An excitable medium is a

nonlinear dynamical system In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in ...

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 A wildfire, forest fire, bushfire, wildland fire or rural fire is an unplanned, uncontrolled and unpredictable fire in an area of Combustibility and flammability, combustible vegetation. Depending on the type of vegetation present, 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 reaction A chemical oscillator is a complex mixture of reacting chemical compounds in which the concentration of one or more components exhibits periodic changes. They are a class of reactions that serve as an example of non-equilibrium thermodynamics with ...

s are excitable media, for example the

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 ...

and the

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 s ...

Cell excitability Membrane potential (also transmembrane potential or membrane voltage) is the difference in electric potential between the interior and the exterior of a biological cell. That is, there is a difference in the energy required for electric charges ...

is the change in

membrane potential Membrane potential (also transmembrane potential or membrane voltage) is the difference in electric potential between the interior and the exterior of a biological cell. That is, there is a difference in the energy required for electric charges ...

that is necessary for cellular responses in various tissues. The

resting potential A relatively static membrane potential which is usually referred to as the ground value for trans-membrane voltage. The relatively static membrane potential of quiescent cells is called the resting membrane potential (or resting voltage), as oppo ...

forms the basis of cell excitability and these processes are fundamental for the generation of graded and

action potentials An action potential occurs when the membrane potential of a specific cell location rapidly rises and falls. This depolarization then causes adjacent locations to similarly depolarize. Action potentials occur in several types of animal cells, c ...

. Normal and 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 The wave (known as a Mexican wave or stadium wave outside of North America) is an example of metachronal rhythm achieved in a packed stadium when successive groups of spectators briefly stand, yell, and raise their arms. Immediately upon str ...

(so-called from its initial appearance in the 1986

World Cup A world cup is a global sporting competition in which the participant entities – usually international teams or individuals representing their countries – compete for the title of world champion. The event most associated with the concept i ...

Mexico Mexico (Spanish: México), officially the United Mexican States, is a country in the southern portion of North America. It is bordered to the north by the United States; to the south and west by the Pacific Ocean; to the southeast by Guatema ...

Modelling excitable media

Excitable media can be modelled using both

partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function. The function is often thought of as an "unknown" to be sol ...

s and

cellular automata A cellular automaton (pl. cellular automata, abbrev. CA) is a discrete model of computation studied in automata theory. Cellular automata are also called cellular spaces, tessellation automata, homogeneous structures, cellular structures, tessel ...

With cellular automata

Cellular automata provide a simple model to aid in the understanding of excitable media. Perhaps the simplest such model is in. See Greenberg-Hastings cellular automaton for this model. 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: ExcitableCellularAutomaton

* 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 the forest fire example the simple rules given in Greenberg-Hastings cellular automaton might be modified 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.

Geometries of waves

One-dimensional waves

It is most common for a one-dimensional medium to form a closed circuit, i.e. a ring. For example, the

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, also called tachyarrhythmia, is a heart rate that exceeds the normal resting rate. In general, a resting heart rate over 100 beats per minute is accepted as tachycardia in adults. Heart rates above the resting rate may be normal (su ...

and

fibrillation Fibrillation is the rapid, irregular, and unsynchronized contraction of muscle fibers. An important occurrence is with regard to the heart. Cardiology There are two major classes of cardiac fibrillation: atrial fibrillation and ventricular fib ...

. Spiral waves constitute one of the mechanisms of fibrillation when they organize in long-lasting reentrant activities named rotors.

Notes

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References

Leon Glass Leon Glass (born 1943) is an American scientist who has studied various aspects of the application of mathematical and physical methods to biology, with special interest in vision, cardiac arrhythmia, and genetic networks. Biography Leon Gl ...

and Daniel Kaplan, ''Understanding Nonlinear Dynamics''.

External links

An introduction to excitable media

CAPOW software by Rudy Rucker contains several excitable media models
Dynamical systems Biophysics Nonlinear systems Mathematical modeling