170
0.29 0.21 0.38 0.41
0.21 0.18 0.29 0.26
0.20 0.46 0.21 0.24
−0.04 −0.12
2194 (uni-directional) 514 (bi-directional) 2184 (uni-directional) 918 (bi-directional)
0.09 −0.01
Fig. 1. Degree distribution (out-degree and in-degree is for the chemical networks, and degree for the gap junction networks) of the networks.
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When there are more than two oscillators, a possible approach to quantify phase synchrony is to compute the order parameter as [45,46]
3
S=
N 1 iϕj (t ) e N j =1
,
(9)
t
6
where ϕj (t ) represents the instantaneous phase of the jth oscillator, and ⟨. . .⟩t makes time averaging. i is the imaginary unit. This index scales as 0 ≤ S ≤ 1, where one has S ∼ 0 for completely independent motion (uncoupled oscillators), and the case S ∼ 1 indicates that the dynamical systems are phase synchronized [47].
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4. Connectivity data
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As real multiplex networks, we consider cortical neural networks of two types of C. elegans: whole-cortex of hermaphrodite type [36] and posterior cortex of male type [37]. In each type, there are two layers of connections between the nodes: chemical synapses and gap junctions. Table 1 shows statistical information of these networks and their degree distributions (in- and out-degree for chemical networks) are shown in Fig. 1. In both types, the chemical layer is denser (i.e., higher average degree) than the gap junctions layer. The gap junction network in the hermaphrodite type shows significantly higher community structure (characterized by the modularity index) than other networks. This network also shows moderate levels of disassortativity.
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5. Results and discussion
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In this section we give the results obtained through numerical simulations in MatLab. The parameters are set as d = 5, b = 4, µ = 0.03, I = 3.6, x0 = −1.6, λ = 100, θs = −0.25, Vs = 2, and a = 3.8 ± 0.38 (10% of variation in this parameter), which leads to spiking pattern for the neurons [39,40]. The neurons will all have spiking behavior but with
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Fig. 2. Spike phase order parameter (S) as a function of coupling strength in gap junction network (δe ) and that of chemical network (δc ) in (A) male C. elegans and (B) its randomized version. The randomized networks are such that the nodes have the same degree as the original network. Graphs show mean values over 50 realizations.
different frequency of spikes. Identical neurons will have high levels of phase synchronization, although for very small coupling strength. The settings used in this work (i.e., different values for parameter a) make the individual neurons to be non-identical for which the synchronization is a more challenging task. Let us denote the unified coupling strength between the neurons in the chemical layer by δc and in the gap junction layer by δe . We use a threshold of th = 1.5 for deciding the spikes, i.e., if the action potential gets a value higher than this threshold, the spike counts. The differential equations are iterated, starting from random initial conditions, using the Heun algorithm [48]. We consider the simulation time as Tf = 1000 time units and the integration time-step as dT = 0.05. Neglecting a number of first iterations in order to eliminate transients, the time series of the action potentials, and hence, the spike trains are obtained. Then, the phase signals are extracted using Eq. (7), and finally, the phase order parameter, as expressed by Eq. (9), is obtained. The simulations are performed 50 times with random initial conditions and the mean values are shown in the graphs. Fig. 2(A) shows the spike phase order parameter S for male C. elegans. We also perform the numerical simulations on properly randomized versions of the original networks. To study whether the synchronization phenomenon was important in the evolution of the networks, we compare the synchronizability of the original networks with those of randomized versions. For each network, the randomized version is constructed such that the nodes degrees (in- and out-degrees for directed networks) are preserved. To this end, two random links without any common source/sink nodes are first chosen. Then, the source or sink nodes are switched for the links. This process is repeated 20,000 times for each network. Degree distribution has a major role in determining networks’ dynamical properties, and this randomization strategy provides an unbiased comparison (through keeping the degrees unchanged). Such a randomization strategy has been previously used for discovering significant motifs in complex networks [49]. Fig. 2(B) shows S as a function of δe and δc in the randomized networks of male C. elegans. In order to better illustrate the interplay between chemical and gap junction layers and their influence on the spike phase synchronization, we show 2D plots of S as a function of δe or δc when one of them is fixed (Fig. 3). The graphs show S as a function of chemical coupling strength δc (gap junction coupling strength δe ) when δe (δc ) is fixed for the corresponding layer. These results show that the original male C. elegans has better phase synchronizability than the randomized networks. When comparing the influence of different layers, the behavior is not the same for different coupling strengths. When δc is fixed at some small values (panels A and B), the gap junction layer has higher contribution on the synchronization (i.e., higher S) than the chemical layer (with the same fixed δe ). This is interesting since the gap junction layer is about 19% sparser (i.e., smaller average degree) than the chemical layer. Often, when average degree increases, the synchronizability of the network also increases [50]. This indicates that in this multiplex neural network, the gap junction network has more significant role than the chemical network in the synchronization phenomenon. Indeed, the structure of the gap junction network has specific properties that make it more synchronizable than the chemical one. This can be linked to its higher global efficiency (Table 1). This is in line with previous results suggesting that gap junctions have the major role in the synchronization process and chemical connections have a complementary role [25,32]. The results for hermaphrodite C. elegans are shown in Figs. 4 and 5. This multiplex network shows somehow different behavior as compared to the male type. In hermaphrodite C. elegans, often the randomized networks have better phase synchronizability than the original ones. Furthermore, for small and high fixed δc , the gap junction layer has less contribution in the synchronization than the chemical layer, while for some medium δc (panel D), the gap junction network contributes more. Note that in hermaphrodite C. elegans, the density of gap junction layer is almost half of the chemical layer, which is one of main factors for its poor synchronizability. Another driving effect on this behavior can be modular structure of the networks. Gap junction network of this type has much higher modularity than the chemical network, and it is well known that modular networks have weak synchronization properties [51,52].
