Gemma Huguet

A Computational and Geometric Approach to Phase Resetting Curves and Surfaces

This work arises from the purpose of applying new tools in dynamical systems to time problems in biological systems. The main aim of this paper is to develop a numerical method to perform the effective computation of the phase advancement when we stimulate an oscillator which has not yet reached the asymptotic state (a limit cycle). That is, we want to extend the computation of the phase resetting curves (PRCs) (the classical tool to compute the phase advancement) to a neighborhood of the limit cycle, obtaining what we call the phase resetting surfaces (PRSs). To achieve this goal we first perform a careful study of the theoretical grounds (the parameterization method for invariant manifolds and another approach using Lie symmetries), which allows us to describe the isochronous sections of the limit cycle and, from them, to obtain the PRSs. In order to make this theoretical framework applicable, we use the numerical algorithms of the parameterization method and other semianalytical tools to extend invaria...

Noise and adaptation in multistable perception: Noise drives when to switch, adaptation determines percept choice

We study the dynamics of perceptual switching in ambiguous visual scenes that admit more than two interpretations/percepts to gain insight into the dynamics of perceptual multistability and its underlying neural mechanisms. We focus on visual plaids that are tristable and we present both experimental and computational results. We develop a firing-rate model based on mutual inhibition and adaptation that involves stochastic dynamics of multiple-attractor systems. The model can account for the dynamic properties (transition probabilities, distributions of percept durations, etc.) observed in the experiments. Noise and adaptation have both been shown to play roles in the dynamics of bistable perception. Here, tristable perception allows us to specify the roles of noise and adaptation in our model. Noise is critical in considering the time of a switch. On the other hand, adaptation mechanisms are critical in considering perceptual choice (in tristable perception, each time a percept ends, there is a possible choice between two new percepts).

Geography of resonances and Arnold diffusion in a priori unstable Hamiltonian systems

In this paper we consider the case of a general perturbation, for r large enough, of an a priori unstable Hamiltonian system of 2 + 1/2 degrees of freedom, and we provide explicit conditions on it, which turn out to be generic and are verifiable in concrete examples, which guarantee the existence of Arnold diffusion.This is a generalization of the result in Delshams et al (2006 Mem. Am. Math. Soc.) where the case of a perturbation with a finite number of harmonics in the angular variables was considered.The method of proof is based on a careful analysis of the geography of resonances created by a generic perturbation and it contains a deep quantitative description of the invariant objects generated by the resonances therein. The scattering map is used as an essential tool to construct transition chains of objects of different topology. The combination of quantitative expressions for both the geography of resonances and the scattering map provides, in a natural way, explicit computable conditions for instability.

Computation of limit cycles and their isochrons: Fast algorithms and their convergence

We present efficient algorithms to compute limit cycles and their isochrons (i.e., the sets of points with the same asymptotic phase) for planar vector fields. We formulate a functional equation for the parameterization of the invariant cycle and its isochrons, and we show that it can be solved by means of a Newton method. Using the right transformations, we can solve the equation of the Newton step efficiently. The algorithms are efficient in the sense that if we discretize the functions using

Phase-Amplitude Response Functions for Transient-State Stimuli

N

Neuroprotective Role of Gap Junctions in a Neuron Astrocyte Network Model

points, a Newton step requires

Computation of Invariant Curves in the Analysis of Periodically Forced Neural Oscillators

O(N)

Quasiperiodic perturbations of heteroclinic attractor networks

storage and

Effects of Thermoregulation on Human Sleep Patterns: A Mathematical Model of Sleep–Wake Cycles with REM–NREM Subcircuit

O(N\log N)

The influence of stationary synaptic activity on the PRC

operations in Fourier discretization or