aa r X i v : . [ phy s i c s . b i o - ph ] S e p How is information transmitted in a nerve?
Michel Peyrard ∗ Laboratoire de Physique de l’Ecole Normale Sup´erieure de Lyon, 46 All´ee d’Italie, 69364 Lyon C´edex 07, France (Dated: October 14, 2020)In the last fifteen years a debate emerged about the validity of the famous Hodgkin-Huxley modelfor nerve impulse. Mechanical models have been proposed. This note reviews the experimentalproperties of the nerve impulse and discusses the proposed alternatives. The experimental data,which rule out some of the alternative suggestions, show that, while the Hodgkin-Huxley modelmay not be complete, it nevertheless includes essential features that should not be overlooked in theattempts made to improve, or supersede, it.
Revised version, accepted for publication in the Journal of Biological Physics.
I. INTRODUCTION
In high-school biology courses a standard experimentshows how a small voltage applied to a dead-frog mus-cle can induce its contraction. Actually it reproducesthe first observation made by Luigi Galvani in the 18 th century. In 1850 Hermann von Helmholtz designed anexperiment to measure the velocity of the signal thatpropagates along the sciatic nerve of a frog [1, 2]. Aquantitative description of the propagation of an elec-trical signal in a nerve was proposed in 1952 by A.L.Hodgkin and A.L. Huxley [3] after a careful series of ex-perimental investigations. For a long time this Hodgkin-Huxley model, recognised in 1963 by the Nobel Prize forPhysiology or Medicine, stayed as the unquestioned basicmodel of this phenomenon, which launched a new fieldof research [4, 5]. According to this model the nerve im-pulse is due to voltage-controlled flows of sodium andpotassium ions through the axon membrane.However the phenomena are more complex. Experi-ments also detect heat transfer and a slight deformationof the axon together with the electrical signal. In the lastfifteen years, this lead some scientists to raise questionsabout the Hodgkin-Huxley model and even to proposean alternative picture in which the propagation of a me-chanical signal is the main feature [6, 7].This article reviews the main experimental data accu-mulated over more than a century on the propagationof the nerve impulse, paying attention to some aspectswhich are particularly relevant for the ongoing discus-sion on the validity of the Hodgkin-Huxley model. Thenit discusses the proposals to replace, or complete, thismodel. The final discussion contains some comments onthese attempts. ∗ E-mail: [email protected]
II. EXPERIMENTAL STUDIES OF THEELECTRICAL PROPERTIES OF AXONS
The German school of “organic physicists” played amajor role in creating modern physiology in the sec-ond half of the 19 th century [8]. Hermann von Hel-moltz (1821–1894), who worked in Heidelberg, was thefirst to measure the velocity of the signal along a nerve,and Julius Bernstein (1839 – 1917), who was trained un-der von Helmholtz, designed a clever apparatus whichallowed him to record the shape of the nerve impulse [9].Then he related the potential difference across the mem-brane to the Nernst theory in a paper which founded thetheoretical analysis of the phenomenon [10]. However, inhis studies Bernstein focused his attention on the neg-ative part of the pulse, which is associated to a potas-sium flux. It is Ernest Overton (1865 – 1933), working inW¨urzburg, who pointed out the essential role of sodiumfor the excitation of a muscle [11].Following this earlier period, the most significant re-sults came from England, with the work of Alan LloydHodgkin (1914 – 1998) (Trinity College, Cambridge)and his co-workers, particularly Andrew Fielding Huxley(1917 – 2012), and from the USA where Kenneth Stew-art Cole (1900 – 1984) and Howard James Curtis (1906 –1972) (Columbia University, New York) developed pow-erful experimental methods. After a first visit there in1939, Hodgkin kept collaborating with this group, whichsignificantly influenced his research. These results areimportant in the present context of the discussion ofthe Hodgkin-Huxley model because they, unambiguously,demonstrated that a flow of ions across the axon mem-brane was determining the shape of the voltage pulsewhich carries information along a nerve. The course ofthe work, which is a mixture of careful planning and ac-cidental observations has been vividly reported in a con-ference of Hodgkin in 1976 [12]. A. Evidence for electrical transmission in nerve(Hodgkin-1937)
The goal of this early study [13] was to determinewhether a local excitation was able to excite a neigh-bouring region, i.e. to get an evidence that a signal wasactually propagating along the nerve. Hodgkin created ablocking region by freezing the axon over 3 to 5 mm, orby pressing it between two blocks of ebonite. Althoughthe electrical pulse could not pass through, he noticedthat some signal was nevertheless leaking through theblocking domain, and was making the nerve highly ex-citable on the other side. Following the arrival of a pulseon one side of the blocked region, on the other side anew signal could be generated by a much smaller elec-trical excitation than in an unperturbed region of theaxon. Although Hodgkin did not discuss any mechanicalresponse of the axon, in the context of the present dis-cussions on the nature of the signal, this example showsthat freezing the mechanical displacements in some re-gion of the axon does not block the transmission of somesignal. This pleads against a purely mechanical drivingmechanism for the nerve impulse.
