
MBW:Period Doubling and other Phenomena in the HodgkinHuxley equationsFrom MathBio(Redirected from Period Doubling and other Phenomena in the HodgkinHuxley equations)
ContentsIntroductionUnder a fixed current, the HogdkinHuxley equations exhibits both stable and oscillatory behavior, and acts in a analogous way to that of a electrical oscillator: namely the Van Der Pol oscillator see. In fact, by introducing a periodic forcing term to the original ODE, we we can duplicate the period doubling route to chaos in the HodgkinHuxley equations [1][2], just as can be done to produce a chaotic system. For the MorrisLecar system, this can be written as:
where can be chosen to be any time dependent function of the applied current . In particular, for the rest of this write up, we choose a periodic stimulus,, where is the amplitude of our forcing current and is the frequency. As we will see, will be our bifurcation parameter in the period doubling route to chaos. Poincare MapBecause the forced HodgkinHuxley is an essentially 5dimensional system it is impossible to fully visualize its higher dimensional phase space. In order to get around this limitation, the technique of a temporal Poincare section is employed that will help us reduce the higher dimensional dynamics and . The idea simple: Let be the flow of the ODE we are trying to study. Fixing and , we define the Poincare map, . Thus the Poincare map can be viewed simply as a periodic sampling of the flow at fixed time steps. To see how the Poincare map might help us understand the dynamic of the of the flow of the HodgkinHuxley equations, imagine a globally attracting limit cycle exits in the phase space of the HH equations a period (the same as the above Poincare map). It is easy to see that the Poincare map, , would approach a fixed point, since eventually all orbits approach the limit cycle and the orbits on the limit cycle itself repeat after second. Likewise, if that same limit cycle had period , then would eventually oscillate between two points, i.e. a period 2 orbit. Lastly, one can also test the stability of the system by determining the Lyapunov exponent of the Poincare map, rather than the flow itself. The advantage of this, over the continuous regime (if an expression of the Poincare map is analytically obtained), is the discrete nature of digital processors and the well know numerical limitation of calculating the Lyapunov exponent under current methods. This is how we will distinguish chaos (positive Lyapunov exponent) from quasiperiodic motion. Period DoublingIn the spirit of [1], we obtain a bifurcation diagram of the HH equations, by projecting the Poincare map onto the first coordinate, voltage:
Unimodal Mapsan analogyGiven the above bifurcation diagram, one might be reminded of other one dimensional map which experience period double, such as the logistic map and the Hassell equation. Check out Extensions to The Dynamics of Arthropod PredatorPrey Systems for some of Hassell's work. Indeed, there is good reason to study unimodal maps, like the logistic map, in order to better understand the period doubling bifurcations of the HH equations. NOTE, however, this is just an analogy, (for the moment at least).
Periodic Chaos in one dimensional mapsAfter the onset of chaos, unimodal maps display an incredible array of both stable and chaotic behavior. In keep with my own work, I introduce to following concept of periodic chaos: Let . We say that is kperiodic chaotic if there exist a homeomorphsim, , such that , i.e. is conjugate to the logistic map for . This phenomena can be seen in the bifurcation diagram where the chaotic branches merge. See [[[project1.pdf]] for a detail discussion of this phenomena.
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