Pattern formation in diffusive excitable systems under magnetic flow effects

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Patterns formation in diffusive excitable systems under magnetic flow effects

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Alain Mvogo

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, Clovis N. Takembo , H.P. Ekobena Fouda , Timoléon C. Kofané

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Laboratory of Biophysics, Department of Physics, Faculty of Science, University of Yaounde I, P.O. Box 812, Cameroon b Laboratory of Mechanics, Department of Physics, Faculty of Science, University of Yaounde I, P.O. Box 812, Cameroon

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Article history: Received 4 March 2017 Received in revised form 11 May 2017 Accepted 12 May 2017 Available online xxxx Communicated by C.R. Doering Keywords: FitzHugh–Nagumo network Electromagnetic induction Memristor coupling Modulational instability Patterns

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We study the spatiotemporal formation of patterns in a diffusive FitzHugh–Nagumo network where the effect of electromagnetic induction has been introduced in the standard mathematical model by using magnetic flux, and the modulation of magnetic flux on membrane potential is realized by using memristor coupling. We use the multi-scale expansion to show that the system equations can be reduced to a single differential-difference nonlinear equation. The linear stability analysis is performed and discussed with emphasis on the impact of magnetic flux. It is observed that the effect of memristor coupling importantly modifies the features of modulational instability. Our analytical results are supported by the numerical experiments, which reveal that the improved model can lead to nonlinear quasi-periodic spatiotemporal patterns with some features of synchronization. It is observed also the generation of pulses and rhythmics behaviors like breathing or swimming which are important in brain researches. © 2017 Elsevier B.V. All rights reserved.

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1. Introduction

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The complexity of the brain makes it the most important system in nature. The brain is made up of a large number of neurons grouped into functional ensembles generally called microcircuits [1]. They have a striking capacity to produce a considerable variety of coordinated patterns in response to their surrounding changes or their behavioral needs. In fact, spiking neurons have attracted the interest because many studies consider this behavior an essential component in biophysics and mathematics information processing in the nerve cell. Excitability is a common property of many physical and biological systems. As now well established, in excitable media nonlinear waves have a great importance for a better understanding of some cooperative behavior including patterns formation and synchronization since such phenomena are related to normal functioning and generation of some neural ailments [2–4]. In general, neuronal systems exemplify the properties of biologically excitable media, although relatively few investigations have been carried out on the initiation, propagation, and pathways of dynamic excitatory waves. The FitzHugh Nagumo (FHN) model is a generic model of excitability and oscillatory dy-

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*

Corresponding author. E-mail address: [email protected] (A. Mvogo).

http://dx.doi.org/10.1016/j.physleta.2017.05.020 0375-9601/© 2017 Elsevier B.V. All rights reserved.

namical behavior. The model has been introduced by FitzHugh [5] with equivalent circuit by Nagumo et al. [6] is a generalization of the Van der Pol oscillator. The FHN mathematical model has been greatly contributing in nonlinear neurodynamics domain and has become prototype model for systems exhibiting excitability. In the past years, mathematical and physical contributions devoted to neurological sciences have improved our understanding of pattern formation using the FHN mathematical model. Just to cite a few, simpler pulse solutions in the discrete FHN model have been constructed asymptotically [7–10]. Taking into account two different time constants, Panfilov and Hogeweg [11] modified the standard FHN model for excitable tissue and showed that a spiral wave can break up into an irregular spatial pattern. Dimitry et al. [12] showed that particle-like behavior can lead to formation of complex periodic and chaotic fractal-like spatiotemporal wave patterns in modified FHN network. Malevanets and Kapral [13] showed that fully developed labyrinthine pattern can be observed in a microscopic reaction model with a FHN mass action law. Very recently, Zhenga and Shena [14] showed that the FHN model has very rich dynamical behaviors, such as spotted, stripe and hexagon patterns. However, the excitability property is much too complex and many factors should be considered as well. According to Faraday´s law of induction, the fluctuation or changes in action potentials in excitable cells (neurons) can generate magnetic field in the media;

