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A multi-base harmonic balance method applied to Hodgkin-Huxley model

  • * Corresponding author: V. Lanza.

    * Corresponding author: V. Lanza. 
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  • Our aim is to propose a new robust and manageable technique, called multi-base harmonic balance method, to detect and characterize the periodic solutions of a nonlinear dynamical system. Our case test is the Hodgkin-Huxley model, one of the most realistic neuronal models in literature. This system, depending on the value of the external stimuli current, exhibits periodic solutions, both stable and unstable.

    Mathematics Subject Classification: Primary: 34C25; Secondary: 65L10.


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  • Figure 1.  Bifurcation diagram of the HH model, showing the stable (solid line) and unstable (dotted line) branches of the periodic solutions of HH model. For each periodic solution the minimum and the maximum values of the potential $V$ over one period are represented. Depending on the values of $I$, two regions with different dynamical behaviors can be identified.

    Figure 2.  Zoom for $I \in~[0, ~I_{2}]$ of Fig 1. HH model exhibits one equilibrium point, one stable limit cycle (solid line) and up to 3 unstable ones (dotted lines).

    Figure 3.  (a) The stable periodic solution detected by the HB method for $I =6.25$ exhibits a sort of Gibbs phenomenon. (b) Zoom showing the small oscillations, sign of a non accurate approximation of the limit cycle, despite the exploitation of 50 harmonics.

    Figure 4.  (a) Stable and (b) unstable periodic solutions for different values of $I$, in a neighborhood of (a) $I_1$ and (b) $I_2$, respectively.

    Figure 5.  Time series of (a) the stable periodic solution for $I = 152.2500$ and (b) the unstable periodic solution for $I = 9.71889$.

    Figure 6.  Stable (solid line) and unstable (dashed line) limit cycles near the first saddle-node of cycles bifurcation, for (a) $I =6.2649$ both solutions are almost coincident, and for (b) $I =6.2716$.

    Figure 7.  Projection of two unstable limit cycles on the $(V, n)$ plane for (a) $I = 7.92198548\lesssim I_3$ and (b) $I = I_3 = 7.92198549$.

    Figure 8.  Projection of two unstable limit cycles on the $(V, n)$ plane for (a) $I = I_4 = 7.84654752$ and (b) $I = 7.84654876\lesssim I_4$.

    Figure 9.  (a)-(b) Floquet multipliers for the stable limit cycles and unstable limit cycles, respectively, associated to the first saddle node of cycles bifurcation for $I\in [6.2792, 6.7872]$. As $I$ increases, in (a) the multiplier $\mu_4$ starts from the value +1 and then enters in the unit circle, while in (b) the multiplier $\mu_4$ starts to the value +1 and becomes bigger and bigger. (c)-(d) Floquet multipliers for the two unstable limit cycles associated to the second saddle node of cycles bifurcation for $I\in [7.921985465, 7.921985491]$. Here, in both cases, the third multiplier is outside the unit circle (this makes the limit cycle unstable) and is not shown, since it takes very high values with respect to the others. As in the previous case, as $I$ decreases, the multiplier $\mu_4$ starts from the value +1 and either (c) enters in the unit circle, or (d) takes higher and higher values. (e)-(f) Floquet multipliers for the two unstable limit cycles associated to the third saddle node of cycles bifurcation for $I\in [7.846557778, 7.846616827]$. Also in this case, for both limit cycles, the third multiplier is not represented. As $I$ increases, the multiplier $\mu_4$ starts from the value +1 and either (e) escapes from, or (f) enters in the unit circle.

    Figure 10.  Floquet multipliers near the period-doubling bifurcation for different values of $I\in [7.92197743, 7.92197799]$. By decreasing $I$, the multiplier $\mu_4$ crosses the unit cycle through $-1$.

    Table 1.  By decreasing the value of I, the multipliers $\mu_4$ decreases, crosses the value -1 for $I =7.92197768$ and enters into the unit circle.

    I $\mu_1$ $\mu_2$ $\mu_3$ $\mu_4$
    7.92197799 1.000 0.000 -2940.687 -1.041
    7.92197793 1.000 -0.000 -2964.042 -1.033
    7.92197787 1.000 0.000 -2987.386 -1.025
    7.92197781 1.000 0.000 -3010.719 -1.017
    7.92197775 1.000 -0.000 -3034.042 -1.009
    7.92197768 1.000 0.000 -3057.354 -1.001
    7.92197762 1.000 -0.000 -3080.655 -0.993
    7.92197756 1.000 0.000 -3103.946 -0.986
    7.92197750 1.000 0.000 -3127.225 -0.978
    7.92197743 1.000 0.000 -3150.494 -0.9713
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