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Perturbations of Hindmarsh-Rose neuron dynamics by fractional operators: Bifurcation, firing and chaotic bursts

The work of EF Doungmo Goufo work was partially supported by the grant No: 114975 from the National Research Foundation (NRF) of South Africa

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  • Studying & understanding the bursting dynamics of membrane potential in neurobiology is captivating in applied sciences, with many features still to be uncovered. In this study, 2D and 3D neuronal activities given by models of Hindmarsh-Rose (HR) neurons with external current input are analyzed numerically with Haar wavelet method, proven to be convergent through error analysis. Our numerical analysis considers two control parameters: the external current $ I^{\text{ext}} $ and the derivative order $ \gamma $, on top of the other seven usual parameters $ a, b, c, d, \nu_1, \nu_2 $ and $ x_{\text{rest}}. $ Bifurcation scenarios for the model show existence of equilibria, both stable and unstable of type saddle and spiral. They also reveal existence of stable limit cycle toward which the trajectories get closer. Numerical approximations of solutions to the 2D model reveals that equilibria remains the same in all cases, irrespective of control parameters'values, but the observed repeated sequences of impulses increase as $ \gamma $ decreases. This inverse proportionability reveals a system likely to be $ \gamma- $controlled with $ \gamma $ varying from 1 down to 0. A similar observation is done for the 3D HR neuron model where regular burst is observed and turns into period-adding chaotic bifurcation (burst with uncountable peaks) as $ \gamma $ changes from 1 down to 0.

    Mathematics Subject Classification: Primary: 26A33, 65P20, 65P30; Secondary: 65L20, 33F05.


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  • Figure 1.  A typical nerve cell with an axon that extends the cell body to the terminal branches and dendrites

    Figure 2.  Phase representation in the plan (x, y) for the model (14) with $ \gamma = 1, \ a = 1, \ b = 3, \ c = 1, \ d = 5. $ The point $ ^1X_0 $ whose abscissa is given by $ ^1e_0 $ is an unstable equilibrium point of spiral type, while $ ^2X_0 $ is an unstable equilibrium point of type saddle point. The dashed line passing through $ ^2X_0 $ represents the saddle line and symbolizes its separatrix. The point $ ^3X_0 $ (with $ ^3e_0 $ as abscissa) is a stable equilibrium point and the trajectories approach the stable limit cycle represented by the solid lines with arrows

    Figure 3.  Numerical solutions showing response of HR neuron 2D-model's membrane potentials for a short current pulse $ I = 1 $ with $ \ a = 1, \ b = 3, \ c = 1, \ d = 5 $ and for $ \gamma = 1.0, \ 0.9,\ $ and $ 0.8 $ respectively. We observe in all three cases a repeated sequences of impulses (periodic firing mode) which happen more rapidly and increasingly as $ \gamma $ decreases, hereby giving $ \gamma $ the status of a suitable parameter for controlling the system

    Figure 4.  Numerical solutions showing response of HR neuron 3D-model's membrane potentials for a short current pulse $ I = 0.5 $ with $ \ a = 1, \ b = 3, \ c = 1, \ d = 5 $ and for $ \gamma = 1.0, \ 0.9,\ $ and $ 0.8 $ respectively. We observe in all three cases regular isolated burst turning into Period-adding chaotic bifurcation (burst with uncountable peaks) as $ \gamma $ decreases. This hereby gives $ \gamma $ the status of a suitable parameter for the system control

    Figure 5.  Numerical solutions showing response of HR neuron 3D-model's membrane potentials for $ I = 2.2 $ with $ \ a = 1, \ b = 3, \ c = 1, \ d = 5 $ and for $ \gamma = 1.0, \ 0.9,\ $ and $ 0.8 $ respectively. Similar to Fig. 4, we observe in all three cases regular but non-isolated burst turning again into Period-adding chaotic bifurcation (burst with uncountable peaks) as $ \gamma $ decreases. This chaos is confirmed by the phase representation in the space $ (x,y,z) $ (on the right). Furthermore, the sequence of repeated bursts happens faster as $ \gamma $ decreases

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