20:
166:
4637:
4617:
200:
also by focusing on analogies with heat conduction. However, it was
Hoorweg who first discovered the analogies with Kelvin's undersea cables in 1898 and then Hermann and Cremer who independently developed the cable theory for neuronal fibers in the early 20th century. Further mathematical theories of
2922:
the slower the nerve impulse can travel. That means, membrane potential (voltage across the membrane) lags more behind current injections. Response times vary from 1â2 milliseconds in neurons that are processing information that needs high temporal precision to 100 milliseconds or longer. A typical
963:
back into the picture is like making holes in a garden hose. The more holes, the faster the water will escape from the hose, and the less water will travel all the way from the beginning of the hose to the end. Similarly, in an axon, some of the current traveling longitudinally through the axoplasm
639:
Several important avenues of extending classical cable theory have recently seen the introduction of endogenous structures in order to analyze the effects of protein polarization within dendrites and different synaptic input distributions over the dendritic surface of a neuron.
1889:
3091:
1687:
1192:
can be expressed with yet another formula, by including the capacitance. The capacitance will cause a flow of charge (a current) towards the membrane on the side of the cytoplasm. This current is usually referred to as displacement current (here denoted
3307:
936:
500:
These relationships make sense intuitively, because the greater the circumference of the axon, the greater the area for charge to escape through its membrane, and therefore the lower the membrane resistance (dividing
3708:
Passive here refers to the membrane resistance being voltage-independent. However recent experiments (Stuart and
Sakmann 1994) with dendritic membranes shows that many of these are equipped with voltage gated
3713:
thus making the resistance of the membrane voltage dependent. Consequently there has been a need to update the classical cable theory to accommodate for the fact that most dendritic membranes are not passive.
3731:
Classical cable theory assumes that the inputs (usually injections with a micro device) are currents which can be summed linearly. This linearity does not hold for changes in synaptic membrane conductance.
842:
1313:
1148:
2625:
1773:
2066:
614:
2229:
2972:
186:(later known as Lord Kelvin) began developing mathematical models of signal decay in submarine (underwater) telegraphic cables. The models resembled the partial differential equations used by
412:
738:
1748:
1064:
3166:
246:, Redman, Rinzel, Idan Segev, Tuckwell, Bell, and Iannella. More recently, cable theory has been applied to model electrical activity in bundled neurons in the white matter of the brain.
1579:
1394:
1473:
2401:
3219:
2830:
482:
2475:
1542:
3209:
3393:
2668:
4511:
2363:
2531:
2158:
2111:
2012:
1992:
1938:
227:. Thus cable theory became important for analyzing data collected from intracellular microelectrode recordings and for analyzing the electrical properties of neuronal
3572:. Studies from the Rockefeller Institute for Medical Research. Reprints. Rockefeller Institute for Medical Research. pp. Part I, 131:1-496, Part II, 132:1-548.
235:, Fatt, Frank, Fuortes and others now relied heavily on cable theory to obtain functional insights of neurons and for guiding them in the design of new experiments.
2962:
2920:
2900:
2873:
2776:
2756:
2736:
2695:
2329:
2272:
2138:
1958:
1569:
1421:
1356:
1245:
1218:
1190:
152:
121:
91:
2557:
2298:
3358:
2511:
643:
To better understand how the cable equation is derived, first simplify the theoretical neuron even further and pretend it has a perfectly sealed membrane (
2160:, the harder it will be for current to travel through the axoplasm, and the shorter the current will be able to travel. It is possible to solve equation (
3759:
3650:
Douglas, PK; Douglas, David B. (2019). "Reconsidering
Spatial Priors in EEG Source Estimation : Does White Matter Contribute to EEG Rhythms?".
4353:
2875:, the more current it takes to charge and discharge a patch of membrane and the longer this process will take. The larger the membrane resistance
1920:) (see later) it is possible to make two important terms appear, namely the length constant (sometimes referred to as the space constant) denoted
868:
1994:(lambda), is a parameter that indicates how far a stationary current will influence the voltage along the cable. The larger the value of
325:(in F/m), which represent the specific resistance and capacitance respectively of one unit area of membrane (in m). Thus, if the radius,
2113:, and the more current will remain inside the axoplasm to travel longitudinally through the axon. The higher the axoplasmic resistance,
781:
3595:
Lazarevich, Ivan A.; Kazantsev, Victor B. (2013). "Dendritic signal transition induced by intracellular charge in inhomogeneties".
1884:{\displaystyle {\frac {1}{r_{l}}}{\frac {\partial ^{2}V}{\partial x^{2}}}=c_{m}{\frac {\partial V}{\partial t}}+{\frac {V}{r_{m}}}}
1255:
1090:
3869:
2567:
3752:
3677:
3515:
2022:
557:
4542:
3086:{\displaystyle {\frac {r_{m}}{r_{l}}}{\frac {\partial ^{2}V}{\partial x^{2}}}=c_{m}r_{m}{\frac {\partial V}{\partial t}}+V}
2174:
4676:
4643:
4194:
3931:
2166:) and arrive at the following equation (which is valid in steady-state conditions, i.e. when time approaches infinity):
636:, the greater the number of paths for the charge to flow through its axoplasm, and the lower the axoplasmic resistance.
4681:
3556:
4455:
4082:
3889:
3745:
3577:
359:
183:
1682:{\displaystyle {\frac {\partial i_{l}}{\partial x}}=-i_{m}={\frac {V}{r_{m}}}+c_{m}{\frac {\partial V}{\partial t}}}
685:
4410:
1708:
1014:
665: = 0 would move along the inside of the fiber unchanged. Moving away from the point of injection and by using
3112:
232:
4597:
4537:
4135:
1361:
528:
127:
forces are acting through the very thin lipid bilayer (see Figure 2). The resistance in series along the fiber
1431:
4130:
3819:
3464:
2373:
1908:
3302:{\displaystyle \lambda ^{2}{\frac {\partial ^{2}V}{\partial x^{2}}}=\tau {\frac {\partial V}{\partial t}}+V}
4572:
3969:
3926:
3879:
3874:
3486:"Reconsidering Spatial Priors in EEG Source Estimation : Does White Matter Contribute to EEG Rhythms?"
