Effect of a modified sinusoidal forcing on spiral wave in a simulated reaction-diffusion system.

Spiral waves are often found in excitable media. In the hearts, they are abnormal forms of action potential propagation. Under an external forcing, the spiral waves drift and are subsequently terminated at the boundary. Spiral waves can be studied in simulations using a discrete reaction-diffusion system; thereby the time step must not exceed a numerical stability limit (ts). In this article, we present the dynamics of spiral waves in a simulated system under an external forcing as a modified sinusoidal function of time. The spiral waves are forced to drift along a straight line with a velocity and an angle depending on the time step. An optimization study provides the optimal time step of 0.2ts, where further reductions of the time step do not alter the drifting of the spiral waves.


Introduction
Spiral waves are often found in excitable media, e.g., cell aggregation in slime mold colonies [1], during CO-oxidation on a platinum surface [2], electrical wave propagation in cardiac tissues [3], and the Belousov-Zabotinsky (BZ) reaction [4]. Action potential propagation in forms of spiral waves in heart tissues related to some cardiac arrhythmias and can lead to sudden cardiac death. Under an external forcing, the spiral waves drift and are subsequently terminated at the boundary of the medium. Drifting of spiral waves can be induced by an applied electric field and the drifting velocity and angle of the spiral waves increase with the field strength [5]. Under a sufficient strong field, the spiral waves are forced to hit the boundary and eventually annihilated [6].
Dynamics of spiral waves and the effect of forcing in excitable media are often studied in a discrete reaction-diffusion system [7,8]. To avoid errors from calculations, the time step used in the simulations must not exceed a numerical stability limit (t s ) [9]. In this article, we present the effect of the time step on the dynamics of spiral waves in a simulated system.

Methods
In our numerical simulations, we use the two-variable Oregonator model [7] to describe the dynamics of the activator u and inhibitor v in excitable media. The advection terms for both u and v account for electric field for sinusoidal function E applied in x-direction. x As in References [8], the parameters are chosen as ε = 0.01, q = 0.002, f = 1.4, the diffusion coefficients D u = 1.0 and D v = 0.6, and the ionic mobilities M u = -1.0 and M v = 2.0. In the absence of an electric field, the tip of a free spiral wave rotates around a circular core (diameter = 0.9 s.u.).
The simulations are performed using the explicit Euler method with 9-point approximation of the two-dimensional Laplacian operator and a centered-space approximation of the gradient term. We use the uniform grid space x = y = 0.1 system unit (s.u.). The time step t is varied between 0.01 -0.9 t s , where the numerical stability limit t s = (3/8) (x) 2 [9]. The dimensionless size of the system is 40 x 40 s.u. (400 x 400 grid points). To create a spiral wave, a planer wave is triggered by setting 5-gridpoint strip at an edge of the medium to an excited state. The wave front is allowed to propagate into the middle of the medium, before half of the medium is reset to an excitable state. Then we apply the forcing E as a sinusoidal function of time t where the amplitude E 0 and the period (T) are kept constant at 0.4, 0.3 t.u. We observe the dynamics of the spiral wave both with and without the applied forcing and search for the optimal time step i.e., the largest time step that further reduction of the time step does not change the spiral wave dynamics.

Results
In the absence of the applied forcing, the free spiral wave, as shown in figure 1a, rotates around a small circular circle. When the time step t is decreased, the wavelength is approximately constant while the wave period T w are increased [ figure 1

Discussion and Conclusion
We have presented an investigation on the effect of the time step on the dynamics of spiral waves at different time step in a simulated system. Both the motion of free spiral waves (in the absence of forcing) and the drifting of spiral waves under an applied sinusoidal forcing depend on the time step. The optimal time step in the case of time varying forcing (0.2t s ) which is smaller than that for the simulations of free spiral waves (0.4t s ). This comes from the fact that the applied time-dependent forcing is discretized and becomes a step-wise function which depends on the time step. Therefore, simulations of spiral waves under an applied forcing as a function of time should be performed with a fine time step to obtain promising results for comparison and confirmation of experimental results. It has been shown that the lifetime of tachycardia is extended by pinning of spiral waves to anatomical obstacles like veins and scars [3] and the elimination of such pinned spiral waves become a tough task especially for large obstacles [10]. Even though we show that the sinusoidal forcing induce