A comparision of Hodgkin-Huxley and soliton neural theories

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1 dv. Radio Sci., 8, 75 79, doi:10.519/ars uthor(s) 010. CC ttribution 3.0 License. dvances in Radio Science comparision Hodgkin-Huxley soliton neural ories R. ppali, S. Petersen, U. van Rienen Institute General Electrical Engineering, Chair Electromagnetic Field Theory, University Rostock, Justus-von-Liebig-Weg, Rostock, Germany bstract. Hodgkin Huxley were pioneers to abstract biological neuron as an electric circuit nerve signal as voltage impulse. The Hodgkin-Huxley ory (Hodgkin Huxley, 195) has set direction defined goals for much ensuing research in biophysics. However, in 005, T. Heimburg. D. Jackson, biophysicists from Copenhagen proposed a new neural ory called Soliton ory (Heimburg Jackson, 005). In this ory, nerve conduction is proposed as a density wave. In this paper, Hodgkin-Huxley Soliton ories are described a oretical comparison has been carried out throughout analysis ories models. 1 Introduction Nerves are information transmitter lines from brain to organs. The nerves are described as bundle discrete individual cells called neurons. These individual cells transmit information as signals. The main parts a neuron are cell body (soma), signal receiving ends (dendrites), signal transmitter (axon), connecting ends to or neurons or gls (synapses). The scientific model neuron was first given in electrical terms by Hodgkin Huxley (Hodgkin Huxley, 195). Hodgkin Huxley conducted experiments on nonmyelinated giant axon squid to study neuronal properties. The Hodgkin-Huxley equations (Hodgkin Huxley, 195) are starting point for detailed neuronal models. The Hodgkin-Huxley ory The ory was summation active experimental oretical association Hodgkin Huxley. Their five publications in 195 in Journal Physiology are quite Correspondence to: R. ppali (revathi.appali@uni-rostock.de) renowned. The four key research developments which led to ory were: (Häusser, 000) Firstly, Cole Curtis demonstrated that action potential is associated with a large increase in membrane conductance (Cole Curtis, 1939). Secondly, Hodgkin Huxley made first intracellular recording an action potential (Hodgkin Huxley, 195). This demonstrated that action potential exceeds zero mv, denying Bernstein s hyposis (Bernstein, 190) that underlying increase in membrane permeability is non-selective. Hodgkin Katz (Hodgkin Katz, 199) explained overshooting action potential as result an increase in sodium permeability approving work Overton (Overton, 190). Finally, Hodgkin, Huxley Katz developed voltage-clamp circuit to facilitate quantitative measurement ionic currents from squid axon. The step depolarization squid axon triggering an inward current followed by an outward current was n proved by Hodgkin Huxley. With aid ionic substitution, y demonstrated that this net current could be separated into two distinct components, a fast inward current carried by Na + ions, a more slowly activating outward current carried by K + ions. Using ingenious voltage-clamp protocols, y concluded that se two currents result from independent permeability mechanisms for Na + K + with conductance changing as a function time membrane potential. This striking conceptual breakthrough was n termed as ionic hyposis (Häusser, 000). The three different types ion current, viz., sodium, potassium, a leak current that consists mainly Cl ions contribute to voltage signal in neuron. The flow se ions through cell membrane is controlled by ir respective voltage-dependent ion channels. The leak current takes care or channel types which are not described in particular. The most remarkable achievement this ory Published by Copernicus Publications on behalf URSI Lesausschuss in der Bundesrepublik Deutschl e.v.

