American Institute of Mathematical Sciences

Advanced
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

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 & Continuous Dynamical Systems - B, 2011, 16 (2) : 607-621. doi: 10.3934/dcdsb.2011.16.607
References:
[1]

O. Sporns and J. C. Honey, Small world inside big brains,, PNAS, 51 (2006), 19219.  doi: 10.1073/pnas.0609523103.  Google Scholar

[2]

R. S. Cajal, "Histology of the Nervous System of Man and Vertebrates,", Oxford Univ. Press, (1995).   Google Scholar

[3]

W. L. Swanson, "Brain Architecture,", Oxford Univ. Press, (2003).   Google Scholar

[4]

E. Bullmore and O. Sporns, Complex brain networks: graph theoretical analysis of structural and functional systems,, Nature, 10 (2009), 186.   Google Scholar

[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.  doi: 10.1016/j.tics.2004.07.008.  Google Scholar

[6]

S. D. Bassett and T. E. Bullmore, Small world brain networks,, Trends Cogn. Sci., 12 (2006), 512.   Google Scholar

[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.  doi: 10.1016/j.clinph.2007.08.010.  Google Scholar

[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.  doi: 10.1070/PU1996v039n04ABEH000141.  Google Scholar

[9]

M. I. Rabinovich, P. Varona, A. I. Selverston and H. D. Abarbanel, Dynamical Principles in Neuroscience,, Reviews of Modern Physics, 78 (2006), 1213.  doi: 10.1103/RevModPhys.78.1213.  Google Scholar

[10]

C. Hauptmann, A. Gail and F. Giannakopoulos, Intermittent burst synchronization in neural networks,, Computational Methods in Neural Modeling, 2686 (2003), 46.  doi: 10.1007/3-540-44868-3_7.  Google Scholar

[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).  doi: 10.1103/PhysRevLett.87.098101.  Google Scholar

[12]

A. Pikovsky, M. Rosenblum and J. Kurths, "Synchronization: A Universal Concept in Nonlinear Sciences,", Cambridge University Press, (2001).  doi: 10.1017/CBO9780511755743.  Google Scholar

[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.  doi: 10.1088/1367-2630/9/10/383.  Google Scholar

[14]

O. Kwon and H.-T. Moon, Coherence resonance in small-world networks of excitable cells,, Phys. Lett. A, 298 (2002), 319.  doi: 10.1016/S0375-9601(02)00575-3.  Google Scholar

[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).  doi: 10.1209/0295-5075/83/50008.  Google Scholar

[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).  doi: 10.1063/1.2148387.  Google Scholar

[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.  doi: 10.1016/j.physa.2006.08.062.  Google Scholar

[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).  doi: 10.1103/PhysRevLett.97.238103.  Google Scholar

[19]

H. Hill, "Ionic Channels of Excitable Membranes,", 3rd edition, (2001).   Google Scholar

[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.  doi: 10.1016/j.physa.2009.09.033.  Google Scholar

[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).  doi: 10.1209/0295-5075/86/40008.  Google Scholar

[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.   Google Scholar

[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).  doi: 10.1103/PhysRevE.71.061904.  Google Scholar

[24]

A. S. Landsman and I. B. Schwartz, Synchronized dynamics of cortical neurons with time-delay feedback,, Nonlinear Biomedical Physics, 1 (2007), 1.  doi: 10.1186/1753-4631-1-2.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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).  doi: 10.1103/PhysRevE.80.026206.  Google Scholar

[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.   Google Scholar

[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).  doi: 10.1103/PhysRevE.69.011909.  Google Scholar

[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.  doi: 10.1002/cphc.200500051.  Google Scholar

[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).  doi: 10.1103/PhysRevE.73.046137.  Google Scholar

[32]

Z. Gao, B. Hu and G. Hu, Stochastic resonance of small-world networks,, Phys. Rev. E., 65 (2001).  doi: 10.1103/PhysRevE.65.016209.  Google Scholar

show all references

References:
[1]

O. Sporns and J. C. Honey, Small world inside big brains,, PNAS, 51 (2006), 19219.  doi: 10.1073/pnas.0609523103.  Google Scholar

