Chimera states in hybrid coupled neuron populations

arXiv:2003.01854v1 [nlin.AO] 4 Mar 2020 Chimera States in Hybrid Coupled Neuron Populations Ali Calim,1, ∗Joaquin J. Torres,2Mahmut Ozer,3, 4, 5 and Muhammet Uzuntarla1 1Department of Biomedical Engineering, Zonguldak Bulent Ecevit University, Zonguldak, Turkey 2Department of Electromagnetism and Physics of the Matter and Institute Carlos I for Theoretical and Computational Physics, University of Granada, Granada, E-18071 Spain 3Ministry of National Education, Ankara, Turkey 4Center for Artificial Intelligence and Data Science, Istanbul Technical University, Istanbul, Turkey 5Department of Electrical and Electronics Engineering, Zonguldak Bulent Ecevit University, Zonguldak, Turkey (Dated: March 5, 2020) Abstract Here we study the emergence of chimera states, a recently reported phenomenon referring to the coexistence of synchronized and unsynchronized dynamical units, in a population of Morris-Lecar neurons which are coupled by both electrical and chemical synapses, constituting a hybrid synaptic architecture, as in actual brain connectivity. This scheme consists of a nonlocal network where the nearest neighbor neurons are coupled by electrical synapses, while the synapses from more distant neurons are of the chemical type. We demonstrate that peculiar dynamical behaviors, including chimera state and traveling wave, exist in such a hybrid coupled neural system, and analyze how the relative abundance of chemical and electrical synapses affects the features of chimera and different synchrony states (i.e. incoherent, traveling wave and coherent) and the regions in the space of relevant parameters for their emergence. Additionally, we show that, when the relative population of chemical synapses increases further, a new intriguing chaotic dynamical behavior appears above the region for chimera states. This is characterized by the coexistence of two distinct synchronized states with different amplitude, and an unsynchronized state, that we denote as a chaotic amplitude chimera. We also discuss about the computational implications of such state. Keywords: Chimera state, hybrid coupling, chaotic population behavior ∗[email protected] 1 I. INTRODUCTION Synchronization is widely considered to be essential for the proper functioning of a large variety of natural and artificial systems, ranging from physical experiments to chemical reactions and physiological processes. Prominent examples include communication networks [1, 2], coupled lasers [3–6], Josephson junctions [7, 8], oxidation and catalytic surface reactions [9–11], power grids [12] as well as circadian oscillators [13, 14] and genetic oscillator networks [15–17]. Apart from these, synchronization in neural systems has remained a very popular research area during the last decades, because it is widely assumed to be a possible underlying mechanism for various behavioral and cognitive functions, e.g., attention, information processing, and neural control of movement [18–22]. Moreover, many findings from both experimental and theoretical research suggest that neural synchronization might be responsible for pathological conditions in brain diseases (i.e., epilepsy and Parkinson), where the synchronized oscillations are the significant difference between healthy and unhealthy conditions [23–30]. Considering such important consequences, understanding the nature and controllability of neuronal synchronization is a critical step in uncovering the bases of many brain functions and diseases. On the other hand, neural synchronization is not always desirable and ubiquitous in the brain [31–34]. It has been found that healthy brain exhibits spontaneous asynchronous activity as well as synchronous patterns [35]. Thus, asynchronous population activity is not an harmful circumstance, it is rather beneficial to the brain. It helps for an efficient information processing and making decision in an excellent way, and also carrying out other vital tasks properly [36, 37]. In particular, the cortex operates in a highly asynchronous state during waking and REM sleep [38]. The subthalamic nucleus, a specific location in the basal ganglia, is another evidence of this inspection. It exhibits asynchronous electrical activity in the beta frequency band as an indicator of movement preparation [39]. Recent experimental and clinical studies have shown that these two common states, namely, synchronous and asynchronous activity, can coexist within the same neuronal circuitry at the same time [40, 41], and such a surprising state occurs, for instance, during unihemispheric sleep, epileptic seizures and bump states [42–47]. In recent years, these evidences have motivated researchers from neurophysics community to study such coexisting states and relate them with physical phenomena observed in nonlinear dynamical systems. In this context, a widely considered representative dynamical phenomenon is the chimera state which was originally described as coexistence of coherent and incoherent system states 2 in a network of coupled identical phase oscillators with nonlocal interactions [48, 49]. This symmetry-breaking physical concept has attracted great attention in determining the biological mechanisms that give rise to coexisting coherent and incoherent population activity in neural circuits. For instance, Omelchenko et al. have shown the emergence of chimera and multichimera – which refers to multiple incoherent domains – states in nonlocal network of electrically coupled Fitzhugh-Nagumo neurons [50]. To test robustness of their results, in [51], authors further investigated chimera states in heterogeneous neuron population considering diversity of intrinsic excitability and coupling, and found that emergence of chimera states is robust for small heterogeneity but, as the heterogeneity increases, multichimeras transform into single chimera. In another work, Bera et al. explored chimera states in nonlocal, global, and local networks of chemically coupled bursting type Hindmarsh-Rose neurons [52], and found that chimera also occurs in population of such model neurons in the presence of chemical synapses at network interactions. In a recent work, we have demonstrated that populations of Morris-Lecar type model neurons also exhibit chimeric behavior with fine tuning of biophysically relevant parameters, i.e. excitability, synaptic strength and network connectivity [53]. Apart from these works, presence of chimera state and its variants (e.g. amplitude chimera, breathing chimera and traveling chimera) have been shown in populations of other types of model neurons which are widely used in theoretical studies of neural circuits [44, 54, 55]. These findings from modeling studies support the idea that emergence of chimera state can indeed be observable in actual neural circuits at the levels of cognitive and functional organizations [56]. In this work, we go a step further in the study and understanding of the appearance of chimera states in neural circuits by introducing another biologically relevant condition, that is the existence of a hybrid synaptic architecture for interneuronal communication. As is well-known from experimental findings, two main types of synapses, namely electrical and chemical ones, take part in synaptic transmission and neuron-to-neuron communication [57]. At an electrical synapse, intercellular channels build a physical connection between cells, called gap junctions, and the signal transmission occurs through these channels directly from one neuron to another bidirectionally. However, information transfer across a chemical synapse take place unidirectionally from preto postsynaptic cell with complex biophysical mechanisms driving the dynamics of excitatory or inhibitory neurotransmitter particles released from presynaptic side which move across the synaptic cleft and activate receptor proteins on the postsynaptic neuron [58]. There have been a large number of works 3 revealing the presence of electrical synapses in different regions of the brain, such as the inferior olive [59], locus coeruleus [60], hypothalamus [61] and spinal cord [62]. On the other hand, chemical synapses are also common through the nervous system [63], and they are extensively found in different regions of cortex, hippocampus and olfactory bulb [64–66]. Nevertheless, under the light of recent reports, it is now known that electrical and chemical synapses coexist in mammalian brain structures. Principal findings from neuroimaging and electrophysiological studies have showed that both forms of transmission can be simultaneously found at the same functional neural circuit, including retina [67], neocortex [68] and spinal cord [69]. So far, generic chimera studies concerning neuron populations have considered network connectivity formed with either solely electrical or chemical synapses. Exceptionally, there only recently appeared a few studies investigating effect of their coexistence on the emergence of chimeric behaviors in networks of networks. In such studies, neurons communicate with each other via one synapse type within a given network and via another type across different networks. For instance, Hizanidis et al. studied chimera states in modular neural networks and showed that chimera-like states spontaneously emerge with a suitable tuning of electrical and chemical coupling strengths within populations and across them, respectively [70]. On the other hand, Majhi et al. recently analyzed the chimera states in a two-layer neural network where connections between neurons are established via electrical synapses in one layer and chemical ones across the other target layer, and demonstrated that the emergence of chimera states depends significantly on coupling strengths of chemical synapses but poorly on the electrical ones [71]. However, these modeling approaches are not sufficient to address aforementioned biological reality, since hybrid synaptic connectivity is considered with lack of physiological findings. In the literature, to our knowledge, no attempts have been made to examine population behavior under consideration of hybridness associated with the connectivity in the same neural medium, except only one recent study carried out to assess the emergence of chimera state in a local community [72]. Although it is evaluated to be more reasonable to consider such a hybrid connectivity, this study has concentrated on only preliminary biophysical relevance, considering just locally electrical synapses as well as nonlocal chemical connections and Hindmarsh-Rose polynomial neuron model as in above mentioned previous works. Thus, it is worth looking at this subject from a wider perspective. In order to analyze in deep the role of coexisting chemical and electrical synapse populations for the emergence of chimera state, and to ensure more relevant and realistic assumptions, 4 we here consider a modeling strategy for comprehensibility, such that chemical synapses are more common within the same neural circuitry and synapses from nearest neighbor neurons are of