Dynamical Systems in Neuroscience:

More documents

Recommendations

Info

202 Bifurcationsc 1c 2xFigure 6.34: Cusp surface.in particular, c 1 = c 2 = 0 at the cusp point. The cusp bifurcation is supercritical whena < 0 and subcritical otherwise. It is explained by the shape of the surfacec 1 + c 2 x + ax 3 = 0depicted in Fig. 6.34.Let us treat c 1 and c 2 as two independent parameters, and check that there aresaddle-node bifurcations in any neighborhood of the cusp point. The bifurcation setsof the topological normal form can easily be found. Differentiating c 1 + c 2 x + ax 3 withrespect to x gives c 2 +3ax 2 . Equating both of these expressions to zero and eliminatingx gives the saddle-node bifurcation curvesc 1 = ± √ 2 ( c2) 3/2,a 3depicted at the bottom of Fig. 6.34.Since c 1 = c 1 (b) and c 2 = c 2 (b), varying the bifurcation parameter b results in apath on the (c 1 , c 2 )-plane. Depending on the shape and location of this path, one canget many 1-dimensional bifurcation diagrams. A summary of some special cases isdepicted in Fig. 6.35 showing that there can be many interesting dynamical regimes inthe vicinity of a cusp bifurcation point.An important special case is when c 1 = 0 and c 2 (b) = b, so that the topologicalnormal form isẋ = bx + ax 3 .This form corresponds to a pitchfork bifurcation, whose diagram is depicted in Fig. 6.36(see also the bottom bifurcation diagram in Fig. 6.35). This bifurcation has an infinitecodimension unless one considers dynamical systems with symmetry, e.g., ẋ = f(x, b)with f(−x, b) = −f(x, b) for all x and b.6.3.3 Bogdanov-TakensCan an equilibrium undergo Andronov-Hopf and saddle-node bifurcations simultaneously?There are two possibilities, illustrated in Fig. 6.37:
Bifurcations 203xc 2 (b)xbbxxbxxbbbc 1 (b)xxxxb b bbFigure 6.35: Summary of special cases for the supercritical cusp bifurcation. Dottedsegments are paths c 1 = c 1 (b), c 2 = c 2 (b), where b is a one-dimensional bifurcationparameter. The corresponding bifurcation diagrams are depicted in boxes. Continuouscurves represent stable solutions, dashed curves represent unstable solutions (modifiedfrom Hoppensteadt and Izhikevich 1997).+b/aunstablexxstable+b/|a|stableunstablebstableunstableb-b/aunstablestable-b/|a|¡ ¡ ¡¡ ¡ ¡¡ ¡ ¡¡ ¡ ¡¡ ¡¡ ¡¡ ¡ ¡¡ ¡ ¡¡ ¡ ¡¡ ¡ ¡subcritical pitchfork, a > 0supercritical pitchfork, a < 0Figure 6.36: Pitchfork bifurcation diagrams.
Page 1:
Eugene M. IzhikevichThe Neuroscienc
Page 6 and 7:
6.3.6 Saddle-node homoclinic orbit
Page 8 and 9:
9.4.2 Integrators vs. Resonators .
Page 10 and 11:
xPrefaceversely, cells having quite
Page 12 and 13:
2 Introductionapical dendritessomar
Page 14 and 15:
4 Introduction(a)spikes(b)spikes cu
Page 16 and 17:
6 Introduction1.1.3 Why are neurons
Page 18 and 19:
8 Introductionconcepts of dynamical
Page 20 and 21:
10 Introductionspikingmoderesting m
Page 22 and 23:
12 Introduction(a)spikinglimitcycle
Page 24 and 25:
14 Introductionco-existence of rest
Page 26 and 27:
16 Introduction1.2.4 Neuro-computat
Page 28 and 29:
18 IntroductionspikeFigure 1.16: Ph
Page 30 and 31:
20 Introductionwith coupled neurons
Page 32 and 33:
22 IntroductionFigure 1.17: John Ri
Page 34 and 35:
24 Introduction
Page 36 and 37:
26 Electrophysiology of NeuronsInsi
Page 38 and 39:
28 Electrophysiology of Neuronsouts
Page 40 and 41:
30 Electrophysiology of NeuronsRest
Page 42 and 43:
32 Electrophysiology of NeuronsFigu
Page 44 and 45:
34 Electrophysiology of Neuronsm (V
Page 46 and 47:
36 Electrophysiology of Neurons10.8
Page 48 and 49:
38 Electrophysiology of Neurons2.3
Page 50 and 51:
Page 52 and 53:
42 Electrophysiology of NeuronsDepo
Page 54 and 55:
44 Electrophysiology of NeuronsV(x,
Page 56 and 57:
46 Electrophysiology of Neurons1m (
Page 58 and 59:
48 Electrophysiology of NeuronsPara
Page 60 and 61:
Page 62 and 63:
52 Electrophysiology of Neuronsvolt
Page 64 and 65:
54 Electrophysiology of Neurons
Page 66 and 67:
56 One-Dimensional SystemsActivatio
Page 68 and 69:
58 One-Dimensional Systemsinwardcur
Page 70 and 71:
60 One-Dimensional Systems40bistabi
Page 72 and 73:
62 One-Dimensional Systemsat every
Page 74 and 75:
64 One-Dimensional Systems?F(V)>0+
Page 76 and 77:
66 One-Dimensional SystemsUnstableE
Page 78 and 79:
¡ ¡ ¡¡ ¡ ¡¡ ¡ ¡¡ ¡ ¡¡
Page 80 and 81:
70 One-Dimensional Systems+ - - + +
Page 82 and 83:
72 One-Dimensional Systemsλ(V-Veq)
Page 84 and 85:
74 One-Dimensional Systemsbifurcati
Page 86 and 87:
76 One-Dimensional Systems100806040
Page 88 and 89:
78 One-Dimensional Systemsbifurcati
Page 90 and 91:
80 One-Dimensional Systems20 mV100
Page 92 and 93:
82 One-Dimensional Systemsmembrane
Page 94 and 95:
84 One-Dimensional Systemssteady-st
Page 96 and 97:
86 One-Dimensional Systemsspike rig
Page 98 and 99:
88 One-Dimensional SystemsVF (V)1VV
Page 100 and 101:
90 One-Dimensional Systems10.80.60.
Page 102 and 103:
92 One-Dimensional Systems
Page 104 and 105:
94 Two-Dimensional SystemsI Na,pneu
Page 106 and 107:
96 Two-Dimensional Systemsdefines a
Page 108 and 109:
98 Two-Dimensional Systems(f(x(t),y
Page 110 and 111:
100 Two-Dimensional Systems0.70acti
Page 112 and 113:
102 Two-Dimensional Systems0.70.6K
Page 114 and 115:
104 Two-Dimensional Systemsy1.510.5
Page 116 and 117:
106 Two-Dimensional SystemsFor exam
Page 118 and 119:
108 Two-Dimensional SystemsIn gener
Page 120 and 121:
110 Two-Dimensional Systemsv 2 v 2v
Page 122 and 123:
112 Two-Dimensional SystemsV-nullcl
Page 124 and 125:
114 Two-Dimensional SystemsV-nullcl
Page 126 and 127:
116 Two-Dimensional Systemsheterocl
Page 128 and 129:
118 Two-Dimensional Systems0.60.5n-
Page 130 and 131:
120 Two-Dimensional Systemseigenval
Page 132 and 133:
122 Two-Dimensional SystemsK + acti
Page 134 and 135:
124 Two-Dimensional Systemsexcitati
Page 136 and 137:
126 Two-Dimensional SystemsReview o
Page 138 and 139:
128 Two-Dimensional SystemsabcdFigu
Page 140 and 141:
130 Two-Dimensional SystemsFigure 4
Page 142 and 143:
132 Two-Dimensional Systems
Page 144 and 145:
134 Conductance-Based Models• If
Page 146 and 147:
136 Conductance-Based Modelsinward(
Page 148 and 149:
138 Conductance-Based Models1I=01I=
Page 150 and 151:
140 Conductance-Based Modelsleak cu
Page 152 and 153:
142 Conductance-Based Modelshyperpo
Page 154 and 155:
144 Conductance-Based Models0I=0ina
Page 156 and 157:
146 Conductance-Based Models0I=9.75
Page 158 and 159:
148 Conductance-Based Models1I=651I
Page 160 and 161:
150 Conductance-Based Modelshave co
Page 162 and 163: 152 Conductance-Based Models1 sec10
Page 164 and 165: 154 Conductance-Based Modelsvoltage
Page 166 and 167: 156 Conductance-Based Models1h=0.89
Page 168 and 169: 158 Conductance-Based Models5.2.2 E
Page 170 and 171: 160 Conductance-Based Models1recove
Page 172 and 173: 162 Conductance-Based Modelscurrent
Page 174 and 175: 164 Conductance-Based Modelsmodels
Page 176 and 177: 166 Conductance-Based Models
Page 178 and 179: 168 Bifurcationssaddle-node bifurca
Page 180 and 181: 170 Bifurcationssaddle-node bifurca
Page 182 and 183: 172 Bifurcations0.5v 2V-nullclineK
Page 184 and 185: 174 BifurcationsBoth types of the b
Page 186 and 187: 176 BifurcationsV-nullcline0.5K + g
Page 188 and 189: 178 Bifurcationsrstable limit cycle
Page 190 and 191: 180 BifurcationsnVstable limit cycl
Page 192 and 193: ¡ ¡¡ ¡¡ ¡ ¡¡ ¡ ¡182 Bifur
Page 194 and 195: ¡ ¡ ¡¡ ¡ ¡¡ ¡ ¡¡ ¡ ¡184
Page 196 and 197: 186 Bifurcationsmembrane potential,
Page 198 and 199: 188 BifurcationsBifurcation of a li
Page 200 and 201: 190 Bifurcationsstableunstablelimit
Page 202 and 203: 192 Bifurcationsamplitude (max-min)
Page 204 and 205: 194 Bifurcationshomoclinic orbithom
Page 206 and 207: 196 Bifurcations0.80.60.4stable lim
Page 208 and 209: 198 Bifurcationsfrequency (Hz)40030
Page 210 and 211: 200 BifurcationsSaddle-Focus Homocl
Page 214 and 215: 204 BifurcationseigenvaluesHopffold
Page 216 and 217: 206 Bifurcationsfast nullclineslow
Page 218 and 219: 208 Bifurcationswith fast and slow
Page 220 and 221: 210 BifurcationsSupercritical Andro
Page 222 and 223: 212 BifurcationsK + conductance tim
Page 224 and 225: 214 BifurcationsIn contrast, if the
Page 226 and 227: 216 Bifurcationsinvariant circlesad
Page 228 and 229: 218 Bifurcationsbifurcationssaddle-
Page 230 and 231: 220 BifurcationsExercises1. (Transc
Page 232 and 233: 222 Bifurcationsv 2v 11a-11-1Figure
Page 234 and 235: 224 Bifurcations19. [M.S.] A leaky
Page 236 and 237: 226 Excitabilityspike?spikerestrest
Page 238 and 239: 228 ExcitabilityAlternatively, the
Page 240 and 241: 230 Excitability50 ms 20 mV 1 ms100
Page 242 and 243: 232 ExcitabilityClass 3 excitable n
Page 244 and 245: 234 Excitability0.20.150.10.05I 1I
Page 246 and 247: 236 Excitabilitysaddle-node bifurca
Page 248 and 249: 238 Excitability(a) resting spiking
Page 250 and 251: 240 Excitabilityproperties integrat
Page 252 and 253: 242 ExcitabilityThe existence of fa
Page 254 and 255: 244 Excitability1coincidencedetecti
Page 256 and 257: 246 Excitabilityspikeexcitatory pul
Page 258 and 259: 248 Excitability(g)9 ms(f)(e)(d)(c)
Page 260 and 261: 250 Excitabilitysquid axonmodel0 mV
Page 262 and 263:
252 Excitability(a) integrator(b) r
Page 264 and 265:
254 Excitability(a) integrator(b) r
Page 266 and 267:
256 Excitabilitymembrane potential,
Page 268 and 269:
258 Excitability0.1saddle-node bifu
Page 270 and 271:
260 Excitability100 ms10 mVoscillat
Page 272 and 273:
262 ExcitabilityBogdanov-Takens bif
Page 274 and 275:
264 Excitabilitya pulse of current.
Page 276 and 277:
266 Excitabilitymembrane potential
Page 278 and 279:
268 Excitability(a)150100I K(M)(b)3
Page 280 and 281:
270 Excitability50 mV300 msFigure 7
Page 282 and 283:
272 Excitability20 mVADP100 msAHP-6
Page 284 and 285:
274 ExcitabilityK + activation gate
Page 286 and 287:
276 ExcitabilityBibliographical Not
Page 288 and 289:
278 Excitability7. Show that the re
Page 290 and 291:
280 Simple Modelsspikemembrane pote
Page 292 and 293:
282 Simple Models1thresholdyz reset
Page 294 and 295:
284 Simple Models1v reset =|b| 1/2b
Page 296 and 297:
286 Simple Modelsbe an integrator o
Page 298 and 299:
288 Simple Modelsintegrate-and-fire
Page 300 and 301:
290 Simple Models(A) tonic spiking(
Page 302 and 303:
292 Simple Modelsgeneral algorithm
Page 304 and 305:
294 Simple Modelscha, cha — real
Page 306 and 307:
296 Simple Modelslayer 5 neuronsimp
Page 308 and 309:
298 Simple Modelsb=-240200saddle-no
Page 310 and 311:
300 Simple Models(a)simple modellay
Page 312 and 313:
302 Simple Models(a)burstingspiking
Page 314 and 315:
304 Simple Models(a)dendriticspike2
Page 316 and 317:
306 Simple Models(a)74 5 6 3210(b)C
Page 318 and 319:
308 Simple Modelschattering neuron
Page 320 and 321:
310 Simple ModelsLTS neuron (in vit
Page 322 and 323:
312 Simple ModelsFS neuron (in vitr
Page 324 and 325:
314 Simple Models(1) fast oscillati
Page 326 and 327:
316 Simple Modelsthrough an appropr
Page 328 and 329:
318 Simple Modelsthey are able to g
Page 330 and 331:
320 Simple Modelsv r = −80 mV, an
Page 332 and 333:
322 Simple Modelsshow in Fig. 7.36.
Page 334 and 335:
324 Simple ModelsBibliographical No
Page 336 and 337:
326 Simple ModelsExercises1. (Integ
Page 338 and 339:
328 Simple Models17. [M.S.] Analyze
Page 340 and 341:
330 Simple Modelsa35 mV350 msc-NAC
Page 342 and 343:
332 Simple Modelsrat RTN neuronsimp
Page 344 and 345:
334 Simple ModelsFigure 8.34: Class
Page 346 and 347:
336 Simple Modelsspiny neuronlatenc
Page 348 and 349:
338 Simple Models(a)stellate cellof
Page 350 and 351:
340 Simple Modelsrat's mitral cell
Page 352 and 353:
342 Bursting(a) cortical chattering
Page 354 and 355:
344 Burstingmembranepotential (mV)-
Page 356 and 357:
346 Burstingvoltage-gatedCa2+-gated
Page 358 and 359:
348 Burstingslow dynamicsneuronvolt
Page 360 and 361:
350 Burstingslow inactivation of in
Page 362 and 363:
352 Bursting9.2.1 Fast-slow burster
Page 364 and 365:
354 Burstingn-nullclinen slow =-0.0
Page 366 and 367:
356 Bursting0maxmembrane potential,
Page 368 and 369:
358 Bursting0membrane potential, V
Page 370 and 371:
360 BurstingI=0I=4.5425 ms 25 mVI=5
Page 372 and 373:
362 Burstingmembrane potential, V (
Page 374 and 375:
364 Bursting9.3 ClassificationIn Fi
Page 376 and 377:
366 Burstingbifurcation of spiking
Page 378 and 379:
368 BurstingFigure 9.26: Putative
Page 380 and 381:
370 Bursting108spikingslow variable
Page 382 and 383:
372 Bursting(a)membrane potential,
Page 384 and 385:
374 Burstingdepending on the type o
Page 386 and 387:
376 BurstingsubcriticalAndronov-Hop
Page 388 and 389:
378 Bursting(a)membrane potential,
Page 390 and 391:
spiking380 Burstingfoldbifurcations
Page 392 and 393:
382 BurstingspikingsupercriticalAnd
Page 394 and 395:
384 Burstingaction potentials cut2
Page 396 and 397:
386 Bursting9.4.3 BistabilitySuppos
Page 398 and 399:
388 BurstingFigure 9.49: The instan
Page 400 and 401:
esting390 Burstingspikesynchronizat
Page 402 and 403:
392 BurstingReview of Important Con
Page 404 and 405:
394 BurstingspikingrestingFigure 9.
Page 406 and 407:
396 Bursting0-10membrane potential,
Page 408 and 409:
398 Bursting0membrane potential, V
Page 410 and 411:
400 BurstingFigure 9.61: A cycle-cy
Page 412 and 413:
402 Bursting28. [Ph.D.] Develop an
Page 414 and 415:
404 Synchronization (see www.izhike
Page 416 and 417:
Page 418 and 419:
408 Solutions to Exercises, Chap. 3
Page 420 and 421:
Page 422 and 423:
Page 424 and 425:
Page 426 and 427:
Page 428 and 429:
Page 430 and 431:
Page 432 and 433:
Page 434 and 435:
Page 436 and 437:
Page 438 and 439:
Page 440 and 441:
Page 442 and 443:
Page 444 and 445:
Page 446 and 447:
Page 448 and 449:
Page 450 and 451:
Page 452 and 453:
442 Referencesterneurons mediated b
Page 454 and 455:
444 ReferencesDickson C.T., Magistr
Page 456 and 457:
446 ReferencesGuckenheimer J. (1975
Page 458 and 459:
448 Referencestional Journal of Bif
Page 460 and 461:
450 ReferencesMarkram H, Toledo-Rod
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:
Page 470 and 471:
Page 472 and 473:
Page 474 and 475:
Page 476 and 477:
Page 478 and 479:
Page 480 and 481:
Page 482 and 483:
Page 484 and 485:
Page 486 and 487:
Page 488 and 489:
Page 490 and 491:
Page 492 and 493:
Page 494 and 495:
Page 496 and 497:
Page 498 and 499:
Page 500 and 501:
Page 502 and 503:
Page 504 and 505:
Page 506 and 507:
Page 508 and 509:
Page 510 and 511:
Page 512 and 513:
Page 514 and 515:
Page 516 and 517:
Page 518 and 519:
Page 520 and 521:
Page 522 and 523:
Page 524 and 525:
Page 526 and 527:
Page 528:
show all

Dynamical Systems in Neuroscience:

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?