Alternating chimeras in networks of ephaptically coupled bursting neurons
AAlternating chimeras in networks of ephaptically coupledbursting neurons
Soumen Majhi and Dibakar Ghosh a) Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B. T. Road,Kolkata-700108, India (Dated: 10 September 2018)
The distinctive phenomenon of chimera state has been explored in neuronalsystems under a variety of different network topologies during the last decade.Nevertheless, in all the works, the neurons are presumed to interact with eachother directly with the help of synapses only. But the influence of ephapticcoupling, particularly magnetic flux across the membrane is mostly unexploredand should essentially be dealt with during the emergence of collective elec-trical activities and propagation of signals among the neurons in a network.Through this article, we report the development of an emerging dynamicalstate, namely, the alternating chimera , in a network of identical neuronal sys-tems induced by an external electromagnetic field. Owing to this interactionscenario, the nonlinear neuronal oscillators are coupled indirectly via electro-magnetic induction with magnetic flux, through which neurons communicatein spite of the absent physical connections among them. The evolution of eachneuron, here, is described by the three-dimensional Hindmarsh-Rose dynam-ics. We demonstrate that the presence of such non-locally and globally inter-acting external environments induce a stationary alternating chimera patternin the ensemble of neurons, whereas in the local coupling limit the networkexhibits transient chimera state whenever the local dynamics of the neuronsis of chaotic square-wave bursting type. For periodic square-wave burstingof the neurons, similar qualitative phenomenon has been witnessed with theexception of the disappearance of cluster states for non-local and global in-teractions. Besides these observations, we advance our work while providingconfirmation of the findings for neuronal ensembles exhibiting plateau burstingdynamics and also put forward the fact that plateau pattern actually favorsthe alternating chimera more than others. These results may deliver betterinterpretations for different aspects of synchronization appearing in networkof neurons through field coupling that also relaxes the prerequisite of synapticconnectivity for realizing chimera state in neuronal networks.PACS numbers: 87.19.Ij, 05.45.Pq, 05.45.Xt, 87.10.-e
Neural networks, among other complex systems, self-organize in such ways thatsynchronous spatiotemporal patterns may appear. Different aspects of synchro-nization are extremely fundamental neural mechanisms. They assist in neuralcommunication, neural plasticity and are important for many cognitive pro-cesses. Chimera-like patterns that deal with co-existence of synchronized andde-synchronized domains in the same system, bear a strong resemblance to sev-eral neuronal developments. The observation of such chimera state in neuronalsystems includes several notable works, all of which contemplate with electri-cal or chemical (or both) synapses as the communicating medium among theneurons. But, the effects of electromagnetic induction must not be neglectedat the time of fluctuation in inter- and extra-cellular ion concentrations. In thiswork, we propose a network model of neurons exposed to external electromag-netic field that in turn is responsible for communication among the neurons, inabsence of any kind of synaptic interactions among them. Noticeably, we en-counter the alternating chimera together with the transient chimera pattern in a) Electronic mail: [email protected] a r X i v : . [ n li n . AO ] S e p the network depending on the coupling radius. The lifetime of transient chimerapatterns may differ for the cases of chaotic and periodic bursting of the localdynamical systems. In addition, cluster states (for chaotic bursting of the neu-rons) along with multicluster oscillation death have been realized in the networkdepending upon the coupling radius and interaction strength. Moreover, theoccurrence of all the dynamical phenomena are confirmed for neuronal networkbased upon plateau bursting dynamics that has been witnessed to broaden thealternating chimera region in the parameter space. I. INTRODUCTION
The brain and nervous system are amazing complex structures whose activities are mod-eled based on the interactions among the neurons. The presence of billions of neurons andthe diverse interaction patterns among them make the cortical network possibly the mostchallenging complex system. Most of the cognitive functions in brain are based on thecoordinated activities in large numbers of neurons distributed over specialized brain areas.Various forms of synchronization in neural oscillations (in low and high frequencies) are quiteessential components for facilitating coordinated activity in the normally functioning brain.On the other hand, in nonlinear dynamics literature, exceptional spatial concurrence of co-herent and incoherent dynamical behaviors arising in network of coupled oscillatory systemsis popularly known as the chimera state . The recognition of such a captivating collectivephenomenon was initiated with Kuramoto’s observation in a nonlocally coupled system ofidentical phase oscillators , since then it has brought a broad research field in the literatureof nonlinear dynamics. It has been well-established that these are not limited to networkof phase oscillators , rather this unique collective state can also appear in a large varietyof other systems including neural networks . Both regular symmetric topology (lo-cal, non-local and global) together with anomalistic interactions ontop of networks have been confronted so far in order to realize chimera-like patterns. Ap-pearance of several variants of the chimera patterns, such as globally clustered chimera ,multi-chimera , traveling chimera , breathing chimera , amplitude mediated chimera ,imperfect chimera , virtual chimera etc. are reported. Besides these, the emergenceof chimeras have been identified in multilayer networks and also been revealedexperimentally .Since the time of its detection, the chimera-like patterns have been strongly connected tovarious neuronal activities, such as the bump states in neural systems , the real worldphenomena of unihemispheric slow-wave sleep of some aquatic animals (e.g. dolphins,eared seals) and of some migrated birds. This also includes various types of pathologicalbrain states such as the Alzheimer’s disease, epilepsy, autism, schizophrenia and braintumors. Due to the existence of such definite correspondences between the chimera-likepatterns and several neuronal evolutions, in recent times there have been efforts tostudy the emergence of such unexpected patterns in neuronal networks with different viewsof interactional form.Nevertheless, the articles noted above deliberately thought of the synapses (electrical orchemical or both) as the only communicating (information transferring) medium in therespective considered neuronal networks. Of course, synapses are essential for neuronalfunctions, that allow a neuron to pass an electrical or chemical signal to another neuron.But, what if the neurons in the network are not coupled through synapses? This is a questionof severe importance and has been a topic of discussion for decades. Researches concerningthis issue attest to the fact that answer to this can be given through the mechanism of the ephaptic communication . It may correspond to the interaction of nerve fibers in contactdue to the exchange of ions between the cells. On the other hand, it refers to the couplingof nerve fibers because of the extracellular local electric fields. Illustrative discussions exist on the issue of non-synaptic ephaptic communication among neurons within the ner-vous system based on the latter one, that essentially differs from the direct communicationprocesses through electrical and chemical synapses. These articles indicate that the externalfields carried by the extracellular medium do plenty of works and that they may, actually,represent an additional form of neuronal communication.Moreover, recent researches suggest that the modeling of neuronal networksas mere an ensemble of physical connections (biochemical or electrical) among neurons isincomplete unless it also includes effusive extra-neuronal phenomena such as electric andmagnetic fields. Indeed, the dynamics of the neuronal activity may get affected because ofthe fluctuation in the inter- and extra- cellular ion concentrations. Consequently, internalvariation in the electromagnetic field may develop and hence the influence of magnetic fluxacross the membrane must be taken into account in order to analyze information transferringand collective electrical activities in neuronal ensembles. Interestingly, the electromagneticfields thus created through neuronal activities can send signals and information to neighbor-ing neurons without following any synaptic information exchange procedures . So, it willbe of great worth studying emergence of diverse collective behaviors arising in network ofneurons communicating through the external electromagnetic fields. Previously, this raisedissue has been dealt with in a few recent works on the basis of origination of possible col-lective synchrony amongst a few neurons . As far as the dynamical consequences in anetwork of indirectly coupled neurons is concerned, one of the previous works suggest thatthe external common noisy field can have a positive influence in the context of inducing orenhancing synchronization among the neurons. Neuron’s membrane potential stimulation can also be a good option regarding this issue of bringing synchrony. The works un-locked the appearance of chimera states in network of uncoupled neurons induced by amultilayer formalism, of course, dealing with synapses (electrical and chemical) after all.At the same time we would like to note that previously, alternating chimeras in whichcoherent and incoherent domains alternate their spatial positions and hence in the levelof synchrony over time, have been realized in only a few works. For instance, in time-static networks of two phase oscillator populations with time delays and another onein time-varying network , alternating chimera has been observed. This pattern is recog-nized in an oscillatory medium with nonlinear uniform global coupling . But this peculiartemporal alternating nature of chimera state that explains the phenomenon of dynamicunihemispheric alternating sleep the best, is still mostly unexplored and yet to be givenits due attention. In the present work, we unravel this unique dynamical phenomenon ofthe alternating chimera patterns in a neuronal network with the neurons interacting viaexternal electromagnetic field. Communication among the neurons through this indirectephaptic formalism can be modeled locally, non-locally or globally. We have