B. Searching for the mechanism of the nerveimpulse
Hodgkin alone or with the help of different collabo-rators in Cambridge, as well as Curtis and Cole in theUSA, carried an impressive series of systematic experi-ments which provided the basis for the development ofthe Hodgkin-Huxley model. The results of these mea-surements should not be overlooked in any theory of thenerve impulse.In an attempt to detect local effects of the membraneexcitation, Hodgkin was able to isolate a single unmyeli-nated nerve fibre of a crab (with a great piece of luck ashe says himself [12]) and this allowed him to detect sub-threshold potentials, i.e. signals which are not sufficientto launch a nerve impulse but nevertheless locally modifythe properties of the membrane [14]. This was the firstevidence of a local response in a nerve fibre.A decisive step was made by Cole and Curtis [15], in-spired by the “membrane theory” of Bernstein [10] whopostulated a decrease of the membrane resistance duringa pulse. They managed to measure the impedance of themembrane of the giant axon of a squid during the passageof a pulse created several centimetres away, using a highlysensitive electrical bridge. The choice of the squid axonwas an element of their success because it has a diame-ter which can reach 0 . − − ◦ C)which reduced the conduction velocity and enhanced thesignal captured by the bridge.The measurements only detected a small change of themembrane capacity but a big drop of its resistance whenthe voltage pulse passed. The change of the capacitydepends on the frequency but never excess 10%, with anaverage of 2%. In the rest state the membrane of the axonis non-conducting, but, during the passage of a pulse,Cole and Curtis found that its resistance dropped by a factor of about 36. The time of rise of the conductancecould not be precisely determined but it was estimatedto be less than 100 µ s.The experiment of Cole and Curtis did not investigatethe role of specific ions. This was further studied in aseries of experiments which varied the ionic content ofthe medium around the nerve fibres.Hodgkin [16] showed that the velocity of the pulse de-creases by about 30% when the outside medium changesfrom sea water to oil, which is an important indicationfor the models.Curtis and Cole [17] made measurements with one elec-trode inside the axon and another outside. For the firsttime it gave a precise view of the membrane potential.The resting potential ( V inside − V outside in Hodgkin-Huxleynotation) is about −
50 mV, while the action potentialis positive and differs by about 110 mV from the rest-ing potential. Replacing the
N a + ions of the outsidemedium by K + causes a dramatic drop of the action po-tential. The values of the resting and action potentialswere later confirmed by Hodgkin and Huxley [18], but, asrecognised by Huxley himself [19] the explanations thatthey proposed for the sign reversal within the pulse werewrong. It is only soon after that they began to con-sider an increase in membrane permeability specific forsodium ions, which was confirmed by a study by Hodgkinand Katz [20]. All these experiments are very challengingbecause the electrodes can become polarised during themeasurement, and thus alter the potentials to be mea-sured. Hodgkin and Katz developed specific electrodesthat they could introduce inside the axon and their paperdescribes several tests that they made to validate theirresults. Using various external solutions, they demon-strated the crucial role of the sodium, and the sharp riseof membrane permeability to sodium when the action po-tential arrives. In agreement with Overton’s observations[11] they could show that lithium ions also show an ef-fect very similar to sodium although, on the long term,lithium damages the axon. Therefore all data started tofit together nicely to set the stage of the Hodgkin-Huxleymodel proposed in 1952 [3].This picture, inferred from experiments immersing ax-ons in various solutions, was later confirmed by a directobservation of the flow of ions through the membrane ina series of papers by Hodgkin and Keynes in 1955 us-ing radioactive tracers [21, 22]. The experiments showedthat, during the nerve impulse, both N a + and K + movedown concentration gradients, i.e. their transport is pas-sive , contrary to the slower transport which brings theaxon back to its rest state in the recovery process. Ituses metabolic energy, it is highly temperature depen-dent, and can be inhibited by dinitrophenol contrary tothe passive transport during the pulse. In these voltageclamp experiment, which can impose a fixed potentialdifference even if the concentration of the ions is mod-ified, Hodgkin and Keynes managed to show that thepotassium flux is not proportional to concentration butincreases more steeply [22]. This convinced them thatthe ions do not move independently from each other, andthey showed that all their observations could be well re-produced by a model in which the K + ions move along achain of potassium selective sites which stretch throughthe membrane, and that all n sites in each chain are occu-pied by a potassium. It is quite remarkable that a carefulanalysis of macroscopic experiments managed to deter-mine the features of the ion channels which were detectedonly years later.Another study by Hodgkin and Katz [23] is also veryimportant in the context of current discussions on thebasic mechanism of the nerve impulse. It is their inves-tigation of the effect of temperature on the electrical ac-tivity of the giant axon of the squid from − ◦ C to 40 ◦ C.The resting potential was practically constant up to 20 ◦ Cand dropped at higher temperature. The action potentialshowed a gradual evolution, with a slight decrease in am-plitude up to 20 ◦ C and then a faster drop above 30 ◦ C.Over the whole temperature range, the change in thetime scale of the pulse is gradual, but very significant. Atlower temperature the nerve pulse becomes very broad,and moreover the rise and fall times have different tem-perature dependencies. The fall time grows much morethan the rise time when temperature decreases. This leadHodgkin and Katz to suggest different mechanisms forthese two processes, which is consistent with the currentknowledge that they involve different ion channels.
