P reface . . ..... . ....... .... .............. . . ......... .... .......... . . . . -.... ix
1: V ector A nalysis .. ..... .. .................... ... . ....... .... .
S1 , Introduction . ... . ..... .. . .......... ... .... .......... ... .............. .. I
1.2. Vectors and Scalars ... ....... ... ... ..... .... ... ...... ........ ....... . 1
13. V ector A lgebra .............. ........... ........... ......... ............ 2
,4. G radient ....... . ............ .. . . ..... ... . .... .. - . .. ..... .... ..... .... 4
1.5. D ivergence ... ............... ... ....... . ...... ... .. .. .. .... 7
1.6. Vector identities ........... .......... .. .............. ....... ... ......... 11
17. Source and Field Points ............ .... ...... ... 12
1,8. Volum es and Surfaces ................. ............ ...................... 14
1.9, The Gradient and Divergence of (1ir) .................... .... ...... - 15
1.10. Solid A ngles . ......... .. .. . ..... .. ........... ........ ............ ...... 18
1.11. Operations Sum m ary .. ............. ...... . ..... .... ...... .... ... ... .. 20
1.12. N otes.... ..... .. ... ................ ............. ... .- .... ............ 20
2: Sources and Fields.. ...... ............... ....... ......... ........... 23
2.1. F ield s .. .. ....... ............ ............ ............. ... ........ 24
2.2. Tissue Resistance and Conductance .......... . .... ... ..... ...... .... ...... 24
2.3. Fields and Currents. ... ..... . ... .... ........... ........... 25
2.4. Fields from Sources and Vice Versa........................................ 27
2.5 . D u ality .................... .... ..................... .. ................ 27
2.6. M onopole Field ...... ...... . .. .... .... ................ ......... ...... 28
12.7 D ipole Field . ....-... .... ........... .......... . ......................... --30
2.8. Evaluating V(l/r) with Respect to Source Variables. . . . . . . . . . . .... . . .. . 32
2.9. M onopole Pairs to Dipoles? .................... .. .... . ..... ....... ...... 33
2.10. C apacitance t..... . . .... ..... ....... ... ..... ......... ...... . .... . 37
2.11. Units for Resistance and Capacitance ....... .... ............ 39
2.12. Units for Some Electrical Quantities .......... .......... ........... ....... 42
2.13. N otes . .... .. .. ........ .... ..... .... .... ...... ............ ... 42
3: B ioelectric Potentials ............. ...... ........ ............. ....... ... ..... 45
3.1. Currents in Solutions... .. .............................. ... ... .. ......... ..... 46
3.2. M oles and Am peres ... .......... . . . .. .... .... ..... .. ............ 46
3.3. Ionic Com position ..l....... ... ....... -.. ... .......... ....... ........ . 47
3.4. N otation for Ion Species ............. ........... ........... .......... ... . 48
3.5. Nernst-Planck E uation.. .... . ..... ............................... 48
3.6. M obility ....... . ............ ..... ...... . ...... ..... ..... ...... .. ...... . 49
3.7. Temperature Variations.. ............ . . ..... .................... ... ..... . 50
38 . F lu x D u e to D iff u s io n P lu s E le c tr ic F ie ld . . . . . . . .. . . ... . . ... . . .. . .. 5 1
3.9. M em brane Structure ... ....................................... ...... ... 54
3.10. N ernst Potential .. .... . . .. .... ..... .. .. ........ ...... .. ....... .. .... 57
3.11. Electrolytes .. .......... . ....... ......... ............. 60
3.12. Sum m ary So Far ....... ................................ ..... .......... 62
3.13. Parallel-Conductance M odel ..................... .... ......... 62
3.14, Contributions from Chloride ......... ........... ................... .... .... 66
3.15. Reference Values ............. ........... ......... ....... ............... 69
3.16. N otes . .. ............................... . ..... ......... ............... . ... 6 9
4 : C h an n els .. ............. .... .. . .................. ..... ........... ... . . 7 1
4.1. Introduction .... .................. ............... .... ................... . 71
4.2. Channel Structure by Electron Microscopy . ........ . . ....... . . . . . 71
4,3. Channel Structure: Molecular Genetics ............ .......... 72
4.4 Ion Channels: Biophysical Methods............................ 75
4.5. M acroscopic Channel Kinetics ........ ............ ............. ... ...... 87
4.6. Channel Statistics ............ .................. ..... ... 89
4.7. The Hodgkin-Huxley Membrane Model .................................. 92
4.8. N otes..... . ...... ....... ......... ......... ... .. ....... ... ......... .... . 94
"5: A ction Potentials ... ..... ....... ............. . ........... ... ........... 97
5.1. Experimental Action Potentials ....... _ ....... ...... ......... 99
5.2. Voltage Clam p ...... .... ...... ... ... ...... ... ......... .. .... ... 110
5.3. Hodgkin-Huxley Conductance Equations .. ., . . . ..... ......... . . . . . .. ... . 119
5.4. Simulation of Membrane Action Potentials ... . . . . . . . .. . . . . .. . . . . . . .. 127
5.5. Beyond H -H M odels.. . ........... ............... ......... ............. 139
5.6. Appendix: GHK Constant-Field Equation ...... 146
5.7. N otes ......... .. . ... .. . ................... . ......... ...... ... .... ...... 15 1
6: Im pulse Propagation .. ........................ ......................... 155
6.1. Core-Conductor M odel .......................... . .......... .. ...... 156
6.2. C able Equations .. .... .. ...... .......... ...... .............. ........... 159
6.3. Propagation ................................................... .. 166
6.4. Propagation in Myelinated Nerve Fibers ............................... 183
6.5. N otes . .. .... i. .... .......... ..................... ... ... ............... . 185
7: Electrical Stim ulation ........ .............. ................ ........ ... 187
7.1. Spherical Cell Stim ulation. .. ..... ...... ........ .................. ...... 188
.2 Stim ulation of Fibers...... ........... ...... .... ...... ......... ........ 194
7.3. Fiber Stim ulation ............... ........ ...... .......... 196
BIOELECTRICITY: A QUANTITATIVE APPROACH Xi..
