Dynamical Systems in Neuroscience:

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478 Synchronization (see www.izhikevich.com)interspike periods, Ti (ms)15010050membranepotentia(a)injected current, 40 pAp(t)0 5 time (sec) 10 15 -T/20 2 4 6time (sec)8 100.3(c)(d) pyramidal neuron8T i T i+1-30 mV010 mV10 pAp(t) -0.10phase of oscillation,Tperiod difference, Ti+1-TiT/2PRC( ), (pA -1 )0(b)averaged spike1(e)Andronov-Hopf (f)1I Na +I K -model (Class 1)0.2023012 3-10 2 4 6phase of oscillation,0 2 4 6phase of oscillation,Figure 10.24: Measuring the infinitesimal PRC experimentally in a layer 5 pyramidalneuron of mouse visual cortex. (a) Interspike periods in response to the injection ofdc-current. (b) Differences between successive periods. (c) Spiking 1 second before andafter the noisy current p(t) is injected. (d) Infinitesimal PRC of the neuron (continuouscurve) obtained from 40 cycles and the MATLAB program in Ex. 13 (first 8 Fourierterms). Averaged voltage trace during the spike (dotted curve) is plotted for reference.The same procedure applied to (e) the Andronov-Hopf oscillator and (f) the I Na,p +I K -model. Numbers in boxes show the number of Fourier terms used to fit the curve,theoretical curves (functions Q 1 (ϑ) from Fig. 10.23) are dashed.
Synchronization (see www.izhikevich.com) 479membranepotential (mV)phasephasedeviationphasedifference200-20-40-60-80T0T0TV 1 V 22122 1100 5 10 15 20 25 30 35 40 45 50time (ms)Figure 10.25: The relationship between membrane potential oscillation of two neurons,V 1 (solid) and V 2 (dashed), their phases, phase deviations, and phase difference. Shownare simulation of two I Na + I K -models with parameters as in Fig. 10.3 and coupledsymmetrically via gap junctions 0.1(V j − V i ) (see Sect. 2.3.4).10.2.5 Phase model for coupled oscillatorsNow consider n weakly coupled oscillators of the formp i (t){ }} {n∑ẋ i = f i (x i ) + ε g ij (x i , x j ) , x i ∈ R m , (10.11)j=1and assume that the oscillators, when uncoupled (ε = 0), have equal free-runningperiods T 1 = · · · = T n = T . Applying any of the three methods above to such a weaklyperturbed system, we obtain the corresponding phase model˙ϑ i = 1 + ε Q i (ϑ i ) ·p i (t){ }} {n∑g ij (x i (ϑ i ), x j (ϑ j )) , (10.12)j=1where each x i (ϑ i ) is the point on the limit cycle having phase ϑ i . Note that (10.11) isdefined in R nm , whereas the phase model (10.12) is defined on the n-torus, denoted asT n .To study collective properties of the network, such as synchronization, it is convenientto represent each ϑ i (t) asϑ i (t) = t + ϕ i , (10.13)
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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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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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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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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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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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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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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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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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158 Conductance-Based Models5.2.2 E
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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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202 Bifurcationsc 1c 2xFigure 6.34:
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204 BifurcationseigenvaluesHopffold
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206 Bifurcationsfast nullclineslow
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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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240 Excitabilityproperties integrat
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242 ExcitabilityThe existence of fa
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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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256 Excitabilitymembrane potential,
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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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318 Simple Modelsthey are able to g
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330 Simple Modelsa35 mV350 msc-NAC
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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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348 Burstingslow dynamicsneuronvolt
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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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spiking380 Burstingfoldbifurcations
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394 BurstingspikingrestingFigure 9.
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396 Bursting0-10membrane potential,
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402 Bursting28. [Ph.D.] Develop an
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404 Synchronization (see www.izhike
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406 Synchronization (see www.izhike
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408 Solutions to Exercises, Chap. 3
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Page 452 and 453: 442 Referencesterneurons mediated b
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Page 462 and 463: 452 ReferencesRosenblum M.G. and Pi
Page 464 and 465: 454 ReferencesTuckwell H.C. (1988)
Page 466 and 467: 456 References9
Page 468 and 469: 458 Synchronization (see www.izhike
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