Eugene Izhikevich, IJB and Chaos, Vol 10, (2000) 1171-1266

1. Download Burst.ode and Burst_FS.ode. Do a quick run of Burst.ode (FULL) and  then plot the 3D trajectory in  (u,w,c) phase space using "View Axes" then "3D". You may choose the (x,y,z) axis as any permutation you like (then "Window" "Fit") but try to find one that looks similar to the figure above.
2. Look at the code in Burst_FS.ode to verifiy (by hand) that by differentiating u' equation and using the w' equation (in part) to eliminate w the new system for (u,u',c) is equivalent to the (FS) we studied in class, namely u''+f(u)u'+g(u)-c=0 for simple functions f(u) and g(u). The reason this system is written this way is that it more closely mimics the form of the equations for "real" models.
3. Until later we'll use the parameters: 
   1. a=0.25, beta=4, ubar=-0.954, eta=0.75, uhat=1.5, epsilon=0.0025. 
4. Execute both codes at the same time: 
   1. xppaut Burst.ode & 
   2. xppaut Burst_FS.ode & 
5. By varying c only, examine how the (FS) phase-space changes. Use a range of c in (-3,6). In particular try to find the value for a) the HB and b) the HC bifurcations. For the latter, for each new c value plot the invariant sets of the middle branch saddle. By refining your c values, try to get the stable and unstable manifolds to coalesce. That is the c value at which the homoclinic bifurcation occurs.
6. By varying c only, examine how the (u,w) phase space of (FS) changes. Use a range of c in (-3,6). In particular try to find the value for:
   1. where the Hopf Bifurcation (HB) occurs.
   2. where the Homoclinic Bifurcation (HC) occurs. To do this, for every new c value plot the invariant sets of the middle branch saddle. By refining your c values, try to get the stable and unstable manifolds to coalesce. That is the c value at which the homoclinic bifurcation occurs.You may need to have a c value accurate to 0.001. 
   3. where the Saddle Node Bifurcations (SN) occurs. Changing values near that c value you can see why its called a (SN) bifurcation, i.e., saddles and nodes coalesce. Again track the invariant sets of the saddle.
      
7. Create a bifurcation diagram of (FS) using "AUTO" under the "FILE" menu. BEFORE you enter AUTO you must make a long time run of (FS) for some c value (recommend c=-3) so that the system approaches (close enough to) a stable equilibria which AUTO can use to continue in c. Make sure you set things up so the "PARAM" is c, stipulate a range of the "Main" param c .....say what we used earlier.       
1. You may need to change the AUTO parameter "DS" and its associated min/max values.
2. First continue "Steady States". The result should include two SN and one HB.
3. Use the "Grab" and "tab" feature to select the "HB"
4. Once selected, continue the periodic orbits. This will should the projection of the periodic manifold of the (FS) onto the (u,c)-plane. 
8. For the changes in (eta,uhat) below, execute (FULL) to see what qualitative changes in u(t) occur. Then repeat 5)-7) to explain/understand them.
1. (eta,uhat)=(1,1.5)
2. (eta,uhat)=(0.75,1)
3. (eta,uhat)=(1.25,1)
4. (eta,uhat)=(0.5,2.25)
9. In 3)-8) above we only altered parameters that occur solely in the (FS) and not the Slow Subsystem (SS). (SS) parameters can also greatly alter the types of solutions seen. Reset the fast parameters (eta,uhat)=(0.75,1.5) in (FULL). Then alter beta in (1,6). You should get roughly three behaviors a) steady state equilibria b) bursting c)  bursting locked into active phase oscillations (no transisition to active phase) and possibly a very narrow window of beta for which chaotic active phase oscillations occur.
10. The earliest electrical activity model for pancreatic beta-cells was due to Chay and Keizer (1983). Here will examine the Chay_Cook_88.ode model. Download the code for (FULL) and make sure you get bursts. This will mean you may need alter numerical methods (GEAR), time steps (DT) , integration duration and tolerances, etc. Create an (FS) bifurcation diagram for the model. What's slow? Plot it AND voltage as a function of time. Do you know (approximately) what each term in the current equation is or means? Original manuscript here.