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Stochastic dynamics of the FitzHugh-Nagumo neuron model through a modified Van der Pol equation with fractional-order term and Gaussian white noise excitation

The work of CBT was supported by the Botswana International University of Science and Technology under the grant DVC/RDI/2/1/16I (25). CBT thanks the Kavli Institute for Theoretical Physics (KITP), University of California Santa Barbara (USA), where this work was supported in part by the National Science Foundation Grant no.NSF PHY-1748958 and NIH Grant no.R25GM067110

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  • The stochastic response of the FitzHugh-Nagumo model is addressed using a modified Van der Pol (VDP) equation with fractional-order derivative and Gaussian white noise excitation. Via the generalized harmonic balance method, the term related to fractional derivative is splitted into the equivalent quasi-linear dissipative force and quasi-linear restoring force, leading to an equivalent VDP equation without fractional derivative. The analytical solutions for the equivalent stochastic equation are then investigated through the stochastic averaging method. This is thereafter compared to numerical solutions, where the stationary probability density function (PDF) of amplitude and joint PDF of displacement and velocity are used to characterized the dynamical behaviors of the system. A satisfactory agreement is found between the two approaches, which confirms the accuracy of the used analytical method. It is also found that changing the fractional-order parameter and the intensity of the Gaussian white noise induces P-bifurcation.

    Mathematics Subject Classification: Primary: 26A33, 92B25; Secondary: 65P30, 37M05.


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  • Figure 1.  The panel shows the stationary probability density function (PDF) of the amplitude for different values of the fractional-order parameter $ \alpha $. Solid lines correspond to analytical results, while symbol ($ \triangle $) corresponds to results from numerical calculations

    Figure 2.  The panels show the joint PDF of the displacement $ X $ and velocity $ Y $, corresponding to Fig. 1, for different values of the fractional-order parameter $ \alpha $. Panels (aj)$ _{j = 1, 2, 3} $ correspond to our analytical calculations, while their corresponding panels (bj)$ _{j = 1, 2, 3} $ are obtained from numerical simulations: (a1)-(b1) $ \alpha = 0.8 $, (a2)-(b2) $ \alpha = 0.6 $ and (a3)-(b3) $ \alpha = 0.3 $

    Figure 3.  The panel shows the stationary probability density function (PDF) of the amplitude for different values of the fractional-order parameter $ \alpha $. Solid lines correspond to analytical results, while symbol ($ \triangle $) corresponds to results from numerical calculations

    Figure 4.  The panels show numerical results for the joint PDF displacement $ X $ and velocity $ Y $. The lines form top to bottom correspond to different values of $ D $, the Gaussian white noise intensity: (aj)$ _{j = 1, 2, 3} $ $ D = 0.1 $, (bj)$ _{j = 1, 2, 3} $ $ D = 0.05 $ and (cj)$ _{j = 1, 2, 3} $ $ D = 0.01 $. The columns from left to right respectively correspond to $ \alpha = 0.8 $, $ \alpha = 0.6 $ and $ \alpha = 0.3 $

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