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Adv. Radio Sci., 8, 75–79, 2010 www.adv-radio-sci.net/8/75/2010/ doi:10.5194/ars-8-75-2010 © Author(s) 2010. CC Attribution 3.0 License. Advances in Radio Science A comparision of Hodgkin-Huxley and soliton neural theories R. Appali, S. Petersen, and U. van Rienen Institute of General Electrical Engineering, Chair of Electromagnetic Field Theory, University of Rostock, Justus-von-Liebig-Weg 2, 18059 Rostock, Germany Abstract. Hodgkin and Huxley were the pioneers to abstract biological neuron as an electric circuit and nerve signal as the voltage impulse. The Hodgkin-Huxley theory (Hodgkin and Huxley, 1952) has set the direction and defined the goals for much of the ensuing research in biophysics. However, in 2005, T. Heimburg and A. D. Jackson, biophysicists from Copenhagen proposed a new neural theory called Soliton theory (Heimburg and Jackson, 2005). In this theory, the nerve conduction is proposed as a density wave. In this paper, Hodgkin-Huxley and Soliton theories are described and a theoretical comparison has been carried out throughout the analysis of the theories and models. 1 Introduction Nerves are the information transmitter lines from the brain to organs. The nerves are described as the bundle of discrete individual cells called neurons. These individual cells transmit the information as signals. The main parts of a neuron are the cell body (soma), signal receiving ends (dendrites), signal transmitter (axon), and connecting ends to other neurons or glands (synapses). The scientific model of the neuron was first given in electrical terms by Hodgkin and Huxley (Hodgkin and Huxley, 1952). Hodgkin and Huxley conducted the experiments on nonmyelinated giant axon of squid to study the neuronal properties. The Hodgkin-Huxley equations (Hodgkin and Huxley, 1952) are the starting point for detailed neuronal models. 2 The Hodgkin-Huxley theory The theory was the summation of active experimental and theoretical association of Hodgkin and Huxley. Their five publications in 1952 in the Journal of Physiology are quite Correspondence to: R. Appali ([email protected]) renowned. The four key research developments which led to the theory were: (Häusser, 2000) – Firstly, Cole and Curtis demonstrated that the action potential is associated with a large increase in membrane conductance (Cole and Curtis, 1939). – Secondly, Hodgkin and Huxley made the first intracellular recording of an action potential (Hodgkin and Huxley, 1952). This demonstrated that the action potential exceeds zero mV, denying Bernstein’s hypothesis (Bernstein, 1902) that the underlying increase in membrane permeability is non-selective. – Hodgkin and Katz (Hodgkin and Katz, 1949) explained the overshooting action potential as the result of an increase in sodium permeability approving the work of Overton (Overton, 1902). – Finally, Hodgkin, Huxley and Katz developed the voltage-clamp circuit to facilitate quantitative measurement of ionic currents from squid axon. The step depolarization of squid axon triggering an inward current followed by an outward current was then proved by Hodgkin and Huxley. With the aid of ionic substitution, they demonstrated that this net current could be separated into two distinct components, a fast inward current carried by Na+ ions, and a more slowly activating outward current carried by K+ ions. Using ingenious voltage-clamp protocols, they concluded that these two currents result from independent permeability mechanisms for Na+ and K+ with conductance changing as a function of time and membrane potential. This striking conceptual breakthrough was then termed as the “ionic hypothesis” (Häusser, 2000). The three different types of ion current, viz., sodium, potassium, and a leak current that consists mainly of Cl− ions contribute to the voltage signal in the neuron. The flow of these ions through the cell membrane is controlled by their respective voltage-dependent ion channels. The leak current takes care of other channel types which are not described in particular. The most remarkable achievement of this theory Published by Copernicus Publications on behalf of the URSI Landesausschuss in der Bundesrepublik Deutschland e.V. the generality of thehowever, approach. Thus, ncentration in extracellular The Nernst Huxley, 1952), liquid. the very first complete indicating respectively. Normally, some of the the H-H model links the microscopic level of ion tential generated by the difference in ion description of the excitability of single