Dynamical Systems in Neuroscience:

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260 Excitability100 ms10 mVoscillations200 ms20 mV-60 mVup-statedown-stateintegratorresonatorn-nullclineV-nullclineup-state(focus)down-state(node)Figure 7.36: Bistability of the up- and down-state of mitral cells in rat main olfactorybulb. The cells are integrators in the down-state and resonators in the up-state. Membranepotential recordings are modified from Heyward et al. (2001). The shaded areadenotes the attraction domain of the up-state.1 mV20 msfast oscillations-60 mV5 mV100 msFigure 7.37: Fast subthreshold oscillations during complex spikes of cerebella Purkinjeneuron of a guinea-pig (data was provided by Yonatan Loewenstein).
Excitability 261integrator(saddle-node bifurcation)?resonator(Andronov-Hopf bifurcation)V-nullclinen-nullcline?Figure 7.38: Is there an intermediate mode between integrators and resonators?7.2.11 Transition between integrators and resonatorsConsider the I Na +I K -model or any other minimal model from Chap. 5 that can exhibitsaddle-node or Andronov-Hopf bifurcation, depending on the parameter values. Let usstart with the I Na +I K -model near saddle-node bifurcation, and hence in the integratormode. The intersection of its nullclines at the left knee is similar to the one in Fig. 7.38,left. Now, let us slowly change the parameters toward the values corresponding to theAndronov-Hopf bifurcation with the nullclines intersecting as in Fig. 7.38, right. Atsome point, the behavior of the model must change from integrator to a resonatormode. Is the change sudden, or is it gradual?Any qualitative change of the behavior of the system is a bifurcation. Such abifurcation should somehow combine the saddle-node and the Andronov-Hopf cases;That is, it should have a zero eigenvalue, and a pair of complex-conjugate eigenvalueswith zero real part. Since the I Na +I K -model is two-dimensional, the only way these twoconditions are satisfied is when the model undergoes the Bogdanov-Takens bifurcationconsidered in Sect. 6.3.3. This bifurcation has codimension 2, that is, it can be reliablyobserved when two parameters are changed, in our case E leak and the half-voltage, V 1/2 ,of n ∞ (V ).The top of Fig. 7.39 depicts the phase portrait of the I Na + I K -model at theBogdanov-Takens bifurcation. Notice that the nullclines are tangent near the left knee,but the tangency is degenerate. A small change of the parameter V 1/2 can result eitherin a saddle and a node (middle of the figure) or in a focus equilibrium (bottom of thefigure). The neuron acts as an integrator in the former case and as a resonator in thelatter case.Due to the proximity to a codimension-2 bifurcation, the behavior of the I Na +I K -model is quite degenerate. That is, it could exhibit features that are normallynot observed. For example, the integrator can exhibit post-inhibitory spiking, as inFig. 7.40. This occurs because the shaded region in the figure, bounded by the stablemanifold of the saddle, goes to the left of the resting state. An inhibitory pulse ofcurrent that hyperpolarizes the membrane potential to V < −65 mV and deactivates
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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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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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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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Page 236 and 237: 226 Excitabilityspike?spikerestrest
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Page 240 and 241: 230 Excitability50 ms 20 mV 1 ms100
Page 242 and 243: 232 ExcitabilityClass 3 excitable n
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Page 246 and 247: 236 Excitabilitysaddle-node bifurca
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Page 280 and 281: 270 Excitability50 mV300 msFigure 7
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Page 288 and 289: 278 Excitability7. Show that the re
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310 Simple ModelsLTS neuron (in vit
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314 Simple Models(1) fast oscillati
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318 Simple Modelsthey are able to g
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320 Simple Modelsv r = −80 mV, an
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322 Simple Modelsshow in Fig. 7.36.
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324 Simple ModelsBibliographical No
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326 Simple ModelsExercises1. (Integ
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328 Simple Models17. [M.S.] Analyze
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330 Simple Modelsa35 mV350 msc-NAC
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332 Simple Modelsrat RTN neuronsimp
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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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