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in that sense, the excitability of neurons will be adjusted under feedback effect [15]. Albeit the satisfactory results reported in the above mentioned studies, the appropriate mechanism and the conditions under which spatiotemporal patterns emerge and spread among coupled neurons under electromagnetic induction have not been investigated. Therefore, it is important to set more reliable FHN models, so that the effect of electromagnetic induction could be considered. This constitutes the strong motivation to this Letter. It is well known that modulational instability (MI) is a process closely related to solitons and wave patterns formation in various physical settings, where amplitude and phase modulations tend to grow exponentially due to the simultaneous effects of nonlinearity and dispersion. We show in this work that the effect of electromagnetic induction can enhance spatiotemporal information through the activation of MI. The improved FHN network can generate rhythmics behaviors like breathing or swimming which are important in brain researches [1]. The rest of the Letter is organized as follows: in Section 2, we introduce the model along with the important mathematical developments and we make use of the semi-discrete approximation to show that the equations of the system can fully be described by a single differential-difference nonlinear equation. In Section 3, the MI of plane wave solution is performed and we discuss the possibility of common regions of instability with emphasis of memristor coupling. In order to confirm our analytical predictions, we perform the numerical simulations from the generic equations of the model. Section 4 concludes the paper.

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2. Model equation and asymptotic expansion

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It is well known that the standard FHN model is described by two variables v (t ) and w (t ), which represent the trans-membrane potential and the slow variable for current, respectively. In this Letter, we introduce the magnetic flux variable φ(t ), which is used to describe the effect of electromagnetic induction. The equations for an improved FHN model for N = 400 identical neurons mutually coupled to their nearest neighbors through the gap junction is now made of three ordinary differential equations for the dimensionless variable v (t ), w (t ) and φ(t ) as follows:

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dv n

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dt

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dw n

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dt dφn

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dt

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= K ( v n+1 − 2v n + v n−1 ) + v n ( v n − a)(2 − v n )

= λ( v n − bw n ),

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the group velocity dispersion. μ is a small deviation from the natural frequency 0 . It is therefore clear that for = 0, the frequency  reduces to the natural frequency 0 of the system. Equation (1) can be summarized in terms of the state vector U n (t ) = { v n (t ), w n (t ), φn (t )}, whose unperturbed expressions can be taken in the generalized form

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 U n (t ) =

=

∂q ∂ is the group velocity and 2C g

=

dUˆ ()e

i (qn+)

(2)

,

ˆ ˆ (), φ()} where Uˆ () = { vˆ (), w . With the expanded ω and q along with change of variables τn = (t + n/ V g ) and ζn = 2 n and the condition C g = 1, the generalized trial solutions take the form (3)

where A (n, t ) = e . The method introduces a new lattice number m to support a large grid [19]. It follows that for a given lattice number n, only the set of lattice points ..., n − N , n, n + N , ... can be indexed in terms of the slow variable m as {..., (n − N ) → (m − 1), n → m, (n + N ) → (m + 1)..., }, where 2 = 1/ N is assumed due to highly pronounced discreteness effects. In so doing, the slow modulation S (ζn , τn ) of the plane wave A (n, t ) can be replaced by the functions U (m, τ ), with τ = τn , and one can easily make use of the Fourier series in power of the parameter ∞ 



p

p 

(4)

l=− p

η10 (m, τ ) = ψ10 (m, τ ) = φ10 (m, τ ) = 0.

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(i  − 2K (cos(q) − 1) + 2a + αk1 )(i  + λb) + λ (i  + k2 ) = 0 (6)

should be satisfied for the system to admit non-trivial solutions in the form

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τ ) = η(m, τ ),

λ τ) = η(m, τ ), i  + λb η(m, τ ) φ11 (m, τ ) = . (i  + k 2 )

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ψ11 (m,

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For l = 1 the dispersion relation

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solutions into Eq. (1) leads to a set of coupled equations to be solved at different orders of the small parameter , with the corresponding harmonics l. For the leading order (1, l), with l = 0, we obtain the solution

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U lp (n, t ).

l l ∗ From (4), we have U − p (m, τ ) = (U p (m, τ )) . Inserting the above



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U n (t ) = A (n, t )U (ζn , τn ),

