On the Dynamics of Small Continuous-Time Recurrent Neural Networks

Abstract

Dynamical neural networks are being increasingly employed in a variety of contexts, including as simple model nervous systems for autonomous agents. For this reason, there is a growing need for a comprehensive understanding of their dynamical properties. Using a combination of elementary analysis and numerical studies, this article begins a systematic examination of the dynamics of continuous-time recurrent neural networks. Specifically, a fairly complete description of the possible dynamical behavior and bifurcations of one- and two-neuron circuits is given, along with a few specific results for larger networks. This analysis provides both qualitative insight and, in many cases, quantitative formulas for predicting the dynamical behavior of particular circuits and how that behavior changes as network parameters are varied. These results demonstrate that even small circuits are capable of a rich variety of dynamical behavior (including chaotic dynamics). An approach to understanding the dynamics of circuits with time-varying inputs is also presented. Finally, based on this analysis, several strategies for focusing evolutionary searches into fruitful regions of network parameter space are suggested.

Timescale and Stability in Adaptive Behaviour

“…Note that τ represents the explicit timescale of each of the units and it is this parameter that we will concern ourselves with in this work. In this formulation, the sigmoidal transfer function is a hyperbolic tangent rather than the more familiar exponential sigmoid (see e.g., REF [3]. Note that, here, activation does not represent the membrane potential of a neuron, but rather the firing rate, or mean number of spiking events per unit time, averaged over some appropriate time window.…”

“…Second, we must calculate the Jacobian of the system at equilibrium,J, given by equation (3), (further details can be found in Refs. [17] and [3]. )…”

“…The ctrnn is commonplace throughout neuroscience (as a leaky integrator) and evolutionary robotics [3].…”

“…Such models tend to be the subject of various different kinds of question. For example, the generation of different classes of dynamic behaviour (fixed, cyclic, complex, chaotic) has been of interest to CA and RBN researchers, e.g., [1,2] whereas those working with ANNs have been interested in questions of evolvability, problem solving and autonomous agent control, amongst others [3]. Interestingly, in answering these questions, the role of timescale within these systems has often been neglected.…”

“…Similarly, while some continuous-time recurrent neural networks (ctrnns) comprise neurons with explicit and varied timescales, this property has not received as much attention as others. For example, Beer [3] presents an extensive examination of the dynamics of recurrent ctrnn neurons, but only briefly mentions the impact of their time constants. This tendency to downplay timescale is somewhat surprising, since the natural adaptive systems that inspired these models typically involve processes and mechanisms that operate at multiple timescales.…”