Equilibrium analysis and phase synchronization of two coupled HR neurons with gap junction

Abstract

The properties of equilibria and phase synchronization involving burst synchronization and spike synchronization of two electrically coupled HR neurons are studied in this paper. The findings reveal that in the non-delayed system the existence of equilibria can be turned into intersection of two odd functions, and two types of equilibria with symmetry and non-symmetry can be found. With the stability and bifurcation analysis, the bifurcations of equilibria are investigated. For the delayed system, the equilibria remain unchanged. However, the Hopf bifurcation point is drastically affected by time delay. For the phase synchronization, we focus on the synchronization transition from burst synchronization to spike synchronization in the non-delayed system and the effect of coupling strength and time delay on spike synchronization in delayed system. In addition, corresponding firing rhythms and spike synchronized regions are obtained in the two parameters plane. The results allow us to better understand the properties of equilibria, multi-time-scale properties of synchronization and temporal encoding scheme in neuronal systems.

A new unidirectional coupling for global synchronization of two Hindmarsh-Rose neurons

“…Hrg (2013) [16] claimed that the conditions (15) and (16) can guarantee the stability of the error system (9). It should be pointed out that there are two mistakes in the above mentioned processing.…”

“…The HR neurons reveal various dynamical behaviors, including spikes, bursting, and chaotic attractors [2], [3]. Similar to master and slave synchronization of neural networks and chaotic systems [4], [5], [6], synchronization of two coupled HR neurons (master and slave neurons) has attracted increasing attention [7], [8], [9]. Several papers consider how to design the external control in one neuron, including linear feedback control, adaptive control, or other nonlinear control to achieve the synchronization of two coupled HR neurons [10], [11], [12], [13], [14], [15].…”

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).…”

Synchronization analysis of coupled calcium oscillators based on two regular coupling schemes

“…Recently, synchronization in coupled electrical systems has been extensively investigated in the context of biological systems. Much work has been done on synchronization behavior of calcium oscillations such as complete synchronization [5,6], phase synchronization [7][8][9], lag synchronization [10], generalized synchronization [11][12][13][14] etc. But in most of the work, the threshold of synchronization is given by the numerical simulations.…”