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Soliton model in neuroscience

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The soliton hypothesis in neuroscience is a model that claims to explain how action potentials are initiated and conducted along axons based on a thermodynamic theory of nerve pulse propagation. It proposes that the signals travel along the cell's membrane in the form of certain kinds of solitary sound (or density) pulses that can be modeled as solitons. The model is proposed as an alternative to the Hodgkin–Huxley model in which action potentials: voltage-gated ion channels in the membrane open and allow sodium ions 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 mainly conserved during propagation except dissipation losses; Measured temperature changes are completely inconsistent with the Hodgkin-Huxley model.

Nonlinear electro-mechanical wave measured in an artificial lipid system

The soliton model (and sound 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 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

Contents

Ichiji Tasaki pioneered a thermodynamic approach to the phenomenon of nerve pulse propagation which identified several phenomena that were not included in the Hodgkin–Huxley model. Along with measuring various non-electrical components of a nerve impulse, Tasaki investigated the physical chemistry of phase transitions 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 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 Niels Bohr Institute of the University of Copenhagen. 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

The model starts with the observation that cell membranes always have a freezing point (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 Hodgkin–Huxley model but does not contradict it, either.

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 polar substances, this will result in an electrical effect, akin to piezoelectricity. 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.

The soliton representing the action potential of nerves is the solution of the partial differential equation

2 Δ ρ t 2 = x [ ( c 0 2 + p Δ ρ + q Δ ρ 2 ) Δ ρ x ] h 4 Δ ρ x 4 , {\displaystyle {\frac {\partial ^{2}\Delta \rho }{\partial t^{2}}}={\frac {\partial }{\partial x}}\left[\left(c_{0}^{2}+p\Delta \rho +q\Delta \rho ^{2}\right){\frac {\partial \Delta \rho }{\partial x}}\right]-h{\frac {\partial ^{4}\Delta \rho }{\partial x^{4}}},}

wheret is time andx is the position along the nerve axon.Δρ is the change in membrane density under the influence of the action potential,c0 is the sound velocity of the nerve membrane,p andq describe the nature of the phase transition and thereby the nonlinearity of the elastic constants of the nerve membrane. The parametersc0,p andq are dictated by the thermodynamic properties of the nerve membrane and cannot be adjusted freely. They have to be determined experimentally. The parameterh describes the frequency dependence of the sound velocity of the membrane (dispersion relation). The above equation does not contain any fit parameters. It is formally related to the Boussinesq approximation 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 refractory periods.

Advocates of the soliton model claim that it explains several aspects of the action potential, which are not explained by the Hodgkin–Huxley model. Since it is of thermodynamic nature it does not address the properties of single macromolecules like ion channel 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 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.

The authors claim that their model explains the previously obscure mode of action of numerous anesthetics. The Meyer–Overton observation holds that the strength of a wide variety of chemically diverse anesthetics is proportional to their lipid solubility, suggesting that they do not act by binding to specific proteins 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 increasing pressure, lowering pH 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 well with experimental observations.

Collision of solitons

The following is a list of some of the disagreements between experimental observations and the "soliton model":

Antidromic invasion of soma from axon
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 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.[citation needed]
Persistence of action potential over wide temperature range
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 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.
Collisions
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 of orthodromic and antidromic 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 and solitons are only a special case of such solitary waves.
Ionic currents under voltage clamp
The voltage clamp, used by Hodgkin and Huxley (1952) (Hodgkin-Huxley Model) 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.[citation needed]
Single channel currents
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 voltage jumps, ionic selectivity) accurately predict the properties of the macroscopic currents measured under conventional voltage clamp.
Selective ionic conductivity
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 abolishes the action potential. The 'soliton model' cannot explain either the ionic selectivity or the responses to voltage changes.
Pharmacology
The drug tetrodotoxin (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.

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 Hady–Machta's model, these co-propagating waves are driven by voltage changes across the membrane caused by the action potential.