gone throughall these topologies while considering three-dimensional Hindmarsh-Rose models as the lo-cal dynamical systems. Chaotic square-wave bursting dynamics of the systems under theabove explained configuration with non-local or global limit has been witnessed to pro-duce stationary alternating chimera patterns as a link between incoherence and coherent(or cluster) states followed by the multi-cluster oscillation death states. In the limit oflocal coupling, transient chimera (leading to fully disordered state over time) is also real-ized. Whenever the dynamical units of the network follow periodic bursting, the transientchimera patterns may be observed to have higher lifetimes but the network does not gothrough any cluster-like states. Furthermore, prominence in our results is substantiatedby providing evidence in case of plateau bursting of the neurons, which makes the pro-posed mechanism of ephaptic communication for realizing the alternating chimera pattern,quite general. Significantly enough, the proposed mechanism thus softens the fundamentalsynaptic connectivity requirement for realizing chimera state in neuronal networks.The remaining part of this paper is organized as follows. In Sec. II, we discuss themathematical model of the network considered here. Sec. IIIA illustrates the case ofemergence of the stationary alternating chimera in interacting chaotic bursting Hindmarsh-Rose models. In Sec. IIIB, we report how alternating (and transient) chimera may appearfor periodic bursting of the neurons in the network followed by a discussion on confirmationof the obtained results for neurons possessing plateau bursting dynamics in Sec. IIIC.Finally, Sec. IV provides the concluding remarks. II. MATHEMATICAL MODEL
This section is devoted to the mathematical description of the considered neuronalnetwork model. The effect of external electromagnetic field through the introduction ofmagnetic flux is presented, where Hindmarsh-Rose dynamics is used to cast each node, asfollows: ˙ x i = y i + bx i − ax i − z i + I − (cid:15)ρ ( φ i ) x i , ˙ y i = α − dx i − y i , ˙ z i = c [ s ( x i − e ) − z i ] , ˙ φ i = − k φ i + k x i + j = i + P (cid:80) j = i − P ( φ j − φ i ) , (1)where ( x i , y i , z i ) ( i = 1 , , · · · , N ) represent the state vectors for the nodes, particularlyvariables x i represent the membrane potentials of the neurons and the variables y i and z i correspond to the transport of ions across the membrane via the fast (associated toNa + or K + ) and slow (associated to Ca ) channels. Here N is the number of neuronsin the network and the parameters b = 3 . a = 1 . α = 1 . d = 5 . s = 4 . e = − . c = 0 .
005 are chosen so that with the externalforcing current I = 3 .
25 and I = 1 .
90, the individual neurons display multi-scale chaoticbursting and periodic bursting respectively. Neuronal bursting is extremely important forneuronal communication, particularly for motor pattern generation, synchronization etc.that facilitates neuro-transmitter release, also helps in overcoming synaptic transmissionfailure.As pointed out above, due to the fluctuation in ion concentrations, electromagnetic in-duction may affect electrical activities of the neurons. In fact, density of magnetic fluxacross membrane gets changed when the neurons are exposed to electromagnetic field thatassists in communication and is also capable of defining the memory effect in neurons. Inthis connection, we should mention that the concept of time delay can be used to explaineffect of memory (while making the system infinite dimensional) in neurons as well. Butmagnetic field can also be very efficient to illustrate the impact of memory in neurons whilegenerating a proper spatial distribution and consequently the fluctuation of electromagneticfield makes information exchange possible. Furthermore, neurons are also considered asintelligent circuits dealing with complex signals in the nervous system. So the function-ality of a proper memristive system can resemble the synaptic interactions in neuronalnetworks and memristors are nonlinear electrical components regulating the current flowin a circuit that links electric charge and magnetic flux and remembers the amount ofcharge that has previously flowed through it. Thus as far as the communication among theneurons is concerned, here (cf. Eq. (1)) φ i defines the magnetic flux across the membraneand ρ ( φ i ) describes the memory conductance (memductance) of a magnetic flux-controlledmemristor that governs the coupling between the flux and membrane potential of neuronswith (cid:15) , k being the strengths controlling the interaction. This memductance is oftenrepresented as ρ ( φ i ) = β + 3 β φ i , (2) β , β being fixed parameters. In fact, through the final term of the fourth equation (cf.Eq. (1)) the magnetic contribution of other j = i − P, i − P + 1 , · · · , i + P neurons to the i -th neuron is described and hence via the magnetic flux φ i , the i -th neuron of the ensemblein turn interacts with P neurons on both sides of an one dimensional ring that signifies anindirect non-local formalism with periodic boundary conditions. The parameter r = P/N is usually termed as the coupling radius in the literature. Further, we have taken k = 0 . k = 0 . and β = 0 . β = 0 .
02, throughout this work. −2−1012 x x (a)(b) FIG. 1. (a) Chaotic square-wave bursting for I = 3 .
25, (b) periodic square-wave bursting (doublet)dynamics for I = 1 .
90, in one of the isolated neurons.