2738:, of an axon changes in response to changes in the current injected into the axoplasm. The time constant,
4623:
3919:
3845:
2786:
435:
4247:
4182:
3783:
3449:
2427:
1494:
976:
is the current escaping through the membrane per length unit, m, then the total current escaping along
3171:
4648:
4506:
4145:
3976:
3799:
3722:
Classical cable theory assumes that the fiber has a constant radius along the distance being modeled.
3337:
3333:
for which many solution methods, such as Green's functions and
Fourier methods, have been developed.
540:, of the axoplasm allows one to calculate the longitudinal intracellular resistance per unit length,
3363:
4671:
4547:
3804:
3434:
2646:
1220:.) The flow will only take place as long as the membrane's storage capacity has not been reached.
4592:
4577:
4230:
4225:
4125:
3993:
3774:
3424:
239:
2342:
217:
4552:
4312:
4031:
4026:
197:
2516:
2143:
2096:
1997:
1977:
1923:
4582:
4567:
4532:
4220:
4120:
3988:
2902:, the harder it is for a current to induce a change in membrane potential. So the higher the
94:
4450:
4602:
4557:
4003:
3948:
3794:
3789:
3614:
3459:
2940:
2905:
2878:
2851:
2761:
2741:
2714:
2673:
2307:
2250:
2116:
1943:
1547:
1399:
1334:
1223:
1196:
1168:
223:
The 1950s saw improvements in techniques for measuring the electric activity of individual
130:
123:
combined in parallel (see Fig. 1). The capacitance of a neuronal fiber comes about because
99:
69:
8:
4177:
4155:
3904:
3899:
3857:
3809:
3419:
3211:
on the right side. The cable equation can now be written in its perhaps best known form:
2536:
2277:
3618:
4562:
4140:
3683:
3655:
3638:
3604:
3521:
3493:
3439:
3343:
3330:
2496:
202:
36:
238:
Later, cable theory with its mathematical derivatives allowed ever more sophisticated
208:
Experimental evidence for the importance of cable theory in modelling the behavior of
4628:
4616:
4420:
4072:
3943:
3936:
3687:
3673:
3630:
3583:
3573:
3552:
3525:
3511:
3642:
19:
4373:
4363:
4170:
3964:
3914:
3909:
3852:
3840:
3665:
3622:
3503:
3444:
281:, as introduced above, are measured per membrane-length unit (per meter (m)). Thus
213:
40:
216:, Rushton, Tasaki and others. Two key papers from this era are those of Davis and
59:
inputs at different sites and times. Estimates are made by modeling dendrites and
4486:
4430:
4252:
3894:
3814:
3669:
3567:
3507:
2711:
Neuroscientists are often interested in knowing how fast the membrane potential,
1969:
159:
4460:
4425:
4415:
4240:
3998:
3824:
3626:
3485:
3414:
3404:
187:
4665:
4405:
4385:
4302:
3981:
3326:
2706:
2014:, the farther the charge will flow. The length constant can be expressed as:
124:
4491:
4322:
3737:
3634:
3587:
1396:
is the change in voltage over time. The current that passes the membrane (
931:{\displaystyle {\frac {1}{r_{l}}}{\frac {\partial V}{\partial x}}=-i_{l}\ }
666:
243:
179:
28:
3652:
2019 7th
International Winter Conference on Brain-Computer Interface (BCI)
3490:
2019 7th
International Winter Conference on Brain-Computer Interface (BCI)
4587:
4358:
4267:
4262:
3884:
3862:
3710:
3454:
756:
where the negative is because current flows down the potential gradient.
64:
201:
nerve fiber conduction based on cable theory were developed by Cole and
4481:
4440:
4435:
4348:
4257:
4165:
4077:
4057:
165:
212:
began surfacing in the 1930s from work done by Cole, Curtis, Hodgkin,
4476:
4445:
4343:
4187:
4150:
4087:
4041:
4036:
4021:
228:
1571:
can be derived if no additional current is added from an electrode:
4378:
4210:
3660:
3498:
3429:
1750:
represents the change per unit length of the longitudinal current.
194:
155:
52:
3609:
4501:
4338:
4292:
4215:
4115:
4110:
4062:
224:
56:
48:
44:
652:=â) with no loss of current to the outside, and no capacitance (
514:); and the more membrane available to store charge (multiplying
4516:
4496:
4368:
4160:
837:{\displaystyle {\frac {\partial V}{\partial x}}=-i_{l}r_{l}\ }
4317:
4297:
4287:
4282:
4277:
4272:
4235:
4067:
1308:{\displaystyle i_{c}=c_{m}{\frac {\partial V}{\partial t}}\ }
1143:{\displaystyle {\frac {\partial i_{l}}{\partial x}}=-i_{m}\ }
304:
209:
60:
23:
Figure. 1: Cable theory's simplified view of a neuronal fiber
2304:
is the exponential constant (approximate value 2.71828) and
4307:
3409:
661: = 0). A current injected into the fiber at position
2926:
2758:, is an index that provides information about that value.
2620:{\displaystyle V_{\lambda }={\frac {V_{0}}{e}}=0.368V_{0}}
763:
go towards zero and having infinitely small increments of
291:
205:(1920sâ1930s), Offner et al. (1940), and Rushton (1951).
1082:
or, using continuous, infinitesimally small increments:
2061:{\displaystyle \lambda ={\sqrt {\frac {r_{m}}{r_{l}}}}}
609:{\displaystyle r_{l}={\frac {\rho _{l}}{\pi a^{2}\ }}}
3366:
3346:
3222:
3174:
3115:
2975:
2943:
2908:
2881:
2854:
2789:
2764:
2744:
2717:
2676:
2649:
2570:
2539:
2519:
2499:
2430:
2376:
2345:
2310:
2280:
2253:
2224:{\displaystyle V_{x}=V_{0}e^{-{\frac {x}{\lambda }}}}
2177:
2146:
2119:
2099:
2025:
2000:
1980:
1946:
1926:
1776:
1711:
1582:
1550:
1497:
1434:
1402:
1364:
1337:
1258:
1226:
1199:
1171:
1093:
1017:
871:
784:
688:
560:
438:
362:
193:
The 1870s saw the first attempts by
Hermann to model
133:
102:
72:
182:
has roots leading back to the 1850s, when
Professor
3594:
329:, of the axon is known, then its circumference is 2
3387:
3352:
3301:
3203:
3160:
3085:
2956:
2914:
2894:
2867:
2824:
2770:
2750:
2730:
2689:
2662:
2619:
2551:
2525:
2505:
2469:
2395:
2357:
2323:
2292:
2266:
2223:
2152:
2132:
2105:
2060:
2006:
1986:
1952:
1932:
1883:
1742:
1681:
1563:
1536:
1467:
1415:
1388:
1350:
1307:
1239:
1212:
1184:
1142:
1058:
930:
836:
732:
632:The greater the cross sectional area of the axon,
608:
476:
406:
146:
115:
85:
4663:
989:. Thus, the change of current in the axoplasm, Î
1960:. The following sections focus on these terms.
