Oscillatory traveling waves in excitable media
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AL, NONLINEAR, AND SOFT MATTER PHYSICS
Oscillatory Traveling Waves in Excitable Media E. P. Zemskova and A. Yu. Loskutovb a
Dorodnitsyn Computing Center, Russian Academy of Sciences, Moscow, 119333 Russia b Moscow State University, Moscow, 119992 Russia e-mail: [email protected] e-mail: [email protected] Received November 27, 2007
Abstract—A new type of waves in an excitable medium, characterized by oscillatory profile, is described. The excitable medium is modeled by a two-component activator–inhibitor system. Reaction–diffusion systems with diagonal and cross diffusion are examined. As an example, a front (kink) represented by a heteroclinic orbit in the phase space is considered. The wave shape and velocity are analyzed with the use of exact analytical solutions for wave profiles. PACS numbers: 82.40.Bj, 82.40.Ck DOI: 10.1134/S1063776108080189
1. INTRODUCTION Distributed active (excitable) media [1], unlike passive media, can transmit signals over large distances without attenuation or distortion. Wave formation and propagation in such media are described by reaction– diffusion equations where kinetics and transport are represented by nonlinear reaction terms and diffusion, respectively [1–3]. The simplest one-dimensional reaction–diffusion equation was originally analyzed in [4] as applied to biological problems. In this paper, we analyze one-dimensional two-component activator–inhibitor systems of the form ∂v ∂ u ∂ ∂u ------ = f ( u, v ) + D u --------2 + h v ------ Q v (u, v ) ------- , ∂x ∂x ∂t ∂x 2
∂u ∂ v ∂ ∂v ------- = g ( u, v ) + D v --------2- + h u ------ Q u(u, v ) ------ , ∂x ∂x ∂t ∂x 2
(1)
where f (u, v) and g(u, v) are reaction terms and Du and Dv are diffusion coefficients. When hu = hv = 0, the model reduces to a reaction–diffusion system with selfdiffusion. When hu, v ≠ 0, we have a cross-diffusion system. The present analysis is restricted to systems with linear cross diffusion (Qu, v(u, v) = const). Waves of various kinds can be observed in reaction– diffusion systems: stationary spatial patterns, spatiotemporal chaos, etc. In this paper, we describe onedimensional traveling waves with oscillatory profiles. The one-component equation ∂ u ∂u ------ = f ( u ) + D u --------2 ∂t ∂x
has oscillatory solutions if at least one of its singular points is a focus. In this case, a nonmonotonic selfsimilar solution is represented by an orbit emanating from the focus. It may describe a rightward- or leftward-propagating wave, and the oscillatory profile may be localized near its leading or trailing front, accordingly [5]. In this paper, we use the FitzHugh–Nagumo model of an excitable medium [6] in a form amenable to exact analytical treatment [7] to show that oscillatory traveling waves can develop in a two-component system, with front corresponding to an orbit joining two saddle points. In the model considered here, wave solutions with oscillatory profiles describe traveling waves (i.e., waves described by functions of a single variable ξ = x – ct, where c is wave velocity). Since the os
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