III. THE NERVE IMPULSE IS NOT ONLY ANELECTRICAL SIGNAL
Although a large part of the efforts to understand thetransmission of signals along nerves focused on the elec-trical aspects, the phenomena are more complex as shownby calorimetric and mechanical measurements.
A. Thermal effects
In 1848 Helmholtz failed to detect any heat effect as-sociated to the nerve impulse, and the data remainedcontroversial for several decades [24]. It is only morethan eighty years later that reliable evidences of a veryweak heat effect associated to the nerve impulse could beobtained but repeated stimulation was necessary so thatthe relative timing between the heat release or absorp-tion and the electrical pulse could not be determined [24].A very small resting heat production could also be de-tected by putting a frog nerve in a nitrogen atmosphere.Depriving the nerve from oxygen appears to stop someoxidative processes, leading to a slow decline of the rest-ing heat production.More modern measurements managed to follow the de-tails of the heat exchanges for a single impulse in non-myelinated nerves [25, 26]. The pulse causes first anemission of heat, and, in a second stage an absorption which almost compensates the emission. The measure-ments show a gradual evolution between 4 ◦ C and 15 ◦ C.The magnitude of the positive heat decreases and theinterval between the stimulus and the negative heat in-creases. Replacing sodium by lithium does not changethe overall picture but the heat emission is reduced byabout 20% while the absorption increases by a similaramount. In the whole temperature range, there is a closeagreement between the heat emission and the rising phaseof the the action potential, and between the absorptionand the falling phase of the potential.It is tempting to connect the heat effects to the en-ergy needed to charge or discharge a capacitance, but thequantitative analysis shows that this simple “condensermodel” does not account for all the observed heat ex-changes. The authors of these measurements speculatedthat a great part of the heat exchanges could come fromchanges in the entropy of the nerve membrane when it isdepolarised and repolarised.However the “condenser model” is clearly oversimpli-fied. For a membrane in a conducting fluid the chargedistribution is not only located at the surface. Very re-cently a new thermodynamics analysis has been carriedout [27]. A full electrostatic model of the charged bi-layer has been established. Assuming that the equilibra-tion of the diffuse layer is sufficiently fast compared tothe dynamics of the action potential, the paper uses aPoisson-Boltzmann approach to derive the charge distri-bution, and then compute the electrostatic energy. Theentropy associated to the electric field takes into accountthe polarisation of the water dipoles in the diffuse layerswhich reduces entropy, as well as the entropy changes in-side the lipid membrane which can be deduced from thetemperature dependence of the membrane capacitance.The results support the idea that the heat exchangesmeasured when the nerve impulse propagates have anelectrostatic origin. The results heavily depend on themembrane surface charge and only a calorimetric mea-surement performed together with the recording of thetransmembrane potential with an electrode inside theaxon might fully confirm the electrostatic origin of theheat of nervous conduction, but, as discussed in detail in[27], this looks highly plausible.