407:{\displaystyle r_{m}={\frac {R_{m}}{2\pi a\ }}}
3649:
3483:
733:{\displaystyle \Delta V=-i_{l}r_{l}\Delta x\ }
249:
3753:
1765:) gives a first version of a cable equation:
1743:{\displaystyle {\partial i_{l}}/{\partial x}}
1059:{\displaystyle \Delta i_{l}=-i_{m}\Delta x\ }
3767:
3336:It is also a special degenerate case of the
3161:{\displaystyle \lambda ^{2}={r_{m}}/{r_{l}}}
3760:
3746:
3716:
3551:. Cambridge : Cambridge University Press.
3360:vanishes and the signal propagation speed
677:) we can calculate the voltage change as:
3659:
3608:
3537:
3497:
2923:response time is around 20 milliseconds.
1389:{\displaystyle {\partial V}/{\partial t}}
158:'s significant resistance to movement of
3725:
3702:
3549:Introduction to theoretical neurobiology
3546:
2964:on both sides of the equal sign we get:
1468:{\displaystyle i_{r}={\frac {V}{r_{m}}}}
307:per meter (F/m). This is in contrast to
242:to be explored by workers such as Jack,
164:
18:
2927:Generic form and mathematical structure
2396:{\displaystyle {\frac {x}{\lambda }}=1}
1914:By a simple rearrangement of equation (
220:(1947) and Hodgkin and Rushton (1946).
190:to describe heat conduction in a wire.
63:as cylinders composed of segments with
4664:
3741:
3565:
2848:The larger the membrane capacitance,
4598:Generative adversarial network (GAN)
3213:
2966:
2780:
2561:
2421:
2367:
2168:
2084:The larger the membrane resistance,
2016:
1767:
1573:
1425:
1249:
1084:
1008:
862:
775:
679:
551:
429:
353:
2825:{\displaystyle \tau =r_{m}c_{m}.\ }
2331:is the voltage at a given distance
477:{\displaystyle c_{m}=C_{m}2\pi a\ }
13:
3284:
3276:
3251:
3237:
3068:
3060:
3018:
3004:
1963:
1852:
1844:
1812:
1798:
1733:
1713:
1670:
1662:
1601:
1586:
1379:
1366:
1358:is the membrane's capacitance and
1293:
1285:
1112:
1097:
1047:
1018:
964:will escape through the membrane.
900:
892:
796:
788:
721:
689:
14:
4693:
3484:Douglas, PK; Douglas, DB (2019).
2493:which means that when we measure
2470:{\displaystyle V_{x}=V_{0}e^{-1}}
1537:{\displaystyle i_{m}=i_{r}+i_{c}}
4636:
4635:
4615:
3204:{\displaystyle \tau =c_{m}r_{m}}
2700:
16:Mathematical model of a dendrite
3654:. Vol. 88. pp. 1â12.
4548:Recurrent neural network (RNN)
4538:Differentiable neural computer
3566:de NĂł, Rafael Lorente (1947).
3477:
3388:{\displaystyle 1/{\sqrt {LC}}}
2300:(point of current injection),
1940:and the time constant denoted
529:specific electrical resistance
178:Cable theory in computational
1:
4593:Variational autoencoder (VAE)
4553:Long short-term memory (LSTM)
3820:Computational learning theory
3542:. San Diego : Academic Press.
3470:
3465:Soliton model in neuroscience
1909:partial differential equation
351:values can be calculated as:
254:Note, various conventions of
4573:Convolutional neural network
3670:10.1109/IWW-BCI.2019.8737307
3538:Poznanski, Roman R. (2013).
3508:10.1109/IWW-BCI.2019.8737307
3315:
3099:
2933:
2931:If one multiplies equation (
2838:
2663:{\displaystyle V_{\lambda }}
2633:
2483:
2409:
2237:
2162:
2074:
1916:
1897:
1761:
1695:
1481:
549:, (in Ω·m) by the equation:
169:Figure. 2: Fiber capacitance
7:
4568:Multilayer perceptron (MLP)
3569:A study of nerve physiology
3547:Tuckwell, Henry C. (1988).
