Dynamical Systems in Neuroscience:

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esting390 Burstingspikesynchronizationspikede-synchronizationspikingABexcitatory couplingburster BinhibitorystimulationexcitatorystimulationABinhibitory couplingFigure 9.52: Burst synchronization and de-synchronization of two coupled“fold/homoclinic” bursters (modified from Izhikevich 2000).the instantaneous interspike frequency, which may vary substantially during a burst.Indeed, a small perturbation of the slow variable may result in large perturbations ofthe interspike frequency in any shaded burster in Fig. 9.50, hence such a burster wouldbe reluctant to exhibit spike synchronization, unless the coupling is strong.Studying burst synchronization of weakly coupled neurons involves the same mathematicalmethods as studying synchronization of strongly coupled relaxation oscillators,which we consider in detail in Chap. 10. The mechanisms of synchronization dependon whether the bursting is of the hysteresis loop type or of the slow wave type, andwhether the resting state is an integrator or a resonator.In Fig. 9.52 we illustrate the geometry of burst synchronization of two coupled“fold/homoclinic” bursters of hysteresis loop type. Burster A is slightly ahead ofburster B so that A starts the spiking phase while B is still resting. If the synapticconnections between the bursters are excitatory, firing of A causes B to jump to thespiking state prematurely, thereby shortening the time difference between the bursts.In addition, the evoked burst of B is shorter, which also speeds up the synchronizationprocess. In contrast, when the connections are inhibitory, firing of A delays thetransition of B to the spiking state, thereby increasing the time difference betweenthe bursts and desynchronizing the bursters. Thus, the “fold/homoclinic” burster behavesaccording to the principle excitation means synchronization, inhibition meansde-synchronization. Since the instantaneous interspike frequency of “fold/homoclinic”bursting decays to zero, small deviations of the slow variable result in large deviationsof the period of oscillation. Typically, the periods of fast oscillations of the two bursterscan slowly diverge from each other. As a result, spikes start synchronized and thende-synchronize during the burst, as we indicate in the figure.If the bursting neuron is a resonator, i.e., it is of the “Hopf/*” or “subHopf/*” type,then both excitation and inhibition may evoke premature spiking, as we have shown
Bursting 391in Chap. 7, and lead to burst synchronization. An important feature here is that theinterspike frequency of one burster be resonant to the subthreshold oscillations of theother one. We study these and other issues related to synchronization in Chap. 10.
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Eugene M. IzhikevichThe Neuroscienc
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6.3.6 Saddle-node homoclinic orbit
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9.4.2 Integrators vs. Resonators .
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xPrefaceversely, cells having quite
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2 Introductionapical dendritessomar
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4 Introduction(a)spikes(b)spikes cu
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6 Introduction1.1.3 Why are neurons
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8 Introductionconcepts of dynamical
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10 Introductionspikingmoderesting m
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12 Introduction(a)spikinglimitcycle
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14 Introductionco-existence of rest
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16 Introduction1.2.4 Neuro-computat
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18 IntroductionspikeFigure 1.16: Ph
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20 Introductionwith coupled neurons
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22 IntroductionFigure 1.17: John Ri
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24 Introduction
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26 Electrophysiology of NeuronsInsi
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28 Electrophysiology of Neuronsouts
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30 Electrophysiology of NeuronsRest
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32 Electrophysiology of NeuronsFigu
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34 Electrophysiology of Neuronsm (V
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36 Electrophysiology of Neurons10.8
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38 Electrophysiology of Neurons2.3
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42 Electrophysiology of NeuronsDepo
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44 Electrophysiology of NeuronsV(x,
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46 Electrophysiology of Neurons1m (
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48 Electrophysiology of NeuronsPara
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52 Electrophysiology of Neuronsvolt
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54 Electrophysiology of Neurons
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56 One-Dimensional SystemsActivatio
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58 One-Dimensional Systemsinwardcur
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60 One-Dimensional Systems40bistabi
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62 One-Dimensional Systemsat every
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66 One-Dimensional SystemsUnstableE
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76 One-Dimensional Systems100806040
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78 One-Dimensional Systemsbifurcati
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80 One-Dimensional Systems20 mV100
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82 One-Dimensional Systemsmembrane
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84 One-Dimensional Systemssteady-st
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86 One-Dimensional Systemsspike rig
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88 One-Dimensional SystemsVF (V)1VV
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90 One-Dimensional Systems10.80.60.
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92 One-Dimensional Systems
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94 Two-Dimensional SystemsI Na,pneu
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96 Two-Dimensional Systemsdefines a
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134 Conductance-Based Models• If
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164 Conductance-Based Modelsmodels
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166 Conductance-Based Models
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168 Bifurcationssaddle-node bifurca
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222 Bifurcationsv 2v 11a-11-1Figure
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224 Bifurcations19. [M.S.] A leaky
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226 Excitabilityspike?spikerestrest
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228 ExcitabilityAlternatively, the
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230 Excitability50 ms 20 mV 1 ms100
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248 Excitability(g)9 ms(f)(e)(d)(c)
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268 Excitability(a)150100I K(M)(b)3
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270 Excitability50 mV300 msFigure 7
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272 Excitability20 mVADP100 msAHP-6
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278 Excitability7. Show that the re
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280 Simple Modelsspikemembrane pote
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308 Simple Modelschattering neuron
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324 Simple ModelsBibliographical No
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330 Simple Modelsa35 mV350 msc-NAC
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332 Simple Modelsrat RTN neuronsimp
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336 Simple Modelsspiny neuronlatenc
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338 Simple Models(a)stellate cellof
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Page 352 and 353: 342 Bursting(a) cortical chattering
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Page 356 and 357: 346 Burstingvoltage-gatedCa2+-gated
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Page 374 and 375: 364 Bursting9.3 ClassificationIn Fi
Page 376 and 377: 366 Burstingbifurcation of spiking
Page 378 and 379: 368 BurstingFigure 9.26: Putative
Page 380 and 381: 370 Bursting108spikingslow variable
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Page 384 and 385: 374 Burstingdepending on the type o
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Page 396 and 397: 386 Bursting9.4.3 BistabilitySuppos
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Page 408 and 409: 398 Bursting0membrane potential, V
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Page 418 and 419: 408 Solutions to Exercises, Chap. 3
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440 Solutions to Exercises, Chap. 9
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442 Referencesterneurons mediated b
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444 ReferencesDickson C.T., Magistr
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446 ReferencesGuckenheimer J. (1975
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448 Referencestional Journal of Bif
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450 ReferencesMarkram H, Toledo-Rod
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452 ReferencesRosenblum M.G. and Pi
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454 ReferencesTuckwell H.C. (1988)
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456 References9
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458 Synchronization (see www.izhike
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