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Noise-induced burst and spike synchronizations in an inhibitory small-world network of subthreshold bursting neurons

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Abstract

We are interested in noise-induced firings of subthreshold neurons which may be used for encoding environmental stimuli. Noise-induced population synchronization was previously studied only for the case of global coupling, unlike the case of subthreshold spiking neurons. Hence, we investigate the effect of complex network architecture on noise-induced synchronization in an inhibitory population of subthreshold bursting Hindmarsh–Rose neurons. For modeling complex synaptic connectivity, we consider the Watts–Strogatz small-world network which interpolates between regular lattice and random network via rewiring, and investigate the effect of small-world connectivity on emergence of noise-induced population synchronization. Thus, noise-induced burst synchronization (synchrony on the slow bursting time scale) and spike synchronization (synchrony on the fast spike time scale) are found to appear in a synchronized region of the \(J\)–\(D\) plane (\(J\): synaptic inhibition strength and \(D\): noise intensity). As the rewiring probability \(p\) is decreased from 1 (random network) to 0 (regular lattice), the region of spike synchronization shrinks rapidly in the \(J\)–\(D\) plane, while the region of the burst synchronization decreases slowly. We separate the slow bursting and the fast spiking time scales via frequency filtering, and characterize the noise-induced burst and spike synchronizations by employing realistic order parameters and statistical-mechanical measures introduced in our recent work. Thus, the bursting and spiking thresholds for the burst and spike synchronization transitions are determined in terms of the bursting and spiking order parameters, respectively. Furthermore, we also measure the degrees of burst and spike synchronizations in terms of the statistical-mechanical bursting and spiking measures, respectively.

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References

  • Achard S, Bullmore E (2007) Efficiency and cost of economical brain functional networks. PLoS Comput Biol 3:e17

    Article  PubMed Central  PubMed  Google Scholar 

  • Bassett DS, Bullmore E (2006) Small-world brain networks. The Neuroscientist 12:512–523

    Article  PubMed  Google Scholar 

  • Batista CAS, Batista AM, de Pontes JAC, Viana RL, Lopes SR (2007) Chaotic phase synchronization in scale-free networks of bursting neurons. Phys Rev E 76:016218

    Article  CAS  Google Scholar 

  • Batista CAS, Lameu EL, Batista AM, Lopes SR, Pereira T, Zamora-Lopez G, Kurths J, Viana RL (2012) Phase synchronization of bursting neurons in clustered small-world networks. Phys Rev E 86:016211

    Article  CAS  Google Scholar 

  • Börgers C, Kopell N (2003) Synchronization in network of excitatory and inhibitory neurons with sparse, random connectivity. Neural Comput 15:509–538

    Article  PubMed  Google Scholar 

  • Börgers C, Kopell N (2005) Effects of noisy drive on rhythms in networks of excitatory and inhibitory neurons. Neural Comput 17:557–608

    Article  PubMed  Google Scholar 

  • Braun HA, Wissing H, Schäfer K, Hirsh MC (1994) Oscillation and noise determine signal transduction in shark multimodal sensory cells. Nature 367:270–273

    Article  CAS  PubMed  Google Scholar 

  • Brunel N, Hakim V (2008) Sparsely synchronized neuronal oscillations. Chaos 18:015113

    Article  PubMed  Google Scholar 

  • Bullmore E, Sporns O (2009) Complex brain networks: graph-theoretical analysis of structural and functional systems. Nat Rev Neurosci 10:186–198

    Article  CAS  PubMed  Google Scholar 

  • Buzsáki G, Wang XJ (2012) Mechanisms of gamma oscillations. Annu Rev Neurosci 35:203–225

    Article  PubMed Central  PubMed  Google Scholar 

  • Buzsáki G, Geisler C, Henze DA, Wang XJ (2004) Interneuron diversity series: circuit complexity and axon wiring economy of cortical interneurons. Trends Neurosci 27:186–193

    Article  PubMed  Google Scholar 

  • Chklovskii DB, Mel BW, Svoboda K (2004) Cortical rewiring and information storage. Nature 431:782–788

    Article  CAS  PubMed  Google Scholar 

  • Coombes S, Bressloff PC (eds) (2005) Bursting: the genesis of rhythm in the nervous system. World Scientific, Singapore

    Google Scholar 

  • Dhamala M, Jirsa V, Ding M (2004) Transitions to synchrony in coupled bursting neurons. Phys Rev Lett 92:028101

