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March  2020, 13(3): 663-682. doi: 10.3934/dcdss.2020036

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

 1 Department of Mathematical Sciences, University of South Africa, Florida, 0003, South Africa 2 Department of Mathematics, Arba Minch University, Ethiopia

* Corresponding author: franckemile2006@yahoo.ca

Received  August 2018 Revised  September 2018 Published  March 2019

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

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.

Citation: Emile Franc Doungmo Goufo, Melusi Khumalo, Patrick M. Tchepmo Djomegni. Perturbations of Hindmarsh-Rose neuron dynamics by fractional operators: Bifurcation, firing and chaotic bursts. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 663-682. doi: 10.3934/dcdss.2020036
##### References:
 [1] A. Atangana and D. Baleanu, New fractional derivatives with non-local and non-singular kerneL, Thermal Science, 20 (2016), 763-769. [2] A. Atangana and I. Koca, Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order, Fractals, 89 (2016), 447-454.  doi: 10.1016/j.chaos.2016.02.012. [3] E. Babolian and A. Shahsavaran, Numerical solution of nonlinear fredholm integral equations of the second kind using haar wavelets, Journal of Computational and Applied Mathematics, 225 (2009), 87-95.  doi: 10.1016/j.cam.2008.07.003. [4] R. Barrio, M. Angeles Martínez, S. Serrano and A. Shilnikov, Macro-and micro-chaotic structures in the Hindmarsh-Rose model of bursting neurons, Chaos: An Interdisciplinary Journal of Nonlinear Science, 24 (2014), 023128, 11pp. doi: 10.1063/1.4882171. [5] D. Brockmann and L. Hufnagel, Front propagation in reaction-superdiffusion dynamics: Taming Lévy flights with fluctuations, Physical Review Letters, 98 (2007), 178-301.  doi: 10.1103/PhysRevLett.98.178301. [6] M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl, 1 (2015), 1-13. [7] M. Caputo, Linear models of dissipation whose Q is almost frequency independent-Ⅱ, Geophysical Journal International, 13 (1967), 529-539.  doi: 10.1111/j.1365-246X.1967.tb02303.x. [8] Y.-Q. Che, J. Wang, K.-M. Tsang and W.-L. Chan, Unidirectional synchronization for Hindmarsh-Rose neurons via robust adaptive sliding mode control, Nonlinear Analysis: Real World Applications, 11 (2010), 1096-1104.  doi: 10.1016/j.nonrwa.2009.02.004. [9] Y. Chen, M. Yi and C. Yu, Error analysis for numerical solution of fractional differential equation by Haar wavelets method, Journal of Computational Science, 3 (2012), 367-373.  doi: 10.1016/j.jocs.2012.04.008. [10] S. Das, Convergence of riemann-liouvelli and caputo derivative definitions for practical solution of fractional order differential equation, International Journal of Applied Mathematics and Statistics, 23 (2011), 64-74. [11] E. F. Doungmo Goufo, Application of the Caputo-Fabrizio fractional derivative without singular kernel to Korteweg-de Vries-Bergers equation, Mathematical Modelling and Analysis, 21 (2016), 188-198.  doi: 10.3846/13926292.2016.1145607. [12] E. F. Doungmo Goufo, Chaotic processes using the two-parameter derivative with non-singular and non-local kernel: Basic theory and applications, Chaos: An Interdisciplinary Journal of Nonlinear Science, 26 (2016), 084305, 10pp. doi: 10.1063/1.4958921. [13] E. F. Doungmo Goufo and A. Atangana, Analytical and numerical schemes for a derivative with filtering property and no singular kernel with applications to diffusion, The European Physical Journal Plus, 131 (2016), 269. [14] E. F. Doungmo Goufo, Stability and convergence analysis of a variable order replicator-mutator process in a moving medium, Journal of Theoretical Biology, 403 (2016), 178-187.  doi: 10.1016/j.jtbi.2016.05.007. [15] E. F. Doungmo Goufo, Solvability of chaotic fractional systems with 3D four-scroll attractors, Fractals, 104 (2017), 443-451.  doi: 10.1016/j.chaos.2017.08.038. [16] E. F. Doungmo Goufo and J. J. Nieto, Attractors for fractional differential problems of transition to turbulent flows, Journal of Computational and Applied Mathematics, 339 (2018), 329-342.  doi: 10.1016/j.cam.2017.08.026. [17] G. Fubini, Opere scelte. Ⅱ, Cremonese, Roma, 1958. [18] A. Gökdoğan, M. Merdan and A. Yildirim, The modified algorithm for the differential transform method to solution of genesio systems, Communications in Nonlinear Science and Numerical Simulation, 17 (2012), 45-51.  