Phase-resetting

After the investigation on the dynamics of the system, we now focus on the analysis of the stability of the bursting pattern. Perturbation via an external current pulse with an appropriate amount at a certain phase angle may influence the phase of the system's periodic motion, as shown by the time series of V in Figure 5.

Figure 5 An external current pulse with a particular strength is applied at the time indicated by the arrow.

The effect of the external current may cause a transient change in the structure of both the bifurcation diagram and the stable manifolds. This affects the organisation of the global periodic attractor around the stable manifolds. We divide the analysis of phase-resetting into two stages; we first consider the fast subsystem and then investigate resetting in the full system. In the fast subsystem, we ignore the dynamics of the Ca and use the phase-plane analysis to find a lower and upper limit of strength of the current pulse that may cause an upward resetting from the silent to the active phase at a particular phase angle. We also construct a two-parameter bifurcation diagram and use the topological normal form equation of a saddle-node bifurcation to estimate the minimum strength and duration for such cases. Although applying the current pulse with such properties may not significantly affect the phase with resetting in the full system, the qualitative criteria for the current pulse to cause a potentially large change in the phase does agree with the analysis of resetting in the fast subsystem.

For the phase-resetting in the full system, we also observe cases where the perturbed orbit stays inside the basin relatively longer than the native phase; an example is shown in Figure 6 where an external current pulse of varying duration is applied with strength 0.5 at Ca = 0.5 in the silent phase.The prolonged active phase can be

Figure 6 A comparison for the duration of the active phase (y-axis) between unperturbed (grey) and perturbed (red) system, based on the duration of pulse (x-axis) when the phase angle is Ca = 0.5 during the silent phase.

explained using the manifolds of the underlying fast subsystem. As illustrated in Figure 7, the perturbed orbit that causes a peak in Figure 6 spirals very close to the basin boundaries after the current pulse has been applied to the system. The prolonged active phase inside the basin enclosed by the stable manifolds can happen when the rate of oscillation is faster then the rate of change of Ca. Towards the end of the silent phase, this phenomenon is hardly observed since the system has a faster rate of change of Ca. The dynamics of Ca shrinks the basin at a faster rate, thus causing more difficulty for the orbit to remain inside the basin without crossing the basin boundaries.

Figure 7 The evolution of the perturbed orbit corresponding to the peak in Figure 6. The current is turned off at the point marked by the yellow dot.

While we believe the restriction is imposed by the geometrical structure of the stable manifolds, we have yet to draw conclusions on the time it takes for the orbit to cross the basin boundary. Further analysis is thus required to deduce the quantitative amount for such case and determine the exact phase change in the system.