Dynamical Systems in Neuroscience:

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388 BurstingFigure 9.49: The instantaneous spike frequency of a trigeminal motor neuron (a) andtrigeminal interneuron (b) of a rodent (modified from Del Negro et al. 1998).bifurcations of limit cyclessaddle-nodeon invariantcirclesaddlehomoclinicorbitsupercriticalAndronov-Hopffoldlimitcyclesaddle-node(fold)fold/circlefold/homoclinicfold/Hopffold/fold cyclebifurcations of equilibriasaddle-nodeon invariantcirclesupercriticalAndronov-HopfsubcriticalAndronov-Hopfcircle/circleHopf/circlesubHopf/circlecircle/homoclinicHopf/homoclinicsubHopf/homocliniccircle/HopfHopf/HopfsubHopf/Hopfcircle/fold cycleHopf/fold cyclesubHopf/fold cycleincreasingfrequencyat thebeginningdecreasingfrequencyat theendFigure 9.50: Topological types of bursters in the shaded regions can produce chirpburststhat sweep a frequency range.
Bursting 389spike synchronizationburst synchronizationFigure 9.51: Various regimes of synchronization of bursters.neurons but not for others. In contrast, other topological types of bursters have widelyvarying instantaneous interspike frequencies, as in Fig. 9.49a, that scan or sweep abroad frequency range going all the way to zero.When the bifurcation from resting to spiking state is of the saddle-node on invariantcircle type, i.e., the system is Class 1 excitable, the frequency of emerging spiking isfirst small, and then increases. Therefore, all “circle/*” bursters generate chirps withinstantaneous interspike frequencies increasing from zero to a relatively large value,at least at the beginning of the burst. Similarly, when the bifurcation of the spikingstate is of the saddle-node on invariant circle or saddle homoclinic orbit type, thefrequency of spiking at the end of the burst decreases to zero, so all “*/circle” and“*/homoclinic” bursters also generate chirps, as in Fig. 9.49a. In summary, all shadedbursters in Fig. 9.50 have sweeping interspike frequencies, so that one part of the burstis resonant for one neuron and another part of the same burst is resonant for anotherneuron.9.4.5 SynchronizationConsider two coupled bursting neurons of the fast-slow type. Since each burster hastwo times scales, one for rhythmic spiking and one for repetitive bursting, there aretwo synchronization regimes:• Spike synchronization, as in Fig. 9.51, left.• Burst synchronization, as in Fig. 9.51, right.One of them does not imply the other. Of course, there is an additional regime whenspikes and bursts are synchronized. We will study synchronization phenomena in detailin Chap. 10; here we just mention how they depend on the topological type of bursting.Let us consider spike synchronization first. Since we are interested in the fast timescale, we neglect the slow variable dynamics for a while and treat two bursters ascoupled oscillators. A necessary condition for synchronization of two weakly coupledoscillators is that they have nearly equal frequencies. How near is “near” dependson the strength of the coupling. Thus, spike synchronization depends crucially on
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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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64 One-Dimensional Systems?F(V)>0+
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66 One-Dimensional SystemsUnstableE
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70 One-Dimensional Systems+ - - + +
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72 One-Dimensional Systemsλ(V-Veq)
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74 One-Dimensional Systemsbifurcati
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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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126 Two-Dimensional SystemsReview o
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128 Two-Dimensional SystemsabcdFigu
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130 Two-Dimensional SystemsFigure 4
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132 Two-Dimensional Systems
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134 Conductance-Based Models• If
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136 Conductance-Based Modelsinward(
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138 Conductance-Based Models1I=01I=
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148 Conductance-Based Models1I=651I
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162 Conductance-Based Modelscurrent
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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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174 BifurcationsBoth types of the b
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¡ ¡¡ ¡¡ ¡ ¡¡ ¡ ¡182 Bifur
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186 Bifurcationsmembrane potential,
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188 BifurcationsBifurcation of a li
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190 Bifurcationsstableunstablelimit
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192 Bifurcationsamplitude (max-min)
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196 Bifurcations0.80.60.4stable lim
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198 Bifurcationsfrequency (Hz)40030
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204 BifurcationseigenvaluesHopffold
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216 Bifurcationsinvariant circlesad
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220 BifurcationsExercises1. (Transc
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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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232 ExcitabilityClass 3 excitable n
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234 Excitability0.20.150.10.05I 1I
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236 Excitabilitysaddle-node bifurca
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240 Excitabilityproperties integrat
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242 ExcitabilityThe existence of fa
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264 Excitabilitya pulse of current.
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266 Excitabilitymembrane potential
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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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282 Simple Models1thresholdyz reset
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284 Simple Models1v reset =|b| 1/2b
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286 Simple Modelsbe an integrator o
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292 Simple Modelsgeneral algorithm
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308 Simple Modelschattering neuron
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328 Simple Models17. [M.S.] Analyze
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330 Simple Modelsa35 mV350 msc-NAC
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336 Simple Modelsspiny neuronlatenc
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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 376 and 377: 366 Burstingbifurcation of spiking
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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 418 and 419: 408 Solutions to Exercises, Chap. 3
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438 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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