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Bifurcation Structure of Two Coupled FHN Neurons with Delay

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Bifurcation Structure of Two Coupled FHN Neurons with Delay Bifurcation structure of two coupled FHN neurons with delay Niloofar Farajzadeh Tehrania,∗, MohammadReza Razvana aDepartment of Mathematical Sciences, Sharif University of Technology, Tehran, Iran. Abstract This paper presents an investigation of the dynamics of two coupled non-identical FitzHugh-Nagumo neurons with delayed synaptic connection. We consider coupling strength and time delay as bifurcation parameters, and try to classify all possible dynamics which is fairly rich. The neural system exhibits a unique rest point or three ones for the different values of coupling strength by employing the pitchfork bifurcation of non-trivial rest point. The asymptotic stability and possible Hopf bifurcations of the trivial rest point are studied by analyzing the corresponding characteristic equation. Homoclinic, fold, and pitchfork bifurcations of limit cycles are found. The delay-dependent stability regions are illustrated in the parameter plane, through which the double-Hopf bifurcation points can be obtained from the intersection points of two branches of Hopf bifurcation. The dynamical behavior of the system may exhibit one, two, or three different periodic solutions due to pitchfork cycle and torus bifurcations (Neimark-Sacker bifurcation in the Poincare map of a limit cycle), of which detection was impossible without exact and systematic dynamical study. In addition, Hopf, double-Hopf, and torus bifurcations of the non trivial rest points are found. Bifurcation diagrams are obtained numerically or analytically from the mathematical model and the parameter regions of different behaviors are clarified. Keywords: FitzHugh-Nagumo neural model, Delay differential equation, Double-Hopf bifurcation, Hopf-Pitchfork bifurcation, Torus bifurcation, Fold of limit cycles 2010 MSC: 34K18, 92C20 1. Introduction in networks in which every neuron connects to all other nearby neurons. Zhou et al [3] modeled a neural net- Neurons and their interactions are generally assumed work as a small sub-network of excitable elements to to be the determinant of the brain performance. The study synchronization dynamics and the hierarchically simplest model to display features of neural interac- clustered organization of excitable neurons in a network tions consists of two coupled neurons or neural systems. of networks. Starting from such simple and reduced networks, larger The FitzHugh-Nagumo (FHN) model with cubic non- networks can be built and their features may be stud- linearity which was derived as a simplified model of the ied. In order to study complicated interaction between famous Hodgkin-Huxley (HH) model [4] by FitzHugh neurons in large neural networks, the neurons are of- [5] and Nagumo et al. [6], is a classic oscillator ex- ten put up into highly connected sub-networks or syn- hibiting variety of nonlinear phenomena in planar au- chronized sub-ensembles. Such neural populations are tonomous systems. The FHN-like systems are of fun- arXiv:1504.07658v2 [math.DS] 6 Oct 2015 usually spatially localized [1]. In this way, the model damental importance for describing the qualitative na- of two mutually coupled neurons may also apply as a ture of nerve impulse propagation and neural activity. framework of two coupled neural sub-ensembles. In fact, this model seems reach enough, and can cap- Destexhe et al. [2], by investigating two interconnected ture neural excitability of original HH equation [7]. In thalamic spindle oscillatory neurons, showed that how this way, Guckenheimer and Kuehn [8] investigated ho- more complex dynamics emanates in ring networks with moclinic orbits of the FitzHughNagumo equation from nearest neighbors and fully reciprocal connectivity, or the view-point of fast-slow dynamical systems. Hoff et al. [9] investigated unidirectional and bidirectional electrical couplings of two identical FHN neurons, in- ∗Corresponding author: Tel.:+982166165626. volving the effects of the four parameters on bifurca- E-mail address: [email protected] Preprint submitted to Mathematical Biosciences September 4, 2018 tions and synchronization. In modeling studies of trans- behavior. Some examples of delayed systems are cou- membrane potential oscillations, Bachelet et al. [10] ex- pled lasers [26], high-speed milling, population dynam- plained that bursting oscillations appear quite naturally ics, epidemiology [27], and gene expression, for two coupled FHN equations. Their coupling model Krupa and Touboul [28] investigated a self-coupled de- is quite different from other investigations. Different fir- layed FHN system to understand the effect of introduc- ing patterns such as chaotic firing is found in a pair of ing delays in a system whose dynamics, in the absence identical FHN elements with phase-repulsive coupling of delays, is characterized by a canard explosion and re- [11]. laxation oscillations. They have shown that delays sig- In addition to study of coupled neurons, the dynamics nificantly enrich the dynamics, leading to mixed-mode of a network of FHN neurons is also studied by re- oscillations, bursting, canard