Excitable medium
Encyclopedia
An excitable medium is a nonlinear dynamical system which has the capacity to propagate a wave of some description, and which cannot support the passing of another wave until a certain amount of time has passed (known as the refractory time).
A forest is an example of an excitable medium: if a wildfire
burns through the forest, no fire can return to a burnt spot until the vegetation has gone through its refractory period and regrown. In Chemistry, oscillating reactions are excitable media, for example the Belousov-Zhabotinsky reaction
and the Briggs-Rauscher reaction
. Pathological activities in the heart and brain can be modelled as excitable media. A group of spectators at a sporting event are an excitable medium, as can be observed in a Mexican wave
(so-called from its initial appearance in the 1986 World Cup in Mexico
).
s and cellular automata.
As in all cellular automata, the state of a particular cell in the next time step depends on the state of the cells around it—its neighbours—at the current time. In an excitable medium the general evolution function that governs the transitions between the three states described above is as follows:
This function can be refined according to the particular medium. For example, the effect of wind can be added to the model of the forest fire.
In the Mexican wave, for example, if for some reason, the originators of the wave are still standing upon its return it will not continue. If the originators have sat back down then the wave can, in theory, continue.
A spreading wave will originate at a single point in the medium and spread outwards. For example a forest fire could start from a lightning strike at the centre of a forest and spread outwards.
A spiral wave will again originate at a single point, but will spread in a spiral circuit. Spiral waves are believed to underlie phenomena such as tachycardia
and fibrillation
.
Spiral waves constitute one of the mechanisms of fibrillation when they organize in long-lasting reentrant activities named rotors.
A forest is an example of an excitable medium: if a wildfire
Wildfire
A wildfire is any uncontrolled fire in combustible vegetation that occurs in the countryside or a wilderness area. Other names such as brush fire, bushfire, forest fire, desert fire, grass fire, hill fire, squirrel fire, vegetation fire, veldfire, and wilkjjofire may be used to describe the same...
burns through the forest, no fire can return to a burnt spot until the vegetation has gone through its refractory period and regrown. In Chemistry, oscillating reactions are excitable media, for example the Belousov-Zhabotinsky reaction
Belousov-Zhabotinsky reaction
A Belousov–Zhabotinsky reaction, or BZ reaction, is one of a class of reactions that serve as a classical example of non-equilibrium thermodynamics, resulting in the establishment of a nonlinear chemical oscillator. The only common element in these oscillating systems is the inclusion of bromine...
and the Briggs-Rauscher reaction
Briggs-Rauscher reaction
The Briggs–Rauscher oscillating reaction is one of a small number of known oscillating chemical reactions. It is especially well suited for demonstration purposes because of its visually striking colour changes: the freshly prepared colourless solution slowly turns an amber colour, suddenly...
. Pathological activities in the heart and brain can be modelled as excitable media. A group of spectators at a sporting event are an excitable medium, as can be observed in a Mexican wave
Audience wave
The wave or the Mexican wave is an example of metachronal rhythm achieved in a packed stadium when successive groups of spectators briefly stand and raise their arms...
(so-called from its initial appearance in the 1986 World Cup in Mexico
Mexico
The United Mexican States , commonly known as Mexico , is a federal constitutional republic in North America. It is bordered on the north by the United States; on the south and west by the Pacific Ocean; on the southeast by Guatemala, Belize, and the Caribbean Sea; and on the east by the Gulf of...
).
Modelling excitable media
Excitable media can be modelled using both partial differential equationPartial differential equation
In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...
s and cellular automata.
With cellular automata
Cellular automata provide a simple model to aid in the understanding of excitable media. Each cell of the automaton is made to represent some section of the medium being modelled (for example, a patch of trees in a forest, or a segment of heart tissue). Each cell can be in one of the three following states:- Quiescent or excitable — the cell is unexcited, but can be excited. In the forest fire example, this corresponds to the trees being unburnt.
- Excited — the cell is excited. The trees are on fire.
- Refractory — the cell has recently been excited and is temporarily not excitable. This corresponds to a patch of land where the trees have burnt and the vegetation has yet to regrow.
As in all cellular automata, the state of a particular cell in the next time step depends on the state of the cells around it—its neighbours—at the current time. In an excitable medium the general evolution function that governs the transitions between the three states described above is as follows:
- If a cell is quiescent, then it remains quiescent unless one or more of its neighbours is excited. In the forest fire example, this means a patch of land only burns if a neighbouring patch is on fire.
- If a cell is excited, it becomes refractory at the next iteration. After trees have finished burning, the patch of land is left barren.
- If a cell is refractory, then its remaining refractory period is lessened at the next period, until it reaches the end of the refractory period and becomes excitable once more. The trees regrow.
This function can be refined according to the particular medium. For example, the effect of wind can be added to the model of the forest fire.
One-dimensional waves
It is most common for a one-dimensional medium to form a closed circuit, i.e. a ring. For example, the Mexican wave can be modeled as a ring going around the stadium. If the wave moves in one direction it will eventually return to where it started. If, upon a wave's return to the origin, the original spot has gone through its refractory period, then the wave will propagate along the ring again (and will do so indefinitely). If, however, the origin is still refractory upon the wave's return, the wave will be stopped.In the Mexican wave, for example, if for some reason, the originators of the wave are still standing upon its return it will not continue. If the originators have sat back down then the wave can, in theory, continue.
Two-dimensional waves
Several forms of waves can be observed in a two-dimensional medium.A spreading wave will originate at a single point in the medium and spread outwards. For example a forest fire could start from a lightning strike at the centre of a forest and spread outwards.
A spiral wave will again originate at a single point, but will spread in a spiral circuit. Spiral waves are believed to underlie phenomena such as tachycardia
Tachycardia
Tachycardia comes from the Greek words tachys and kardia . Tachycardia typically refers to a heart rate that exceeds the normal range for a resting heart rate...
and fibrillation
Fibrillation
Fibrillation is the rapid, irregular, and unsynchronized contraction of muscle fibers. An important occurrence is with regards to the heart.-Cardiology:There are two major classes of cardiac fibrillation: atrial fibrillation and ventricular fibrillation....
.
Spiral waves constitute one of the mechanisms of fibrillation when they organize in long-lasting reentrant activities named rotors.