Multistate and Multistage Synchronization of Hindmarsh-Rose Neurons With Excitatory Chemical and Electrical Synapses

The new phenomena of the multistate synchronization of Hindmarsh-Rose (HR) neurons with both excitatory chemical and electrical synapses over the complex network are analytically studied. The regions for coupling strengths to achieve local synchronization are explicitly obtained. Such regions are characterized by the second largest eigenvalue λ2 of the electrical connection matrix and the number k of chemical signals each neuron receives. The dynamics of the multistate synchronization includes the coexistence of stable regular bursting and periodic/steady-state behaviors. Our theory predicts that recurrent networks formed by a certain cell types in layers 4 and 6 in cat area 17 could lead to multistate synchronization. These are in contrast with coupled oscillator systems or coupled map lattices where only single-state synchronization is found. It should also be noted that if the parameters of HR neurons are chosen resulting in an irregular (chaotic) bursting, then the coexistence state would contain chaotic attractor. Our method employed here is quite general. For instance, it can be immediately applied to other coupled nervous systems such as FitzHugh-Nagumo and Morris-Lecar nervous systems. The analytical tools and concepts needed include coordinate transformations, matrix measures, monotone dynamics and time averaging estimates.

[1]  Enno de Lange,et al.  The Hindmarsh-Rose neuron model: bifurcation analysis and piecewise-linear approximations. , 2008, Chaos.

[2]  G. Wei,et al.  Mathematical analysis of the wavelet method of chaos control , 2006 .

[3]  C. Stam,et al.  Small-world networks and functional connectivity in Alzheimer's disease. , 2006, Cerebral cortex.

[4]  Jonq Juang,et al.  Coordinate transformation and matrix measure approach for synchronization of complex networks. , 2009, Chaos.

[5]  Kevan A. C. Martin,et al.  Topology and dynamics of the canonical circuit of cat V1 , 2009, Neural Networks.

[6]  C. A. Desoer,et al.  Nonlinear Systems Analysis , 1978 .

[7]  Bojan Mohar,et al.  Eigenvalues, diameter, and mean distance in graphs , 1991, Graphs Comb..

[8]  Eugene M. Izhikevich,et al.  Which model to use for cortical spiking neurons? , 2004, IEEE Transactions on Neural Networks.

[9]  Alessandro Torcini,et al.  Dynamical phases of the Hindmarsh-Rose neuronal model: studies of the transition from bursting to spiking chaos. , 2007, Chaos.

[10]  L. Glass.  Synchronization and rhythmic processes in physiology , 2001, Nature.

[11]  Mahdi Jalili,et al.  Synchronizing Hindmarsh-Rose neurons over Newman-Watts networks. , 2009, Chaos.

[12]  T. Bliss,et al.  Controlling the Ups and Downs of Synaptic Strength , 2004, Science.

[13]  R. Genesio,et al.  On the dynamics of chaotic spiking-bursting transition in the Hindmarsh-Rose neuron. , 2009, Chaos.

[14]  P. Hartman.  Ordinary Differential Equations , 1965 .

[15]  Qiankun Song,et al.  Multistability in Networks With Self-Excitation and High-Order Synaptic Connectivity , 2010, IEEE Transactions on Circuits and Systems I: Regular Papers.

[16]  Jean-Jacques E. Slotine,et al.  On partial contraction analysis for coupled nonlinear oscillators , 2004, Biological Cybernetics.

[17]  G. Buzsáki.  Rhythms of the brain , 2006 .

[18]  Chai Wah Wu,et al.  Synchronization in Complex Networks of Nonlinear Dynamical Systems , 2008 .

[19]  C. M. Place,et al.  Ordinary Differential Equations , 1982 .

[20]  Sze-Bi Hsu,et al.  ORDINARY DIFFERENTIAL EQUATIONS WITH APPLICATIONS , 2005 .

[21]  Guanrong Chen,et al.  Chaos synchronization of coupled neurons with gap junctions , 2006 .

[22]  R. Dzakpasu,et al.  Discriminating differing types of synchrony in neural systems , 2005 .

[23]  Sho Hashimoto,et al.  A Novel Hybrid Spiking Neuron: Bifurcations, Responses, and On-Chip Learning , 2010, IEEE Transactions on Circuits and Systems I: Regular Papers.

[24]  Jonq Juang,et al.  Synchronous Chaos in Coupled Map Lattices with General Connectivity Topology , 2008, SIAM J. Appl. Dyn. Syst..

[25]  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.

[26]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[27]  W. Singer,et al.  Oscillatory responses in cat visual cortex exhibit inter-columnar synchronization which reflects global stimulus properties , 1989, Nature.

[28]  B. Ermentrout,et al.  Chemical and electrical synapses perform complementary roles in the synchronization of interneuronal networks. , 2004, Proceedings of the National Academy of Sciences of the United States of America.

[29]  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.

[30]  Mario di Bernardo,et al.  Contraction Theory and Master Stability Function: Linking Two Approaches to Study Synchronization of Complex Networks , 2009, IEEE Transactions on Circuits and Systems II: Express Briefs.

[31]  Guanrong Chen,et al.  Synchronous Bursts on Scale-Free Neuronal Networks with Attractive and Repulsive Coupling , 2010, PloS one.

[32]  Marco Righero,et al.  Synchronization in Networks of Hindmarsh–Rose Neurons , 2008, IEEE Transactions on Circuits and Systems II: Express Briefs.

[33]  Martin Hasler,et al.  Synchronization of bursting neurons: what matters in the network topology. , 2005, Physical review letters.

[34]  W. Singer,et al.  Long-range synchronization of oscillatory light responses in the cat retina and lateral geniculate nucleus , 1996, Nature.

[35]  W. Singer,et al.  Neural Synchrony in Brain Disorders: Relevance for Cognitive Dysfunctions and Pathophysiology , 2006, Neuron.

[36]  Wolf Singer,et al.  Neuronal Synchrony: A Versatile Code for the Definition of Relations? , 1999, Neuron.

[37]  Martin Hasler,et al.  Predicting single spikes and spike patterns with the Hindmarsh–Rose model , 2008, Biological Cybernetics.

[38]  Timothy K. Horiuchi,et al.  A Neuromorphic VLSI Head Direction Cell System , 2011, IEEE Transactions on Circuits and Systems I: Regular Papers.

[39]  John Rinzel,et al.  Dynamics of Spiking Neurons Connected by Both Inhibitory and Electrical Coupling , 2003, Journal of Computational Neuroscience.