Vector fields, global flows and the calculus of pulses
Reuben Rabi · Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences · 2001
The point of this paper is to show a remarkable property of functional expansions in the form of iterated integrals in capturing the dynamics of digital computation when realized in the form of pulse/spike–driven controlled smooth dynamical systems. The key features of the structure thus captured by the formalism are (a) the combinatorics of spike trains in various input channels and their effect on the discrete–time dynamics of the realized automaton; and (b) the relative timing precedence of the spike trains in various input channels. We do this by providing globally valid functional expansions for a certain kind of hybrid dynamical system when subject to inputs which resemble pulse trains. We give these functional expansions in the form of a product of a nonlinear holonomy factor and a local Fliess functional expansion. We also obtain a computational scheme to determine the response of finite automata to input sequences, which is meant as a substitute for the naive iteration of the next–state maps. We obtain this by embedding finite automata in suitable controlled dynamical systems on the special orthogonal group. Incidentally, we show that our realization of automata can also be modified to realize hybrid machines in which a finite automaton is interacting with an external system, some of whose analog variables are converted into interspike time-intervals which, in turn, are simultaneously tokenized and acted upon by the modified realization of the finite automaton. The latter is achieved intrinsically by exploiting the dynamics of the realization.