electrical type, whereas farther ones are of chemical type in a nonlocal network. Our main contribution in this work is to analyze emergence of chimera state in more physiological Morris-Lecar neuron populations coupled by abundant hybrid connections. We show that chemical synapses are essential for chimera-like behaviors whereas electrical ones are surprisingly a key component for emergence of new intriguing behavior in hybrid coupled network, namely chaotic amplitude chimera. The rest of the paper is organized as follows: In the next section, we introduce the neural population model, that is a set of N= 1000 spiking Morris-Lecar neurons which are electrically and excitatory-chemically coupled in a nonlocal network, and the method used to characterize chimeric behavior, i.e. mean firing frequency. In the results section, we will first investigate how critical is the role that the relative number of each synapse type can play for the emergence of chimera state with given synaptic strengths. It is obvious that among different system features affecting the possible emergence of chimeric behaviors, synaptic coupling strength is one of the most significant factors in interneuronal communication since it dramatically affects the dynamics of the population. Consequently, as a next step, we will explore the influence of coupling strengths on the appearance of chimera-like states with a controlled variation for electrical and chemical synapses. After that, we also analyze the emergent intriguing chaotic behavior caused by the presence of hybrid synaptic interactions. Finally, our main findings and analysis are summarized in the conclusion section. II. MODELS AND METHODS Lets consider a network of coupled neurons placed in the nodes of a ring as it is depicted in Fig. 1. The dynamics of the membrane potential of each neuron in the network is modeled using the two-variable Morris-Lecar equations [73–76]: CdVi dt =I0+gCam∞ i(ECa −Vi) + gKwi(EK−Vi) + gL(EL−Vi) + Isyn i(1) dwi dt =φ(w∞ i−wi) cosh Vi−βw 2γw(2) m∞ i(Vi) = 0.51 + tanh Vi−βm γm (3) w∞ i(Vi) = 0.51 + tanh Vi−βw γw,(4) 5 Figure 1: Figure shows the scheme of nonlocal hybrid connectivity used in the present study. In the plotted example there are 13 neurons in the network in the form of a ring where each one is connected to R= 2 neighbors via electrical synapses (red solid lines) and to S= 3 neighbors via chemical connections (blue dashed lines). where i= 1,2,...,N denotes the neuron index. Viand wirepresent the membrane potential and activation dynamics of potassium channels for neuron i, respectively. I0is a constant bias current externally applied to all neurons in the network, which is fixed to I0= 10 µA/cm2 providing regularly spiking individual cells in the population. The parameters w∞ iand m∞ i are the steady-state functions of activated potassium and calcium channels, respectively. The constants gCa = 1 mS/cm2,gK= 2 mS/cm2and gL= 0.5mS/cm2are maximal conductance values for calcium, potassium and leak channels, respectively. Accordingly, ECa = 100 mV, EK=−70 mV and EL=−50 mV represent the corresponding ionic equilibrium potentials. Other system parameters are set as C= 1 µF/cm2(the cell membrane capacitance), φ= 1/3, βm=−1mV,γm= 15 mV,βw= 10 mV and γw= 14.5mV. In Eq. (1), Isyn idenotes the total synaptic current received by neuron ifrom its neighbors in the ring. In order to connect neurons, we consider here a hybrid coupling scheme with electrical and chemical synapses incorporated into a nonlocal network as shown in Fig. 1. More precisely, we consider that each neuron in such networked ring is electrically connected with its 2Rnearest neighbors neurons in the ring and excitatory chemically coupled with 2S more distant neurons. This strategy results in totally 2(R+S) connections for each neuron coupled electrically to Rand chemically to Sneighbors in both directions as illustrated in Fig. 1. Then, the total synaptic current a neuron is receiving from its neighbors can be written as Isyn i=IE i+IC iwith IE i=1 2R j=i+R X j=i−R ge(Vj−Vi) (5) 6 IC i= j=i+R+S, j6=i+R X j=i−R−S, j6=i−R gcyj(6) where geis the electrical coupling strength and gcis the maximum postsynaptic current which can be generated at the synapse by activating all synaptic resources. When a spike arrives at a chemical synapse jat time t, there is an instantaneous release of a fraction uj= 0.9 of neurotransmitter resources that then becomes active to transmit the spike. Active resources, namely yj(t),then deactivate over a time on the order of a few milliseconds, characterized by the time constant τin. We fixed it as τin = 10 ms for whole subsequent study, which is within the physiological range for excitatory synapses [77–79]. Using standard synaptic transmission modeling [80, 81], we assume that the dynamical behavior of the fraction of active neurotransmitter resources yj(t) is governed by the following dynamics: dyj dt =−yj τin +ujδ(t−tAP j) (7) where the delta function refers to the arrival time of a spike at synapse jat t=tAP j, which is defined by the upward crossing of the membrane potential past a threshold of 10 mV . To quantitatively determine the population activity behavior and characterize the existence of chimera states, in the following we will monitor the behavior of the mean firing frequency of all neurons in the ring, which is defined as fi=Fi/∆Tfor any given parameter set. Here, Fiis the number of spikes fired by neuron iwithin a period of time ∆Tcomputed after a sufficient transient time. The initial conditions for