State space reconstruction using interspike intervals

Essentially all the applications of nonlinear dynamical systems theory to signal processing rely on the ability to reconstruct a dynamical system from a time series of measurements made on it. We to consider a class of systems in which information about a dynamical system is encoded in a sequence of time intervals, rather than a sequence of values of some measurement. We also consider cases where the observations are measurements of some variable, as in Takens´ (1981) theorem, but the measurements are not taken at a uniform rate; instead the times between measurements are a function of the system´s state. (We call this situation `state dependent sampling´.) The basic idea is that there is a real-valued function, τ, on the state space of the system, which at each point in the state space gives the time interval after which the next measurement is taken. For state dependent sampling we thus consider two values at each sampling time: the first is the value of the measurement (which we record) the second is the value of τ which tells us how long to wait before making the next observation. In the interspike intervals scenario, we only consider τ, which we record as well as using to determine the next sampling time. Sauer´s `integrate and fire´ model is a special case of this, in the sense that `integrate and fire´ is essentially a way of defining τ. One of our motivations for considering the state dependent sampling case is the hope that nonuniform sampling may make easier the analysis of time series using embedding techniques