# Excitable media

Post-publication activity

Curator: Vladimir S. Zykov

## Definition

An excitable medium is a dynamical system distributed continuously in space, each elementary segment of which possesses the property of excitability. Neighboring segments of an excitable medium interact with each other by diffusion-like local transport processes. In an excitable medium it is possible for excitation to be passed from one segment to another by means of local coupling. Thus, an excitable medium is able to support propagation of undamped solitary excitation waves, as well as wave trains.

Originally, the term excitability referred to the property of living organisms (or of their constituent cells) to respond strongly to the action of a relatively weak external stimulus. A typical example of excitability is the generation of a spike of transmembrane potential (action potential) by a cardiac cell, induced by a short depolarizing electrical perturbation of a resting state. Usually, the shape of the generated action potential does not depend on the perturbation strength, as long as the perturbation exceeds some threshold (all-or-nothing principle). After the generation of this strong response, the system returns to its initial resting state. A subsequent excitation can be generated after a suitable length of time, called the refractory period, has passed. This obviously non-linear, dynamical behavior is characteristic of a large class of systems in biology, chemistry and physics (Winfree, 2000; Krinsky, Swinney 1991; Kapral, Showalter 1995; Zykov, 1987; Mikhailov, 1990; Kaplan, Glass, 1995; Izhikevich, 2007).

## Examples

The most prominent examples of excitable media are

• propagation of electrical excitation in various biological tissues, including nerve fiber and myocardium
• waves of spreading depression in the retina of eye
• concentration waves in yeast extract doing glycolysis
• calcium waves within frog eggs

## Mathematical models

In many applications the mathematical description of the dynamical processes in an excitable medium can be represented in the form of a reaction-diffusion system$\frac{\partial E_i}{\partial t}=F_i(\nabla E_i,\vec{E}) + \nabla(D_i\nabla E_i) + I_i(\vec{r},t).$

Here the vector $$\vec{E}$$ determines the state of the system, the $$F_i$$ are nonlinear functions of $$\vec{E}$$ and, perhaps, $$\nabla E_i\ ,$$ the $$D_i$$ are diffusion coefficients. The $$I_i$$ are external actions varying in space and time, which can be used for initiation of excitation waves. The most famous example of such type of descriptions are the Oregonator model, Hodgkin-Huxley model, Noble model, other models of cardiac cell, etc.

The basic features of the self-sustained dynamics in excitable media can be reproduced by the relatively simple and widely used two-component activator-inhibitor system

$${\partial u \over \partial t} = \nabla^2 u + f(u,v),$$

$${\partial v \over \partial t} = \sigma \nabla^2 v + \epsilon g(u,v).$$

Here $$u(\vec {r},t)$$ and $$v(\vec {r},t)\ ,$$ describe the state of the system, nonlinear functions $$f(u,v)$$ and $$g(u,v)$$ specify the local dynamics and $$\sigma$$ determines the ratio between two diffusion constants. If parameter $$\epsilon << 1$$ this reaction-diffusion system exhibit relaxational dynamics with intervals of fast and slow motion. Depending on the particular shape of the nonlinear functions $$f(u,v)$$ and $$g(u,v)$$ this system is refered to as the Brusselator, Fitzhugh-Nagumo model, Rinzel-Keller model, Barkley model, Morris-Lecar model, etc.

These generic systems of partial differential equations can be used to simulate the dynamics of one-, two- or three-dimensional media (Tyson, Keener, 1988). They can be adjusted to particular applications. For instance, the diffusion flows can be anisotropic or can include cross-diffusion terms. Some local and/or global feedback loops can be induced to reproduce naturally existing or artificially created stabilizing or destabilizing circuits.

