Conference abstracts

Session B1 - Computational Dynamics

July 13, 16:00 ~ 16:30 - Room B1

Topological changes in slow-fast systems: chaotic neuron models

Roberto Barrio

University of Zaragoza, Spain   -   rbarrio@unizar.es

The wide-range assessment of brain dynamics is one of the pivotal challenges of this century. To understand how an incredibly sophisticated system such as the brain functions dynamically, it is imperative to study the dynamics of its constitutive elements -- neurons. Therefore, the design of mathematical models for neurons has arisen as a trending topic in science for a few decades. Mathematical neuron models are examples of fast-slow systems, and they exhibit several typical behaviours and bifurcations, like chaotic dynamics, spike-adding bifurcations and so on.

Two key open questions are how the chaotic behavior is organized [1, 2] and how spike-adding bifurcations influence chaotic behavior. In this talk we show how the orbit-flip (OF) codimension-2 bifurcation points, placed in homoclinic bifurcation curves and related with the spike-adding bifurcations [1, 2, 3], originate countable pencils of period-doubling and saddle-node (of limit cycles) bifurcation lines, but also of symbolic-flip bifurcations [4]. These bifurcations appear interlaced and generate the different symbolic sequences of periodic orbits, constituting the skeleton of the different chaotic attractors and determining their topological structure [4]. The study of the changes in the chaotic invariants is done by analyzing the gradual change in the spectrum of periodic orbits embedded in the invariant, and the onion-like structure in parameter space can be understood directly in terms of symbolic dynamics. The use of several numerical techniques, as continuation techniques, Lyapunov exponents, detection of unstable periodic orbits foliated to the chaotic invariants, template analysis and so on, has played a relevant role in the complete analysis of the problem [4].

[1] Barrio R., Martinez M. A. , Serrano S., Shilnikov A. (2014) Macro and micro-chaotic structures in the Hindmarsh-Rose model of bursting neurons. Chaos 24(2):023128.

[2] Barrio R., Lefranc M., Martinez M. A., Serrano S. (2015) Symbolic dynamical unfolding of spike-adding bifurcations in chaotic neuron models. EPL 109:20002.

[3] Barrio R., Ibanez S., Perez L. (2017) Hindmarsh-Rose model: close and far to the singular limit. Physics Letters A 381(6), 597–603.

[4] Barrio R., Lefranc M., Serrano S. (2017). Topological changes in chaotic neuron models. Preprint.

Joint work with Marc Lefranc (Universite Lille I, France), M. Angeles Martinez (University of Zaragoza, Spain) and Sergio Serrano (University of Zaragoza, Spain).

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