Breakdown of fast-slow analysis in an excitable system with channel noise.

We consider a stochastic version of an excitable system based on the Morris-Lecar model of a neuron, in which the noise originates from stochastic sodium and potassium ion channels opening and closing. One can analyze neural excitability in the deterministic model by using a separation of time scales involving a fast voltage variable and a slow recovery variable, which represents the fraction of open potassium channels. In the stochastic setting, spontaneous excitation is initiated by ion channel noise. If the recovery variable is constant during initiation, the spontaneous activity rate can be calculated using Kramer's rate theory. The validity of this assumption in the stochastic model is examined using a systematic perturbation analysis. We find that, in most physically relevant cases, this assumption breaks down, requiring an alternative to Kramer's theory for excitable systems with one deterministic fixed point. We also show that an exit time problem can be formulated in an excitable system by considering maximum likelihood trajectories of the stochastic process.

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

[2]  D. Ludwig.  Persistence of Dynamical Systems under Random Perturbations , 1975 .

[3]  L Schimansky-Geier,et al.  Analytical approach to the stochastic FitzHugh-Nagumo system and coherence resonance. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[4]  Graham,et al.  Nonequilibrium potentials and their power-series expansions. , 1989, Physical review. A, General physics.

[5]  Robert S. Maier,et al.  Limiting Exit Location Distributions in the Stochastic Exit Problem , 1994, SIAM J. Appl. Math..

[6]  James P. Keener,et al.  Mathematical physiology , 1998 .

[7]  A. V. Polovinkin,et al.  Noise-induced escape in an excitable system , 2013 .

[8]  Z. Schuss.  Theory and Applications of Stochastic Processes: An Analytical Approach , 2009 .

[9]  James P Keener,et al.  Perturbation analysis of spontaneous action potential initiation by stochastic ion channels. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[10]  Idan Segev,et al.  Ion Channel Stochasticity May Be Critical in Determining the Reliability and Precision of Spike Timing , 1998, Neural Computation.

[11]  D. Luchinsky,et al.  Corrals and Critical Behavior of the Distribution of Fluctuational Paths. , 1996, Physical review letters.

[12]  강희정,et al.  17 , 1995, The Hatak Witches.

[13]  Jay M Newby,et al.  Isolating intrinsic noise sources in a stochastic genetic switch , 2011, Physical biology.

[14]  C. Morris,et al.  Voltage oscillations in the barnacle giant muscle fiber. , 1981, Biophysical journal.

[15]  S. Swain.  Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences , 1984 .

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

[17]  ScienceDirect.  Bulletin of mathematical biology , 1973 .

[18]  K. Vahala.  Handbook of stochastic methods for physics, chemistry and the natural sciences , 1986, IEEE Journal of Quantum Electronics.

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

[20]  B. M. Fulk.  MATH , 1992 .

[21]  James P. Keener,et al.  An Asymptotic Analysis of the Spatially Inhomogeneous Velocity-Jump Process , 2011, Multiscale Model. Simul..