7.4. Axial Curreniet Transient . .... ..... ... ........... ....... .......... . ..... . 203
7.5. Field Stimulus of an Individual Fiber ......... .................... ....... 206
7.6. Stimulus, then Suprathreshold Response ................................ 214
77. Fiber Input Impedance .. ......... .... .... ............... .... .. ..... ... . 217
7. . M agnetic Field Stimulation.. . ... ... ..................... ....... ..... . 221
7.9. N otes ...... . . .... .... ... ........ ... ..... . . .. ... ... .......... .. .. 22 1
8: E xtracellular Fields . .............................................. ..... . 223
8.1. Background . . ....... .... ..... .... ..... . ...... ..... .......... ........ 223
82. Extracellular Potentials from Fibers ................. 228
8.3. Potentials from a Cell .............................................. 252
8.4 . N otes... ... . ......... .. .............. ........ ....... .. .............. .. 264
9: Cardiac Electrophysiology ......................... ............... 267
91. Intercellular Communication ................................. ........ .... 268
9.2. Cardiac Cellular M odels ............... ........... ....... ............. 290
9.3. .Electrocardiography ..... ......... ............ . .... . ... 303
9.4. N otes . .... ........ ....... ... . . .. .... .......... 320
10: The Neuromuscular Junction ...... ...... ................. ........ . 325
10.1. Introduction.. ........... . . ...... .......... .. ... ............. .......... 325
10.2. Neuromuscular Junction . ...... ..... ............ ... . 326
10.3. Quantal Transmitter Release ............... ................ .............. 329
10.4. Transmitter Release, Poisson Statistics ...... .. ............... ........ 331
10.5. Transmitter Release, Ca++ and Mg-+ .. .. .............................. 333
10.6. Post-Junctional Response to Transmitter ... . ... . .. . . . .. . . . . . .. ... 335
11: Skeletal M uscle ......... ...... ........ .. .......... ....... .. .... . ...... 341
1.1. M uscle Structure ....... . ........................................ ....... 341
11.2. M uscle Contraction .. ............ ...... ........ ............... ...... 342
11.3. Sliding Filam ent Theory ......... ... ........ ........ .............. . 348
11.4. Excitation-Contratonn .... . .......... ............... .... ....... .......... 353
11.5. N otes........................................................ ......... .. . . . . ... .....354
12: Functional Electrical Stimulation.. ............. 355
12.1. Introduction . .. .... ........... . . .. ................ ....... .......... . 355
12.2. Electrode Considerations............... 355
12.3. Operating Outside Reversible Region ............ ... ..................... 362
12.4. Electrode M aterials ..... .......... ........ .... .............. .. ..... . . 363
12.5. N erve Excitation .. .. ............... ... . ..... ............. .. . 365
12.6. Stimulating Electrode Types .. .... . .... . . . ........ ......... 372
12.7. Analysis of Electrode Performance ........ .............. ........ 374
12.8. Clinical Applications .......................... ... ........... .. . ..... . 385
12.9. Frankenhaeuser-Huxley Membrane ......... . .... .... ..... 386
12.10. FES Outlook o.................. . ....... .. ............................. .. 387
12.1 1. N otes ..... ......... ................... .................... ....... ... 388
13 : E xercises . ........................................ ......... ......... ..... 39 1
13.. Exercises, Chapter 1: Vector Exercises.. ............... ...... ... .. .... 391
13.2. Exercises, Chapter 2: Sources and Fields .............. .. 397
13.3. Exercises, Chapter 3: Bioelectric Potentials . . . . . . . . . . . ... . .. .. .. . . . . . 406
13 . Exercises Chapter 4: Channels ..... ............... ....... 417
135. Exercises, Chapter : Action Potentials .............. ... .......... 427
13.6. Exercises, Chapter 6: Impulse Propagation.................................. 449
13.7. Exercises, Chapter 7: Electrical Stimulation of Excitable Tissue ..... ......... 462
13.8. Exercises, Chapter 8: Extracellular Fields ............. ............. ......... 478
13.9. Exercises, Chapter 9: Cardiac Electrophysiology . ... . . . . . . . . . . . .. . . .... 491
13.10. Exercises, Chapter 10: The Neuromuscular Junction . .. . .. .............. 514
13.11. Exercises, Chapter 11: Skeletal M uscle .... ....... ... ..... . ... ...... ..... 516
13.12. Exercises, Chapter 12: Functional Electrical Stimulation .......... - 517