III. RESULTS
Whenever there is no interaction among the neurons (i.e., for (cid:15) = 0), with I = 3 . I = 1 . , all the neurons display periodicbursting, as in Fig.1(b). Here we note that in systems having directly interacting dynamicalunits with P nearest neighbors on each side of a ring, in order to isolate the units one canset either the coupling strength (cid:15) = 0 or coupling range P = 0. But here, in our model (Eq.1), the units (the neurons) do not really interact directly through the membrane potentials x i . Rather, they communicate indirectly via different variables representing magnetic flux φ i across the membrane. Here (cid:15) and k are the two parameters that respectively governsthe mutual effects between membrane potential and magnetic flux and so even if one sets P = 0 (or equivalently r = 0), there would be terms remaining in the system that mayaffect the original uncoupled dynamics of the neurons because of their correlations to themagnetic flux. But still the neurons remain completely uncoupled with each other as theassociated fields are not interacting then.Let us now start by looking at the dynamical behavior of the neurons by changing theinteraction strength (cid:15) . For this, we will primarily concentrate on chaotic square-wave burst-ing dynamics of the neurons in subsection IIIA followed by its periodic counterpart in IIIBand finally plateau bursting in IIIC. A. Chaotic square-wave bursting of the neurons
Initially for a fixed coupling radius r = 0 . (cid:15) turned on, the networked system (Eq. (1)) remains in incoherent (desynchronized)state until (cid:15) reaches (cid:15) = 0 . . Beyond this value of (cid:15) , the network starts realizing chimerapatterns, as shown by the snapshots of the membrane potentials in Fig. 2 where theinteraction strength is (cid:15) = 0 .
5. Snapshot of the membrane potentials x i are shown in Figs.2(a), (b) and (c) respectively at three different times t = 1600, t = 1700 and t = 1800. Thepresence of two spatially coherent domains mediated by an incoherent domain is conspicuous x i x i x i ω i ω i ω i (a) (b) (c)(f)(e)(d) FIG. 2. Snapshot of the membrane potentials x i depicting chimera state at (a) t = 1600, (b) t = 1700 and (c) t = 1800. The corresponding instantaneous angular frequencies ω i are respectivelyshown in (d), (e) and (f). Here r = 0 . from Fig. 2(a). But these locations of the coherent and incoherent domains get interchangedin the next snapshot (cf. Fig. 2(b)). However, a similar spatial arrangement in the snapshotas in Fig. 2(a) is again observed in Fig. 2(c). This indicates that there may be some sortof periodic alteration in the coherent (incoherent) domain formation of the system overtime, readily signifying an alternating nature of the chimera profile. Before going into thedetailed understanding of this behavior, we firstly validate the appearance of this chimerapattern by computing instantaneous angular frequencies ω i of all the neurons as ω i = ˙ ψ i = x i ˙ y i − ˙ x i y i x i + y i , (3)where ψ i ( t ) = tan − [ y i ( t ) x i ( t ) ] is the geometric phase associated to the fast variables x i and y i of the i -th neuron, which is a good approximation as long as c is small ( << ω i corresponding to the snapshots of Figs. 2(a), 2(b) and2(c) are respectively plotted in Figs. 2(d), 2(e) and 2(f). This makes the coexistence ofcoherence and incoherence quite clear in which coherent cluster has the same ω i whereas inincoherent domain neurons possess different frequencies.But these snapshots portray the chimera profiles only at fixed time instants. So, fur-ther in pursuance of characterizing the chimera patterns obtained through the numericalexperiments in more detail, we perform the analysis of temporal evolutions of the spatialdomains possessed by coherent oscillators in terms of local curvature . Local curvature ateach point in space is computed while operating discrete Laplacian l i on each snapshot ofthe membrane potentials x i ( i = 1 , , ..., N ). For instance, l i (and hence L i ) applied on asnapshot of x i at time t is defined as L i ( t ) = | l i ( t ) | = | x i +1 ( t ) + x i − ( t ) − x i ( t ) | , (4)smooth profile of which represents spatial coherence and significant non-zero curvature por-traying incoherence. Now due to the dependence of amount of coherence on the individualdynamical systems and since this coherence may not be absolute complete synchrony, soone needs to go through the further estimation procedure by defining some threshold (say, δ ) based upon the maximal curvature possessed by the system. Wherefore, we define thefunction f : L → { , } where L ≡ ( L , L , ..., L N ), such that f ( L i ) = (cid:26) , if L i ≤ δ , otherwise . (5)The spatial correlation measure C sp ( t ) is thus described as C sp ( t ) = N N (cid:80) i =1 f ( L i ) . (6) C sp ( t ) basically reflects the spatial extent of coherence exhibited by the networked systemat the time instant t (with the threshold δ being around 1% of the maximal value of L i ). Particularly, C sp ( t ) = 0 refers to the situation in which none of the L i ’s are zerodepicting desynchronized dynamical behavior. Unit value of C sp ( t ) resembles the coherentstate whereas the range 0 < C sp ( t ) < C sp ( t ) is, in general, time-dependent. Static coherence in the chimeric patternwill then be identified by a non-zero constant C sp ( t ) while its (non-zero, non-unit) time-varying character signifies the existence of non-static chimera.Besides this spatial correlation measure C sp ( t ), in order to quantify the extent of time-correlated nodes in the network, we will be calculating the time-correlation measure C tm inthe following way C tm = (cid:115) N ( N − N (cid:80) i,j =1( i (cid:54) = j ) g ( | σ ij | ) . (7)Here σ corresponds to the time-correlation coefficient and the function g of | σ | is defined as g ( | σ ij | ) = (cid:26) , if | σ ij | > δ , otherwise . (8)In fact, for two membrane potentials (time-series) x i and x j with m i , m j and s i , s j re-spectively being their temporal means and standard deviations, the pairwise correlationcoefficients σ ij are defined as σ ij = (cid:104) ( x i − m i )( x j − m j ) (cid:105) s i s j . (9)Then, x i and x j are linearly time-correlated whenever σ ij (cid:39) σ ij (cid:39) − C tm serves as a time-correlation measure. We have chosen the values δ = 0 .
