Consider in Figure 7 the I-interval between A and D. There is a stable rest state around -35 mV and a stable (more depolarized) limit cycle. Suppose the current I slowly varies back and forth across this interval. Then because of the bistability, it is easy to see how a hysteresis loop is formed in which the membrane is alternately at rest and alternately firing repetitively. This provides a simple mechanism, and geometric interpretation, for square-wave bursting. However, since the current I is externally imposed, this is forced rather than autonomous bursting. To achieve the latter, one could (as in ) redefine I as a dynamic dependent variable in such a way that I decreases when the membrane is depolarized and firing repetitively, and I increases when the membrane is resting. Although artificial, this example demonstrates the basic principle that (very) slow negative feedback together with hysteresis in the fast dynamics underlie square-wave bursting. Many different ionic current mechanisms could likewise produce the slow negative feedback. For further illustration we employ a calcium-dependent potassium current, analogous to that studied by others (see ). We assume the current activates instantaneously in response to calcium and that the calcium handling dynamics are slow. Thus, we add to (6) the current given by
where is the maximal conductance for this current and z is the gating variable with a Hill-like dependence on Ca (the near-membrane calcium concentration scaled by its dissociation constant for activating the gate, ):
(For simplicity, we set the Hill exponent p=1, although this is not required.) The balance equation for Ca is:
where the parameter is for converting current into a concentration flux and involves the ratio of the cell's surface area to the calcium compartment's volume. The parameter is a product of the calcium removal rate and the ratio of free to total calcium in the cell. Since calcium is highly buffered, is small so that the calcium dynamics is slow. This is a greatly simplified model, for example, one could have more complicated calcium handling, including diffusion of calcium in the cytoplasm, nonlinear removal of calcium by pumps/exchangers, perhaps even release of calcium from intracellular pools. If the conductance of this outward current is large, the membrane is hyperpolarized and if it is small, then the membrane can fire. Thus, when a bifurcation curve is drawn as a function of this conductance, it is reversed from that of Figure 7 which plots the behavior as a function of an inward current. When the membrane is firing, intracellular calcium slowly accumulates, turning on this outward conductance and thereby terminating the firing. Figure 9a shows a bursting solution to the three variable model, eqns. 4-6 coupled with the slow calcium dynamics, eqn. 22.
(A,B) Run this to see the projection in the z-V plane. Plot
V versus t. Change the current, I and the
conductance of the AHP current, gkca to see what happens.
When is there no longer bursting? Plot the calcium as a function of
time. (C) Run the simulation longer (say, 4500)
with I=45, gkca=0.25 and
now changing Ca0=12. There is no longer bursting but the
repetitive firing is not regular. Now, if you want, you can look at
this chaotic behavior. Compute the Poincare map
Set V=-22.63, w=0.018, ca=18.53 as
initial data; ca0=12 . Now set the total simulation time to
50000 and set Dt=0.25 From the (nUmerics) menu, choose
(Poincare map) (Section) The following should be filled in:
Projecting the solution onto the z-V plane, where z is defined in eq. 21, shows (Figure 9B) how the burst's trajectory slowly tracks the attracting branches of the fast subsystem. Rapid transitions occur when the branches terminate at bifurcation points and turning points. We note that any number of alternate mechanisms could provide the slow negative feedback for bursting including a slow gating kinetics for z with fast calcium handling, or slow inactivation of , driven by V or Ca.