Hopf bifurcation and bursting synchronization in an excitable systems with chemical delayed coupling

Abstract

In this paper we consider the Hopf bifurcation and synchronization in the two coupled Hindmarsh-Rose excitable systems with chemical coupling and time-delay. We surveyed the conditions for Hopf bifurcations by means of dynamical bifurcation analysis and numerical simulation. The results show that the coupled excitable systems with no delay have supercritical Hopf bifurcation, while the delayed system undergoes Hopf bifurcations at critical time delays when coupling strength lies in a particular region. We also investigated the effect of the delay on the transition of bursting synchronization in the coupled system. The results are helpful for us to better understand the dynamical properties of excitable systems and the biological mechanism of information encoding and cognitive activity.

Dynamic behavior analysis of fractional-order Hindmarsh–Rose neuronal model

“…Most previous theoretical studies on the neuronal firing rhythm and modes have been in the form of integer-order neuronal models. Many useful conclusions consistent with the results of biological experiments can be drawn (Guang-jun and Jian-xue 2005;Perc 2005;Perc and Marhl 2005;Yang et al 2006;Sun et al 2011;Perc and Marhl 2007;Wang et al 2011;Duan et al 2013;Zhang et al 2013;Wang et al 2013;Yamada and Kashimori 2013). Brian et al (REF) analyzed the dynamics of firing rate with a range of stimulus dynamics in their neurophysiology experiment (Lundstrom et al 2008).…”

“…Previous reports have revealed the mechanism for different modes of neuronal bursting or spiking from the point of view of dynamics. These investigators discovered nonlinear dynamic characteristics involved in bursting or spiking modes of neurons (Yong and Yong 2010;Liu 2010;Guang-jun and Jian-xue 2005;Perc 2005;Perc and Marhl 2005;Yang et al 2006;Sun et al 2011;Perc and Marhl 2007;Wang et al 2011;Duan et al 2013;Zhang et al 2013;Wang et al 2013;Yamada and Kashimori 2013). Yang et al evaluated various bursting and spiking modes through the analysis of fast and slow dynamics and bifurcation, and they observed a series of ISI bifurcation modes (Yang et al 2006).…”

“…For this reason, investigation of these dynamic characteristics of the fractional-order neuronal model should be performed. Although the firing characteristics of the integer-order neuronal model have been studied extensively (Guang-jun and Jian-xue 2005;Perc 2005;Perc and Marhl 2005;Yang et al 2006;Sun et al 2011;Perc and Marhl 2007;Wang et al 2011;Duan et al 2013;Zhang et al 2013;Wang et al 2013;Yamada and Kashimori 2013), there have been few reports on fractional-order neuronal models. Xie and Liu analyzed the dynamic characteristics of the fractional-order FitzHugh-Nagumo neuronal model (Yong and Yong 2010), but their study only dealt with the case of a change in external current intensity and not with the transition of firing mode with a change in the order of the fractional-order model.…”

Global Attractors for Hindmarsh-Rose Equations in Neurodynamics

“…[1,9,10,18,19,28,30,32,36]. The more interesting study is on the behavior of neurons coupling and synchronization [8,10,23,26].…”

Frequency-domain order parameters for the burst and spike synchronization transitions of bursting neurons

“…Burst synchronization on the slow bursting timescale refers to a coherence between the active phase onset or offset times of bursting neurons, while spike synchronization on the fast spike timescale characterizes a coherence between intraburst spikes fired by bursting neurons (Rubin 2007;Omelchenko et al 2010). Many recent studies on the burst and spike synchronizations have been made in several aspects (e.g., chaotic phase synchronization, transitions between different states of burst synchronization, effect of network topology, effect on information transmission, suppression of bursting synchronization, and effect of noise and coupling on the burst and spike synchronization) (van Vreeswijk and Hansel 2001;Dhamala et al 2004;Ivanchenko et al 2004;Belykh et al 2005;Duan et al 2013;Lu 2005, 2009;Tanaka et al 2006;Batista et al 2007Batista et al , 2012Pereira et al 2007;Sun et al 2011;Yu et al 2011;Lameu et al 2012;Kim et al 2012;Kim and Lim 2013;Meng et al 2013;Wang et al 2013).…”