816 research outputs found
The effects of noise on binocular rivalry waves: a stochastic neural field model
We analyse the effects of extrinsic noise on traveling waves of visual perception in a competitive neural field model of binocular rivalry. The model consists of two one-dimensional excitatory neural fields, whose activity variables represent the responses to left-eye and right-eye stimuli, respectively. The two networks mutually inhibit each other, and slow adaptation is incorporated into the model by taking the network connections to exhibit synaptic depression. We first show how, in the absence of any noise, the system supports a propagating composite wave consisting of an invading activity front in one network co-moving with a retreating front in the other network. Using a separation of time scales and perturbation methods previously developed for stochastic reaction-diffusion equations, we then show how multiplicative noise in the activity variables leads to a diffusive–like displacement (wandering) of the composite wave from its uniformly translating position at long time scales, and fluctuations in the wave profile around its instantaneous position at short time scales. The multiplicative noise also renormalizes the mean speed of the wave. We use our analysis to calculate the first passage time distribution for a stochastic rivalry wave to travel a fixed distance, which we find to be given by an inverse Gaussian. Finally, we investigate the effects of noise in the depression variables, which under an adiabatic approximation leads to quenched disorder in the neural fields during propagation of a wave
Spatiotemporal dynamics of continuum neural fields
We survey recent analytical approaches to studying the spatiotemporal dynamics of continuum neural fields. Neural fields model the large-scale dynamics of spatially structured biological neural networks in terms of nonlinear integrodifferential equations whose associated integral kernels represent the spatial distribution of neuronal synaptic connections. They provide an important example of spatially extended excitable systems with nonlocal interactions and exhibit a wide range of spatially coherent dynamics including traveling waves oscillations and Turing-like patterns
Large Deviations for Nonlocal Stochastic Neural Fields
We study the effect of additive noise on integro-differential neural field
equations. In particular, we analyze an Amari-type model driven by a -Wiener
process and focus on noise-induced transitions and escape. We argue that
proving a sharp Kramers' law for neural fields poses substanial difficulties
but that one may transfer techniques from stochastic partial differential
equations to establish a large deviation principle (LDP). Then we demonstrate
that an efficient finite-dimensional approximation of the stochastic neural
field equation can be achieved using a Galerkin method and that the resulting
finite-dimensional rate function for the LDP can have a multi-scale structure
in certain cases. These results form the starting point for an efficient
practical computation of the LDP. Our approach also provides the technical
basis for further rigorous study of noise-induced transitions in neural fields
based on Galerkin approximations.Comment: 29 page
Stochastic neural field equations: A rigorous footing
We extend the theory of neural fields which has been developed in a
deterministic framework by considering the influence spatio-temporal noise. The
outstanding problem that we here address is the development of a theory that
gives rigorous meaning to stochastic neural field equations, and conditions
ensuring that they are well-posed. Previous investigations in the field of
computational and mathematical neuroscience have been numerical for the most
part. Such questions have been considered for a long time in the theory of
stochastic partial differential equations, where at least two different
approaches have been developed, each having its advantages and disadvantages.
It turns out that both approaches have also been used in computational and
mathematical neuroscience, but with much less emphasis on the underlying
theory. We present a review of two existing theories and show how they can be
used to put the theory of stochastic neural fields on a rigorous footing. We
also provide general conditions on the parameters of the stochastic neural
field equations under which we guarantee that these equations are well-posed.
In so doing we relate each approach to previous work in computational and
mathematical neuroscience. We hope this will provide a reference that will pave
the way for future studies (both theoretical and applied) of these equations,
where basic questions of existence and uniqueness will no longer be a cause for
concern
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