Untitled

Abstract

An analytical approach is presented for determining the response of a neuron or of the activity in a network of connected neurons, represented by systems of nonlinear ordinary stochastic differential equations--the Fitzhugh-Nagumo system with Gaussian white noise current. For a single neuron, five equations hold for the first- and second-order central moments of the voltage and recovery variables. From this system we obtain, under certain assumptions, five differential equations for the means, variances, and covariance of the two components. One may use these quantities to estimate the probability that a neuron is emitting an action potential at any given time. The differential equations are solved by numerical methods. We also perform simulations on the stochastic Fitzugh-Nagumo system and compare the results with those obtained from the differential equations for both sustained and intermittent deterministic current inputs with superimposed noise. For intermittent currents, which mimic synaptic input, the agreement between the analytical 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 probability that a neuron is spiking as obtained from the empirical distribution of the potential in the simulations gives a result almost identical to that obtained using the analytical approach. However, when there is sustained large-amplitude noise, the analytical method is only accurate for short time intervals. Using the simulation method, we study the distribution of the interspike interval directly from simulated sample paths. We confirm that noise extends the range of input currents over which (nonperiodic) spike trains may exist and investigate the dependence of such firing on the magnitude of the mean input current and the noise amplitude. For networks we find the differential equations for the means, variances, and covariances of the voltage and recovery variables and show how solving them leads to an expression for the probability that a given neuron, or given set of neurons, is firing at time t. Using such expressions one may implement dynamical rules for changing synaptic strengths directly without sampling. The present analytical method applies equally well to temporally nonhomogeneous input currents and is expected to be useful for computational studies of information processing in various nervous system centers.

Dynamical mean-field theory of noisy spiking neuron ensembles: Application to the Hodgkin-Huxley model

“…Numerical simulations have shown that for weak noises, the distribution of v(t) of the membrane potential of a single HH neuron nearly obeys the Gaussian distribution, although for strong noises, the distribution of v(t) deviates from the Gaussian, taking a bimodal form [22] [37]. Similar behavior of the membrane-potential distribution has been reported also in a FN neuron model [18] [38]. By using Eq.…”

“…Similar population density approaches have been recently developed for a large-scale neuronal clusters [15,16]. The moment method initiated by Rodriguez and Tuckwell (RT) has been applied to single FN [17,18] and HH neurons [19,20]. When the moment method is applied to a single neuron model with K variables, K-dimensional stochastic DEs are replaced by (1/2)K(K + 3)-dimensional deterministic DEs.…”

How Synchronization Protects from Noise

“…In particular, such a property has been demonstrated in the case of FN oscillators, with representing a white noise process [22].…”

“…At the other end of the spectrum, the effect of noise on nonlinear contracting systems is bounded by where is the noise intensity – which can be arbitrarily large – and is the contraction rate of the system [21]. Between these two extremes, it has been shown analytically that some limit-cycle oscillators commonly used as simplified neuron models, such as FitzHugh-Nagumo (FN) oscillators, are basically unperturbed when they are subject to a small amount of white noise [22]. Yet, a larger amount of noise breaks this “resistance”, both in the state space and in the frequency space [Figures 1(A)–(D)].…”

Augmented moment method for stochastic ensembles with delayed couplings. II. FitzHugh-Nagumo model

“…We assume that the noise intensity β is weak and that the distribution of state variables takes the Gaussian form concentrated near the means of (µ 1 , µ 2 ), as previously adopted [21] [22]. Numerical simulations for a single FN neuron have shown that for weak noises, the distribution of x(t) of the membrane potential nearly obeys the Gaussian distribution, although for strong noises, the distribution of x(t) deviates from the Gaussian, taking a bimodal form [26] [27]. Similar behavior of the membrane-potential distribution has been reported also in the HH neuron model [28] [29].…”