Chaos in neural networks with a nonmonotonic transfer function

Abstract

Time evolution of diluted neural networks with a nonmonotonic transfer function is analytically described by flow equations for macroscopic variables. The macroscopic dynamics shows a rich variety of behaviors: fixed-point, periodicity, and chaos. We examine in detail the structure of the strange attractor and in particular we study the main features of the stable and unstable manifolds, the hyperbolicity of the attractor, and the existence of homoclinic intersections. We also discuss the problem of the robustness of the chaos and we prove that in the present model chaotic behavior is fragile (chaotic regions are densely intercalated with periodicity windows), according to a recently discussed conjecture. Finally we perform an analysis of the microscopic behavior and in particular we examine the occurrence of damage spreading by studying the time evolution of two almost identical initial configurations. We show that for any choice of the parameters the two initial states remain microscopically distinct.

Instabilities in attractor networks with fast synaptic fluctuations and partial updating of the neurons activity

“…In our case, however, a non-monotonic type of gain function occurs for some values of Φ and ρ (see comparison in figure 4). This has been reported to be important to originate a chaotic dynamics among the attractors (Dominguez and Theumann, 1997;Caroppo et al, 1999).…”

Chaotic hopping between attractors in neural networks

“…In contrast, some related works, in order to deepen more directly on the possible origin of chaos, use the gain function itself as a parameter. It is also remarkable that, e.g., in (Dominguez and Theumann, 1997) and some related work (Caroppo et al, 1999;Mainieri and Jr., 2002;Katayama et al, 2003), the gain function is phenomenologically controlled by tuning the neuron threshold for firing, θ i . The threshold function thus becomes a relevant parameter, and it ensues that any meaningful chaos in this context requires non-zero threshold.…”

“…olfactory stimuli (Ashwin and Timme, 2005). Consequently, there has recently been some effort in incorporating constructive chaos in neural network modeling (Wang et al, 1990;Bolle and Vink, 1996;Dominguez and Theumann, 1997;Caroppo et al, 1999;Poon and Barahona, 2001;Mainieri and Jr., 2002;Katayama et al, 2003). Concluding on the significance of chaos in neurobiological systems is still an open issue (Rabinovich and Abarbanel, 1998;Faure and Korn, 2001;Korn and Faure, 2003), however.…”

Chaos in Diluted Networks With Continuous Neurons

“…the function that gives the state of the neuron as a function of the postsynaptic potential) have been presented [12][13][14][15]. The physiological justification of a nonmonotonic transfer function is ascribed to the fatigue of neurons after being exposed to a large post-synaptic input [16].…”