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August  2020, 25(8): 3233-3256. doi: 10.3934/dcdsb.2020060

Random attractor for stochastic Hindmarsh-Rose equations with multiplicative noise

University of South Florida, Department of Mathematics and Statistics, Tampa, FL 33620, USA

Received  August 2019 Revised  October 2019 Published  August 2020 Early access  February 2020

The longtime and global pullback dynamics of stochastic Hind-marsh-Rose equations with multiplicative noise on a three-dimensional bounded domain in neurodynamics is investigated in this work. The existence of a random attractor for this random dynamical system is proved through the exponential transformation and uniform estimates showing the pullback absorbing property and the pullback asymptotically compactness of this cocycle in the $ L^2 $ Hilbert space.

Citation: Chi Phan. Random attractor for stochastic Hindmarsh-Rose equations with multiplicative noise. Discrete and Continuous Dynamical Systems - B, 2020, 25 (8) : 3233-3256. doi: 10.3934/dcdsb.2020060
References:
[1]

L. Arnold, Random Dynamical Systems, Springer, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.

[2]

P. W. BatesK. Lu and B. Wang, Random attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations, 246 (2009), 845-869.  doi: 10.1016/j.jde.2008.05.017.

[3]

R. BertramM. J. ButteT. Kiemel and A. Sherman, Topologica and phenomenological classification of bursting oscillations, Bulletin of Mathematical Biology, 57 (1995), 413-439. 

[4]

R. J. ButersJ. Rinzel and J. C. Smith, Models respiratory rhythm generation in the pre-Bötzinger complex, I. Bursting pacemaker neurons, J. Neurophysiology, 81 (1999), 382-397. 

[5]

C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Math., Vol. 580, Springer-Verlag, Berlin-New York, 1977.

[6]

T. CaraballoM. J. Garrido-AtienzaB. Schmalfuss and J. Valero, Asymptotic behavior of a stochastic semilinear dissipative functional equations without uniqueness of solutions, Discrete and Continuous Dynamical Systems, Series B, 14 (2010), 439-455.  doi: 10.3934/dcdsb.2010.14.439.

[7]

T. R. Chay and J. Keizer, Minimal model for membrane oscillations in the pancreatic beta-cell, Biophysiology Journal, 42 (1983), 181-189.  doi: 10.1016/S0006-3495(83)84384-7.

[8]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, AMS Colloquium Publications, 49, AMS, Providence, RI, 2002.

[9]

I. Chueshov, Monotone Random Systems Theory and Applications, Lect. Notes of Math., Vol. 1779, Springer-Verlag, Berlin, 2002. doi: 10.1007/b83277.

[10]

L. N. CornelisseW. J. ScheenenW. J. KoopmanE. W. Roubos and S. C. Gielen, Minimal model for intracellular calcium oscillations and electrical bursting in melanotrope cells of Xenopus Laevis, Neural Computations, 13 (2000), 113-137. 

[11]

H. CrauelA. Debusche and F. Flandoli, Random attractors, J. Dynamics and Differential Equations, 9 (1997), 307-341.  doi: 10.1007/BF02219225.

[12]

H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probability Theory and Related Fields, 100 (1994), 365-393.  doi: 10.1007/BF01193705.

[13]

M. DhamalaV. K. Jirsa and M. Ding, Transitions to synchrony in coupled bursting neurons, Physical Review Letters, 92 (2004), 028101.  doi: 10.1103/PhysRevLett.92.028101.

[14]

M. Efendiev and S. Zelik, Upper and lower bounds for the Kolmogorov entropy of the attractor for an RDE in an unbounded domain, J. Dynamics and Differential Equations, 14 (2002), 369-403.  doi: 10.1023/A:1015130904414.

[15]

G. B. Ementrout and D. H. Terman, Mathematical Foundations of Neurosciences, Springer, 2010. doi: 10.1007/978-0-387-87708-2.

[16]

R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophysical Journal, 1 (1961), 445-466.  doi: 10.1016/S0006-3495(61)86902-6.

