October  2013, 33(10): 4693-4729. doi: 10.3934/dcds.2013.33.4693

Global asymptotic dynamics of a class of nonlinearly coupled neural networks with delays

1. 

Department of Applied Mathematics, National Pingtung University of Education, Pingtung 900, Taiwan

Received  September 2012 Revised  January 2013 Published  April 2013

This work presents an effective approach to the study of the global asymptotic dynamics of general coupled systems. Under the developed framework, the problem of establishing global synchronization or global convergence reduces to solving a corresponding system of linear equations. We illustrate this approach with a class of neural networks that consist of a pair of sub-networks under various types of nonlinear and delayed couplings. We study both the synchronization and the asymptotic synchronous phases of the dynamics, including global convergence to zero, global convergence to multiple synchronous equilibria, and global synchronization with nontrivial synchronous periodic solutions. Our investigation also provides theoretical support to some numerical findings, and improves or extend some results in the literature.
Citation: Jui-Pin Tseng. Global asymptotic dynamics of a class of nonlinearly coupled neural networks with delays. Discrete and Continuous Dynamical Systems, 2013, 33 (10) : 4693-4729. doi: 10.3934/dcds.2013.33.4693
References:
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F. M. Atay, Oscillator death in coupled functional differential equations near Hopf bifurcation, J. Differential Equations, 221 (2003), 190-209. doi: 10.1016/j.jde.2005.01.007.

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I. Belykh, M. Hasler, M. Lauret and H. Nijmeijer, Synchronization and graph topology, Internat. J. Bifur. Chaos, 15 (2005), 3423-3433. doi: 10.1142/S0218127405014143.

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N. Buric and D. Todorovic, Dynamics of Fitzhugh-Nagumo excitable systems with delayed coupling, Phys. Rev. E, 67 (2003), 066222. doi: 10.1103/PhysRevE.67.066222.

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S. A. Campbell, Time delays in neural systems, Handbook of Brain Connectivity, 65-90, Underst. Complex Syst., Springer, Berlin, (2007), 65-90. doi: 10.1007/978-3-540-71512-2_2.

[5]

S. A. Campbell, R. Edwards and P. van den Driessche, Delayed coupling between two neural networks loops, SIAM J. Appl. Math., 65 (2004), 316-335. doi: 10.1137/S0036139903434833.

[6]

S. A. Campbell, I. Ncube and J. Wu, Multistability and stable asynchronous periodic oscillations in a multiple-delayed neural system, Phys. D, 214 (2006), 101-119. doi: 10.1016/j.physd.2005.12.008.

[7]

S. A. Campbell, Y. Yuan and S. D. Bungay, Equivariant Hopf bifurcation in a ring of identical cells with delayed coupling, Nonlinearity, 18 (2005), 2827-2846. doi: 10.1088/0951-7715/18/6/022.

[8]

Y. C. Chang and J. Juang, Stable synchrony in globally-coupled integrate-and-fire oscillators, SIAM Appl. Dynam. Systems, 7 (2008), 1445-1476. doi: 10.1137/070709220.

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S. M. Crook, G. B. Ermentrout, M. C. Vanier and J. M. Bower, The role of axonal delay in the synchronization of networks of coupled cortical oscillators, J. Comput. Neurosci., 4 (1997), 161-172.

[10]

P. Grosse, M. J. Cassidy and P. Brown, MEG-EMG and EMG-EMG frequency analysis: Physiological principles and clinical applications, Clin. Neurophysiol, 113 (2002), 1523-1531. doi: 10.1016/S1388-2457(02)00223-7.

[11]

S. Guo, Spatio-temporal patterns of nonlinear oscillations in an excitatory ring network with delay, Nonlinearity, 18 (2005), 2391-2407. doi: 10.1088/0951-7715/18/5/027.

[12]

B. D. Hassard, N. D. Kazarinoff and Y. H. Wan, "Theory and Applications of Hopf Bifurcation," London Math. Socl. Lecture Note Series 41, Cambridge University Press, Cambridge, 1981.

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H. S. Hsu and T. S Yang, Periodic oscillations arising and death in delay-coupled neural loops, Intern. J. of Bifurc. and Chaos, 17 (2007), 4015-4032. doi: 10.1142/S0218127407019834.

[14]

J. Juang, C.-L. Li and Y.-H. Liang, Global synchronization in lattices of coupled chaotic systems, Chaos, 17 (2007), 033111. doi: 10.1063/1.2754668.

