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Chou, Mo-Hong

Computer-aided experiments on the Hopf bifurcation of the FitzHugh-Nagumo nerve model. (English) Zbl 0820.92003

Comput. Math. Appl. 29, No. 10, 19-33 (1995).
Summary: A space-clamped FitzHugh-Nagumo (FHN) nerve model subjected to a stimulating electrical current, \(I\), is investigated by a combination of perturbation and numerical methods. Our goal is to trace out the path of periodic solutions initiated by a , especially when the FHN model presents a slow recovery mechanism denoted here by the small control parameter \(\beta\).
It is shown in the computed period diagram, that in addition to the two Hopf bifurcation points \(I^ -\) and \(I^ +\), there are another two critical points \(I_ M\) and \(I_ N\) satisfying \(I^ - < I_ M < I_ N < I^ +\) and forming the points of maximum period for FHN models with single steady state, while satisfying \(I_ M < I^ - < I^ + < I_ N\) and forming the turning points for models with multiple steady states. If \(\beta\) is sufficiently small, the results are accompanied with cusp formation at \(I_ M\) and \(I_ N\). This fact indicates a discontinuous transition between oscillations of different characters. Further evidences are given by other bifurcation diagrams. For FHN models with multiple steady states, a similar hysteresis phenomenon is also observed for periodic solutions.

Cited in 4 Documents

MSC:

92C20 Neural biology
65L99 Numerical methods for ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
65L07 Numerical investigation of stability of solutions to ordinary differential equations
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C25 Periodic solutions to ordinary differential equations

Keywords:

space-clamped FitzHugh-Nagumo nerve model; relaxation oscillation; periodic solutions; Hopf bifurcation; period diagram; critical points; multiple steady states; cusp formation; hysteresis

Software:

Mathematica; AUTO
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References:

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