A bifurcation analysis of the four dimensional generalized Hopfield neural network

Abstract A four neuron Hopfield neural network with asymmetric weights and self-connection is analyzed. Its stable steady state and periodic attractors are identified and a complete bifurcation diagram is constructed. A center manifold reduction is undertaken and by means of normal form theory, the characteristics of the limit cycles are obtained. The network is seen to have a large memory storage capacity with both fixed point and periodic memories coexisting. The onset of chaos as reported previously is considered.

[1]  L. Glass,et al.  STEADY STATES, LIMIT CYCLES, AND CHAOS IN MODELS OF COMPLEX BIOLOGICAL NETWORKS , 1991 .

[2]  Y. Sandler.  Model of neural networks with selective memorization and chaotic behavior , 1990 .

[3]  W. Martienssen,et al.  Lyapunov exponents and dimensions of chaotic neural networks , 1991 .

[4]  Wolfgang Kinzel,et al.  Freezing transition in asymmetric random neural networks with deterministic dynamics , 1989 .

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

[6]  J. Crawford.  Introduction to bifurcation theory , 1991 .

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

[8]  P. Das,et al.  Two neuron dynamics and adiabatic elimination , 1993 .

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

[10]  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.

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

[12]  J J Hopfield,et al.  Neural networks and physical systems with emergent collective computational abilities. , 1982, Proceedings of the National Academy of Sciences of the United States of America.

[13]  F. R. Waugh,et al.  Fixed-point attractors in analog neural computation. , 1990, Physical review letters.

[14]  W. Martienssen,et al.  Quasi-Periodicity Route to Chaos in Neural Networks , 1989 .

[15]  Xin Wang,et al.  Period-doublings to chaos in a simple neural network , 1991, IJCNN-91-Seattle International Joint Conference on Neural Networks.

[16]  S. Renals,et al.  A study of network dynamics , 1990 .

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

[18]  W. Kinzel,et al.  Hopfield network with directed bonds , 1989 .

[19]  Charles M. Marcus,et al.  Nonlinear dynamics and stability of analog neural networks , 1991 .