An Exact Jitter Method using Dynamic Programming

We are given a neural spike train with spike times t1, . . . , tn and we want to discover something about the resolution over which these spike times were generated. A heuristic approach is to fix a statistic of the spike train, say f(t1, . . . , tn), and then Monte Carlo jitter each of the spike times by some small random amount, computing the statistic over and over again for each jitter. Presumably, if the jitter amount is smaller than the meaningful resolution of the spike train, then the original value of the statistic will not be an outlier in the distribution of jittered statistics. The jitter heuristic is designed to create a lot of “similar” spike trains in the sense that it preserves the “rate of spiking” measured over windows larger than the jitter amount. Failing to account for a nonconstant firing rate has been the downfall of many statistical methods which look at the time resolution of spike trains. A close analysis of the jitter method reveals several drawbacks. Perhaps the easiest to see is that the jittered spike trains may not obey known physiological constraints, such as the absolute refractory period. Furthermore, spikes may swap positions, which is intuitively unsettling at the very least. Han Amarasingham’s thesis [1] contains a detailed discussion of various jitter methods, including an exact (not Monte Carlo) method that includes physiological constraints in an agnostic way. Unfortunately, this exact method is computationally intensive and may not be appropriate for online physiological experiments where a jitter method is applied to data from one trial in order to generate stimuli for the next trial. Here we describe a fast and exact jitter method that allows for some predefined physiological constraints. We feel that it is a reasonable compromise between the well formulated and agnostic methods in [1] and the obviously deficient original jitter heuristic. We will first describe a Monte Carlo formulation of our method. Then we will show that it can be solved exactly and quickly using dynamic programming. That our method is exact and is not a Monte Carlo method is important for reducing the variance of our estimates. Reduced variance translates into reduced numbers of trials in physiological experiments.