Bursting Oscillations Induced by Small Noise

We consider a model of a square-wave bursting neuron residing in the regime of tonic spiking. Upon introduction of small stochastic forcing, the model generates irregular bursting. The statistical properties of the emergent bursting patterns are studied in the present work. In particular, we identify two principal statistical regimes associated with the noise-induced bursting. In the first case, type I, bursting oscillations are created mainly due to the fluctuations in the fast subsystem. In the alternative scenario, type II bursting, the random perturbations in the slow dynamics play a dominant role. We propose two classes of randomly perturbed slow-fast systems that realize type I and type II scenarios. For these models, we derive the Poincare maps. The analysis of the linearized Poincare maps of the randomly perturbed systems explains the distributions of the number of spikes within one burst and reveals their dependence on the small and control parameters present in the models. The mathematical analy...

[1]  Georgi S. Medvedev,et al.  Reduction of a model of an excitable cell to a one-dimensional map , 2005 .

[2]  Mark Freidlin,et al.  ON STOCHASTIC PERTURBATIONS OF DYNAMICAL SYSTEMS WITH FAST AND SLOW COMPONENTS , 2001 .

[3]  Eric Vanden-Eijnden,et al.  Two distinct mechanisms of coherence in randomly perturbed dynamical systems. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

[5]  J. Steele.  Stochastic Calculus and Financial Applications , 2000 .

[6]  Philip Hartman,et al.  Ordinary differential equations, Second Edition , 2002, Classics in applied mathematics.

[7]  M. Freidlin,et al.  Random Perturbations of Dynamical Systems , 1984 .

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

[9]  Karin Hinzer,et al.  Encoding with Bursting, Subthreshold Oscillations, and Noise in Mammalian Cold Receptors , 1996, Neural Computation.

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

[11]  D. Jordan,et al.  Nonlinear Ordinary Differential Equations: An Introduction for Scientists and Engineers , 1979 .

[12]  J. Hale,et al.  Ordinary Differential Equations , 2019, Fundamentals of Numerical Mathematics for Physicists and Engineers.

[13]  Peter F. Rowat,et al.  State-Dependent Effects of Na Channel Noise on Neuronal Burst Generation , 2004, Journal of Computational Neuroscience.

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

[15]  J. Rubin,et al.  Effects of noise on elliptic bursters , 2004 .

[16]  Gregory D. Smith.  Modeling the Stochastic Gating of Ion Channels , 2002 .

[17]  David Terman,et al.  Uniqueness and stability of periodic bursting solutions , 1999 .

[18]  C. Goldie.  IMPLICIT RENEWAL THEORY AND TAILS OF SOLUTIONS OF RANDOM EQUATIONS , 1991 .

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

[20]  W. Vervaat.  On a stochastic difference equation and a representation of non–negative infinitely divisible random variables , 1979, Advances in Applied Probability.

[21]  H. Kesten.  Random difference equations and Renewal theory for products of random matrices , 1973 .

[22]  Fox,et al.  Emergent collective behavior in large numbers of globally coupled independently stochastic ion channels. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[23]  J. Doob.  Stochastic processes , 1953 .

[24]  Desmond J. Higham,et al.  An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations , 2001, SIAM Rev..

[25]  Michael Friendly,et al.  Discrete Distributions , 2005, Probability and Bayesian Modeling.

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

[27]  Emil Vitásek.  Approximate Solution of Ordinary Differential Equations , 1994 .

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

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

[30]  Yu. N. Blagoveshchenskii.  Diffusion Processes Depending on a Small Parameter , 1962 .

[31]  Mark Freidlin,et al.  On Stable Oscillations and Equilibriums Induced by Small Noise , 2001 .

[32]  Neil Fenichel.  Persistence and Smoothness of Invariant Manifolds for Flows , 1971 .

[33]  José M. Casado,et al.  Bursting behaviour of the FitzHugh-Nagumo neuron model subject to quasi-monochromatic noise , 1998 .

[34]  R. Fox.  Stochastic versions of the Hodgkin-Huxley equations. , 1997, Biophysical journal.

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

[36]  H. Piaggio.  Mathematical Analysis , 1955, Nature.

[37]  J. White,et al.  Channel noise in neurons , 2000, Trends in Neurosciences.

[38]  S. Kwapień,et al.  Random Series and Stochastic Integrals: Single and Multiple , 1992 .

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

[40]  J. Lu,et al.  A Model of a Segmental Oscillator in the Leech Heartbeat Neuronal Network , 2001, Journal of Computational Neuroscience.

[41]  Charles M. Goldie,et al.  Perpetuities and Random Equations , 1994 .

[42]  Feller William,et al.  An Introduction To Probability Theory And Its Applications , 1950 .

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

[44]  Georgi S Medvedev,et al.  Transition to bursting via deterministic chaos. , 2006, Physical review letters.

[45]  N. Berglund,et al.  Noise-Induced Phenomena in Slow-Fast Dynamical Systems: A Sample-Paths Approach , 2005 .

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

[47]  Carson C. Chow,et al.  Aperiodic stochastic resonance in excitable systems. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

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

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

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

[51]  Carson C. Chow,et al.  Spontaneous action potentials due to channel fluctuations. , 1996, Biophysical journal.