2493:
1012:
32:
577:
2258:
646:
1287:
1962:
3061:
221:
2488:{\displaystyle {\begin{aligned}E_{X}&\approx {\frac {0.0267\ \mathrm {V} }{z}}\ln {\frac {X_{o}}{X_{i}}}\\&={\frac {26.7\ \mathrm {mV} }{z}}\ln {\frac {X_{o}}{X_{i}}}\\&\approx {\frac {61.5\ \mathrm {mV} }{z}}\log {\frac {X_{o}}{X_{i}}}&{\text{ since }}\ln 10\approx 2.303\end{aligned}}}
1007:{\displaystyle E_{m,\mathrm {K} _{x}\mathrm {\text{Na}} _{1-x}\mathrm {Cl} }={\frac {RT}{F}}\ln {\left({\frac {P_{\text{Na}}_{\mathrm {out} }+P_{\text{K}}_{\mathrm {out} }+P_{\text{Cl}}_{\mathrm {in} }}{P_{\text{Na}}_{\mathrm {in} }+P_{\text{K}}_{\mathrm {in} }+P_{\text{Cl}}_{\mathrm {out} }}}\right)}}
3272:
Current density has units of (Amperes/m). Molar flux has units of (mol/(s m)). Thus, to get current density from molar flux one needs to multiply by
Faraday's constant F (Coulombs/mol). F will then cancel from the equation below. Since the valence has already been accounted for above, the charge
3868:
3679:
1967:
The ionic charge determines the sign of the membrane potential contribution. During an action potential, although the membrane potential changes about 100mV, the concentrations of ions inside and outside the cell do not change significantly. They are always very close to their respective
1027:
2754:
1735:
3280:
There is such a current associated with every type of ion that can cross the membrane; this is because each type of ion would require a distinct membrane potential to balance diffusion, but there can only be one membrane potential. By assumption, at the
Goldman voltage
2893:
2109:
572:{\displaystyle E_{m}={\frac {RT}{F}}\ln {\left({\frac {\sum _{i}^{n}P_{M_{i}^{+}}_{\mathrm {out} }+\sum _{j}^{m}P_{A_{j}^{-}}_{\mathrm {in} }}{\sum _{i}^{n}P_{M_{i}^{+}}_{\mathrm {in} }+\sum _{j}^{m}P_{A_{j}^{-}}_{\mathrm {out} }}}\right)}}
3362:, not considered here which serve to balance each individual ion's current, so that the ion concentrations on either side of the membrane do not change over time in equilibrium.) If all the ions are monovalent—that is, if all the
2035:
3687:
3498:
1282:{\displaystyle E_{m,{\text{Na}}}={\frac {RT}{F}}\ln {\left({\frac {P_{\text{Na}}_{\mathrm {out} }}{P_{\text{Na}}_{\mathrm {in} }}}\right)}={\frac {RT}{F}}\ln {\left({\frac {_{\mathrm {out} }}{_{\mathrm {in} }}}\right)}}
1727:
2610:
2247:
638:
3181:
2164:
1957:{\displaystyle E_{X}=61.5\,\mathrm {mV} \cdot \log {\left({\frac {_{\mathrm {out} }}{_{\mathrm {in} }}}\right)}=-61.5\,\mathrm {mV} \cdot \log {\left({\frac {_{\mathrm {out} }}{_{\mathrm {in} }}}\right)}}
3487:
3267:
2263:
1677:
3117:
3353:
1366:
1417:
3056:{\displaystyle j_{\mathrm {A} }=\mu n_{\mathrm {A} }P_{\mathrm {A} }{\frac {\left_{\mathrm {out} }-\left_{\mathrm {in} }e^{n_{\mathrm {A} }\mu }}{1-e^{n_{\mathrm {A} }\mu }}}}
1473:
1586:
3412:
1588:
is approximately 26.7 mV at human body temperature (37 °C); when factoring in the change-of-base formula between the natural logarithm, ln, and logarithm with base 10
2040:
1317:
2566:
According to
Goldman's model, only two factors influence the motion of ions across a permeable membrane: the average electric field and the difference in ionic
1547:
1525:
1495:
213:
189:
1979:
3863:{\displaystyle v=\sum _{\mathrm {cations\ C} }P_{\mathrm {C} }\left_{\mathrm {in} }+\sum _{\mathrm {anions\ A} }P_{\mathrm {A} }\left_{\mathrm {out} }}
3674:{\displaystyle w=\sum _{\mathrm {cations\ C} }P_{\mathrm {C} }\left_{\mathrm {out} }+\sum _{\mathrm {anions\ A} }P_{\mathrm {A} }\left_{\mathrm {in} }}
2786:
due to the electric field, which increases linearly with the electric field; Formally, it is multiplied by the drift velocity of the ions, with the
2570:
from one side of the membrane to the other. The electric field is assumed to be constant across the membrane, so that it can be set equal to
2749:{\displaystyle j_{\mathrm {A} }=-D_{\mathrm {A} }\left({\frac {d\left}{dz}}-{\frac {n_{\mathrm {A} }F}{RT}}{\frac {E_{m}}{L}}\left\right)}
1682:
20:
2172:
585:
3135:
2114:
3423:
96:
4192:
The membrane voltage is calculated interactively as the number of ions are changed between the inside and outside of the cell.
4175:
4110:
4061:
4019:
3977:
68:
3218:
75:
4215:
2791:
3277:
of each ion in the equation above, therefore, should be interpreted as +1 or -1 depending on the polarity of the ion.
4148:
115:
49:
82:
3358:(Although the current for each ion type considered here is nonzero, there are other pumps in the membrane, e.g.
1591:
3072:
2844:
53:
3296:
64:
2604:—in other words, the number of ions crossing per time and per area of the membrane—is given by the formula
4210:
2523:
4184:
4090:
4041:
3999:
3957:
1336:
2760:
1373:
144:
across a cell's membrane, taking into account all of the ions that are permeant through that membrane.
