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

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.

[1]  T. Sideris.  Ordinary Differential Equations and Dynamical Systems , 2013 .

[2]  Ling Hong,et al.  Hopf bifurcation analysis in synaptically coupled HR neurons with two time delays , 2010 .

[3]  J. Hindmarsh,et al.  A model of neuronal bursting using three coupled first order differential equations , 1984, Proceedings of the Royal Society of London. Series B. Biological Sciences.

[4]  Johan A K Suykens,et al.  Introduction to Focus Issue: synchronization in complex networks. , 2008, Chaos.

[5]  Zhonghuai Hou,et al.  Transition to burst synchronization in coupled neuron networks. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[6]  Eugene M. Izhikevich,et al.  Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting , 2006 .

[7]  P. Fries.  Neuronal gamma-band synchronization as a fundamental process in cortical computation. , 2009, Annual review of neuroscience.

[8]  C. Rocsoreanu,et al.  Hopf bifurcation in a system of two coupled advertising oscillators , 2005 .

[9]  Y. Kuznetsov.  Elements of Applied Bifurcation Theory , 2023, Applied Mathematical Sciences.

[10]  D. Ranković.  BIFURCATIONS OF FITZHUGH-NAGUMO EXCITABLE SYSTEMS WITH CHEMICAL DELAYED COUPLING , 2011 .

[11]  陆启韶,et al.  Rhythm Synchronization of Coupled Neurons with Temporal Coding Scheme , 2007 .

[12]  Lu Qi-Shao,et al.  Time Delay-Enhanced Synchronization and Regularization in Two Coupled Chaotic Neurons , 2005 .

[13]  S. Boccaletti,et al.  Synchronization of chaotic systems , 2001 .

[14]  I. Kobelevskiy.  Bifurcation analysis of a system of Morris-Lecar neurons with time delayed gap junctional coupling , 2008 .

[15]  R. D. Driver.  Introduction to Delay Differential Equations , 1977 .

[16]  Xianfa Jiao,et al.  Analysis of stability of neural network with inhibitory neurons , 2010, Cognitive Neurodynamics.

[17]  Kurths,et al.  Phase synchronization of chaotic oscillators. , 1996, Physical review letters.

[18]  Xin Biao Lu,et al.  Synchronization in Complex Networks , 2011 .

[19]  A. Aertsen,et al.  Dynamics of neuronal interactions in monkey cortex in relation to behavioural events , 1995, Nature.

[20]  Rubin Wang,et al.  Dynamic phase synchronization characteristics of variable high-order coupled neuronal oscillator population , 2010, Neurocomputing.

[21]  Lu Qi-Shao,et al.  Phase synchronization and its transition in two coupled bursting neurons: theoretical and numerical analysis , 2010 .

[22]  Rubin Wang,et al.  Stochastic model and neural coding of large-scale neuronal population with variable coupling strength , 2006, Neurocomputing.

[23]  Nikola Burić,et al.  Bursting neurons with coupling delays , 2007 .

[24]  Kristina Todorović,et al.  Synchronization of bursting neurons with delayed chemical synapses. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[25]  Mingzhou Ding,et al.  Enhancement of neural synchrony by time delay. , 2004, Physical review letters.

[26]  Mingzhou Ding,et al.  Transitions to synchrony in coupled bursting neurons. , 2004, Physical review letters.

[27]  Rubin Wang,et al.  Nonlinear stochastic models of neurons activities , 2001, Neurocomputing.

[28]  Lu Qi-Shao,et al.  Rhythm Synchronization of Coupled Neurons with Temporal Coding Scheme , 2007 .

[29]  G. Teschl.  Ordinary Differential Equations and Dynamical Systems , 2012 .

[30]  Yubing Gong,et al.  Single or multiple synchronization transitions in scale-free neuronal networks with electrical or chemical coupling , 2011 .

[31]  J. Dostrovsky,et al.  High-frequency Synchronization of Neuronal Activity in the Subthalamic Nucleus of Parkinsonian Patients with Limb Tremor , 2000, The Journal of Neuroscience.

[32]  Yanhong Zheng,et al.  Spatiotemporal patterns and chaotic burst synchronization in a small-world neuronal network , 2008 .

[33]  Mahdi Jalili,et al.  Phase synchronizing in Hindmarsh-Rose neural networks with delayed chemical coupling , 2011, Neurocomputing.

[34]  Zhaosheng Feng,et al.  Synchronization transition in gap-junction-coupled leech neurons , 2008 .

[35]  Zhaosheng Feng,et al.  Synchrony and lag synchrony on a neuron model coupling with time delay , 2010 .

[36]  Dominique M. Durand,et al.  Phase synchronization in two coupled chaotic neurons , 1999 .

[37]  Z. Duan,et al.  Synchronization transitions on scale-free neuronal networks due to finite information transmission delays. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[38]  Thomas Nowotny,et al.  Neuronal synchrony: peculiarity and generality. , 2008, Chaos.

[39]  Eugene M. Izhikevich,et al.  Neural excitability, Spiking and bursting , 2000, Int. J. Bifurc. Chaos.

[40]  Haixia Wang,et al.  In-Phase Burst Synchronization and Rhythm Dynamics of Complex neuronal Networks , 2012, Int. J. Bifurc. Chaos.

[41]  W. Singer,et al.  Stimulus-specific neuronal oscillations in orientation columns of cat visual cortex. , 1989, Proceedings of the National Academy of Sciences of the United States of America.

[42]  Qishao Lu,et al.  Burst synchronization of electrically and chemically coupled map-based neurons , 2009 .