    Article  PubMed  Google Scholar 

  • Duan L, Fan D, Lu Q (2013) Hopf bifurcation and bursting synchronization in an excitable systems with chemical delayed coupling. Cogn Neurodyn 7:341–349

    Article  PubMed Central  PubMed  Google Scholar 

  • Erdös P, Rényi A (1959) On random graph. Publicationes Mathematicae Debrecen 6:290–297

    Google Scholar 

  • Golomb D, Rinzel J (1994) Clustering in globally coupled inhibitory neurons. Physica D 72:259–282

    Article  Google Scholar 

  • Guare J (1990) Six degrees of separation: a play. Random House, New York

    Google Scholar 

  • Guo D, Wang Q, Perc M (2012) Complex synchronous behavior in interneuronal networks with delayed inhibitory and fast electrical synapses. Phys Rev E 85:061905

    Article  Google Scholar 

  • Hansel D, Mato G, Meunier C (1995) Synchrony in excitatory neural networks. Neural Comput 7:307–337

    Article  CAS  PubMed  Google Scholar 

  • Hindmarsh JL, Rose RM (1982) A model of the nerve impulse using two first-order differential equations. Nature 296:162–164

    Article  CAS  PubMed  Google Scholar 

  • Hindmarsh JL, Rose RM (1984) A model of neuronal bursting using three coupled first order differential equations. Proc R Soc Lond Ser B 221:87–102

    Article  CAS  Google Scholar 

  • Hong DG, Kim SY, Lim W (2011) Effect of sparse random connectivity on the stochastic spiking coherence of inhibitory subthreshold neurons. J Korean Phys Soc 59:2840–2846

    Article  Google Scholar 

  • Hu B, Zhou C (2000) Phase synchronization in coupled nonidentical excitable systems and array-enhanced coherence resonance. Phys Rev E 61:R1001–R1004

    Article  CAS  Google Scholar 

  • Huber MT, Braun HA (2006) Stimulus-response curves of a neuronal model for noisy subthreshold oscillations and related spike generation. Phys Rev E 73:041929

    Article  Google Scholar 

  • Ivanchenko MV, Osipov GV, Shalfeev VD, Kurths J (2004) Phase synchronization in ensembles of bursting oscillators. Phys Rev Lett 93:134101

    Article  PubMed  Google Scholar 

  • Izhikevich EM (2006) Bursting. Scholarpedia 1(3):1300

    Article  Google Scholar 

  • Izhikevich EM (2007) Dynamical systems in neuroscience. MIT Press, Cambridge

    Google Scholar 

  • Izhikevich EH, Edelman GM (2008) Large-scale model of mammalian thalamocortical systems. Proc Natl Acad Sci USA 105:3593–3598

    Article  PubMed Central  CAS  PubMed  Google Scholar 

  • Kaiser M, Hilgetag CC (2006) Nonoptimal component placement, but short processing paths, due to long-distance projections in neural systems. PLoS Comput Biol 2:e95

    Article  PubMed Central  PubMed  Google Scholar 

  • Kim SY, Lim W (2013a) Sparsely-synchronized brain rhythm in a small-world neural network. J Korean Phys Soc 63:104–113

    Article  CAS  Google Scholar 

  • Kim SY, Lim W (2013b) Coupling-induced population synchronization in an excitatory population of subthreshold Izhikevich neurons. Cogn Neurodyn 7:495–503

    Article  PubMed Central  PubMed  Google Scholar 

  • Kim SY, Lim W (2014a) Realistic thermodynamic and statistical-mechanical measures for neural synchronization. J Neurosci Methods 226:161–170

    Article  PubMed  Google Scholar 

  • Kim SY, Lim W (2014b) Thermodynamic and statistical-mechanical measures for characterization of the burst and spike synchronizations of bursting neurons. e-print: arXiv:1403.3994 [q-bio.NC]

  • Kim SY, Lim W (2014c) Effect of small-world connectivity on fast sparsely synchronized cortical rhythms. e-print: arXiv:1403.1034 [q-bio.NC]

  • Kim SY, Kim Y, Hong DG, Kim J, Lim W (2012) Stochastic bursting synchronization in a population of subthreshold Izhikevich neurons. J Korean Phys Soc 60:1441–1447

    Article  Google Scholar 

  • Kreuz T, Haas JS, Morelli A, Abarbanel HDI, Politi A (2007) Measuring spike train synchrony. J Neurosci Methods 165:151–161

    Article  PubMed  Google Scholar 

  • Kreuz T, Chicharro D, Greschner M, Andrzejak RG (2011) Time-resolved and time-scale adaptive measures of spike train synchrony. J Neurosci Methods 195:92–106