1432:
1559:
3922:
3917:
3375:
2104:{\displaystyle F\approx {\frac {9.6\times 10^{4}\ \mathrm {J} }{\mathrm {mol} \cdot \mathrm {V} }}}
42:
3956:
Enderle, John (2005-01-01), Enderle, John D.; Blanchard, Susan M.; Bronzino, Joseph D. (eds.),
3927:
2530:
direction being perpendicular to the membrane. Assuming that the system is symmetrical in the
2166:
and the fact that one volt is equal to one joule of energy per coulomb of charge, the equation
89:
4140:
4133:
3937:
3912:
1295:
8:
2802:
152:
4195:
4102:
4053:
3969:
2779:
1532:
1510:
1498:
1480:
198:
174:
156:
2030:{\displaystyle R\approx {\frac {8.3\ \mathrm {J} }{\mathrm {K} \cdot \mathrm {mol} }}}
4144:
4106:
4057:
4015:
4011:
3973:
3890:
141:
4172:
4098:
4097:, Woodhead Publishing Series in Biomaterials, Woodhead Publishing, pp. 13–43,
4049:
4007:
3965:
3882:
3187:
1550:
148:
4179:
3932:
3190:
2799:
2795:
2499:
1018:
3902:
2830:
2787:
2560:
4204:
4040:
Enderle, John D. (2012-01-01), Enderle, John D.; Bronzino, Joseph D. (eds.),
4000:"Chapter 2 – Mechanisms of Ion Transport Across Cell Membranes and Epithelia"
3889:; in this case, the formula for the Goldman equation can be solved using the
2768:
2567:
4189:
3907:
2823:
160:
2587:
is the thickness of the membrane. For a given ion denoted A with valence
4169:
Includes a well-explained derivation of the
Goldman-Hodgkin-Katz equation
2812:
1420:
4139:(2nd ed.). Sunderland, Massachusetts: Sinauer Associates. pp.
1475:= the intracellular concentration of that ion (in moles per cubic meter)
3998:
Reuss, Luis (2008-01-01), Alpern, Robert J.; Hebert, Steven C. (eds.),
582:
This results in the following if we consider a membrane separating two
137:
4196:
Potential, Impedance, and
Rectification in Membranes by Goldman (1943)
3359:
2764:
4048:, Biomedical Engineering, Boston: Academic Press, pp. 747–815,
3964:, Biomedical Engineering, Boston: Academic Press, pp. 627–691,
31:
4166:
2837:
2552:
2811:
of the ion A (e.g., +1 for K, +2 for Ca and −1 for Cl), the
1722:{\displaystyle 26.7\,\mathrm {mV} \cdot 2.303=61.5\,\mathrm {mV} }
3874:
2511:
1328:
2242:{\displaystyle E_{X}={\frac {RT}{zF}}\ln {\frac {X_{o}}{X_{i}}}}
1968:
concentrations when the membrane is at their resting potential.
633:{\displaystyle \mathrm {K} _{x}\mathrm {Na} _{1-x}\mathrm {Cl} }
16:
Generalization of the Nernst equation for the membrane potential
2819:
2538:
directions (around and along the axon, respectively), only the
1502:
3176:{\displaystyle P_{\mathrm {A} }={\frac {D_{\mathrm {A} }}{L}}}
2771:
gradient, i.e., from high to low concentration. The constant
2159:{\displaystyle T=37\ ^{\circ }\mathrm {C} =310\ \mathrm {K} }
1324:
3482:{\displaystyle {\frac {FE_{m}}{RT}}=\mu =\ln {\frac {w}{v}}}
2783:
2595:
1320:
192:
4091:"2 - Physiological principles of electrical stimulation"
3962:
Introduction to
Biomedical Engineering (Second Edition)
1424:
4046:
Introduction to
Biomedical Engineering (Third Edition)
3262:{\displaystyle J_{A}=q_{\mathrm {A} }j_{\mathrm {A} }}
4089:
Bhadra, Narendra (2015-01-01), Kilgore, Kevin (ed.),
3690:
3501:
3426:
3378:
3299:
3221:
3138:
3075:
2896:
2613:
2261:
2175:
2117:
2043:
1982:
1738:
1685:
1594:
1562:
1535:
1513:
1483:
1435:
1376:
1368:= the selectivity for that ion (in meters per second)
1339:
1298:
1030:
649:
588:
224:
201:
177:
3369:equal either +1 or -1—this equation can be written
56:. Unsourced material may be challenged and removed.
4132:
4095:Implantable Neuroprostheses for Restoring Function
3862:
3673:
3481:
3406:
3347:
3261:
3175:
3111:
3055:
2748:
2487:
2241:
2158:
2103:
2029:
1956:
1721:
1671:
1580:
1541:
1519:
1489:
1467:
1419:= the extracellular concentration of that ion (in
1411:
1360:
1311:
1281:
1006:
632:
571:
207:
183:
4004:Seldin and Giebisch's The Kidney (Fourth Edition)
4202:
2542:direction need be considered; thus, the voltage
166:
4006:, San Diego: Academic Press, pp. 35–56,
1971:
1021:-like" but has a term for each permeant ion:
2782:of the ion A. The second term reflects the
3129:is the ionic permeability, defined here as
2510:Goldman's equation seeks to determine the
1672:{\displaystyle (^{-1}=\ln(10)=2.30258...)}
3112:{\displaystyle \mu ={\frac {FE_{m}}{RT}}}
2836:, which is the total charge of a mole of
2526:is used to describe the system, with the
1859:
1755:
1710:
1689:
116:Learn how and when to remove this message
4185:Goldman-Hodgkin-Katz Equation Calculator
3348:{\displaystyle J_{tot}=\sum _{A}J_{A}=0}
4130:
4039:
3955:
3417:whose solution is the Goldman equation
4203:
4190:Nernst/Goldman interactive Java applet
4088:
1729:, a value often used in neuroscience.
