Reduction of a model of an excitable cell to a one-dimensional map

We use qualitative methods for singularly perturbed systems of differential equations and the principle of averaging to compute the first return map for the dynamics of a slow variable (calcium concentration) in the model of an excitable cell. The bifurcation structure of the system with continuous time endows the map with distinct features: it is a unimodal map with a boundary layer corresponding to the homoclinic bifurcation in the original model. This structure accounts for different periodic and aperiodic regimes and transitions between them. All parameters in the discrete system have biophysical meaning, which allows for precise interpretation of various dynamical patterns. Our results provide analytical explanation for the numerical studies reported previously.

[1]  G. de Vries,et al.  Multiple Bifurcations in a Polynomial Model of Bursting Oscillations , 1998 .

[2]  Philip Holmes,et al.  Minimal Models of Bursting Neurons: How Multiple Currents, Conductances, and Timescales Affect Bifurcation Diagrams , 2004, SIAM J. Appl. Dyn. Syst..

[3]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

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

[5]  X. Wang Fast burst firing and short-term synaptic plasticity: A model of neocortical chattering neurons , 1999, Neuroscience.

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

[7]  Christopher Jones,et al.  Geometric singular perturbation theory , 1995 .

[8]  G. Ermentrout,et al.  Parabolic bursting in an excitable system coupled with a slow oscillation , 1986 .

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

[10]  Robert M. Miura,et al.  Perturbation techniques for models of bursting electrical activity in pancreatic b-cells , 1992 .

[11]  Michael A. Arbib,et al.  The handbook of brain theory and neural networks , 1995, A Bradford book.

[12]  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.

[13]  John Rinzel,et al.  A one-variable map analysis of bursting in the Belousov-Zhabotinskii reaction , 1983 .

[14]  E. Izhikevich,et al.  Weakly connected neural networks , 1997 .

[15]  John Guckenheimer,et al.  Bifurcation, Bursting, and Spike Frequency Adaptation , 1997, Journal of Computational Neuroscience.

[16]  A. N. Sharkovskiĭ,et al.  Difference Equations and Their Applications , 1993 .

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

[18]  P Holmes,et al.  A model for the periodic synaptic inhibition of a neuronal oscillator. , 1987, IMA journal of mathematics applied in medicine and biology.

[19]  Philip Holmes,et al.  A Minimal Model of a Central Pattern Generator and Motoneurons for Insect Locomotion , 2004, SIAM J. Appl. Dyn. Syst..

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

[21]  J. Keizer,et al.  Minimal model for membrane oscillations in the pancreatic beta-cell. , 1983, Biophysical journal.

[22]  J. Rinzel,et al.  Oscillatory and bursting properties of neurons , 1998 .

[23]  Georgi S. Medvedev,et al.  Multimodal regimes in a compartmental model of the dopamine neuron , 2004 .

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

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

[26]  Yoshikazu Giga,et al.  Nonlinear Partial Differential Equations , 2004 .

[27]  Andrey Shilnikov,et al.  Mechanism of bistability: tonic spiking and bursting in a neuron model. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[28]  S. D. Chatterji Proceedings of the International Congress of Mathematicians , 1995 .

[29]  M. Krupa,et al.  Relaxation Oscillation and Canard Explosion , 2001 .

[30]  Nancy Kopell,et al.  Timing regulation in a network reduced from voltage-gated equations to a one-dimensional map , 1999, Journal of mathematical biology.

[31]  John Rinzel,et al.  A Formal Classification of Bursting Mechanisms in Excitable Systems , 1987 .

[32]  T. Morrison,et al.  Dynamical Systems , 2021, Nature.

[33]  Shuji Yoshizawa,et al.  Pulse sequences generated by a degenerate analog neuron model , 1982, Biological Cybernetics.

[34]  John Rinzel,et al.  Intrinsic and network rhythmogenesis in a reduced traub model for CA3 neurons , 1995, Journal of Computational Neuroscience.

[35]  X J Wang,et al.  Calcium coding and adaptive temporal computation in cortical pyramidal neurons. , 1998, Journal of neurophysiology.

[36]  A. Y. Kolesov,et al.  Asymptotic Methods in Singularly Perturbed Systems , 1994 .

[37]  G. Ermentrout,et al.  Analysis of neural excitability and oscillations , 1989 .

[38]  Idan Segev,et al.  Methods in Neuronal Modeling , 1988 .

[39]  J. C. Smith,et al.  Models of respiratory rhythm generation in the pre-Bötzinger complex. I. Bursting pacemaker neurons. , 1999, Journal of neurophysiology.

[40]  Andrey Shilnikov,et al.  Transition between tonic spiking and bursting in a neuron model via the blue-sky catastrophe. , 2005, Physical review letters.

[41]  J. Gillis,et al.  Asymptotic Methods in the Theory of Non‐Linear Oscillations , 1963 .

[42]  Helwig Löffelmann,et al.  GEOMETRY OF MIXED-MODE OSCILLATIONS IN THE 3-D AUTOCATALATOR , 1998 .

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

[44]  Arthur Sherman,et al.  Calcium and Membrane Potential Oscillations in Pancreatic-Cells , 2000 .

[45]  H. Othmer,et al.  Case Studies in Mathematical Modeling: Ecology, Physiology, and Cell Biology , 1997 .

[46]  Teresa Ree Chay,et al.  BURSTING, SPIKING, CHAOS, FRACTALS, AND UNIVERSALITY IN BIOLOGICAL RHYTHMS , 1995 .

[47]  J. Alexander,et al.  On the dynamics of bursting systems , 1991, Journal of mathematical biology.

[48]  A. Hodgkin,et al.  A quantitative description of membrane current and its application to conduction and excitation in nerve , 1990 .

[49]  Stephen Wiggins,et al.  Global Bifurcations and Chaos , 1988 .

[50]  Mark Levi,et al.  A period-adding phenomenon , 1990 .

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

[52]  Yue-Xian Li,et al.  Whole—Cell Models , 2002 .

[53]  John Rinzel,et al.  Bursting phenomena in a simplified Oregonator flow system model , 1982 .

[54]  P. J. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

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

[56]  E Mosekilde,et al.  Bifurcation structure of a model of bursting pancreatic cells. , 2001, Bio Systems.