[15]

R. E. Kalaba and K. Spingarn, A criterion for the convergence of the Gauss-Seidel method, Appl. Math. Comput., 4 (1978), 359-367. doi: 10.1016/0096-3003(78)90004-8.

[16]

J. Karbowski and N. Kopell, Multispikes and synchronization in a large neural network with temporal delays, Neural Comput., 12 (2000), 1573-1606. doi: 10.1162/089976600300015277.

[17]

N. Kopell, G. B. Ermentrout, M. A. Whittington and R. Traub, Gamma rhythms and beta rhythms have different synchronization properties, Proc. Natl. Acad. Sci. USA, 97 (2000), 1867-1872. doi: 10.1073/pnas.97.4.1867.

[18]

K.-L. Liao, C.-W. Shih and J.-P. Tseng, Synchronized oscillations in a mathematical model of segmentation in zebrafish, Nonlinearity, 25 (2012), 869-904. doi: 10.1088/0951-7715/25/4/869.

[19]

X. Liu and T. Chen, Exponential synchronization of nonlinear coupled dynamical networks with a delayed coupling, Phys. A, 381 (2007), 82-92. doi: 10.1016/j.physa.2007.03.026.

[20]

A. C. Marti and C. Masoller, Delay-induced synchronization phenomena in an array of globally coupled logistic maps, Phys. Rev. E, 67 (2003), 056219. doi: 10.1103/PhysRevE.67.056219.

[21]

C. S. Peskin, "Mathematical Aspects of Heart Physiology," Courant Institute of Mathematical Science, New York, 1975.

[22]

M. Porfiri and R. Pigliacampo, Master-slave global stochastic synchronization of chaotic oscillators, SIAM Appl. Dynam. Systems, 7 (2008), 825-842. doi: 10.1137/070688973.

[23]

C.-W. Shih and J.-P.Tseng, Convergent dynamics for multistable delayed neural networks, Nonlinearity, 21 (2008), 2361-2389. doi: 10.1088/0951-7715/21/10/009.

[24]

C.-W.Shih and J.-P. Tseng, Global synchronization and asymptotic phases for a ring of identical cells with delayed coupling, SIAM J. Math. Anal., 43 (2011), 1667-1697. doi: 10.1137/10080885X.

[25]

J.-J. E. Slotine, W. Wang and K. E. Rifai, Contraction analysis of synchronization in networks of nonlinearly coupled oscillators, in "Proc. 16th Int. Symp.: Mathematical Theory of Networks and Systems," Brussels, Belgium, (2004).

[26]

Y. Song, M. Tade and T. Zhang, Bifurcation analysis and spatio-temporal patterns of nonlinear oscillations in a delayed neural network with unidirectional coupling, Nonlinearity, 22 (2009), 975-1001. doi: 10.1088/0951-7715/22/5/004.

[27]

Y. Song, T. Zhang and M. O. Tade, Stability switches, Hopf bifurcations, and spatio-temporal patterns in a delayed neural model with bidirectional coupling, J. Nonlinear Sci., (2009), 597-632. doi: 10.1007/s00332-009-9046-1.

[28]

E. Steur, I. Tyukin and H. Nijmeijer, Semi-passivity and synchronization of diffusively coupled neuronal oscillators, Phys. D, 238 (2009), 2119-2128. doi: 10.1016/j.physd.2009.08.007.

[29]

S. H. Strogatz and I. Stewart, Coupled oscillators and biological synchronization, Scientific American, 269 (1993), 102-109. doi: 10.1038/scientificamerican1293-102.

[30]

X.-J. Wang and G. Buzsaki, Gamma oscillation by synaptic inhibition in a hippocampal interneuronal network model, J. Neuroscience, 16 (1998), 6-16.

[31]

X.-F. Wang and G. Chen, Synchronization in small-world dynamical networks, Internat. J. Bifur. Chaos, 12 (2002), 187-192. doi: 10.1142/S0218127402004292.

[32]

J. White, C. Chow, J. Ritt, C. Soto-Trenivo and N. Kopell White, Synchronization and oscillaory dynamics in heterogeneous, mutually inhibited neurons, J. Comput. Neurosci., 5 (1998), 5-16.

[33]

J. Wu, Symmetric functional differential equations and neural networks with memory, Tansactions of American Mathematical Society, 350 (1998), 4799-4838. doi: 10.1090/S0002-9947-98-02083-2.

