Onstochastic models of the activity of single neurons.

Stein (1965, 1967) considered a realistic model of nerve membrane potential which had jumps either up or down when excitatory or inhibitory synaptic events occurred and which at other times decayed exponentially to the resting level. In view of the “quantal” nature of synaptic transmission it is probably best to present Stein’s model in its general form with a discrete distribution of post-synaptic potential (p.s.p.) amplitudes. Let the level of depolarization at time t be X(t) and suppose there are n independent synaptic inputs with p.s.p. amplitudes sI (which may be positive or negative) and with Poisson rate parameters &. If we set A = C rZ, 1 then the conditional distribution of the p.s.p. amplitude E, given that a p.s.p. occurred, is Pr (E = si) = &/A. The model can be presented as the stochastic differential equation of a Markov process. We let II, denote a Poisson process of rate parameter 1. Then we have dX(t) = -OX(t) dt+Csi dII,,(t), (0 I where c is the reciprocal of the time constant of the membrane. The moments of X(t) in the absence of threshold can be readily found without finding characteristic functions. If Y(t) = e”‘X(t), then this process satisfies dY(t) = eQfCsi dIIJt). (21