Antonio Carlos Roque da Silva Filho
This thesis deals with the mathematical modelling of neural networks. Its aim is to explore an approximation whereby a discrete net of neurons is idealized as a continuous field. continuous approaches to neural networks have been proposed in the past and one of these, Amari's model, has been chosen to provide the general framework for the model studied here. Several new features are introduced into Amari's original model and their implications regarding equilibrium properties of neural maps and dynamic behaviours of uniform fields are analytically studied. Among the new features considered, the most important are: 1. The explicit division of neurons into to two basic types, namely excitatory and inhibitory; 2. The existence of all possible kinds of connections between the two types of neurons, i.e. intra-field connections between excitatory neurons and between inhibitory neurons, and inter-field connections linking excitatory neurons and inhibitory neurons in both directions; and 3. The time modifiability of the weights of all connections according to two different types of Hebbian rule. One type is based on correlation of activities for the two types of neurons, and the other type is based on correlation of activities for excitatory neurons, and on anti-correlation of activities for inhibitory neurons.
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