2 centration Huxley, in extracellular 195), liquid. very The Nernst first complete indicating respectively. generality Normally, however, approach. some Thus, ential generated description by excitability difference single ion neuron. H-H channels model links are blocked. microscopic The probability level ion that a centrations is represented by a battery. The channels channel to macroscopic is open is level described currents by additional ivalent circuit 76 R. ppali et al.: comparision Hodgkin-Huxley soliton neural ories.1 The is shown Model: in fig.. action potentials variables m, (see n, fig. ). h. The combined action m h controls Na + channels. The K + gates I Extracellular are controlled by n. Hodgkin Huxley Outside Na + formulated three ionic components as du 3 C = gnam h(u ENa ) + g n (u E ) K K dt g L g Na g k + gl(u EL ) + I(t) V (1) Inside K + The parameters E Na, E K, E L are reversal potentials. Reversal potentials conductances E L E Na E k are empirical parameters. So, it is demonstrated Fig. 1. Schematic model Hodgkin-Huxley ory (modified from Figure 1: Schematic model Hodgkin-Huxley that neurons transmit information by firing Gerstner et al., 00). ory (modified from (Gerstner et al., 00)) propagating electrical action potentials along Intracellular ir axons. Usually, motor neurons possess a The Hodgkin-Huxley model (H-H model) can be Figure myelin 3: ction sheath potential around ir adapted axons from acting as an Fig. 3. ction explained with help fig. 1. The semipermeable Huxley, cell 195) membrane separates interior transmission. insulator potential adapted enhancing from original paper speed Hodgkin ure : Equivalent circuit H-H model signal original Huxley. paper Hodgkin Huxley dgkin cell from extracellular liquid acts The model can reproduce explain a wide as a capacitor. If an input current I(t) is injected acterized range by ir data resistance from or, squid equivalently, axon like by ir shape conductance. propagation The leakage channel action is described potential by (see a voltage- fig. 3), its into cell, it may add furr charge on capacitor, or leak through channels in independent cell sharp conductance threshold, refractory conductance period, anode-break or ion channels is voltage time dependent. Because active ion transport excitation, accommodation sub threshold If all channels are open, y transmit currents with a maximum conductance oscillations. g Na through cell membrane, ion concentration The or g K model, respectively. can also Normally, describe how-manever, some channel channels types with are blocked. small The parameter probability that changes inside cell is different from that ion concentration in extracellular liquid. The Nernst a channel indicating is open is described generality by additional variables approach. m, n, Thus, potential generated by difference in ion h. TheH-H combined model action links m microscopic h controls level Na + ion channels. concentrations is represented by a battery. The channels The K + to gates are macroscopic controlled bylevel n. Hodgkin currents Huxley formulated three ionic components as equivalent circuit is shown in fig.. action potentials (see fig. ). Fig.. 195). I Extracellular Equivalent circuit H-H model (Hodgkin Huxley, C du dt = g Na m 3 h(u E Na )+g K n (u E K ) +g L (u E L )+I (t). (1) C was empirical representation experimental data in a quantitative g L model (Hodgkin g Na Huxley, g k 195), very first complete description excitability single neuron. V.1 The model E L E Na The Hodgkin-Huxley model (H-H model) can be explained with help Fig. 1. The semi-permeable cell membrane separates interior cell from extracellular liquid acts as a capacitor. If an input current I (t) is injected into Intracellular cell, it may add furr charge on capacitor C, or leak through channels in cell Because active ion Figure transport : through Equivalent cell membrane, circuit ion H-H concentration (Hodgkin inside cellhuxley, is different 195) from that ion concen- model tration in extracellular liquid. The Nernst potential generated by difference in ion concentrations is represented by a battery. The equivalent circuit is shown in Fig.. s mentioned above, Hodgkin-Huxley model describes three types channels. ll channels may be char- E k The parameters E Na, E K, E L are reversal potentials. Reversal potentials conductances are empirical parameters. So, it is demonstrated that neurons transmit information by firing propagating electrical action potentials along ir axons. Usually, motor neurons possess a myelin sheath around ir axons acting as an insulator enhancing speed signal transmission. The model can reproduce explain a wide range data from squid axon like shape propagation action potential (see Fig. 3), its sharp threshold, refractory period, anode-break excitation, accommodation sub threshold oscillations. The model can also describe many channel Figure 3: ction potential adapted from original paper Hodgkin Huxley types with small parameter changes indicating generality approach. Thus, H-H model links microscopic level ion channels to macroscopic level currents action potentials (see Fig. ). dv. Radio Sci., 8, 75 79, 010

