Delay-induced synchronization transition in small-world Hodgkin-Huxley neuronal networks with channel blocking

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September 2011, 16(2): 607-621. doi: 10.3934/dcdsb.2011.16.607

Delay-induced synchronization transition in small-world Hodgkin-Huxley neuronal networks with channel blocking

Qingyun Wang ^1, , Xia Shi ^2, and Guanrong Chen ^3,

1.	Department of Dynamics and Control, Beihang University, Beijing 100191, China
2.	School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China
3.	Department of Electronic Engineering, City University of Hong Kong, Hong Kong

Received December 2009 Revised September 2010 Published June 2011

Abstract
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We study the evolution of spatiotemporal dynamics and synchronization transition on small-world Hodgkin-Huxley (HH) neuronal networks that are characterized with channel noises, ion channel blocking and information transmission delays. In particular, we examine the effects of delay on spatiotemporal dynamics over neuronal networks when channel blocking of potassium or sodium is involved. We show that small delays can detriment synchronization in the network due to a dynamic clustering anti-phase synchronization transition. We also show that regions of irregular and regular wave propagations related to synchronization transitions appear intermittently as the delay increases, and the delay-induced synchronization transitions manifest as well-expressed minima in the measure for spatial synchrony. In addition, we show that the fraction of sodium or potassium channels can play a key role in dynamics of neuronal networks. Furthermore, We found that the fraction of sodium and potassium channels has different impacts on spatiotemporal dynamics of neuronal networks, respectively. Our results thus provide insights that could facilitate the understanding of the joint impact of ion channel blocking and information transmission delays on the dynamical behaviors of realistic neuronal networks.

Keywords: synchronization., time delay, Neural network.

Mathematics Subject Classification: 76R10, 76F7.

Citation: Qingyun Wang, Xia Shi, Guanrong Chen. Delay-induced synchronization transition in small-world Hodgkin-Huxley neuronal networks with channel blocking. Discrete and Continuous Dynamical Systems - B, 2011, 16 (2) : 607-621. doi: 10.3934/dcdsb.2011.16.607

References:

