Dynamical Systems in Neuroscience:

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216 Bifurcationsinvariant circlesaddle-nodenodesaddlesaddle-node on invariant circle (SNIC) bifurcationsupercritical Andronov-Hopf bifurcationstableunstablelimit cyclelimitcyclefoldlimitcyclefold limit cycle bifurcationhomoclinic orbitstablelimit cyclesaddle saddle saddlesaddle homoclinic orbit bifurcationFigure 6.47: Summary of all codimension-1 bifurcations of a stable limit cycle (tonicspiking state) on a plane.
Bifurcations 217Figure 6.48: Richard FitzHugh with analog computer, National Institute of Health,Bethesda, Maryland, ca. 1960 (photograph provided by R. FitzHugh in 2005).early as 1955, when Richard FitzHugh concluded his paper on mathematical modelingof threshold phenomena saying that many neuronal properties“... are invariant under continuous, one-to-one transformations of the coordinates ofphase space and fall within the domain of topology, a branch of mathematics whichmay be intrinsically better fitted for the preliminary description and classification ofbiological systems than analysis, which includes differential equations. This suggestionis of little practical value at present, since too little is known of the topology ofvector fields in many-dimensional spaces, at least to those interested in theoreticalbiology. Nevertheless, the most logical procedure in the description of a complexbiological system might be to characterize the topology of its phase space, then toestablish a set of physically identifiable coordinates in the space, and finally to fitdifferential equations to the trajectories, instead of trying to reach this final goal atone leap.”It is remarkable that FitzHugh was explicitly talking about topological equivalence andbifurcations, though never called them such, years before these mathematical notionswere firmly established. This book continues the line of research initiated by FitzHughand further developed by Rinzel and Ermentrout (1989).In this chapter we provided a fairly detailed exposition of bifurcation theory. Whatwe covered should be sufficient not only for understanding the rest of the book, butalso for navigating through bifurcation papers concerned with computational neuro-
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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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Page 178 and 179: 168 Bifurcationssaddle-node bifurca
Page 180 and 181: 170 Bifurcationssaddle-node bifurca
Page 182 and 183: 172 Bifurcations0.5v 2V-nullclineK
Page 184 and 185: 174 BifurcationsBoth types of the b
Page 186 and 187: 176 BifurcationsV-nullcline0.5K + g
Page 188 and 189: 178 Bifurcationsrstable limit cycle
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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
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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 218 and 219: 208 Bifurcationswith fast and slow
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Page 224 and 225: 214 BifurcationsIn contrast, if the
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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 236 and 237: 226 Excitabilityspike?spikerestrest
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
Page 252 and 253: 242 ExcitabilityThe existence of fa
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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
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Page 268 and 269: 258 Excitability0.1saddle-node bifu
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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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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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