Abstract
We consider a scale-free network of inhibitory Hindmarsh–Rose (HR) bursting neurons, and make a computational study on coupling-induced cluster burst synchronization by varying the average coupling strength \(J_0\). For sufficiently small \(J_0\), non-cluster desynchronized states exist. However, when passing a critical point \(J^*_c~(\simeq 0.16)\), the whole population is segregated into 3 clusters via a constructive role of synaptic inhibition to stimulate dynamical clustering between individual burstings, and thus 3-cluster desynchronized states appear. As \(J_0\) is further increased and passes a lower threshold \(J^*_l~(\simeq 0.78)\), a transition to 3-cluster burst synchronization occurs due to another constructive role of synaptic inhibition to favor population synchronization. In this case, HR neurons in each cluster make burstings every 3rd cycle of the instantaneous burst rate \(R_w(t)\) of the whole population, and exhibit burst synchronization. However, as \(J_0\) passes an intermediate threshold \(J^*_m~(\simeq 5.2)\), HR neurons fire burstings intermittently at a 4th cycle of \(R_w(t)\) via burst skipping rather than at its 3rd cycle, and hence they begin to make intermittent hoppings between the 3 clusters. Due to such intermittent intercluster hoppings via burst skippings, the 3 clusters become broken up (i.e., the 3 clusters are integrated into a single one). However, in spite of such break-up (i.e., disappearance) of the 3-cluster states, (non-cluster) burst synchronization persists in the whole population, which is well visualized in the raster plot of burst onset times where bursting stripes (composed of burst onset times and indicating burst synchronization) appear successively. With further increase in \(J_0\), intercluster hoppings are intensified, and bursting stripes also become dispersed more and more due to a destructive role of synaptic inhibition to spoil the burst synchronization. Eventually, when passing a higher threshold \(J^*_h~(\simeq 17.8)\) a transition to desynchronization occurs via complete overlap between the bursting stripes. Finally, we also investigate the effects of stochastic noise on both 3-cluster burst synchronization and intercluster hoppings.
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References
Albert R, Barabási AL (2002) Statistical mechanics of complex networks. Rev Mod Phys 74:47–97
Aronson DG, Golubitsky M, Krupa M (1991) Coupled arrays of Josephson junctions and bifurcation of maps with \(S_N\) symmetry. Nonlinearity 4:861–902
Barabási AL, Albert R (1999) Emergence of scaling in random networks. Science 286:509–512
Bassett DS, Bullmore E (2006) Small-world brain networks. Neuroscientist 12:512–523
Batista CAS, Batista AM, de Pontes JAC, Viana RL, Lopes SR (2007) Chaotic phase synchronization in scale-free networks of bursting neuron. Phys Rev E 76:016218
Batista CAS, Batista AM, de Pontes JAC, Lopes SR, Viana RL (2009) Bursting synchronization in scale-free networks. Chaos Soliton Fract 41:2220–2225
Batista CAS, Lopes SR, Viana RL, Batista AM (2010) Delayed feedback control of bursting synchronization in a scale-free neuronal network. Neural Netw 23:114–124
Batista CA, 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
Bazhenov M, Timofeev I (2006) Thalamocortical oscillations. Scholarpedia 1(6):1319
Belykh I, Hasler M (2011) Mesoscale and clusters of synchrony in networks of bursting neurons. Chaos 21:016106
Belykh VN, Osipov GV, Petrov VS, Suykens JAK, Vandewalle J (2008) Cluster synchronization in oscillatory networks. Chaos 18:037106
Bevan M, Magill P, Terman D, Bolam J, Wilson C (2002) Move to the rhythm: oscillations in the subthalamic nucleus-external globus pallidus network. Trends Neurosci 25:525–531
Bonifazi P, Goldin M, Picardo MA, Jorquera I, Cattani A, Bianconi G, Represa A, Ben-Ari Y, Cossart R (2009) GABAergic hub neurons orchestrate synchrony in developing hippocampal networks. Science 326:1419–1424
Brown P (2007) Abnormal oscillatory synchronisation in the motor system leads to impaired movement. Cur Opin Neurobiol 17:656–664