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[2]	R. S. Cajal, "Histology of the Nervous System of Man and Vertebrates," Oxford Univ. Press, New York, 1995.
[3]	W. L. Swanson, "Brain Architecture," Oxford Univ. Press, New York, 2003.
[4]	E. Bullmore and O. Sporns, Complex brain networks: graph theoretical analysis of structural and functional systems, Nature, 10 (2009), 186-198.
[5]	O. Sporns, D. Chialvo, M. Kaiser and C. C. Hilgetag, Organization, development and function of complex brain networks, Trends Cogn. Sci., 8 (2004), 418-425. doi: 10.1016/j.tics.2004.07.008.
[6]	S. D. Bassett and T. E. Bullmore, Small world brain networks, Trends Cogn. Sci., 12 (2006), 512-523.
[7]	C. J. Reijneveld, S. C. Ponten, H. W. Berendse and J. C. Stam, The application of graph theoretical analysis to complex networks in the brain, Clin. Neurophysiol., 118 (2007), 2317-2331. doi: 10.1016/j.clinph.2007.08.010.
[8]	H. D. I. Abarbanel, M. I. Rabinovich, A. I. Selverston, M. V. Bazhenov, R. Huerta, M. M. Suschchik and L. L. Rubchinskii, Synchronisation in neural networks, Phys. Usp., 39 (1996), 337-362. doi: 10.1070/PU1996v039n04ABEH000141.
[9]	M. I. Rabinovich, P. Varona, A. I. Selverston and H. D. Abarbanel, Dynamical Principles in Neuroscience, Reviews of Modern Physics, 78 (2006), 1213-1265. doi: 10.1103/RevModPhys.78.1213.
[10]	C. Hauptmann, A. Gail and F. Giannakopoulos, Intermittent burst synchronization in neural networks, Computational Methods in Neural Modeling, 2686 (2003), 46-53. doi: 10.1007/3-540-44868-3_7.
[11]	C. Zhou, J. Kurths and B. Hu, Array-enhanced coherence resonance: nontrivial effects of heterogeneity and spatial independence of noise, Phys. Rev. Lett., 87 (2001), 098101. doi: 10.1103/PhysRevLett.87.098101.
[12]	A. Pikovsky, M. Rosenblum and J. Kurths, "Synchronization: A Universal Concept in Nonlinear Sciences," Cambridge University Press, Cambridge, 2001. doi: 10.1017/CBO9780511755743.
[13]	Q. Y. Wang, Z. S. Duan, L. Huang, G. R. Chen and Q. S. Lu, Pattern formation and firing synchronization in networks of map neurons, New J. Phys, 9 (2007), 1-11. doi: 10.1088/1367-2630/9/10/383.
[14]	O. Kwon and H.-T. Moon, Coherence resonance in small-world networks of excitable cells, Phys. Lett. A, 298 (2002), 319-324. doi: 10.1016/S0375-9601(02)00575-3.
[15]	Q. Y. Wang, Z. S. Duan, M. Perc and G. R. Chen, Synchronization transitions on small-world neuronal networks: Effects of information transmission delay and rewiring probability, Europhys. Lett., 83 (2008), 50008. doi: 10.1209/0295-5075/83/50008.
[16]	G. Tanakaa, B. Ibarz, M. A. F. Sanjuan and K. Aihara, Synchronization and propagation of bursts in networks of coupled map neurons, Chaos, 16 (2006), 013113. doi: 10.1063/1.2148387.
[17]	Q. Y. Wang, Q. S. Lu and G. R. Chen, Ordered bursting synchronization and complex wave propagation in a ring neuronal network, Physica A, 374 (2007), 869-878. doi: 10.1016/j.physa.2006.08.062.
[18]	C. S. Zhou, L. Zemanová, G. Zamora, C. C. Hilgetag and J. Kurths, Hierarchical organization unveiled by functional connectivity in complex brain networks, Phys. Rev. Lett., 97 (2006), 238103. doi: 10.1103/PhysRevLett.97.238103.
[19]	H. Hill, "Ionic Channels of Excitable Membranes," 3rd edition, Sinauer Associates, Sundrland, MA, 2001.
[20]	Y. B. Gong, Y. H. Hao and Y. H. Xie, Channel blocking-optimized spiking activity of Hodgkin-Huxley neurons on random networks, Physica A, 389 (2010), 349-357. doi: 10.1016/j.physa.2009.09.033.
[21]	M. Ozer, M. Perc and M. Uzuntarl, Controlling the spontaneous spiking regularity via channel blockinging on Newman-Watts networks of Hodgkin-Huxley neurons, Europhys. Lett., 86 (2009), 40008. doi: 10.1209/0295-5075/86/40008.
[22]	Q. Y. Wang and Q. S. Lu, Time Delay-Enhanced Synchronization and Regularization in Two Coupled Chaotic Neurons, Chin. Phys. Lett., 3 (2005), 543-546.
[23]	E. Rossoni, Y. H. Chen, M. Z. Ding and J. F. Feng, Stability of synchronous oscillations in a system of Hodgkin-Huxley neurons with delayed diffusive and pulsed coupling, Phys. Rev. E, 71 (2005), 061904. doi: 10.1103/PhysRevE.71.061904.
[24]	A. S. Landsman and I. B. Schwartz, Synchronized dynamics of cortical neurons with time-delay feedback, Nonlinear Biomedical Physics, 1 (2007), 1-9. doi: 10.1186/1753-4631-1-2.
[25]	S. Q. Ma, Z. S. Feng and Q. S. Lu, A two-parameter geometrical criteria for delay differential equations, Discrete and Continuous Dynamical Systems-Series B, 9 (2008), 397-413.
[26]	F. K. Wu and Y. Z. Hu, Stochastic Lotka-Volterra system with unbounded distributed delay, Discrete and Continuous Dynamical Systems-Series B, 14 (2010), 275-288.
[27]	Q. Y. Wang, M. Perc, Z. S. Duan and G. R. Chen, Synchronization transitions on scale-free neuronal networks due to finite information transmission delays, Phys. Rev. E, 80 (2009), 026206. doi: 10.1103/PhysRevE.80.026206.
[28]	A. L. Hodgkin and A. F. Huxley, Quantitative description of membrane and its application to conduction and excitation in nerve, J Physiol, 117 (1952), 500-544.
[29]	S. T. Wang, F. Liu, W. Wang and Y. G. Yu, Impact of spatially correlated noise on neuronal firing, Phys. Rev. E., 69 (2004), 011909. doi: 10.1103/PhysRevE.69.011909.
[30]	Y. B. Gong, M. S. Wang, Z. H. Hou and H. W. Xin, Optimal Spike Coherence and Synchronization on Complex Hodgkin-Huxley Neuron Networks, Chem. Phys. Chem., 6 (2005), 1042-1047. doi: 10.1002/cphc.200500051.
[31]	Y. B. Gong, B. Xu, Q. Xu, C. L. Yang, T. Q. Ren, Z. H. Hou and H. W Xin, Ordering spatiotemporal chaos in complex thermosensitive neuron networks, Phys. Rev. E, 73 (2006), 046137. doi: 10.1103/PhysRevE.73.046137.
[32]	Z. Gao, B. Hu and G. Hu, Stochastic resonance of small-world networks, Phys. Rev. E., 65 (2001), 016209. doi: 10.1103/PhysRevE.65.016209.