    Article  PubMed  Google Scholar 

  • Kreuz T, Chicharro D, Houghton C, Andrzejak RG, Mormann F (2013) Monitoring spike train synchrony. J Neurophysiol 109:1457–1472

    Article  PubMed  Google Scholar 

  • Kuramoto Y (2003) Chemical oscillations, waves and turbulence. Dover, New York

    Google Scholar 

  • Kwon O, Moon HT (2002) Coherence resonance in small-world networks of excitable cells. Phys Lett A 298:319–324

    Article  CAS  Google Scholar 

  • Lago-Fernández LF, Huerta R, Corbacho F, Sigüenza JA (2000) Fast response and temporal coherent oscillations in small-world networks. Phys Rev Lett 84:2758–2761

    Article  PubMed  Google Scholar 

  • Lameu EL, Batista CAS, Batista AM, Larosz K, Viana RL, Lopes SR, Kurths J (2012) Suppression of bursting synchronization in clustered scale-free (rich-club) neural networks. Chaos 22:043149

    Article  CAS  PubMed  Google Scholar 

  • Larimer P, Strowbridge BW (2008) Nonrandom local circuits in the dentate gyrus. J Neurosci 28:12212–12223

    Article  CAS  PubMed  Google Scholar 

  • Latora V, Marchiori M (2001) Efficient behavior of small-world networks. Phys Rev Lett 87:198701

    Article  CAS  PubMed  Google Scholar 

  • Latora V, Marchiori M (2003) Economic small-world behavior in weighted networks. Eur Phys J B 32:249–263

    Article  CAS  Google Scholar 

  • Liang X, Tang M, Dhamala M, Liu Z (2009) Phase synchronization of inhibitory bursting neurons induced by distributed time delays in chemical coupling. Phys Rev E 80:066202

    Article  Google Scholar 

  • Lim W, Kim SY (2011) Statistical-mechanical measure of stochastic spiking coherence in a population of inhibitory subthreshold neuron. J Comput Neurosci 31:667–677

    Article  PubMed  Google Scholar 

  • Lizier JT, Pritam S, Prokopenko M (2011) Information dynamics in small-world Boolean networks. Artif Life 17:293–314

    Article  PubMed  Google Scholar 

  • Longtin A (1997) Autonomous stochastic resonance in bursting neurons. Phys Rev E 55:868–876

    Article  Google Scholar 

  • Longtin A, Hinzer K (1996) Encoding with bursting, subthreshold oscillations, and noise in mammalian cold receptors. Neural Comput 8:217–255

    Article  Google Scholar 

  • Lu Q, Tian J (2014) Synchronization and stochastic resonance of the small-world neural network based on the CPG. Cogn Neurodyn 8:217–226

    Article  PubMed  Google Scholar 

  • Meng P, Wang Q, Lu Q (2013) Bursting synchronization dynamics of pancreatic \(\beta\)-cells with electrical and chemical coupling. Cogn Neurodyn 7:197–212

    Article  PubMed Central  PubMed  Google Scholar 

  • Milgram S (1967) The small-world problem. Psychol Today 1:61–67

    Google Scholar 

  • Neiman A (2007) Coherence resonance. Scholarpedia 2(11):1442

    Article  Google Scholar 

  • Omelchenko I, Rosenblum M, Pikovsky A (2010) Synchronization of slow-fast systems. Eur Phys J Special Top 191:3–14

    Article  CAS  Google Scholar 

  • Ozer M, Perc M, Uzuntarla M (2009) Stochastic resonance on Newman–Watts networks of Hodgkin–Huxley neurons with local periodic driving. Phys Lett A 373:964–968

    Article  CAS  Google Scholar 

  • Perc M (2007) Stochastic resonance on excitable small-world networks via a pacemaker. Phys Rev E 76:066203

    Article  Google Scholar 

  • Pereira T, Baptista M, Kurths J (2007) Multi-time-scale synchronization and information processing in bursting neuron networks. Eur Phys J Special Top 146:155–168

    Article  Google Scholar 

  • Qu J, Wang R, Yan C, Du Y (2014) Oscillations and synchrony in a cortical neural network. Cogn Neurodyn 8:157–166

    Article  PubMed  Google Scholar 

  • Riecke H, Roxin A, Madruga S, Solla S (2007) Multiple attractors, long chaotic transients, and failure in small-world networks of excitable neurons. Chaos 17:026110