(1)

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with n = 1, 2, ..., N. The parameter K is the coupling parameter between cells, while the parameter λ represents the ratio of the time scales for v n (t ) and w n (t ). The function ρ (φn ) = α + 3βφn2 is the conductance developed from memristor [16,17] and used for memory associated with magnetic field. According to Faraday’s law of electromagnetic induction and description about memristor [18], the term k1 ρ (φn ) v n could be regarded as additive induction current on the membrane. φext is the external electromagnetic radiation which for simplicity is taken as a periodical function φext = A cos(2π f t ). The ion currents of sodium, potassium contribute the membrane potential and also the magnetic flux across the membrane; thus, a negative feedback term −k2 φn has been introduced in the third equation of (1). I ext represents the external forcing current. The parameter values used in this work are: a = 0.3, b = 0.5, λ = 0.01, k2 = 1, α = 0.1 and β = 0.02. The parameters K , k1 , I ext and φext will be selected so to display formation of complex patterns of the action potential.

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Nonlinear equations are, owing to their complexity, typically not accessible to an analytic approach. Sometimes, the analysis involve the use of asymptotic expansions such as the multiple-scale expansion, the Fourier series expansion and the semi-discrete approximation, just to cite a few. In this Letter, we use the multiple scale analysis expansions, which implies that the first node of the network, i.e., n = 0, is excited at the natural frequency 0 . Due to nonlinear effects, that naturally affect real systems, the natural frequency will deviate, with actual frequency  and wave numμ ber q, to become  = 0 + μ and q = K + V + 2 C g μ2 + ...,

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The order (2, l), when l = 0, has the solutions

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η20 (m, τ ) = ψ20 (m,

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|η(m, τ )|2 .

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sin(q)

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η21 (m, τ ) = δ(m, τ ),

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where δ(m, τ ) is an arbitrary function. At the same order, but for l = 2, the solutions are derived in the form

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(11)

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where

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 = (2i  − 2K (cos 2q − 1) + 2a + αk1 )(2i  + λb) + λ = 0. Finally, we solve the system for η31 (m,τ ), ψ31 (m,τ ) and φ31 (m,τ ), which is obtained for p = 3 and l = 1. This yields after making use of the previous solutions, the amplitude equation in ηm in the form

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Fig. 1. The panels display the critical amplitude η0,cr , vs. the wave number q, as given by the instability criterion (20). Panel (a) displays the critical amplitude for k1 = 1.5 and K = 0.05, while panel (b) shows how η0,cr is an increasing function of the memristor coupling parameter k1 for K = 0.5.

μ2 =

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The global form of (12) is now similar to the one obtained by Leon and Manna [19] in the case of electrical transmission lines, according to the same analytical procedure. This equation will be used only for the linear stability analysis, while the direct numerical experiments will be performed on the generic equations (1). 3. Linear stability analysis and spatiotemporal patterns In other to perform the linear stability analysis, we assume a solution of (12) to be of the form ηm (τ ) = η0 e i (ν m−μτ ) , where the complex amplitude and frequency are related to the perturbation wavenumber ν via the nonlinear dispersion relation

if

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sin(ν )].

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The plane-wave solution will be unstable if μ2 < 0. This latter condition depends simultaneously on the signs of the terms R / Q 2K sin(q) sin(ν ). In fact, sin(ν ) should be bounded, then and |η0 |2 − R we fix sin(ν ) = 1 so that

where

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for the solution to be found in the form

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The case l = 1, with a zero determinant, should satisfy the Fredholm condition

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= η02,cr .

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When the above condition is satisfied, the plane wave is unstable under modulation, and wave patterns will be expected in the system. Otherwise, the plane wave will be said to be stable. Along the same line, if QR > 0, instability will manifest if

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wave number q in Fig. 1(a), where the stable and unstable regions are clearly displayed. We take the memristor coupling parameter k1 = 1.5 and the coupling parameter between cells K = 0.1. However, since the main purpose is to bring out the effect of the magnetic flux through the memristor coupling parameter, we plot in Fig. 1(b) the critical amplitude η0,cr for three different values of the memristor coupling parameter k1 , namely k1 = 0.5, k1 = 1.5 and k1 = 2.5. We take the coupling parameter K = 0.1. It is observed that the critical amplitude is an increasing function of k1 , thus showing how the electromagnetic induction could be important in the process of intercellular communication in neural net-

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works via MI activation. Comparing Figs. 1(a) and (b) for k1 = 1.5, it is observed that the increasing of the coupling parameter K also increases the critical amplitude η0,cr . The linear stability analysis does not give any information about the long-time evolution of the modulated waves, since it is mainly based on the linearization around an unperturbed plane wave. We therefore intend to perform full numerical simulations on the generic system (1). Results are obtained via the fourth-order Runge–Kutta computational scheme with periodic boundary conditions and time-step t = 10−2 . The number of FHN neurons forming the network has been chosen in order to avoid wave reflexion, while initial conditions correspond to the combination of the respective solutions used both in section 2 and through the linear stability analysis. The wavenumber q has been chosen in regions of instability as q = 0.15π , while we have considered ν = 0.1π . In the rest of the paper, the memristor coupling is investigated through the parameter k1 and the features of the subsequent waves patterns of the membrane potential. In Fig. 2, we perform the numerical simulations for K = 0.05. That is to say we have considered two nearest neighbors coupling in a weak coupling regime. A weak coupling between neighboring cells is also related with a situation that arises in the study of nonlinear oscillating waves in the β -cell islets of the pancreas, which secrete insulin in response to glucose in the blood [20,21]. In the detected regions of instability, one should expect different behaviors of the system, depending on the value of k1 , which indubitably influences the instability features. Therefore, we can control the feature of pattern through the variation of parameter k1 . Then, we fix all the other parameters and play on the values of k1 for the suitable phenomena of memristor coupling in pattern formation to be described. It is well known that time and space are of high importance in the process of interneuronal communication, some good examples being visual, sensory motors and olfactory cortices [26]. Therefore, we represent the spatiotemporal dynamics of the membrane potential in Figs. 2 and 3 under low dispersive effect. We use two different values of k1 , namely k1 = 1.5 [Fig. 2] and k1 = 2.5 [Fig. 3]. As one can remark, the system presents multiple states comprised of patterns with different topologies. The obtained patterns confirm our analytical expectations as one observes patterns of waves. This once more confirms the fact that wave patterns are possible in all systems where dispersion and nonlinearity are present. In Fig. 2 that is for k1 = 1.5, we observe the formation of quasi-periodic wave patterns with swimming behavior. As interestingly observed in Fig. 2(b), the X , Z plane presents breathing modes. Such generated rhythmic behaviors like breathing or swimming have been observed in central pattern generators (CPGs) [1, 27,28] and are particularly important in brain researches due to their striking capacity to produce a considerable variety of coordinated patterns in response to their surrounding changes and their behavior needs. Inspired by the fact the CPGs have some similarity and are able to share their morphological dynamical properties, Yuste et al. [29] introduced and developed the idea of considering CPGs as the theoretical framework to understand cortical microcircuits. The oscillations connected to the rate of breathing are very tied to bringing sensory data together as perceptions [30]. In fact, breathing is highly connected to altering perceptions related to emotions and different physical activities [30]. Still, we do not change the value of k2 , but as the memristor coupling parameter k1 increases, we expect the formed patterns to change their features as this is the case for Fig. 3, where we have fixed k1 = 2.5. The features observed in Fig. 2 are affected, as swimming patterns are now more visible. Then, feedbacks can also modulate the traveling waves and can produce complex spatial patterns in various developmental contexts [22]. We observe also in Figs. 2(c) and 3(c) that each structure generated is well separated from each other. Also, the amplitude of pat-

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Fig. 2. The panels show the spatiotemporal features of patterns for the memristor coupling parameter k1 = 1.5 and the coupling parameter K = 0.05. (a) the 3D feature, (b) the X , Z plane and (c) the X , Y plane.

