Antonio E. Teruel

Existence of a Reversible T-Point Heteroclinic Cycle in a Piecewise Linear Version of the Michelson System

The proof of the existence of a global connection in differential systems is generally a difficult task. Some authors use numerical techniques to show this existence, even in the case of continuous piecewise linear systems. In this paper we give an analytical proof of the existence of a reversible T-point heteroclinic cycle in a continuous piecewise linear version of the widely studied Michelson system. The principal ideas of this proof can be extended to other piecewise linear systems.

Existence of homoclinic connections in continuous piecewise linear systems

Numerical methods are often used to put in evidence the existence of global connections in differential systems. The principal reason is that the corresponding analytical proofs are usually very complicated. In this work we give an analytical proof of the existence of a pair of homoclinic connections in a continuous piecewise linear system, which can be considered to be a version of the widely studied Michelson system. Although the computations developed in this proof are specific to the system, the techniques can be extended to other piecewise linear systems.

Canards, Folded Nodes, and Mixed-Mode Oscillations in Piecewise-Linear Slow-Fast Systems

Canard-induced phenomena have been extensively studied in the last three decades, from both the mathematical and the application viewpoints. Canards in slow-fast systems with (at least) two slow variables, especially near folded-node singularities, give an essential generating mechanism for mixed-mode oscillations (MMOs) in the framework of smooth multiple timescale systems. There is a wealth of literature on such slow-fast dynamical systems and many models displaying canard-induced MMOs, particularly in neuroscience. In parallel, since the late 1990s several papers have shown that the canard phenomenon can be faithfully reproduced with piecewise-linear (PWL) systems in two dimensions, although very few results are available in the three-dimensional case. The present paper aims to bridge this gap by analyzing canonical PWL systems that display folded singularities, primary and secondary canards, with a similar control of the maximal winding number as in the smooth case. We also show that the singular phase portraits are compatible in both frameworks. Finally, we show using an example how to construct a (linear) global return and obtain robust PWL MMOs.

Noose bifurcation and crossing tangency in reversible piecewise linear systems

The structure of the periodic orbits analysed in this paper was first reported in the Michelson system by Kent and Elgin (1991 Nonlinearity 4 1045–61), who gave it the name noose bifurcation. Recently, it has been found in a piecewise linear system with two linearity zones separated by a plane, which is called the separation plane. In this system, the orbits that take part in the noose bifurcation have two and four points of intersection with the separation plane, and they are arranged in two curves that are connected by a point where the periodic orbit has a crossing tangency with the separation plane. In this work, we analytically prove the local existence of the curve of periodic orbits with four intersections that emerges from the point corresponding to the crossing tangency. Moreover, we add a numerical study of the stability and bifurcations of the periodic orbits involved in the noose curve for the piecewise linear system and check that they exhibit the same configuration as that of the Michelson system.

Canard solutions in planar piecewise linear systems with three zones

ABSTRACT In this work, we analyze the existence and stability of canard solutions in a class of planar piecewise linear systems with three zones, using a singular perturbation theory approach. To this aim, we follow the analysis of the classical canard phenomenon in smooth planar slow–fast systems and adapt it to the piecewise-linear framework. We first prove the existence of an intersection between repelling and attracting slow manifolds, which defines a maximal canard, in a non-generic system of the class having a continuum of periodic orbits. Then, we perturb this situation and we prove the persistence of the maximal canard solution, as well as the existence of a family of canard limit cycles in this class of systems. Similarities and differences between the piecewise linear case and the smooth one are highlighted.

Noose Structure and Bifurcations of Periodic Orbits in Reversible Three-Dimensional Piecewise Linear Differential Systems

The main goal of this work is to describe the periodic behavior of a class of three-dimensional reversible piecewise linear continuous systems. More concretely, we study an interesting structure called the noose bifurcation that was previously detected by Kent and Elgin in the Michelson system. We numerically obtain the curves of periodic orbits that appear from the bifurcations at the noose curve, where other phenomena related to different types of tangencies with the separation plane arise. Besides that, we show that some of these curves of periodic orbits wiggle around global connections when the period increases. The complete structure of periodic orbits, including the stability and bifurcations, coincides with the one observed in the Michelson system. However, we also point out the relevance of the crossing tangency and the small loop that emerges from it in the existence of the noose bifurcation.

Global dynamics of a family of 3-D Lotka–Volterra systems

In this article, we analyse the flow of a family of three-dimensional Lotka–Volterra systems restricted to an invariant and bounded region. The behaviour of the flow in the interior of this region is simple: either every orbit is a periodic orbit or orbits move from one boundary to another. Nevertheless, the complete study of the limit sets in the boundary allows one to understand the bifurcations which take place in the region as a global bifurcation that we denote by focus-centre-focus bifurcation.

Estimation of Synaptic Conductance in the Spiking Regime for the McKean Neuron Model

In this work, we aim at giving a first proof of concept to address the estimation of synaptic conductances when a neuron is spiking, a complex inverse nonlinear problem which is an open challenge in neuroscience. Our approach is based on a simplified model of neuronal activity, namely, a piecewise linear version of the FitzHugh--Nagumo model. This simplified model allows precise knowledge of the nonlinear f-I curve by using standard techniques of nonsmooth dynamical systems. In the regular firing regime of the neuron model, we obtain an approximation of the period which, in addition, improves previous approximations given in the literature to date. By knowing both this expression of the period and the current applied to the neuron, and then solving an inverse problem with a unique solution, we are able to estimate the steady synaptic conductance of the cells oscillatory activity. Moreover, the method gives also good estimations when the synaptic conductance varies slowly in time.

Nonlinear Estimation of Synaptic Conductances via Piecewise Linear Systems

We use the piecewise linear McKean model to present a proof-of-concept to address the estimation of synaptic conductances when a neuron is spiking. Using standard techniques of non-smooth dynamical systems, we obtain an approximation of the period in terms of the parameters of the system which allows to estimate the steady synaptic conductance of the spiking neuron. The method gives also fairly good estimations when the synaptic conductances vary slowly in time.

Estimation of the synaptic conductance in a McKean-model neuron

Estimating the synaptic conductances impinging on a single neuron directly from its membrane potential is one of the open problems to be solved in order to understand the flow of information in the brain. Despite the existence of some computational strategies that give circumstantial solutions ([1-3] for instance), they all present the inconvenience that the estimation can only be done in subthreshold activity regimes. The main constraint to provide strategies for the oscillatory regimes is related to the nonlinearity of the input-output curve and the difficulty to compute it. In experimental studies it is hard to obtain these strategies and, moreover, there are no theoretical indications of how to deal with this inverse non-linear problem. In this work, we aim at giving a first proof of concept to address the estimation of synaptic conductances when the neuron is spiking. For this purpose, we use a simplified model of neuronal activity, namely a piecewise linear version of the Fitzhugh-Nagumo model, the McKean model ([4], among others), which allows an exact knowledge of the nonlinear f-I curve by means of standard techniques of non-smooth dynamical systems. As a first step, we are able to infer a steady synaptic conductance from the cells oscillatory activity. As shown in Figure ​Figure1,1, the model shows the relative errors of the conductances of order C, where C is the membrane capacitance (C<<1), notably improving the errors obtained using filtering techniques on the membrane potential plus linear estimations, see numerical tests performed in [5]. Figure 1 Goodness of fit of the synaptic conductance parameter. Panel A represents the relative error versus the applied current for a fixed value of C = 10-4. Red points represent the values of I1 (left points) and I2 (right points) for each gsyn. Panel B represents ...