[17]

F. Flandoli and B. Schmalfuss, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative noise, Stoch. Stoch. Rep., 59 (1996), 21-45.  doi: 10.1080/17442509608834083.

[18]

X. HanW. Shen and S. Zhou, Random attractors for stochastic lattice dynamical systems in weighted spaces, J. Differential Equations, 250 (2011), 1235-1266.  doi: 10.1016/j.jde.2010.10.018.

[19]

J. L. Hindmarsh and R. M. Rose, A model of the nerve impulse using two first-order differential equations, Nature, 206 (1982), 162-164.  doi: 10.1038/296162a0.

[20]

J. L. Hindmarsh and R. M. Rose, A model of neuronal bursting using three coupled first-order differential equations, Proceedings of the Royal Society London, Ser. B: Biological Sciences, 221 (1984), 87-102. 

[21]

A. Hodgkin and A. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, J. Physiology, Ser. B, 117 (1952), 500-544. 

[22]

G. Innocenti and R. Genesio, On the dynamics of chaotic spiking-bursting transition in the Hindmarsh-Rose neuron, Chaos, 19 (2009), 023124, 8pp. doi: 10.1063/1.3156650.

[23] E.M. Izhikecich, Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting, MIT Press, Cambridge, Massachusetts, 2007. 
[24]

S. Q. MaZ. Feng and Q. Lu, Dynamics and double Hopf bifurcations of the Rose-Hindmarsh model with time delay, International Journal of Bifurcation and Chaos, 19 (2009), 3733-3751.  doi: 10.1142/S0218127409025080.

[25]

L. H. Nguyen and K.-S. Hong, Lyapunov-based synchronization of two coupled chaotic Hindmarsh-Rose neurons, Journal of Computer Science and Cybernetics, 30 (2014), 335.  doi: 10.14736/kyb-2015-5-0784.

[26]

B. Øksendal, Stochastic Differential Equations, 6th edition, Springer-Verlag, Berlin, 2003. doi: 10.1007/978-3-642-14394-6.

[27]

C. Phan, Y. You and J. Su, Global attractors for Hindmarsh-Rose equations in neurodynamics, arXiv: 1907.13225.

[28]

J. Rinzel, A formal classification of bursting mechanism in excitable systems, Proceedings of International Congress of Mathematics, 1 (1987), 1578-1593. 

[29]

J. Rubin, Bursting induced by excitatory synaptic coupling in nonidentical conditional relaxation oscillators or square-wave bursters, Physics Review E, 74 (2006), 021917, 15pp. doi: 10.1103/PhysRevE.74.021917.

[30]

K. R. Schenk-Hoppé, Random attractors - general properties, existence and applications to stochastic bifurcation theory, Discrete and Continuous Dynamical Systems, 4 (1998), 99-130.  doi: 10.3934/dcds.1998.4.99.

[31]

B. Schmalfuss, Backward cocycles and attractors of stochastic differential equations, International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractors Approximation and Global Behavior, Dresden, (1992), 185–192.

[32]

G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Applied Mathematical Sciences, 143, Springer, New York, 2002. doi: 10.1007/978-1-4757-5037-9.

[33]

A. ShapiroR. CurtuJ. Rinzel and N. Rubin, Dynamical characteristics common to neuronal competition models, J. Neurophysiology, 97 (2007), 462-473.  doi: 10.1152/jn.00604.2006.

[34]

L. ShiR. WangK. Lu and B. Wang, Asymptotic behavior of stochastic FitzHugh-Nagumo systems on unbounded thin domains, J. Differential Equations, 267 (2019), 4373-4409.  doi: 10.1016/j.jde.2019.05.002.

[35]

J. Su, H. Perez-Gonzalez and M. He, Regular bursting emerging from coupled chaotic neurons, Discrete and Continuous Dynamical Systems, Supplement, 2007,946–955.

[36]

D. Terman, Chaotic spikes arising from a model of bursting in excitable membrane, J. Appl. Math., 51 (1991), 1418-1450.  doi: 10.1137/0151071.