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terns decreases when the memristor coupling parameter increases, however this leads to a more observation of swimming patterns in the second case [Fig. 3]. Synchronization can be enhanced at different levels, that is, the constraints on which the synchronization appears. Those can be on the trajectory amplitude, requiring the amplitudes of oscillators to be equal, giving place to complete synchronization [23–25]. Figs. 2(a), (c) and 3(a), (c) give an example of this where patterns of membrane potential display synchronized states described by the obtained quasi-periodic patterns. Such synchronized patterns may represent a generic property of globally coupled neural networks, which might be of interest for experimental studies. There are evidences that such patterns may be related to a series of processes in the brain such as movement control and cognition [31–33]. Although quasi-periodic-like behaviors are significative for brain functions, they should be controlled to avoid some pathological brain rhythms such as the Parkinson’s

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Fig. 4. The panels show the features of patterns for the memristor coupling parameter k1 = 2.5 and the coupling parameter K = 0.5. (a) the 3D feature and (b) the X , Y plane.

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Fig. 3. The panels show the spatiotemporal features of patterns for the memristor coupling parameter k1 = 2.5 and the coupling parameter K = 0.05. Set up is the same as in Fig. 2.

disease and epileptic seizure, which have identical physiological anatomic structures [34,35]. In Fig. 4, the numerical simulations are performed for k1 = 2.5 under strong coupling between cells K = 0.5, while the other parameters remain unchanged. It is observed that the increasing of the coupling strength K more depicts the breathing in swimming of patterns as time progresses. In fact, as one can observe in Fig. 4(a), the initial wave gives rise to additional vagueness that tend to form visible swimming patterns. More interesting, such dynamical properties are seen in many forms of cortical activity [28]. On the other hand, strong coupling between neurons under the effect of electromagnetic radiation can give rise to visible swimming or breathing patterns in excitable systems. We have plotted in Fig. 5, the spatial evolution of patterns for K = 0.5 and k1 = 2.5. These patterns, being plotted at dif-

ferent times, see their behaviors change as time increases. Remarkably, at t = 2500, series of spikes are obtained [this can be seen in Fig. 5(b)], and their behaviors clearly picture memristor coupling effects [15,36]. We clearly notice that cells display the same dynamics. These oscillations depicted in Fig. 5 are nonlinear oscillations with some features of synchronization as already mentioned above. Synchronous oscillations from groups of neurons have been shown to correlate with quickly changing mental states, such as absorbing and analyzing new information [37]. Along the same line, synchronized firing of neurons also forms the basis of periodic motor commands for rhythmic movements [27]. We should however stress that the observed behaviors reinforce the efficiency in describing waves flowing in discrete networks of coupled cells which can behave as excitable media. In Fig. 6, distinct patterns for membrane potential are observed. The transmembrane current is taken in the form of a periodical function I ext = I 0 sin(ωt ) and I 0 is selected to detect the mode response of electrical activities. In Fig. 6(a) for I 0 = 0.01, it is observed the generation of pulses. However, when I 0 is increased, these pulses are linked each other from the time t = 800 for I 0 = 1 and from t = 600 for I 0 = 2.5. In fact, patterns are individually localized at the beginning, but after a given time, there appear linked structures which suggest connectivity between cells. Interestingly, the linked structures, when the transmembrane current is appropriate should tend to become linear as displayed by Fig. 6(c). In the latter, information is effectively shared between all the cells from t = 600, so that as time increases, intercellular communication tends to become perfect. Then Fig. 6 reveals that various

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Fig. 5. The panels show the spatial features of waves for k1 = 2.5 and K = 0.5 at: (a) t = 1500, (b) t = 2500 and (c) t = 3000.

Fig. 6. The panels show the features of patterns by setting different transmembrane currents at fixed parameters K = 0.5, ω = 0.05, k1 = 1.5 for (a) I 0 = 0.01, (b) I 0 = 1 and (c) I 0 = 2.5.

