310:
the proposal that nerve impulses are an adiabatic phenomenon much like sound waves. Synaptically evoked action potentials in the electric organ of the electric eel are associated with substantial positive (only) heat production followed by active cooling to ambient temperature. In the garfish olfactory nerve, the action potential is associated with a biphasic temperature change; however, there is a net production of heat. These published results are inconsistent with the
Hodgkin-Huxley Model and the authors interpret their work in terms of that model: The initial sodium current releases heat as the membrane capacitance is discharged; heat is absorbed during recharge of the membrane capacitance as potassium ions move with their concentration gradient but against the membrane potential. This mechanism is called the "Condenser Theory". Additional heat may be generated by membrane configuration changes driven by the changes in membrane potential. An increase in entropy during depolarization would release heat; entropy increase during repolarization would absorb heat. However, any such entropic contributions are incompatible with Hodgkin and Huxley model
239:
350:(the temperature below which the consistency changes from fluid to gel-like) only slightly below the organism's body temperature, and this allows for the propagation of solitons. An action potential traveling along a mixed nerve results in a slight increase in temperature followed by a decrease in temperature. Soliton model proponents claim that no net heat is released during the overall pulse and that the observed temperature changes are inconsistent with the Hodgkin-Huxley model. However, this is untrue: the Hodgkin Huxley model predicts a biphasic release and absorption of heat. In addition, the action potential causes a slight local thickening of the membrane and a force acting outwards; this effect is not predicted by the HodgkinâHuxley model but does not contradict it, either.
763:) to experimentally dissect the action potential in the squid giant axon, uses electronic feedback to measure the current necessary to hold membrane voltage constant at a commanded value. A silver wire, inserted into the interior of the axon, forces a constant membrane voltage along the length of the axon. Under these circumstances, there is no possibility of a traveling 'soliton'. Any thermodynamic changes are very different from those resulting from an action potential. Yet, the measured currents accurately reproduce the action potential.
715:
330:
two dimensional sound waves in the membrane by nonlinear elastic properties near a phase transition. The initial impulse can acquire a stable shape under such circumstances, in general known as a solitary wave. Solitons are the simplest solution of the set of nonlinear wave equations governing such phenomenon and were applied to model nerve impulse in 2005 by Thomas
Heimburg and Andrew D. Jackson, both at the
165:
63:
22:
362:. Indeed, such nonlinear sound waves have now been shown to exist at lipid interfaces that show superficial similarity to action potentials (electro-opto-mechanical coupling, velocities, biphasic pulse shape, threshold for excitation etc.). Furthermore, the waves remain localized in the membrane and do not spread out in the surrounding due to an impedance mismatch.
676:
those reported for ion channel proteins. They are thought to be caused by lipid membrane pores spontaneously generated by the thermal fluctuations. Such thermal fluctuations explain the specific ionic selectivity or the specific time-course of the response to voltage changes on the basis of their effect on the macroscopic susceptibilities of the system.
793:(TTX) blocks action potentials at extremely low concentrations. The site of action of TTX on the sodium channel has been identified. Dendrotoxins block the potassium channels. These drugs produce quantitatively predictable changes in the action potential. The 'soliton model' provides no explanation for these pharmacological effects.
