Cross-correlation functions for a neuronal model

Abstract

Cross-correlation functions, R(XY)(t,tau), are obtained for a neuron model which is characterized by constant threshold theta, by resetting to resting level after an output, and by membrane potential U(t) which results from linear summation of excitatory postsynaptic potentials h(t). The results show that: (1) Near time lag tau = 0, R(XY)(t,tau) = f(U) [theta-h(tau), t + tau] {h'(tau) + E(U) [u'(t + tau)]} for positive values of this quantity, where f(U)(u,t) is the probability density function of U(t) and E(U) [u'(t + tau)] is the mean value function of U'(t + tau). (2) Minima may appear in R(XY)(t,tau) for a neuron subjected only to excitation. (3) For large tau, R(XY)(t,tau) is given approximately by the convolution of the input autocorrelation function with the functional of point (1). (4) R(XY)(t,tau) is a biased estimator of the shape of h(t), generally over-estimating both its time to peak and its rise time.