neuron. channels are blocked. The probability that a channelschannel to the macroscopic of currents and ncentrations is represented by a battery. The is open is level described by additional 76 R. Appali et al.: A comparision of Hodgkin-Huxley and soliton neural theories action potentials (see variables m, n, fig. and 4). h. The combined action of m uivalent circuit is shown 2.1 The Model:in fig. 2. and h controls the Na+ channels. The K+ gates are controlled by n. Hodgkin and Huxley formulated the three ionic components as Outside Na+ du C = gNam3h( u ENa ) + gK n4 ( u EK ) + C + + + + + dt g g Na k gL + gL ( u EL ) + I( t ). V (1) Inside K+ The parameters ENa, EK, and EL are the reversal potentials. Reversal potentials and conductances are empirical parameters. So, it is demonstrated EL ENa Ek Fig. 1. Schematic model of Hodgkin-Huxley theory (modified from that the neurons transmit information by firing Figure 1: Schematic model of Hodgkin-Huxley Gerstner et al., 2002). and propagating electrical action potentials along theory (modified from (Gerstner et al., 2002)) their axons. Usually, motor neurons possess a Intracellular sheathpotential around their axonsfrom actingthe as an The Hodgkin-Huxley model (H-H model) can be Figure myelin 3: Action adapted Fig. 3. Action potential adapted from the original paper of Hodgkin insulator and enhancing the speed of signal the help of fig.model 1. The semi- original gure 2: explained Equivalentwith circuit of H-H paper of Hodgkin and Huxley and Huxley. transmission. permeable cell membrane separates the interior odgkin and Huxley, 1952) The model can reproduce and explain a wide of the cell from the extracellular liquid and acts acterized by their resistance equivalently, by their con- and range of data fromor, squid axon like the shape as a capacitor. If an input current I(t) is injected ductance. The leakage channel is described by a voltagepropagation of the action potential (see fig. 3), its into the cell, it may add further charge on the independent the conductance of the other 2 sharpconductance threshold,andrefractory period, anode-break capacitor, or leak through the channels in the cell ion channels is voltage and time dependent. excitation, accommodation and sub threshold membrane. Because of active ion transport If all channels are open, they transmit currents with a maxoscillations. can also describe through the cell membrane, the ion concentration imum conductance gNaThe or gKmodel , respectively. Normally, how- many channel types with smallTheparameter changes of the channels are blocked. probability that inside the cell is different from that of the ion ever, some a channel is open is described by additional variables m, n,Thus, indicating the generality of the approach. concentration in extracellular liquid. The Nernst + and h. The combined action of m and h controls the Na the H-H +model links the microscopic level of ion potential generated by the difference in ion channels. The K togates controlled bylevel n. Hodgkin and and the are macroscopic of currents concentrations is represented by a battery. The Huxleychannels formulated the three ionic components as action potentials (see fig. 4). equivalent circuit is shown in fig. 2. I Extracellular Extracellular I of H-H model (Hodgkin and Huxley, Fig. 2. Equivalent circuit 1952). C was the empirical representation of the experimental data in gNa gk 1952), the very a quantitativegLmodel (Hodgkin and Huxley, first complete description of the excitability of single neuron. V 2.1 The model EL ENa (H-H model) Ecan The Hodgkin-Huxley model k be explained with the help of Fig. 1. The semi-permeable cell membrane separates the interior of the cell from the extracellular liquid and acts as a capacitor. If an input current I (t) is injected into Intracellular the cell, it may add further charge on the capacitor C, or leak through the channels in the cell membrane. Because of active Figure Equivalent circuit of H-H model ion transport2:through the cell membrane, the ion concentra(Hodgkin 1952) tion inside theand cellHuxley, is different from that of the ion concentration in extracellular liquid. The Nernst potential generated by the difference in ion concentrations is represented by a battery. The equivalent circuit is shown in Fig. 2. As mentioned above, the Hodgkin-Huxley model describes three types of channels. All channels may be charAdv. Radio Sci., 8, 75–79, 2010 C du = gNa m3 h(u − ENa ) + gK n4 (u − EK ) dt +gL (u − EL ) + I (t). (1) The parameters ENa , EK , and EL are the reversal