B. Mechanical and structural changes
Early measurements showed signs that the excitationpotential is accompanied by some mechanical and struc-tural changes in the axon [28]. Small changes in turbid-ity and birefringence were observed. Axons immersed inanilinonaphtalene sulfonic acid, the fluorescence of whichis extremely sensitive to conformational changes of vari-ous macromolecules, also showed fluorecence changes as-sociated to nerve excitation [28].Laser interferometry managed to detect rapid changesin the diameter of an axon, which take place when anaction potential progresses along the giant axon of thecrayfish [29]. The recorded deformation starts 250 µ s after the excitation by the pulse and starts by a contrac-tion which peaks 400 µ s later. Then the diameter returnsto normal before showing a slow expansion to finally re-cover its initial size after about 4 ms. The simultaneousrecording of the electrical and mechanical pulses showsthat, besides the delay of the mechanical deformation,one also observes that this deformation occurs on a sig-nificantly longer time scale. The overall displacement isvery small, ranging from 3 ˚A to 25 ˚A in different nervepreparations. It decreases when the nerve deteriorates.These experiments could also show that the deformationis directly linked to the action potential. If the electricalstimulation of the axon is reduced to 90% of the valuethat triggers a pulse, the mechanical deformation of theaxon is not observed.This result was later confirmed by optical measure-ments using the near field at the end of an optical fibrebrought in close proximity of the axon and by piezo-electric measurements of the pressure at the axon sur-face [30]. The mechanical displacement reaches 50 to100 ˚A for crab nerves [31]. Clues on the mechanismof the swelling of axons were provided by studying vol-ume transitions observed in synthetic and natural ionicgels by varying the ratio of monovalent and bivalent ions[32]. Changing the concentration in N a + , Li + , K + ionsof a medium containing gel beads showed sharp transi-tions of the bead diameters, associated to thermal effects,which could be understood as structural transitions in thegel. Similar observations were made with the squid giantaxon [33, 34], suggesting that the change in ionic con-centrations around the membrane, induced by the nerveimpulse, could lead to transitions in the membrane struc-ture responsible for the mechanical deformation associ-ated to the pulse. Heat exchanges, which accompaniesthese transitions could contribute to the thermal effectsrecorded when a pulse passes. IV. THE DEBATE ON THEHODGKIN-HUXLEY MODELA. The Hodgkin-Huxley model
In the famous article which introduced their model [3],Hodgkin and Huxley explain that it is built upon a seriesof measurements of the flow of electric currents throughthe surface membrane of the squid giant axon. Fromtheir experiments, they deduced that the main featuresof the nerve impulse result from a transient increase ofthe sodium conductance of the membrane, which leads toa strong flow of
N a + ions towards the inside of the axongenerated by the gradient of sodium concentration acrossthe membrane. This step, leading to a rise of the actionpotential V = V inside − V outside , is followed by a slowerbut maintained increase in potassium conductance. Inthe rest state the potassium concentration is greater in- side the axon than outside, so that potassium tends toflow out of the axon, causing a decrease of the actionpotential, which, after a small overshoot, finally comesback to its resting value. However Hodgkin and Hux-ley went well beyond this qualitative picture. From theirmeasurements they managed to propose a set of differ-ential equations which model the sodium and potassiumconductances. These equations could not be solved ana-lytically but Hodgkin and Huxley relied on a pioneeringapproach to obtain solutions that they could quantita-tively compare with their experiments. They performedwhat was probably the first numerical simulation in bio-logical physics. However, as the the EDSAC (Electronicdelay storage automatic calculator), built in the Cam-bridge Mathematical Laboratory, which they later usedfor their studies, was not immediately available, they hadto carry lengthy calculations on a manual calculator [5].Their article [3] contains a detailed section on the nu-merical methods, which explains for instance how an it-erative scheme could be used to determine one unknownparameter, the propagation speed of the impulse. Thisapproach allowed a thorough test of the model, not onlyfor the pulse but also for subthreshold responses.The model is clearly only focused on the electrical as-pects of signal propagation along nerves. Hodgkin andHuxley took a great care to discuss the limitations oftheir model, but did not discuss other physical phenom-ena, such as thermal effects. although they were certainlyaware of their existence. Presumably they considered theaction potential to be the dominant phenomenon. More-over the model has not been established from the basicprinciple of physics and chemistry, which would have nat-urally introduced other phenomena, for instance througha thermodynamics analysis. In the discussion of the pa-per the authors wrote that “the agreement [with exper-iments] must not be taken as evidence that our equa-tions are anything more than an empirical description ofthe time course of the changes in permeability to sodiumand potassium”. Therefore it is not surprising that themodel can be discussed and completed. However, if themodel stood out as the main model to describe the nerveimpulse for about 70 years, it is because Hodgkin andHuxley based their conclusions on a large set of detailedexperiments that they thoroughly analysed. This is prob-ably why the model, established