04 and δ = 0 .
90 which are sufficient here in order to discriminate respectively thespatially correlated and time-correlated units.As far as the evolution of this correlation measure recognizing different dynamical be-haviors is concerned, the value of C tm must be non-zero in the regime of any non-transientchimera no matter which time-span is taken into account for the calculation of temporalmeans and standard deviations. Of course, one should define δ depending on the level ofcoherence observed in the system. It does not necessarily reflect the size of coherent do-main, rather for non-static (alternating) coherent clusters its values are smaller than C sp ( t ).However, for the static chimeras where no coherence is present in the incoherent domain, C tm yields same values as C sp ( t ). In brief, for the system in the stationary (non-transient)alternating chimera state, one has 0 < C sp ( t ) < , ∀ t with an oscillatory behavior in C sp ( t )having C tm >
0. On the other hand, the scenario: ∃ T : 0 < C sp ( t ) < , ∀ t < T and C sp ( t ) = 0 or C sp ( t ) = 1 , ∀ t ≥ T is characterized as the transient chimera. Regular repet-itive variations of C sp ( t ) may also signify the breathing or traveling chimera like states.But in our case, the clear occurrence of periodically alternating coherent and incoherentdomains (cf. Fig. 2 and Fig. 3 ) in the chimera pattern readily implies the emergence ofalternating chimera states .For a better perception on the dynamics of the system while exhibiting the chimera state,we plot the spatiotemporal evolution ( t ∈ [2000 , x i (at time t = 1600) as in Fig.2(a) and obtain the FIG. 3. (a) Spatio-temporal plot associated to the chimeric evolution in Fig. 2; Absolute values L i of the local curvature obtained through discrete Laplacian l i on the data of Fig. 2(a) and (b)are respectively shown in (b) and (c) here. C s p ( t ) , C t m R ( t ) C sp C tm (b)(a) FIG. 4. (a) Spatial correlation measure C sp ( t ) as a function of time and the time-correlation mea-sure C tm calculated over time interval t ∈ [2000 , R ( t ) characterizingalternating nature of the chimera state. profile of absolute value L i of the local curvature, shown in Fig. 3(b). The profile illustratesthe spatial concurrence of coherence and incoherence from another aspect here. Similarly, L i applied on the snapshot at t = 1700 (cf. Fig. 2(b)) is figured out in Fig. 3(c) the profileof which is having a contrasting nature to the previous one.Figure 4(a) shows the spatial correlation measure C sp ( t ) as a function of time ( t ∈ [2000 , C sp ( t ) over time can be easily observed and this variation elaborates the emergence of the“alternating chimera” pattern in the ephaptically coupled neuronal network. The time cor-relation measure C tm calculated over the entire time interval t ∈ [2000 , C tm (cid:39) .
18 which is also shown in Fig. 4(a) implying that the alternating chimera isof stationary character. For further validation of the observed result on the alternatingchimera, we compute the order parameter R ( t ) as R ( t ) = | N N (cid:80) j =1 e iψ j | , (10)where i = √− ψ j ( t ) is calculated in ( x j , y j ) plane as ψ j ( t ) = tan − [ y j ( t ) x j ( t ) ]for j = 1 , , ..., N . This order parameter R ( t ) basically quantifies the level of synchronypresent in the system at the time instant t . Looking at the oscillating C sp ( t ) characterizingthe alternating chimera, one expects a similar kind of behavior in the order parameter R ( t )as well. Figure 4(b) depicts the variation in R ( t ) with respect to time, that oscillates andconsequently the chimera breathes. Regarding this nature of R ( t ), the oscillating behaviorin the order parameter is also observed previously in networks of phase oscillators withtwo small populations . Formation and destruction of traveling fronts in periodicallymodulated neural field model could also lead to oscillations in the order parameter. But,here, the regular alternating coexistence of coherence and incoherence is quite conspicuousfrom Fig. 2 and Fig. 3 (b, c) that demonstrate the origination of alternating chimerapatterns which is valid even for large network sizes (cf. Appendix section).So far, we have concentrated on the dynamical behavior of the non-locally coupled neu-ronal network for a single value of P = 30 (i.e., r = 0 .
