Dynamical Systems in Neuroscience:

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284 Simple Models1v reset =|b| 1/2bistabilitysaddle-node bifurcationparameter v reset0saddle homoclinic orbit bifurcationv resetv resetv resetsaddle-node homoclinicorbit bifurcationexcitablev resetv resetv resetsaddle-node on invariantcircle bifurcationperiodicv reset-1-1 0 1parameter bFigure 8.3: Bifurcation diagram of the quadratic integrate-and-fire neuron (8.2).current or their combination. Typically, the fast variable has an N-shaped nullcline andthe slower variable has a sigmoid-shaped nullcline. The resting state in such modelsis the intersection of the nullclines near the left knee, as we illustrate in Fig. 8.4a.There, V and u denote the fast and the slow variables, respectively. In Chap. 7 weshowed that many computational properties of biological neurons could be explainedby considering dynamics at the left knee.In Sect. 5.2.4 we derive a simple model that captures the subthreshold behavior ina small neighborhood of the left knee confined to the shaded square in Fig. 8.4 andthe initial segment of the up-stroke of an action potential. In many cases, especiallyinvolving large-scale simulations of spiking models, the shape of the action potentialis less important than the subthreshold dynamics leading to this action potential. So,retaining detailed information about the left knee and its neighborhood and simplifyingthe vector field outside the neighborhood is justified.The simple model˙v = I + v 2 − u if v ≥ 1, then (8.3)˙u = a(bv − u) v ← c, u ← u + d. (8.4)has only four dimensionless parameters. Depending on the values of a and b, it can
Simple Models 2851recovery variable, u0.80.60.40.2u-nullclineV-nullcline0.20.10-80 -60 -40 -20 0 20membrane potential, V (mV)a0-70 -60 -50 -40membrane potential, V (mV)bFigure 8.4: Phase portrait (a) and its magnification (b) of a typical neuronal modelhaving voltage variable V and a recovery variable u.integratorresonator0.0250thresholdmanifoldsmallEPSPu=I+v2u=bvspikerecovery variable, u0.10smallEPSPu=bvu=I+v2spike-0.1thresholdset-0.2 -0.1 0 0.1 0.2 0.3membrane potential, v-0.5 -0.25 0 0.25 0.5 0.75membrane potential, vFigure 8.5: The simple model (8.3, 8.4) can be an integrator or a resonator. Comparewith Fig. 7.27.
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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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Page 246 and 247: 236 Excitabilitysaddle-node bifurca
Page 248 and 249: 238 Excitability(a) resting spiking
Page 250 and 251: 240 Excitabilityproperties integrat
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Page 254 and 255: 244 Excitability1coincidencedetecti
Page 256 and 257: 246 Excitabilityspikeexcitatory pul
Page 258 and 259: 248 Excitability(g)9 ms(f)(e)(d)(c)
Page 260 and 261: 250 Excitabilitysquid axonmodel0 mV
Page 262 and 263: 252 Excitability(a) integrator(b) r
Page 264 and 265: 254 Excitability(a) integrator(b) r
Page 266 and 267: 256 Excitabilitymembrane potential,
Page 268 and 269: 258 Excitability0.1saddle-node bifu
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Page 272 and 273: 262 ExcitabilityBogdanov-Takens bif
Page 274 and 275: 264 Excitabilitya pulse of current.
Page 276 and 277: 266 Excitabilitymembrane potential
Page 278 and 279: 268 Excitability(a)150100I K(M)(b)3
Page 280 and 281: 270 Excitability50 mV300 msFigure 7
Page 282 and 283: 272 Excitability20 mVADP100 msAHP-6
Page 284 and 285: 274 ExcitabilityK + activation gate
Page 286 and 287: 276 ExcitabilityBibliographical Not
Page 288 and 289: 278 Excitability7. Show that the re
Page 290 and 291: 280 Simple Modelsspikemembrane pote
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Page 330 and 331: 320 Simple Modelsv r = −80 mV, an
Page 332 and 333: 322 Simple Modelsshow in Fig. 7.36.
Page 334 and 335: 324 Simple ModelsBibliographical No
Page 336 and 337: 326 Simple ModelsExercises1. (Integ
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334 Simple ModelsFigure 8.34: Class
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336 Simple Modelsspiny neuronlatenc
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338 Simple Models(a)stellate cellof
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340 Simple Modelsrat's mitral cell
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342 Bursting(a) cortical chattering
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344 Burstingmembranepotential (mV)-
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346 Burstingvoltage-gatedCa2+-gated
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348 Burstingslow dynamicsneuronvolt
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350 Burstingslow inactivation of in
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352 Bursting9.2.1 Fast-slow burster
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354 Burstingn-nullclinen slow =-0.0
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356 Bursting0maxmembrane potential,
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358 Bursting0membrane potential, V
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360 BurstingI=0I=4.5425 ms 25 mVI=5
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362 Burstingmembrane potential, V (
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364 Bursting9.3 ClassificationIn Fi
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366 Burstingbifurcation of spiking
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368 BurstingFigure 9.26: Putative
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370 Bursting108spikingslow variable
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372 Bursting(a)membrane potential,
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374 Burstingdepending on the type o
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376 BurstingsubcriticalAndronov-Hop
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378 Bursting(a)membrane potential,
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spiking380 Burstingfoldbifurcations
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382 BurstingspikingsupercriticalAnd
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384 Burstingaction potentials cut2
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386 Bursting9.4.3 BistabilitySuppos
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388 BurstingFigure 9.49: The instan
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esting390 Burstingspikesynchronizat
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392 BurstingReview of Important Con
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394 BurstingspikingrestingFigure 9.
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396 Bursting0-10membrane potential,
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398 Bursting0membrane potential, V
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400 BurstingFigure 9.61: A cycle-cy
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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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