Abstract
We study the role of network architecture and synaptic inputs in the formation of synchronous clusters in synaptically coupled networks of bursting neurons. Through analysis and numerics, we show that the stability of the completely synchronous state, representing the largest cluster, only depends on the number of synaptic inputs each neuron receives, independent from all other details of the network topology. We also give a simple combinatorial algorithm that finds synchronous clusters from the network topology. We demonstrate that networks with a certain degree of internal symmetries are likely to have cluster decompositions with relatively large clusters, leading potentially to cluster synchronization at the mesoscale network level. We address the asymptotic stability of cluster synchronization in excitatory networks of bursting neurons and derive explicit thresholds for the coupling strength that guarantees stable cluster synchronization.
Dedicated to Valentin S. Afraimovich on the occasion of his 65th birthday
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Sporns, O.: Brain connectivity. Scholarpedia 2(10), 469–5 (2007)
Watts, D.J., Strogatz, S.H.: Collective dynamics of 'small-world` networks. Nature 393, 440–442 (1998)
Strogatz, S.H.: Exploring complex networks. Nature 410(6825), 268–276 (2001)
Afraimovich, V.S., Verichev, N.N., Rabinovich, M.I.: Stochastic synchronization of oscillations in dissipative systems. Izv. Vuzov. Radiofiz. 29, 795 (1986)
Pecora, L.M., Carroll, T.L.: Synchronization in chaotic systems. Phys. Rev. Lett. 64(8), 821–824 (1990)
Pecora, L.M., Carroll, T.L.: Master stability functions for synchronized coupled systems. Phys. Rev. Lett. 80(10), 210–9 (1998)
Barahona, M., Pecora, L.M.: Synchronization in small-world systems. Phys. Rev. Lett. 89(5), 05410–1 (2002)
Nishikawa, T., Motter, A.E., Lai, Y.-C., Hoppensteadt, F.C.: Heterogeneity in oscillator networks: Are smaller worlds easier to synchronize? Phys. Rev. Lett. 91(1), 01410–1 (2003)
Belykh, V.N., Belykh, I.V., Hasler, M.: Connection graph stability method for synchronized coupled chaotic systems. Physica (Amsterdam) 195D, 159 (2004)
Almendral, J., Leyva, I., Daqing, L., Sendina-Nadal, I., Shlomo, H., Boccaletti, S.: Dynamics of overlapping structures in modular networks. Phys. Rev. E 82, 01611–5 (2010)
Rodriguez-Caso, C., Corominas-Murtra, B., Sole, R.V.: On the basic computational structure of gene regulatory networks. Mol. Biosyst. 5(12), 1617–1629 (2009)
Newman, M.E.: Modularity and community structure in networks. Proc. Natl. Acad. Sci. USA 103(23), 8577–8582 (2006)
Boccaletti, S., Latora, V., Moreno, Y., Chavez, M., Hwang, D.-U.: Complex networks: Structure and dynamics. Phys. Rep. 424(4–5), 175–308 (2006)
Afraimovich, V.S., Bunimovich, L.A.: Dynamical networks: Interplay of topology, interactions and local dynamics. Nonlinearity 20(7), 176–1 (2007)
Afraimovich, V.S., Zhigulin, V.P., Rabinovich, M.I.: On the origin of reproducible sequential activity in neural circuits. Chaos 14(4), 1123–1129 (2004)
Afraimovich, V.S., Yong, T., Muezzinoglu, M.K., Rabinovich, M.I.: Nonlinear dynamics of emotion-cognition interaction: When emotion does not destroy cognition? Bull Math. Biol. 73(2), 266–284 (2011)
Gray, C.M., Singer, W.: Stimulus-specific neuronal oscillations in orientation columns of cat visual cortex. Proc. Natl. Acad. Sci. USA 86(5), 1698–1702 (1989). (Bazhenov, M., Stopfer, M., Rabinovich, M.I., Huerta, R., Abarbanel, H.D.I., Sejnowski, T.J., and Laurent, G.: Neuron 30, 553 (2001); Mehta, M. R., Lee, A. K., and Wilson, M. A.: Nature 417, 741 (2002))
Ermentrout, G.B., Kopell, N.: Parabolic bursting in an excitable system coupled with a slow oscillation. SIAM J. Appl. Math. 46(2) 233–253 (1986)
Sherman, A., Rinzel, J.: Rhythmogenic effects of weak electrotonic coupling in neuronal models. Proc. Natl. Acad. Sci. USA 89(6), 2471–2474 (1994)
Sherman, A.: Anti-phase, asymmetric and aperiodic oscillations in excitable cells–I. Coupled bursters. Bull. Math. Biol. 56, 811–835 (1994)
Terman, D., Wang, D.: Global competition and local cooperation in a network of neural oscillators. Phys. (Amsterdam) 81D, 148–176 (1995)
Rabinovich, M.I., Torres, J.J., Varona, P., Huerta, R., Weidman, P.: Origin of coherent structures in a discrete chaotic medium. Phys. Rev E 60, R1130–R1133 (1999)
Izhikevich, E.M.: Synchronization of Elliptic Bursters. SIAM Rev. 43, 315–344 (2001)
Rubin, J., Terman, D.: Synchronized bursts and loss of synchrony among heterogeneous conditional oscillators. SIAM J. Appl. Dyn. Sys. 1, 146 (2002)
Dhamala, M., Jirsa, V.K., Ding, M.: Enhancement of neural synchrony by time delay. Phys. Rev. Lett. 92, 07410–4 (2004)