3997:
1423:per cubic meter, to match the other
130:Goldman–Hodgkin–Katz voltage equation
4084:
4082:
4080:
4078:
4042:"Chapter 12 – Bioelectric Phenomena"
3290:, the total current density is zero
54:adding citations to reliable sources
25:
2875:with the boundary conditions (0) =
155:, and the Medicine Nobel laureates
13:
4103:10.1016/b978-1-78242-101-6.00002-1
4054:10.1016/b978-0-12-374979-6.00012-5
3970:10.1016/b978-0-12-238662-6.50013-6
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3253:
3241:
3206:of the ion multiplied by the flux
3162:
3145:
3066:where ÎĽ is a dimensionless number
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1381:
1361:{\displaystyle P_{\mathrm {ion} }}
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404:
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331:
328:
21:Goldman–Hodgkin–Katz flux equation
14:
4227:
4173:Nernst/Goldman Equation Simulator
4160:
4075:
1412:{\displaystyle _{\mathrm {out} }}
4012:10.1016/b978-012088488-9.50005-x
1468:{\displaystyle _{\mathrm {in} }}
30:
4167:Subthreshold membrane phenomena
1581:{\displaystyle {\frac {RT}{F}}}
41:needs additional citations for
4124:
4033:
3991:
3949:
3877:are considered, terms such as
2863:; integrating both sides from
2763:, which gives the flux due to
2759:The first term corresponds to
2111:, (assuming body temperature)
1932:
1918:
1899:
1885:
1828:
1814:
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1620:
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1206:
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759:
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544:
525:
471:
452:
399:
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323:
304:
1:
3943:
3407:{\displaystyle w-ve^{\mu }=0}
2798:. The constants here are the
2505:
1319:= the membrane potential (in
171:The GHK voltage equation for
1527:= the temperature in kelvins
167:Equation for monovalent ions
147:The discoverers of this are
7:
4135:Nerve and Muscle Excitation
3896:
2524:Cartesian coordinate system
136:, is used in cell membrane
10:
4232:
1972:Calculating the first term
18:
4216:Electrochemical equations
3873:If divalent ions such as
2887:, one gets the solution
2796:electrophoretic mobility
2792:Stokes–Einstein relation
19:Not to be confused with
3958:"Bioelectric Phenomena"
2761:Fick's law of diffusion
132:, sometimes called the
3864:
3675:
3483:
3408:
3349:
3263:
3177:
3113:
3057:
2843:This is a first-order
2750:
2522:across a membrane. A
2489:
2243:
2160:
2105:
2031:
1958:
1723:
1673:
1582:
1543:
1521:
1491:
1469:
1413:
1362:
1313:
1283:
1008:
634:
573:
502:
429:
357:
281:
209:
185:
3881:appear, which is the
3865:
3676:
3484:
3409:
3350:
3264:
3178:
3114:
3058:
2751:
2563:across the membrane.
2490:
2244:
2161:
2106:
2032:
1959:
1724:
1674:
1583:
1544:
1522:
1492:
1470:
1414:
1363:
1314:
1312:{\displaystyle E_{m}}
1284:
1009:
635:
574:
488:
415:
343:
267:
210:
186:
3938:Saltatory conduction
3923:Hodgkin–Huxley model
3918:Hindmarsh–Rose model
3913:GHK current equation
3688:
3499:
3424:
3376:
3297:
3219:
3136:
3073:
2894:
2790:expressed using the
2611:
2259:
2173:
2115:
2041:
1980:
1736:
1683:
1592:
1560:
1533:
1511:
1481:
1433:
1374:
1337:
1296:
1028:
647:
586:
222:
199:
191:monovalent positive
175:
50:improve this article
1553:(coulombs per mole)
542:
522:
469:
449:
397:
377:
321:
301:
153:Columbia University
4211:Physical chemistry
4178:2010-08-08 at the
3928:Morris–Lecar model
3860:
3812:
3732:
3671:
3626:
3543:
3479:
3404:
3345:
3328:
3259:
3199:equals the charge
3173:
3109:
3053:
2780:diffusion constant
2746:
2485:
2483:
2252:can be reduced to
2239:
2156:
2101:
2027:
1954:
1719:
1669:
1578:
1551:Faraday's constant
1539:
1517:
1499:ideal gas constant
1487:
1465:
1409:
1358:
1309:
1279:
1004:
630:
569:
528:
508:
455:
435:
383:
363:
307:
287:
205:
181:
157:Alan Lloyd Hodgkin
65:"Goldman equation"
4112:978-1-78242-101-6
4063:978-0-12-374979-6
4021:978-0-12-088488-9
3979:978-0-12-238662-6
3891:quadratic formula
3806:
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3726:
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3620:
3594:
3537:
3508:
3477:
3452:
3319:
3171:
3107:
3051:
2726:
2709:
2677:
2559:component of the
2464:
2463: since
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2414:
2394:
2364:
2351:
2331:
2301:
2291:
2237:
2207:
2150:
2130:
2099:
2071:
2025:
1997:
1947:
1843:
1576:
1542:{\displaystyle F}
1520:{\displaystyle T}
1490:{\displaystyle R}
1272:
1248:
1213:
1190:
1167:
1143:
1133:
1098:
1088:
1065:
1045:
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842:
832:
796:
786:
750:
740:
717:
677:
562:
251:
208:{\displaystyle A}
184:{\displaystyle M}
142:Resting potential
140:to determine the
126:
125:
118:
100:
4223:
4155:
4154:
4138:
4131:Junge D (1981).
