Quasiperiodic perturbations of heteroclinic attractor networks

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 quasiperiodic perturbations. For this purpose, we present a methodology based on the separatrix map to model the dynamics close to heteroclinic networks with quasiperiodic perturbations. Our methodology unifies two different approaches, one based on Melnikov integrals and the other 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 instabilitiesPeer ReviewedPostprint (author's final draft

Study of perturbations of an oscillating neuronal network via phase-amplitude response functions

Phase reduction is a powerful tool for understanding the behavior of perturbed oscillators. It allows for the description of high-dimensional oscillatory systems in terms of a single variable, the phase. Alternatively, mean-field models are a viable option to make the analysis of large systems more tractable. In this work, we apply a phase-amplitude technique on a mean-field model for a network of quadratic integrate-and-fire neurons, which is exact in the thermodynamic limit. This methodology allows us to compute the global isochrons and isostables of the system, and a generalization of the phase response curve beyond the limit cycle constraint: the phase and amplitude response functions. We compare the perturbed dynamics of the oscillating mean-field system with its N-dimensional counterpart, which also exhibits synchronized spiking, and observe how the response functions are able to predict accurately the evolution of the network. Moreover, since the model exhibits slow-fast dynamics, the method yields a dimensionality reduction restricted to the slow stable manifold of the system