3) and we have found persistent(stationary) alternating chimera for a fixed interaction strength (cid:15) = 0 .
5. Next we focuson the local (nearest neighbor) configuration of the network in which P = 1 ( r = 0 . (cid:15) restrain the neuronal ensemblein the incoherent (disordered) state. This is true only up to (cid:15) = 2 .
45, beyond which thenetwork starts experiencing chimera-like state. For instance, an exemplary snapshot of themembrane potentials for (cid:15) = 2 .
45 at the time t = 1600 is shown in Fig. 5(a) from whichcoexistence of coherence and incoherence can be realized. But, as one looks into a snapshotat higher time t = 2500, this chimera profile completely dies out and rather the networkbehaves in a disordered fashion, as depicted in Fig. 5(b). This implies the emergence of atypical transient chimera in the neuronal network. This circumstance is then identified bycalculating instantaneous angular frequencies ω i particularly at these times in Figs. 5(c)and (d) respectively. The profiles of ω i are quite self-explanatory as in the chimera state, thecoherent clusters have similar frequencies while possessing no correlations in the incoherentdomains.This impermanence in the chimera state is thereafter justified by plotting the spa-tiotemporal evolutions of the network for two different time ranges t ∈ [1500 , t ∈ [2500 , (cid:15) = 2 .
45. From Fig. 6(a),one can relate the scenario of chimera to the snapshot obtained earlier (cf. Figs. 5(a) and(c)). However, there is no continuity of this phenomenon (of chimera) in the spatio-temporalplot of Fig.6(b) plotted in higher time limit. Moreover, Fig. 6(c) shows variation in thespatial correlation measure C sp ( t ) that decreases over time and eventually happens to bezero. This feature of the transient chimera temporally leading to incoherence is in contrastto the observation reported in in which a transition to full coherence was identified.These observations are further confirmed for larger size of the network, namely N = 500neurons by taking nonlocal ( P = 150) and local ( P = 1) interactions [results are illustratedin Appendix].This way we have realized stationary alternating chimera and transient chimera patternsin the network for r = 0 . r = 0 .
01 respectively, i.e. with non-local and local con-figurations. Now the time is for scrutinizing what is actually happening for other valuesof r and what role is (cid:15) playing as far as the possible emanation of other dynamical statesis concerned. In order to reveal this, we rigorously plot the phase diagram in the (cid:15) − r parameter plane in Fig. 7 while keeping the other parameters fixed as before. In orderto produce this phase diagram, we discriminate different dynamical states in the followingway: for each and every value of (cid:15) and r , first we calculate the ‘ D factor’, from which wecan say whether the system exhibits oscillatory behavior or a steady state. We define the0 x i x i ω i ω i (a) (b)(c) (d) FIG. 5. Snapshot of the membrane potentials x i reflecting (a) chimera and (b) incoherent statesat t = 1600 and t = 2500 respectively. The corresponding instantaneous angular frequencies ω i arerespectively plotted in (c) and (d). The coupling strength (cid:15) = 2 .
45 and r = 0 .
01 are chosen here.FIG. 6. Spatio-temporal evolution over the time range (a) t ∈ [1500 , t ∈ [2500 , (cid:15) = 2 .
45 with r = 0 .
01; (c) Spatial correlation measure C sp ( t ) as a function of time over theinterval t ∈ [1500 , ε MCOD COHCLTICH TC AC
FIG. 7. Two parameter phase diagram in (cid:15) − r plane where brown, blue, magenta, black, greenand red colors respectively correspond to the incoherent (ICH), alternating chimera (AC), cluster(CLT), transient chimera (TC), coherent (COH) and multicluster oscillation death (MCOD) states. factor D as D = 1 N N (cid:88) i =1 Θ (cid:18) T (cid:88) l =1 | x i,t l − x i,t l − | − δ (cid:19) , (11)where x i,t l , x i,t l − are the states of the i th oscillator at time iteration t l and its previousiteration t l − with T sufficiently large time and Θ( x ) being the Heaviside step function. Ifthe system goes to a steady state, value of D will be ‘0’and if there is any sort of oscillation,then D assumes value unity (i.e., ‘1’) for proper choice of δ . Whenever D = 1, thenwe go for computing the values of C sp ( t ) and C tm over a sufficiently high time range. Infact, C sp ( t ) (cid:39) (cid:0) C sp ( t ) (cid:39) (cid:1) represents incoherence (cid:0) coherence (cid:1) , the periodically repeatingvalues with 0 < C sp ( t ) < < C sp ( t ) < C sp ( t ) such that 0 < C sp ( t ) < D = 0 represents oscillation suppression state that happens to bethe multicluster oscillation death state here. We have fixed the value δ = 0 . . ≤ r ≤ .
