## 2 The Model of the System  
The general idea of cable like transmissions is implemented on a discrete grid as follows. Three states of cells are considered: the resting state, the excited state and the refractory state. Resting cells and excited cells interact in two directions with different probabilities--probability 1 for the longitudinal direction and probability \(\
u\) for the latitudinal (transversal) direction. Right after the cell is excited it enters the refractory state where it remains for the time period equal to \(\tau\) (as illustrated in Figure 1).  
In order to recreate some basic topological features of the atrial muscle periodic boundary conditions in the transversal direction can be combined with open boundary conditions in the longitudinal direction. This corresponds to the topology of a cylinder that is useful in order to investigate the effects of spontaneous heterogeneity that take place on the front of the propagating wave.  
In general, once the initial grid is fixed, all the interactions that can be numerically simulated depend only on system parameters \(\
u\) (the probability of interactions along the transversal direction) and \(\tau\) (the time spent in the refractory state).  
However, the interactions between cells can be seriously complicated by dysfunctional cells. That can be illustrated by a simple computational experiment in Figure 2. Pacemaker cells are placed at the left boundary of the digital image and initiate the wave front which propagates to the right in all longitudinal cables (Figure 2a). However, because of dysfunctional cells, this is not always the only wave direction. The dysfunctional cell may block the propagation wave front (Figure 2b). The interplay between the wave fronts can form different patterns as shown in Figure 2c. Note that the boundary conditions are periodic.  
However, such computational setup as illustrated in Figure 2 requires many time-forward interactions for the formation of a well-developed patterns. The wave front must propagate over all plane (and preferably more than once); it has to collide with all dysfunctional cells.  
Figure 1: The diagram of interaction between cells within the network. Resting cells (black) and excited cells (white) interact in two directions with different probabilities. The resting cell becomes excited if it interacts with another excited cell. An excited cell (white) enters a refractory state (gray) for a time period \(\tau\).  
Figure 2: Initial plane vertical wave is induced at left side by pacemaker cells (**a**). However, one cell is dysfunctional and blocks the propagation of the wave front (**b**). Due to the vertical connections between longitudinal cables, the resulting process forms a pattern (**c**).  
Let us consider an alternative computational setup where pacemaker cells are randomly distributed across 2D plane. Now, wave propagation continues to all directions and collide with each other. The benefit of such AF model implementation is that it requires less than \(1\%\) cells to be initially excited in order to form self-organized patterns (Figure 3).  
The computational efficiency of the numerical AF scheme can be enhanced even further by decreasing real pacing times \(\tau\) of the cells. Optimal pacing time \(\tau\) depends on the number of initially excited cells and the length of simulation. If \(\tau\) is too small not all plane cells become excited, thus pattern is not fully formed. Otherwise, a too long refractory period results in relatively static pattern, because process cannot evaluate it is stuck in the same place.