3398:
2140:, the smaller the value of
2093:, the greater the value of
1755:
1544:the following equation for
1321:
1156:
1072:
944:
850:
769:
746:
622:
490:
420:
250:Deriving the cable equation
10:
4698:
4677:Computational neuroscience
4644:Artificial neural networks
4558:Gated recurrent unit (GRU)
3784:Differentiable programming
3627:10.1103/PhysRevE.88.062718
2704:
2670:is always 36.8 percent of
2358:{\displaystyle x=\lambda }
1967:
1247:can then be expressed as:
231:. Scientists like Coombs,
173:
4682:Cardiac electrophysiology
4611:
4525:
4469:
4398:
4331:
4203:
4103:
4096:
4050:
4014:
3977:Artificial neural network
3957:
3833:
3800:Automatic differentiation
3773:
3540:Mathematical Neuroscience
2274:is the depolarization at
3805:Neuromorphic engineering
3768:Differentiable computing
3695:
2526:{\displaystyle \lambda }
2153:{\displaystyle \lambda }
2106:{\displaystyle \lambda }
2007:{\displaystyle \lambda }
1987:{\displaystyle \lambda }
1933:{\displaystyle \lambda }
1907:which is a second-order
4578:Residual neural network
3994:Artificial Intelligence
3425:Biological neuron model
3340:, where the inductance
1423:) can be expressed as:
198:electrotonic potentials
3450:NernstâPlanck equation
3389:
3354:
3338:Telegrapher's equation
3303:
3205:
3162:
3087:
2958:
2916:
2896:
2869:
2826:
2778:can be calculated as:
2772:
2752:
2732:
2691:
2664:
2621:
2553:
2527:
2507:
2471:
2397:
2359:
2325:
2294:
2268:
2225:
2154:
2134:
2107:
2062:
2008:
1988:
1954:
1934:
1885:
1744:
1683:
1565:
1538:
1469:
1417:
1390:
1352:
1309:
1241:
1214:
1186:
1144:
1060:
1006:=0 can be written as:
932:
838:
734:
610:
478:
408:
170:
148:
117:
87:
24:
4533:Neural Turing machine
4121:Human image synthesis
3390:
3355:
3304:
3206:
3168:on the left side and
3163:
3088:
2959:
2957:{\displaystyle r_{m}}
2917:
2915:{\displaystyle \tau }
2897:
2895:{\displaystyle r_{m}}
2870:
2868:{\displaystyle c_{m}}
2827:
2773:
2771:{\displaystyle \tau }
2753:
2751:{\displaystyle \tau }
2733:
2731:{\displaystyle V_{m}}
2692:
2690:{\displaystyle V_{0}}
2665:
2622:
2554:
2528:
2508:
2472:
2398:
2360:
2326:
2324:{\displaystyle V_{x}}
2295:
2269:
2267:{\displaystyle V_{0}}
2226:
2155:
2135:
2133:{\displaystyle r_{l}}
2108:
2063:
2009:
1989:
1974:The length constant,
1955:
1953:{\displaystyle \tau }
1935:
1886:
1753:Combining equations (
1745:
1684:
1566:
1564:{\displaystyle i_{m}}
1539:
1470:
1418:
1416:{\displaystyle i_{r}}
1391:
1353:
1351:{\displaystyle c_{m}}
1310:
1242:
1240:{\displaystyle i_{c}}
1215:
1213:{\displaystyle i_{c}}
1187:
1185:{\displaystyle i_{m}}
1145:
1061:
933:
839:
735:
611:
479:
409:
168:
149:
147:{\displaystyle r_{l}}
118:
116:{\displaystyle r_{m}}
88:
86:{\displaystyle c_{m}}
22:
4624:Computer programming
4603:Graph neural network
4178:Text-to-video models
4156:Text-to-image models
4004:Large language model
3989:Scientific computing
3795:Statistical manifold
3790:Information geometry
3460:Saltatory conduction
3435:HodgkinâHuxley model
3364:
3344:
3220:
3172:
3113:
2973:
2941:
2906:
2879:
2852:
2787:
2762:
2742:
2715:
2674:
2647:
2568:
2537:
2517:
2497:
2428:
2374:
2343:
2308:
2278:
2251:
2175:
2144:
2117:
2097:
2023:
1998:
1978:
1944:
1924:
1774:
1709:
1580:
1548:
1495:
1432:
1400:
1362:
1335:
1256:
1224:
1197:
1169:
1091:
1015:
869:
782:
686:
558:
436:
360:
131:
100:
70:
3970:In-context learning
3810:Pattern recognition
3619:2013PhRvE..88f2718L
3420:Bioelectrochemistry
2552:{\displaystyle x=0}
2293:{\displaystyle x=0}
51:, particularly the
37:mathematical models
4563:Echo state network
4451:JĂŒrgen Schmidhuber
4146:Facial recognition
4141:Speech recognition
4051:Software libraries
3440:Membrane potential
3385:
3350:
3331:diffusion equation
3299:
3201:
3158:
3083:
2954:
2912:
2892:
2865:
2822:
2768:
2748:
2728:
2687:
2660:
2617:
2549:
2523:
2503:
2467:
2393:
2355:
2321:
2290:
2264:
2221:
2150:
2130:
2103:
2058:
2004:
1984:
1950:
1930:
1881:
1740:
1679:
1561:
1534:
1465:
1413:
1386:
1348:
1305:
1237:
1210:
1182:
1140:
1056:
928:
834:
730:
606:
474:
404:
294:·meters (Ω·m) and
171:
144:
113:
83:
43:(and accompanying
25:
4659:
4658:
4421:Stephen Grossberg
4394:
4393:
3679:978-1-5386-8116-9
3517:978-1-5386-8116-9
3492:. pp. 1â12.
