Rulkov neural maps cross-coupling proves chaos with fractal attractors

Mathematician Stefano Disca has published a paper on arXiv (2607.22318) introducing a novel cross-coupling scheme for Rulkov neural maps—a class of discrete-time dynamical systems used to model neuronal firing. The key contribution is a rigorous analytical proof that this coupling preserves boundedness of motion and, critically, the existence of a snap-back repeller. Via the Marotto theorem, this implies Devaney chaos: sensitive dependence on initial conditions, topological transitivity, and dense periodic orbits. Disca also provides a heuristic biological interpretation, explaining how perturbations on slow variables can transition to non-small values.

Numerical simulations for two coupled chaotic Rulkov maps confirm the theoretical predictions. Orbits converge to a global strange attractor whose fractal structure is strongly indicated by a non-integer Kaplan-Yorke dimension. Standard chaos diagnostics—Lyapunov exponent spectra, bifurcation diagrams, and basins of attraction—all corroborate the emergence of complex dynamics. The paper also briefly generalizes the coupling to an arbitrary number of neurons, opening the door for scalable studies of neural network chaos. With 26 pages and 19 figures, this work bridges rigorous nonlinear dynamics and computational neuroscience.