[2]

R. S. Cajal, "Histology of the Nervous System of Man and Vertebrates,", Oxford Univ. Press, (1995).   Google Scholar

[3]

W. L. Swanson, "Brain Architecture,", Oxford Univ. Press, (2003).   Google Scholar

[4]

E. Bullmore and O. Sporns, Complex brain networks: graph theoretical analysis of structural and functional systems,, Nature, 10 (2009), 186.   Google Scholar

[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.  doi: 10.1016/j.tics.2004.07.008.  Google Scholar

[6]

S. D. Bassett and T. E. Bullmore, Small world brain networks,, Trends Cogn. Sci., 12 (2006), 512.   Google Scholar

[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.  doi: 10.1016/j.clinph.2007.08.010.  Google Scholar

[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.  doi: 10.1070/PU1996v039n04ABEH000141.  Google Scholar

[9]

M. I. Rabinovich, P. Varona, A. I. Selverston and H. D. Abarbanel, Dynamical Principles in Neuroscience,, Reviews of Modern Physics, 78 (2006), 1213.  doi: 10.1103/RevModPhys.78.1213.  Google Scholar

[10]

C. Hauptmann, A. Gail and F. Giannakopoulos, Intermittent burst synchronization in neural networks,, Computational Methods in Neural Modeling, 2686 (2003), 46.  doi: 10.1007/3-540-44868-3_7.  Google Scholar

[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).  doi: 10.1103/PhysRevLett.87.098101.  Google Scholar

[12]

A. Pikovsky, M. Rosenblum and J. Kurths, "Synchronization: A Universal Concept in Nonlinear Sciences,", Cambridge University Press, (2001).  doi: 10.1017/CBO9780511755743.  Google Scholar

[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.  doi: 10.1088/1367-2630/9/10/383.  Google Scholar

[14]

O. Kwon and H.-T. Moon, Coherence resonance in small-world networks of excitable cells,, Phys. Lett. A, 298 (2002), 319.  doi: 10.1016/S0375-9601(02)00575-3.  Google Scholar

[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).  doi: 10.1209/0295-5075/83/50008.  Google Scholar

[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).  doi: 10.1063/1.2148387.  Google Scholar

[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.  doi: 10.1016/j.physa.2006.08.062.  Google Scholar

[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).  doi: 10.1103/PhysRevLett.97.238103.  Google Scholar

[19]

H. Hill, "Ionic Channels of Excitable Membranes,", 3rd edition, (2001).   Google Scholar

[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.  doi: 10.1016/j.physa.2009.09.033.  Google Scholar

[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).  doi: 10.1209/0295-5075/86/40008.  Google Scholar

[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.   Google Scholar

[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).  doi: 10.1103/PhysRevE.71.061904.  Google Scholar

[24]

A. S. Landsman and I. B. Schwartz, Synchronized dynamics of cortical neurons with time-delay feedback,, Nonlinear Biomedical Physics, 1 (2007), 1.  doi: 10.1186/1753-4631-1-2.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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).  doi: 10.1103/PhysRevE.80.026206.  Google Scholar

[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.   Google Scholar

[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).  doi: 10.1103/PhysRevE.69.011909.  Google Scholar

[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.  doi: 10.1002/cphc.200500051.  Google Scholar

[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).  doi: 10.1103/PhysRevE.73.046137.  Google Scholar

[32]

Z. Gao, B. Hu and G. Hu, Stochastic resonance of small-world networks,, Phys. Rev. E., 65 (2001).  doi: 10.1103/PhysRevE.65.016209.  Google Scholar

[1]

Tingting Su, Xinsong Yang. Finite-time synchronization of competitive neural networks with mixed delays. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3655-3667. doi: 10.3934/dcdsb.2016115
[2]

Rui Hu, Yuan Yuan. Stability, bifurcation analysis in a neural network model with delay and diffusion. Conference Publications, 2009, 2009 (Special) : 367-376. doi: 10.3934/proc.2009.2009.367
[3]

