# Emergence of Asynchronous Local Clocks in Excitable Media

## TL;DR

This paper explores how asynchronous local clocks emerge in excitable media like the brain or heart, even when individual elements act independently.

## Contribution

The study reveals a novel synchronization mechanism in excitable media without phase-aligned oscillators.

## Key findings

- A global clock period T emerges as the most probable waiting time for each element.
- Fluctuations around T decrease with increasing system size.
- Asynchronous local clocks also form in non-homogeneous systems with varying self-excitation rates.

## Abstract

Excitable media such as the myocardium or the brain consist of arrays of coupled excitable elements, in which the local excitation of a single element can propagate to its neighbors in the form of a non-linear autowave. Since each element has to pass through a refractory period immediately after excitation, the frequency of autowaves is self-limiting. In this work, we consider the case where each element is spontaneously excited at a fixed average rate and thereby initiates a new autowave. Although these spontaneous self-excitation events are modelled as independent Poisson point processes with exponentially distributed waiting times, the travelling autowaves lead collectively to a non-exponential, unimodal waiting time distribution for the individual elements. With increasing system size, a global ‘clock’ period T emerges as the most probable waiting time for each element, which fluctuates around T with an increasingly small but non-zero variance. This apparent synchronization between asynchronous, temporally uncorrelated point processes differs from synchronization effects between perfect oscillators interacting in a phase-aligning manner. Finally, we demonstrate that asynchronous local clocks also emerge in non-homogeneous systems in which the rates of self-excitation are different for all individuals, suggesting that this novel mechanism can occur in a wide range of excitable media.