de Vries, G., Sherman, A.: Beyond synchronization: Modulatory and emergent effects of coupling in square-wave bursting. In: Coombes, S., Bressloff, P.C. (eds.) Bursting: The Genesis of Rhythm in the Nervous System, pp. 243–272. World Scientific Publishing, London (2005)
Belykh, I., de Lange, E., Hasler, M.: Synchronization of bursting neurons: what matters in the network topology. Phys. Rev. Lett. 94(18), 18810–1 (2005)
Belykh, I., Shilnikov, A.: When weak inhibition synchronizes strongly desynchronizing networks of bursting neurons. Phys. Rev. Lett. 101(7), 07810–2 (2008)
Shilnikov, A., Gordon, R., Belykh, I.: Polyrhythmic synchronization in bursting networking motifs. Chaos 18(3), 03712–0 (2008)
Jalil, S., Belykh, I., Shilnikov, A.: Fast reciprocal inhibition can synchronize bursting neurons. Phys. Rev. E 81, R04520–1 (2010)
Rinzel, J.: Lecture Notes in Biomathematics, vol. 71, pp. 251–291. Springer, Berlin (1987)
Terman, D.: Chaotic spikes arising from a model of bursting in excitable membranes. SIAM J. Appl. Math. 51, 1418 (1991)
Bertram, R., Butte, M.J., Kiemel, T., Sherman, A.: Topological and phenomenological classification of bursting oscillations. Bull. Math. Biol. 57(3), 413–439 (1995)
Belykh, V.N., Belykh, I.V., Colding-Joergensen, M., Mosekilde, E.: Homoclinic bifurcations leading to bursting oscillations in cell models. Eur. Phys. J. E 3(3), 205–219 (2000)
Izhikevich, E.M.: Neural excitability, spiking, and bursting. Int. J. Bifurc. Chaos 10, 1171–1266 (2000)
Shilnikov, A., Cymbalyuk, G.: Transition between tonic spiking and bursting in a neuron model via the blue-sky catastrophe. Phys. Rev. Lett. 94(4), 04810–1 (2005)
Shilnikov, A., Calabrese, R., Cymbalyuk, G.: Mechanism of bistability: Tonic spiking and bursting in a neuron model. Phys. Rev. E 71, 05621–4 (2005)
Frohlich, F., Bazhenov, M.: Coexistence of tonic firing and bursting in cortical neurons. Phys. Rev. E 74, 03192–2 (2006)
Pogromsky, A.Yu., Nijmeijer, H.: Cooperative oscillatory behavior of mutually coupled dynamical systems. IEEE Trans. Circuits Syst., I: Fundam. Theory Appl. 48(2), 15–2 (2001).
Belykh, V., Belykh, I., Hasler, M.: Hierarchy and stability of partially synchronous oscillations of diffusively coupled dynamical systems. Phys. Rev. E 62, 6332–6345 (2000)
Belykh, I., Belykh, V., Nevidin, K., Hasler, M.: Persistent clusters in lattices of coupled nonidentical chaotic systems. Chaos 13(1), 165–178 (2003)
Stewart, I., Golubitsky, M., Pivato, M.: Symmetry groupoids and patterns of synchrony in coupled cell networks. SIAM J. Appl. Dynam. Sys. 2(4), 609–646 (2003)
Golubitsky, M., Stewart, I., Torok, A.: Patterns of synchrony in coupled cell networks with multiple arrows. SIAM J. Appl. Dynam. Sys. 4(1), 78–100 (2005)
Golubitsky, M., Stewart, I.: Nonlinear dynamics of networks: The groupoid formalism. Bull. Am. Math. Soc. 43, 305–364 (2006)
Wang, Y., Golubitsky, M.: Two-color patterns of synchrony in lattice dynamical systems. Nonlinearity 18, 631–657 (2005)
Belykh, I., Hasler, M.: Mesoscale and clusters of synchrony in networks of bursting neurons. Chaos 21(1), 01610–6 (2011)
Bollobas, B.: Modern Graph Theory. Springer, New York (1998)
Hindmarsh, J.L., Rose, M.: A model of neuronal bursting using three coupled first order differential equations. Proc. R Soc. Lond. B Biol. Sci. 221(1222), 87–102 (1984)
Wang, X.-J.: Genesis of bursting oscillations in the Hindmarsh-Rose model and homoclinicity to a chaotic saddle. Physica (Amsterdam) 62D, 263 (1993)
Shilnikov, A.L., Kolomiets, M.L.: Methods of the qualitative theory for the Hindmarsh–Rose model: A case study. Int. J. Bifurc. Chaos 18(8) 1 (2008)
Storace, M., Linaro, D., de Lange, E.: The Hindmarsh-Rose neuron model: bifurcation analysis and piecewise-linear approximations. Chaos 18(3), 03312–8 (2008)
Bautin, N.N.: Behavior of Dynamical Systems Near the Boundary of Stability. Nauka Publ. House, Moscow (1984)
Somers, D., Kopell, N.: Rapid synchronization through fast threshold modulation. Biol. Cybern. 68, 393–407 (1993)
Wang, X.F., Chen, G.: Synchronization in scale free dynamical networks: Robustness and fragility. IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 49(1), 54–62 (2002)
Acknowledgment
This work was supported by the National Science Foundation under Grant DMS-1009744, the GSU Brains and Behavior program, and RFFI Grants N 2100-065268 and N 09-01-00498-a (to I.B.).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Belykh, I., Hasler, M. (2015). Patterns of Synchrony in Neuronal Networks: The Role of Synaptic Inputs. In: González-Aguilar, H., Ugalde, E. (eds) Nonlinear Dynamics New Directions. Nonlinear Systems and Complexity, vol 12. Springer, Cham. https://doi.org/10.1007/978-3-319-09864-7_1
Download citation
DOI: https://doi.org/10.1007/978-3-319-09864-7_1
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-09863-0
Online ISBN: 978-3-319-09864-7
eBook Packages: EngineeringEngineering (R0)