Mean-field equations, bifurcation map and route to chaos in discrete time neural networks

Results regarding probable bifurcations from fixed points are presented in the context of general dynamical systems (real, random matrices), time-delay dynamical systems (companion matrices), and a set of mappings known for their properties as universal approximators (neural networks). The eigenvalue spectra is considered both numerically and analytically using previous work of Edelman et. al. Based upon the numerical evidence, various conjectures are presented. The conclusion is that in many circumstances, most bifurcations from fixed points of large dynamical systems will be due to complex eigenvalues. Nevertheless, surprising situations are presented for which the aforementioned conclusion is not general, e.g. real random matrices with Gaussian elements with a large positive mean and finite variance.

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“…al. [10], and Sompolinsky et. al [38] have arrived at similar conclusions in a variety of circumstances.…”

Section: Beyond Fixed Pointsmentioning

confidence: 99%

Probability of Local Bifurcation Type from a Fixed Point: A Random Matrix Perspective

Albers

Sprott

2006

J Stat Phys

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“…What happens if one increases the size ? The statistical behavior of this model has been widely studied in [8,9,11]. It has been shown that chaos generically occurs provided g is sufficiently large.…”

Section: General Setting and Purposesmentioning

confidence: 99%

Transmitting a signal by amplitude modulation in a chaotic network

Cessac¹,

Sepulchre²

2006

Chaos: An Interdisciplinary Journal of Nonlinear Science

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We discuss the ability of a network with non linear relays and chaotic dynamics to transmit signals, on the basis of a linear response theory developed by Ruelle [1] for dissipative systems. We show in particular how the dynamics interfere with the graph topology to produce an effective transmission network, whose topology depends on the signal, and cannot be directly read on the "wired" network. This leads one to reconsider notions such as "hubs". Then, we show examples where, with a suitable choice of the carrier frequency (resonance), one can transmit a signal from a node to another one by amplitude modulation, in spite of chaos. Also, we give an example where a signal, transmitted to any node via different paths, can only be recovered by a couple of specific nodes. This opens the possibility for encoding data in a way such that the recovery of the signal requires the knowledge of the carrier frequency and can be performed only at some specific node.

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“…E is the fixed external pattern applied as a bias to the network, and a* is the strength with which the external pattern is applied. This network has previously been studied and theoretically considered ( [DCQS93], [DCQS95], [QDS95], [CDQS94], [SCS88]). The P.I.…”

Section: Aj (T + L)= Fand^gwjifiiit)) + Ae {mentioning

confidence: 99%

Summer Research Program - 1996. Summer Faculty Research Program Final Reports. Volume 3A, Phillips Laboratory

Moore¹

1996

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Mean-field equations, bifurcation map and route to chaos in discrete time neural networks

Cited by 62 publications

References 25 publications

Probability of Local Bifurcation Type from a Fixed Point: A Random Matrix Perspective

Probability of Local Bifurcation Type from a Fixed Point: A Random Matrix Perspective

Transmitting a signal by amplitude modulation in a chaotic network

Summer Research Program - 1996. Summer Faculty Research Program Final Reports. Volume 3A, Phillips Laboratory

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