Brunel N, Wang XJ (2003) What determines the frequency of fast network oscillations with irregular neural discharges? I. Synaptic dynamics and excitation-inhibition balance. J Neurophysiol 90:415–430
Bullmore E, Sporns O (2009) Complex brain networks: graph theoretical analysis of structural and functional systems. Nat Rev Neurosci 10:186–198
Butera RJ, Rinzel J, Smith JC (1999) Models of respiratory rhythm generation in the pre-Botzinger complex. I. Bursting pacemaker neurons. J Neurophysiol 82:382–397
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
Chay TR, Keizer J (1983) Minimal model for membrane oscillations in the pancreatic \(\beta \)-cell. Biophys J 42:181–190
Chik DTW, Coombes S, Wang ZD (2004) Clustering through postinhibitory rebound in synaptically coupled neurons. Phys Rev E 70:011908
Chklovskii DB, Mel BW, Svoboda K (2004) Cortical rewiring and information storage. Nature 431:782–788
Connors BW, Gutnick MJ (1990) Intrinsic firing patterns of diverse neocortical neurons. Trends Neurosci 13:99–104
Coombes S, Bressloff PC (eds) (2005) Bursting: the genesis of rhythm in the nervous system. World Scientific, Singapore
Del Negro CA, Hsiao CF, Chandler SH, Garfinkel A (1998) Evidence for a novel bursting mechanism in rodent trigeminal neurons. Biophys J 75:174–182
Dhamala M, Jirsa V, Ding M (2004) Transitions to synchrony in coupled bursting neurons. Phys Rev Lett 92:028101
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
Eguíluz VM, Chialvo DR, Cecchi GA, Baliki M, Apkarian AV (2005) Scale-free brain functional networks. Phys Rev Lett 94:018102
Elson RC, Selverston AI, Huerta R, Rulkov NF, Rabinovich MI, Abarbanel HDI (1998) Synchronous behavior of two coupled biological neurons. Phys Rev Lett 81:5691–5695
Felleman DJ, Van Essen DC (1991) Distributed hierarchical processing in the primate cerebral cortex. Cereb Cortex 1:1–47
Ferrari FAS, Viana RL, Lopes SR, Stoop R (2015) Phase synchronization of coupled bursting neurons and the generalized Kuramoto model. Neural Netw 66:107–118
Fisher R, van Emde BW, Blume W, Elger C, Genton P, Lee P, Engel J (2005) Epileptic seizures and epilepsy: definitions proposed by the International League Against Epilepsy (ILAE) and the International Bureau for Epilepsy (IBE). Epilepsia 46:470–472
Gais S, Plihal W, Wagner U, Born J (2000) Early sleep triggers memory for early visual discrimination skills. Nat Neurosci 3:1335–1339
Golomb D, Rinzel J (1994) Clustering in globally coupled inhibitory neurons. Phys D 72:259–282
Gray CM, McCormick DA (1996) Chattering cells: superficial pyramidal neurons contributing to the generation of synchronous oscillations in the visual cortex. Science 274:109–113
Hammond C, Bergman H, Brown P (2007) Pathological synchronization in Parkinson’s disease: networks, models and treatments. Trends Neurosci 30:357–364
Hindmarsh JL, Rose RM (1982) A model of the nerve impulse using two first-order differential equations. Nature 296:162–164
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
Ivanchenko MV, Osipov GV, Shalfeev VD, Kurths J (2004) Phase synchronization in ensembles of bursting oscillators. Phys Rev Lett 93:134101
Izhikevich EM (2000) Neural excitability, spiking and bursting. Int J Bifurcat Chaos 10:1171–1266
Izhikevich EM (2004) Which model to use for cortical spiking neurons? IEEE Trans Neural Netw 15:1063–1070
Izhikevich EM (2006) Bursting. Scholarpedia 1(3):1300
Izhikevich EM (2007) Dynamical systems in neuroscience. MIT Press, Cambridge
Izhikevich EM, Desai NS, Walcott EC, Hoppensteadt FC (2003) Bursts as a unit of neural information: selective communication via resonance. Trends Neurosci 26:161–167
Kaiser M, Martin R, Andras P, Young MP (2007) Simulation of robustness against lesions of cortical networks. Eur J Neurosci 25:3185–3192
Kim SY, Lim W (2015a) Noise-induced burst and spike synchronizations in an inhibitory small-world network of subthreshold bursting neurons. Cogn Neurodyn 9:179–200
Kim SY, Lim W (2015b) Thermodynamic order parameters and statistical-mechanical measures for characterization of the burst and spike synchronizations of bursting neurons. Phys A 438:544–559
Kim SY, Lim W (2016) Effect of network architecture on burst and spike synchronization in a scale-free network of bursting neurons. Neural Netw 79:53–77