potentials. Reversal potentials and conductances are empirical parameters. So, it is demonstrated that the neurons transmit information by firing and propagating electrical action potentials along their axons. Usually, motor neurons possess a myelin sheath around their axons acting as an insulator and enhancing the speed of signal transmission. The model can reproduce and explain a wide range of data from squid axon like the shape and propagation of the action potential (see Fig. 3), its sharp threshold, refractory period, anode-break excitation, accommodation and sub threshold oscillations. The model can also describe many channel Figure 3: Action potential adapted from types with small parameter changes indicating the generality original Thus, paperthe ofH-H Hodgkin and the Huxley of the approach. model links microscopic level of ion channels to the macroscopic level of currents and action potentials (see Fig. 4). 2 www.adv-radio-sci.net/8/75/2010/ the for the generation and propagation of solitons. Pure lipid membranes exhibit “non-linearity” (: T. Heimburg and A. D. Jackson proposed an maximally compressible close to the transition alternative theory in 2005. The theory is a and decreased compressibility with increasing theoryof Hodgkin-Huxley of nerve and pulse R. thermodynamic Appali et al.: A comparision soliton neural theories 77 density or pressure) and “dispersion” (: lipid propagation. This theory is based on the lipid compressibility is frequency dependent both for characteristic of transition from a fluid to a gel area and volume compressibility) (Heimburg and state over a range of several degrees, slightly Jackson, 2005; Heimburg and Jackson, 2007a). below the body temperature (Heimburg and Both the effects of non-linearity and dispersion Jackson, 2005 and references there in). That is, produce a self-sustaining and localized density when a biological membrane is compressed, the pulse with a moving segment of the nerve fluid membrane gradually reaches to a denser gel membrane in the gel state (see fig. 6). This pulse state. Now, a sufficient pressure/ compression is called a soliton (Heimburg and Jackson, either in membrane area or volume will bring a 2007a). The solitary wave maintains its shape transition to the gel state (Heimburg, 2007a). while it travels at a constant speed less than the This transition is associated with the release of sound velocity in the lipid membrane. Solitons heat (Heimburg, 2007a; Heimburg, 2007b). The Figure 4: Separation of ionic conductances underlying the action in H-H model. can potential propagate over longModified distances without loss of relation between heat underlying capacity,the action density, Fig. 4. Separation of ionic conductances potential in H-H model. Modified from Hodgkin and Huxley (1952). from (Hodgkin and Huxley, 1952) energy (Heimburg and Jackson, 2005). The pulse compression and temperature are shown in fig. 5. is also called adiabatic pulse, because no Lipid membranes have the as properties required 3. Soliton Theory for the energy generation is and propagation lost to of thesolitons. environment during Pure lipid membranes exhibit “non-linearity” (: T. Heimburg and A. D. Jackson proposed an propagation (Heimburg and Jackson, 2007b). maximally compressible close to the transition alternative theory in 2005. The theory is a The soliton modewithofincreasing pulse propagation is and decreased compressibility thermodynamic theory of nerve pulse density or pressure) and “dispersion” (: propagation. This theory is based on the lipid basically different from the lipid AP given by the H-H compressibility is frequency dependent both for characteristic of transition from a fluid to a gel model (Anderson et al., 2009). area and volume compressibility) (Heimburg and state over a range of several degrees, slightly is associated with a transient Jackson,The 2005;solitary Heimburgwave and Jackson, 2007a). below the body temperature (Heimburg and Both the effects of non-linearity and dispersion Jackson, 2005 and references there in). That is, voltage change due to the transient change in produce a self-sustaining and localized density when a biological membrane is compressed, the statea of membrane. This the membrane pulse with moving segment of the affects nerve fluid membrane gradually reaches to a denser gel potential difference. Especially, it changes both membrane in the gel state (see fig. 6). This pulse state. Now, a sufficient pressure/ compression is called a soliton (Heimburg and Jackson, either in membrane area or volume