We modified the FitzHugh-Nagumo model of an excitable medium so that it describes adequately the dynamics of pulse propagation in the canine myocardium. The modified model is simple enough to be used for intensive three-dimensional computations of the whole heart. It simulates the pulse shape and the restitution property of the canine myocardium with good precision. In 1952 Hodgkin and Huxley proposed the first quantitative mathematical model of wave propagation in squid nerve [1]. This work has had a great impact on modeling of various nonlinear phenomena in biology. On the basis of this model Noble in 1962 developed the first physiological model of cardiac E-mail: rubin@wave.biol.ruu.nl; permanent address: Institute of Theoretical and Experimental Biophysics, Puschino, Moscow Region, 142292 Russia A simple model of cardiac excitation 2 tissue [2]. Further studies in this field resulted in the development of several realistic ionic models of cardiac tissue which were derived from ...

Abstract. A formal theory of symmetries of networks of coupled dynamical systems, stated in terms of the group of permutations of the nodes that preserve the network topology, has existed for some time. Global network symmetries impose strong constraints on the corresponding dynamical systems, which affect equilibria, periodic states, heteroclinic cycles, and even chaotic states. In particular, the symmetries of the network can lead to synchrony, phase relations, resonances, and synchronous or cycling chaos. Symmetry is a rather restrictive assumption, and a general theory of networks should be more flexible. A recent generalization of the group-theoretic notion of symmetry replaces global symmetries by bijections between certain subsets of the directed edges of the network, the ‘input sets’. Now the symmetry group becomes a groupoid, which is an algebraic structure that resembles a group, except that the product of two elements may not be defined. The groupoid formalism makes it possible to extend group-theoretic methods to more general networks, and in particular it leads to a complete classification of ‘robust ’ patterns of synchrony in terms of the combinatorial structure of the network. Many phenomena that would be nongeneric in an arbitrary dynamical system can become generic when constrained by a particular network topology. A network of dynamical systems is not just a dynamical system with a high-dimensional phase space. It is also equipped with a canonical set of observables—the states of the individual nodes of the network. Moreover, the form of the underlying ODE is constrained by the network topology—which variables occur in which component equations, and how those equations relate to each other. The result is a rich and new range of phenomena, only a few of which are yet properly understood. Contents 1.

A relaxation oscillator model of neural spiking dynamics is applied to the task of finding downbeats in rhythmical patterns. The importance of downbeat discovery or beat induction is discussed, and the relaxation oscillator model is compared to other oscillator models. In a set of computer simulations the model is tested on 35 rhythmical patterns from Povel and Essens (1985). The model performs well, making good predictions in 34 of 35 cases. In an analysis we identify some shortcomings of the model and relate model behavior to dynamical properties of relaxation oscillators.

In a network of dynamical systems, concurrent synchronization is a regime where multiple groups of fully synchronized elements coexist. In the brain, concurrent synchronization may occur at several scales, with multiple “rhythms ” interacting and functional assemblies combining neural oscillators of many different types. Mathematically, stable concurrent synchronization corresponds to convergence to a flow-invariant linear subspace of the global state space. We derive a general condition for such convergence to occur globally and exponentially. We also show that, under mild conditions, global convergence to a concurrently synchronized regime is preserved under basic system combinations such as negative feedback or hierarchies, so that stable concurrently synchronized aggregates of arbitrary size can be constructed. Simple applications of these results to classical questions in systems neuroscience and robotics are discussed. 1

Abstract. Cardiac arrhythmias can develop complex electrophysiological patterns which complexify the planning and control of therapies, especially in the context of radio-frequency ablation. The development of electrophysiology models aims at testing different therapy strategies. However, current models are computationally expensive and often too complex to be adjusted with limited clinical data. In this paper, we propose a real-time method to simulate cardiac electrophysiology on triangular meshes. This model is based on a multi-front integration of the Fast Marching Method. This efficient approach opens new possibilities, including the ability to directly integrate modelling in the interventional room. 1

If a traveling wave is stable or unstable depends essentially on the spectrum of a differential operator P obtained by linearization. We investigate how spectral properties of the all--line operator P are related to eigenvalues of finite--interval BVPs Pu(x) = su(x); x \Gamma x x+ ; Ru = 0. Here R is a linear boundary operator, for which we will derive determinant conditions, and the x--interval is assumed to be sufficiently large. Under suitable assumptions, we show (a) resolvent estimates for large s; (b) if s is in the resolvent of the all--line operator P , then s is also in the resolvent for finite--interval BVPs; (c) eigenvalues of P lead to approximating eigenvalues on finite intervals. These results allow to study the stability question for traveling waves by investigating eigenvalues of finite--interval problems. We give applications to the FitzHugh--Nagumo system with small diffusion and to the complex Ginzburg--Landau equations. Key words: Traveling waves, stability, expo...

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In this article we address the problem of estimating the parameters of an electrophysiological model of the heart from a set of electrical recordings. The chosen model is the reaction-diffusion model on the transmembrane potential proposed by Aliev and Panfilov. For this model, we estimate a local apparent 2D conductivity from a measured depolarization time distribution. First, we perform an initial adjustment including the choice of initial conditions and of a set of global parameters. We then propose a local estimation by minimizing the quadratic error between the depolarization time computed by the model and the measures. As a first step we address the problem on the epicardial surface in the case of an isotropic version of the Aliev and Panfilov model. The minimization is performed using Brent method without computing the derivative of the error. The feasibility of the approach is demonstrated on synthetic electrophysiological measurements. A proof of concept is obtained on real electrophysiological measures of normal and infarcted canine hearts.

The search for chaotic patterns has occupied numerous investigators in neuroscience, as in many other fields of science. Their results and main conclusions are reviewed in the light of the most recent criteria that need to be satisfied since the first descriptions of the surrogate strategy. The methods used in each of these studies have almost invariably combined the analysis of experimental data with simulations using formal models, often based on modified Huxley and Hodgkin equations and/or of the Hindmarsh and Rose models of bursting neurons. Due to technical limitations, the results of these simulations have prevailed over experimental ones in studies on the nonlinear properties of large cortical networks and higher brain functions. Yet, and although a convincing proof of chaos (as defined mathematically) has only been obtained at the level of axons, of single and coupled cells, convergent results can be interpreted as compatible with the notion that signals in the brain are distributed according to chaotic patterns at all levels of its various forms of hierarchy. This chronological account of the main landmarks of nonlinear neurosciences follows an earlier publication [Faure, Korn, C. R. Acad. Sci. Paris, Ser. III 324 (2001) 773–793] that was focused on the basic concepts of nonlinear dynamics and methods of investigations which allow chaotic processes to be distinguished from stochastic ones and on the rationale for envisioning their control using external perturbations. Here we present the data and main arguments that support the existence of chaos at all levels from the simplest to the most complex forms of organization of the nervous system.

We study networks of relaxation oscillators coupled with time delay synapses. A pair of oscillators is analyzed and shown to attain loosely synchronous solutions for a wide range of initial conditions and time delays. Simulations of one and two dimensional oscillator networks indicate that locally coupled oscillators are also loosely synchronous. Desynchronous solutions are possible when system parameters are varied. To characterize loosely synchronous networks, we introduce a measure of synchrony, the maximum time difference between any two oscillators. In locally excitatory globally inhibitory oscillator networks with time delays, we find that desynchronous solutions for different groups of oscillators are maintained, and the number of groups that can be segregated is related to the maximum time difference within each group. To examine the maximum time difference, we display its histograms for oscillator networks in one and two dimensions. Also, a range of initial conditions is given...