Bursting and two-parameter bifurcation in the Chay neuronal model

In this paper, we study and classify the firing patterns in the Chay neuronal model by the fast/slow decomposition and the two-parameter bifurcations analysis. We show that the Chay neuronal model can display complex bursting oscillations, including the "fold/fold" bursting, the "Hopf/Hopf" bursting and the "Hopf/homoclinic" bursting. Furthermore, dynamical properties of different firing activities of a neuron are closely related to the bifurcation structures of the fast subsystem. Our results indicate that the codimension-2 bifurcation points and the related codimension-1 bifurcation curves of the fast-subsystem can provide crucial information to predict the existence and types of bursting with changes of parameters.

[1]  John Rinzel,et al.  Analysis of bursting in a thalamic neuron model , 1994, Biological Cybernetics.

[2]  Shigui Ruan,et al.  Bifurcation analysis in models of tumor and immune system interactions , 2009 .

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

[4]  QiShao Lu,et al.  Different types of bursting in Chay neuronal model , 2008 .

[5]  Qishao Lu,et al.  Two-parameter bifurcation analysis of firing activities in the Chay neuronal model , 2008, Neurocomputing.

[6]  U. Kummer,et al.  Fold–Hopf Bursting in a Model for Calcium Signal Transduction , 2003, q-bio/0310018.

[7]  P. Shorten,et al.  A Hodgkin-Huxley model exhibiting bursting oscillations , 2000, Bulletin of mathematical biology.

[8]  Xiao-Jing Wang,et al.  Genesis of bursting oscillations in the Hindmarsh-Rose model and homoclinicity to a chaotic saddle , 1993 .

[9]  Wei Ren,et al.  Gwn-Induced bursting, Spiking, and Random subthreshold Impulsing oscillation before Hopf bifurcations in the Chay Model , 2004, Int. J. Bifurc. Chaos.

[10]  John Rinzel,et al.  Bursting oscillations in an excitable membrane model , 1985 .

[11]  R. Bertram,et al.  Topological and phenomenological classification of bursting oscillations. , 1995, Bulletin of mathematical biology.

[12]  V. N. Belykh,et al.  Homoclinic bifurcations leading to the emergence of bursting oscillations in cell models , 2000 .

[13]  Daizhan Cheng,et al.  Bursting of Morris-Lecar neuronal model with current-feedback control , 2009 .

[14]  Adam Kepecs,et al.  Information encoding and computation with spikes and bursts , 2003, Network.

[15]  Xiao-Jing Wang,et al.  Bursting Neurons Signal Input Slope , 2002, The Journal of Neuroscience.

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

[17]  Matjaž Perc,et al.  Different types of bursting calcium oscillations in non-excitable cells , 2003 .

[18]  Allan R. Willms,et al.  BIFURCATIONS IN THE FAST DYNAMICS OF NEURONS: IMPLICATIONS FOR BURSTING , 2005 .

[19]  Li Li,et al.  Dynamics of autonomous stochastic resonance in neural period adding bifurcation scenarios , 2003 .

[20]  Teresa Ree Chay,et al.  Generation of periodic and chaotic bursting in an excitable cell model , 1994, Biological Cybernetics.

[21]  Bard Ermentrout,et al.  Simulating, analyzing, and animating dynamical systems - a guide to XPPAUT for researchers and students , 2002, Software, environments, tools.

[22]  Liancheng Wang,et al.  Hopf Bifurcation of a Class of Two Coupled Relaxation Oscillators of the van der Pol Type with Delay , 2009 .

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