## Self synchronous cellular automata

Authors:David Cornforth

The Clock scheme (Thomas 1979, Low and Lapsley 1999) assigns a timer to each cell. The period of each timer is set at random, as is the initial phase of the timer. The model is evolved by incrementing the values of all timers at each time step, then checking the values. Those timers that have exceeded the value of their period variable are updated in order, with new states being implemented immediately. After a cell has been updated, its timer is set to zero.

In the Cyclic scheme, a node is chosen at each time step according to a fixed update order. This order is decided at random during initialisation of the model. This follows Kanada (1994).

In the Random Independent scheme (Harvey and Bossomaier, 1997), one cell is randomly selected for update at each time step. Subsequent selections are independent of each other.

The Random Order scheme (Harvey and Bossomaier, 1997), makes a list of all cells at each time step, and sorts them into a random order. It then uses this list to decide which cell to update. In this variant, each cell is updated once only for every time step, but the order of updating is random.

The Synchronous scheme is included for comparison. It has been described in many papers, and its properties are well-known (Wolfram, 1984). At each time step, the state of each cell is calculated, but held in a temporary store until all states have been calculated. Then the cells are all updated to their new state synchronously.

The Self-sync scheme is a modification to the clocked scheme to show how self-synchrony, observed in ant colonies and neural tissue, may be easily modeled. Our model is similar to the Kuramotos model of self-synchronising oscillators (Strogatz 2000), but each cell is connected only to its two adjacent neighbours. At each time step, the frequency omega of each oscillator is updated according to the phase ? of its neighbours, using a gain term beta (see attached equation image).

These five schemes are implemented as a one dimensional CA having 250 cells and 2 states, with each cell connected to its two neighbours and itself. For each rule, the five models were initialised to random states.