3 T. Heimburg. D. Jackson proposed an alternative ory in 005. The ory is a R. rmodynamic ppali et al.: comparision ory Hodgkin-Huxley nerve pulse soliton neural ories decreased compressibility with increasing 77 propagation. This ory is based on lipid density or pressure) dispersion (: lipid characteristic transition from a fluid to a gel compressibility is frequency dependent both for state over a range several degrees, slightly area volume compressibility) (Heimburg below body temperature (Heimburg Jackson, 005; Heimburg Jackson, 007a). Jackson, 005 references re in). That is, Both effects non-linearity dispersion when a biological membrane is compressed, produce a self-sustaining localized density fluid membrane gradually reaches to a denser gel pulse with a moving segment nerve state. Now, a sufficient pressure/ compression membrane in gel state (see fig. 6). This pulse eir in membrane area or volume will bring a is called a soliton (Heimburg Jackson, transition to gel state (Heimburg, 007a). 007a). The solitary wave maintains its shape This transition is associated with release while it travels at a constant speed less than heat (Heimburg, 007a; Heimburg, 007b). The sound velocity in lipid Solitons relation between heat capacity, density, compression temperature are shown in fig. 5. for generation propagation solitons. Pure lipid membranes exhibit non-linearity (: maximally compressible close to transition can propagate over long distances without loss energy (Heimburg Jackson, 005). The pulse is also called as adiabatic pulse, because no Figure : Separation ionic conductances underlying action potential in H-H model. Modified Fig.. Separation from ionic (Hodgkin conductances Huxley, underlying 195) action potential in H-H model. Modified from Hodgkin Huxley (195). 3. Soliton Theory Lipid membranes have properties required for energy generation is propagation lost to solitons. environment during T. Heimburg. D. Jackson proposed an Pure lipid membranes exhibit non-linearity (: propagation (Heimburg Jackson, 007b). alternative ory in 005. The ory is a maximally compressible close to transition rmodynamic ory nerve pulse decreased The soliton compressibility mode with increasing pulse propagation is propagation. This ory is based on lipid density basically or pressure) different dispersion from (: lipid P given by H-H characteristic transition from a fluid to a gel compressibility is frequency dependent both for model (nderson et al., 009). state over a range several degrees, slightly area volume compressibility) (Heimburg below body temperature (Heimburg Jackson, The 005; solitary Heimburg wave Jackson, is associated 007a). with a transient Jackson, 005 references re in). That is, Both voltage effects change non-linearity due to dispersion transient change in when a biological membrane is compressed, produce a self-sustaining localized density fluid membrane gradually reaches to a denser gel pulse with state a moving segment This affects nerve membrane state. Now, a sufficient pressure/ compression membrane potential in gel difference. state (see fig. 6). Especially, This pulse it changes both eir in membrane area or volume will bring a is called a area soliton (Heimburg thickness Jackson, membrane reby transition to gel state (Heimburg, 007a). 007a). The solitary wave maintains its shape This transition is associated with release while it capacitance travels at a constant (Heimburg speed less than Jackson, 007a). heat (Heimburg, 007a; Heimburg, 007b). The sound velocity These in changes lipid seem to be Solitons as a voltage pulse relation between heat capacity, density, can propagate leads over to long a distances capacitive without loss current due to compression temperature are shown in fig. 5. energy (Heimburg Jackson, 005). The pulse is also unbalanced called as adiabatic charge pulse, because no The reversible Figure 6: Schematic model membrane lipid rmodynamic ory (adapted from (nderso energy Fig. 6. heat Schematic is lost observed to model during environment membrane action lipid during potential rmodynamic is proposed ory propagation (Heimburg Jackson, 007b). Figure 5: Heat capacity, lateral density Thus heat flow (adapted is correlated as from a nderson result with et changes al., 009). lipid melting This ory enthalpy also explains i.e. effect Fig. 5. Heat capacity, lateral density lateral compressibility in membrane density The soliton potential. mode pulse propagation on is nerve pulse propagation. cco as lateral a function compressibility temperature (adapted from as Heimburg a function Jackson, basically production different from latent P given heat by during H-H lipid transition ory, anestics in memb 005). temperature (adapted from (Heimburg 3.1 Soliton equation model (nderson from fluid et al., to 009). gel state fluid-gel reabsorption transition temp The solitary wave is associated with a transient Jackson, 005) Lipid heat membranes as have system properties returns phenomenon required to for known fluid generation as state freezing poin Hydrodynamic voltage equation change due to transient