File:Nullclines.png

The Barkley model is a system of reaction-diffusion equation modeling excitable media and oscillatory media. It is very similar to the FitzHugh-Nagumo (FHN) model and for many purposes the Barkley model can be considered to be same as the FHN model. The advantage of the Barkley model over other models of excitable media is that the reaction terms permit very fast numerical simulations of spiral waves in two dimensions and scroll waves in three dimensions.

In the simplest case the equations for the Barkley model are \[ \frac{\partial u }{ \partial t} = f(u,v) + \nabla^2 u, \] \[ \frac{\partial v }{ \partial t} = g(u,v), \] where two reaction terms take the form \[ f(u,v) = \frac{1}{\epsilon}u (1-u)(u-\frac{v+b}{a}), \] \[ g(u,v) = u - v. \] A more general form of the model is possible (see below). \(\epsilon, a, b \) are model parameters whose role is discussed in the next section.

Figures <ref>F1</ref> and <ref>F2</ref> show typical waves in two and three dimensions from simulations of the model.

Figure 1: Spiral wave solution of the Barkley model.

Figure 2: Scroll wave solution of the Barkley model.

The text below is awaiting further revision.

Nullclines

Figure 3: Nullclines for the Barkley model. Bold blue shows the N-shaped \(u\)-nullcline and green shows the \(v\)-nullcline.

The nullclines for the Barkley model are displaced in figure <ref>F3</ref> where one see the defining feature of the model -- the nullclines for the nonlinear \(u\) reaction kinetics are straight lines. The \(u\)-nullclines are given by \(f(u,v)=0\) so that the three branches are \[ u = \begin{cases} 0, \\ (v + b)/a, \\ 1 \end{cases} \]

In practice only the N-shaped portion shown in bold in figure <ref>F3</ref> is relevant since for spiral and scroll wave solutions the system do not pass through the corners of the nullclines. It should be emphasized that while the nullclines are piecewise linear, the function \(f(u,v)\) is a cubic polynomial in \(u\) and hence everywhere smooth.

The nullclines in figure <ref>F3</ref> should be compared with the S-shaped nullclines for the FitzHugh-Nagumo model.

Parameters

\(\epsilon\) sets the timescale separation between the fast \(u\)-equation and the slow \(v\)-equation, is thus taken to be small. \(a\), and \(b\) collectively control the excitability threshold, and the duration of excitation.

Why are simulations fast?

The are two independent factors which make numerical time stepping of the Barkley fast.

Semi-implicit time stepping of the reaction-kinetics

Efficient treatment of diffusion

It is difficult to compare with other simulations since issue of implementation etc are important (and also vary with time due to computer architecture).

General form of the model

In the most general case the equations for the Barkley model are \[ \frac{\partial u }{ \partial t} = f(u,v) + \nabla^2 u, \] \[ \frac{\partial v }{ \partial t} = g(u,v) + D_v \nabla^2 v, \] where \[ f(u,v) = \frac{h(v)}{\epsilon}u (1-u)(u-u_{th}(v)) \] and the function \(g(u,v)\) is essentially arbitrary.

in the model, although the standard choice is

Software

Programs EZ-Spiral and EZ-Scroll have been available for some time

References

Barkley D. (1991) A model for fast computer simulation of waves in excitable media. Physica 49D, 61--70.

Dowle M., Mantel R.M., and Barkley D. (1997) Fast simulations of waves in three-dimensional excitable media. Int. J. Bif. Chaos 7, 2529--2545 (1997).