Application of a Two-Dimensional Hindmarsh-Rose Type Model for bifurcation Analysis

In this study, we examine the bifurcation scenarios of a two-dimensional Hindmarsh–Rose type model [Tsuji et al., 2007] with four parameters and simulate some resemblances of neurophysiological features for this model using spike-and-reset conditions. We present possible classifications based on the results of the following assessments: (1) the number and stability of the equilibria are analyzed in detail using a table to demonstrate the matter in which the stability of the equilibrium changes and to determine which two equilibria collapse through the saddle-node bifurcation; (2) the sufficient conditions for an Andronov–Hopf bifurcation and a saddle-node bifurcation are mathematically confirmed; and (3) we elaborately evaluate the sufficient conditions for the Bogdanov–Takens (BT) and Bautin bifurcations. Several numerical simulations for these conditions are also presented. In particular, two types of bistable behaviors are numerically demonstrated: the BT and Bautin bifurcations. Notably, all of the bifurcation curves in the domain of the remaining parameters are similar when the time scale is large. Additionally, to show the potential for a limit cycle, the existence of a trapping region is demonstrated. These results present a variety of diverse behaviors for this model. The results of this study will be helpful in assessing suitable parameters for fitting the resemblances of experimental observations.

[1]  A. Hodgkin.  The local electric changes associated with repetitive action in a non‐medullated axon , 1948, The Journal of physiology.

[2]  Joaquín Delgado,et al.  Analysis of the Takens-Bogdanov bifurcation on M-Parameterized Vector Fields , 2010, Int. J. Bifurc. Chaos.

[3]  J. Rinzel,et al.  Bursting, beating, and chaos in an excitable membrane model. , 1985, Biophysical journal.

[4]  Yong-Bin Kim,et al.  Low power CMOS electronic central pattern generator design for a biomimetic underwater robot , 2007, Neurocomputing.

[5]  J. Rinzel,et al.  Emergence of organized bursting in clusters of pancreatic beta-cells by channel sharing. , 1988, Biophysical journal.

[6]  C. Stevens,et al.  Inward and delayed outward membrane currents in isolated neural somata under voltage clamp , 1971, The Journal of physiology.

[7]  S. Yoshizawa,et al.  An Active Pulse Transmission Line Simulating Nerve Axon , 1962, Proceedings of the IRE.

[8]  J. Connor,et al.  Neural repetitive firing: modifications of the Hodgkin-Huxley axon suggested by experimental results from crustacean axons. , 1977, Biophysical journal.

[9]  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.

[10]  David Terman,et al.  Chaotic spikes arising from a model of bursting in excitable membranes , 1991 .

[11]  Hiroshi Kawakami,et al.  Bifurcations in Two-Dimensional Hindmarsh-rose Type Model , 2007, Int. J. Bifurc. Chaos.

[12]  J. M. Gonzalez-Miranda.  Complex bifurcation Structures in the Hindmarsh-rose Neuron Model , 2007, Int. J. Bifurc. Chaos.

[13]  J. Hindmarsh,et al.  The assembly of ionic currents in a thalamic neuron I. The three-dimensional model , 1989, Proceedings of the Royal Society of London. B. Biological Sciences.

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

[15]  Zhaosheng Feng,et al.  Fold-Hopf bifurcations of the Rose-Hindmarsh Model with Time Delay , 2011, Int. J. Bifurc. Chaos.

[16]  J. M. Gonzalez-Miranda,et al.  Observation of a continuous interior crisis in the Hindmarsh-Rose neuron model. , 2003, Chaos.

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

[18]  Michael Denker,et al.  A network of electronic neural oscillators reproduces the dynamics of the periodically forced pyloric pacemaker group , 2005, IEEE Transactions on Biomedical Engineering.

[19]  Teresa Ree Chay,et al.  Chaos in a three-variable model of an excitable cell , 1985 .

[20]  Jonathan Touboul,et al.  Bifurcation Analysis of a General Class of Nonlinear Integrate-and-Fire Neurons , 2008, SIAM J. Appl. Math..

[21]  J. Hindmarsh,et al.  A model of the nerve impulse using two first-order differential equations , 1982, Nature.

[22]  D. Terman,et al.  The transition from bursting to continuous spiking in excitable membrane models , 1992 .

[23]  T. Chay.  Glucose response to bursting-spiking pancreatic β-cells by a barrier kinetic model , 1985, Biological Cybernetics.

[24]  R. FitzHugh.  Impulses and Physiological States in Theoretical Models of Nerve Membrane. , 1961, Biophysical journal.

[25]  A. Hodgkin,et al.  A quantitative description of membrane current and its application to conduction and excitation in nerve , 1952, The Journal of physiology.

[26]  J. Keizer,et al.  Theory of the effect of extracellular potassium on oscillations in the pancreatic beta-cell. , 1985, Biophysical journal.

[27]  J. Hindmarsh,et al.  The assembly of ionic currents in a thalamic neuron III. The seven-dimensional model , 1989, Proceedings of the Royal Society of London. B. Biological Sciences.

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

[29]  R. Robinson,et al.  An Introduction to Dynamical Systems: Continuous and Discrete , 2004 .

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