Dynamical Systems in Neuroscience:

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226 Excitabilityspike?spikerestrestexcitableoscillatoryFigure 7.1: Left: An abstract definition of excitability. There is a spike trajectory thatstarts near a stable equilibrium and returns to it. Right: Excitable systems are nearbifurcations. A modification of the vector field in the small shaded region can resultin a periodic trajectory.that push the state of the neuron to or near the beginning of the large trajectory(small square in Fig. 7.1), thereby initiating the spike. These inputs can be injected byexperimenter via an attached electrode, or they can represent the total synaptic inputfrom the other neurons in the network, or both.7.1.1 BifurcationsThe definition in Fig. 7.1 is quite general, and it does not make any assumptions regardingthe details of the vector field inside or outside of the small shaded neighborhood.Using the theory presented in the previous chapter, we can show that such an excitablesystem is near a bifurcation from resting to oscillatory dynamics.• Bifurcation of a limit cycle. The vector field in the small shaded neighborhoodof the equilibrium can be modified slightly so that the spike trajectory enters thesquare and becomes periodic, as in Fig. 7.1, right. That is, the dynamical systemgoes through a bifurcation resulting in the appearance of a limit cycle.What happens to the stable equilibrium, denoted as “?” in the figure? Depending onthe type of the bifurcation of the limit cycle, the equilibrium may disappear or may losestability. This happens when the limit cycle appears via saddle-node on invariant circleor supercritical Andronov-Hopf bifurcations, respectively. Both cases are depicted inFig. 7.2.
Excitability 227excitable bistable oscillatorysaddle-node on invariantcircle bifurcationsaddle-homoclinicorbit bifurcationsaddlenodebifurcationsupercritical Andronov-Hopfbifurcationfold limit cyclebifurcationsubcriticalAndronov-HopfbifurcationFigure 7.2: Excitable dynamical systems bifurcate into oscillatory ones either directlyor indirectly, via bistable systems.
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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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Page 192 and 193: ¡ ¡¡ ¡¡ ¡ ¡¡ ¡ ¡182 Bifur
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Page 196 and 197: 186 Bifurcationsmembrane potential,
Page 198 and 199: 188 BifurcationsBifurcation of a li
Page 200 and 201: 190 Bifurcationsstableunstablelimit
Page 202 and 203: 192 Bifurcationsamplitude (max-min)
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Page 206 and 207: 196 Bifurcations0.80.60.4stable lim
Page 208 and 209: 198 Bifurcationsfrequency (Hz)40030
Page 210 and 211: 200 BifurcationsSaddle-Focus Homocl
Page 212 and 213: 202 Bifurcationsc 1c 2xFigure 6.34:
Page 214 and 215: 204 BifurcationseigenvaluesHopffold
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Page 232 and 233: 222 Bifurcationsv 2v 11a-11-1Figure
Page 234 and 235: 224 Bifurcations19. [M.S.] A leaky
Page 238 and 239: 228 ExcitabilityAlternatively, the
Page 240 and 241: 230 Excitability50 ms 20 mV 1 ms100
Page 242 and 243: 232 ExcitabilityClass 3 excitable n
Page 244 and 245: 234 Excitability0.20.150.10.05I 1I
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 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
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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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304 Simple Models(a)dendriticspike2
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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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310 Simple ModelsLTS neuron (in vit
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312 Simple ModelsFS neuron (in vitr
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314 Simple Models(1) fast oscillati
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316 Simple Modelsthrough an appropr
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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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