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Soliton model in neuroscience
Soliton model in neuroscience Language Watch Edit This article has multiple issues Please help improve it or discuss these issues on the talk page Learn how and when to remove these template messages This article needs additional citations for verification Please help improve this article by adding citations to reliable sources Unsourced material may be challenged and removed Find sources Soliton model in neuroscience news newspapers books scholar JSTOR May 2015 Learn how and when to remove this template message The template below POV is being considered for merging See templates for discussion to help reach a consensus The neutrality of this article is disputed Relevant discussion may be found on the talk page Please do not remove this message until conditions to do so are met May 2015 Learn how and when to remove this template message Learn how and when to remove this template message The soliton hypothesis in neuroscience is a model that claims to explain how action potentials are initiated and conducted along axons based on a thermodynamic theory of nerve pulse propagation 1 It proposes that the signals travel along the cell s membrane in the form of certain kinds of solitary sound or density pulses that can be modeled as solitons The model is proposed as an alternative to the Hodgkin Huxley model 2 in which action potentials voltage gated ion channels in the membrane open and allow sodium ions 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 mainly conserved during propagation except dissipation losses Measured temperature changes are completely inconsistent with the Hodgkin Huxley model 3 4 Nonlinear electro mechanical wave measured in an artificial lipid system The soliton model and sound 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 5 6 were the basis of 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 7 In the garfish olfactory nerve the action potential is associated with a biphasic temperature change however there is a net production of heat 8 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 9 Contents 1 History 2 Justification 3 Formalism 4 Role of ion channels 5 Application to anesthesia 6 Differences between model predictions and experimental observations 7 Action waves 8 See also 9 Sources 10 ReferencesHistory EditIchiji Tasaki pioneered a thermodynamic approach to the phenomenon of nerve pulse propagation which identified several phenomena that were not included in the Hodgkin Huxley model 10 Along with measuring various non electrical components of a nerve impulse Tasaki investigated the physical chemistry of phase transitions 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 11 The basic idea at the core of the soliton model is the balancing of intrinsic dispersion of the 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 12 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 13 14 15 both at the Niels Bohr Institute of the University of Copenhagen 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 16 17 18 19 Justification EditThe model starts with the observation that cell membranes always have a freezing point 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 20 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 9 In addition the action potential causes a slight local thickening of the membrane and a force acting outwards 21 this effect is not predicted by the Hodgkin Huxley model but does not contradict it either 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 polar substances this will result in an electrical effect akin to piezoelectricity 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 17 Furthermore the waves remain localized in the membrane and do not spread out in the surrounding due to an impedance mismatch 22 Formalism EditThe soliton representing the action potential of nerves is the solution of the partial differential equation 2 D r t 2 x c 0 2 p D r q D r 2 D r x h 4 D r x 4 displaystyle frac partial 2 Delta rho partial t 2 frac partial partial x left left c 0 2 p Delta rho q Delta rho 2 right frac partial Delta rho partial x right h frac partial 4 Delta rho partial x 4 where t is time and x is the position along the nerve axon Dr is the change in membrane density under the influence of the action potential c0 is the sound velocity of the nerve membrane p and q describe the nature of the phase transition and thereby the nonlinearity of the elastic constants of the nerve membrane The parameters c0 p and q are dictated by the thermodynamic properties of the nerve membrane and cannot be adjusted freely They have to be determined experimentally The parameter h describes the frequency dependence of the sound velocity of the membrane dispersion relation The above equation does not contain any fit parameters It is formally related to the Boussinesq approximation 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 refractory periods 23 Role of ion channels EditAdvocates of the soliton model claim that it explains several aspects of the action potential