Mean field theory of disordered neural networks with multiple neuron interactions

In this thesis we investigate the effect of including neuron interactions of orders greater than the traditional two (pairwise interactions). To this end we first survey neural network models of the Hopfield type which use mean field theory in their analysis. We then analyze a Hopfield type network model generalized to include higher order interactions and investigate the model numerically and analytically. The numerical analysis consists of the utilization of optimization curves; a technique developed in this thesis. The analytic investigation is by means of a mean field theory. Higher order interactions must be included in neural networks to achieve more accurate physiological realism. Structurally, real neural networks are highly interconnected and neurons are extensively arborized. We find that the storage capacity increases greatly when the order of the interactions is increased. Furthermore, the mean field theory results indicate there is a discontinuous jump in the order parameter from the trivial solution to a solution which has an overlap with only one memory state for orders of interaction greater than three. This makes the memory states of the model very stable against the simultaneous flipping of a large number of neurons. Our results accord with observations on the architecture of actual neural systems in achieving functional efficiency, viz. the 'cost' to make a neuron is very high, whereas the cost to make a connection is very much lower. Real networks can thus efficiently increase their storage capacity and their memory stability by increasing the order of their interactions, without a costly increase in the number of neurons.