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.
Abraham, R.H., & Shaw, C.D. (1992). Dynamics—the geometry of behavior. Redwood City, CA: Addison-Wesley
2.
Arnol'd, V I. (Ed.). (1994). Dynamical systems V: Bifurcation theory and catastrophe theory. New York: Springer-Verlag.
3.
Atiya, A., & Baldi, P. (1989). Oscillations and synchronizations in neural networks: An exploration of the labeling hypothesis. International Journal of Neural Systems, 1 (2), 103-124.
4.
Beer, R.D. (1990). Intelligence as adaptive behavior: An experiment in computational neuroethology. San Diego: Academic Press.
5.
Beer, R.D. (1995a). A dynamical systems perspective on agent-environment interaction. Artificial Intelligence, 72, 173-215.
6.
Beer, R.D. (1995b). Computational and dynamical languages for autonomous agents. In R. Port and T. van Gelder (Eds.), Mind as motion: Explorations in the dynamics of cognition. Cambridge, MA: MIT Press.
Blum, E.K., & Wang, X. (1992). Stability of fixed points and periodic orbits and bifurcations in analog neural networks. Neural Networks , 5, 577-587.
9.
Cliff, D., Harvey, I., & Husbands, P. (1993). Explorations in evolutionary robotics. Adaptive Behavior, 2(1), 73-110.
10.
Cliff, D., Husbands, P., & Harvey, I. (1993). Analysis of evolved sensory-motor controllers . In Proceedings of the Second European Conference on Artificial Life.
11.
Cohen, M.A., & Grossberg, S. (1983). Absolute stability of global pattern formation and parallel memory storage by competitive neural networks. IEEE Transactions on Systems, Man and Cybernetics, 13, 813-825.
12.
Collins, J.J., & Stewart, I. (1994). A group-theoretic approach to rings of coupled biological oscillators. Biological Cybernetics, 71, 95-103.
13.
Collins, R.J., and Jefferson, D.R. (1991). Representations for artificial organisms. In J.-A. Meyer, H. Roitblat, & S. Wilson (Eds.), From animals to animats 2: Proceedings of the Second International Conference on the Simulation of Adaptive Behavior. Cambridge, MA: MIT Press.
14.
Cowan, J.D., & Ermentrout, G.B. (1978). Some aspects of the eigenbehavior of neural nets. In S. A. Levin (Ed.), Studies in mathematical biology 1: Cellular behavior and the development of pattern. Washington, DC: The Mathematical Association of America.
15.
Das, P.K., Schieve, W.C. & Zeng, Z. (1991). Chaos in an effective four-neuron neural network . Physics Letters [A], 161, 60-66.
16.
de Garis, H. (1992). Steerable GenNets: The genetic programming of steerable behaviors in GenNets. In F. J. Varela and P. Bourgine (Eds.), Toward a practice of autonomous systems: Proceedings of the First European Conference on Artificial Life. Cambridge, MA: MIT Press.
17.
Erlanson, R., & Abu-Mostafa, Y. (1991). Analog neural networks as decoders. In R. P. Lippman, J. E. Moody, & D. S. Touretzky (Eds.), Advances in neural information processing systems 3. San Mateo, CA: Morgan Kaufmann.
18.
Ermentrout, B. (1995). Phase plane analysis of neural activity In M.A. Arbib (Ed.), The handbook of brain theory and neural networks. Cambridge, MA: MIT Press.
19.
Floreano, D., & Mondada, F. (1994). Automatic creation of an autonomous agent: Genetic evolution of a neural-network driven robot. In D. Cliff, P. Husbands, J. Meyer, & S. Wilson (Eds.), From animals to animats 3: Proceedings of the Third International Conference on the Simulation of Adaptive Behavior. Cambridge, MA: MIT Press.
20.
Funahashi, K., & Nakamura, Y (1993). Approximation of dynamical systems by continuous time recurrent neural networks. Neural Networks , 6, 801-806.
21.
Gallagher, J.C., & Beer, R.D. (1993). A qualitative dynamical analysis of evolved locomotion controllers. In J.-A. Meyer, H. Roitblat, & S. Wilson (Eds.), From animals to animats 2: Proceedings of the Second International Conference on the Simulation of Adaptive Behavior. Cambridge, MA: MIT Press.
22.
