Stochastic neural field models of binocular rivalry waves
Keywords: applied mathematics
stochastic field models
neural fields

Binocular rivalry is an interesting phenomenon where perception oscillates between different images presented to the two eyes. This thesis is primarily concerned with modelling travelling waves of visual perception during transitions between these perceptual states. In order to model this effect in such a way that we retain as much analytical insight into the mechanisms as possible we employed neural field theory. That is, rather than modelling individual neurons in a neural network we treat the cortical surface as a continuous medium and establish integro-differential equations for the activity of a neural population.
Our basic model which has been used by many previous authors both within and outside of neural field theory is to consider a one dimensional network of neurons for each eye. It is assumed that each network responds maximally to a particular feature of the underlying image, such as orientation. Recurrent connections within each network are taken to be excitatory and connections between the networks are taken to be inhibitory. In order for such a topology to exhibit the oscillations found in binocular rivalry there needs to be some form of slow adaptation which weakens the cross-connections under continued firing. By first considering a deterministic version of this model, we will show that, in fact, this slow adaptation also serves as a necessary "symmetry breaking" mechanism. Using this knowledge to make some mild assumptions we are then able to derive an expression for the shape of a travelling wave and its wave speed. We then go on to show that these predictions of our model are consistent not only with numerical simulations but also experimental evidence.
It will turn out that it is not acceptable to completely ignore noise as it is a fundamental part of the underlying biology. Since methods for analyzing stochastic neural fields did not exist before our work, we first adapt methods originally intended for reaction-diffusion PDE systems to a stochastic version of a simple neural field equation. By regarding the motion of a stochastic travelling wave as being made up of two distinct components, firstly, the drift-diffusion of its overall position, secondly, fast fluctuations in its shape around some average front shape, we are able to derive a stochastic differential equation for the front position with respect to time. It is found that the front position undergoes a drift-diffusion process with constant coefficients. We then go on to show that our analysis agrees with numerical simulation. The original problem of stochastic binocular rivalry is then re-visited with this new toolkit and we are able to predict that the first passage time of a perceptual wave hitting a fixed barrier should be an inverse Gaussian distribution, a result which could potentially be experimentally tested.
We also consider the implications of our stochastic work on different types of neural field equation to those used for modelling binocular rivalry. In particular, for neural fields which support pulled fronts propagating into an unstable state, the stochastic version of such an equation has wave fronts which undergo subdiffusive motion as opposed to the standard diffusion in the binocular rivalry case.