4128:
4122:
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3188:electric current
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2165:
2163:
2162:
2157:
2155:
2148:
2141:
2136:
2135:
2128:
2110:
2108:
2107:
2102:
2100:
2098:
2097:
2089:
2077:
2076:
2069:
2068:
2067:
2051:
2036:
2034:
2033:
2028:
2026:
2024:
2023:
2009:
2003:
2002:
1995:
1990:
1963:
1961:
1960:
1955:
1953:
1952:
1948:
1946:
1945:
1944:
1943:
1930:
1929:
1916:
1915:
1914:
1913:
1897:
1896:
1883:
1867:
1849:
1848:
1844:
1842:
1841:
1840:
1839:
1826:
1825:
1812:
1811:
1810:
1809:
1793:
1792:
1779:
1763:
1748:
1747:
1728:
1726:
1725:
1720:
1718:
1697:
1678:
1676:
1675:
1670:
1641:
1640:
1610:
1609:
1587:
1585:
1584:
1579:
1577:
1572:
1564:
1548:
1546:
1545:
1540:
1526:
1524:
1523:
1518:
1496:
1494:
1493:
1488:
1474:
1472:
1471:
1466:
1464:
1463:
1462:
1449:
1418:
1416:
1415:
1410:
1408:
1407:
1406:
1390:
1367:
1365:
1364:
1359:
1357:
1356:
1355:
1323:, equivalent to
1318:
1316:
1315:
1310:
1308:
1307:
1288:
1286:
1285:
1280:
1278:
1277:
1273:
1271:
1270:
1269:
1268:
1255:
1254:
1249:
1246:
1239:
1238:
1237:
1236:
1220:
1219:
1214:
1211:
1204:
1191:
1186:
1178:
1173:
1172:
1168:
1166:
1165:
1164:
1163:
1150:
1149:
1144:
1141:
1135:
1134:
1131:
1124:
1123:
1122:
1121:
1105:
1104:
1099:
1096:
1090:
1089:
1086:
1079:
1066:
1061:
1053:
1048:
1047:
1046:
1043:
1013:
1011:
1010:
1005:
1003:
1002:
998:
996:
995:
994:
993:
977:
976:
971:
968:
962:
961:
958:
949:
948:
947:
934:
933:
928:
925:
919:
918:
915:
906:
905:
904:
891:
890:
885:
882:
876:
875:
872:
865:
864:
863:
862:
849:
848:
843:
840:
834:
833:
830:
821:
820:
819:
803:
802:
797:
794:
788:
787:
784:
775:
774:
773:
757:
756:
751:
748:
742:
741:
738:
731:
718:
713:
705:
700:
699:
698:
690:
689:
678:
675:
672:
671:
666:
639:
637:
636:
631:
629:
621:
620:
609:
600:
599:
594:
578:
576:
575:
570:
568:
567:
563:
561:
560:
559:
558:
541:
536:
524:
523:
521:
516:
501:
496:
484:
483:
482:
468:
463:
451:
450:
448:
443:
428:
423:
413:
412:
411:
410:
396:
391:
379:
378:
376:
371:
356:
351:
339:
338:
337:
320:
315:
303:
302:
300:
295:
280:
275:
265:
252:
247:
239:
234:
233:
214:
212:
211:
206:
190:
188:
187:
182:
149:David E. Goldman
134:Goldman equation
121:
114:
110:
107:
101:
99:
58:
34:
26:
4231:
4230:
4226:
4225:
4224:
4222:
4221:
4220:
4201:
4200:
4180:Wayback Machine
4163:
4158:
4151:
4129:
4125:
4117:
4115:
4113:
4087:
4076:
4068:
4066:
4064:
4038:
4034:
4026:
4024:
4022:
3996:
3992:
3984:
3982:
3980:
3954:
3950:
3946:
3933:Nernst equation
3899:
3847:
3846:
3836:
3831:
3830:
3826:
3825:
3818:
3817:
3813:
3785:
3784:
3767:
3766:
3756:
3751:
3750:
3746:
3745:
3738:
3737:
3733:
3702:
3701:
3689:
3686:
3685:
3661:
3660:
3650:
3645:
3644:
3640:
3639:
3632:
3631:
3627:
3599:
3598:
3578:
3577:
3567:
3562:
3561:
3557:
3556:
3549:
3548:
3544:
3513:
3512:
3500:
3497:
3496:
3469:
3444:
3437:
3433:
3429:
3427:
3425:
3422:
3421:
3392:
3388:
3377:
3374:
3373:
3368:
3333:
3329:
3323:
3304:
3300:
3298:
3295:
3294:
3289:
3276:
3252:
3251:
3247:
3240:
3239:
3235:
3226:
3222:
3220:
3217:
3216:
3212:
3205:
3198:
3161:
3160:
3156:
3154:
3144:
3143:
3139:
3137:
3134:
3133:
3128:
3099:
3092:
3088:
3084:
3082:
3074:
3071:
3070:
3038:
3037:
3033:
3032:
3028:
3021:
3008:
3007:
3003:
3002:
2998:
2988:
2987:
2978:
2974:
2973:
2957:
2956:
2947:
2943:
2942:
2941:
2939:
2932:
2931:
2927:
2920:
2919:
2915:
2902:
2901:
2897:
2895:
2892:
2891:
2886:
2878:
2810:
2777:
2732:
2728:
2717:
2713:
2711:
2701:
2690:
2689:
2685:
2684:
2682:
2669:
2659:
2655:
2651:
2649:
2648:
2644:
2637:
2636:
2632:
2619:
2618:
2614:
2612:
2609:
2608:
2603:
2593:
2578:
2550:
2521:
2508:
2500:Nernst equation
2482:
2481:
2461:
2459:
2451:
2447:
2441:
2437:
2435:
2415:
2408:
2406:
2397:
2396:
2388:
2384:
2378:
2374:
2372:
2352:
2345:
2343:
2334:
2333:
2325:
2321:
2315:
2311:
2309:
2292:
2285:
2283:
2276:
2270:
2266:
2262:
2260:
2257:
2256:
2231:
2227:
2221:
2217:
2215:
2199:
2191:
2189:
2180:
2176:
2174:
2171:
2170:
2151:
2137:
2131:
2127:
2116:
