Nonlinear Dynamics of Point Process Systems and Data

Nonlinear Dynamics has been applied to Point Process Systems profitably. Such systems manifest themselves by Series of Events in space or time that are then assimilated to Point Processes, i.e. countable collections of points in continua. Arguments are illustrated here with univariate one-dimension Point Processes representing spike trains from nerve cells. These are described fully by their "timing", i.e. the instants {… ti-1<ti<ti+1 …} when events occur. Timings are described as fully by the time series of the intervals between events {…, Ti-1, Ti, Ti+1, …}; this portrayal is the most common one in Dynamics. This communication's goal is to call attention to the peculiarities of Point Process Systems most relevant to dynamic approaches. Previously, explicit, deliberate statements to that effect have been bypassed because approaches, though justified, were extensions from Continuous Systems. The Point Process notion essential for this goal is that the ith event at ti exclusively reflects the most recent finite epoch and the timing of the finite number of events within it. This is because real world events always have transitory consequences and cannot be repeated at intervals shorter than some dead time. Those finite quantities, though varying from event to event, have maxima (or least upper bounds), S for the most recent epochs and D for the numbers of events. Accordingly, the most involved will be spans from ti-S to ti and events i-D; i-D+1: S is called "integration period" and the D events "influential". Critical is the timing of the influential events plus the ith, described by the D intervals Ti-D+1, …, Ti-1 and Ti also called "influential". Therefore, it makes sense to assign to a Point Process the D-dimension state space whose coordinates are the influential intervals. The ith event is represented by point {Ti-D+1, …, Ti-1, Ti} whose location implies the timing of events i-D, i-D+1, …, i-1 and i. Successive point positions compose a discrete trajectory with countable points. Point Process trajectories have exhibited the same kinds of behaviors and attractors as continuous cases. Point Process data are analyzed along the same steps (a–d) prescribed for continuous data but peculiarities affect the first two. Data consist, not of several variables, but of the single variable Ti. Hence, in step (a) where a representative variable is selected, Ti is the only choice. In (b) a discrete trajectory is extracted, with Point Processes simply that of the original data Ti: points stagger along it according to the system's dynamics, and need not be periodic or uniformly distributed. With Continuous Systems, discrete trajectories arise from sampling at investigator-chosen invariant time-lags; points stagger periodically and uniformly. The remaining stages are similar for Point Process and Continuous Systems: namely, (c) trajectory embedding in a space with suitable dimensionality and (d) behavior and attractor cataloging by dimensionality, nonlinearity and predictability. The examples listed illustrate the many ways Dynamic approaches to Point Processes, as embodied by spike trains, has and will continue to contribute to the Neurosciences.