A mechanism for elliptic-like bursting and synchronization of bursts in a map-based neuron network

A system consisting of two Rulkov map-based neurons coupled through reciprocal electrical synapses as a simple phenomenological example is discussed. When the electrical coupling is excitatory, the square-wave bursting can be well predicted by the bifurcation analysis of a comparatively simple low-dimensional subsystem embedded in the invariant manifold. While, when the synapses are inhibitory due to the artificial electrical coupling, a fast–slow analysis is carried out by treating the two slow variables as two different bifurcation parameters. The main result of this paper is to present a mechanism for the occurrence of a kind of special elliptic bursting. The mechanism for this kind of elliptic-like bursting is due to the interaction between two chaotic oscillations with different amplitudes. Moreover, the generation of antiphase synchronization of networks lies in the different switching orders between two pairs of different chaotic oscillations corresponding to the first neuron and the second neuron, respectively.

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

[2]  Miguel A F Sanjuán,et al.  Patterns in inhibitory networks of simple map neurons. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[3]  Nikolai F. Rulkov,et al.  Oscillations in Large-Scale Cortical Networks: Map-Based Model , 2004, Journal of Computational Neuroscience.

[4]  N. Rulkov.  Regularization of synchronized chaotic bursts. , 2000, Physical review letters.

[5]  E. Teramoto,et al.  Mathematical Topics in Population Biology, Morphogenesis and Neurosciences , 1987 .

[6]  Gerda de Vries,et al.  Bursting as an emergent phenomenon in coupled chaotic maps. , 2001 .

[7]  Hiroshi Nozawa,et al.  A neural network model as a globally coupled map and applications based on chaos. , 1992, Chaos.

[8]  A. Selverston,et al.  Synchronous Behavior of Two Coupled Biological Neurons , 1998, chao-dyn/9811010.

[9]  Alla Borisyuk,et al.  The Dynamic Range of Bursting in a Model Respiratory Pacemaker Network , 2005, SIAM J. Appl. Dyn. Syst..

[10]  Miguel A. F. Sanjuán,et al.  Winnerless Competition in Networks of Coupled Map Neurons , 2004 .

[11]  Nikolai F. Rulkov,et al.  Subthreshold oscillations in a map-based neuron model , 2004, q-bio/0406007.

[12]  Gouhei Tanaka,et al.  Sensitivity versus resonance in two-dimensional spiking-bursting neuron models. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[13]  A. Sherman,et al.  Contributions of modeling to understanding stimulus-secretion coupling in pancreatic beta-cells. , 1996, The American journal of physiology.

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

[15]  Wulfram Gerstner,et al.  Spiking Neuron Models , 2002 .

[16]  Miguel A. F. Sanjuán,et al.  Map‐based neuron networks , 2007 .

[17]  Dante R. Chialvo,et al.  Generic excitable dynamics on a two-dimensional map , 1995 .

[18]  Gouhei Tanaka,et al.  Synchronization and propagation of bursts in networks of coupled map neurons. , 2006, Chaos.

[19]  J. Casado,et al.  Transient activation in a network of coupled map neurons. , 2003, Physical review letters.