    Article  PubMed  Google Scholar 

  • Rinzel J (1985) Bursting oscillations in an excitable membrane model. In: Sleeman BD, Jarvis RJ (eds) Ordinary and partial differential equations. Lecture notes in mathematics, vol 1151. Springer, Berlin, pp 304–316

  • Rinzel J (1987) A formal classification of bursting mechanisms in excitable systems. In: Teramoto E, Yamaguti M (eds) Mathematical topics in population biology, morphogenesis, and neurosciences. Lecture Notes in biomathematics, vol 71. Springer, Berlin, pp 267–281

  • Rose RM, Hindmarsh JL (1985) A model of a thalamic neuron. Proc R Soc Lond Ser B 225:161–193

    Article  CAS  Google Scholar 

  • Roxin A, Riecke H, Solla SA (2004) Self-sustained activity in a small-world network of excitable neurons. Phys Rev Lett 92:198101

    Article  PubMed  Google Scholar 

  • Rubin JE (2007) Burst synchronization. Scholarpedia 2(10):1666

    Article  Google Scholar 

  • San Miguel M, Toral R (2000) Stochastic effects in physical systems. In: Martinez J, Tiemann R, Tirapegui E (eds) Instabilities and nonequilibrium structures VI. Kluwer, Dordrecht, pp 35–130

    Chapter  Google Scholar 

  • Shanahan M (2008) Dynamical complexity in small-world networks of spiking neurons. Phys Rev E 78:041924

    Article  Google Scholar 

  • Shi X, Lu Q (2005) Firing patterns and complete synchronization of coupled Hindmarsh–Rose neurons. Chin Phys 14:77–85

    Article  Google Scholar 

  • Shi X, Lu Q (2009) Burst synchronization of electrically and chemically coupled map-based neurons. Physica A 388:2410–2419

    Article  CAS  Google Scholar 

  • Shimazaki H, Shinomoto S (2010) Kernel band width optimization in spike rate estimation. J Comput Neurosci 29:171–182

    Article  PubMed Central  PubMed  Google Scholar 

  • Shinohara Y, Kanamaru T, Suzuki H, Horita T, Aihara K (2002) Array-enhanced coherence resonance and forced dynamics in coupled FitzHugh–Nagumo neurons with noise. Phys Rev E 65:051906

    Article  Google Scholar 

  • Song S, Sjöström PJ, Reigl M, Nelson S, Chklovskii DB (2005) Highly nonrandom features of synaptic connectivity in local cortical circuits. PLoS Biol 3:e68

    Article  PubMed Central  PubMed  Google Scholar 

  • Sporns O (2011) Networks of the brain. MIT Press, Cambridge

    Google Scholar 

  • Sporns O, Honey CJ (2006) Small worlds inside big brains. Proc Natl Acad Sci USA 103:19219–19220

    Article  PubMed Central  CAS  PubMed  Google Scholar 

  • Sporns O, Tononi G, Edelman GM (2000) Theoretical neuroanatomy: relating anatomical and functional connectivity in graphs and cortical connection matrices. Cereb Cortex 10:127–141

    Article  CAS  PubMed  Google Scholar 

  • Strogatz SH (2001) Exploring complex networks. Nature 410:268–276

    Article  CAS  PubMed  Google Scholar 

  • Sun X, Lei J, Perc M, Kurths J, Chen G (2011) Burst synchronization transitions in a neuronal network of subnetworks. Chaos 21:016110

    Article  PubMed  Google Scholar 

  • Tanaka G, Ibarz B, Sanjuan MA, Aihara K (2006) Synchronization and propagation of bursts in networks of coupled map neurons. Chaos 16:013113

    Article  PubMed  Google Scholar 

  • Tiesinga PHE, Fellous JM, Jose JV, Sejnowski TJ (2001) Computational model of carbachol-induced delta, theta, and gamma oscillations in the hippocampus. Hippocampus 11:251–274

    Article  CAS  PubMed  Google Scholar 

  • van Rossum MCW (2001) A novel spike distance. Neural Comput 13:751–763

    Article  PubMed  Google Scholar 

  • van Vreeswijk C, Hansel D (2001) Patterns of synchrony in neural networks with adaptation. Neural Comput 13:959–992

    Article  PubMed  Google Scholar 

  • van Vreeswijk C, Abbott LF, Ermentrout GB (1994) When inhibition not excitation synchronizes neural firing. J Comput Neurosci 1:313–321

    Article  PubMed  Google Scholar 

  • Victor JD, Purpura KP (1996) Nature and precision of temporal coding in visual cortex: a metric-space analysis. J Neurophysiol 76:1310–1326