[37]

B. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differential Equations, 253 (2012), 1544-1583.  doi: 10.1016/j.jde.2012.05.015.

[38]

B. Wang, Random attractors for non-autonomous stochastic wave equations, Discrete and Continuous Dynamical Systems, Series A, 34 (2014), 269-300.  doi: 10.3934/dcds.2014.34.269.

[39]

R. Wang and B. Wang, Asymptotic behavior of non-autonomous fractional stochastic $p$-Laplacian equations, Computers and Mathematics with Applications, 78 (2019), 3527-3543.  doi: 10.1016/j.camwa.2019.05.024.

[40]

Z. L. Wang and X. R. Shi, Chaotic bursting lag synchronization of Hindmarsh-Rose system via a single controller, Applied Mathematics and Computation, 215 (2009), 1091-1097.  doi: 10.1016/j.amc.2009.06.039.

[41]

Y. You, Global dynamics and robustness of reversible autocatalytic reaction-diffusion systems, Nonlinear Analysis, Series A, 75 (2012), 3049-3071.  doi: 10.1016/j.na.2011.12.002.

[42]

Y. You, Random attractors and robustness for stochastic reversible reaction-diffusion systems, Discrete and Continuous Dynamical Systems, Series A, 34 (2014), 301-333.  doi: 10.3934/dcds.2014.34.301.

[43]

Y. You, Random attractors for stochastic reversible Schnackenberg equations, Discrete and Continuous Dynamical Systems, Series S, 7 (2014), 1347-1362.  doi: 10.3934/dcdss.2014.7.1347.

[44]

Y. You, Random dynamics of stochastic reaction-diffusion systems with additive noise, Journal of Dynamics and Differential Equations, 29 (2017), 83-112.  doi: 10.1007/s10884-015-9431-4.

[45]

F. ZhangA. LubbeQ. Lu and J. Su, On bursting solutions near chaotic regimes in a neuron model, Discrete and Continuous Dynamical Systems, Series S, 7 (2014), 1363-1383.  doi: 10.3934/dcdss.2014.7.1363.

[46]

S. Zhou, Random exponential attractor for stochastic reaction-diffusion equations with multiplicative noise in $\mathbb{R}^3$, J. Differential Equations, 263 (2017), 6347-6383.  doi: 10.1016/j.jde.2017.07.013.

show all references

References:
[1]

L. Arnold, Random Dynamical Systems, Springer, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.

[2]

P. W. BatesK. Lu and B. Wang, Random attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations, 246 (2009), 845-869.  doi: 10.1016/j.jde.2008.05.017.

[3]

R. BertramM. J. ButteT. Kiemel and A. Sherman, Topologica and phenomenological classification of bursting oscillations, Bulletin of Mathematical Biology, 57 (1995), 413-439. 

[4]

R. J. ButersJ. Rinzel and J. C. Smith, Models respiratory rhythm generation in the pre-Bötzinger complex, I. Bursting pacemaker neurons, J. Neurophysiology, 81 (1999), 382-397. 

[5]

C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Math., Vol. 580, Springer-Verlag, Berlin-New York, 1977.

[6]

T. CaraballoM. J. Garrido-AtienzaB. Schmalfuss and J. Valero, Asymptotic behavior of a stochastic semilinear dissipative functional equations without uniqueness of solutions, Discrete and Continuous Dynamical Systems, Series B, 14 (2010), 439-455.  doi: 10.3934/dcdsb.2010.14.439.

[7]

T. R. Chay and J. Keizer, Minimal model for membrane oscillations in the pancreatic beta-cell, Biophysiology Journal, 42 (1983), 181-189.  doi: 10.1016/S0006-3495(83)84384-7.

[8]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, AMS Colloquium Publications, 49, AMS, Providence, RI, 2002.

[9]

I. Chueshov, Monotone Random Systems Theory and Applications, Lect. Notes of Math., Vol. 1779, Springer-Verlag, Berlin, 2002. doi: 10.1007/b83277.

