On the Dynamics of Small Continuous-Time Recurrent Neural Networks

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.

[1]  S. Grossberg.  On learning and energy-entropy dependence in recurrent and nonrecurrent signed networks , 1969 .

[2]  J. Cowan,et al.  Excitatory and inhibitory interactions in localized populations of model neurons. , 1972, Biophysical journal.

[3]  Nils J. Nilsson,et al.  Artificial Intelligence , 1974, IFIP Congress.

[4]  Simon A Levin,et al.  Cellular behavior and the development of pattern , 1978 .

[5]  Stephen Grossberg,et al.  Absolute stability of global pattern formation and parallel memory storage by competitive neural networks , 1983, IEEE Transactions on Systems, Man, and Cybernetics.

[6]  R. Abraham,et al.  Dynamics--the geometry of behavior , 1983 .

[7]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[8]  J J Hopfield,et al.  Neurons with graded response have collective computational properties like those of two-state neurons. , 1984, Proceedings of the National Academy of Sciences of the United States of America.

[9]  M. Golubitsky,et al.  Singularities and groups in bifurcation theory , 1985 .

[10]  Stephen Grossberg,et al.  Nonlinear neural networks: Principles, mechanisms, and architectures , 1988, Neural Networks.

[11]  Sommers,et al.  Chaos in random neural networks. , 1988, Physical review letters.

[12]  Morris W. Hirsch,et al.  Convergent activation dynamics in continuous time networks , 1989, Neural Networks.

[13]  G. Kass-simon,et al.  Excitatory and Inhibitory Interactions in the Opener Muscle of Lobster Claws , 1989 .

[14]  Leon O. Chua,et al.  Practical Numerical Algorithms for Chaotic Systems , 1989 .

[15]  S. Wiggins.  Introduction to Applied Nonlinear Dynamical Systems and Chaos , 1989 .

[16]  Amir F. Atiya,et al.  Oscillations and Synchronizations in Neural Networks: an Exploration of the Labeling Hypothesis , 1991, Int. J. Neural Syst..

[17]  R. Beer,et al.  Intelligence as Adaptive Behavior: An Experiment in Computational Neuroethology , 1990 .

[18]  Michael I. Jordan,et al.  Advances in Neural Information Processing Systems 30 , 1995 .

[19]  Barak A. Pearlmutter.  Dynamic recurrent neural networks , 1990 .

[20]  Yaser S. Abu-Mostafa,et al.  Analog Neural Networks as Decoders , 1990, NIPS.

[21]  M. Čadek.  Singularities and groups in bifurcation theory, volume I , 1990 .

[22]  Terrence J. Sejnowski,et al.  A Dynamic Neural Network Model of Sensorimotor Transformations in the Leech , 1990, Neural Computation.

[23]  Yann LeCun,et al.  Reverse TDNN: An Architecture For Trajectory Generation , 1991, NIPS.

[24]  P. Das,et al.  Chaos in an effective four-neuron neural network , 1991 .

[25]  David R. Jefferson,et al.  Representations for artificial organisms , 1991 .

[26]  C. Lee Giles,et al.  Neural Network Routing for Random Multistage Interconnection Networks , 1991, NIPS.

[27]  Schieve,et al.  Single effective neuron. , 1991, Physical Review A. Atomic, Molecular, and Optical Physics.

[28]  J. Hale,et al.  Dynamics and Bifurcations , 1991 .

[29]  G. M. Werner.  Evolution of Communication in Artificial Organisms, Artifial Life II , 1991 .

[30]  Charles E. Taylor,et al.  Artificial Life II , 1991 .

[31]  Xin Wang,et al.  Stability of fixed points and periodic orbits and bifurcations in analog neural networks , 1992, Neural Networks.

[32]  Randall D. Beer,et al.  Evolving Dynamical Neural Networks for Adaptive Behavior , 1992, Adapt. Behav..

[33]  P. Husbands,et al.  Analysis of Evolved Sensory-motor Controllers Analysis of Evolved Sensory-motor Controllers , 1992 .

[34]  Frank C. Hoppensteadt,et al.  Analysis and simulation of chaotic systems , 1992 .

[35]  Inman Harvey,et al.  Explorations in Evolutionary Robotics , 1993, Adapt. Behav..

[36]  Yuichi Nakamura,et al.  Approximation of dynamical systems by continuous time recurrent neural networks , 1993, Neural Networks.

[37]  Randall D. Beer,et al.  A qualitative dynamical analysis of evolved locomotion controllers , 1993 .

[38]  Piero Mussio,et al.  Toward a Practice of Autonomous Systems , 1994 .

[39]  Steven H. Strogatz,et al.  Nonlinear Dynamics and Chaos , 2024 .

[40]  Dave Cliff,et al.  Protean behavior in dynamic games: arguments for the co-evolution of pursuit-evasion tactics , 1994 .

[41]  Francesco Mondada,et al.  Automatic creation of an autonomous agent: genetic evolution of a neural-network driven robot , 1994 .

[42]  Randall D. Beer,et al.  Sequential Behavior and Learning in Evolved Dynamical Neural Networks , 1994, Adapt. Behav..

[43]  V. I. Arnol'd,et al.  Dynamical Systems V , 1994 .

[44]  V. Arnold.  Dynamical systems V. Bifurcation theory and catastrophe theory , 1994 .

[45]  D. Signorini,et al.  Neural networks , 1995, The Lancet.

[46]  Peter Tiňo,et al.  Fixed points in two-neuron discrete time recurrent networks: stability and bifurcation considerations , 1995 .

[47]  Randall D. Beer,et al.  A Dynamical Systems Perspective on Agent-Environment Interaction , 1995, Artif. Intell..

[48]  Randall D. Beer,et al.  Computational and dynamical languages for autonomous agents , 1996 .

[49]  T. Gelder,et al.  Mind as Motion: Explorations in the Dynamics of Cognition , 1995 .

[50]  Inman Harvey,et al.  Circle in the round: State space attractors for evolved sighted robots , 1995, Robotics Auton. Syst..

[51]  Stanley J. Rosenschein,et al.  A dynamical systems perspective on agent-environment interaction , 1996 .

[52]  D. Burrows.  A Dynamical Systems Perspective on Music , 1997 .