08 the network retains itsdisordered state even upto (cid:15) = 2 .
45, above which transient chimera may appear (dependingon the value of r ) as a link between incoherence and coherence followed by multiclusteroscillation death (MCOD) state . Although a different picture of transition is observedwith 0 . ≤ r ≤ .
49 for which the alternating chimera states appear quite early at (cid:15) ≥ . (cid:15) = 2 .
60 (depending on r ). Beyond this value, the networkexperiences coherent and MCOD states respectively for increasing (cid:15) . Interestingly, for thecoupling range 0 . < r ≤ .
49, in addition to the earlier noted states, synchronized cluster(CLT) states appear in between the alternating chimera and coherent patterns. But forthe rest of r ∈ [0 . , .
49] (even for global interaction), MCOD states emerge right afterthe appearance of cluster states for increasing (cid:15) . B. Neurons exhibiting periodic square-wave bursting
Whenever neurons in the network are inferred to exhibit regular (periodic) square-wavebursting dynamics (cf. Fig. 1(b)) with the external stimulus I = 1 .
90, a more or lesssimilar qualitative phenomenon has been witnessed. As in the earlier observation for chaoticsquare-wave bursting of the neurons with N = 100, here also for lower coupling radius( r ∈ [0 . , . C sp ( t ) over time t ∈ [3000 , r = 0 .
01 and (cid:15) = 1 .
90. The chimera2 C s p ( t ) C s p ( t ) ε (b)(a) (c) MCODCOHICH TC AC FIG. 8. Spatial correlation measure C sp ( t ) characterizing (a) transient chimera (for r = 0 .
01 and (cid:15) = 1 .
90) and (b) alternating chimera (for r = 0 . (cid:15) = 0 .
45) states; (c) Phase diagram in (cid:15) − r coupling parameter plane where brown, blue, black, green and red colors respectively correspondto the incoherent (ICH), alternating chimera (AC), transient chimera (TC), coherent (COH) andmulticluster oscillation death (MCOD) states. states obtained for non-local coupling ( r ∈ [0 . , . C sp ( t ) profile is shown in Fig. 8(b) for r = 0 . (cid:15) = 0 .
45. In Fig. 8(c), we figure out the transition scenario of the dynamical network forsimultaneous variation in the coupling parameters (cid:15) ∈ [0 ,
3] and r . As pointed out, with thecoupling radius r ∈ [0 . , . r alternating chimera appears. But in contrast to the chaotic bursting case, herethe neurons become directly coherent from the state of alternating chimera as a result ofincreasing coupling strength without experiencing the cluster states and finally reaches theoscillation quenching state (MCOD). C. Plateau bursting of the neurons
In this subsection, we evidence the emergence of alternating chimera states even whenthe neurons in the ensemble ensue a different sort of bursting dynamics, namely plateaubursting. Exemplary time evolution of the membrane potentials corresponding to plateaubursting is shown in Fig. 9(a). In this context, for the local coupling limit, transient chimerapattern arises within a very narrow range of the interaction strength (cid:15) . For instance,with r = 0 .
01 and (cid:15) = 3 .
60, the chimera states observed are essentially short-lived (intime). This dynamical feature of the network is explained in terms of decaying profile ofthe spatial correlation measure C sp ( t ) within the time interval [1500 , r lies in therange [0 . , . (cid:15) ≥ .
05. Periodically pulsating C sp ( t ) ∈ (0 ,
1) profile identifying alternating chimera has been presented in the Fig. 9(c)for t ∈ [6500 , r = 0 . (cid:15) = 0 .
2. As the next step, phase diagram in the (cid:15) − r coupling parameter plane is plotted in Fig. 9(d) for a complete understanding of theimpacts of these parameters on the network. As stated earlier, within the coupling radius0 . ≤ r ≤ .
06, the network realizes chimera pattern which is impermanent and leadsto incoherence over time (cf. black region in the figure). For these coupling radii, withincreasing (cid:15) , the network experiences coherent state followed by MCOD state. For higher3 x C s p ( t ) C s p ( t ) ε MCOD CLTCOHTC (a) (b)(c) (d)
ACICH
FIG. 9. (a) Plateau bursting dynamics of the neurons with I = 3 . b = 2 .
50; Spatial correlationmeasure C sp ( t ) characterizing (b) transient chimera (for r = 0 .