[34]

K. Xiao and S. Guo, Synchronization for two coupled oscillators with inhibitory connection, Math. Methods Appl. Sci., 33 (2010), 892-903. doi: 10.1002/mma.1225.

[35]

Y. Yuan and S. A. Campbell, Stability and synchronization of a ring of identical cells with delayed coupling, J. Dynam. Differential Equations, 16 (2004), 709-744. doi: 10.1007/s10884-004-6114-y.

[36]

W. Yu, J. Cao and J. Lu, Global synchronization of linearly hybrid coupled networks with time-varying delay, SIAM Appl. Dynam. Systems, 7 (2007), 108-133. doi: 10.1137/070679090.

show all references

References:
[1]

F. M. Atay, Oscillator death in coupled functional differential equations near Hopf bifurcation, J. Differential Equations, 221 (2003), 190-209. doi: 10.1016/j.jde.2005.01.007.

[2]

I. Belykh, M. Hasler, M. Lauret and H. Nijmeijer, Synchronization and graph topology, Internat. J. Bifur. Chaos, 15 (2005), 3423-3433. doi: 10.1142/S0218127405014143.

[3]

N. Buric and D. Todorovic, Dynamics of Fitzhugh-Nagumo excitable systems with delayed coupling, Phys. Rev. E, 67 (2003), 066222. doi: 10.1103/PhysRevE.67.066222.

[4]

S. A. Campbell, Time delays in neural systems, Handbook of Brain Connectivity, 65-90, Underst. Complex Syst., Springer, Berlin, (2007), 65-90. doi: 10.1007/978-3-540-71512-2_2.

[5]

S. A. Campbell, R. Edwards and P. van den Driessche, Delayed coupling between two neural networks loops, SIAM J. Appl. Math., 65 (2004), 316-335. doi: 10.1137/S0036139903434833.

[6]

S. A. Campbell, I. Ncube and J. Wu, Multistability and stable asynchronous periodic oscillations in a multiple-delayed neural system, Phys. D, 214 (2006), 101-119. doi: 10.1016/j.physd.2005.12.008.

[7]

S. A. Campbell, Y. Yuan and S. D. Bungay, Equivariant Hopf bifurcation in a ring of identical cells with delayed coupling, Nonlinearity, 18 (2005), 2827-2846. doi: 10.1088/0951-7715/18/6/022.

[8]

Y. C. Chang and J. Juang, Stable synchrony in globally-coupled integrate-and-fire oscillators, SIAM Appl. Dynam. Systems, 7 (2008), 1445-1476. doi: 10.1137/070709220.

[9]

S. M. Crook, G. B. Ermentrout, M. C. Vanier and J. M. Bower, The role of axonal delay in the synchronization of networks of coupled cortical oscillators, J. Comput. Neurosci., 4 (1997), 161-172.

[10]

P. Grosse, M. J. Cassidy and P. Brown, MEG-EMG and EMG-EMG frequency analysis: Physiological principles and clinical applications, Clin. Neurophysiol, 113 (2002), 1523-1531. doi: 10.1016/S1388-2457(02)00223-7.

[11]

S. Guo, Spatio-temporal patterns of nonlinear oscillations in an excitatory ring network with delay, Nonlinearity, 18 (2005), 2391-2407. doi: 10.1088/0951-7715/18/5/027.

[12]

B. D. Hassard, N. D. Kazarinoff and Y. H. Wan, "Theory and Applications of Hopf Bifurcation," London Math. Socl. Lecture Note Series 41, Cambridge University Press, Cambridge, 1981.

[13]

H. S. Hsu and T. S Yang, Periodic oscillations arising and death in delay-coupled neural loops, Intern. J. of Bifurc. and Chaos, 17 (2007), 4015-4032. doi: 10.1142/S0218127407019834.

[14]

J. Juang, C.-L. Li and Y.-H. Liang, Global synchronization in lattices of coupled chaotic systems, Chaos, 17 (2007), 033111. doi: 10.1063/1.2754668.

[15]

R. E. Kalaba and K. Spingarn, A criterion for the convergence of the Gauss-Seidel method, Appl. Math. Comput., 4 (1978), 359-367. doi: 10.1016/0096-3003(78)90004-8.

[16]

J. Karbowski and N. Kopell, Multispikes and synchronization in a large neural network with temporal delays, Neural Comput., 12 (2000), 1573-1606. doi: 10.1162/089976600300015277.

[17]

N. Kopell, G. B. Ermentrout, M. A. Whittington and R. Traub, Gamma rhythms and beta rhythms have different synchronization properties, Proc. Natl. Acad. Sci. USA, 97 (2000), 1867-1872. doi: 10.1073/pnas.97.4.1867.

