On a cross-coupling of Rulkov neural maps

Abstract

We introduce a novel coupling of Rulkov neural maps, proposing a heuristic biological interpretation for the transition to non-small values of the perturbations acting on the slow variables. We analytically prove that the coupling of two identical neurons preserves the emergence of Devaney chaos through the existence of a generalized snap-back repeller, provided that a snap back repeller exists for the original system. We present numerical simulations for the coupling of two different neurons showing the arising of a potential global strange attractor, whose fractal structure is strongly suggested by the computation of a non-integer Kaplan-Yorke dimension. Furthermore, we perform standard numerical studies concerning time series, Lyapunov exponents spectra, bifurcation diagrams and basins of attraction. Finally, we briefly propose a generalization of the coupling to an arbitrary number of neurons.

arXiv comment29 pages, 19 figures. This version contains substantial revisions to the analytical part of the paper. Lemma 1 and Theorem 3 from the previous version have been removed as incorrect. Theorem 4 has been reformulated, while preserving an analogous main result. The numerical results and simulations are unchanged