2131:
802:
A recent theoretical model, proposed by Ahmed El Hady and
Benjamin Machta, proposes that there is a mechanical surface wave which co-propagates with the electrical action potential. These surface waves are called "action waves". In the El HadyâMachta's model, these co-propagating waves are driven by
738:
An important assumption of the soliton model is the presence of a phase transition near the ambient temperature of the axon ("Formalism", above). Then, rapid change of temperature away from the phase transition temperature would necessarily cause large changes in the action potential. Below the phase
705:
or lowering temperature, this difference can be restored back to normal, which should cancel the action of anesthetics: this is indeed observed. The amount of pressure needed to cancel the action of an anesthetic of a given lipid solubility can be computed from the soliton model and agrees reasonably
780:
The current underlying the action potential depolarization is selective for sodium. Repolarization depends on a selective potassium current. These currents have very specific responses to voltage changes which quantitatively explain the action potential. Substitution of non-permeable ions for sodium
581:
745:
Nerve impulses traveling in opposite directions annihilate each other on collision. On the other hand, mechanical waves do not annihilate but pass through each other. Soliton model proponents have attempted to show that action potentials can pass through a collision; however, collision annihilation
729:
An action potential initiated anywhere on an axon will travel in an antidromic (backward) direction to the neuron soma (cell body) without loss of amplitude and produce a full-amplitude action potential in the soma. As the membrane area of the soma is orders of magnitude larger than the area of the
771:
The patch clamp technique isolates a microscopic patch of membrane on the tip of a glass pipette. It is then possible to record currents from single ionic channels. There is no possibility of propagating solitons or thermodynamic changes. Yet, the properties of these channels (temporal response to
750:
action potentials is a routinely observed phenomenon in neuroscience laboratories and are the basis of a standard technique for identification of neurons. Solitons pass each other on collision (Figure--"Collision of
Solitons"), solitary waves in general can pass, annihilate or bounce of each other
329:
in nerve fibers and its importance for nerve pulse propagation. Based on Tasaki's work, Konrad
Kaufman proposed sound waves as a physical basis for nerve pulse propagation in an unpublished manuscript. The basic idea at the core of the soliton model is the balancing of intrinsic dispersion of the
309:
waves in general) depends on adiabatic propagation in which the energy provided at the source of excitation is carried adiabatically through the medium, i.e. plasma membrane. The measurement of a temperature pulse and the claimed absence of heat release during an action potential were the basis of
700:
such as ion channels but instead by dissolving in and changing the properties of the lipid membrane. Dissolving substances in the membrane lowers the membrane's freezing point, and the resulting larger difference between body temperature and freezing point inhibits the propagation of solitons. By
301:
to enter the cell (inward current). The resulting decrease in membrane potential opens nearby voltage-gated sodium channels, thus propagating the action potential. The transmembrane potential is restored by delayed opening of potassium channels. Soliton hypothesis proponents assert that energy is
675:
proteins on a molecular scale. It is rather assumed that their properties are implicitly contained in the macroscopic thermodynamic properties of the nerve membranes. The soliton model predicts membrane current fluctuations during the action potential. These currents are of similar appearance as
661:
for solitons in water canals. The solutions of the above equation possess a limiting maximum amplitude and a minimum propagation velocity that is similar to the pulse velocity in myelinated nerves. Under restrictive assumptions, there exist periodic solutions that display hyperpolarization and
730:
axon, conservation of energy requires that an adiabatic mechanical wave decrease in amplitude. Since the absence of heat production is one of the claimed justifications of the 'soliton model', this is particularly difficult to explain within that model.
670:
Advocates of the soliton model claim that it explains several aspects of the action potential, which are not explained by the
HodgkinâHuxley model. Since it is of thermodynamic nature it does not address the properties of single macromolecules like
739:
transition temperature, the soliton wave would not be possible. Yet, action potentials are present at 0 °C. The time course is slowed in a manner predicted by the measured opening and closing kinetics of the
Hodgkin-Huxley ion channels.
353:
The soliton model attempts to explain the electrical currents associated with the action potential as follows: the traveling soliton locally changes density and thickness of the membrane, and since the membrane contains many charged and
338:. Heimburg heads the institute's Membrane Biophysics Group. The biological physics group of Matthias Schneider has studied propagation of two-dimensional sound waves in lipid interfaces and their possible role in biological signalling
379:
2643:
Gonzalez, Alfredo; Budvytyte, Rima; Mosgaard, Lars D; Nissen, Søren; Heimburg, Thomas (10 Sep 2014). "Penetration of Action
Potentials During Collision in the Median and Lateral Giant Axons of Invertebrates".
238:
2342:
Villagran Vargas, E., Ludu, A., Hustert, R., Gumrich, P., Jackson, A.D., Heimburg, T. (2011). "Periodic solutions and refractory periods in the soliton theory for nerves and the locust femoral nerve".
2069:
Griesbauer, J; Bossinger, S; Wixforth, A; Schneider, M (19 Dec 2012). "Simultaneously propagating voltage and pressure pulses in lipid monolayers of pork brain and synthetic lipids".
