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 -cell (see Chapter 6 of Keener and Sneyd, Mathematical Biology, on reserve in the library).
Consider the system of ODEs
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])
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];
Figure 4 shows timeseries for generated using these programs for . 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).
In the following, we try to understand mathematically how such bursting behavior occurs.