Bursting that differs from the square-wave pattern above
can arise from slow
modulation of bistable fast dynamics, since bistability can
arise in a number of ways.
Consider a parameter regime for eqs 4-6 in which the
onset of oscillations is via a subcritical Hopf bifurcation, such as
shown in Figure 2 and Figure 3. As for the square-wave burster,
there is a regime where the
spike-generating dynamics is bistable; a limit cycle and a fixed point coexist.
However, unlike the bistability shown in
Figure 7 , here,
the limit cycle ``surrounds'' the fixed point. Figure 10 shows an example
of a so-called * elliptic burst* pattern generated when the fast
dynamics of Figures 2 and 3 are coupled with the slow calcium-
dependent potassium current used in the previous bursting model
(equation 22).
Here (Figure 10A)
the envelope of the spikes is ``elliptical'' in shape as the amplitude
gradually waxes and wanes. The silent phase is characterized by
damping and growing oscillations as the trajectory slowly drifts through
the Hopf bifurcation of the fast subsystem.
This type of activity pattern has been seen
in sleep spindles and a cellular model related to
it involves this mathemetical mechanism
[7] (see also, [36]).
As with a square-wave burster, this type of
bursting can also have complex dynamics such as quasi-periodic
behavior and chaos.

*
Integrate this equation, look at V versus t , and
look at the calcium concentration, Ca . Change the current,
I and study what happens.
*

Mon Jul 29 17:47:46 EDT 1996