A bifurcation theory approach to vector field programming for periodic attractors

Analytic methods of bifurcation theory are used to design algorithms for determining synaptic weights in recurrent analog neural network architectures. Nonnumerical formulas, as well as numerical learning algorithms using hidden units for the storage of static and periodic attractors, are introduced. These algorithms allow programming of the network vector field regardless of the patterns to be stored. Stability of patterns and cycles, basin geometry, and rates of convergence may be controlled. For a network of n nodes, n static of n/2 periodic attractors can be stored by a projection algorithm. For orthogonal patterns, this learning operation reduces to a kind of periodic outer product rule that allows local, additive, and commutative incremental learning. Standing- or traveling-wave cycles may be stored to mimic the kind of oscillating spatial patterns that appear in the neural activity of the olfactory bulb and prepyriform cortex during inspiration, and suffice, in the bulb, to predict the pattern recognition behavior of rabbits in classical conditioning experiments. These attractors arise, during simulated inspiration, through a multiple Hopf bifurcation, which can act as a critical decision point for their selection by very small input patterns.<<ETX>>

[1]  M. Hirsch,et al.  Differential Equations, Dynamical Systems, and Linear Algebra , 1974 .

[2]  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.

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

[4]  M. Golubitsky,et al.  Singularities and Groups in Bifurcation Theory: Volume I , 1984 .

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

[6]  B. Baird.  Nonlinear dynamics of pattern formation and pattern recognition in the rabbit olfactory bulb , 1986 .

[7]  W. Freeman,et al.  Spatial patterns of visual cortical fast EEG during conditioned reflex in a rhesus monkey , 1987, Brain Research.

[8]  B. Baird,et al.  Relation of olfactory EEG to behavior: spatial analysis. , 1987, Behavioral neuroscience.

[9]  C. L. Giles,et al.  High order correlation model for associative memory , 1987 .

[10]  B. Baird,et al.  Bifurcation theory methods for programming static or periodic attractors and their bifurcations in dynamic neural networks , 1988, IEEE 1988 International Conference on Neural Networks.

[11]  Bill Baird.  A Bifurcation Theory Approach to the Programming of Periodic Attractors in Network Models of Olfactory Cortex , 1988, NIPS.

[12]  Barak A. Pearlmutter.  Learning State Space Trajectories in Recurrent Neural Networks , 1988, Neural Computation.

[13]  W. Singer,et al.  Stimulus-specific neuronal oscillations in orientation columns of cat visual cortex. , 1989, Proceedings of the National Academy of Sciences of the United States of America.

[14]  Ronald J. Williams,et al.  A Learning Algorithm for Continually Running Fully Recurrent Neural Networks , 1989, Neural Computation.

[15]  Michael I. Jordan.  Attractor dynamics and parallelism in a connectionist sequential machine , 1990 .