859:
2760:
866:
1686:
1956:
Griesbauer, J; Bossinger, S; Wixforth, A; Schneider, M (9 May 2012). "Propagation of 2D Pressure Pulses in Lipid
Monolayers and Its Possible Implications for Biology".
2956:
2537:
2395:
2271:
2214:
1942:
1885:
1818:
1094:
1702:
80:
35:
576:{\displaystyle {\frac {\partial ^{2}\Delta \rho }{\partial t^{2}}}={\frac {\partial }{\partial x}}\left-h{\frac {\partial ^{4}\Delta \rho }{\partial x^{4}}},}
647:
are dictated by the thermodynamic properties of the nerve membrane and cannot be adjusted freely. They have to be determined experimentally. The parameter
916:
302:
mainly conserved during propagation except dissipation losses; Measured temperature changes are completely inconsistent with the
Hodgkin-Huxley model.
321:
pioneered a thermodynamic approach to the phenomenon of nerve pulse propagation which identified several phenomena that were not included in the
658:
2708:
1669:
127:
772:
voltage jumps, ionic selectivity) accurately predict the properties of the macroscopic currents measured under conventional voltage clamp.
99:
2981:
626:
describe the nature of the phase transition and thereby the nonlinearity of the elastic constants of the nerve membrane. The parameters
2163:
41:
106:
2831:
2698:
781:
abolishes the action potential. The 'soliton model' cannot explain either the ionic selectivity or the responses to voltage changes.
225:
207:
146:
49:
2022:"Evidence for two-dimensional solitary sound waves in a lipid controlled interface and its implications for biological signalling"
930:
174:
113:
2976:
689:
84:
2550:
Rall, W and Shepherd, GM (1968) Theoretical reconstructions of dendrodendritic synaptic interactions in the olfactory bulb.
852:
714:
325:. Along with measuring various non-electrical components of a nerve impulse, Tasaki investigated the physical chemistry of
1832:
Heimburg, T., Jackson, A.D. (2007). "On the action potential as a propagating density pulse and the role of anesthetics".
371:
95:
2147:
271:
1899:
Andersen, S.S.L., Jackson, A.D., Heimburg, T. (2009). "Towards a thermodynamic theory of nerve pulse propagation".
2161:
Abbott, B.C., Hill, A.V., Howarth, J.V. (1958). "The positive and negative heat associated with a nerve impulse".
836:
The 60th anniversary of the Hodgkin-Huxley model: a critical assessment from a historical and modelerâs viewpoint
294:
266:
based on a thermodynamic theory of nerve pulse propagation. It proposes that the signals travel along the cell's
817:
722:
The following is a list of some of the disagreements between experimental observations and the "soliton model":
322:
286:
963:
812:
335:
255:
2826:
Hille, Bertil (2001). Ion channels of excitable membranes (3. ed. ed.). Sunderland, Massachusetts: Sinauer.
73:
120:
1424:"Mechanical and Thermal Changes in the Torpedo Electric Organ Associated with Its Postsynaptic Potentials"
1257:"A quantitative description of membrane current and its application to conduction and excitation in nerve"
846:
178:
2950:
2895:
2531:
2389:
2265:
2228:
Iwasa, K., Tasaki I., Gibbons, R. (1980). "Swelling of nerve fibres associated with action potentials".
2208:
1936:
1879:
1812:
1512:"Rapid heat production associated with electrical excitation of the electric organs of the electric eel"
1088:
873:
985:
1551:"Rapid thermal and mechanical changes in garfish olfactory nerve associated with a propagated impulse"
2914:
2782:
2733:
2663:
2493:
2428:
2298:
2237:
2172:
2088:
1975:
1851:
1772:
1725:
1618:
1562:
1376:
1317:
889:
760:
692:
holds that the strength of a wide variety of chemically diverse anesthetics is proportional to their
331:
1463:"The heat production associated with the passage of a single impulse in pike olfactory nerve fibres"
185:
2986:
2893:
El Hady, A., Machta, B. (2015). "Mechanical surface waves accompany action potential propagation".
