Wiener-Rosenblueth model

Fig. 2. Geometrical construction of the spiral wave in the Wiener-Rosenblueth model.

The detailed mathematical analysis of this model and its derivation for the reaction-diffusion systems are given in the review [29]. The aim of the present paper is to provide a summary of various effects in the complex dynamics of spiral waves which could be explained and were predicted in the framework of this kinematical model. We start with a discussion of the original Wiener-Rosenblueth (WR) model which lies in the basis of our constructions. 

The first kinematical model for the description of processes in excitable media was proposed in 1946 by Wiener and Rosenblueth [6]. In this model it is assumed that each small segment of an oriented curve, representing the excitation front, moves in its normal direction with the same constant velocity. It was shown in [6] that such a curve rotating around an obstacle forms a spiral which constitutes an involute of this obstacle and approaches the Archimedean spiral far from it. 

At the time when Wiener and Rosenblueth formulated their model no reaction-diffusion equations of excitable media were available. Therefore, they simply postulated that the medium consists of elements which can be found in three distinct states of rest, excitation and recovery. After a perturbation, an element goes from the state of rest into the state of excitation, stays there for a fixed time, Tg, and goes then into the state of recovery. The latter lasts for a fixed time % during which the element could not be forced back into the excited state even by strong perturbations. When the recovery is completed, the element returns to the initial rest state and could be again excited. The perturbation, transferring the element into the excited state, could be provided by the neighbouring element if it is currently in the state of excitation. 

Formulated on a lattice and with discrete time, the WR model becomes a variant of a cellular automaton (it was later used in the construction of more complex cellular automata for excitable media, see [30-32]). However, Wiener and Rosenblueth actually assumed in [6] that both time and space were continuous. In their model smooth excitation fronts propagate into the regions where the medium is in the state of rest. The duration Tg of excitation is taken... 

---