before the knowledge ofthe existence and structure of ionic channels could pro-pose equations for the variation of the membrane con-duction which turned out to match structural data ofthe ionic channels that were discovered much later. Forinstance, for potassium channels Hodgkin and Huxleynoticed that, for the opening, the conductance versustime needed a third or fourth-order equation to be fit-ted, while the closing could be described by a first or-der equation. This lead them to a model in which thepotassium channel opening is controlled by 4 sites whichshould simultaneously occupy a certain position in themembrane. Molecular dynamics simulations of a voltagegated potassium channel, carried 60 years later [35], areperfectly consistent with this view because they confirmthat 4 voltage sensitive domains must be up before thepore can reopen after closure. Thus, although Hodgkinand Huxley took great care in stressing that “the inter-pretation given is unlikely to provide a correct pictureof the membrane”, they managed to extract a lot fromtheir voltage clamp measurements, which strengthen thecredibility of the model that they proposed. The struc-ture and basic function of sodium and potassium ionicchannels were discovered about 30 years after the pro-posal of Hodgkin and Huxley [36], and this confirmedtheir insight.Of course the Hodgkin-Huxley model is not perfect,however the precise analysis of the voltage clamp tech-niques that underlie its equations lead to a wide accep-tance of the hypothesis of channels mediating ionic flowacross the membrane and the Hodgkin-Huxley model ki-netics describes most of the classic experiments fairly wellalthough some experiments showed that the picture maybe oversimplified. In spite of its successes, as noticed inSec. III, the model does not describe all phenomena asso-ciated to nerve signalling, so that some scientists noticedthat “given the many experimental features not explainedwithin the Hodgkin-Huxley theory, it is surprising thatit remains an unchallenged dogma” [6]. B. A mechanical model for the propagation of thenerve impulse
And indeed the dogma has been challenged, in par-ticular with proposals that the dominant effect in nervesignalling might be mechanical rather than electrical [6].T. Heimburg and A. Jackson suggest that the nerve im-pulse could actually propagate as a localised deforma-tion of the axon [6]. In most of the physical systemsa localised deformation, which, in Fourier space, is aninfinite combination of signals of different wavelengths,tends to spread as it propagates due to dispersion be-cause the different wavelengths propagate at differentspeeds. However, in some systems, the effect of dis-persion can be compensated by nonlinearity, leading tosolitary waves, which may have particle-like properties,which gave them their name of “solitons” [37]. Heimburgand Jackson noticed that lipid membranes generally dis-play order-disorder phase transitions in a temperaturerange which is not far from physiological temperatures.Heating can destroy the lateral order of the molecules,which absorbs heat and leads to a swelling of the struc-ture. This structural change modifies the volume- andarea-compressibility of the membrane. As a result, inthe vicinity of the transition, the speed of sound dependson ρ A , its mass per unit area. Therefore the equation forthe propagation of a mechanical disturbance ∆ ρ A con-tains a nonlinear contribution associated to the variationof the lateral compressibility near the transition. Andbecause the thermal exchanges occurring at the transi-tion are slow processes, the speed of sound, which is also a function of the specific heat as shown by the thermody-namics Maxwell relations, depends on the frequency (andwavelength) of the mechanical disturbance, so that theequation for the propagation of the mechanical distur-bance also includes dispersion. Heimburg and Jacksonshow that the signs of the contributions are such thatnonlinearity can compensate dispersion. Using standardexpansions, they derive an equation for ∆ ρ A which hassome similarities with the Boussinesq equation, a stan-dard equation in soliton theory [37].This analysis suggests that, in the vicinity of the order-disorder transition of lipid membranes, a mechanical per-turbation can therefore propagate as a quasi-soliton. Ow-ing to the exceptional properties of solitons, in particu-lar their ability to move by preserving their shapes in thepresence of perturbations, or even in collisions with othersolitons, its is tempting to conclude that the main phe-nomenon that lies behind the propagation of the nerveimpulse is the compensation between nonlinearity anddispersion which allows the motion of narrow mechanicaldisturbances. In this view the electrical signal studiedby Hodgkin and Huxley is not the dominant mechanismbut a secondary effect that would be slaved to the me-chanical disturbance. The deformation of the membranecould affect the proteins that form the ion channels, andtherefore induce the ionic flow through the membrane.The idea of solitons propagating along lipid membranesis interesting and it would deserve studies to confirm itin some experiments. This is probably why it soundedsufficiently attractive to appear as an alternative to theHodgkin-Huxley model of nerve impulse. It is supportedby the experiments that recorded a variation of the diam-eter of the axon in the region of the pulse [29–31]. Theheat exchange at the transition also appeared as a candi-date to explain the discrepancy between the measuredthermal exchanges that accompany the nerve impulseand the evaluations deduced from the condenser theory[25, 26]. Moreover an experiment which showed that ac-tion potential launched towards each other in some axonswhich allow orthodromic (normal) and antidromic (in-verse) propagation can pass through each other [38], inanalogy with a remarkable property of solitons, soundsas a strong argument supporting the mechanical solitonpicture for the nerve impulse.However, while it might play a role in cell mechanics,the theory proposed by Heimburg and Jackson does notstand up to close scrutiny regarding the propagation ofnerve impulse when it is confronted to experiments. (i) The link to a phase transition in the axon mem-brane, occurring at a particular temperature T c close tophysiological temperature, is a serious constraint. First,while such a transition has been observed in unilamellarvesicles, bovine lung surfactant, and two bacterial mem-branes, it has not been reported for the axon membrane[6]. But the main problem is that it gives a specific role toa temperature T c while experimental investigations of theeffect of temperature on the nerve impulse, from − ◦ Cto 40 ◦ C [23] do not show any discontinuity or qualita-tive change around a specific temperature, but rather agradual evolution (Sec. II B). (ii)
The argument of a disagreement between the mea-sured thermal effects and the evaluations of the condensertheory, that could be explained by the thermal exchangesassociated to the latent heat of the transition is not verystrong because the condenser theory is oversimplified andmore careful evaluations of the energy exchange associ-ated to ion transfers [27] do not conclude to such a dis-agreement, although the conclusion on this point maystill be open. (iii)
Similarly the conclusion drawn from the possibil-ity of nerve pulses to pass through each other should betaken with caution because other experiments, using dif-ferent axons, contradict this result [39]. Moreover theHodgkin-Huxley model does not preclude such a cross-ing of the pulses for some values of its parameters [40],which could explain why some axons show the survivalof colliding nerve pulses while others do not. (iv)
In [6] the mechanical properties of the axonmembrane are those of a lipid layer, but actually themembrane is much more complex. As shown in [41],actin, spectrin and associated proteins form a cytoskele-tal structure in axons. This must make the membranemuch more rigid than a simple lipid bilayer, drasticallyreducing nonlinear effects in its mechanical distortions. (v)
The strongest argument against the theory of He-imburg and Jackson [6] is actually provided by the mea-surements of the deformation of the axon [29], which theypresent as supporting their idea (Sec. III B). As they usedan interferometric method and simultaneously record theelectrical signal, Hill et al. could determine the precisetiming of the two phenomena. The mechanical signalsbegins by a contraction which starts 250 µ s after the ar-rival of the electrical pulse, and therefore it cannot be themechanical signal that causes the electrical signal. Ow-ing to the time scale of the electrical nerve impulse, thedelay of 250 µ s is really significant. Moreover the timetracks displayed in Fig. 3 of [29] show that the mechanicalsignal lasts significantly longer than the electrical pulse.Rather than supporting the idea that a mechanical couldinduce the electrical pulse, as proposed in [6], the dataof [29] support instead the proposal that Tasaki [32] pre-sented after a series of measurements [30, 31, 33, 34].The swelling of the axon might be related to a structuraltransition, but, instead of a transition induced by a tem-perature change, it would be a transition caused by thechange of the ionic environment around the membrane,which follows the electrical pulse.Thus, although the idea of a soliton-like propagationof the nerve impulse might look attractive, a precise con-frontation with experimental facts cannot support thisproposal.Another recent view of the nerve impulse [42] con-siders another type of soliton, belonging to the class of envelope solitons [37] in which a carrier wave is modu-lated by a localised envelope function to generate a lo-calised wavepacket. In this model the carrier wave wouldbe an oscillation of the dipolar orientation of the watermolecules in the vicinity of the membrane. The dipolarinteractions would lead to forces applied to the mem-brane, generating a coupling between a mechanical dis-turbance and the dipolar reorientations so that the dipo-lar signal would “surf on the capillary waves propagatingalong the axon”. This picture appears to suffer from sev-eral weaknesses: (i) The picture is only qualitative and no quantita-tive evaluation has been made to explain how such acombined dipolar-mechanical signal could stay localised.Capillary waves are dispersive and would tend to spread.A quantitative mechanism has to be presented to show,in a convincing way, that the coupling tends to maintainthe necessary localisation, and that it has the strengthto do it. (ii)
The dissipation in the mechanical signal would pre-sumably be very high. At the macroscale, capillary wavescan propagate with a rather small dissipation. But, asshown by Purcell in a a beautiful article “Life at lowReynolds number” [43], at the microscopic scale phenom-ena are very different, and the viscosity of water plays amuch stronger role. Moreover, considering the frequen-cies of the order of 100 kHz considered as plausible in[42] for the capillary waves, at the scale of the axon dis-turbances, dissipation would damp out the motion veryquickly due to water viscosity but also the losses withinthe membrane itself. (iii)
But the main objection to the scheme proposed in[42] is that it assumes that, once the signal is launched bysome ion transfers, the action potential moves on with-out charge currents. This contradicts the observationsof local ionic currents through the membrane [44] andthe evidence of the ionic flows using radioactive tracers[21, 22] (Sect. II B).