change in state (nderson for propagation This et al., affects 009). solitons. a Pure (Heimburg lipid membranes Jackson, 007c). So i 3 Soliton ory membrane density pulse in potential exhibit presence non-linearity dispersion difference. Especially, (maximally for a it changes compressible to force both close membrane to through th cylindrical membrane transition along area x-axis thickness decreased is (Heimburg membrane compressibility transition, reby with increasing reby density or pressure) (Heimburg dispersion Jackson, 007a). (lipid compressibility is action potentia T. Heimburg. D. Jackson proposed an alternative Jackson, ory in 005. The ory is a rmodynamic ory nerve depression is a general r 005) given by is suppressed. It is noted that fre capacitance These 3frequency changes dependent seem to be both as a for voltage areapulse pulse propagation. This ory is based on lipid characteristic transition from a fluid to a gel state over a range unbalanced [ ity) phenomenon volume compressibil- leads to a capacitive not chemical current due to (Heimburg charge ] Jackson, () 005; Heimburg The reversible (nderson et al., Jackson, 009). Δ ρ = c x Δ ρ - h Δ ρ t x 007a). x several degrees, slightly below body temperature (Heimburg Jackson, 005, references re in). That heat observed during action potential is proposed Figure 5: Heat capacity, lateral density as a result lipid melting enthalpy i.e. Both effects non-linearity. Comparison dispersion produce lateral compressibility as a function fter co-ordinate production transformation latent heat (z= during x-v.t), lipid transition is, when a biological membrane is compressed, fluid a self-sustaining localized density pulse with a moving temperature (adapted from (Heimburg by introducing from propagation fluid to gel velocity, state time reabsorption The major differences between both membrane gradually Jackson, 005) reaches to a denser gel state. independent Now, segment nerve membrane in gel state (see Fig. 6). equation heat as describing system returns propagation to fluid ories state are: (Hodgkin Hux a sufficient pressure/ compression eir in membrane density area excitation (nderson This is pulse given et is al., as called 009). (Heimburg a soliton (HeimburgHeimburg Jackson, 007a). Jackson, 005; He or volume will bring a transition to gel state Jackson, (Heimburg, 007a). This transition is associated with release 007a) The solitary wave maintains its shapejackson, while it 007a; travels nderson at a et al., 009 constant speed less than sound velocity in lipid mem- Solitons can propagate ρ ] H-H The action potential is due to th heat (Heimburg, 007a; Heimburg, 007b). The v relation Δρ = 3 [( c + pδρ + q( Δρ ) +...) 0brane. z z z Δ over long distances without current conducted across m between heat capacity, density, compression temperature loss energy (Heimburg Jackson, 005). The pulse is - h Δρ (3) ion channels that act as resisto are shown in Fig. 5. also called as adiabatic pulse, because no energy is lost to Where z environment during propagation (Heimburg ν is propagation velocity S The nerve Jackson, pulse is a sol ρ is lateral membrane 007b). density The soliton mode pulse propagation generated is basically by lipid transi c is sound velocity in fluid phase membrane dv. Radio p q are parameters determined H-H Sci., Nerve 8, 75 79, pulse propagation 010 is n from sound velocity density as a moving self regenerating p dependence capacitive discharges through th h is parameter to set linear scale resistors followed by capacitor

4 78 R. ppali et al.: comparision Hodgkin-Huxley soliton neural ories different from P given by H-H model (nderson et al., 009). The solitary wave is associated with a transient voltage change due to transient change in state This affects membrane potential difference. Especially, it changes both area thickness membrane reby capacitance (Heimburg Jackson, 007a). These changes seem to be as a voltage pulse leads to a capacitive current due to unbalanced charge The reversible heat observed during action potential is proposed as a result lipid melting enthalpy i.e. production latent heat during lipid transition from fluid to gel state reabsorption heat as system returns to fluid state (nderson et al., 009). Thus heat flow is correlated with changes in membrane density potential. 3.1 Soliton equation Hydrodynamic equation for propagation a density pulse in presence dispersion for a cylindrical membrane along x-axis is (Heimburg Jackson, 005) given by t ρ = [c / x ρ ] h x x ρ () fter co-ordinate transformation (z = x v t), by introducing propagation velocity, time independent equation describing propagation density excitation is given as (Heimburg Jackson, 007a) v z ρ = [( ) c0 +p ρ +q( ρ ) +... z /z ρ] h z ρ (3) Where ν is propagation velocity ρ is lateral membrane density c is sound velocity in fluid phase membrane p q are parameters determined from sound velocity density dependence h is parameter to set linear scale propagating pulse. This ory also explains effect anestics on nerve pulse propagation. ccording to this ory, anestics in membrane lower fluid-gel transition temperature, a phenomenon known as freezing point depression (Heimburg Jackson, 007c). So it is difficult to force membrane through fluid-gel transition, reby action potential generation is suppressed. It is noted that freezing point depression is a general rmodynamic phenomenon not chemical in nature (nderson et al., 009). Comparison The major differences between both neural ories are (Hodgkin Huxley, 195; Heimburg Jackson, 005; Heimburg Jackson, 007a; nderson et al., 009): H-H The action potential is due to electric current conducted across membrane by ion channels that act as resistors. S The nerve pulse is a solitary wave generated by lipid transition in H-H Nerve pulse propagation is n explained as a moving self regenerating pulse local capacitive discharges through resistors followed by capacitor recharging. S The propagation is due to combined effect nonlinearity dispersion. H-H The ory is pure electrical with ionic hyposis. S The ory is physical/ mechanical i.e. based on rmodynamics. H-H The nerve signal or action potential is electrical pulse. S The nerve signal is propagating adiabatic electromechanical pulse. H-H The pulse propagation dissipates energy in form heat. S The propagating pulse does not dissipate heat energy. H-H The nerve s physical changes are not explained. S The nerve changes dimensions gets thicker during nerve pulse. The voltage pulse accompanies propagating signal, as a result se transient geometric changes Both ories explain existence voltage pulse in nerve pulse propagation. However, as an information transmitting pulse in H-H model as a by-product signal in soliton ory. 5 Discussion Hodgkin-Huxley model nerve pulse propagation is undoubtedly a remarkable achievement in field physiology. However, ory could not explain physical phenomena such as reversible heat changes, density changes geometrical changes observed in experiments (Iwasa et al., 1980; Tasaki, 198, 1999; Tasaki et al., 1989, Tasaki Byrne, 199). Soliton model is an alternative model, convincingly explaining some unexplained observations in experiments. However, re are several or questions that this dv. Radio Sci., 8, 75 79, 010

5 R. ppali et al.: comparision Hodgkin-Huxley soliton neural ories 79 has to answer like ion flow involvement in nerve signal propagation as stated by H-H model also faster propagation in myelinated nerves than in unmyelinated. 6 Context This is a oretical comparison both ories. This has been carried out to analyze choose a detailed neuron model which fits in for goals our project. We are examining soliton ory to model simulate action potential coupling electrode for a detailed study neuron. cknowledgements. We are grateful to DFG (German Science Foundation) for funding our project in Research Training Group 1505/1. References ndersen, S., Jackson,. D., Heimburg, T.: Towards a rmodynamic ory nerve pulse propagation, Progr. Neurobiol., 88, , 009. Bernstein, J.: Studies on rmodynamics bioelectric currents, Pflügers rch. Ges. Physiol., 9, 51 56, 190. Cole, K. S. Curtis, H. J.: Electric impedance squid giant axon during activity, J. Gen. Physiol.,, , Gerstner, W. Kistler, M. W.: Spiking neuron models. Single neurons, populations, plasticity, Cambridge Univ. Press, p. 3, 00. Häusser, M.: The Hodgkin-Huxley ory action potential, Nature neuroscience, 3, 1165, doi: /816, 000. Heimburg, T. Jackson,. D.: On soliton propagation in biomembranes nerves, Proc. Natl. cad. Sci. US., 10, , 005. Heimburg, T.: Chapter 6: lipid melting, in: Thermal Biophysics Membranes, Wiley VCH, pp , 007a. Heimburg, T. Jackson,. D.: On action potential as a propagating density pulse role anestics, Biophys. Rev. Lett.,, 57 78, 007a. Heimburg, T.: Chapter 7: phase diagrams, in: Thermal Biophysics Membranes, Wiley VCH, pp. 99 1, 007b. Heimburg, T. Jackson,. D.: Thermodynamics nervous impulse. In: Nag, K. (Ed.), Structure Function Membranous Interfaces, Wiley Sons, 007b. Heimburg, T. Jackson,. D.: The rmodynamics general anessia Biophys. J., 9, , 007c. Hodgkin,. L. Huxley,. F.: quantitative description membrane current its application to conduction excitation in nerve, J. Physiol., 117, 500 5, 195. Hodgkin,. L.: The ionic basis nervous conduction, Nobel Lectures, Physiology or Medicine, Hodgkin,. L. Katz, B.: The effect sodium ions on electrical activity giant axon squid, J. Physiol. (Lond.), 108, 37 77, 199. Iwasa, K., Tasaki, I., Gibbons, R. C.: Swelling nerve fibers associated with action potentials, Science, 10, , Overton, E.: Contributions to general muscle nerve physiology II. Indispensability sodium (or lithium) ions for muscle contraction, Pflügers rch. Ges. Physiol., 9, , 190. Tasaki, I.: Physiology Electrochemistry Nerve Fibers, cademic Press, New York, 198. Tasaki, I.: Evidence for phase transitions in nerve fibers, cells synapses, Ferroelectrics, 0, , Tasaki, I. Byrne, P. M.: Heat production associated with a propagated impulse in bullfrog myelinated nerve fibers, Jpn. J. Physiol.,, , 199. Tasaki, I., Kusano, K., Byrne, P. M.: Rapid mechanical rmal changes in garfish olfactory nerve associated with a propagated impulse, Biophys. J., 55, , dv. Radio Sci., 8, 75 79, 010