doi: 10.1016/j.cnsns.2011.03.039. [19] J. Gómez-Aguilar and A. Atangana, New insight in fractional differentiation: Power, exponential decay and Mittag-Leffler laws and applications, The European Physical Journal Plus, 132 (2017), p13. [20] R. Hilfer, Application of Fractional Calculus in Physics, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. doi: 10.1142/9789812817747. [21] J. Hindmarsh and R. Rose, A model of the nerve impulse using two first-order differential equations, Nature, 296 (1982), 162-164.  doi: 10.1038/296162a0. [22] J. L. Hindmarsh and R. Rose, A model of neuronal bursting using three coupled first order differential equations, Proceedings of the Royal Society of London B: Biological Sciences, 221 (1984), 87-102. [23] A. Hodgkin and A. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, Bulletin of Mathematical Biology, 52 (1990), 25-71. [24] A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, The Journal of Physiology, 117 (1952), 500-544. [25] G. Innocenti, A. Morelli, R. Genesio and A. Torcini, Dynamical phases of the Hindmarsh-Rose neuronal model: Studies of the transition from bursting to spiking chaos, Chaos: An Interdisciplinary Journal of Nonlinear Science, 17 (2007), 043128, 11pp. doi: 10.1063/1.2818153. [26] D. Jun, Z. Guang-jun, X. Yong, Y. Hong and W. Jue, Dynamic behavior analysis of fractional-order Hindmarsh-Rose neuronal model, Cognitive Neurodynamics, 8 (2014), 167-175.  doi: 10.1007/s11571-013-9273-x. [27] E. R. Kandel, J. H. Schwartz, T. M. Jessell, S. A. Siegelbaum, A. J. Hudspeth and others, Principles of Neural Science, (McGraw-hill New York, 2000). [28] A. A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, (Elsevier Science Limited, 2006). [29] Ü. Lepik and H. Hein, Haar Wavelets: With Applications, (Springer Science & Business Media, 2014). doi: 10.1007/978-3-319-04295-4. [30] N. K. Logothetis, J. Pauls, M. Augath, T. Trinath and A. Oeltermann, Neurophysiological investigation of the basis of the fMRI signal, Nature, 412 (2001), 150-157.  doi: 10.1038/35084005. [31] D. Matignon, Stability results for fractional differential equations with applications to control processing, Computational Engineering in Systems Applications, 2 (1996), 963-968. [32] J. E. Misiaszek, The H-reflex as a tool in neurophysiology: Its limitations and uses in understanding nervous system function, nerve, 28 (2003), 144-160.  doi: 10.1002/mus.10372. [33] S. Ostojic, N. Brunel and V. Hakim, Synchronization properties of networks of electrically coupled neurons in the presence of noise and heterogeneities, Journal of Computational Neuroscience, 26 (2009), 369-392.  doi: 10.1007/s10827-008-0117-3. [34] L. D. Partridge and C. Stevens, A mechanism for spike frequency adaptation, The Journal of Physiology, 256 (1976), 315-332.  doi: 10.1113/jphysiol.1976.sp011327. [35] S. Pooseh, H. S. Rodrigues and D. F. Torres, Fractional derivatives in dengue epidemics, AIP Conference Proceedings, 1389 (2011), 739-742.  doi: 10.1063/1.3636838. [36] H.-P. Ren and C. Bai, M. S. Baptista and C. Grebogi, Weak connections form an infinite number of patterns in the brain, Scientific Reports, 2017. [37] G. Rizzolatti and L. Craighero, The mirror-neuron system, Annu. Rev. Neurosci, 27 (2004), 169-192.  doi: 10.1002/9780470478509.neubb001017. [38] M. Storace, D. Linaro and E. de Lange, The Hindmarsh-Rose neuron model: bifurcation analysis and piecewise-linear approximations, Chaos: An Interdisciplinary Journal of Nonlinear Science, 18 (2008), 033128, 10pp. doi: 10.1063/1.2975967. [39] R. F. Thompson and W. A. Spencer, Habituation: A model phenomenon for the study of neuronal substrates of behavior, Psychological Review, 73 (1966), 16-43.  doi: 10.1037/h0022681. [40] L. Tonelli, Sullintegrazione per parti, Rend. Acc. Naz. Lincei, 5 (1909), 246-253. [41] X.-J. Yang, H. Srivastava, J.-H. He and D. Baleanu, Cantor-type cylindrical-coordinate method for differential equations with local fractional derivatives, Physics Letters A, 377 (2013), 1696-1700.  doi: 10.1016/j.physleta.2013.04.012. [42] Y. Yamada and Y. Kashimori, Neural mechanism of dynamic responses of neurons in inferior temporal cortex in face perception, Cognitive Neurodynamics, 7 (2013), 23-38.  doi: 10.1007/s11571-012-9212-2. [43] A. Yildirim, A. Gökdoğan and M. Merdan, Chaotic systems via multi-step differential transformation method, Canadian Journal of Physics, 90 (2012), 391-406.