C. Completing the Hodgkin-Huxley model
While the picture of a nerve pulse dominated by a me-chanical signal does not stand up in front of a criticalexamination, it does not mean that the Hodgkin-Huxleymodel cannot be completed to include some phenomenathat were deliberately left out by Hodgkin and Huxley,who focused their attention on the electrical phenomenaonly. Several attempts have been made around this idea,and they probably point out to a direction that couldimprove our understanding of nerve signalling.A. El Hady and B.B. Machta [45] studied the mechan-ical surface waves which accompany the propagation ofthe action potential. Their viewpoint is completely dif-ferent from those of Heimburg [6] and Kotthaus [42].They don’t consider that the mechanical signal is themain signalling pathway. Instead they assume that theaxon carries an electrical pulse, defining the action po-tential, without making any hypothesis on the origin ofthis pulse, which could be the Hodgkin-Huxley pulse, orhave a different origin. This pulse drives the membranedeformation and they compute the response of the axonto this driving. In this approach the axon is viewed as anelastic and dielectric tube filled by a viscous fluid. Theelastic energy is stored in the deformation of the tube,and the kinetic energy is carried by the motion of the ax-oplasmic and extracellular viscous fluid, which moves ac-cording to the Navier-Stokes equation. The displacementof the membrane is expressed as a linear function of theforces due to the action potential. Therefore this modeldoes not consider any mechanical nonlinearity, which isprobably legitimate if the membrane is actually strength-ened by a cytoskeleton, as observed in [41]. However thisapproach does not require nonlinearity to localise the me-chanical distortion because the shape of the signal is im-posed by that of the action potential, which is spatiallylocalised. Besides the energy exchanges due to chargetransfer through the membrane, this model predicts anadditional heat effect due to the isothermal distortionof the membrane because its free energy depends on itsarea, which is locally modified. Using parameters esti-mated from what we know of the axon, the calculationleads to results in the range of the experimental obser-vations. And, in contrast to the model of [6], as themechanical distortion is a consequence of the electricalpulse, it is natural that it follows the action potentialarrival with a small delay as observed in [29].The approach of Engelbrecht et al. [46] takes a simi-lar viewpoint that the electrical signals are the carriersof information in nerves and trigger all other processes,but it is more ambitious because it tries to describe thecoupling between the mechanical aspects (fluid flows andmembrane deformation) and the action potential, insteadof assuming that the mechanical component is slaved tothe electrical one. In their view the channels in themembrane can be open and closed under the influenceof the electrical signals but also by mechanical inputs.The shape of the action potential is therefore not as-sumed, but instead described by a simplified version ofthe Hodgkin-Huxley model, the FitzHug-Nagumo model,initially proposed by FitzHugh as a model for the axon[47] and then built and further studied by Nagumo etal. [48] as a transmission line using tunnel diodes, forpossible applications in electronics and signal processing.The mechanical signal is described by an equation whichincludes nonlinearity and dispersion, in the spirit of [6],and a phenomenological coupling between the electricaland mechanical components is added. This coupling isexpressed as a function of the variation of the fields ratherthan their instantanous values, which puts emphasis ondynamical effects.The motivation of this approach is interesting becauseit should provide some understanding of the mechanisms that link the electrical and mechanical signals. Howeverthe experimental data on this coupling, which appearsto be too complex to be described from first principlesof physics and chemistry, are still sparse, and little isknown on the effect of mechanical constraints to controlthe ion channels. As a result the assumptions made byEngelbrecht et al. are difficult to validate so that, in itspresent stage, this approach does not actually bring afurther understanding of nerve signalling.Finding the actual mechanisms behind the couplingbetween the electrical and mechanical components is achallenge to reach a meaningful extension of the Hodgkin-Huxley model. An interesting suggestion has been madeby Krichen and Sharma [49]. Piezoelectricity is a wellknown coupling between mechanical strain and electri-cal polarization, but, for uniform strains it only appliesto materials which lack a mirror symmetry. However,as pointed out by Krichen and Sharma, if the strain it-self does not have a mirror symmetry, even a centro-symmetric medium such as a fluid or a membrane canexhibit an electromechanical coupling. This is flexoelec-tricity. Moreover, as membranes are highly flexible, theycan show very large strain gradients. Even if the couplingcoefficient is small, the resulting effect can be large. Chenet. al. [50] used this idea to propose an axon model whichcouples the distortion of the axon and the action poten-tial. It is a two-way coupling because a strain gradientcan induce an electrical polarization, and the action po-tential can cause a local distortion. Their approach cantreat myelinated and unmyelinated axons. This is an at-tractive idea which would deserve further experimentaland theoretical studies to be fully validated.
V. DISCUSSION
In spite of alternatives introduced in the last 15 years,the answer to the question raised in the title “How isinformation transmitted in a nerve ?” still appears to be“as an electrical pulse”, approximately described by theHodgkin-Huxley model proposed in 1952. Of course theanswer that it provides is oversimplified. This model wasdeveloped mostly from data recorded on the giant squidaxon, and it should be amended to describe other axons,but it is nevertheless very likely to describe the essenceof the phenomena involved in nerve signalling.From the start the model was not designed to be com-plete as Hodgkin and Huxley only investigated the elec-trical component of the signal. Experiments have shownthat, as for most biological phenomena, nerve signallingis complex, also involving a local deformation of the axonand thermal exchanges. This lead some authors to chal-lenge the main ideas of the Hodgkin-Huxley model andeven to put forward mechanisms in which the electricalsignal is not dominant, such as a mechanical soliton as