Dynamical Systems in Neuroscience:

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354 Burstingn-nullclinen slow =-0.03V-nullclinen slow =0.0033membrane potential, V (mV)0-20-40-60-80-0.030slow K + gating variable, n slow0.030.060.0900.5fast K + gating variable, n1saddlenodebifurcationn slow =0.03n slow =0.06n slow =0.066saddlehomoclinic orbitbifurcationspikingn slow =0.0910.8thresholdrestingVfastnn slow0.60.40.20fast K+ activation gate, n-80 -60 -40 -20 0membrane potential, V (mV)Figure 9.13: Bursting trajectory of the I Na,p +I K +I K(M) -model in three-dimensionalphase space and its slices n slow = const.
Bursting 355to the stable equilibrium corresponding to the resting state. This jumping correspondsto the termination of the active phase of bursting and transition to resting. While atrest, K + current deactivates, n slow decreases, and so on.Figure 9.13 presents the inner structure of the geometrical mechanism of burstingof the I Na,p +I K +I K(M) -model with parameters as in Fig. 9.4. Other values of theparameters can result in different geometrical mechanisms, summarized in Sect. 9.3.In all cases, our approach is the same: freeze the slow subsystem by setting µ = 0;analyze phase portraits of the fast subsystem treating the slow variable as a bifurcationparameter; then glue the phase portraits, let µ ≠ 0 but small, and see how the evolutionof the slow subsystem switches the fast subsystem between spiking and resting states.The method usually breaks down if µ is not small enough, because evolution of the“slow” variable starts to interfere with that of the fast variable. How small is smalldepends on the particulars of the equations describing bursting activity. One shouldworry when µ is greater than 0.1.9.2.3 AveragingWhat governs the evolution of the slow variable u? To study this question, we describea well-known and widely used method that reduces the fast-slow system (9.1) to its slowcomponent. In fact, we have already used this method in Chapters 3 and 4 to reducethe dimension of neuronal models via substitution m = m ∞ (V ). Using essentially thesame ideas, we take advantage of the two time scales in (9.1) and get rid of the fastsubsystem by means of a substitution x = x(u).When the neuron is resting, its membrane potential is at an equilibrium and allfast gating variables are at their steady-state values, so that x = x rest (u). Using thisfunction in the slow equation in (9.1) we obtain˙u = µg(x rest (u), u) (reduced slow subsystem) , (9.2)which can easily be studied using the geometrical methods presented in Chapters 3 or4.Let us illustrate all the steps involved using the I Na,p +I K +I K(M) -model with n slowbeing the gating variable of the slow K + M-current. First, we freeze the slow subsystem,i.e., set τ slow (V ) = ∞ so that µ = 1/τ slow = 0, and determine numerically the restingpotential V rest as a function of the slow variable n slow . The function V = V rest (n slow )is depicted in Fig. 9.14, top, and it coincides with the solid half-parabola in Fig. 9.13.Then, we use this function in the gating equation for the M-current to obtain (9.2)ṅ slow = (n ∞,slow (V rest (n slow )) − n slow )/τ slow (V rest (n slow )) = ḡ(n slow )depicted in Fig. 9.14, bottom. Notice that ḡ < 0, meaning that n slow decreases whilethe fast subsystem rests. The rate of decrease is fairly small when n slow ≈ 0.A similar method of reduction, with an extra step, can be used when the fastsubsystem fires spikes. Let x(t) = x spike (t, u) be a periodic function corresponding to
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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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¡ ¡ ¡¡ ¡ ¡¡ ¡ ¡¡ ¡ ¡¡
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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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104 Two-Dimensional Systemsy1.510.5
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106 Two-Dimensional SystemsFor exam
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108 Two-Dimensional SystemsIn gener
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110 Two-Dimensional Systemsv 2 v 2v
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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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146 Conductance-Based Models0I=9.75
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148 Conductance-Based Models1I=651I
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150 Conductance-Based Modelshave co
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152 Conductance-Based Models1 sec10
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154 Conductance-Based Modelsvoltage
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156 Conductance-Based Models1h=0.89
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158 Conductance-Based Models5.2.2 E
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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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¡ ¡ ¡¡ ¡ ¡¡ ¡ ¡¡ ¡ ¡184
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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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238 Excitability(a) resting spiking
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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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250 Excitabilitysquid axonmodel0 mV
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252 Excitability(a) integrator(b) r
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254 Excitability(a) integrator(b) r
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256 Excitabilitymembrane potential,
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258 Excitability0.1saddle-node bifu
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260 Excitability100 ms10 mVoscillat
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262 ExcitabilityBogdanov-Takens bif
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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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290 Simple Models(A) tonic spiking(
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292 Simple Modelsgeneral algorithm
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294 Simple Modelscha, cha — real
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296 Simple Modelslayer 5 neuronsimp
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298 Simple Modelsb=-240200saddle-no
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300 Simple Models(a)simple modellay
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302 Simple Models(a)burstingspiking
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Page 334 and 335: 324 Simple ModelsBibliographical No
Page 336 and 337: 326 Simple ModelsExercises1. (Integ
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Page 350 and 351: 340 Simple Modelsrat's mitral cell
Page 352 and 353: 342 Bursting(a) cortical chattering
Page 354 and 355: 344 Burstingmembranepotential (mV)-
Page 356 and 357: 346 Burstingvoltage-gatedCa2+-gated
Page 358 and 359: 348 Burstingslow dynamicsneuronvolt
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Page 362 and 363: 352 Bursting9.2.1 Fast-slow burster
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Page 372 and 373: 362 Burstingmembrane potential, V (
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
Page 382 and 383: 372 Bursting(a)membrane potential,
Page 384 and 385: 374 Burstingdepending on the type o
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Page 394 and 395: 384 Burstingaction potentials cut2
Page 396 and 397: 386 Bursting9.4.3 BistabilitySuppos
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Page 402 and 403: 392 BurstingReview of Important Con
Page 404 and 405: 394 BurstingspikingrestingFigure 9.
Page 406 and 407: 396 Bursting0-10membrane potential,
Page 408 and 409: 398 Bursting0membrane potential, V
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Page 412 and 413: 402 Bursting28. [Ph.D.] Develop an
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404 Synchronization (see www.izhike
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408 Solutions to Exercises, Chap. 3
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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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