and chaos. searchers. Oscillatory chaotic behaviors and the mech- The research for coupled FHN systems with delay has anism of dynamic change from chaotically spiking to attracted many authors’ attention. Dahlem et al. [29] firing death is induced in [12], by considering the cou- studied an electrically delay-coupled neural system. pling of FHN oscillators in a network. Cattani [13] They showed that, for sufficiently large delay and cou- proposed a network model of instantaneous propagation pling strength bi-stability of a fixed point and limit cy- of signals between neurons in which the neurons need cle oscillations occur, even though the single excitable not be physically close to each other. The dynamics of elements display only a stable fixed point. Buric and the action potential and the recovery variable within a Todorovic [30], and also Jia et al. [31] investigated neural network is described by proposed network. The Hopf bifurcation of coupled FHN neurons with delayed oscillation patterns in a symmetric network of modi- coupling, they concluded that the time delay can in- fied FHN neurons is discussed by Murza [14]. The au- duce Hopf bifurcation. Buric et al. [32], observed dif- thor described the building block structure of a three- ferent synchronization states in a coupled FHN system dimensional lattice shaped as a torus, and identified the with delay and by variation of the coupling strength and symmetry group acting on the electrically coupled dif- time delay. Also they showed that the stability and the ferential systems located at the nodes of the lattice. patterns of exactly synchronous oscillations depend on It is known that signal transmission in coupled neurons the type of excitability and type of coupling. In [33] is not instantaneous in general [15]. Hence a time de- by employing the center manifold theory and normal lay can occur in the coupling between neurons or in a form method, the Fold-Hopf bifurcation in a coupled self-feedback loop. For example, synaptic communi- identical FHN neural system with delay is investigated. cation between neurons depends on the propagation of The synchronization transitions between inphase and action potentials across axons. The finite conduction antiphase oscillations for delay-coupled FHN system is speed and the information processing time in synapses studied in [34]. Wang et al. [35] reported that time lead to a conduction delay. The speed of signal trans- delay can control the occurrence of some bifurcations mission through unmyelinated axonal fibers is in the in two synaptically coupled FHN neurons. However, order of 1 m/s, which results in existence of time de- only codimension-1 bifurcations are discussed in these lays up to 80 ms for propagation through the cortical papers. Fan and Hong [36], and also Xu et al. [37], network [15, 16]. Moreover, the interaction delay be- considered the stability and Hopf bifurcation of dou- tween the oscillators can be as large as the oscillation ble delay coupled FHN system with different coupling period [17]. Therefore, time delay is inevitable in sig- strength. Yao and Tu [38] discussed the combined effect nal transmission for real neurons. On the other hand, of coupling strength and multiple delays on the stabil- time-delayed feedback mechanisms might be intention- ity of the rest point, and they obtained stability switches ally implemented to control neural disturbances, e.g. in the coupled FHN neural system with multiple delays. to suppress alpha rhytm [18] and undesired synchrony In addition to the study of Hopf bifurcation, Zhen and of firing neurons in Parkinson’s disease or epilepsy Xu [39], and also Rankovic [40] discussed the effects [19, 20, 21]. Various delayed feedback loops have been of the coupling strength and delay on steady state bifur- proposed as effective and powerful therapy of neuro- cations in coupled systems with identical neurons. The logical diseases which are caused by synchronization of Bautin bifurcation due to small delay, was analyzed by neurons [20, 21, 22, 23, 24, 25]. Buric and Todorovic [41]. Nevertheless, in that paper From the modeling viewpoint, delay in dynamical sys- the time-delayed argument of the coupling function was tems is exhibited whenever the system’s behavior is de- replaced by its Taylor expansion, so that their delay- pendent at least in part on its history. There are many differential system was approximated by ordinary dif- technological and biological systems which exhibit such ferential equations. Zhen and Xu [42] studied Bautin bi- 2 furcation of completely synchronous FHN neurons. Li neurons in the work of Zhen and Xu [39], and also Li and Jiang [43], investigated Hopf and Bogdanov-Takens and Jiang [43] are assumed identical which leads to a bifurcations in a coupled FHN neural system with gap symmetry for their system of equations, that can not be junction for identical neurons. In [44] it was shown that found in our system. Instead the symmetry in our solu- in a simple motif of two coupled FHN systems with two tions is due to the inherent symmetry of the FHN neu- different delay times and with self-feedback, complex rons, namely our system of equations is equivariant un- dynamics arise.
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