847:
A comparison of the Hodgkin-Huxley model and the Soliton theory for the Action Potential in Nerves
654:
2133:
NON-LINEAR SOLITARY SOUND WAVES IN LIPID MEMBRANES AND THEIR POSSIBLE ROLE IN BIOLOGICAL SIGNALING
922:
2938:
2904:
2806:
2772:
2679:
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2483:
2452:
2418:
2377:
2351:
2196:
2141:
2112:
2078:
1999:
1965:
1924:
1867:
1841:
1696:
1634:
1248:
1199:
1150:
1101:
1034:
1016:
1110:"Currents carried by sodium and potassium ions through the membrane of the giant axon of Loligo"
1047:"Currents carried by sodium and potassium ions through the membrane of the giant axon of Loligo"
2930:
2875:
2827:
2798:
2704:
2598:
2519:
2444:
2369:
2324:
2253:
2188:
2104:
2051:
1991:
1916:
1800:
1741:
1688:
Action Potentials and Electrochemical Coupling in the Macroscopic Chiral Phospholipid Membrane
1665:
1642:
1588:
1531:
1492:
1443:
1404:
1345:
1286:
1237:
1188:
1139:
1076:
1008:
905:
867:
Action Potentials and Electrochemical Coupling in the Macroscopic Chiral Phospholipid Membrane
842:
684:
The authors claim that their model explains the previously obscure mode of action of numerous
355:
259:
1716:
Xin-Yi, Wang (1985). "Solitary wave and nonequilibrium phase transition in liquid crystals".
2922:
2865:
2857:
2790:
2741:
2671:
2625:
2588:
2580:
2509:
2501:
2436:
2361:
2314:
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2180:
2096:
2041:
2033:
1983:
1908:
1859:
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1733:
1626:
1578:
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1523:
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1435:
1394:
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1335:
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1219:
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1066:
1058:
1000:
897:
835:
359:
326:
290:
2724:
Eckl, C; Mayer, A P; Kovalev, A S (3 August 1998). "Do Surface Acoustic Solitons Exist?".
1208:"The dual effect of membrane potential on sodium conductance in the giant axon of Loligo"
657:). The above equation does not contain any fit parameters. It is formally related to the
2918:
2786:
2737:
2667:
2555:
2497:
2432:
2302:
2241:
2176:
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1979:
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893:
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2514:
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2286:
2046:
2021:
1795:
1760:
1583:
1550:
1487:
1462:
1399:
1364:
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1256:
1232:
1207:
1183:
1158:
1134:
1109:
1071:
1046:
822:
1574:
1340:
1305:
945:
2970:
2629:
2569:"The Effect of Temperature on the Electrical Activity of the Giant Axon of the Squid"
1252:
1203:
1154:
1105:
1038:
347:
318:
267:
2942:
2810:
2761:"Solitary shock waves and adiabatic phase transition in lipid interfaces and nerves"
2683:
2381:
2116:
1912:
1004:
2861:
2584:
2200:
2003:
1987:
1928:
1871:
1478:
1272:
1223:
1174:
1125:
1062:
1042:
1020:
790:
251:
189:
2456:
925:, Eurekalert, according to a press release University of Copenhagen, 6 March 2007
2505:
672:
605:
is the change in membrane density under the influence of the action potential,
62:
2794:
2745:
2100:
370:
The soliton representing the action potential of nerves is the solution of the
2846:"Structure and function of voltage-gated sodium channels at atomic resolution"
2675:
2440:
2365:
2310:
1863:
747:
685:
2616:
Tasaki, Ichiji (1949). "Collision of two nerve impulses in the nerve fiber".
1737:
901:
2249:
1785:
1389:
2934:
2879:
2802:
2602:
2523:
2448:
2373:
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2192:
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2055:
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1995:
1920:
1804:
1630:
1527:
1439:
1408:
1330:
1290:
1241:
1192:
1143:
1080:
1012:
964:
Solitary acoustic waves observed to propagate at a lipid membrane interface
2257:
1745:
1592:
1535:
1496:
1447:
909:
653:
describes the frequency dependence of the sound velocity of the membrane (
2700:
Peripheral Nerve Diseases: Handbook of Clinical Neurophysiology, Volume 7
1646:
880:
Pradip Das; W.H. Schwarz (4 November 1994). "Solitons in cell membrane".
2488:
1846:
1349:
346:
The model starts with the observation that cell membranes always have a
242:
Nonlinear electro-mechanical wave measured in an artificial lipid system
2926:
2759:
Shrivastava, Shamit; Kang, Kevin; Schneider, Matthias F (30 Jan 2015).
1638:
697:
282:
278:
1365:"Metabolic cost as a unifying principle governing neuronal biophysics"
1461:
Howarth, J V; Keynes, R D; Ritchie, J M; Muralt, A von (1 Jul 1975).
1511:
1423:
1159:"The components of membrane conductance in the giant axon of Loligo"
803:
voltage changes across the membrane caused by the action potential.
1609:
Howarth, J. V. (1975). "Heat Production in Non-Myelinated Nerves".
710:
Differences between model predictions and experimental observations
696:
solubility, suggesting that they do not act by binding to specific
2909:
2777:
2658:
2423:
2356:
2083:
1970:
693:
306:
274:
237:
1306:"Molecular events and energy changes during the action potential"
263:
2136:(1st ed.). Boston, MA 02215 US: Thesis, Boston University.
358:
substances, this will result in an electrical effect, akin to
298:
158:
56:
15:
751:
and solitons are only a special case of such solitary waves.
986:"Towards a thermodynamic theory of nerve pulse propagation"
735:
Persistence of action potential over wide temperature range
702:
2285:
Griesbauer, J; Wixforth, A; Schneider, M F (15 Nov 2009).
1369:
Proceedings of the National Academy of Sciences of the USA
1363:
Hasenstaub, A; Callaway, E; Otte, S; Sejnowski, T (2010).
2020:
Shrivastava, Shamit; Schneider, Matthias (18 June 2014).
874:
Towards a thermodynamic theory of nerve pulse propagation
759:
The voltage clamp, used by Hodgkin and Huxley (1952) (
2822:
2820:
2556:
http://jn.physiology.org/content/jn/31/6/884.full.pdf
1604:
1602:
382:
1664:. Bethesda, Maryland: Academic Press Inc. (London).
1761:"On soliton propagation in biomembranes and nerves"
1428:
Biochemical and Biophysical Research Communications
919:, Princeton University Journal watch, 1 April 2015.
87:. Unsourced material may be challenged and removed.
575:
285:. The model is proposed as an alternative to the
917:Revisiting the mechanics of the action potential
1662:Physiology and Electrochemistry of Nerve Fibers
1611:Philosophical Transactions of the Royal Society
860:Physiology and Electrochemistry of Nerve Fibers
2015:
2013:
984:Andersen, S; Jackson, A; Heimburg, T (2009).
614:is the sound velocity of the nerve membrane,
8:
2955:: CS1 maint: multiple names: authors list (
2536:: CS1 maint: multiple names: authors list (
2394:: CS1 maint: multiple names: authors list (
2270:: CS1 maint: multiple names: authors list (
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1941:: CS1 maint: multiple names: authors list (
1884:: CS1 maint: multiple names: authors list (
1817:: CS1 maint: multiple names: authors list (
1759:Heimburg, T., Jackson, A.D. (12 July 2005).
1093:: CS1 maint: multiple names: authors list (
931:"Function of Nerves â Action of Anesthetics"
2409:Heimburg, T. (2010). "Lipid Ion Channels".
1304:Margineanu, D.-G; Schoffeniels, E. (1977).
50:Learn how and when to remove these messages
2472:"The thermodynamics of general anesthesia"
1701:: CS1 maint: location missing publisher (
188:. Please do not remove this message until
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226:Learn how and when to remove this message
208:Learn how and when to remove this message
147:Learn how and when to remove this message
1549:Tasaki, K; Kusano, K; Byrne, PM (1989).
713:
184:Relevant discussion may be found on the
976:
2948:
2529:
2387:
2287:"Wave Propagation in Lipid Monolayers"
2263:
2206:
2139:
2026:Journal of the Royal Society Interface
1934:
1877:
1810:
1694:
1086:
598:is the position along the nerve axon.
297:in the membrane open and allow sodium
726:Antidromic invasion of soma from axon
706:well with experimental observations.
7:
2470:Heimburg, T., Jackson, A.D. (2007).
85:adding citations to reliable sources
923:On the (sound) track of anesthetics
2164:Proceedings of the Royal Society B
1422:Tasaki, Ichiji (13 October 1995).
756:Ionic currents under voltage clamp
554:
546:
537:
513:
505:
502:
481:
469:
429:
425:
404:
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262:are initiated and conducted along
14:
31:This article has multiple issues.
2130:Shrivastava, Shamit (Jan 2014).
1510:Tasaki, I; Byrne, P. M. (1993).
281:) pulses that can be modeled as
270:in the form of certain kinds of
163:
61:
20:
1913:10.1016/j.pneurobio.2009.03.002
1005:10.1016/j.pneurobio.2009.03.002
872:Andersen, Jackson and Heimburg"
96:"Soliton model in neuroscience"
72:needs additional citations for
39:or discuss these issues on the
2862:10.1113/expphysiol.2013.071969
2585:10.1113/jphysiol.1949.sp004388
1988:10.1103/PhysRevLett.108.198103
1479:10.1113/jphysiol.1975.sp011019
1273:10.1113/jphysiol.1952.sp004764
1224:10.1113/jphysiol.1952.sp004719
1175:10.1113/jphysiol.1952.sp004718
1126:10.1113/jphysiol.1952.sp004717
1063:10.1113/jphysiol.1952.sp004717
701:increasing pressure, lowering
1:
1575:10.1016/s0006-3495(89)82902-9
372:partial differential equation
2703:. Elsevier Health Sciences.
2630:10.1016/0006-3002(49)90121-3
1765:Proc. Natl. Acad. Sci. U.S.A
777:Selective ionic conductivity
2506:10.1529/biophysj.106.099754
960:An elementary introduction.
855:, The Guardian, 1 May 2015.
258:that claims to explain how
190:conditions to do so are met
3003:
2982:Computational neuroscience
2795:10.1103/PhysRevE.91.012715
2746:10.1103/PhysRevLett.81.983
2697:Kimura, Jun (2006-06-08).
2101:10.1103/PhysRevE.86.061909
1516:Biochem Biophys Res Commun
845:, Thomas Heimburg (2012) "
295:voltage-gated ion channels
2676:10.1103/PhysRevX.4.031047
2441:10.1016/j.bpc.2010.02.018
2366:10.1016/j.bpc.2010.11.001
2311:10.1016/j.bpj.2009.07.049
1864:10.1142/S179304800700043X
1685:Kaufmann, Konrad (1989).
1467:The Journal of Physiology
853:Action Waves in the Brain
690:MeyerâOverton observation
680:Application to anesthesia
2146:: CS1 maint: location (
1738:10.1103/PhysRevA.32.3126
993:Progress in Neurobiology
966:, Phys.org June 20, 2014
902:10.1103/PhysRevE.51.3588
834:Federico Faraci (2013) "
813:Biological neuron models
659:Boussinesq approximation
336:University of Copenhagen
2850:Experimental Physiology
2726:Physical Review Letters
2250:10.1126/science.7423196
1958:Physical Review Letters
1786:10.1073/pnas.0503823102
1660:Tasaki, Ichiji (1982).
1390:10.1073/pnas.0914886107
929:Kaare GrÌsbøll (2006).
865:Konrad Kaufman (1989) "
768:Single channel currents
305:The soliton model (and
2844:Catterall, WA (2014).
2567:Hodgkin; Katz (1949).
2185:10.1098/rspb.1958.0012
2038:10.1098/rsif.2014.0098
1631:10.1098/rstb.1975.0020
1528:10.1006/bbrc.1993.2565
1440:10.1006/bbrc.1995.2514
1331:10.1073/pnas.74.9.3810
858:Ichiji Tasaki (1982) "
719:
577:
243:
2977:Cellular neuroscience
2896:Nature Communications
2344:Biophysical Chemistry
1114:Journal of Physiology
1051:Journal of Physiology
718:Collision of solitons
717:
578:
241:
2618:Biochim Biophys Acta
818:HodgkinâHuxley model
761:Hodgkin-Huxley Model
666:Role of ion channels
662:refractory periods.