Eqs. (1-7) are randomly selected with uniform probability within fixed intervals of (−40 mV,30 mV) for Vi, (0,0.4) for wi and (0,1) for yi. Numerical integration of our system is performed using the fourth-order Runge-Kutta algorithm with a fixed time step of 10 µs. III. RESULTS In the following, we systematically investigate the emergent dynamical behaviors, especially chimera-like states, in hybrid coupled spiking neural populations as described in the previous section. As a first step, we begin by demonstrating the appearance of several distinct population behaviors when the chemical connection density Sis varied for a particular fixed number of electrical connections (that we set to R= 100) with maximal conductances for electrical and chemical synaptic current being, respectively, ge= 10−7mS/cm2 and gc= 10−2mS/cm2. The corresponding obtained results are illustrated in Fig. 2 where 7 A −40 30 Vi 1 500 1000 70 76 110 i fi −40 0 40 0 0.5 V W B −40 30 Vi 1 500 1000 70 80 110 i fi −40 0 40 0 0.5 V W C −40 30 Vi 1 500 1000 90 100 i fi −40 0 40 0 0.5 V W D −40 30 Vi 1 500 1000 70 102 110 i fi −40 0 40 0 0.5 V W Figure 2: Emergence of different dynamical behaviors in a hybrid coupled neural population as described in Fig. 1 with variation of the number Sof chemical connections with a fixed number of electrical connections R. Each row shows spatiotemporal activity patterns, snapshots of membrane potentials, mean firing frequency profiles, and periodic orbits with instantaneous positions of two neighboring neurons in the networked ring (marked with red and green arrows) and one distant neuron (marked with blue arrow) on V−wphase plane, respectively. Number of chemical connections are set as S= 5 (A), S= 125 (B), S= 250 (C) and S= 350 (D). Other system parameters are fixed as gc= 10−2mS/cm2,ge= 10−7mS/cm2and R= 100. panels in each column, from top to bottom, show spatiotemporal patterns, snapshots of membrane potentials, mean firing frequencies, and periodic orbits with instantaneous positions (marked with colored arrows) of selected three neurons projected on V-wphase plane, respectively. By visual inspection of the spatiotemporal patterns and membrane potential snapshots, it is obvious that hybrid coupled population exhibits four different dynamical behaviors as Sincreases. First one is the incoherent state where neurons fire independently. Each neuron evolves with regard to its initial position in parameter space without waiting any response from neighboring neurons. Second behavior is the intriguing activity pattern of traveling wave, which consists of spatially coherent oscillations that propagate progressively 8 across the population. This type of activity widely occurs in different oscillatory brain states and under different sensory conditions, and is associated with transmission of neural information across different functional brain regions, for example, during propagation of theta and alpha band rhythms [82] and spread of epileptic seizures [83]. Next, third one is the intriguing population behavior of chimera state. This state describes the occurrence of synchronous and asynchronous electrical activity in the same functional healthy or diseased brain regions [84]. Finally, the fourth population behavior corresponds to a coherent state where neurons fire in a synchronous and phase-locked manner. This last behavior is widely assumed to be a critical mechanism for various vital functions of nervous system, such as information processing and transmission [85, 86], movement control [87] and many other different cognitive or behavioral tasks [88]. To quantitatively characterize these different behaviors, we compute mean firing frequencies of individual neurons in the population as shown in third panels of each column in Fig. 2. We observe that all neurons, for a given population state, fire at a constant frequency, except for chimera state which has a characteristic bell-shaped mean firing frequency profile indicating the coexistence of two different subpopulations, coherent and incoherent, within the same network. It is also worth to note that mean firing frequency of the hybrid coupled population increases with Sregardless of the existing dynamical state in which the system operate. For a more clear understanding of the above-mentioned emergent behaviors, we also perform a phase plane analysis (for each one of the illustrated cases) of the activity trajectories of three particular neurons from the population, which are selected as two neighbors i= 1,2 in the networked ring and a distant neuron i= 200. This is depicted in the bottom panels of each column of Fig. 2 where it is seen that these three neurons move on a single orbit in incoherent, traveling wave, chimera state and coherent states, respectively when Sis increased. One can easily distinguish these states by following the trajectories of each cell (marked with arrows) in V-wphase plane. Although the phase plane behavior of three neurons in the traveling wave and chimera state seems to be similar, we observe that, in the chimera state, the phase profile of neighboring neurons from coherent group differs from that of the distant one belonging to the incoherent group, in such a way that coherent and incoherent group trajectories move on two different periodic orbits. We see more distinct periodic orbits in the phase plane when additional different neurons from incoherent subpopulation are considered (not shown for simplicity). This is a clear indicator for the presence of coherent 9 chimera emergence has