Cooperative behavior in periodically driven noisy
Integrate-Fire Models of Neuronal Dynamics
A.R. 
,
T.C. 
,
Charles R. 
,
S.B. 
,
and
K. 
Command, Control and Ocean Surveillance Center,
Research, Development, Test, and Evaluation Division, Code 364,
San Diego, CA 92152;
for Nonlinear Studies and Theoretical
Division, MS-B258,
Los Alamos National
Laboratory, Los Alamos, NM 87545;
Engineering Department, Columbia University,
New York, NY 10027;
Department B014, University of California
at San Diego, La Jolla, CA 920903
The response of nonlinear systems to weak periodic stimuli and noise has recently been of interest to the statistical physics community. One of the most intriguing cooperative effects that arises out of the coupling between deterministic and random dynamics in a nonlinear system is ``Stochastic Resonance`` (SR). This effect consists of a noise-induced enhancement in the signal-to-noise ratio measured at the frequency of the external driving force.
There has been considerable speculation about the possible role played by SR in the response of sensory neurons, and recent experiments have indeed shown SR-like behavior in these neurons. SR was first studied in the context of bistable systems. Since many mathematical models of neuron dynamics are not bistable, the question arises as to whether or not similar cooperative effects can be seen in the response of simpler systems, quantified solely in terms of threshold crossing events.
To address this question, we study the dynamics of one of the better-known models of neuron dynamics, the so-called ``integrate-fire model'', in the presence of external driving. While this model does not provide a complete description of real neurons, it does capture many of their statistical properties. Using the method of images, we construct an approximation to the first passage time probability density for the integrate-fire model. The approximate solutions are shown to agree very well with results from numerical simulations, and more importantly, the noise-induced critical behavior is accurately captured by the method of images.
We also investigate the dynamics of a simpler model for neural dynamics that is qualitatively similar to the integrate-fire model. One advantage of this model is that in the absence of periodic driving, solutions of the Fokker-Planck equation governing this process have a relatively simple analytic form. The solution to the simpler model is expressed in terms of eigenfunctions of the Fokker-Planck operator, and using perturbation techniques we are then able to construct an asymptotic solution for the case of weak low-frequency periodic forcing.