show all references

References:

[1]	O. Sporns and J. C. Honey, Small world inside big brains, PNAS, 51 (2006), 19219-19220. doi: 10.1073/pnas.0609523103.
[2]	R. S. Cajal, "Histology of the Nervous System of Man and Vertebrates," Oxford Univ. Press, New York, 1995.
[3]	W. L. Swanson, "Brain Architecture," Oxford Univ. Press, New York, 2003.
[4]	E. Bullmore and O. Sporns, Complex brain networks: graph theoretical analysis of structural and functional systems, Nature, 10 (2009), 186-198.
[5]	O. Sporns, D. Chialvo, M. Kaiser and C. C. Hilgetag, Organization, development and function of complex brain networks, Trends Cogn. Sci., 8 (2004), 418-425. doi: 10.1016/j.tics.2004.07.008.
[6]	S. D. Bassett and T. E. Bullmore, Small world brain networks, Trends Cogn. Sci., 12 (2006), 512-523.
[7]	C. J. Reijneveld, S. C. Ponten, H. W. Berendse and J. C. Stam, The application of graph theoretical analysis to complex networks in the brain, Clin. Neurophysiol., 118 (2007), 2317-2331. doi: 10.1016/j.clinph.2007.08.010.
[8]	H. D. I. Abarbanel, M. I. Rabinovich, A. I. Selverston, M. V. Bazhenov, R. Huerta, M. M. Suschchik and L. L. Rubchinskii, Synchronisation in neural networks, Phys. Usp., 39 (1996), 337-362. doi: 10.1070/PU1996v039n04ABEH000141.
[9]	M. I. Rabinovich, P. Varona, A. I. Selverston and H. D. Abarbanel, Dynamical Principles in Neuroscience, Reviews of Modern Physics, 78 (2006), 1213-1265. doi: 10.1103/RevModPhys.78.1213.
[10]	C. Hauptmann, A. Gail and F. Giannakopoulos, Intermittent burst synchronization in neural networks, Computational Methods in Neural Modeling, 2686 (2003), 46-53. doi: 10.1007/3-540-44868-3_7.
[11]	C. Zhou, J. Kurths and B. Hu, Array-enhanced coherence resonance: nontrivial effects of heterogeneity and spatial independence of noise, Phys. Rev. Lett., 87 (2001), 098101. doi: 10.1103/PhysRevLett.87.098101.
[12]	A. Pikovsky, M. Rosenblum and J. Kurths, "Synchronization: A Universal Concept in Nonlinear Sciences," Cambridge University Press, Cambridge, 2001. doi: 10.1017/CBO9780511755743.
[13]	Q. Y. Wang, Z. S. Duan, L. Huang, G. R. Chen and Q. S. Lu, Pattern formation and firing synchronization in networks of map neurons, New J. Phys, 9 (2007), 1-11. doi: 10.1088/1367-2630/9/10/383.
[14]	O. Kwon and H.-T. Moon, Coherence resonance in small-world networks of excitable cells, Phys. Lett. A, 298 (2002), 319-324. doi: 10.1016/S0375-9601(02)00575-3.
[15]	Q. Y. Wang, Z. S. Duan, M. Perc and G. R. Chen, Synchronization transitions on small-world neuronal networks: Effects of information transmission delay and rewiring probability, Europhys. Lett., 83 (2008), 50008. doi: 10.1209/0295-5075/83/50008.
[16]	G. Tanakaa, B. Ibarz, M. A. F. Sanjuan and K. Aihara, Synchronization and propagation of bursts in networks of coupled map neurons, Chaos, 16 (2006), 013113. doi: 10.1063/1.2148387.
[17]	Q. Y. Wang, Q. S. Lu and G. R. Chen, Ordered bursting synchronization and complex wave propagation in a ring neuronal network, Physica A, 374 (2007), 869-878. doi: 10.1016/j.physa.2006.08.062.
