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A Simple Model Showing Bursting Behavior

Here we consider a three-dimensional system of ODEs which has no biological motivation, but does display bursting behavior very similar to models for bursting in the pancreatic $\beta$-cell (see Chapter 6 of Keener and Sneyd, Mathematical Biology, on reserve in the library).

Consider the system of ODEs

$\displaystyle \frac{dx}{dt}$ $\textstyle =$ $\displaystyle y - x^3 + 3 x^2 + I - z$ (3)
$\displaystyle \frac{dy}{dt}$ $\textstyle =$ $\displaystyle 1 - 5 x^2 - y$ (4)
$\displaystyle \frac{dz}{dt}$ $\textstyle =$ $\displaystyle r (s (x-x_1) - z).$ (5)

We'll take $x_1 = -\frac{1+\sqrt{5}}{2}, r = 0.001$, and $s=4$. Equations (3-5) may be numerically solved in Matlab using the following code; first bursts.m:


global r s x1 I

r = 0.001;
s = 4;
x1 = -(1+sqrt(5))/2;
I = 2;

[T,Y] = ode23('func_bursts',[0,1000],[-1.4485,-9.2475,2.1105]);

figure(1)
subplot(3,1,1)
plot(T,Y(:,1));
xlabel('t');
ylabel('x');

subplot(3,1,2);
plot(T,Y(:,2));
xlabel('t');
ylabel('y');

subplot(3,1,3);
plot(T,Y(:,3));
xlabel('t');
ylabel('z');
axis([0 1000 1.5 2.5])

Text version of this program

Next, func_bursts.m:


function df = func_bursts(t,y)

global r s x1 I

xx = y(1);
yy = y(2);
zz = y(3);

dx = yy - xx^3 + 3*xx^2 + I - zz;
dy = 1 - 5*xx^2 - yy;
dz = r*(s*(xx-x1)-zz);

df = [dx;dy;dz];

Text version of this program


Figure 4 shows timeseries for $(x,y,z)$ generated using these programs for $I=2$. Such a state is described as bursting because of the switching of the voltage between an active state (characterized by rapid oscillations) and a rest state (characterized by the lack of oscillations).

Figure 4: Timeseries for $I=2$.
\begin{figure}\begin{center}
\leavevmode
\epsfbox{burstsI2.eps}\end{center}\end{figure}

In the following, we try to understand mathematically how such bursting behavior occurs.


next up previous
Next: Treating as Fixed in Up: APC591 Tutorial 3: The Previous: Bifurcations
Jeffrey M. Moehlis 2001-10-03