    CAS  PubMed  Google Scholar 

  • Victor JD, Purpura KP (1997) Metric-space analysis of spike trains: theory, algorithms, and application. Network 8:127–164

    Article  Google Scholar 

  • Wang XJ (2010) Neurophysiological and computational principles of cortical rhythms in cognition. Physiol Rev 90:1195–1268

    Article  PubMed Central  PubMed  Google Scholar 

  • Wang XJ, Buzsáki G (1996) Gamma oscillations by synaptic inhibition in a hippocampal interneuronal network. J Neurosci 16:6402–6413

    CAS  PubMed  Google Scholar 

  • Wang XJ, Rinzel J (1992) Alternating and synchronous rhythms in reciprocally inhibitory model neurons. Neural Comput 4:84–97

    Article  Google Scholar 

  • Wang Y, Chik DTW, Wang ZD (2000) Coherence resonance and noise-induced synchronization in globally coupled Hodgkin–Huxley neurons. Phys Rev E 61:740–746

    Article  CAS  Google Scholar 

  • Wang Q, Duan Z, Perc M, Chen G (2008) Synchronization transitions on small-world neuronal networks: effects of information transmission delay and rewiring probability. Europhys Lett 83:50008

    Article  Google Scholar 

  • Wang Q, Perc M, Duan Z, Chen G (2010) Impact of delays and rewiring on the dynamics of small-world neuronal networks with two types of coupling. Physica A 389:3299–3306

    Article  Google Scholar 

  • Wang QY, Murks A, Perc M, Lu QS (2011a) Taming desynchronized bursting with delays in the Macaque cortical network. Chin Phys B 20:040504

    Article  Google Scholar 

  • Wang Q, Chen G, Perc M (2011b) Synchronous bursts on scale-free neuronal networks with attractive and repulsive coupling. PLoS One 6:e15851

    Article  PubMed Central  CAS  PubMed  Google Scholar 

  • Wang H, Wang Q, Lu Q, Zheng Y (2013) Equilibrium analysis and phase synchronization of two coupled HR neurons with gap junction. Cogn Neurodyn 7:121–131

    Article  PubMed Central  PubMed  Google Scholar 

  • Watts DJ (2003) Small worlds: the dynamics of networks between order and randomness. Princeton University Press, Princeton

    Google Scholar 

  • Watts DJ, Strogatz SH (1998) Collective dynamics of ‘small-world’ networks. Nature 393:440–442

    Article  CAS  PubMed  Google Scholar 

  • White J, Chow CC, Ritt J, Soto-Trevino C, Kopell N (1998) Synchronization and oscillatory dynamics in heterogeneous, mutually inhibited neurons. J Comput Neurosci 5:5–16

    Article  CAS  PubMed  Google Scholar 

  • Whittington MA, Traub RD, Kopell N, Ermentrout B, Buhl EH (2000) Inhibition-based rhythms: experimental and mathematical observations on network dynamics. Int J Psychophysiol 38:315336

    Article  Google Scholar 

  • Yu S, Huang D, Singer W, Nikolie D (2008) A small world of neuronal synchrony. Cereb Cortex 18:2891–2901

    Article  PubMed Central  PubMed  Google Scholar 

  • Yu H, Wang J, Deng B, Wei X, Wong YK, Chan WL, Tsang KM, Yu Z (2011) Chaotic phase synchronization in small world networks of bursting neurons. Chaos 21:013127

    Article  PubMed  Google Scholar 

  • Zheng Y, Wang Q, Danca MF (2014) Noise induced complexity: patterns and collective phenomena in a small-world neuronal network. Cogn Neurodyn 8:143–149

    Article  PubMed  Google Scholar 

  • Zhou C, Kurths J (2002) Spatiotemporal coherence resonance of phase synchronization in weakly coupled chaotic oscillators. Phys Rev E 65:040101

    Article  Google Scholar 

  • Zhou C, Kurths J, Hu B (2001) Array-enhanced coherence resonance: nontrivial effects of heterogeneity and spatial independence of noise. Phys Rev Lett 87:098101

    Article  CAS  PubMed  Google Scholar 

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Acknowledgments

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (Grant No. 2013057789).

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  1. Computational Neuroscience Lab., Department of Science Education, Daegu National University of Education, Daegu, 705-115, Korea

    Sang-Yoon Kim & Woochang Lim

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Kim, SY., Lim, W. Noise-induced burst and spike synchronizations in an inhibitory small-world network of subthreshold bursting neurons. Cogn Neurodyn 9, 179–200 (2015). https://doi.org/10.1007/s11571-014-9314-0

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