Dynamical Systems in Neuroscience:

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174 BifurcationsBoth types of the bifurcation can occur in the I Na,p +I K -model as we show in Fig. 6.7.The difference between top and bottom of the figure is the time constant τ(V ) of theK + current. Since the K + current has high threshold, the time constant does not affectdynamics at rest, but it makes a huge difference when action potential is generated.If the current is fast (top), it activates during the upstroke thereby decreasing theamplitude of action potential, and deactivates during the downstroke thereby resultingin overshoot and another action potential. In contrast, slower K + current (bottom)does not have time to deactivate during the downstroke, thereby resulting in undershoot(short after-hyperpolarization), with V going below the resting state.From the geometrical point of view, the phase portraits in Fig. 6.6b and in Fig. 6.7,bottom, have the same topological structure: there is a homoclinic trajectory (aninvariant circle) that originates at the saddle-node point, leaves its small neighborhood(to fire an action potential), then reenters the neighborhood again, and terminatesat the saddle-node point. This homoclinic trajectory is a limit cycle attractor withinfinite period, which corresponds to firing with zero frequency. This and other neurocomputationalfeatures of saddle-node bifurcations are discussed in the next chapter.Below we only explore how the frequency of oscillation depends on the bifurcationparameter, e.g., on the injected dc-current I.A remarkable fact is that we can estimate the frequency of the large-amplitude limitcycle attractor by considering a small neighborhood of the saddle-node point. Indeed,a trajectory on the limit cycle generates a fast spike from point B to A in Fig. 6.8 andthen slowly moves from A to B (shaded region in the figure) because the vector field(the velocity) in the neighborhood between A and B is very small. The duration of thestereotypical action potentials, denoted here as T 1 , is relatively constant and does notdepend much on the injected current I. In contrast, the time spent in the neighborhood(A, B) depends significantly on I. Since the behavior in the neighborhood is describedby the topological normal form (6.2), we can estimate the time the trajectory spendsthere in terms of the parameters a, b and c (see Ex. 3). This yieldsT 2 =π√ac(b − bsn ) ,where the parameters a, b, and c are those defined in the previous section. So theperiod of one oscillation is T = T 1 + T 2 .In Fig. 6.8, top, we illustrate the accuracy of this estimation using the I Na,p +I K -model, whose topological normal form (6.3) was derived earlier. The duration of theaction potential is T 1 = 4.7 ms, and the duration of time the voltage variable spendsin the shaded neighborhood (A,B) (here −61 ± 11 mV) is approximated byT 2 =π√0.1887(I − 4.51)(ms)The analytical curveω = 1000T 1 + T 2(Hz)
Bifurcations 175fast K+ current0.60.5n-nullclineK + activation variable, n0.40.30.2V-nullcline0.10saddle-node0.60.5slow K+ currentn-nullcline0.4K + activation variable, n0.30.20.1V-nullclineinvariant circle0saddle-nodeon invariant circle-80 -70 -60 -50 -40 -30 -20 -10 0 10 20membrane voltage, V (mV)Figure 6.7: Saddle-node bifurcation in the I Na,p +I K -model with high-threshold K +current can be off the limit cycle (top) or on the invariant circle (bottom). Parametersas in Fig. 4.1a with τ(V ) = 0.152 (top) or τ(V ) = 1 (bottom).
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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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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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294 Simple Modelscha, cha — real
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298 Simple Modelsb=-240200saddle-no
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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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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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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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346 Burstingvoltage-gatedCa2+-gated
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348 Burstingslow dynamicsneuronvolt
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352 Bursting9.2.1 Fast-slow burster
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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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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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392 BurstingReview of Important Con
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396 Bursting0-10membrane potential,
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398 Bursting0membrane potential, V
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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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