Dynamical Systems in Neuroscience:

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150 Conductance-Based Modelshave comparable time constants or inactivation is much faster than activation.Even though none of the experimentally measured A-currents show fast inactivationand a relatively slower activation, this case is still interesting from the pure theoreticalpoint of view, since it shows how a single K + current can give rise to oscillations.Assuming instantaneous inactivation and using h = h ∞ (V ) in the voltage equation, weobtain a two-dimensional systemC ˙V = I −leak I L{ }} {g L (V − E L ) −ṁ = (m ∞ (V ) − m)/τ m (V ) ,I A , inst. inactivation{ }} {g A m h ∞ (V )(V − E K ) ,whose nullclines can easily be found:m = I − g L(V −E L )g A h ∞ (V )(V − E K )(V -nullcline)andm = m ∞ (V ) (m-nullcline) .Two typical cases are depicted in Fig. 5.14a and b. We start with the simpler case inFig. 5.14b.Figure 5.14b depicts nullclines when the A-current has low activation threshold.There is only one intersection of the nullclines, hence there is only one equilibrium,which is a stable focus when the injected dc-current I is not strong enough (Fig. 5.14b,top). Increasing I makes the equilibrium lose stability via a supercritical Andronov-Hopf bifurcation that gives birth to a small amplitude limit cycle attractor (not shownin the figure). A further increase of I increases the amplitude of oscillations, and e.g.,when I = 10 (middle of Fig. 5.14b), the attractor corresponds to periodic firing of actionpotentials. When I = 10.5, the attractor disappears and the equilibrium becomesstable (via Andronov-Hopf bifurcation) again. The model, however, becomes excitable.A small hyperpolarization does not change significantly the A-current, and the voltagereturns back to rest resulting in a “subthreshold response”. A sufficiently largehyperpolarization deinactivates enough I A to open the K + current and hyperpolarizethe membrane even further. This regenerative process produces the downstroke andbrings V close to E K . During the state of hyperpolarization, the A-current deactivates(m → 0), and the dc-current I brings V back to rest. Notice that the action potentialis directed downward.In Fig. 5.14a we consider the I A -model with exactly the same parameters exceptthat we shift the half-voltage activation V 1/2 of I A by 10 mV so that the A-currenthas higher activation threshold. This does not affect much the behavior of the systemwhen I is small. However, when I ≈ 10.7 the spiking limit cycle attractor undergoesa new kind of bifurcation — saddle-node bifurcation — resulting in the appearance oftwo new equilibria: a stable node and a saddle. If the reader looks at Fig. 5.14a upsidedown,he will notice that this figure resembles figures 5.4a, 5.6a, or 5.12a, with all the
Conductance-Based Models 151K + activation, m10.80.60.40.20 time, ms 200m-nullclineI=8K + activation, m10.80.60.40.2V-nullcline0 time, ms 200I=6m-nullclineV-nullclineK + activation, mK + activation, m0-80 -60 -40 -20 0membrane voltage, V10.80.60.40.20 time, ms 5000-80 -60 -40 -20 0membrane voltage, V10.80.60.40.2I=10.6m-nullclineI=10.8m-nullcline0 time, ms 2000-80 -60 -40 -20 0membrane voltage, VaV-nullclineV-nullclineK + activation, mK + activation, m0-80 -60 -40 -20 0membrane voltage, V10.80.60.40.20-80 -60 -40 -20 0membrane voltage, V10.80.60.40.20 time, ms 500m-nullcline0 time, ms 200m-nullclineI=10.50-80 -60 -40 -20 0membrane voltage, VbI=10V-nullclineV-nullclineFigure 5.14: Possible intersections of nullclines in the I A -model. Parameters: E K =−80 mV, E L = −60 mV, g A = 5, g L = 0.2. Instantaneous inactivation kinetics withV 1/2 = −66 mV and k = −10. Activation of the A-current with k = 10, τ(V ) = 20 ms,and V 1/2 = −45 mV (in a) or V 1/2 = −35 mV (in b).
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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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Page 174 and 175: 164 Conductance-Based Modelsmodels
Page 176 and 177: 166 Conductance-Based Models
Page 178 and 179: 168 Bifurcationssaddle-node bifurca
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Page 182 and 183: 172 Bifurcations0.5v 2V-nullclineK
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Page 196 and 197: 186 Bifurcationsmembrane potential,
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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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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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