Kim SY, Lim W (2018) Effect of spike-timing-dependent plasticity on stochastic burst synchronization in a scale-free neuronal network. Cogn Neurodyn 12:315–342
Kinard TA, de Vries G, Sherman A, Satin LS (1999) Modulation of the bursting properties of single mouse pancreatic \(\beta \)-cells by artificial conductances. Biophys J 76:1423–1435
Kiss IZ, Zhai Y, Hudson H (2005) Predicting mutual entrainment of oscillators with experiment-based phase models. Phys Rev Lett 94:248301
Krahe R, Gabbian F (2004) Burst firing in sensory system. Nat Rev Neurosci 5:13–23
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) neuronal networks. Chaos 22:043149
Langdon AJ, Breakspear M, Coombes S (2012) Phase-locked cluster oscillations in periodically forced integrate-and-fire-or-burst neuronal populations. Phys Rev E 86:061903
Larimer P, Strowbridge BW (2008) Nonrandom local circuits in the dentate gyrus. J Neurosci 28:12212–12223
Lee SH, Govindaiah G, Cox CL (2007) Heterogeneity of firing properties among rat thalamic reticular nucleus neurons. J Physiol 582:195–208
Li X, Ouyang G, Usami A, Ikegaya Y, Sik A (2010) Scale-free topology of the CA3 hippocampal network: a novel method to analyze functional neuronal assemblies. Biophys J 98:1733–1741
Lisman J (1997) Bursts as a unit of neural information: making unreliable synapse reliable. Trends Neurosci 20:38–43
Llinás RL, Jahnsen H (1982) Electrophysiology of mammalian thalamic neurons in vitro. Nature 297:406–408
Longtin A (1997) Autonomous stochastic resonance in bursting neurons. Phys Rev E 55:868–876
McCormick DA, Huguenard JR (1992) A Model of the electrophysiological properties of thalamocortical relay neurons. J Neurophysiol 8:1384–1400
Meng P, Wang Q, Lu Q (2013) Bursting synchronization dynamics of pancreatic \(\beta \)-cells with electrical and chemical coupling. Cogn Neurodyn 7:197–212
Miyakawa K, Okano T, Yamazaki S (2013) Cluster synchronization in a chemical oscillator network with adaptive coupling. J Phy Soc Jpn 82:034005
Moon SJ, Cook KA, Rajendran K, Kevrekidis KG, Cisternas J, Liang CR (2015) Coarse-grained clustering dynamics of heterogeneously coupled neurons. J Math Neurosci 5:2
Morgan RJ, Soltesz I (2008) Nonrandom connectivity of the epileptic dentate gyrus predicts a major role for neuronal hubs in seizures. Proc Natl Acad Sci USA 105:6179–6184
Park C, Worth RM, Rubchinsky LL (2010) Fine temporal structure of beta oscillations synchronization in subthalamic nucleus in Parkinson’s disease. J Neurophysiol 103:2707–2716
Pereira T, Baptista M, Kurths J (2007) Multi-time-scale synchronization and information processing in bursting neuron networks. Eur Phys J Spec Top 146:155–168
Pernarowski M, Miura RM, Kevorkian J (1992) Perturbation techniques for models of bursting electrical activity in pancreatic \(\beta \)-cells. SIAM J Appl Math 52:1627–1650c
Plant RE (1981) Bifurcation and resonance in a model for bursting nerve cells. J Math Biol 11:15–32
Prado TdeL, Lopes SR, Batista CAS, Kurths J, Viana RL (2014) Synchronization of bursting Hodgkin-Huxley-type neurons in clustered networks. Phys Rev E 90:032818
Rose RM, Hindmarsh JL (1985) A model of a thalamic neuron. Proc R Soc Lond Ser B 225:161–193
Rinzel J (1985) Bursting oscillations in an excitable membrane model. In: Sleeman BD, Jarvis RJ (eds) Lecture notes in mathematics, vol 1151. Ordinary and partial differential equations. Springer, Berlin, pp 304–316
Rinzel J (1987) A formal classification of bursting mechanisms in excitable systems. In: Teramoto E, Yamaguti M (eds) Lecture notes in biomathematics, vol 71. Mathematical topics in population biology, morphogenesis, and neurosciences. Springer, Berlin, pp 267–281
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 Academic Publisher, Dordrecht, pp 35–130
Scannell JW, Blakemore C, Young MP (1995) Analysis of connectivity in the cat cerebral cortex. J Neurosci 15:1463–1483
Scannell JW, Burns GAPC, Hilgetag CC, O’Neil MA, Young MP (1999) The connectional organization of the cortico-thalamic system of the cat. Cereb Cortex 9:277–299
Sejnowski TJ, Destexhe A (2000) Why do we sleep? Brain Res 886:208–223