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states for patterning can be observed by varying the transmembrane current. These results confirm those recently obtained by Ma et al. [38] who found that transmembrane current can be responsible for complex modes of the action potential in excitable media. It is found in Fig. 7 that the electrical activity of cells can present multiple types of pattern with increasing the angular frequency. In Fig. 7(a) that is for ω = 0.1, as displayed in Fig. 6, from t = 600 intercellular communication is favored due to the emergence of linked structures with solitonic features. In Fig. 7(b), that is for ω = 0.5, patterns are robust and periodically separated in time while in Fig. 7(c), that is for ω = 1.5, they appear as individual entities in the form of lobes. Spatiotemporal pattern and synchronization dynamics are very crucial in understanding normal function and dysfunction of neuronal systems, but they rely on specific frequencies. Experimentally, such frequencies are known and lots of contributions have been devoted to their subsequent wave patterns, especially those related to pathological rhythmic

brain activity in Parkinson’s disease, essential tremor, and epilepsies. In Fig. 8, different feedback gains k1 are applied by selecting ω = 0.5 and I 0 = 1. It is observed various patterns depending of the choice of k1 . As k1 increases, the formed patterns display rows which are obviously separated. The linear stability analysis could not predict the emergence of separated structures, which in present work is new and appears to be a consequence of discreteness effects. This is the main characteristic of MI, which involves sequences of lattice cells in collective oscillations. These results confirm the fact that modulated traveling waves and complex spatial patterns can be produced by feedbacks in various developmental contexts [22]. The numerical results reported in this Letter can be extended to account for these processes. In Figs. 9 and 10, periodical type of radiation is imposed to change the distribution of magnetic field and also the magnetic flux. To discern the effect of electromagnetic radiation, the forcing current I ext is set to zero and the external electromagnetic radia-

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Fig. 7. The panels show the features of patterns by setting different values of ω for transmembrane currents at fixed parameters K = 0.5, k1 = 2.5, I 0 = 1 for (a) ω = 0.1, (b) ω = 0.5 and (c) ω = 1.5.

Fig. 8. The panels show the features of patterns when induction current k1 ρ (φn ) v n is generated with different intensities. The parameters are selected by K = 0.5, ω = 0.5, I 0 = 1 for (a) k1 = 0.1, (b) k1 = 0.5 and (c) k1 = 2.5.

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tion is defined by φext = A cos(2π f t ). In Fig. 9, different feedback gains k1 are applied to observe the pattern formation for membrane potential of the cells. In Fig. 9(a) that is for the value k1 = 1, it is observed a generation of stable wave trains or pulses, while in Fig. 9(b) increasing the feedback gain k1 leads to the formation of impulses which degenerate after a short time into pulses. This once more prove that MI is an excellent and a direct mechanism that lead to the formation of solitons and train of waves in systems where there are competitive effects between nonlinearity and dispersion [39–41]. In Fig. 10, the external electromagnetic radiation is increased and the results also show the generation of traveling pulses with compact support as found in Fig. 9(a). Their amplitude increases with increasing of the intensity A. Figs. 9 and 10 also reveal also that the features of patterns remain the same when electromagnetic radiation is applied with different frequencies.

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This paper was devoted to the study of pattern formation via the modulational instability analysis in excitable media under the effect of electromagnetic radiation. We have shown that a simple and physically-motivated modification of the FitzHugh– Nagumo equations can mimic the coexistence of dynamical motifs seen as swimming or breathing patterns. Analytically, it has been shown that increasing the value of the feedback gain considerably modifies the instability features. Numerical simulations have been carried out to confirm analytical predictions. The results have suggested that the effect of electromagnetic induction through the memristor coupling in excitable cells enhances efficient information transfer among coupled cells, therefore can lead to memory signaling within the brain. The phenomenon of wave modulation observed in the present letter is straightly related to the concept of frequency, and both are useful in the understanding of the EEG-

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brain oscillations related to emotional, basic cognitive and motivational as well as normal and pathological conditions. We hope that this work principally stimulates further investigations in other areas connected with neuron processes, such as the combined effect of chemical and electrical couplings, time delay induced transitions, phase control, temporal and spatial coexistence of periodic, aperiodic, and chaotic states for patterning.

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Fig. 10. The panels show the features of patterns when electromagnetic radiation is applied with different intensities. The parameters are selected by K = 0.5, f = 1, k1 = 1 for (a) A = 0.1 and (b) A = 10.

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