Golubitsky, M., & Schaefer, D.G. (1985). Singularities and groups in bifurcation theory, Vol. 1. New York: Springer-Verlag.
23.
Goudreau, M.W., & Giles, C.L. (1992). Neural network routing for random multistage interconnection networks. In J. E. Moody, S. J. Hanson & R. P. Lippmann (Eds.), Advances in neural information processing systems 4. San Mateo, CA: Morgan Kaufmann.
24.
Grossberg, S. (1969). On learning and energy-entropy dependence in recurrent and nonrecurrent signed networks. Journal of Statistical Physics , 1, 319-350. Grossberg , S. (1988). Nonlinear neural networks: Principles, mechanisms and architectures. Neural Networks, 1, 17-61.
25.
Guckenheimer, J., & Holmes, P (1983). Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. New York: Springer-Verlag.
26.
Hale, J.K., & Koçak, H. (1991). Dynamics and bifurcations. New York: Springer-Verlag.
27.
Hirsch, M. (1989). Convergent activation dynamics in continuous time networks. Neural Networks, 2, 331-349.
28.
Hopfield, J.J. (1984). Neurons with graded response properties have collective computational properties like those of two-state neurons. Proceedings of the National Academy of Sciences, 81, 3088-3092.
29.
Hopfield, J.J., & Tank, D.W. (1985). Neural computation of decisions in optimization problems. Biological Cybernetics, 52, 141-152.
30.
Hoppensteadt, F.C. (1993). Analysis and simulation of chaotic systems. New York: Springer-Verlag.
31.
Husbands, P., Harvey, I., & Cliff, D. (in press). Circle in the round: State space attractors for evolved sighted robots. Robotics and Autonomous Systems.
32.
Lockery, S.R., Fang, Y., & Sejnowski, T.J. (1990). A dynamic neural network model of sensorimotor transformations in the leech. Neural Computation, 2(3), 274-282.
33.
Miller, G.F., & Cliff, D. (1994). Protean behavior in dynamic games: Arguments for the co-evolution of pursuit-evasion tactics. In D. Cliff, P. Husbands, J. Meyer, & S. Wilson (Eds.), From animals to animats 3: Proceedings of the Third International Conference on the Simulation of Adaptive Behavior. Cambridge, MA: MIT Press. Parker, T.S., & Chua, L.O. (1989). Practical numerical algorithms for chaotic systems . New York: Springer-Verlag.
Simard, P., & Cun, Y.L. (1992). Reverse TDNN: An architecture for trajectory generation . In J. E. Moody, S. J. Hanson, & R. P. Lippman (Eds.), Advances in neural information processing systems 4. San Mateo, CA: Morgan Kaufmann.
36.
Sompolinsky, H., & Crisanti, A. (1988). Chaos in random neural networks. Physical Review Letters, 61(3), 259-262.
37.
Spiessens, P., & Torreele, J. (1992). Massively parallel evolution of recurrent networks : An approach to temporal processing. In F. J. Varela & P. Bourgine (Eds.), Toward a practice of autonomous systems: Proceedings of the First European Conference on Artificial Life. Cambridge, MA: MIT Press.
38.
Strogatz, S.H. (1994). Nonlinear dynamics and chaos. Reading, MA: Addison-Wesley.
39.
Tino, P., Horne, B.G., & Giles, C.L. (1995). Fixed points in two-neuron discrete time recurrent networks: Stability and bifurcation considerations (Tech. Rep. UMIACS-TR-95-51 & CS-TR-3461). College Park, MD: University of Maryland, Institute for Advanced Computer Studies.
40.
Werner, G.M., & Dyer, M.G. (1991). Evolution of communication in artificial organisms . In C. G. Langton, C. Taylor, J. D. Farmer, & S. Rasmussen (Eds.), Artificial life II. Reading, MA: Addison-Wesley
41.
Wiggins, S. (1990). Introduction to applied nonlinear dynamical systems and chaos. New York: Springer-Verlag .
42.
Wilson, H.R., & Cowan, J.D. (1972). Excitatory and inhibitory interactions in localized populations of model neurons. Biophysical Journal, 12, 1-24.
43.
Yamauchi, B., & Beer, R.D. (1994). Sequential behavior and learning in evolved dynamical neural networks. Adaptive Behavior, 2(3), 219-246.
