Computation in Dynamically Bounded Asymmetric Systems

Previous explanations of computations performed by recurrent networks have focused on symmetrically connected saturating neurons and their convergence toward attractors. Here we analyze the behavior of asymmetrical connected networks of linear threshold neurons, whose positive response is unbounded. We show that, for a wide range of parameters, this asymmetry brings interesting and computationally useful dynamical properties. When driven by input, the network explores potential solutions through highly unstable ‘expansion’ dynamics. This expansion is steered and constrained by negative divergence of the dynamics, which ensures that the dimensionality of the solution space continues to reduce until an acceptable solution manifold is reached. Then the system contracts stably on this manifold towards its final solution trajectory. The unstable positive feedback and cross inhibition that underlie expansion and divergence are common motifs in molecular and neuronal networks. Therefore we propose that very simple organizational constraints that combine these motifs can lead to spontaneous computation and so to the spontaneous modification of entropy that is characteristic of living systems.Author Summary: Biological systems are obviously able to process abstract information on the states of neuronal and molecular networks. However, the concepts and principles of such biological computation are poorly understood by comparison with technological computing. A key concept in models of biological computation has been the attractor of dynamical systems, and much progress has been made in describing the conditions under which attractors exist, and their stability. Instead, we show here for a broad class of asymmetrically connected networks that it is the unstable dynamics of the system that drive its computation, and we develop new analytical tools to describe the kinds of unstable dynamics that support this computation in our model. In particular we explore the conditions under which networks will exhibit unstable expansion of their dynamics, and how these can be steered and constrained so that the trajectory implements a specific computation. Importantly, the underlying computational elements of the network are not themselves stable. Instead, the overall boundedness of the system is provided by the asymmetrical coupling between excitatory and inhibitory elements commonly observed in neuronal and molecular networks. This inherent boundedness permits the network to operate with the unstably high gain necessary to continually switch its states as it searches for a solution. We propose that very simple organizational constraints can lead to spontaneous computation, and thereby to the spontaneous modification of entropy that is characteristic of living systems.