Dynamical Systems in Neuroscience:

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516 Solutions to Exercises, Chap. 10The following program must be in the file prcerror.m.function err=prcerror(coeff)global spikes p tau na = coeff(1:n); % Fourier coefficients for sinb = coeff(n+1:2*n); % Fourier coefficients for cosb0= coeff(2*n+1); % dc termT = coeff(2*n+2); % period of oscillationerr=0;i=1;clf;for s=2:length(spikes)theta=0;while i*tau
Solutions to Exercises, Chap. 10 51716. Winfree approach: Using results of Ex. 4,at A = 0.Z(ϑ) =∂∂A acos 1 + A cos ϑ√ = − sin ϑ1 + 2A cos ϑ + A2Kuramoto approach: Since grad ϑ(x) is orthogonal to the contour line of the function ϑ(x) atx, i.e., the isochron of x, and the results of Ex. 1 that isochrons are radial, we get grad (ϑ) =(− sin ϑ, cos ϑ) using purely geometrical considerations. Since p(t) is real, we need to keep onlythe first component.Malkin approach: Let us work in the complex domain. On the circle z(t) = e it we get Df = i.Since Df ⊤ is equivalent to complex-conjugation in the complex domain, we get ˙Q = i · Q,which has the solution Q(t) = Ce it . The free constant C = i is found from the normalizationcondition Q(0) ∗ i = 1, where ∗ means complex-conjugate.Alternatively, on the circle z(t) = e it , we have f(z(t)) = f(e it ) = ie it . From the normalizationcondition Q(t) ∗ f(z(t)) = 1 we find Q(t) = ie it = − sin ϑ + i cos ϑ.17. Rescaling the state variable z = √ bu and the time, τ = ɛt, we obtain the reduced systemu ′ = (1 + i)u + (−1 + di)u|u| 2 + εp(t) .We can apply the theory only when ε is small. That is, the theory is guaranteed to work in avery weak limit ɛ ≪ b √ b ≪ 1. As it is often the case, numerical simulations suggest that thetheory works well outside the guaranteed interval. Substituting u = re iθ into this equation,r ′ e iθ + re iθ iθ ′ = (1 + i)re iθ + (−1 + di)r 3 e iθ + εp(t) ,dividing by e iθ and separating real and imaginary terms, we represent the oscillator in polarcoordinatesr ′ = r − r 3 + ε Re p(t)e −iθθ ′ = 1 + dr 2 + ε Im r −1 p(t)e −iθ .When ε = 0, this system has a limit cycle attractor r(t) = 1 and θ(t) = (1 + d)t, provided thatd ≠ −1. On the attractor, the solution to Malkin’s adjoint equation (10.10),(Q ′ −2 0= −2d 0) ⊤ (0Q with Q(t) ·1 + d)= 1 ,is Q(t) = (d, 1)/(1+d). Indeed, the normalization condition results in Q 2 (t) = 1/(1+d). Hence,unique periodic solution of the first equation, Q ′ 1 = 2Q 1 − 2d/(1 + d), is Q 1 (t) = d/(1 + d). Onecan also use Kuramoto’s approach and the results of Ex. 2. The corresponding phase model,ϑ ′ = 1 + ε{d Re p(t)e −i(1+d)ϑ + Im p(t)e −i(1+d)ϑ }/(d + 1) ,can be simplified via θ = (1 + d)ϑ (notice the font difference) to get the result.18. (Delayed coupling) Let ϑ(t) = t + ϕ(τ), where τ = εt is the slow time. Since ϑ(t − d) =t − d + ϕ(τ − εd) = t − d + ϕ(τ) + O(ε), we have x j (ϑ i (t − d ij )) = x j (t − d ij + ϕ(τ)) so thatwe can proceed as in Sect. 10.2.5 except that there is an extra term, −d ij , in (10.16). See alsoIzhikevich (1998).
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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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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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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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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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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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240 Excitabilityproperties integrat
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242 ExcitabilityThe existence of fa
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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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264 Excitabilitya pulse of current.
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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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296 Simple Modelslayer 5 neuronsimp
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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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316 Simple Modelsthrough an appropr
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318 Simple Modelsthey are able to g
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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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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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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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