will bring a the area and thickness of membrane and thereby 2007a). The solitary wave maintains its shape transition to the gel state (Heimburg, 2007a). (Heimburg 2007a). while itcapacitance travels at a constant speed lessand thanJackson, the This transition is associated with the release of These inchanges to be as a voltage pulse and sound velocity the lipid seem membrane. Solitons heat (Heimburg, 2007a; Heimburg, 2007b). The can propagate without loss of relation between heat capacity, density, leads overtolonga distances capacitive current due to the energy (Heimburg and Jackson, 2005). The pulse compression and temperature are shown in fig. 5. unbalanced charge of membrane. The reversible is also model called ofasmembrane adiabatic lipid pulse, because no theory (adapted Figure 6: Schematic thermodynamic from (Anderso observed during action potential is proposed energy is lost tomodel the of environment during Fig. 6.heat Schematic membrane lipid thermodynamic theory propagation This theoryenthalpy also explainsi.e. the effect of Thus the heat flow is correlated with theand changes (adapted Anderson et 2009). as from a(Heimburg result ofal.,Jackson, lipid 2007b). melting Figure Heatlateral capacity, lateral and Fig. 5. Heat 5: capacity, density and lateral density compressibility The soliton mode of pulse propagationon is the nerve pulse propagation. Acco in membrane density and potential. production of latent heat during lipid transition as Heimburg a function of basically different from the AP given by the theory, as lateral a function ofcompressibility temperature (adapted from and Jackson, H-H the anesthetics in the memb from fluid to gel state and the fluid-gel reabsorption of temp 2005). model (Anderson et al., 2009). temperature (adapted from (Heimburg and the transition 3.1 Soliton equation TheLipid solitary wave isthe associated with a transient membranes have the properties required for the genphenomenon known as freezing poin heat as system returns to the fluid state Jackson, 2005) change due to the transient in lipid membranes (Heimburg and Jackson, 2007c). So i eration(Anderson and propagation of2009). Pure Hydrodynamic voltage equation for the propagation ofsolitons. a change et al., state of membrane. This affects the membrane to force the membrane density pulse in exhibit the presence of dispersion for a (maximally compressible close to the through th potential “non-linearity” difference. Especially, it changes both transition, thereby action potentia cylindrical membrane along x-axis is (Heimburg transition decreased compressibility with increasing the denthe area andand thickness of membrane and thereby is suppressed. It is noted that fre T. Heimburg and A. D. Jackson proposed an alternative the- 2005) and Jackson, given by sity or pressure) and (lipid compressibility is capacitance (Heimburg and“dispersion” Jackson, 2007a). depression is a general ther ory in 2005. The theory is a thermodynamic theory of nerve These changes seem to beboth as a for voltage and frequency dependent areapulse and phenomenon volume compressibil3 and not chemical 2 4 pulse propagation. This theory is based on the lipid leads to a capacitive current due to the ∂ charac∂ ∂ ity) and Jackson, and Jackson, 2 ∂ (Heimburg A A A (2) 2005; Heimburg (Anderson et al., 2009). [ ] Δ ρ = c Δ ρ h Δ ρ ∂x unbalanced charge of membrane. The reversible teristic of transition from a fluid to a gel state over ∂ t 2 a range ∂ x 2007a). ∂x 4 heat observed during action potential is proposed of several degrees, slightly below the body temperature (He4.i.e. Comparison as Both a result lipidofmelting enthalpy the of effects non-linearity and dispersion produce Figure 5: Heat capacity, lateral density and imburg and Jackson, 2005, and references there After in). That transformation x-v.t),lipid production of latent and heat(z= during transition lateral compressibility as a function ofthe co-ordinate a self-sustaining localized density pulse with a moving is, when a biological membrane is compressed,bythe fluid the propagation thethe time The differences between both from fluid of to the gelvelocity, state and reabsorption of major temperature (adapted from (Heimburg introducing and segment nerve membrane in the gel state (see Fig. 6). membrane gradually reaches to a denser gel state. Now, equation independent thereturns propagation are: (Hodgkin and Hux heat asdescribing the system to the fluid theories state Jackson, 2005) This called a solitonand (HeimburgHeimburg and Jackson, densityarea excitation is pulse