thecarrier of information. However, as shown in Sec. IV B,this idea cannot stand in front of a careful examination ofthe experimental facts. In the case of a phenomenon ascomplex as the nerve impulse, a first-principle model isstill out of reach and one must rely on phenomenologicalmodels. This can nevertheless be fruitful if this approachis built on a detailed analysis of the experimental obser-vations, without neglecting some data which, at a firstglance, may seem non-essential. This is exactly whatHodgkin and Huxley did. They established their modelafter years of experiments and thoughts, which allowedthem to predict phenomena or properties which had notyet been observed, such as some features of the ion chan-nels. Instead the models assuming a dominant mechani-cal signal, although they were motivated by the observedchange of the diameter of the axon that accompanies theaction potential, neglect some elements (Sec. IV B). Thedeformation starts after the rise of the electrical pulse sothat causality excludes that it could generate it.The mechanisms of anesthesia have been discussed asa possible test of the models for the propagation of thenerve impulse. One could wonder whether their studycould help decide between the two alternatives, electricalor mechanical. This is questionable because anesthet-ics probably exert their action on synaptic transmissionrather than axonal conduction [51]. Nevertheless thereare many evidences that anesthetics act on ion chan-nels [52], which is a hint that nerve signals are electricalrather than mechanical. Various studies have shown thatanesthetics act directly on proteins rather than on lipids[51, 53]. There are however cases in which the mem-brane is involved, as demonstrated recently for inhaledanesthetics [54], but the actual target is nevertheless anion channel and the membrane lipids only play an inter-mediate role.Thermal effects have also been suggested as a mean tosolve the dilemma between the electrical and mechani-cal views of nerve signaling. This has two aspects, firstwhether the pulse is adiabatic, i.e. exothermic and en-dothermic contributions are equal, and second what isthe magnitude of thermal effects. Adiabaticity cannotmake the difference between the two alternatives. Theview that the Hodgkin-Huxley action potential must bedissipative because it involves currents through resis-tors is oversimplified. Experiments [25, 26] show thatexothermic effects are followed by an endothermic processof about the same magnitude. In the Hodgkin-Huxleymodel this is understandable because during the rise ofthe action potential the
N a + ions move down the po-tential gradient giving rise to heat release, while, in thesecond stage the K + ions move up the potential gradientbut down their concentration gradient, converting heatinto capacitive energy [45]. In the soliton picture, ther-mal effects come from the latent heat of a reversible phasetransition in the membrane, so that the overall thermaleffect vanishes. The second aspect, the magnitude of thethermal effect, could, in principle, lead to a conclusionbecause the thermal effect due to a phase transition inthe membrane should be significantly greater than themagnitude measured by experiments, ruling out the soli-ton model. However the measurements are very difficult and, for the fastest pulses, they may not have a suf-ficient temporal resolution to catch the full magnitudeof the thermal exchanges [55]. Nevertheless, as experi-ments observe that replacing sodium by lithium modifiesthe magnitude of the thermal effect, and as theories thatgo beyond the simple condenser model conclude that aproper thermodynamic analysis of the processes involvedin the Hodgkin-Huxley model is compatible with the ex-perimental observations, the studies of the thermal effectsappear to favor the electrical view. As pointed out in[45], an additional contribution to thermal effects couldnevertheless come from the stretching of the membranewhich accompanies the action potential in recent models.Its order of magnitude is well below the contribution oflatent heat in the soliton picture.This does not mean that the proposals challenging theHodgkin-Huxley model have been useless. They stimu-lated further thoughts and lead to models that combineelectrical and mechanical effects. The goal of a model isnot to reproduce experimental facts but to show what arethe underlying phenomena which lead to these observa-tions and allow further developments. Including mechan-ical distortion in a nerve-impulse model makes sense if itactually contributes to the process, which would be thecase if ion channels are not only sensitive to voltage butalso to forces or membrane distortions, as suggested bysome studies [56]. However, establishing a reliable modelof the axon coupling electrical and mechanical signals willcertainly need further experimental investigations at thescale of ion channels.A model may also be useful to understand additionalphenomena. For nerve signalling, the role of the mye-line layer deserves attention. Within the Hodgkin-Huxleymodel, or its simplified version the FitzHugh-Nagumomodel, it is easy to show how an extra layer reducingthe capacitance of the membrane can speed-up the sig-nal, but the role of myeline also introduces constraintson the ionic flow so that its effect is not straightforwardto predict [57].The Hodgkin-Huxley model has many parameters, andthey could vary from cell to cell or with changes in theexternal medium. A promising line of investigations is totry to reduce the parameter space by looking how someparameter combinations may be enough to determine themain features of the model [58], and what is the stabilityof the results when the parameters or environment con-ditions change. Recent investigations [58] suggest thatviewing the Hodgkin-Huxley model in a low-dimensionalspace may bring a deeper understanding of cell excitabil-ity.The validity of a model also depends on the scale ofinterest. The Hodgkin-Huxley model smoothes out theeffect of individual ion channels in its continuous equa-tions. When the noise due to individual channels be-comes relevant, discrete stochastic models may be moreappropriate [59]. On the other hand the Hodgkin-Huxleymodel has also been challenged for studies at the scale ofa full neural network [60]. Therefore this model is cer-tainly not the ultimate model for nerve signalling. It canbe completed to include additional physical phenomenaor modified for special purposes, but it does not deserveto be discarded simply because it does not describe allthe complexities of the nerve impulse. Conflict of interest
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