which are not explained by the Hodgkin Huxley model Since it is of thermodynamic nature it does not address the properties of single macromolecules like ion channel 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 those reported for ion channel proteins 24 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 Application to anesthesia EditThe authors claim that their model explains the previously obscure mode of action of numerous anesthetics The Meyer Overton observation holds that the strength of a wide variety of chemically diverse anesthetics is proportional to their lipid solubility suggesting that they do not act by binding to specific proteins 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 25 By increasing pressure lowering pH 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 well with experimental observations Differences between model predictions and experimental observations Edit Collision of solitons The following is a list of some of the disagreements between experimental observations and the soliton model Antidromic invasion of soma from axon 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 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 26 citation needed Persistence of action potential over wide temperature range 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 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 27 Collisions Nerve impulses traveling in opposite directions annihilate each other on collision 28 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 29 however collision annihilation of orthodromic and antidromic action potentials is a routinely observed phenomenon in neuroscience laboratories and are the basis of a standard technique for identification of neurons 30 Solitons pass each other on collision Figure Collision of Solitons solitary waves in general can pass annihilate or bounce of each other 31 and solitons are only a special case of such solitary waves 32 Ionic currents under voltage clamp The voltage clamp used by Hodgkin and Huxley 1952 Hodgkin Huxley Model 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 citation needed Single channel currents 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 voltage jumps ionic selectivity accurately predict the properties of the macroscopic currents measured under conventional voltage clamp 33 Selective ionic conductivity 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 abolishes the action potential The soliton model cannot explain either the ionic selectivity or the responses to voltage changes Pharmacology The drug tetrodotoxin TTX blocks action potentials at extremely low concentrations The site of action of TTX on the sodium channel has been identified 34 Dendrotoxins block the potassium channels These drugs produce quantitatively predictable changes in the action potential 33 The soliton model provides no explanation for these pharmacological effects Action waves EditA 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 35 In the El Hady Machta s model these co propagating waves are driven by voltage changes across the membrane caused by the action potential See also EditBiological neuron models Hodgkin Huxley model Vector solitonSources EditFederico Faraci 2013 The 60th anniversary of the Hodgkin Huxley model a critical assessment from a historical and modeler s viewpoint Revathi Appali Ursula van Rienen Thomas Heimburg 2012 A comparison of the Hodgkin Huxley model and the Soliton theory for the Action Potential in Nerves Action Waves in the Brain The Guardian 1 May 2015 Ichiji Tasaki 1982 Physiology and Electrochemistry of Nerve Fibers Konrad Kaufman 1989 Action Potentials and Electrochemical Coupling in the Macroscopic Chiral Phospholipid Membrane Andersen Jackson and Heimburg Towards a thermodynamic theory of nerve pulse propagation Pradip Das W H Schwarz 4 November 1994 Solitons in cell membrane Physical Review E 51 4 3588 3612 Bibcode 1995PhRvE 51 3588D doi 10 1103 PhysRevE 51 3588 PMID 9963042 Revisiting the mechanics of the action potential Princeton University Journal watch 1 April 2015 On the sound track of anesthetics Eurekalert according to a press release University of Copenhagen 6 March 2007 Kaare Graesboll 2006 Function of Nerves Action of Anesthetics PDF Gamma 143 An elementary introduction Solitary acoustic waves observed to propagate at a lipid membrane interface Phys org June 20 2014References Edit Andersen S Jackson A Heimburg T 2009 Towards a thermodynamic theory of nerve pulse propagation PDF Progress in Neurobiology 88 2 104 113 doi 10 1016 j pneurobio 2009 03 002 PMID 19482227 S2CID 2218193 Hodgkin AL Huxley AF Katz B 1952 Currents carried by sodium and potassium ions through the membrane of the giant axon of Loligo Journal of Physiology 116 4 424 448 doi 10 1113 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Heimburg T Jackson A D 12 July 2005 On soliton propagation in biomembranes and nerves Proc Natl Acad Sci U S A 102 2 9790 9795 Bibcode 2005PNAS 102 9790H doi 10 1073 pnas 0503823102 PMC 1175000 PMID 15994235 CS1 maint multiple names authors list link Heimburg T Jackson A D 2007 On the action potential as a propagating density pulse and the role of anesthetics Biophys Rev Lett 2 57 78 arXiv physics 0610117 Bibcode 2006physics 10117H doi 10 1142 S179304800700043X S2CID 1295386 CS1 maint multiple names authors list link Andersen S S L Jackson A D Heimburg T 2009 Towards a thermodynamic theory of nerve pulse propagation Prog Neurobiol 88 2 104 113 doi 10 1016 j pneurobio 2009 03 002 PMID 19482227 S2CID 2218193 CS1 maint multiple names authors list link dead link 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 Physical Review Letters 108 19 198103 arXiv 1211 4104 Bibcode 2012PhRvL 108s8103G doi 10 1103 PhysRevLett 108 198103 PMID 23003093 a b Shrivastava Shamit Schneider Matthias 18 June 2014 Evidence for two dimensional solitary sound waves in a lipid controlled interface and its implications for biological signalling Journal of the Royal Society Interface 11 97 20140098 doi 10 1098 rsif 2014 0098 PMC 4078894 PMID 24942845 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 Physical Review E 86 6 061909 arXiv 1211 4105 Bibcode 2012PhRvE 86f1909G doi 10 1103 PhysRevE 86 061909 PMID 23367978 S2CID 25259498 Shrivastava Shamit Jan 2014 NON LINEAR SOLITARY SOUND WAVES IN LIPID MEMBRANES AND THEIR POSSIBLE ROLE IN BIOLOGICAL SIGNALING 1st ed Boston MA 02215 US Thesis Boston University CS1 maint location link Abbott B C Hill A V Howarth J V 1958 The positive and negative heat associated with a nerve impulse Proceedings of the Royal Society B 148 931 149 187 Bibcode 1958RSPSB 148 149A doi 10 1098 rspb 1958 0012 PMID 13518134 S2CID 2252017 CS1 maint multiple names authors list link Iwasa K Tasaki I Gibbons R 1980 Swelling of nerve fibres associated with action potentials Science 210 4467 338 9 Bibcode 1980Sci 210 338I doi 10 1126 science 7423196 PMID 7423196 CS1 maint multiple names authors list link Griesbauer J Wixforth A Schneider M F 15 Nov 2009 Wave Propagation in Lipid Monolayers Biophysical Journal 97 10 2710 2716 Bibcode 2009BpJ 97 2710G doi 10 1016 j bpj 2009 07 049 PMC 2776282 PMID 19917224 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 Biophysical Chemistry 153 2 3 159 167 arXiv 1006 3281 doi 10 1016 j bpc 2010 11 001 PMID 21177017 S2CID 15106768 CS1 maint multiple names authors list link Heimburg T 2010 Lipid Ion Channels Biophys Chem 150 1 3 2 22 arXiv 1001 2524 Bibcode 2010arXiv1001 2524H doi 10 1016 j bpc 2010 02 018 PMID 20385440 S2CID 926828 Heimburg T Jackson A D 2007 The thermodynamics of general anesthesia Biophys J 92 9 3159 65 arXiv physics 0610147 Bibcode 2007BpJ 92 3159H doi 10 1529 biophysj 106 099754 PMC 1852341 PMID 17293400 CS1 maint multiple names authors list link Rall W and Shepherd GM 1968 Theoretical reconstructions of dendrodendritic synaptic interactions in the olfactory bulb J Neurophysiol 31 884 915 http jn physiology org content jn 31 6 884 full pdf Hodgkin Katz 1949 The Effect of Temperature on the Electrical Activity of the Giant Axon of the Squid J Physiol 109 1 2 240 249 doi 10 1113 jphysiol 1949 sp004388 PMC 1392577 PMID 15394322 Tasaki Ichiji 1949 Collision of two nerve impulses in the nerve fiber Biochim Biophys Acta 3 494 497 doi 10 1016 0006 3002 49 90121 3 Gonzalez Alfredo Budvytyte Rima Mosgaard Lars D Nissen Soren Heimburg Thomas 10 Sep 2014 Penetration of Action Potentials During Collision in the Median and Lateral Giant Axons of Invertebrates Physical Review X 4 3 031047 arXiv 1404 3643 Bibcode 2014PhRvX 4c1047G doi 10 1103 PhysRevX 4 031047 S2CID 17503341 Sander HW Collision Testing in J Kimura Peripheral Nerve Diseases https books google com books id jp05zU9vxo8C amp pg PA359 amp lpg PA359 amp dq collision test neurophysiology amp source bl amp ots PTz3H5Mn t amp sig LUtPKvs1ad8q0wX8zIQ712mNG7E amp hl en amp sa X amp ei dE9iVfvlENOMyASFz4OQAg amp ved 0CB0Q6AEwADgK v onepage amp q collision 20test 20neurophysiology amp f false Eckl C Mayer A P Kovalev A S 3 August 1998 Do Surface Acoustic Solitons Exist Physical Review Letters 81 5 983 986 Bibcode 1998PhRvL 81 983E doi 10 1103 PhysRevLett 81 983 Shrivastava Shamit Kang Kevin Schneider Matthias F 30 Jan 2015 Solitary shock waves and adiabatic phase transition in lipid interfaces and nerves Physical Review E 91 12715 012715 arXiv 1411 2454 Bibcode 2015PhRvE 91a2715S doi 10 1103 PhysRevE 91 012715 PMID 25679650 S2CID 12034915 a b Hille Bertil 2001 Ion channels of excitable membranes 3 ed ed Sunderland Massachusetts Sinauer ISBN 9780878933211 Catterall WA 2014 Structure and function of voltage gated sodium channels at atomic resolution Experimental Physiology 99 1 35 51 doi 10 1113 expphysiol 2013 071969 PMC 3885250 PMID 24097157 El Hady A Machta B 2015 Mechanical surface waves accompany action potential propagation Nature Communications 6 6697 arXiv 1407 7600 Bibcode 2015NatCo 6 6697E doi 10 1038 ncomms7697 PMID 25819404 S2CID 17462621 CS1 maint multiple names authors list link Retrieved from https en wikipedia org w index php title Soliton model in neuroscience amp oldid 1045377422, wikipedia, wiki, book,

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