Wild oscillations in a nonlinear neuron model with resets: (II) Mixed-mode oscillations

This work continues the analysis of complex dynamics in a class of bidimensional nonlinear hybrid dynamical systems with resets modeling neuronal voltage dynamics with adaptation and spike emission. We show that these models can generically display a form of mixed-mode oscillations (MMOs), which are trajectories featuring an alternation of small oscillations with spikes or bursts (multiple consecutive spikes). The mechanism by which these are generated relies fundamentally on the hybrid structure of the flow: invariant manifolds of the continuous dynamics govern small oscillations, while discrete resets govern the emission of spikes or bursts, contrasting with classical MMO mechanisms in ordinary differential equations involving more than three dimensions and generally relying on a timescale separation. The decomposition of mechanisms reveals the geometrical origin of MMOs, allowing a relatively simple classification of points on the reset manifold associated to specific numbers of small oscillations. We show that the MMO pattern can be described through the study of orbits of a discrete adaptation map, which is singular as it features discrete discontinuities with unbounded left- and right-derivatives. We study orbits of the map via rotation theory for discontinuous circle maps and elucidate in detail complex behaviors arising in the case where MMOs display at most one small oscillation between each consecutive pair of spikes

Topological-numerical analysis of a two-dimensional discrete neuron model

We conduct computer-assisted analysis of the two-dimensional model of a neuron introduced by Chialvo in 1995 (Chaos, Solitons & Fractals 5, 461-479). We apply the method for rigorous analysis of global dynamics based on a set-oriented topological approach, introduced by Arai et al. in 2009 (SIAM J. Appl. Dyn. Syst. 8, 757-789) and improved and expanded afterwards. Additionally, we introduce a new algorithm to analyze the return times inside a chain recurrent set. Based on this analysis, together with the information on the size of the chain recurrent set, we develop a new method that allows one to determine subsets of parameters for which chaotic dynamics may appear. This approach can be applied to a variety of dynamical systems, and we discuss some of its practical aspects. The data and the software described in the paper are available at http://www.pawelpilarczyk.com/neuron/

Analysis of Interspike-Intervals for the General Class of Integrate-and-Fire Models with Periodic Drive

International audienc

Wild oscillations in a nonlinear neuron model with resets: (II) Mixed-mode oscillations

International audienceThis work continues the analysis of complex dynamics in a class of bidimensional nonlinear hybrid dynamical systems with resets modeling neuronal voltage dynamics with adaptation and spike emission. We show that these models can generically display a form of mixed-mode oscillations (MMOs), which are trajectories featuring an alternation of small oscillations with spikes or bursts (multiple consecutive spikes). The mechanism by which these are generated relies fundamentally on the hybrid structure of the flow: invariant manifolds of the continuous dynamics govern small oscillations, while discrete resets govern the emission of spikes or bursts, contrasting with classical MMO mechanisms in ordinary differential equations involving more than three dimensions and generally relying on a timescale separation. The decomposition of mechanisms reveals the geometrical origin of MMOs, allowing a relatively simple classification of points on the reset manifold associated to specific numbers of small oscillations. We show that the MMO pattern can be described through the study of orbits of a discrete a daptation map, which is singular as it features discrete discontinuities with unbounded left-and right-derivatives. We study orbits of the map via rotation theory for discontinuous circle maps and elucidate in detail complex behaviors arising in the case where MMOs display at most one small oscillation between each consecutive pair of spikes

Wild oscillations in a nonlinear neuron model with resets: (I) Bursting, spike adding and chaos

International audienceIn a series of two papers, we investigate the mechanisms by which complex oscillations are generated in a class of nonlinear dynamical systems with resets modeling the voltage and adaptation of neurons. This first paper presents mathematical analysis showing that the system can support bursts of any period as a function of model parameters, and that these are organized in a period-incrementing structure. In continuous dynamical systems with resets, such structures are complex to analyze. In the present context, we use the fact that bursting patterns correspond to periodic orbits of the adaptation map that governs the sequence of values of the adaptation variable at the resets. Using a slow-fast approach, we show that this map converges towards a piecewise linear discontinuous map whose orbits are exactly characterized. That map shows a period-incrementing structure with instantaneous transitions. We show that the period-incrementing structure persists for the full system with non-constant adaptation, but the transitions are more complex. We investigate the presence of chaos at the transitions

Wild oscillations in a nonlinear neuron model with resets: (I) Bursting, spike-adding and chaos

Orientador: Professor Dr. Ennio LuzDissertaçao (mestrado) -Universidade Federal do Paraná, Curso de Pós-Graduação em EntomologiaInclui referências: p. 41-46Resumo: São apresentadas observações feitas durante um ano de capturas de tabanídeos no Litoral e Primeiro Planalto paranaenses. Procurou-se, em principio, observar o comportamento dos dípteros visando informações sobre os representantes desse grupo distribuídos no leste paranaense. E relacionou-se as espécies capturada, épocas de maior atividade dos insetos e as espécies que ocorrem em cada área. Através de uma coleta de treze horas seguidas observou-se, também, a atividade diária das mutucas. Nas capturas foram utilizados, como isca, equinos e muares, para a verificação da preferência pelas cores das pelagens e pelas regiões do corpo dos animais.Abstract: Information on tabanids collected throughout a period of one year in the coastal and first plateau of eastern Parana, Brazil are presented. Observations were made on the behavior of tabanids - occurring in Parana to provide data on the representative fauna. The species caught related of greatest activity in each of two regions. Activity of these tabanids was observed troughout a thirteen hour continuous collecting period. Horses and mules were used as bait for catching the tabanids, and preference for different skin color and body regions of the animal were studied