[18]	C. S. Zhou, L. Zemanová, G. Zamora, C. C. Hilgetag and J. Kurths, Hierarchical organization unveiled by functional connectivity in complex brain networks, Phys. Rev. Lett., 97 (2006), 238103. doi: 10.1103/PhysRevLett.97.238103.
[19]	H. Hill, "Ionic Channels of Excitable Membranes," 3rd edition, Sinauer Associates, Sundrland, MA, 2001.
[20]	Y. B. Gong, Y. H. Hao and Y. H. Xie, Channel blocking-optimized spiking activity of Hodgkin-Huxley neurons on random networks, Physica A, 389 (2010), 349-357. doi: 10.1016/j.physa.2009.09.033.
[21]	M. Ozer, M. Perc and M. Uzuntarl, Controlling the spontaneous spiking regularity via channel blockinging on Newman-Watts networks of Hodgkin-Huxley neurons, Europhys. Lett., 86 (2009), 40008. doi: 10.1209/0295-5075/86/40008.
[22]	Q. Y. Wang and Q. S. Lu, Time Delay-Enhanced Synchronization and Regularization in Two Coupled Chaotic Neurons, Chin. Phys. Lett., 3 (2005), 543-546.
[23]	E. Rossoni, Y. H. Chen, M. Z. Ding and J. F. Feng, Stability of synchronous oscillations in a system of Hodgkin-Huxley neurons with delayed diffusive and pulsed coupling, Phys. Rev. E, 71 (2005), 061904. doi: 10.1103/PhysRevE.71.061904.
[24]	A. S. Landsman and I. B. Schwartz, Synchronized dynamics of cortical neurons with time-delay feedback, Nonlinear Biomedical Physics, 1 (2007), 1-9. doi: 10.1186/1753-4631-1-2.
[25]	S. Q. Ma, Z. S. Feng and Q. S. Lu, A two-parameter geometrical criteria for delay differential equations, Discrete and Continuous Dynamical Systems-Series B, 9 (2008), 397-413.
[26]	F. K. Wu and Y. Z. Hu, Stochastic Lotka-Volterra system with unbounded distributed delay, Discrete and Continuous Dynamical Systems-Series B, 14 (2010), 275-288.
[27]	Q. Y. Wang, M. Perc, Z. S. Duan and G. R. Chen, Synchronization transitions on scale-free neuronal networks due to finite information transmission delays, Phys. Rev. E, 80 (2009), 026206. doi: 10.1103/PhysRevE.80.026206.
[28]	A. L. Hodgkin and A. F. Huxley, Quantitative description of membrane and its application to conduction and excitation in nerve, J Physiol, 117 (1952), 500-544.
[29]	S. T. Wang, F. Liu, W. Wang and Y. G. Yu, Impact of spatially correlated noise on neuronal firing, Phys. Rev. E., 69 (2004), 011909. doi: 10.1103/PhysRevE.69.011909.
[30]	Y. B. Gong, M. S. Wang, Z. H. Hou and H. W. Xin, Optimal Spike Coherence and Synchronization on Complex Hodgkin-Huxley Neuron Networks, Chem. Phys. Chem., 6 (2005), 1042-1047. doi: 10.1002/cphc.200500051.
[31]	Y. B. Gong, B. Xu, Q. Xu, C. L. Yang, T. Q. Ren, Z. H. Hou and H. W Xin, Ordering spatiotemporal chaos in complex thermosensitive neuron networks, Phys. Rev. E, 73 (2006), 046137. doi: 10.1103/PhysRevE.73.046137.
[32]	Z. Gao, B. Hu and G. Hu, Stochastic resonance of small-world networks, Phys. Rev. E., 65 (2001), 016209. doi: 10.1103/PhysRevE.65.016209.

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