Excitable Media Seminar

The simulation data presented here, and the conceptual framework developed for their interpretation are, both, in need of substantial refinement and extension. However, granting that they are initial pointers of some merit, and elementary indicators of general principles, several implications follow: the activity patterns of neurons and their assemblies are\ud interdependent with the extracellular milieu in which they are embedded, and to whose time varying composition they contribute. The complexity of this interdependence in the temporal dimension forecloses any time and context invariant relation between what the experimenter may consider stimulus input and its representation in neural activity. Hence, ideas of coding by (quasi)-digital neurons are called in question by the mutual interdependence of neurons and their\ud humoral milieu. Instead, concepts of 'mass action' in the Nervous system gain a new perspective: this time augmented by including the chemical medium surrounding neurons as part of the dynamics of the system as a whole. Accordingly, a meaningful way to describe activity in a neuron assembly would be in terms of a state space in which it can move along an infinite number of trajectories.\u

Burridge-Knopoff Models as Elastic Excitable Media

We construct a model of an excitable medium with elastic rather than the usual diffusive coupling. We explore the dynamics of elastic excitable media, which we find to be dominated by low dimensional structures, including global oscillations, period-doubled pacemakers, and propagating fronts. We suggest that examples of elastic excitable media are to be found in such diverse physical systems as Burridge-Knopoff models of frictional sliding, electronic transmission lines, and active optical waveguides

A normal form for excitable media

We present a normal form for travelling waves in one-dimensional excitable media in form of a differential delay equation. The normal form is built around the well-known saddle-node bifurcation generically present in excitable media. Finite wavelength effects are captured by a delay. The normal form describes the behaviour of single pulses in a periodic domain and also the richer behaviour of wave trains. The normal form exhibits a symmetry preserving Hopf bifurcation which may coalesce with the saddle-node in a Bogdanov-Takens point, and a symmetry breaking spatially inhomogeneous pitchfork bifurcation. We verify the existence of these bifurcations in numerical simulations. The parameters of the normal form are determined and its predictions are tested against numerical simulations of partial differential equation models of excitable media with good agreement.Comment: 22 pages, accepted for publication in Chao

Dynamics of Elastic Excitable Media

The Burridge-Knopoff model of earthquake faults with viscous friction is equivalent to a van der Pol-FitzHugh-Nagumo model for excitable media with elastic coupling. The lubricated creep-slip friction law we use in the Burridge-Knopoff model describes the frictional sliding dynamics of a range of real materials. Low-dimensional structures including synchronized oscillations and propagating fronts are dominant, in agreement with the results of laboratory friction experiments. Here we explore the dynamics of fronts in elastic excitable media.Comment: Int. J. Bifurcation and Chaos, to appear (1999

Multiarmed Spirals in Excitable Media

Numerical studies of the properties of multiarmed spirals show that they can form spontaneously in low excitability media. The maximum number of arms in a multiarmed spiral is proportional to the ratio of the single spiral period to the refractoriness of the medium. Multiarmed spirals are formed due to attraction of single spirals if these spirals rotate in the same direction and their tips are less than one wavelength apart, i.e., a spiral broken not far from its tip can evolve into a 2-armed spiral. We propose this mechanism to be responsible for the formation of multiarmed spirals in mounds of the slime mold Dictyostelium discoideum

Propagation Failure in Excitable Media

We study a mechanism of pulse propagation failure in excitable media where stable traveling pulse solutions appear via a subcritical pitchfork bifurcation. The bifurcation plays a key role in that mechanism. Small perturbations, externally applied or from internal instabilities, may cause pulse propagation failure (wave breakup) provided the system is close enough to the bifurcation point. We derive relations showing how the pitchfork bifurcation is unfolded by weak curvature or advective field perturbations and use them to demonstrate wave breakup. We suggest that the recent observations of wave breakup in the Belousov-Zhabotinsky reaction induced either by an electric field or a transverse instability are manifestations of this mechanism.Comment: 8 pages. Aric Hagberg: http://cnls.lanl.gov/~aric; Ehud Meron:http://www.bgu.ac.il/BIDR/research/staff/meron.htm

On propagation failure in 1 and 2 dimensional excitable media

We present a non-perturbative technique to study pulse dynamics in excitable media. The method is used to study propagation failure in one-dimensional and two-dimensional excitable media. In one-dimensional media we describe the behaviour of pulses and wave trains near the saddle node bifurcation, where propagation fails. The generalization of our method to two dimensions captures the point where a broken front (or finger) starts to retract. We obtain approximate expressions for the pulse shape, pulse velocity and scaling behavior. The results are compared with numerical simulations and show good agreement.Comment: accepted for publication in Chao

Dynamic range of hypercubic stochastic excitable media

We study the response properties of d-dimensional hypercubic excitable networks to a stochastic stimulus. Each site, modelled either by a three-state stochastic susceptible-infected-recovered-susceptible system or by the probabilistic Greenberg-Hastings cellular automaton, is continuously and independently stimulated by an external Poisson rate h. The response function (mean density of active sites rho versus h) is obtained via simulations (for d=1, 2, 3, 4) and mean field approximations at the single-site and pair levels (for all d). In any dimension, the dynamic range of the response function is maximized precisely at the nonequilibrium phase transition to self-sustained activity, in agreement with a reasoning recently proposed. Moreover, the maximum dynamic range attained at a given dimension d is a decreasing function of d.Comment: 7 pages, 4 figure