Dynamical Systems in Neuroscience:

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370 Bursting108spikingslow variable, u fast variable, v64homoclinic2reset0fold3 0.5 1 1.5 2 2.52homoclinicslow variable, u1fold00 50time100 150thresholdrestingFigure 9.29: “Fold/homoclinic” bursting in the canonical model (9.7) with parametersµ = 0.02, I = 1 and d = 0.2.saddle-node oninvariant circle bifurcationspikingxurestsaddle-node on invariantcircle bifurcationsaddle-nodeon invariantcircle bifurcationsaddle-nodeon invariantcircle bifurcationFigure 9.30: “Circle/circle” bursting: The resting state disappears via saddle-node oninvariant circle bifurcation and the spiking limit cycle disappears via saddle-node oninvariant circle bifurcation.
Bursting 371Figure 9.31: Putative “circle/circle” bursting pacemaker activity of neuron R 15 inabdominal ganglion of the mollusk Aplysia (modified from Levitan and Levitan 1988).interspike period (sec)21.510.500 5 10 15spike numberinterspike frequency (Hz)2.521.510.500 5 10 15spike numberFigure 9.32: The interspike period in the “circle/circle” bursting in Fig. 9.31 resemblesa parabola, which motivates the name “parabolic bursting” used in earlier studies.9.3.2 circle/circleWhen the equilibrium corresponding to the resting state disappears via a saddle-nodeon invariant circle bifurcation, and the limit cycle attractor corresponding to spikingstate disappears via another saddle-node on invariant circle bifurcation, the burster issaid to be of the “circle/circle” type shown in Fig. 9.30. Since the bifurcation doesnot produce a co-existence of attractors, there is usually no hysteresis loop, and thebursting is of the slow-wave type with at least two slow variables. (An example of“circle/circle” hysteresis loop bursting in a “2+1” system is provided by Izhikevich(2000)).“Circle/circle” bursting is a prominent feature of the R 15 cells in the abdominal ganglionof the mollusk Aplysia, shown in Fig. 9.31 (Plant 1981). It was called “parabolic”bursting in earlier studies because the interspike period depicted in Fig. 9.32 was erroneouslythought to be a parabola. In Sect. 6.1.2 we showed that when a systemundergoes a saddle-node on invariant circle bifurcation, its period scales as 1/ √ λ,where λ is the distance to the bifurcation. Two pieces of this function, put togetheras in Fig. 9.32, do indeed resemble a parabola. But so does the interspike period of a“circle/homoclinic” burster.To transform the I Na,p +I K -model to a “circle/circle” burster, we take the parametersas in Fig. 4.1a so that there is a saddle-node on invariant circle bifurcation whenI = 4.51 (see Sect. 6.1.2). Its nullclines and phase portrait look similar to those inFig. 9.30. Then, we add one amplifying and one resonant current with gating variablesṁ slow = (m ∞,slow (V ) − m slow )/τ Na,slow (V ) (slow I Na,slow ),ṅ slow = (n ∞,slow (V ) − n slow )/τ K(M) (V ) (slow I K(M) ),
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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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98 Two-Dimensional Systems(f(x(t),y
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100 Two-Dimensional Systems0.70acti
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102 Two-Dimensional Systems0.70.6K
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106 Two-Dimensional SystemsFor exam
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112 Two-Dimensional SystemsV-nullcl
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114 Two-Dimensional SystemsV-nullcl
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116 Two-Dimensional Systemsheterocl
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118 Two-Dimensional Systems0.60.5n-
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120 Two-Dimensional Systemseigenval
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122 Two-Dimensional SystemsK + acti
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124 Two-Dimensional Systemsexcitati
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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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140 Conductance-Based Modelsleak cu
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142 Conductance-Based Modelshyperpo
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144 Conductance-Based Models0I=0ina
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148 Conductance-Based Models1I=651I
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152 Conductance-Based Models1 sec10
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154 Conductance-Based Modelsvoltage
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160 Conductance-Based Models1recove
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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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170 Bifurcationssaddle-node bifurca
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172 Bifurcations0.5v 2V-nullclineK
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174 BifurcationsBoth types of the b
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176 BifurcationsV-nullcline0.5K + g
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178 Bifurcationsrstable limit cycle
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180 BifurcationsnVstable limit cycl
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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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194 Bifurcationshomoclinic orbithom
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196 Bifurcations0.80.60.4stable lim
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198 Bifurcationsfrequency (Hz)40030
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200 BifurcationsSaddle-Focus Homocl
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202 Bifurcationsc 1c 2xFigure 6.34:
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204 BifurcationseigenvaluesHopffold
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206 Bifurcationsfast nullclineslow
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208 Bifurcationswith fast and slow
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210 BifurcationsSupercritical Andro
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212 BifurcationsK + conductance tim
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214 BifurcationsIn contrast, if the
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216 Bifurcationsinvariant circlesad
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218 Bifurcationsbifurcationssaddle-
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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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244 Excitability1coincidencedetecti
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246 Excitabilityspikeexcitatory pul
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248 Excitability(g)9 ms(f)(e)(d)(c)
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254 Excitability(a) integrator(b) r
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258 Excitability0.1saddle-node bifu
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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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274 ExcitabilityK + activation gate
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276 ExcitabilityBibliographical Not
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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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288 Simple Modelsintegrate-and-fire
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292 Simple Modelsgeneral algorithm
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294 Simple Modelscha, cha — real
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298 Simple Modelsb=-240200saddle-no
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306 Simple Models(a)74 5 6 3210(b)C
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308 Simple Modelschattering neuron
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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
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Page 378 and 379: 368 BurstingFigure 9.26: Putative
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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 412 and 413: 402 Bursting28. [Ph.D.] Develop an
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Page 418 and 419: 408 Solutions to Exercises, Chap. 3
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420 Solutions to Exercises, Chap. 4
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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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