3383:
3353:{\displaystyle L}
3323:
3322:
3291:
3265:
3107:
3106:
3075:
3032:
2998:
2846:
2845:
2821:
2641:
2640:
2599:
2506:{\displaystyle V}
2491:
2490:
2417:
2416:
2385:
2245:
2244:
2217:
2082:
2081:
2056:
2055:
1905:
1904:
1879:
1859:
1826:
1792:
1703:
1702:
1677:
1644:
1608:
1489:
1488:
1463:
1329:
1328:
1304:
1300:
1164:
1163:
1139:
1119:
1080:
1079:
1055:
952:
951:
927:
907:
887:
858:
857:
833:
803:
767:, one can write (
754:
753:
729:
630:
629:
604:
602:
498:
497:
473:
428:
427:
402:
400:
39:to calculate the
4689:
4649:Machine learning
4639:
4638:
4619:
4374:Action selection
4364:Self-driving car
4171:Stable Diffusion
4136:Speech synthesis
4101:
4100:
3965:Machine learning
3841:Gradient descent
3762:
3755:
3748:
3739:
3738:
3732:
3729:
3723:
3720:
3714:
3706:
3691:
3663:
3646:
3612:
3591:
3562:
3543:
3530:
3529:
3501:
3481:
3445:Monodomain model
3394:
3392:
3391:
3386:
3384:
3376:
3374:
3359:
3357:
3356:
3351:
3317:
3308:
3306:
3305:
3300:
3292:
3290:
3282:
3274:
3266:
3264:
3263:
3262:
3249:
3245:
3244:
3234:
3232:
3231:
3214:
3210:
3208:
3207:
3202:
3200:
3199:
3190:
3189:
3167:
3165:
3164:
3159:
3157:
3156:
3155:
3145:
3140:
3139:
3138:
3125:
3124:
3101:
3092:
3090:
3089:
3084:
3076:
3074:
3066:
3058:
3056:
3055:
3046:
3045:
3033:
3031:
3030:
3029:
3016:
3012:
3011:
3001:
2999:
2997:
2996:
2987:
2986:
2977:
2967:
2963:
2961:
2960:
2955:
2953:
2952:
2921:
2919:
2918:
2913:
2901:
2899:
2898:
2893:
2891:
2890:
2874:
2872:
2871:
2866:
2864:
2863:
2840:
2831:
2829:
2828:
2823:
2819:
2815:
2814:
2805:
2804:
2781:
2777:
2775:
2774:
2769:
2757:
2755:
2754:
2749:
2737:
2735:
2734:
2729:
2727:
2726:
2696:
2694:
2693:
2688:
2686:
2685:
2669:
2667:
2666:
2661:
2659:
2658:
2635:
2626:
2624:
2623:
2618:
2616:
2615:
2600:
2595:
2594:
2585:
2580:
2579:
2562:
2558:
2556:
2555:
2550:
2532:
2530:
2529:
2524:
2512:
2510:
2509:
2504:
2485:
2476:
2474:
2473:
2468:
2466:
2465:
2453:
2452:
2440:
2439:
2422:
2411:
2402:
2400:
2399:
2394:
2386:
2378:
2368:
2364:
2362:
2361:
2356:
2330:
2328:
2327:
2322:
2320:
2319:
2299:
2297:
2296:
2291:
2273:
2271:
2270:
2265:
2263:
2262:
2239:
2230:
2228:
2227:
2222:
2220:
2219:
2218:
2210:
2200:
2199:
2187:
2186:
2169:
2159:
2157:
2156:
2151:
2139:
2137:
2136:
2131:
2129:
2128:
2112:
2110:
2109:
2104:
2076:
2067:
2065:
2064:
2059:
2057:
2054:
2053:
2044:
2043:
2034:
2033:
2017:
2013:
2011:
2010:
2005:
1993:
1991:
1990:
1985:
1959:
1957:
1956:
1951:
1939:
1937:
1936:
1931:
1899:
1890:
1888:
1887:
1882:
1880:
1878:
1877:
1865:
1860:
1858:
1850:
1842:
1840:
1839:
1827:
1825:
1824:
1823:
1810:
1806:
1805:
1795:
1793:
1791:
1790:
1778:
1768:
1749:
1747:
1746:
1741:
1739:
1731:
1726:
1725:
1724:
1697:
1688:
1686:
1685:
1680:
1678:
1676:
1668:
1660:
1658:
1657:
1645:
1643:
1642:
1630:
1625:
1624:
1609:
1607:
1599:
1598:
1597:
1584:
1574:
1570:
1568:
1567:
1562:
1560:
1559:
1543:
1541:
1540:
1535:
1533:
1532:
1520:
1519:
1507:
1506:
1483:
1474:
1472:
1471:
1466:
1464:
1462:
1461:
1449:
1444:
1443:
1426:
1422:
1420:
1419:
1414:
1412:
1411:
1395:
1393:
1392:
1387:
1385:
1377:
1372:
1357:
1355:
1354:
1349:
1347:
1346:
1323:
1314:
1312:
1311:
1306:
1302:
1301:
1299:
1291:
1283:
1281:
1280:
1268:
1267:
1250:
1246:
1244:
1243:
1238:
1236:
1235:
1219:
1217:
1216:
1211:
1209:
1208:
1191:
1189:
1188:
1183:
1181:
1180:
1158:
1149:
1147:
1146:
1141:
1137:
1136:
1135:
1120:
1118:
1110:
1109:
1108:
1095:
1085:
1074:
1065:
1063:
1062:
1057:
1053:
1046:
1045:
1030:
1029:
1009:
1002:, from position
998:, at distance, Î
946:
937:
935:
934:
929:
925:
924:
923:
908:
906:
898:
890:
888:
886:
885:
873:
863:
852:
843:
841:
840:
835:
831:
830:
829:
820:
819:
804:
802:
794:
786:
776:
748:
739:
737:
736:
731:
727:
720:
719:
710:
709:
680:
624:
615:
613:
612:
607:
605:
603:
600:
599:
598:
585:
584:
575:
570:
569:
552:
492:
483:
481:
480:
475:
471:
461:
460:
448:
447:
430:
422:
413:
411:
410:
405:
403:
401:
398:
387:
386:
377:
372:
371:
354:
214:Sir Bernard Katz
153:
151:
150:
145:
143:
142:
122:
120:
119:
114:
112:
111:
92:
90:
89:
84:
82:
81:
47:) along passive
41:electric current
4697:
4696:
4692:
4691:
4690:
4688:
4687:
4686:
4672:Neurophysiology
4662:
4661:
4660:
4655:
4607:
4521:
4487:Google DeepMind
4465:
4431:Geoffrey Hinton
4390:
4327:
4253:Project Debater
4199:
4097:Implementations
4092:
4046:
4010:
3953:
3895:Backpropagation
3829:
3815:Tensor calculus
3769:
3766:
3736:
3735:
3730:
3726:
3721:
3717:
3707:
3703:
3698:
3680:
3580:
3559:
3534:
3533:
3518:
3482:
3478:
3473:
3401:
3375:
3370:
3365:
3362:
3361:
3345:
3342:
3341:
3283:
3275:
3273:
3258:
3254:
