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MBW:Period Doubling and other Phenomena in the Hodgkin-Huxley equations

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Under a fixed current, the Hogdkin-Huxley 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 Hodgkin-Huxley equations [1][2], just as can be done to produce a chaotic system. For the Morris-Lecar system, this can be written as:

I_{{ext}}(t)=C{\dot  {V}}+g_{L}(V_{L})+g_{{Ca}}M(V-V_{{Ca}})+g_{K}N(V-V_{k})

{\dot  {M}}=\lambda _{M}(V)(M_{{\infty }}(V)-M)

{\dot  {N}}=\lambda _{N}(V)(N_{{\infty }}(V)-N)

The Morris-Lecar equations under forcing

where I_{{ext}} 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,I_{{ext}}(t)=A\sin(2\pi ft), where A is the amplitude of our forcing current and f is the frequency. As we will see, A will be our bifurcation parameter in the period doubling route to chaos.

Poincare Map

Because the forced Hodgkin-Huxley is an essentially 5-dimensional 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 F({\vec  {x}},t) be the flow of the ODE we are trying to study. Fixing T and {\vec  {x}}, we define the Poincare map, P_{n}({\vec  {x}})=F({\vec  {x}},nT). 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 Hodgkin-Huxley equations, imagine a globally attracting limit cycle exits in the phase space of the HH equations a period T (the same as the above Poincare map). It is easy to see that the Poincare map, P_{n}({\vec  {x}}), would approach a fixed point, since eventually all orbits approach the limit cycle and the orbits on the limit cycle itself repeat after T second. Likewise, if that same limit cycle had period T/2, then P_{n}({\vec  {x}}) 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 quasi-periodic motion.

Period Doubling

In the spirit of [1], we obtain a bifurcation diagram of the HH equations, by projecting the Poincare map onto the first coordinate, voltage:


Here we can see the Period doubling cascade: A=50.42 starts as a fixed point, transitioning to a period-2 orbit around 50.3, to a period- orbit around 50.28, and so on and so forth, until the HH equations experience regions of chaotic behavior as well as narrow regions of stable periodic orbits of all orders.

Unimodal Maps--an analogy

Given 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 Predator-Prey 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).


Unimodal maps undergoes what is call a flip or tangent bifurcation, where 2^{n} losses stability as its derivative of passes through -1 and a stable 2^{{n+1}} orbit is born. (the derivative of f^{2^{n+1}} of these orbits are 1, hence the name tangent bifurcation). In the figure above, we can see the birth of a period 2 orbit and the loss of stability of the fixed point. In fact, we can determine the bifurcation parameter at the onset of each flip bifurcation by solving the system with the tangent condition as an additional constraint. [ see].

Periodic Chaos in one dimensional maps

After 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 f:X\to X. We say that f is k-periodic chaotic if there exist a homeomorphsim, h, such that h(f(h^{{-1}}(x)))=4x(1-x), i.e. f is conjugate to the logistic map for r=4.

This phenomena can be seen in the bifurcation diagram where the chaotic branches merge. See [[[project1.pdf]] for a detail discussion of this phenomena.