2113:
2112:
2093:
2079:
2078:
2072:
2063:
2059:
2052:
2050:
2042:
2039:
2038:
2013:
2005:
2004:
1998:
1991:
1989:
1981:
1978:
1977:
1974:
1936:
1935:
1931:
1925:
1921:
1917:
1903:
1902:
1898:
1892:
1888:
1884:
1882:
1878:
1877:
1860:
1832:
1831:
1827:
1821:
1817:
1813:
1799:
1798:
1794:
1788:
1784:
1780:
1778:
1774:
1773:
1756:
1743:
1739:
1737:
1734:
1733:
1711:
1690:
1684:
1681:
1680:
1633:
1629:
1605:
1601:
1593:
1590:
1589:
1565:
1563:
1561:
1558:
1557:
1534:
1531:
1530:
1512:
1509:
1508:
1482:
1479:
1478:
1455:
1454:
1450:
1439:
1434:
1431:
1430:
1396:
1395:
1391:
1380:
1375:
1372:
1371:
1345:
1344:
1340:
1338:
1335:
1334:
1303:
1299:
1297:
1294:
1293:
1261:
1260:
1256:
1250:
1245:
1244:
1240:
1226:
1225:
1221:
1215:
1210:
1209:
1205:
1203:
1199:
1198:
1179:
1177:
1156:
1155:
1151:
1145:
1140:
1139:
1130:
1126:
1125:
1111:
1110:
1106:
1100:
1095:
1094:
1085:
1081:
1080:
1078:
1074:
1073:
1054:
1052:
1042:
1035:
1031:
1029:
1026:
1025:
983:
982:
978:
972:
967:
966:
957:
953:
940:
939:
935:
929:
924:
923:
914:
910:
897:
896:
892:
886:
881:
880:
871:
867:
866:
855:
854:
850:
844:
839:
838:
829:
825:
809:
808:
804:
798:
793:
792:
783:
779:
763:
762:
758:
752:
747:
746:
737:
733:
732:
730:
726:
725:
706:
704:
691:
679:
674:
673:
667:
662:
661:
654:
650:
648:
645:
644:
622:
610:
602:
601:
595:
590:
589:
587:
584:
583:
548:
547:
543:
537:
532:
517:
512:
507:
503:
497:
492:
475:
474:
470:
464:
459:
444:
439:
434:
430:
424:
419:
414:
403:
402:
398:
392:
387:
372:
367:
362:
358:
352:
347:
327:
326:
322:
316:
311:
296:
291:
286:
282:
276:
271:
266:
264:
260:
259:
240:
238:
229:
225:
223:
220:
219:
200:
197:
196:
176:
173:
172:
169:
122:
111:
105:
102:
59:
57:
47:
35:
24:
17:
12:
11:
5:
4229:
4219:
4218:
4213:
4199:
4198:
4193:
4187:
4182:
4170:
4162:
4161:External links
4159:
4157:
4156:
4149:
4123:
4111:
4074:
4062:
4032:
4020:
3990:
3978:
3947:
3945:
3942:
3941:
3940:
3935:
3930:
3925:
3920:
3915:
3910:
3905:
3903:Bioelectronics
3898:
3895:
3871:
3870:
3856:
3853:
3850:
3844:
3839:
3834:
3829:
3821:
3816:
3809:
3803:
3800:
3797:
3794:
3791:
3788:
3783:
3779:
3773:
3770:
3764:
3759:
3754:
3749:
3741:
3736:
3729:
3723:
3720:
3717:
3714:
3711:
3708:
3705:
3700:
3696:
3693:
3682:
3681:
3667:
3664:
3658:
3653:
3648:
3643:
3635:
3630:
3623:
3617:
3614:
3611:
3608:
3605:
3602:
3597:
3593:
3587:
3584:
3581:
3575:
3570:
3565:
3560:
3552:
3547:
3540:
3534:
3531:
3528:
3525:
3522:
3519:
3516:
3511:
3507:
3504:
3490:
3489:
3476:
3473:
3468:
3465:
3462:
3459:
3456:
3450:
3447:
3440:
3436:
3432:
3415:
3414:
3403:
3400:
3395:
3391:
3387:
3384:
3381:
3366:
3356:
3355:
3344:
3341:
3336:
3332:
3326:
3322:
3318:
3313:
3310:
3307:
3303:
3285:
3274:
3270:
3269:
3255:
3250:
3243:
3238:
3234:
3229:
3225:
3210:
3203:
3196:
3184:
3183:
3170:
3164:
3159:
3153:
3147:
3142:
3126:
3120:
3119:
3105:
3102:
3095:
3091:
3087:
3081:
3078:
3064:
3063:
3047:
3041:
3036:
3031:
3027:
3024:
3017:
3011:
3006:
3001:
2994:
2991:
2985:
2981:
2977:
2972:
2966:
2963:
2960:
2954:
2950:
2946:
2935:
2930:
2923:
2918:
2914:
2911:
2905:
2900:
2884:
2876:
2808:
2788:drift velocity
2775:
2757:
2756:
2744:
2739:
2735:
2731:
2725:
2720:
2716:
2707:
2704:
2699:
2693:
2688:
2681:
2675:
2672:
2666:
2662:
2658:
2654:
2647:
2640:
2635:
2631:
2628:
2622:
2617:
2601:
2591:
2574:
2561:electric field
2546:
2517:
2507:
2504:
2496:
2495:
2480:
2477:
2474:
2471:
2468:
2460:
2454:
2450:
2444:
2440:
2434:
2431:
2426:
2421:
2418:
2411:
2405:
2402:
2400:
2398:
2391:
2387:
2381:
2377:
2371:
2368:
2363:
2358:
2355:
2348:
2342:
2339:
2337:
2335:
2328:
2324:
2318:
2314:
2308:
2305:
2300:
2295:
2288:
2282:
2279:
2277:
2273:
2269:
2265:
2264:
2250:
2249:
2234:
2230:
2224:
2220:
2214:
2211:
2205:
2202:
2197:
2194:
2188:
2183:
2179:
2154:
2147:
2144:
2140:
2134:
2126:
2123:
2120:
2096:
2092:
2088:
2085:
2082:
2075:
2066:
2062:
2058:
2055:
2049:
2046:
2022:
2019:
2016:
2012:
2008:
2001:
1994:
1988:
1985:
1973:
1970:
1965:
1964:
1951:
1942:
1939:
1934:
1928:
1924:
1920:
1912:
1909:
1906:
1901:
1895:
1891:
1887:
1881:
1876:
1873:
1870:
1866:
1863:
1858:
1855:
1852:
1847:
1838:
1835:
1830:
1824:
1820:
1816:
1808:
1805:
1802:
1797:
1791:
1787:
1783:
1777:
1772:
1769:
1766:
1762:
1759:
1754:
1751:
1746:
1742:
1717:
1714:
1709:
1706:
1703:
1700:
1696:
1693:
1688:
1668:
1665:
1662:
1659:
1656:
1653:
1650:
1647:
1644:
1639:
1636:
1632:
1628:
1625:
1622:
1619:
1616:
1613:
1608:
1604:
1600:
1597:
1575:
1571:
1568:
1555:
1554:
1538:
1528:
1516:
1506:
1486:
1476:
1461:
1458:
1453:
1448:
1445:
1442:
1438:
1428:
1405:
1402:
1399:
1394:
1389:
1386:
1383:
1379:
1369:
1354:
1351:
1348:
1343:
1332:
1306:
1302:
1290:
1289:
1276:
1267:
1264:
1259:
1253:
1243:
1235:
1232:
1229:
1224:
1218:
1208:
1202:
1197:
1194:
1189:
1185:
1182:
1176:
1171:
1162:
1159:
1154:
1148:
1138:
1129:
1120:
1117:
1114:
1109:
1103:
1093:
1084:
1077:
1072:
1069:
1064:
1060:
1057:
1051:
1041:
1038:
1034:
1015:
1014:
1001:
992:
989:
986:
981:
975:
965:
956:
952:
946:
943:
938:
932:
922:
913:
909:
903:
900:
895:
889:
879:
870:
861:
858:
853:
847:
837:
828:
824:
818:
815:
812:
807:
801:
791:
782:
778:
772:
769:
766:
761:
755:
745:
736:
729:
724:
721:
716:
712:
709:
703:
697:
694:
688:
685:
682:
670:
665:
660:
657:
653:
628:
625:
619:
616:
613:
608:
605:
598:
593:
580:
579:
566:
557:
554:
551:
546:
540:
535:
531:
527:
520:
515:
511:
506:
500:
495:
491:
487:
481:
478:
473:
467:
462:
458:
454:
447:
442:
438:
433:
427:
422:
418:
409:
406:
401:
395:
390:
386:
382:
375:
370:
366:
361:
355:
350:
346:
342:
336:
333:
330:
325:
319:
314:
310:
306:
299:
294:
290:
285:
279:
274:
270:
263:
258:
255:
250:
246:
243:
237:
232:
228:
204:
180:
168:
165:
124:
123:
38:
36:
29:
15:
9:
6:
4:
3:
2:
4228:
4217:
4214:
4212:
4209:
4208:
4206:
4197:
4194:
4191:
4188:
4186:
4183:
4181:
4177:
4174:
4171:
4168:
4165:
4164:
4152:
4150:0-87893-410-3
4146:
4142:
4137:
4136:
4127:
4114:
4108:
4104:
4100:
4096:
4092:
4085:
4083:
4081:
4079:
4065:
4059:
4055:
4051:
4047:
4043:
4036:
4023:
4017:
4013:
4009:
4005:
4001:
3994:
3981:
3975:
3971:
3967:
3963:
3959:
3952:
3948:
3939:
3936:
3934:
3931:
3929:
3926:
3924:
3921:
3919:
3916:
3914:
3911:
3909:
3906:
3904:
3901:
3900:
3894:
3892:
3888:
3884:
3880:
3876:
3842:
3837:
3827:
3814:
3781:
3777:
3762:
3757:
3747:
3734:
3698:
3694:
3691:
3684:
3683:
3656:
3651:
3641:
3628:
3595:
3591:
3573:
3568:
3558:
3545:
3509:
3505:
3502:
3495:
3494:
3493:
3474:
3471:
3466:
3463:
3460:
3457:
3454:
3448:
3445:
3438:
3434:
3430:
3420:
3419:
3418:
3401:
3398:
3393:
3389:
3385:
3382:
3379:
3372:
3371:
3370:
3365:
3361:
3342:
3339:
3334:
3330:
3324:
3320:
3316:
3311:
3308:
3305:
3301:
3293:
3292:
3291:
3288:
3284:
3278:
3248:
3236:
3232:
3227:
3223:
3215:
3214:
3213:
3209:
3202:
3195:
3192:
3189:
3168:
3157:
3151:
3140:
3132:
3131:
3130:
3125:
3103:
3100:
3093:
3089:
3085:
3079:
3076:
3069:
3068:
3067:
3045:
3034:
3029:
3025:
3022:
3015:
3004:
2999:
2983:
2975:
2970:
2952:
2944:
2928:
2916:
2912:
2909:
2898:
2890:
2889:
2888:
2882:
2874:
2870:
2866:
2862:
2858:
2854:
2850:
2846:
2841:
2839:
2835:
2832:
2828:
2825:
2822:), the molar
2821:
2817:
2814:
2807:
2804:
2801:
2797:
2793:
2789:
2785:
2781:
2774:
2770:
2769:concentration
2766:
2762:
2742:
2737:
2729:
2723:
2718:
2714:
2705:
2702:
2697:
2686:
2679:
2673:
2670:
2664:
2656:
2652:
2645:
2633:
2629:
2626:
2615:
2607:
2606:
2605:
2600:
2597:
2590:
2586:
2582:
2577:
2573:
2569:
2568:concentration
2564:
2562:
2558:
2554:
2549:
2545:
2541:
2537:
2533:
2529:
2525:
2520:
2516:
2513:
2503:
2501:
2498:which is the
2478:
2475:
2472:
2469:
2466:
2452:
2448:
2442:
2438:
2432:
2429:
2424:
2409:
2403:
2401:
2389:
2385:
2379:
2375:
2369:
2366:
2361:
2346:
2340:
2338:
2326:
2322:
2316:
2312:
2306:
2303:
2298:
2286:
2280:
2278:
2271:
2267:
2255:
2254:
2253:
2232:
2228:
2222:
2218:
2212:
2209:
2203:
2200:
2195:
2192:
2186:
2181:
2177:
2169:
2168:
2167:
2145:
2142:
2132:
2124:
2121:
2118:
2090:
2064:
2060:
2056:
2053:
2047:
2044:
2010:
1992:
1986:
1983:
1969:
1949:
1926:
1922:
1893:
1889:
1879:
1874:
1871:
1868:
1856:
1853:
1850:
1845:
1822:
1818:
1789:
1785:
1775:
1770:
1767:
1764:
1752:
1749:
1744:
1740:
1732:
1731:
1730:
1707:
1704:
1701:
1698:
1686:
1679:, it becomes
1663:
1660:
1654:
1648:
1645:
1642:
1637:
1634:
1623:
1617:
1614:
1611:
1606:
1602:
1573:
1569:
1566:
1552:
1536:
1529:
1514:
1507:
1504:
1500:
1484:
1477:
1429:
1426:
1422:
1370:
1341:
1333:
1330:
1326:
1322:
1304:
1300:
1292:
1291:
1274:
1251:
1216:
1200:
1195:
1192:
1187:
1183:
1180:
1174:
1169:
1146:
1127:
1101:
1082:
1075:
1070:
1067:
1062:
1058:
1055:
1049:
1039:
1036:
1032:
1024:
1023:
1022:
1020:
999:
973:
954:
950:
930:
911:
907:
887:
868:
845:
826:
822:
799:
780:
776:
753:
734:
727:
722:
719:
714:
710:
707:
701:
686:
683:
680:
668:
658:
655:
651:
643:
642:
641:
617:
614:
611:
596:
564:
538:
533:
529:
518:
513:
509:
504:
498:
493:
489:
485:
465:
460:
456:
445:
440:
436:
431:
425:
420:
416:
393:
388:
384:
373:
368:
364:
359:
353:
348:
344:
340:
317:
312:
308:
297:
292:
288:
283:
277:
272:
268:
261:
256:
253:
248:
244:
241:
235:
230:
226:
218:
217:
216:
202:
194:
178:
164:
162:
158:
154:
150:
145:
143:
139:
135:
131:
120:
117:
109:
98:
95:
91:
88:
84:
81:
77:
74:
70:
67: –
66:
62:
61:Find sources:
55:
51:
45:
44:
39:This article
37:
33:
28:
27:
22:
4134:
4126:
4116:, retrieved
4094:
4067:, retrieved
4045:
4035:
4025:, retrieved
4003:
3993:
3983:, retrieved
3961:
3951:
3908:Cable theory
3886:
3878:
3872:
3491:
3416:
3363:
3357:
3286:
3282:
3279:
3271:
3207:
3200:
3193:
3185:
3123:
3121:
3065:
2880:
2872:
2868:
2864:
2860:
2856:
2852:
2848:
2847:of the form
2842:
2833:
2826:
2824:gas constant
2815:
2805:
2772:
2758:
2598:
2588:
2584:
2580:
2575:
2571:
2565:
2556:
2547:
2543:
2539:
2535:
2531:
2527:
2518:
2514:
2509:
2497:
2251:
1975:
1966:
1556:
1501:(joules per
1016:
640:-solutions:
581:
195:species and
170:
161:Bernard Katz
146:
133:
129:
127:
112:
103:
93:
86:
79:
72:
60:
48:Please help
43:verification
40:
3360:Na/K-ATPase
2849:y' = ay + b
2813:temperature
2794:applied to
4205:Categories
4118:2020-10-23
4069:2020-10-23
4027:2020-10-23
3985:2020-10-23
3944:References
2829:, and the
2506:Derivation
1664:2.30258...
215:negative:
138:physiology
76:newspapers
3838:−
3782:∑
3699:∑
3652:−
3596:∑
3510:∑
3467:
3458:μ
3394:μ
3383:−
3321:∑
3077:μ
3046:μ
3026:−
3016:μ
2971:−
2913:μ
2838:electrons
2767:down the
2765:diffusion
2680:−
2630:−
2476:≈
2470:
2433:
2404:≈
2370:
2307:
2281:≈
2213:
2133:∘
2091:⋅
2057:×
2048:≈
2011:⋅
1987:≈
1927:−
1894:−
1875:
1869:⋅
1854:−
1771:
1765:⋅
1699:⋅
1649:
1635:−
1618:
1612:
1505:per mole)
1196:
1071:
974:−
846:−
723:
684:−
615:−
539:−
519:−
490:∑
417:∑
394:−
374:−
345:∑
269:∑
257:
106:June 2010
4176:Archived
3897:See also
2583:, where
2553:integral
3875:calcium
3191:density
2855:= and
2851:, with
2831:faraday
2820:kelvins
2803:valence
2778:is the
2555:of the
2551:is the
2512:voltage
1329:coulomb
1017:It is "
90:scholar
4147:
4109:
4060:
4018:
3976:
3883:square
3805:
3725:
3619:
3536:
3492:where
2867:=0 to
2800:charge
2594:, its
2413:
2350:
2290:
2287:0.0267
2149:
2129:
2070:
1996:
1976:Using
1503:kelvin
1497:= the
1427:units)
1325:joules
1019:Nernst
92:
85:
78:
71:
63:
4141:33–37
2879:and (
2859:= d/d
2479:2.303
1702:2.303
1421:moles
1321:volts
193:ionic
97:JSTOR
83:books
4145:ISBN
4107:ISBN
4058:ISBN
4016:ISBN
3974:ISBN
3186:The
3122:and
2883:) =
2818:(in
2784:flux
2596:flux
2534:and
2410:61.5
2347:26.7
1857:61.5
1753:61.5
1708:61.5
1687:26.7
1327:per
159:and
128:The
69:news
4099:doi
4050:doi
4008:doi
3966:doi
3885:of
2885:out
2845:ODE
2430:log
2146:310
2054:9.6
1993:8.3
1872:log
1768:log
1615:exp
1603:log
151:of
52:by
4207::
4143:.
4105:,
4093:,
4077:^
4056:,
4044:,
4014:,
4002:,
3972:,
3960:,
3893:.
3464:ln
2877:in
2857:y'
2840:.
2502:.
2473:10
2467:ln
2367:ln
2304:ln
2210:ln
2125:37
2061:10
2037:,
1655:10
1646:ln
1607:10
1549:=
1425:SI
1247:Na
1212:Na
1193:ln
1142:Na
1132:Na
1097:Na
1087:Na
1068:ln
1044:Na
969:Cl
959:Cl
883:Na
873:Na
841:Cl
831:Cl
749:Na
739:Na
720:ln
676:Na
254:ln
163:.
4153:.
