Nonparametric inference for Markovian interval processes

Consider a p-variate counting process N = (N(i)) with jump times {[tau](i)1, [tau](i)2, ...}. Suppose that the intensity of jumps [lambda](i) of N(i) at time t depends on the time since its last jump as well as on the times since the last jumps of the other components, i.e. [lambda](i)(t) = [alpha](i)(t - [tau](1)N(1)(t -), ... t - [tau](p)N(p)(t -)), where the [alpha](i)s are unknown, nonrandom functions. From observing one single trajectory of the process N over an increasing 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)(x1, ..., xp) does not depend on x2, ..., xp, i.e. that N(1) is a renewal process. The results obtained are applied in the analysis of simultaneously recorded neuronal spike train series. In the example given, inhibition of one neuron (target) through another neuron (trigger) is characterized and identified as a geometric feature in the graphical representation of the estimate of [alpha](i) as a surface. Estimating the intensity of the target as a function of time of only the most recent trigger firing the estimate is displayed as a planar curve with a sharp minimum. This leads to a new method of assessing neural connectivity which is proposed as an alternative to existing cross-correlation-based methods.