380:
332:Niels Bohr Institute
323:HodgkinâHuxley model
287:HodgkinâHuxley model
81:improve this article
2919:2015NatCo...6.6697E
2787:2015PhRvE..91a2715S
2738:1998PhRvL..81..983E
2668:2014PhRvX...4c1047G
2498:2007BpJ....92.3159H
2433:2010arXiv1001.2524H
2303:2009BpJ....97.2710G
2291:Biophysical Journal
2242:1980Sci...210..338I
2177:1958RSPSB.148..149A
2093:2012PhRvE..86f1909G
1980:2012PhRvL.108s8103G
1856:2006physics..10117H
1777:2005PNAS..102.9790H
1730:1985PhRvA..32.3126X
1623:1975RSPTB.270..425H
1567:1989BpJ....55.1033T
1381:2010PNAS..10712329H
1375:(27): 12329â12334.
1322:1977PNAS...74.3810M
894:1995PhRvE..51.3588D
746:of orthodromic and
655:dispersion relation
462:
177:of this article is
2927:10.1038/ncomms7697
1834:Biophys. Rev. Lett
1691:. Caruaru, Brazil.
720:
573:
448:
248:soliton hypothesis
244:
2771:(12715): 012715.
2765:Physical Review E
2710:978-0-444-51358-8
2646:Physical Review X
2297:(10): 2710â2716.
2071:Physical Review E
1718:Physical Review A
1671:978-0-12-683780-3
882:Physical Review E
843:Ursula van Rienen
568:
520:
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327:phase transitions
291:action potentials
260:action potentials
236:
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823:Vector soliton
820:
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342:Justification
341:
339:
337:
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319:Ichiji Tasaki
313:
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92:Find sources:
86:
82:
76:
75:
70:This article
68:
64:
59:
58:
53:
51:
44:
43:
38:
37:
32:
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18:
17:
2951:cite journal
2900:
2894:
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2856:(1): 35â51.
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2849:
2839:
2768:
2764:
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2551:
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2532:cite journal
2479:
2475:
2465:
2414:
2410:
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2390:cite journal
2347:
2343:
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2266:cite journal
2233:
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2209:cite journal
2168:
2162:
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2132:
2125:
2074:
2070:
2064:
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1904:
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1837:
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1813:cite journal
1768:
1764:
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1089:cite journal
1054:
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996:
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979:
953:. Retrieved
946:the original
941:
937:
885:
881:
801:
798:Action waves
791:tetrodotoxin
786:Pharmacology
721:
683:
669:
649:
643:
637:
628:
622:
616:
607:
601:
594:
592:is time and
588:
585:
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352:
345:
317:
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252:neuroscience
247:
245:
222:
204:
195:
173:
143:
134:
124:
117:
110:
103:
91:
79:Please help
74:verification
71:
47:
40:
34:
33:Please help
30:
2624:: 494â497.
686:anesthetics
673:ion channel
2987:Biophysics
2971:Categories
2573:J. Physiol
2476:Biophys. J
1249:Hodgkin AL
1200:Hodgkin AL
1151:Hodgkin AL
1102:Hodgkin AL
1035:Hodgkin AL
971:References
955:2007-03-11
748:antidromic
742:Collisions
175:neutrality
107:newspapers
36:improve it
2910:1407.7600
2778:1411.2454
2659:1404.3643
2424:1001.2524
2357:1006.3281
2142:cite book
2084:1211.4105
1971:1211.4104
1840:: 57â78.
1697:cite book
1555:Biophys J
1261:J Physiol
1253:Huxley AF
1212:J Physiol
1204:Huxley AF
1163:J Physiol
1155:Huxley AF
1106:Huxley AF
1039:Huxley AF
789:The drug
555:∂
550:ρ
547:Δ
538:∂
528:−
514:∂
509:ρ
506:Δ
503:∂
486:ρ
482:Δ
473:ρ
470:Δ
430:∂
426:∂
405:∂
400:ρ
397:Δ
388:∂
366:Formalism
289:in which
186:talk page
42:talk page
2943:17462621
2935:25819404
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