Ruoxia Li, Huaiqin Wu, Xiaowei Zhang, Rong Yao. Adaptive projective synchronization of memristive neural networks with time-varying delays and stochastic perturbation. Mathematical Control & Related Fields, 2015, 5 (4) : 827-844. doi: 10.3934/mcrf.2015.5.827
[4]

Fang Han, Bin Zhen, Ying Du, Yanhong Zheng, Marian Wiercigroch. Global Hopf bifurcation analysis of a six-dimensional FitzHugh-Nagumo neural network with delay by a synchronized scheme. Discrete & Continuous Dynamical Systems - B, 2011, 16 (2) : 457-474. doi: 10.3934/dcdsb.2011.16.457
[5]

Jianfeng Feng, Mariya Shcherbina, Brunello Tirozzi. Stability of the dynamics of an asymmetric neural network. Communications on Pure & Applied Analysis, 2009, 8 (2) : 655-671. doi: 10.3934/cpaa.2009.8.655
[6]

Stefano Fasani, Sergio Rinaldi. Local stabilization and network synchronization: The case of stationary regimes. Mathematical Biosciences & Engineering, 2010, 7 (3) : 623-639. doi: 10.3934/mbe.2010.7.623
[7]

Ying Sue Huang, Chai Wah Wu. Stability of cellular neural network with small delays. Conference Publications, 2005, 2005 (Special) : 420-426. doi: 10.3934/proc.2005.2005.420
[8]

King Hann Lim, Hong Hui Tan, Hendra G. Harno. Approximate greatest descent in neural network optimization. Numerical Algebra, Control & Optimization, 2018, 8 (3) : 327-336. doi: 10.3934/naco.2018021
[9]

Shyan-Shiou Chen, Chih-Wen Shih. Asymptotic behaviors in a transiently chaotic neural network. Discrete & Continuous Dynamical Systems - A, 2004, 10 (3) : 805-826. doi: 10.3934/dcds.2004.10.805
[10]

Ndolane Sene. Fractional input stability and its application to neural network. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 853-865. doi: 10.3934/dcdss.2020049
[11]

Benedetta Lisena. Average criteria for periodic neural networks with delay. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 761-773. doi: 10.3934/dcdsb.2014.19.761
[12]

Hui-Qiang Ma, Nan-Jing Huang. Neural network smoothing approximation method for stochastic variational inequality problems. Journal of Industrial & Management Optimization, 2015, 11 (2) : 645-660. doi: 10.3934/jimo.2015.11.645
[13]

Yixin Guo, Aijun Zhang. Existence and nonexistence of traveling pulses in a lateral inhibition neural network. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1729-1755. doi: 10.3934/dcdsb.2016020
[14]

Jianhong Wu, Ruyuan Zhang. A simple delayed neural network with large capacity for associative memory. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 851-863. doi: 10.3934/dcdsb.2004.4.851
[15]

Sanjay K. Mazumdar, Cheng-Chew Lim. A neural network based anti-skid brake system. Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 321-338. doi: 10.3934/dcds.1999.5.321
[16]

K. L. Mak, J. G. Peng, Z. B. Xu, K. F. C. Yiu. A novel neural network for associative memory via dynamical systems. Discrete & Continuous Dynamical Systems - B, 2006, 6 (3) : 573-590. doi: 10.3934/dcdsb.2006.6.573
[17]

Honggang Yu. An efficient face recognition algorithm using the improved convolutional neural network. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 901-914. doi: 10.3934/dcdss.2019060
[18]

Lidong Liu, Fajie Wei, Shenghan Zhou. Major project risk assessment method based on BP neural network. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1053-1064. doi: 10.3934/dcdss.2019072
[19]

Zhuwei Qin, Fuxun Yu, Chenchen Liu, Xiang Chen. How convolutional neural networks see the world --- A survey of convolutional neural network visualization methods. Mathematical Foundations of Computing, 2018, 1 (2) : 149-180. doi: 10.3934/mfc.2018008
[20]

Pierre Guiraud, Etienne Tanré. Stability of synchronization under stochastic perturbations in leaky integrate and fire neural networks of finite size. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 5183-5201. doi: 10.3934/dcdsb.2019056

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (19)
  • HTML views (0)
  • Cited by (14)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences