Quasi-periodic perturbations of heteroclinic attractor networks

TL;DR

This paper demonstrates that quasi-periodic perturbations can produce Gamma-distributed dominance times in heteroclinic attractor networks, offering a noise-free alternative to explain bistable perception phenomena like binocular rivalry.

Contribution

It introduces a unified methodology based on the separatrix map to model heteroclinic networks with quasi-periodic perturbations, applicable to both Hamiltonian and non-Hamiltonian systems.

Findings

Gamma distributions of dominance times are achievable without noise.

The separatrix map simplifies modeling and reduces computational costs.

Chaotic behavior observed in perturbed systems aligns with experimental data.

Abstract

We consider heteroclinic attractor networks motivated by models of competition between neural populations during binocular rivalry. We show that Gamma distributions of dominance times observed experimentally in binocular rivalry and other forms of bistable perception, commonly explained by means of noise in the models, can be achieved with quasi-periodic perturbations. For this purpose, we present a methodology based on the separatrix map to model the dynamics close to heteroclinic networks with quasi-periodic perturbations. Our methodology unifies two different approaches, one based on Melnikov integrals and another one based on variational equations. We apply it to two models: first, to the Duffing equation, which comes from the perturbation of a Hamiltonian system and, second, to a heteroclinic attractor network for binocular rivalry, for which we develop a suitable method based on Melnikov integrals for non-Hamiltonian systems. In both models, the perturbed system shows chaotic behavior while dominance times achieve good agreement with Gamma distributions. Moreover, the separatrix map provides a new (discrete) model for bistable perception which, in addition, replaces the numerical integration of time-continuous models and, consequently, reduces the computational cost and avoids numerical instabilities.