3250:
3240:
3236:
3235:
3233:
3227:
3223:
3221:
3218:
3217:
3195:
3191:
3185:
3181:
3173:
3170:
3169:
3151:
3147:
3146:
3141:
3134:
3130:
3129:
3120:
3116:
3114:
3111:
3110:
3067:
3059:
3057:
3051:
3047:
3041:
3037:
3025:
3021:
3017:
3007:
3003:
3002:
3000:
2992:
2988:
2982:
2978:
2976:
2974:
2971:
2970:
2948:
2944:
2942:
2939:
2938:
2929:
2907:
2904:
2903:
2886:
2882:
2880:
2877:
2876:
2859:
2855:
2853:
2850:
2849:
2810:
2806:
2800:
2796:
2788:
2785:
2784:
2763:
2760:
2759:
2743:
2740:
2739:
2722:
2718:
2716:
2713:
2712:
2709:
2703:
2681:
2677:
2675:
2672:
2671:
2654:
2650:
2648:
2645:
2644:
2611:
2607:
2590:
2586:
2584:
2575:
2571:
2569:
2566:
2565:
2538:
2535:
2534:
2518:
2515:
2514:
2498:
2495:
2494:
2458:
2454:
2448:
2444:
2435:
2431:
2429:
2426:
2425:
2377:
2375:
2372:
2371:
2344:
2341:
2340:
2315:
2311:
2309:
2306:
2305:
2279:
2276:
2275:
2258:
2254:
2252:
2249:
2248:
2209:
2205:
2201:
2195:
2191:
2182:
2178:
2176:
2173:
2172:
2145:
2142:
2141:
2124:
2120:
2118:
2115:
2114:
2098:
2095:
2094:
2092:
2049:
2045:
2039:
2035:
2032:
2024:
2021:
2020:
1999:
1996:
1995:
1979:
1976:
1975:
1972:
1970:Length constant
1966:
1964:Length constant
1945:
1942:
1941:
1925:
1922:
1921:
1873:
1869:
1864:
1851:
1843:
1841:
1835:
1831:
1819:
1815:
1811:
1801:
1797:
1796:
1794:
1786:
1782:
1777:
1775:
1772:
1771:
1732:
1727:
1720:
1716:
1712:
1710:
1707:
1706:
1669:
1661:
1659:
1653:
1649:
1638:
1634:
1629:
1620:
1616:
1600:
1593:
1589:
1585:
1583:
1581:
1578:
1577:
1555:
1551:
1549:
1546:
1545:
1528:
1524:
1515:
1511:
1502:
1498:
1496:
1493:
1492:
1457:
1453:
1448:
1439:
1435:
1433:
1430:
1429:
1407:
1403:
1401:
1398:
1397:
1378:
1373:
1365:
1363:
1360:
1359:
1342:
1338:
1336:
1333:
1332:
1292:
1284:
1282:
1276:
1272:
1263:
1259:
1257:
1254:
1253:
1231:
1227:
1225:
1222:
1221:
1204:
1200:
1198:
1195:
1194:
1176:
1172:
1170:
1167:
1166:
1131:
1127:
1111:
1104:
1100:
1096:
1094:
1092:
1089:
1088:
1041:
1037:
1025:
1021:
1016:
1013:
1012:
997:
988:
975:
962:
919:
915:
899:
891:
889:
881:
877:
872:
870:
867:
866:
825:
821:
815:
811:
795:
787:
785:
783:
780:
779:
715:
711:
705:
701:
687:
684:
683:
660:
651:
594:
590:
586:
580:
576:
574:
565:
561:
559:
556:
555:
548:
539:
522:
509:
456:
452:
443:
439:
437:
434:
433:
388:
382:
378:
376:
367:
363:
361:
358:
357:
350:
341:
324:
315:
302:
290:is measured in
289:
280:
271:
262:
252:
184:William Thomson
176:
160:electric charge
138:
134:
132:
129:
128:
107:
103:
101:
98:
97:
77:
73:
71:
68:
67:
17:
12:
11:
5:
4695:
4685:
4684:
4679:
4674:
4657:
4656:
4654:
4653:
4652:
4651:
4646:
4633:
4632:
4631:
4626:
4612:
4609:
4608:
4606:
4605:
4600:
4595:
4590:
4585:
4580:
4575:
4570:
4565:
4560:
4555:
4550:
4545:
4540:
4535:
4529:
4527:
4523:
4522:
4520:
4519:
4514:
4509:
4504:
4499:
4494:
4489:
4484:
4479:
4473:
4471:
4467:
4466:
4464:
4463:
4461:Ilya Sutskever
4458:
4453:
4448:
4443:
4438:
4433:
4428:
4426:Demis Hassabis
4423:
4418:
4416:Ian Goodfellow
4413:
4408:
4402:
4400:
4396:
4395:
4392:
4391:
4389:
4388:
4383:
4382:
4381:
4371:
4366:
4361:
4356:
4351:
4346:
4341:
4335:
4333:
4329:
4328:
4326:
4325:
4320:
4315:
4310:
4305:
4300:
4295:
4290:
4285:
4280:
4275:
4270:
4265:
4260:
4255:
4250:
4245:
4244:
4243:
4233:
4228:
4223:
4218:
4213:
4207:
4205:
4201:
4200:
4198:
4197:
4192:
4191:
4190:
4185:
4175:
4174:
4173:
4168:
4163:
4153:
4148:
4143:
4138:
4133:
4128:
4123:
4118:
4113:
4107:
4105:
4098:
4094:
4093:
4091:
4090:
4085:
4080:
4075:
4070:
4065:
4060:
4054:
4052:
4048:
4047:
4045:
4044:
4039:
4034:
4029:
4024:
4018:
4016:
4012:
4011:
4009:
4008:
4007:
4006:
3999:Language model
3996:
3991:
3986:
3985:
3984:
3974:
3973:
3972:
3961:
3959:
3955:
3954:
3952:
3951:
3949:Autoregression
3946:
3941:
3940:
3939:
3929:
3927:Regularization
3924:
3923:
3922:
3917:
3912:
3902:
3897:
3892:
3890:Loss functions
3887:
3882:
3877:
3872:
3867:
3866:
3865:
3855:
3850:
3849:
3848:
3837:
3835:
3831:
3830:
3828:
3827:
3825:Inductive bias
3822:
3817:
3812:
3807:
3802:
3797:
3792:
3787:
3779:
3777:
3771:
3770:
3765:
3764:
3757:
3750:
3742:
3734:
3733:
3724:
3715:
3700:
3699:
3697:
3694:
3693:
3692:
3678:
3647:
3592:
3578:
3563:
3558:978-0521350969
3557:
3544:
3532:
3531:
3516:
3475:
3474:
3472:
3469:
3468:
3467:
3462:
3457:
3452:
3447:
3442:
3437:
3432:
3427:
3422:
3417:
3415:Bidomain model
3412:
3407:
3405:Nanophysiology
3400:
3397:
3382:
3379:
3373:
3369:
3349:
3321:
3320:
3311:
3309:
3298:
3295:
3289:
3286:
3281:
3278:
3272:
3269:
3261:
3257:
3253:
3248:
3243:
3239:
3230:
3226:
3198:
3194:
3188:
3184:
3180:
3177:
3154:
3150:
3144:
3137:
3133:
3128:
3123:
3119:
3109:and recognize
3105:
3104:
3095:
3093:
3082:
3079:
3073:
3070:
3065:
3062:
3054:
3050:
3044:
3040:
3036:
3028:
3024:
3020:
3015:
3010:
3006:
2995:
2991:
2985:
2981:
2951:
2947:
2928:
2925:
2911:
2889:
2885:
2862:
2858:
2844:
2843:
2834:
2832:
2818:
2813:
2809:
2803:
2799:
2795:
2792:
2767:
2747:
2725:
2721:
2705:Main article:
2702:
2699:
2684:
2680:
2657:
2653:
2639:
2638:
2629:
2627:
2614:
2610:
2606:
2603:
2598:
2593:
2589:
2583:
2578:
2574:
2548:
2545:
2542:
2522:
2502:
2489:
2488:
2479:
2477:
2464:
2461:
2457:
2451:
2447:
2443:
2438:
2434:
2415:
2414:
2405:
2403:
2392:
2389:
2384:
2381:
2354:
2351:
2348:
2318:
2314:
2289:
2286:
2283:
2261:
2257:
2243:
2242:
2233:
2231:
2216:
2213:
2208:
2204:
2198:
2194:
2190:
2185:
2181:
2149:
2127:
2123:
2102:
2088:
2080:
2079:
2070:
2068:
2052:
2048:
2042:
2038:
2031:
2028:
2003:
1983:
1968:Main article:
1965:
1962:
1949:
1929:
1903:
1902:
1893:
1891:
1876:
1872:
1868:
1863:
1857:
1854:
1849:
1846:
1838:
1834:
1830:
1822:
1818:
1814:
1809:
1804:
1800:
1789:
1785:
1781:
1738:
1735:
1730:
1723:
1719:
1715:
1701:
1700:
1691:
1689:
1675:
1672:
1667:
1664:
1656:
1652:
1648:
1641:
1637:
1633:
1628:
1623:
1619:
1615:
1612:
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4470:Organizations
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4386:Robot control
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4303:Chinchilla AI
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3982:Deep learning
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3870:Hallucination
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3395:is infinite.
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3327:heat equation
3325:This is a 1D
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316:(in Ω·m) and
314:
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288:
284:
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275:
270:
266:
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257:
247:
245:
241:
240:neuron models
236:
234:
230:
226:
221:
219:
218:Lorente de NĂł
215:
211:
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125:electrostatic
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55:that receive
54:
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46:
42:
38:
34:
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4492:Hugging Face
4456:David Silver
4104:Audioâvisual
3958:Applications
3937:Augmentation
3782:
3727:
3718:
3711:ion channels
3704:
3651:
3600:
3597:Phys. Rev. E
3596:
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2513:at distance
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1491:and because
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295:
286:
282:
277:
273:
268:
264:
263:exist. Here
259:
255:
253:
237:
222:
207:
192:
180:neuroscience
177:
65:capacitances
33:cable theory
32:
31:, classical
29:neuroscience
26:
4640:Categories
4588:Autoencoder
4543:Transformer
4411:Alex Graves
4359:OpenAI Five
4263:IBM Watsonx
3885:Convolution
3863:Overfitting
3455:Patch clamp
95:resistances
4666:Categories
4629:Technology
4482:EleutherAI
4441:Fei-Fei Li
4436:Yann LeCun
4349:Q-learning
4332:Decisional
4258:IBM Watson
4166:Midjourney
4058:TensorFlow
3905:Activation
3858:Regression
3853:Clustering
3661:2111.08939
3499:2111.08939
3471:References
342:, and its
333:, and its
4512:MIT CSAIL
4477:Anthropic
4446:Andrew Ng
4344:AlphaZero
4188:VideoPoet
4151:AlphaFold
4088:MindSpore
4042:SpiNNaker
4037:Memristor
3944:Diffusion
3920:Rectifier
3900:Batchnorm
3880:Attention
3875:Adversary
3688:195064621
3610:1308.0821
3526:195064621
3285:∂
3277:∂
3271:τ
3252:∂
3238:∂
3225:λ
3176:τ
3118:λ
3069:∂
3061:∂
3019:∂
3005:∂
2910:τ
2791:τ
2766:τ
2746:τ
2656:λ
2577:λ
2521:λ
2460:−
2383:λ
2353:λ
2339:=0. When
2215:λ
2207:−
2148:λ
2101:λ
2027:λ
2002:λ
1982:λ
1948:τ
1928:λ
1853:∂
1845:∂
1813:∂
1799:∂
1734:∂
1714:∂
1671:∂
1663:∂
1614:−
1602:∂
1587:∂
1380:∂
1367:∂
1294:∂
1286:∂
1125:−
1113:∂
1098:∂
1048:Δ
1035:−
1019:Δ
954:Bringing
913:−
901:∂
893:∂
809:−
797:∂
789:∂
759:Letting Î
722:Δ
699:−
690:Δ
667:Ohm's law
588:π
578:ρ
466:π
393:π
229:dendrites
53:dendrites
4620:Portals
4379:Auto-GPT
4211:Word2vec
4015:Hardware
3932:Datasets
3834:Concepts
3643:13353454
3635:24483497
3430:Dendrite
3399:See also
195:neuronal
156:axoplasm
57:synaptic
49:neurites
4502:Meta AI
4339:AlphaGo
4323:PanGu-ÎŁ
4293:ChatGPT
4268:Granite
4216:Seq2seq
4195:Whisper
4116:WaveNet
4111:AlexNet
4083:Flux.jl
4063:PyTorch
3915:Sigmoid
3910:Softmax
3775:General
3615:Bibcode
3588:6217290
2559:we get
1911:(PDE).
1759:) and (
527:). The
225:neurons
203:Hodgkin
188:Fourier
174:History
45:voltage
4517:Huawei
4497:OpenAI
4399:People
4369:MuZero
4231:Gemini
4226:Claude
4161:DALL-E
4073:Theano
3686:
3676:
3641:
3633:
3586:
3576:
3555:
3524:
3514:
2820:
2247:Where
1705:where
1331:where
1303:
1138:
1054:
926:
832:
773:) as:
728:
601:
472:
399:
305:farads
233:Eccles
4583:Mamba
4354:SARSA
4318:LLaMA
4313:BLOOM
4298:GPT-J
4288:GPT-4
4283:GPT-3
4278:GPT-2
4273:GPT-1
4236:LaMDA
4068:Keras
3696:Notes
3684:S2CID
3656:arXiv
3639:S2CID
3605:arXiv
3522:S2CID
3494:arXiv
2937:) by
2643:Thus
2605:0.368
2533:from
2365:then
2335:from
210:axons
61:axons
35:uses
4507:Mila
4308:PaLM
4241:Bard
4221:BERT
4204:Text
4183:Sora
3674:ISBN
3631:PMID
3584:OCLC
3574:ISBN
3553:ISBN
3512:ISBN
3410:Axon
2419:and
523:by 2
510:by 2
272:and
244:Rall
93:and
4248:NMT
4131:OCR
4126:HWR
4078:JAX
4032:VPU
4027:TPU
4022:IPU
3846:SGD
3666:doi
3623:doi
3504:doi
3329:or
982:y·i
967:If
860:or
303:in
292:ohm
27:In
4668::
3682:.
3672:.
3664:.
3637:.
3629:.
3621:.
3613:.
3601:88
3599:.
3582:.
3520:.
3510:.
3502:.
3488:.
3316:20
3100:19
2934:12
2839:18
2697:.
2634:17
2484:16
2410:15
2238:14
2163:12
2075:13
1917:12
1898:12
1762:11
1696:11
1482:10
675:IR
634:Ïa
531:,
525:Ïa
512:Ïa
331:Ïa
162:.
3761:e
3754:t
3747:v
3690:.
3668::
3658::
3645:.
3625::
3617::
3607::
3590:.
3561:.
3528:.
3506::
3496::
3381:C
3378:L
3372:/
3368:1
3348:L
3318:)
3314:(
3297:V
3294:+
3288:t
3280:V
3268:=
3260:2
3256:x
3247:V
3242:2
3229:2
3197:m
3193:r
3187:m
3183:c
3179:=
3153:l
3149:r
3143:/
3136:m
3132:r
3127:=
3122:2
3102:)
3098:(
3081:V
3078:+
3072:t
3064:V
3053:m
3049:r
3043:m
3039:c
3035:=
3027:2
3023:x
3014:V
3009:2
2994:l
2990:r
2984:m
2980:r
2950:m
2946:r
2888:m
2884:r
2861:m
2857:c
2841:)
2837:(
2817:.
2812:m
2808:c
2802:m
2798:r
2794:=
2724:m
2720:V
2683:0
2679:V
2652:V
2636:)
2632:(
2613:0
2609:V
2602:=
2597:e
2592:0
2588:V
2582:=
2573:V
2547:0
2544:=
2541:x
2501:V
2486:)
2482:(
2463:1
2456:e
2450:0
2446:V
2442:=
2437:x
2433:V
2412:)
2408:(
2391:1
2388:=
2380:x
2350:=
2347:x
2337:x
2333:x
2317:x
2313:V
2302:e
2288:0
2285:=
2282:x
2260:0
2256:V
2240:)
2236:(
2212:x
2203:e
2197:0
2193:V
2189:=
2184:x
2180:V
2126:l
2122:r
2090:m
2086:r
2077:)
2073:(
2051:l
2047:r
2041:m
2037:r
2030:=
1900:)
1896:(
1875:m
1871:r
1867:V
1862:+
1856:t
1848:V
1837:m
1833:c
1829:=
1821:2
1817:x
1808:V
1803:2
1788:l
1784:r
1780:1
1756:6
1737:x
1729:/
1722:l
1718:i
1698:)
1694:(
1674:t
1666:V
1655:m
1651:c
1647:+
1640:m
1636:r
1632:V
1627:=
1622:m
1618:i
1611:=
1605:x
1595:l
1591:i
1557:m
1553:i
1530:c
1526:i
1522:+
1517:r
1513:i
1509:=
1504:m
1500:i
1484:)
1480:(
1459:m
1455:r
1451:V
1446:=
1441:r
1437:i
1409:r
1405:i
1383:t
1375:/
1370:V
1344:m
1340:c
1324:)
1322:9
1320:(
1297:t
1289:V
1278:m
1274:c
1270:=
1265:c
1261:i
1233:c
1229:i
1206:c
1202:i
1178:m
1174:i
1159:)
1157:8
1155:(
1133:m
1129:i
1122:=
1116:x
1106:l
1102:i
1075:)
1073:7
1071:(
1051:x
1043:m
1039:i
1032:=
1027:l
1023:i
1004:x
1000:x
995:l
991:i
986:m
978:y
973:m
969:i
960:m
956:r
947:)
945:6
943:(
921:l
917:i
910:=
904:x
896:V
883:l
879:r
875:1
853:)
851:5
849:(
827:l
823:r
817:l
813:i
806:=
800:x
792:V
770:4
765:x
761:x
749:)
747:4
745:(
725:x
717:l
713:r
707:l
703:i
696:=
693:V
671:V
669:(
663:x
658:m
654:c
649:m
645:r
625:)
623:3
621:(
596:2
592:a
582:l
572:=
567:l
563:r
546:l
542:r
537:l
533:Ï
520:m
516:C
507:m
503:R
493:)
491:2
489:(
469:a
463:2
458:m
454:C
450:=
445:m
441:c
423:)
421:1
419:(
396:a
390:2
384:m
380:R
374:=
369:m
365:r
348:m
344:c
339:m
335:r
327:a
322:m
318:C
313:m
309:R
300:m
296:c
287:m
283:r
278:m
274:c
269:m
265:r
260:m
256:r
140:l
136:r
109:m
105:r
79:m
75:c
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