Dynamical Systems in Neuroscience:

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74 One-Dimensional Systemsbifurcationbut no change of behaviorchange of behavior but no bifurcationpulsedinputFigure 3.23: Bifurcations are not equivalent to qualitative change of behavior if thesystem is started with the same initial condition or subject to external input.bistablebistablemonostablemembrane potential, mV10 mV1 ms-40 mVdc-current I=0 dc-current I=16dc-current I=60pulses pulses pulsesFigure 3.24: Qualitative change of the up-stroke dynamics of a layer 5 pyramidal neuronfrom rat visual cortex (the same neuron as in Fig. 3.15).
One-Dimensional Systems 75result in upstroke of the action potential, while others do not. When I = 60 pA, allinitial conditions result in the generation of an action potential. Clearly, a change ofqualitative behavior occurs for some I between 0 and 60.To understand the qualitative dynamics in Fig. 3.24, we consider the one-dimensionalI Na,p -model (3.5) having different values of the parameter I and depict its trajectoriesin Fig. 3.25. One can clearly see that the qualitative behavior of the model dependson whether I is greater or less than 16. When I = 0 (top of Fig. 3.25), the system isbistable. The rest and the excited states coexist. When I is large (bottom of Fig. 3.25)the rest state no longer exists because the leak outward current cannot balance thelarge injected dc-current I and the inward Na + current.What happens when we change I past 16? The answer lies in the details of thegeometry of the right-hand side function F (V ) of (3.5) and how it depends on theparameter I. Increasing I elevates the graph of F (V ). The higher the graph of F (V )is, the closer its intersections with the V -axis are, as we illustrate in Fig. 3.26 depictingonly the low-voltage range of the system. When I approaches 16, the distance betweenthe stable and unstable equilibria vanishes; the equilibria coalesce and annihilate eachother. The value I = 16 at which the equilibria coalesce is called the bifurcation value.This value separates two qualitatively different regimes: When I is near but less than16, the system has three equilibria and bistable dynamics. The quantitative features,such as the exact locations of the equilibria, depend on the particular values of I, butthe qualitative behavior remains unchanged no matter how close I to the bifurcationvalue is. In contrast, when I is near but greater than 16 the system has only oneequilibrium and monostable dynamics.In general, a dynamical system may depend on a vector of parameters, say p. Apoint in the parameter space, say p = a, is said to be a regular or non-bifurcation point,if the system’s phase portrait at p = a is topologically equivalent to the phase portraitat p = c for any c sufficiently close to a. For example, the value I = 13 in Fig. 3.26 isregular, since the system has topologically equivalent phase portraits for all I near 13.Similarly, the value I = 18 is also regular. Any point in the parameter space that isnot regular is called a bifurcation point. Namely, a point p = b is a bifurcation point,if the system’s phase portrait at p = b is not topologically equivalent to the phaseportrait at a point p = c no matter how close c to b is. The value I = 16 in Fig. 3.26 isa bifurcation point. It corresponds to the saddle-node (also known as fold or tangent)bifurcation for reasons described later. It is one of the simplest bifurcations consideredin this book.3.3.4 Saddle-node (fold) bifurcationIn general, a one-dimensional system˙V = F (V, I)having an equilibrium point V = V sn for some value of the parameter I = I sn (i.e.,F (V sn , I sn ) = 0) is said to be at a saddle-node bifurcation (sometimes called a fold
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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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Page 36 and 37: 26 Electrophysiology of NeuronsInsi
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Page 40 and 41: 30 Electrophysiology of NeuronsRest
Page 42 and 43: 32 Electrophysiology of NeuronsFigu
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Page 48 and 49: 38 Electrophysiology of Neurons2.3
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Page 66 and 67: 56 One-Dimensional SystemsActivatio
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Page 80 and 81: 70 One-Dimensional Systems+ - - + +
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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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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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¡ ¡ ¡¡ ¡ ¡¡ ¡ ¡¡ ¡ ¡184
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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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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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