Spiral dynamics of excitable media

Spiral waves represent a very prominent example of a self-organization in quite different active distributed systems including the colonies of Dictyostelium discoideum, the chemical Belousov-Zhabotinsky reaction, the heart muscle, the eye retina, the CO oxidation on platinum single crystal surface and many others.
The most general features of the spiral wave dynamics can be reproduced by a two-component reaction-diffusion system [1]


where the local kinetics of an activator u and an inhibitor v is specified by the nonlinear functions F(u, v) and G(u, v). In the case of excitable dynamics a supra-threshold stimulus generates a propagation of the excitation front at a velocity cp which depends on the inhibitor value v near the front and on its local curvature k. As an example, the spiral front obtained numerically for specific values of cp and for the selected value of ω as shown in Fig. 1. Based on the assumption that these two relationships are linear, the existence of two limits of the spiral wave dynamics has been demonstrated relating to the trigger-trigger (TT) and trigger-phase (TP) waves [2, 3]. In both of the limits, a dimensionless parameter B = 2D/(ducp²), where du is the excitation pulse duration, completely determines the spiral wave parameters [3, 4, 5]. However, there is a strong quantitative difference between the relationships predicted for TT and TP waves [3]. Therefore, we are working on a model, where a free boundary approach is applied to demonstrate the existence of a smooth transition between TT and TP spiral waves. We introduce a modified Barkley model, which enables us to study this transition by varying of a single model parameter. We also want to show that this parameter can be determined experimentally.

Figure 1:

The selected shape of a spiral wave rigidly rotating at ω obtained as a solution of the free-boundary problem (taken from (3)).

Contact: Vladimir Zykov, Eberhard Bodenschatz

[1] A.T. Winfree, Chaos, 1, 303 (1991).
[2] A. Karma, Phys. Rev. Lett. 66, 2274 (1991).
[3] V.S. Zykov, N. Oikawa, and E. Bodenschatz, Phys. Rev. Lett. 107, 254101 (2011).
[4] V. Hakim and A. Karma, Phys. Rev. E, 60, 5073 (1999).
[5] V.S. Zykov Physica (Amsterdam) 238D, 931 (2009).