Dynamical Systems in Neuroscience:

More documents

Recommendations

Info

214 BifurcationsIn contrast, if the equilibrium loses stability via subcritical Andronov-Hopf bifurcation,the state of the system diverges from it, which results in an immediate spike orsome kind of a large-amplitude jump. Such a loss of stability is called hard: neither theequilibrium nor its neighborhood are attractive. The hard loss of stability usually leadsto noticeable or catastrophic changes in systems behavior, and the stability boundaryis called dangerous (Bautin 1949). Changing the bifurcation parameter in the oppositedirection will make the equilibrium stable again, but may not bring the state of thesystem back to it. Saddle-node bifurcation is hard unless it is on an invariant circle.In this case, the loss of stability is catastrophic, i.e., leading to noticeable spikes, butreversible. Saddle homoclinic orbit bifurcation is hard, regardless of whether it is subcriticalor supercritical. In general, most bifurcations in neurons or at least in neuronalmodels are hard.Review of Important Concepts• Stable equilibrium (resting state) in a typical neuronal model caneither– disappear via saddle-node bifurcation, which can be off or oninvariant circle, or– lose stability via Andronov-Hopf bifurcation, which can be supercriticalor subcritical.These four cases are summarized in Fig. 6.46.• Stable limit cycle (periodic spiking state) in a typical twodimensionalneuronal model can either– be cut by saddle-node on invariant circle bifurcation,– shrink to a point via supercritical Andronov-Hopf bifurcation,– disappear via fold limit cycle bifurcation, or– disappear via saddle homoclinic orbit bifurcation.These four cases are summarized in Fig. 6.47.• Some atypical (codimension-2) bifurcations may play importantroles in neuronal dynamics.• Bogdanov-Takens bifurcation separates integrators from resonators.Bibliographical NotesThough bifurcation theory can be traced back to Poincare and Andronov, it is a relativelynew branch of mathematics. The first attempt to apply it to neuroscience was as
Bifurcations 215nodesaddlesaddle-nodesaddle-node bifurcationinvariant circlenode saddle saddle-nodesaddle-node on invariant circle (SNIC) bifurcationsupercritical Andronov-Hopf bifurcationsubcritical Andronov-Hopf bifurcationFigure 6.46: Summary of all codimension-1 bifurcations of a stable equilibrium (restingstate).
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 212 and 213: 202 Bifurcationsc 1c 2xFigure 6.34:
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 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?