show all references

##### References:
 [1] A. Atangana and D. Baleanu, New fractional derivatives with non-local and non-singular kerneL, Thermal Science, 20 (2016), 763-769. [2] A. Atangana and I. Koca, Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order, Fractals, 89 (2016), 447-454.  doi: 10.1016/j.chaos.2016.02.012. [3] E. Babolian and A. Shahsavaran, Numerical solution of nonlinear fredholm integral equations of the second kind using haar wavelets, Journal of Computational and Applied Mathematics, 225 (2009), 87-95.  doi: 10.1016/j.cam.2008.07.003. [4] R. Barrio, M. Angeles Martínez, S. Serrano and A. Shilnikov, Macro-and micro-chaotic structures in the Hindmarsh-Rose model of bursting neurons, Chaos: An Interdisciplinary Journal of Nonlinear Science, 24 (2014), 023128, 11pp. doi: 10.1063/1.4882171. [5] D. Brockmann and L. Hufnagel, Front propagation in reaction-superdiffusion dynamics: Taming Lévy flights with fluctuations, Physical Review Letters, 98 (2007), 178-301.  doi: 10.1103/PhysRevLett.98.178301. [6] M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl, 1 (2015), 1-13. [7] M. Caputo, Linear models of dissipation whose Q is almost frequency independent-Ⅱ, Geophysical Journal International, 13 (1967), 529-539.  doi: 10.1111/j.1365-246X.1967.tb02303.x. [8] Y.-Q. Che, J. Wang, K.-M. Tsang and W.-L. Chan, Unidirectional synchronization for Hindmarsh-Rose neurons via robust adaptive sliding mode control, Nonlinear Analysis: Real World Applications, 11 (2010), 1096-1104.  doi: 10.1016/j.nonrwa.2009.02.004. [9] Y. Chen, M. Yi and C. Yu, Error analysis for numerical solution of fractional differential equation by Haar wavelets method, Journal of Computational Science, 3 (2012), 367-373.  doi: 10.1016/j.jocs.2012.04.008. [10] S. Das, Convergence of riemann-liouvelli and caputo derivative definitions for practical solution of fractional order differential equation, International Journal of Applied Mathematics and Statistics, 23 (2011), 64-74. [11] E. F. Doungmo Goufo, Application of the Caputo-Fabrizio fractional derivative without singular kernel to Korteweg-de Vries-Bergers equation, Mathematical Modelling and Analysis, 21 (2016), 188-198.  doi: 10.3846/13926292.2016.1145607. [12] E. F. Doungmo Goufo, Chaotic processes using the two-parameter derivative with non-singular and non-local kernel: Basic theory and applications, Chaos: An Interdisciplinary Journal of Nonlinear Science, 26 (2016), 084305, 10pp. doi: 10.1063/1.4958921. [13] E. F. Doungmo Goufo and A. Atangana, Analytical and numerical schemes for a derivative with filtering property and no singular kernel with applications to diffusion, The European Physical Journal Plus, 131 (2016), 269. [14] E. F. Doungmo Goufo, Stability and convergence analysis of a variable order replicator-mutator process in a moving medium, Journal of Theoretical Biology, 403 (2016), 178-187.  doi: 10.1016/j.jtbi.2016.05.007. [15] E. F. Doungmo Goufo, Solvability of chaotic fractional systems with 3D four-scroll attractors, Fractals, 104 (2017), 443-451.  doi: 10.1016/j.chaos.2017.08.038. [16] E. F. Doungmo Goufo and J. J. Nieto, Attractors for fractional differential problems of transition to turbulent flows, Journal of Computational and Applied Mathematics, 339 (2018), 329-342.  doi: 10.1016/j.cam.2017.08.026. [17] G. Fubini, Opere scelte. Ⅱ, Cremonese, Roma, 1958. [18] A. Gökdoğan, M. Merdan and A. Yildirim, The modified algorithm for the differential transform method to solution of genesio systems, Communications in Nonlinear Science and Numerical Simulation, 17 (2012), 45-51.  doi: 10.1016/j.cnsns.2011.03.039. [19] J. Gómez-Aguilar and A. Atangana, New insight in fractional differentiation: Power, exponential decay and Mittag-Leffler laws and applications, The European Physical Journal Plus, 132 (2017), p13. [20] R. Hilfer, Application of Fractional Calculus in Physics, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. doi: 10.1142/9789812817747. [21] J. Hindmarsh and R. Rose, A model of the nerve impulse using two first-order differential equations, Nature, 296 (1982), 162-164.  