[10]

L. N. CornelisseW. J. ScheenenW. J. KoopmanE. W. Roubos and S. C. Gielen, Minimal model for intracellular calcium oscillations and electrical bursting in melanotrope cells of Xenopus Laevis, Neural Computations, 13 (2000), 113-137. 

[11]

H. CrauelA. Debusche and F. Flandoli, Random attractors, J. Dynamics and Differential Equations, 9 (1997), 307-341.  doi: 10.1007/BF02219225.

[12]

H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probability Theory and Related Fields, 100 (1994), 365-393.  doi: 10.1007/BF01193705.

[13]

M. DhamalaV. K. Jirsa and M. Ding, Transitions to synchrony in coupled bursting neurons, Physical Review Letters, 92 (2004), 028101.  doi: 10.1103/PhysRevLett.92.028101.

[14]

M. Efendiev and S. Zelik, Upper and lower bounds for the Kolmogorov entropy of the attractor for an RDE in an unbounded domain, J. Dynamics and Differential Equations, 14 (2002), 369-403.  doi: 10.1023/A:1015130904414.

[15]

G. B. Ementrout and D. H. Terman, Mathematical Foundations of Neurosciences, Springer, 2010. doi: 10.1007/978-0-387-87708-2.

[16]

R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophysical Journal, 1 (1961), 445-466.  doi: 10.1016/S0006-3495(61)86902-6.

[17]

F. Flandoli and B. Schmalfuss, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative noise, Stoch. Stoch. Rep., 59 (1996), 21-45.  doi: 10.1080/17442509608834083.

[18]

X. HanW. Shen and S. Zhou, Random attractors for stochastic lattice dynamical systems in weighted spaces, J. Differential Equations, 250 (2011), 1235-1266.  doi: 10.1016/j.jde.2010.10.018.

[19]

J. L. Hindmarsh and R. M. Rose, A model of the nerve impulse using two first-order differential equations, Nature, 206 (1982), 162-164.  doi: 10.1038/296162a0.

[20]

J. L. Hindmarsh and R. M. Rose, A model of neuronal bursting using three coupled first-order differential equations, Proceedings of the Royal Society London, Ser. B: Biological Sciences, 221 (1984), 87-102. 

[21]

A. Hodgkin and A. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, J. Physiology, Ser. B, 117 (1952), 500-544. 

[22]

G. Innocenti and R. Genesio, On the dynamics of chaotic spiking-bursting transition in the Hindmarsh-Rose neuron, Chaos, 19 (2009), 023124, 8pp. doi: 10.1063/1.3156650.

[23] E.M. Izhikecich, Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting, MIT Press, Cambridge, Massachusetts, 2007. 
[24]

S. Q. MaZ. Feng and Q. Lu, Dynamics and double Hopf bifurcations of the Rose-Hindmarsh model with time delay, International Journal of Bifurcation and Chaos, 19 (2009), 3733-3751.  doi: 10.1142/S0218127409025080.

[25]

L. H. Nguyen and K.-S. Hong, Lyapunov-based synchronization of two coupled chaotic Hindmarsh-Rose neurons, Journal of Computer Science and Cybernetics, 30 (2014), 335.  doi: 10.14736/kyb-2015-5-0784.

[26]

B. Øksendal, Stochastic Differential Equations, 6th edition, Springer-Verlag, Berlin, 2003. doi: 10.1007/978-3-642-14394-6.

[27]

C. Phan, Y. You and J. Su, Global attractors for Hindmarsh-Rose equations in neurodynamics, arXiv: 1907.13225.

[28]

J. Rinzel, A formal classification of bursting mechanism in excitable systems, Proceedings of International Congress of Mathematics, 1 (1987), 1578-1593. 

[29]

J. Rubin, Bursting induced by excitatory synaptic coupling in nonidentical conditional relaxation oscillators or square-wave bursters, Physics Review E, 74 (2006), 021917, 15pp. doi: 10.1103/PhysRevE.74.021917.