Shi X, Lu Q (2005) Firing patterns and complete synchronization of coupled Hindmarsh-Rose neurons. Chin Phys 14:77–85
Shi X, Lu Q (2009) Burst synchronization of electrically and chemically coupled map-based neurons. Phys A 388:2410–2419
Shilnikov A, Cymbalyuk G (2005) Transition between tonic spiking and bursting in a neuron model via the blue-sky catastrophe. Phys Rev Lett 94:048101
Shimazaki H, Shinomoto S (2010) Kernel bandwidth optimization in spike rate estimation. J Comput Neurosci 29:171–182
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
Sporns O (2011) Networks of the brain. MIT Press, Cambridge
Sporns O, Honey CJ (2006) Small worlds inside big brains. Proc Natl Acad Sci USA 103:19219–19220
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
Sporns O, Chialvo DR, Kaiser M, Hilgetag CC (2004) Organization, development and function of complex brain networks. Trends Cogn Sci 8:418–425
Steriade M, McCormick DA, Sejnowski TJ (1993) Thalamocortical oscillations in the sleeping and aroused brain. Science 262:679–685
Stern EA, Jaeger D, Wilson CJ (1998) Membrane potential synchrony of simultaneously recorded striatal spiny neurons in vivo. Nature 394:475–478
Su H, Alroy G, Kirson ED, Yaari Y (2001) Extracellular calcium modulates persistent sodium current-dependent burst-firing in hippocampal pyramidal neurons. J Neurosci 21:4173–4182
Sun X, Lei J, Perc M, Kurths J, Chen G (2011) Burst synchronization transitions in a neuronal network of subnetworks. Chaos 21:016110
Tanaka G, Ibarz B, Sanjuan MA, Aihara K (2006) Synchronization and propagation of bursts in networks of coupled map neurons. Chaos 16:013113
Taylor AF, Kapetanopoulos P, Whitaker BJ, Toth R, Bull L, Tinsley MR (2008) Clusters and switchers in globally coupled photochemical oscillators. Phys Rev Lett 100:214101
Uhlhaas PJ, Singer W (2006) Neural synchrony in brain disorders: relevance for cognitive dysfunctions and pathophysiology. Neuron 52:155–168
Varona P, Torres JJ, Abarbanel HDI, Rabinovych MI, Elson RC (2001) Dynamics of two electrically coupled chaotic neurons: experimental observations and model analysis. Biol Cybern 84:91–101
van Vreeswijk C, Hansel D (2001) Patterns of synchrony in neural networks with spike adaptation. Neural Comput 13:959–992
Wang Q, Perc M, Duan Z, Chen G (2009) Synchronization transitions on scale-free neuronal networks due to finite information transmission delays. Phys Rev E 80:026206
Wang QY, Murks A, Perc M, Lu QS (2011a) Taming desynchronized bursting with delays in the Macaque cortical network. Chin Phys B 20:040504
Wang Q, Chen G, Perc M (2011b) Synchronous bursts on scale-free neuronal networks with attractive and repulsive coupling. PLoS One 6:e15851
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
Watts DJ, Strogatz SH (1998) Collective dynamics of ‘small-world’ networks. Nature 393:440–442
Wiedemann C (2010) Neuronal networks: a hub of activity. Nat Rev Neurosci 11:74
Wisenfeld K, Colet P, Wisenfeld S (1996) Synchronization transitions in a disordered Josephson series array. Phys Rev Lett 76:404–407
Womack MD, Khodakhah K (2002) Active contribution of dendrites to the tonic and trimodal patterns of activity in cerebellar Purkinje neurons. J Neurosci 22:10603–10612
Young MP (1993) The organization of neural systems in the primate cerebral cortex. Philos Trans R Soc 252:13–18
Young MP, Scannell JW, Burns GA, Blakemore C (1994) Analysis of connectivity: neural systems in the cerebral cortex. Rev Neurosci 5:227–250
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
Zhang J, Yuan Z, Zhou T (2009) Synchronization and clustering of synthetic genetic networks: a role for cis-regulatory modules. Phys Rev E 79:041903
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This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (Grant No. 20162007688).
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Kim, SY., Lim, W. Cluster burst synchronization in a scale-free network of inhibitory bursting neurons. Cogn Neurodyn 14, 69–94 (2020). https://doi.org/10.1007/s11571-019-09546-9
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DOI: https://doi.org/10.1007/s11571-019-09546-9