01 and (cid:15) = 3 .
60) and (c) alternatingchimera (for r = 0 . (cid:15) = 0 .
20) states; (d) Phase diagram in the (cid:15) − r parameter plane withbrown, blue, black, magenta, green and red colors respectively depicting the incoherent (ICH),alternating chimera (AC), transient chimera (TC), cluster (CLT), coherent (COH) and multiclusteroscillation death (MCOD) states. coupling radius r ∈ [0 . , . . ≤ (cid:15) ≤ .
50 depending on r . For (cid:15) beyondthis range, cluster synchronization arises in the ensemble similarly as in case of chaoticsquare-wave bursting neurons. Coherent states appear with higher interaction strength (cid:15) followed by the MCOD state. One more important thing this figure explicitly shows is theprolongation of the alternating chimera region in the (cid:15) − r parameter plane. In this sense,the plateau bursts promote alternating chimera pattern more than the other two burstingdynamics considered here. IV. CONCLUSIONS
The concurrence of dynamical coherence and incoherence in a symmetrically configurednetwork has strong resemblance to several neuronal developments and so far networks ofneurons based upon synaptic communication have been treated to demonstrate possibleappearance of such chimera-like states. But synaptic coupling is not the only interactingmedium among the neurons, rather external fields in the form of magnetic flux across themembrane can act as a supplementary mode of information exchange. So in contrast, inthis work we have presented the emergence of chimera patterns in a network of neurons inabsence of any sort of synaptic connectivity among them. Specifically, neurons are assumedto be connected through ephaptic coupling of nerve fibers by virtue of local electric fields.Whenever chaotic bursting dynamics arising from Hindmarsh-Rose neuron model has beenused to cast the local dynamics of the nodes, stationary alternating chimera is realizedfor both non-local and global interactions. However, transient chimeras are identified forlocal coupling. Besides these dynamical states, coherent patterns along with cluster andmulticluster oscillation death states are also observed in the network depending on thevalues of coupling radius and interaction strength. Nevertheless, strong coupling inducesmulticluster oscillation death state irrespective of the value of coupling radius. For periodic4 x i x i ω i ω i C s p ( t ) C s p ( t ) (c) (b)(a) (d)(e) (f) FIG. 10. (a,b) Snapshot of the membrane potentials at t = 1500 showing chimera patterns for P =150, (cid:15) = 0 . P = 1, (cid:15) = 2 .
45 respectively. Corresponding instantaneous angular frequenciesare shown in (c,d). Spatial correlation measure C sp ( t ) reflecting alternating and transient naturesof chimera states are respectively plotted in (e) and (f). bursting of the neurons, the lifetime of the transient chimera states may increase. Moreoverincreasing interaction strength guides the network towards the coherent state directly fromstate of alternating chimera without going through cluster states, if the neurons interactnon-locally or globally. We have validated our results on obtained dynamical phenomena forneurons exhibiting plateau bursting dynamics and also observed that this plateau burstingactually expands the alternating chimera region in the coupling parameter plane quitecomprehensively.We would like to further mention that the alternating chimera patterns in our workare realized in a network of indirectly coupled neuronal systems through electromagneticfield whereas the previous works , on the same phenomena were mainly investigatedin directly connected networks of phase oscillators. Besides, the transient chimera persistirrespective of the neuronal network size and remarkably this transitory feature appears evenin purely locally interacting ensemble. The acquired outcomes thus raise the significance ofephaptic communication in neuronal ensembles from the perspective of several dynamicalconsequences and further relaxes the requirement of synaptic communication in order toexperience chimera patterns in the network. V. APPENDIX
In order to demonstrate that the observed qualitative results on the alternating (non-localinteraction) and transient (local coupling limit) chimeras do not depend on the networksize and persist even for larger networks, we went for analyzing the possible network behav-iors for non-local and local interactions in which the number of neurons in the network is N = 500. For the sake of simplicity, we kept all other parameters fixed as above (in case of N = 100). Figure 10(a) shows snapshot of the membrane potentials x i at t = 1500 that de-picts coexistence of coherence and incoherence for r = 0 . (cid:15) = 0 .
5. On the other hand,5with r = 0 .
002 (local coupling) and (cid:15) = 2 .
45, snapshot at the same time-instant is plottedin Fig. 10(b) that also depicts chimera state. In addition, as a confirmation of these states,we have shown the angular frequencies ω i corresponding to these snapshots in Figs. 10(c)and (d) respectively. To have a perception about the lifetime of these chimera patterns,we further plot the spatial correlation measure C sp ( t ) in Figs. 10(e) and (f). Figure 10(e)shows (periodically) pulsating C sp ( t ) throughout t ∈ [1500 , ACKNOWLEDGMENTS
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