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7 In fact, as there exists only a single fixed point, this orbit is more accurately now a "homocline".

Fig. 18. Limit cycle dynamics for I = 4.75 (top row) and I = 6 (bottom row)

Fig. 18. Limit cycle dynamics for I = 4.75 (top row) and I = 6 (bottom row)

through a very narrow gap between them. The vector field in this gap bears the "memory" of the fixed points - namely it is very slow. Hence the orbits in this vicinity are near-stationary, as can be seen in the time domain. As I increases this influence diminishes and the frequency hence increases. Note that in both cases, however, there is virtually no change in the morphology of the depolarization, which is not related to this phenomenon.

### Hopf bifurcation

Through a slight change in the parameters relating to the potassium channels, however, the transition from steady state (fixed point) to periodic (limit cycle) dynamics can occur through a different type of bifurcation. In the above scenario the potassium channels had values consistent with a "high threshold", namely the mean threshold potential of the K+ potassium channels Vn = — 25 mV. Lowering Vn to —45mV and changing the Nernst potential to VK = — 78 mV yields the phase space portraits and time series plotted in Fig. 19.

Firstly, there is only one interception of the nullclines for these parameter values, and hence only one fixed point. For I < 19 this is a spiral inset, hence yielding damped oscillations (panels a,b). For I> 19 the fixed point has undergone a (supercritical) Hopf bifurcation, hence yielding a small amplitude limit cycle, coinciding with sustained but subthreshold voltage oscillations. For I^26, the amplitude of these oscillations grows smoothly but rapidly so that with I = 27 the system exhibits sustained suprathreshold oscillations. However, note that the damped, subthreshold and suprathreshold oscillations

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