doi: 10.1038/296162a0. [22] J. L. Hindmarsh and R. Rose, A model of neuronal bursting using three coupled first order differential equations, Proceedings of the Royal Society of London B: Biological Sciences, 221 (1984), 87-102. [23] A. Hodgkin and A. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, Bulletin of Mathematical Biology, 52 (1990), 25-71. [24] A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, The Journal of Physiology, 117 (1952), 500-544. [25] G. Innocenti, A. Morelli, R. Genesio and A. Torcini, Dynamical phases of the Hindmarsh-Rose neuronal model: Studies of the transition from bursting to spiking chaos, Chaos: An Interdisciplinary Journal of Nonlinear Science, 17 (2007), 043128, 11pp. doi: 10.1063/1.2818153. [26] D. Jun, Z. Guang-jun, X. Yong, Y. Hong and W. Jue, Dynamic behavior analysis of fractional-order Hindmarsh-Rose neuronal model, Cognitive Neurodynamics, 8 (2014), 167-175.  doi: 10.1007/s11571-013-9273-x. [27] E. R. Kandel, J. H. Schwartz, T. M. Jessell, S. A. Siegelbaum, A. J. Hudspeth and others, Principles of Neural Science, (McGraw-hill New York, 2000). [28] A. A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, (Elsevier Science Limited, 2006). [29] Ü. Lepik and H. Hein, Haar Wavelets: With Applications, (Springer Science & Business Media, 2014). doi: 10.1007/978-3-319-04295-4. [30] N. K. Logothetis, J. Pauls, M. Augath, T. Trinath and A. Oeltermann, Neurophysiological investigation of the basis of the fMRI signal, Nature, 412 (2001), 150-157.  doi: 10.1038/35084005. [31] D. Matignon, Stability results for fractional differential equations with applications to control processing, Computational Engineering in Systems Applications, 2 (1996), 963-968. [32] J. E. Misiaszek, The H-reflex as a tool in neurophysiology: Its limitations and uses in understanding nervous system function, nerve, 28 (2003), 144-160.  doi: 10.1002/mus.10372. [33] S. Ostojic, N. Brunel and V. Hakim, Synchronization properties of networks of electrically coupled neurons in the presence of noise and heterogeneities, Journal of Computational Neuroscience, 26 (2009), 369-392.  doi: 10.1007/s10827-008-0117-3. [34] L. D. Partridge and C. Stevens, A mechanism for spike frequency adaptation, The Journal of Physiology, 256 (1976), 315-332.  doi: 10.1113/jphysiol.1976.sp011327. [35] S. Pooseh, H. S. Rodrigues and D. F. Torres, Fractional derivatives in dengue epidemics, AIP Conference Proceedings, 1389 (2011), 739-742.  doi: 10.1063/1.3636838. [36] H.-P. Ren and C. Bai, M. S. Baptista and C. Grebogi, Weak connections form an infinite number of patterns in the brain, Scientific Reports, 2017. [37] G. Rizzolatti and L. Craighero, The mirror-neuron system, Annu. Rev. Neurosci, 27 (2004), 169-192.  doi: 10.1002/9780470478509.neubb001017. [38] M. Storace, D. Linaro and E. de Lange, The Hindmarsh-Rose neuron model: bifurcation analysis and piecewise-linear approximations, Chaos: An Interdisciplinary Journal of Nonlinear Science, 18 (2008), 033128, 10pp. doi: 10.1063/1.2975967. [39] R. F. Thompson and W. A. Spencer, Habituation: A model phenomenon for the study of neuronal substrates of behavior, Psychological Review, 73 (1966), 16-43.  doi: 10.1037/h0022681. [40] L. Tonelli, Sullintegrazione per parti, Rend. Acc. Naz. Lincei, 5 (1909), 246-253. [41] X.-J. Yang, H. Srivastava, J.-H. He and D. Baleanu, Cantor-type cylindrical-coordinate method for differential equations with local fractional derivatives, Physics Letters A, 377 (2013), 1696-1700.  doi: 10.1016/j.physleta.2013.04.012. [42] Y. Yamada and Y. Kashimori, Neural mechanism of dynamic responses of neurons in inferior temporal cortex in face perception, Cognitive Neurodynamics, 7 (2013), 23-38.  doi: 10.1007/s11571-012-9212-2. [43] A. Yildirim, A. Gökdoğan and M. Merdan, Chaotic systems via multi-step differential transformation method, Canadian Journal of Physics, 90 (2012), 391-406.
A typical nerve cell with an axon that extends the cell body to the terminal branches and dendrites
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
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
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
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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