Real Neurons and Mathematical Models

Abstract

In these notes, the spontaneous activity of several kinds of neurons has been modelled as a stochastic point process that results from the interaction of inputs to a neuron that are also stochastic processes. First related theory on which the models are based is presented and these are then enlarged in stages. Thus models with only one kind of input-excitatory — (Models 8.1, 8.2, 9.1 and 9.2) are frequently studied first before constructing more general models with two kinds of input — excitatory and inhibitory — (Models 8.3 to 8.6 and 9.3 to 9.5). Since the neuron receives inputs at its several synapses, spatial summation occurs at the membrane. This effect is modelled by superposition of input sequences (Models 4.1 to 4.3). The introduction of inhibition leads to a more ambitious modelling approach. Thus different kinds of interaction of excitatory and inhibitory inputs have been considered. This interaction may be (a) pre-synaptic (Models 5.1 to 5.4 and 9.2) or post-synaptic (Models 6.1, 6.2, 8.3 to 8.6 and 9.3 to 9.5) (b) independent (Models 5.1, 5.2, 6.1, 6.2, 8.3 to 8.5, 9.3 and 9.4) or dependent (Models 5.3, 5.4, 8.6 and 9.5).