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Fig. 3. Spike phase order parameter (S) as a function of δe (δc ) in male C. elegans. In order to plot S as a function of δe (δc ), the value of δc (δe ) is fixed at (A) 0.03, (B) 0.06, (C) 0.105, (D) 0.21, (E) 0.3, and (F) 0.45. With a fixed coupling strength in gap junction (chemical) network, S is shown as a function of δc (δe ). The graphs show the mean values (with bars corresponding to the standard error in randomized networks) over 50 realizations.
Fig. 4. Spike phase order parameter (S) as a function of δe and δc in (A) hermaphrodite C. elegans and (B) its randomized version. Other designations are as Fig. 2.
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Fig. 6 shows the synchronizability of the multiplex male and hermaphrodite C. elegans when the coupling type is FTM (Eq. (6)), mimicking connections made by excitatory chemical synapses in neural networks. As it is seen, in male C. elegans, the gap junction network (although being sparser) has better synchronizability than the chemical network under this coupling fashion, which is similar to the observation obtained under linear coupling (Fig. 3). Also similar to the previous case, in hermaphrodite C. elegans, the chemical network has better synchronizability. Note that the coupling through chemical synapses (Eq. (6)) has been considered to be instantaneous in this work; however, transmission time-delay is often incorporated in real neural networks. Synchronization has a complex dependency on the time delay [23,53] and it has been shown that the delay can support transition between synchronization and desynchronization regimes [54]. This simplification is made to allow relating the synchronization to solely on the topology of the layers. In summary, these results show that layers’ contribution on the synchronization of multiplex networks is complex that highly depends on the specific network type. To have a further understanding of the synchronization patterns in
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Fig. 5. Spike phase order parameter (S) as a function of δe (δc ) in hermaphrodite C. elegans. Other designations are as Fig. 3.
Fig. 6. S as a function of δe (δc ) in (A) male and (B) hermaphrodite C. elegans. In order to plot S as a function of δe (δc ), the value of δc (δe ) is fixed at 0.05. With a fixed coupling strength in gap junction (chemical) network, S is shown as a function of δc (δe ). The graphs show the mean values (with bars corresponding to the standard error in randomized networks) over 50 realizations. The neurons are coupled through fast threshold modulation, a nonlinear coupling frequently used to model chemical synapses, as expressed by Eq. (6).
multiplex networks, the experiments should be repeated in more real examples of multiplex networks. In order to have better understanding on the role of electrical (i.e., gap junction) and chemical connections in the synchronization process, similar experiments should be repeated in vitro and in vivo conditions.
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6. Conclusions
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Synchronization phenomenon plays an important role in the process of information binding in neuronal systems. Neurons often get into synchrony when they are involved in some kinds of information processing. Here we considered two real multiplex networks (i.e., networks for which the connections exist in distinct layers between a group of nodes): connectome of whole-cortex in hermaphrodite C. elegans and that of posterior cortex in male C. elegans. In these multiplex networks the connections are formed in two layers: chemical synapses and gap junctions. Considering Hindmarsh–Rose neuron model on the nodes, we studied spike phase synchronization in the networks. First, the networked differential equations were numerically solved and time series of the action potentials were extracted. Then, the spike trains were obtained, and finally phase order parameter was computed quantifying spike phase synchrony in the networks. The simulation results showed that the way the layers (chemical and gap junction) contribute in the phase synchronization process is different. Having lower edge density, the gap junction layer showed higher or equal contribution in the synchronization than the chemical layer in male C. elegans. Whereas in hermaphrodite type, the chemical layer often showed higher contribution, which is due to its much higher density (edge density in the chemical layer is 2.14 times more as compared to the gap junction layer) and significantly weaker community structure. Furthermore, male cortical network showed higher synchronization level than the randomized networks, while this was not the case in hermaphrodite type.
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Acknowledgments
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This research was supported by Australian Research Council through project No DE140100620. The author would like to thank Dr. Mark Damian McDonnell for sharing the connectivity matrices of the cortical networks.
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