recently been investigated with different hybrid coupling schemes [70–72], such studies are very preliminary and make too simplistic assumptions that provide inadequate results when confronted with actual physiological conditions. With the motivation to provide a deeper understanding for the emergence of chimera states in actual neural systems, we here present a comprehensive analysis of how such intriguing dynamical behavior can emerge in a neural system including hybrid synaptic coupling. We have first reported the occurrence of chimera-like behaviors in a population of MorrisLecar neurons, which are coupled by electrical and chemical synapses in a regular network constituting a hybrid coupling scheme. We have explored how dynamical behavior of system changes as a function of the different features of the hybrid connectivity. In particular, we concentrated our analysis on the role of the connection type densities and coupling strengths of electrical and chemical synapses on emergent behavior. It is shown that hybrid coupled populations exhibit variety of dynamical behaviors as a function of electrical and chemical synapse densities in the network. Our findings reveal that chemical synapses, compared to electrical ones, play more significant roles in determining richness of dynamical behavior of the population. Despite this, we also observed that such behavioral variety can only occur in the presence of relatively weak electrical connections. In fact, when electrical coupling strength increases, population exhibits more synchronized behavior as well as the probability to see chimera-like behaviors dramatically decreases. On the other hand, evaluating the effect of chemical coupling strength on population behavior, we found a different trend when the chemical synapse density increases further. In cases of large chemical synaptic strength, the neural population exhibits a new behavior that we have called chaotic amplitude chimera state which has not been reported before. We also observed a pronounced change in membrane potentials of Morris-Lecar neurons for highly intense chemical interaction within hybrid coupled population in such a way that the coherent regular spiking behavior changes to a coherent bursting state. Since pure chemically connected Morris-Lecar neuron population exhibits incoherent, traveling wave, chimera and coherent (spiking) states, we conclude that presence of electrical connections gives rise the emergence of these new intriguing dynamical states. Understanding the underlying mechanisms of cognitive processes in actual brains is crucial for appropriate design of the artificially intelligent systems. There are many experimental and theoretical findings that have shown the relation between observed dynamical states in this paper and various cognitive processes [90–94]. For instance, synchronization is widely 16 assumed to be a essential mechanism for selective attention [19] and memory processes [95]. Also, it has been shown that traveling waves are closely associated with cognitive processes, ranging from long-term memory consolidation to processing of dynamic visual stimuli [96, 97]. On the other hand, as chimera state is a recently discovered population behavior, the knowledge of its current role in cognitive processing is still lacking. However, chimera state can naturally appear in brain which satisfies the minimal requirements for its emergence. A well-known example is the unihemispheric sleep activity observed in some marine mammals where their half brain exhibits coherent electrical activity while the other half is incoherent [98, 99]. In terms of cognition, chimera state may represent pattern recognition, episodic and spatial memory, similarly to the localized patterns of excitation or “bump states” which can also be interpreted as one of dynamical attractors of the working memory [100, 101]. Moreover, one can associate chimeric behavior with event-related synchronization, task switching or multitasking functional states applied in artificially-intelligent systems [102, 103]. Neural chimera studies may also provide different insights for the understanding of pathological conditions, particularly seizure-related, originating from impairment of balance between synchronous and asynchronous activity. Given the diversity of emergent states reported in our work and the knowledge of the critical conditions for formation and dissolution of chimera states, this knowledge can be useful for an appropriate design of cure strategies of those diseases. For the future studies to investigate chimera state, hybrid coupling concept can be extended to networks including synaptic plasticity with combination of excitatory/inhibitory synapses and gap junctions, and perhaps with different network topologies, i.e. scale-free, small-world and multilayered networks. Acknowledgments MU acknowledges Bulent Ecevit University Research Foundation under Project No. BAP2018-39971044-01. JJT acknowledges the Spanish Ministry for Science and Technology and the “Agencia Espa˜nola de Investigaci´on” (AEI) for financial support under grant FIS2017-84256-P (FEDER funds). AC acknowledges financial support from the Scientific and Technological Research Council of Turkey (TUBITAK) BIDEB-2214/A International Research Fellowship Program, and the hospitality of the Institute Carlos I for Theoretical 17 and Computational Physics at University of Granada. [1] A. A. Nasir, S. Durrani, H. Mehrpouyan, S. D. Blostein, and R. A. 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