Final Year Project: Influence of Stable Manifolds in Phase-Resetting of Excitable Cells
by Siti Hasmah Mazlan
Supervisors: Dr Hinke Osinga and Dr Krasimira Tsaneva-Atanasova
1 - Home
2 - Cell Model
3 - Bifurcation Analysis
4 - Stable Manifolds
5 - Phase-resetting
6 - References
Stable Manifolds
The stable manifolds are associated with the points on the branch of saddle equilibria in Figure 3. These manifolds are defined as the one-dimensional curves of points that approach the branch asymptotically as time grows to infinity. The combination of the manifolds forms a continuous surface that partially encloses the global attractor and the bifurcation diagram. The family of manifolds is portrayed as the blue surface in Figure 4.
Figure
4
The global attractor is controlled by the bifurcation curves and the
stable manifolds.
The stable
manifolds act as the basin boundaries between the branches of
stable spirals and stable nodes. The global attractor spirals closer to
the
basin boundaries as Ca
increases during the oscillations around
the branch of spiral equilibria. At the same time, the dynamics of Ca
shrinks the basin enclosed by the stable manifolds. As the size of
the basin becomes smaller, the
global attractor ultimately passes through the basin boundary and
escapes to the back side of the manifolds. The global attractor
converges to the lower branch of attracting equilibria, thus ending the
active phase. We
observe the similar phenomenon of termination of the active phase after
the global attractor crosses the basin boundary when some of the ionic
channels in the cell are blocked to obtain a shorter burst period. In the later study of phase-resetting,
we show that the restriction imposed by the stable manifolds can indeed
be extended to any orbit in the system that satisfies the
same model equation.
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