Ciències Matemàtiques i Informàtica
http://hdl.handle.net/11201/256
2020-02-20T11:27:58ZCanards, Folded Nodes and Mixed-Mode Oscillations in Piecewise-Linear Systems
http://hdl.handle.net/11201/150868
Canards, Folded Nodes and Mixed-Mode Oscillations in Piecewise-Linear Systems
Desroches, M.; Guillamont, A.; Ponce, E.; Prohens, R.; Rodrigues, S.; Teruel, A.E.
[eng] 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.
Estimation of Synaptic Conductances in the Spiking Regime for the McKean Neuron Model
http://hdl.handle.net/11201/150867
Estimation of Synaptic Conductances in the Spiking Regime for the McKean Neuron Model
Guillamon, A.; Prohens, R.; Teruel A.E.; Vich, C.
[eng] 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 cell's oscillatory activity. Moreover, the method gives also good estimations when the synaptic conductance varies slowly in time.
Existence of a Reversible T-Point Heteroclinic Cycle in a Piecewise Linear Version of the Michelson System
http://hdl.handle.net/11201/150866
Existence of a Reversible T-Point Heteroclinic Cycle in a Piecewise Linear Version of the Michelson System
Carmona, V.; Fernández-Sánchez, F.; Teruel, A.E.
[eng] 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.
Canard trajectories in 3D piecewise linear systems
http://hdl.handle.net/11201/150865
Canard trajectories in 3D piecewise linear systems
Prohens, R.; Teruel, A.E.
[eng] We present some results on singularly perturbed piecewise linear systems, similar to those obtained by the Geometric Singular Perturbation Theory. Unlike the differentiable case, in the piecewise linear case we obtain the global expression of the slow manifold Sε. As a result, we characterize the existence of canard orbits in such systems. Finally, we apply the above theory to a specific case where we show numerical evidences of the existence of a canard cycle.