4101::
4052::
4010::
3968::
3887:e
3879:e
3855:t
3852:u
3849:o
3843:]
3833:A
3828:[
3820:A
3815:P
3808:A
3802:s
3799:n
3796:o
3793:i
3790:n
3787:a
3778:+
3772:n
3769:i
3763:]
3758:+
3753:C
3748:[
3740:C
3735:P
3728:C
3722:s
3719:n
3716:o
3713:i
3710:t
3707:a
3704:c
3695:=
3692:v
3666:n
3663:i
3657:]
3647:A
3642:[
3634:A
3629:P
3622:A
3616:s
3613:n
3610:o
3607:i
3604:n
3601:a
3592:+
3586:t
3583:u
3580:o
3574:]
3569:+
3564:C
3559:[
3551:C
3546:P
3539:C
3533:s
3530:n
3527:o
3524:i
3521:t
3518:a
3515:c
3506:=
3503:w
3475:v
3472:w
3461:=
3455:=
3449:T
3446:R
3439:m
3435:E
3431:F
3402:0
3399:=
3390:e
3386:v
3380:w
3367:A
3364:n
3343:0
3340:=
3335:A
3331:J
3325:A
3317:=
3312:t
3309:o
3306:t
3302:J
3287:m
3283:E
3275:A
3273:q
3254:A
3249:j
3242:A
3237:q
3233:=
3228:A
3224:J
3211:A
3208:j
3204:A
3201:q
3197:A
3194:J
3169:L
3163:A
3158:D
3152:=
3146:A
3141:P
3127:A
3124:P
3104:T
3101:R
3094:m
3090:E
3086:F
3080:=
3040:A
3035:n
3030:e
3023:1
3010:A
3005:n
3000:e
2993:n
2990:i
2984:]
2980:A
2976:[
2965:t
2962:u
2959:o
2953:]
2949:A
2945:[
2934:A
2929:P
2922:A
2917:n
2910:=
2904:A
2899:j
2881:L
2873:L
2871:=
2869:z
2865:z
2861:z
2853:y
2834:F
2827:R
2816:T
2809:A
2806:n
2776:A
2773:D
2743:)
2738:]
2734:A
2730:[
2724:L
2719:m
2715:E
2706:T
2703:R
2698:F
2692:A
2687:n
2674:z
2671:d
2665:]
2661:A
2657:[
2653:d
2646:(
2639:A
2634:D
2627:=
2621:A
2616:j
2602:A
2599:j
2592:A
2589:n
2585:L
2581:L
2579:/
2576:m
2572:E
2557:z
2548:m
2544:E
2540:z
2536:y
2532:x
2528:z
2519:m
2515:E
2453:i
2449:X
2443:o
2439:X
2425:z
2420:V
2417:m
2390:i
2386:X
2380:o
2376:X
2362:z
2357:V
2354:m
2341:=
2327:i
2323:X
2317:o
2313:X
2299:z
2294:V
2272:X
2268:E
2233:i
2229:X
2223:o
2219:X
2204:F
2201:z
2196:T
2193:R
2187:=
2182:X
2178:E
2153:K
2143:=
2139:C
2122:=
2119:T
2095:V
2087:l
2084:o
2081:m
2074:J
2065:4
2045:F
2021:l
2018:o
2015:m
2007:K
2000:J
1984:R
1950:)
1941:n
1938:i
1933:]
1923:X
1919:[
1911:t
1908:u
1905:o
1900:]
1890:X
1886:[
1880:(
1865:V
1862:m
1851:=
1846:)
1837:n
1834:i
1829:]
1823:+
1819:X
1815:[
1807:t
1804:u
1801:o
1796:]
1790:+
1786:X
1782:[
1776:(
1761:V
1758:m
1750:=
1745:X
1741:E
1716:V
1713:m
1705:=
1695:V
1692:m
1667:)
1661:=
1658:)
1652:(
1643:=
1638:1
1631:]
1627:)
1624:1
1621:(
1599:[
1596:(
1574:F
1570:T
1567:R
1537:F
1515:T
1485:R
1460:n
1457:i
1452:]
1447:n
1444:o
1441:i
1437:[
1404:t
1401:u
1398:o
1393:]
1388:n
1385:o
1382:i
1378:[
1353:n
1350:o
1347:i
1342:P
1331:)
1305:m
1301:E
1275:)
1266:n
1263:i
1258:]
1252:+
1242:[
1234:t
1231:u
1228:o
1223:]
1217:+
1207:[
1201:(
1188:F
1184:T
1181:R
1175:=
1170:)
1161:n
1158:i
1153:]
1147:+
1137:[
1128:P
1119:t
1116:u
1113:o
1108:]
1102:+
1092:[
1083:P
1076:(
1063:F
1059:T
1056:R
1050:=
1040:,
1037:m
1033:E
1000:)
991:t
988:u
985:o
980:]
964:[
955:P
951:+
945:n
942:i
937:]
931:+
926:K
921:[
916:K
912:P
908:+
902:n
899:i
894:]
888:+
878:[
869:P
860:n
857:i
852:]
836:[
827:P
823:+
817:t
814:u
811:o
806:]
800:+
795:K
790:[
785:K
781:P
777:+
771:t
768:u
765:o
760:]
754:+
744:[
735:P
728:(
715:F
711:T
708:R
702:=
696:l
693:C
687:x
681:1
669:x
664:K
659:,
656:m
652:E
627:l
624:C
618:x
612:1
607:a
604:N
597:x
592:K
565:)
556:t
553:u
550:o
545:]
534:j
530:A
526:[
514:j
510:A
505:P
499:m
494:j
486:+
480:n
477:i
472:]
466:+
461:i
457:M
453:[
446:+
441:i
437:M
432:P
426:n
421:i
408:n
405:i
400:]
389:j
385:A
381:[
369:j
365:A
360:P
354:m
349:j
341:+
335:t
332:u
329:o
324:]
318:+
313:i
309:M
305:[
298:+
293:i
289:M
284:P
278:n
273:i
262:(
249:F
245:T
242:R
236:=
231:m
227:E
203:A
179:M
119:)
113:(
108:)
104:(
94:·
87:·
80:·
73:·
46:.
23:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