[30]

K. R. Schenk-Hoppé, Random attractors - general properties, existence and applications to stochastic bifurcation theory, Discrete and Continuous Dynamical Systems, 4 (1998), 99-130.  doi: 10.3934/dcds.1998.4.99.

[31]

B. Schmalfuss, Backward cocycles and attractors of stochastic differential equations, International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractors Approximation and Global Behavior, Dresden, (1992), 185–192.

[32]

G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Applied Mathematical Sciences, 143, Springer, New York, 2002. doi: 10.1007/978-1-4757-5037-9.

[33]

A. ShapiroR. CurtuJ. Rinzel and N. Rubin, Dynamical characteristics common to neuronal competition models, J. Neurophysiology, 97 (2007), 462-473.  doi: 10.1152/jn.00604.2006.

[34]

L. ShiR. WangK. Lu and B. Wang, Asymptotic behavior of stochastic FitzHugh-Nagumo systems on unbounded thin domains, J. Differential Equations, 267 (2019), 4373-4409.  doi: 10.1016/j.jde.2019.05.002.

[35]

J. Su, H. Perez-Gonzalez and M. He, Regular bursting emerging from coupled chaotic neurons, Discrete and Continuous Dynamical Systems, Supplement, 2007,946–955.

[36]

D. Terman, Chaotic spikes arising from a model of bursting in excitable membrane, J. Appl. Math., 51 (1991), 1418-1450.  doi: 10.1137/0151071.

[37]

B. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differential Equations, 253 (2012), 1544-1583.  doi: 10.1016/j.jde.2012.05.015.

[38]

B. Wang, Random attractors for non-autonomous stochastic wave equations, Discrete and Continuous Dynamical Systems, Series A, 34 (2014), 269-300.  doi: 10.3934/dcds.2014.34.269.

[39]

R. Wang and B. Wang, Asymptotic behavior of non-autonomous fractional stochastic $p$-Laplacian equations, Computers and Mathematics with Applications, 78 (2019), 3527-3543.  doi: 10.1016/j.camwa.2019.05.024.

[40]

Z. L. Wang and X. R. Shi, Chaotic bursting lag synchronization of Hindmarsh-Rose system via a single controller, Applied Mathematics and Computation, 215 (2009), 1091-1097.  doi: 10.1016/j.amc.2009.06.039.

[41]

Y. You, Global dynamics and robustness of reversible autocatalytic reaction-diffusion systems, Nonlinear Analysis, Series A, 75 (2012), 3049-3071.  doi: 10.1016/j.na.2011.12.002.

[42]

Y. You, Random attractors and robustness for stochastic reversible reaction-diffusion systems, Discrete and Continuous Dynamical Systems, Series A, 34 (2014), 301-333.  doi: 10.3934/dcds.2014.34.301.

[43]

Y. You, Random attractors for stochastic reversible Schnackenberg equations, Discrete and Continuous Dynamical Systems, Series S, 7 (2014), 1347-1362.  doi: 10.3934/dcdss.2014.7.1347.

[44]

Y. You, Random dynamics of stochastic reaction-diffusion systems with additive noise, Journal of Dynamics and Differential Equations, 29 (2017), 83-112.  doi: 10.1007/s10884-015-9431-4.

[45]

F. ZhangA. LubbeQ. Lu and J. Su, On bursting solutions near chaotic regimes in a neuron model, Discrete and Continuous Dynamical Systems, Series S, 7 (2014), 1363-1383.  doi: 10.3934/dcdss.2014.7.1363.

[46]

S. Zhou, Random exponential attractor for stochastic reaction-diffusion equations with multiplicative noise in $\mathbb{R}^3$, J. Differential Equations, 263 (2017), 6347-6383.  doi: 10.1016/j.jde.2017.07.013.

Figure 1.  Time responses of the membrane potential for various value of the stimulated current: (a) resting state when J = 0, (b) tonic spiking when J = 1.2, (c) regular bursting when J = 2.2, (d) chaotic bursting when J = 3.1, (e) the x-z phase portrait when J = 2.2, (f) the x-z phase portrait when J = 3.1. Source: [25]
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