ANALYTICAL AND SIMULATION RESULTS FOR STOCHASTIC FITZHUGH-NAGUMO NEURONS AND NEURAL NETWORKS

An analytical approach is presented for determining the response of a neuron or of the activity in a network of connected neurons, represent ed by systems of nonlinear ordinary stochastic differential equations- the Fitzhugh-Nagumo system with Gaussian white noise current. For a si ngle neuron, five equations hold for the first-and second-order centra l moments of the voltage and recovery variables. From this system we o btain, under certain assumptions, five differential equations for the means, variances, and covariance of the two components. One may use th ese quantities to estimate the probability that a neuron is emitting a n action potential at any given time. The differential equations are s olved by numerical methods. We also perform simulations on the stochas tic Fitzugh-Nagumo system and compare the results with those obtained from the differential equations for both sustained and intermittent de terministic current inputs with superimposed noise. For intermittent c urrents, which mimic synaptic input, the agreement between the analyti cal and simulation results for the moments is excellent. For sustained input, the analytical approximations perform well for small noise as there is excellent agreement for the moments. In addition, the probabi lity that a neuron is spiking as obtained from the empirical distribut ion of the potential in the simulations gives a result almost identica l to that obtained using the analytical approach. However, when there is sustained large-amplitude noise, the analytical method is only accu rate for short time intervals. Using the simulation method, we study t he distribution of the interspike interval directly from simulated sam ple paths. We confirm that noise extends the range of input currents o ver which (nonperiodic) spike trains may exist and investigate the dep endence of such firing on the magnitude of the mean input current and the noise amplitude. For networks we find the differential equations f or the means, variances, and covariances of the voltage and recovery v ariables and show how solving them leads to an expression for the prob ability that a given neuron, or given set of neurons, is firing at tim e t. Using such expressions one may implement dynamical rules for chan ging synaptic strengths directly without sampling. The present analyti cal method applies equally well to temporally nonhomogeneous input cur rents and is expected to be useful for computational studies of inform ation processing in various nervous system centers.