A continuous time dynamical Turing machine

Abstract

Continuous time recurrent neural networks (CTRNN) are systems of coupled ordinary differential equations inspired by the structure of neural networks in the brain. CTRNN are known to be universal dynamical approximators: given a large enough system, the parameters of a CTRNN can be tuned to produce output that is arbitrarily close to that of any other dynamical system. However, in practise, both designing systems of CTRNN to have a certain output, and the reverse—understanding the dynamics of a given system of CTRNN—can be non-trivial. In this paper, we describe a method for embedding any specified Turing machine in its entirety into a CTRNN. As such, we describe in detail a continuous-time dynamical system that performs arbitrary discrete-state computations. We suggest that in acting as both a continuous-time dynamical system and as a computer, the study of such systems can help refine and advance the debate concerning the Computational Hypothesis that cognition is a form of computation and the Dynamical Hypothesis that cognitive systems are dynamical systems.