[18]

K.-L. Liao, C.-W. Shih and J.-P. Tseng, Synchronized oscillations in a mathematical model of segmentation in zebrafish, Nonlinearity, 25 (2012), 869-904. doi: 10.1088/0951-7715/25/4/869.

[19]

X. Liu and T. Chen, Exponential synchronization of nonlinear coupled dynamical networks with a delayed coupling, Phys. A, 381 (2007), 82-92. doi: 10.1016/j.physa.2007.03.026.

[20]

A. C. Marti and C. Masoller, Delay-induced synchronization phenomena in an array of globally coupled logistic maps, Phys. Rev. E, 67 (2003), 056219. doi: 10.1103/PhysRevE.67.056219.

[21]

C. S. Peskin, "Mathematical Aspects of Heart Physiology," Courant Institute of Mathematical Science, New York, 1975.

[22]

M. Porfiri and R. Pigliacampo, Master-slave global stochastic synchronization of chaotic oscillators, SIAM Appl. Dynam. Systems, 7 (2008), 825-842. doi: 10.1137/070688973.

[23]

C.-W. Shih and J.-P.Tseng, Convergent dynamics for multistable delayed neural networks, Nonlinearity, 21 (2008), 2361-2389. doi: 10.1088/0951-7715/21/10/009.

[24]

C.-W.Shih and J.-P. Tseng, Global synchronization and asymptotic phases for a ring of identical cells with delayed coupling, SIAM J. Math. Anal., 43 (2011), 1667-1697. doi: 10.1137/10080885X.

[25]

J.-J. E. Slotine, W. Wang and K. E. Rifai, Contraction analysis of synchronization in networks of nonlinearly coupled oscillators, in "Proc. 16th Int. Symp.: Mathematical Theory of Networks and Systems," Brussels, Belgium, (2004).

[26]

Y. Song, M. Tade and T. Zhang, Bifurcation analysis and spatio-temporal patterns of nonlinear oscillations in a delayed neural network with unidirectional coupling, Nonlinearity, 22 (2009), 975-1001. doi: 10.1088/0951-7715/22/5/004.

[27]

Y. Song, T. Zhang and M. O. Tade, Stability switches, Hopf bifurcations, and spatio-temporal patterns in a delayed neural model with bidirectional coupling, J. Nonlinear Sci., (2009), 597-632. doi: 10.1007/s00332-009-9046-1.

[28]

E. Steur, I. Tyukin and H. Nijmeijer, Semi-passivity and synchronization of diffusively coupled neuronal oscillators, Phys. D, 238 (2009), 2119-2128. doi: 10.1016/j.physd.2009.08.007.

[29]

S. H. Strogatz and I. Stewart, Coupled oscillators and biological synchronization, Scientific American, 269 (1993), 102-109. doi: 10.1038/scientificamerican1293-102.

[30]

X.-J. Wang and G. Buzsaki, Gamma oscillation by synaptic inhibition in a hippocampal interneuronal network model, J. Neuroscience, 16 (1998), 6-16.

[31]

X.-F. Wang and G. Chen, Synchronization in small-world dynamical networks, Internat. J. Bifur. Chaos, 12 (2002), 187-192. doi: 10.1142/S0218127402004292.

[32]

J. White, C. Chow, J. Ritt, C. Soto-Trenivo and N. Kopell White, Synchronization and oscillaory dynamics in heterogeneous, mutually inhibited neurons, J. Comput. Neurosci., 5 (1998), 5-16.

[33]

J. Wu, Symmetric functional differential equations and neural networks with memory, Tansactions of American Mathematical Society, 350 (1998), 4799-4838. doi: 10.1090/S0002-9947-98-02083-2.

[34]

K. Xiao and S. Guo, Synchronization for two coupled oscillators with inhibitory connection, Math. Methods Appl. Sci., 33 (2010), 892-903. doi: 10.1002/mma.1225.

[35]

Y. Yuan and S. A. Campbell, Stability and synchronization of a ring of identical cells with delayed coupling, J. Dynam. Differential Equations, 16 (2004), 709-744. doi: 10.1007/s10884-004-6114-y.

[36]

W. Yu, J. Cao and J. Lu, Global synchronization of linearly hybrid coupled networks with time-varying delay, SIAM Appl. Dynam. Systems, 7 (2007), 108-133. doi: 10.1137/070679090.

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