NONPARAMETRIC-INFERENCE FOR MARKOVIAN INTERVAL PROCESSES

Consider a p-variate counting process N = (N-(i)) with jump times {tau (1)((i)), tau(2)((i)), ...}. Suppose that the intensity of jumps lambd a((i)) of N-(i) at time t depends on the time since its last jump as w ell as on the times since the last jumps of the other components, i.e. lambda((i))(t) = alpha((i))(t - tau(N(1)(t -))((1)), ... t - tau(N(p) (t -))((p))), where the alpha((i))s are unknown, nonrandom functions. From observing one single trajectory of the process N over an increasi ng interval of time we estimate nonparametrically the functions alpha( (i)). The estimators are shown to be uniformly consistent over compact subsets. We derive a nonparametric asymptotic test for the hypothesis that alpha((1))(x(1), ..., x(p)) does not depend on x(2), ..., x(p), i.e. that N-(1) is a renewal process. The results obtained are applied in the analysis of simultaneously recorded neuronal spike train serie s. In the example given, inhibition of one neuron (target) through ano ther neuron (trigger) is characterized and identified as a geometric f eature in the graphical representation of the estimate of alpha((i)) a s a surface. Estimating the intensity of the target as a function of t ime of only the most recent trigger firing the estimate is displayed a s a planar curve with a sharp minimum. This leads to a new method of a ssessing neural connectivity which is proposed as an alternative to ex isting cross-correlation-based methods.