An important example of possible modification represents the bidomain model describing the cardiac tissue as consisting of two colocated continuous media termed the intracellular and extracellular domain. The intracellular and extracellular potentials $$\phi_i$$ and $$\phi_e$$ are specified by the bidomain equations with conductivity tensors $$G_i$$ and $$G_e\ :$$

$$\nabla \cdot (G_i\nabla \phi_i)= I_m,$$

$$\nabla \cdot (G_e\nabla \phi_e)= -I_m,$$

where the transmembrane current density $$I_m$$ is determined as the following

$$I_m=\beta (C_m {\partial V_m \over \partial t} +I _{ion} + I_s).$$

Here $$\beta$$ is the membrane surface-to-volume ratio, $$C_m$$ is the membrane capacitance per unit area, $$I_{ion}$$ is the ionic current density generated by the cell membrane and $$I_s$$ is an imposed stimulation current density. The ionic current $$I_{ion}$$ depends on the transmembrane potential $$V_m=\phi_i-\phi_e$$ and on the vector $$\vec{m}$$ of gate variables in accordance to one of the models of cardiac cell. Each gate variable obeys first order ordinary differential equation as was firstly specified by the Hodgkin-Huxley model${d m_i \over d t}= \alpha_i(v_m)(1-m_i)-\beta_i(V_m)m_i.$

In the limit $$G_e \rightarrow \infty$$ the extracellular potential becomes to be uniform in space and the bidomain description is reduced to a multi-component reaction-diffusion system reproducing many important features of cardiac dynamics. However, the bidomain model allows for to consider the effects of, and the effects on, the surrounding extracellular electrical field (Keener, Sneyd, 1998).

Cellular automata and simplified descriptions of kinematics of excitation waves represent additional effective tools to describe different aspect of excitable-medium dynamics. They minimize computation time and under some approximations allow for analytical results.

## Self-organized patterns

The waves in passive, linear media differ considerably from waves in excitable media, which are nonlinear and driven by a distributed energy source. One-dimensional excitable media are able to support a single traveling wave and also wave trains propagating without decrement. If the medium size is sufficiently large, the propagation velocity and the wave profile do not depend on the initial and/or boundary conditions. This distinguishes the waves in excitable media from solitons propagating in nonlinear conservative systems, where velocities and pulse shapes strongly depend on the initial conditions.

In two-dimensional excitable media, one can observe expanding target patterns and rotating spiral waves. In three dimensions, rotating scroll waves are possible. All these dynamic phenomena represent well-known examples of self-organization in complex systems resulting in pattern formation.

The structure and the properties of the self-organized patterns are established due to a balance between the energy influx from internal distributed sources and the energy dissipation. Thus, the spatio-temporal patterns in excitable media belong to self-organization processes, which occur far away from the thermodynamical equilibrium.

## References

• A.T. Winfree, The Geometry of Biological Time (Springer, Berlin, Heidelberg, 2000).
• V. Krinsky and H. Swinney (eds), Wave and patterns in biological and chemical excitable media, (North-Holland, Amsterdam, 1991).
• R. Kapral and K. Showalter (eds.), Chemical Waves and Patterns, (Kluwer, Dordrecht, 1995).
• V.S. Zykov, Simulation of Wave Processes in Excitable Media, (Manchester Univ. Press, Manchester, 1987).
• A.S. Mikhailov, Foundation of Synergetics, (Springer, New York, 1990).
• D. Kaplan and L. Glass, Understanding Nonlinear Dynamics, (Springer, New York, 1995).
• E.M. Izhikevich, Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting, (MTI Press, 2007).
• J.P. Keener and J. Sneyd, Mathematical Physiology, (Springer, New York, 1998).

Internal references

• Gregoire Nicolis and Catherine Rouvas-Nicolis (2007) Complex systems. Scholarpedia, 2(11):1473.
• Eugene M. Izhikevich (2007) Equilibrium. Scholarpedia, 2(10):2014.
• Martin Fink and Denis Noble (2008) Noble model. Scholarpedia, 3(2):1803.
• Richard J. Field (2007) Oregonator. Scholarpedia, 2(5):1386.
• John Dowling (2007) Retina. Scholarpedia, 2(12):3487.