given as (Heimburg and 2007a). Jackson, 2005; He (Anderson et is al., 2009). a sufficient pressure/ compression either in membrane The solitary wave maintains its shape while it travels at a et al., 2009 Jackson, 2007a) Jackson, 2007a; Anderson or volume will bring a transition to the gel state (Heim- 3 Soliton theory constant speed less than the sound velocity in the lipid memburg, 2007a). This transition is associated with the ∂release 2 ∂ 2 A 2 A H-H distances The actionwithout potential is due to th over long ] Δρ =3 [(c 0brane. v relation + pΔρ A +Solitons q( Δρ A ) 2 +can ...) ∂ ∂propagate z Δρ of heat (Heimburg, 2007a; Heimburg, 2007b). The ∂z ∂z 2 current conducted loss of energy (Heimburg and Jackson, 2005). The pulse is across the m between heat capacity, density, compression and temperature ∂4 A (3) because no energy ion channels that act as resisto -as h adiabatic Δ ρ also called pulse, is lost to are shown in Fig. 5. ∂z 4 Where the environment during propagation (Heimburg Jackson, S The and nerve pulse is a sol ν is propagation velocity A 2007b). The soliton mode of pulse propagation is basically generated by the lipid transi ρ is lateral membrane density www.adv-radio-sci.net/8/75/2010/ membrane. c is sound velocity in fluid phase of membrane Adv. Radio Sci.,Nerve 8, 75–79, 2010 H-H pulse propagation is then p and q are the parameters determined as a moving self regenerating p from sound velocity and density capacitive discharges through th dependence resistors followed by capacitor h is the parameter to set the linear scale 78 R. Appali et al.: A comparision of Hodgkin-Huxley and soliton neural theories different from the AP given by the H-H model (Anderson et al., 2009). The solitary wave is associated with a transient voltage change due to the transient change in state of membrane. This affects the membrane potential difference. Especially, it changes both the area and thickness of membrane and thereby capacitance (Heimburg and Jackson, 2007a). These changes seem to be as a voltage pulse and leads to a capacitive current due to the unbalanced charge of membrane. The reversible heat observed during action potential is proposed as a result of lipid melting enthalpy i.e. production of latent heat during lipid transition from fluid to gel state and the reabsorption of heat as the system returns to the fluid state (Anderson et al., 2009). Thus the heat flow is correlated with the changes in membrane density and potential. 3.1 4 Comparison The major differences between both the neural theories are (Hodgkin and Huxley, 1952; Heimburg and Jackson, 2005; Heimburg and Jackson, 2007a; Anderson et al., 2009): H-H The action potential is due to the electric current conducted across the membrane by ion channels that act as resistors. S The nerve pulse is a solitary wave generated by the lipid transition in the membrane. H-H Nerve pulse propagation is then explained as a moving self regenerating pulse of local capacitive discharges through the resistors followed by capacitor recharging. S The propagation is due to the combined effect of nonlinearity and dispersion. Soliton equation H-H The theory is pure electrical with ionic hypothesis. Hydrodynamic equation for the propagation of a density pulse in the presence of dispersion for a cylindrical membrane along x-axis is (Heimburg and Jackson, 2005) given by i ∂2 ∂ h 2∂ ∂4 A A 1ρ = c / 1ρ − h 1ρ A ∂x ∂x ∂t 2 ∂x 4 S The theory is physical/ mechanical i.e. based on thermodynamics. H-H The nerve signal or action potential is electrical pulse. S The nerve signal is propagating adiabatic electromechanical pulse. (2) After the co-ordinate transformation (z = x − v · t), by introducing the propagation velocity, the time independent equation describing the propagation density excitation is given as (Heimburg and Jackson, 2007a) H-H The pulse propagation dissipates energy in the form of heat. S The propagating pulse does not dissipate heat energy. H-H The nerve’s physical changes are not explained. i ∂2 ∂ h 2 v 2 2 1ρ A = c0 + p1ρ A + q(1ρ A )2 + ... ∂ ∂z1ρ A ∂z ∂z ∂4 (3) −h 4 1ρ A ∂z Where ν is propagation velocity ρ A is lateral membrane density c is sound velocity in fluid phase of membrane p and q are the parameters determined from sound velocity and density dependence h is the parameter to set the linear scale of propagating pulse. This theory also explains the effect of anesthetics on the nerve pulse propagation. According to this theory, the anesthetics in the membrane lower the fluid-gel transition temperature, a phenomenon known as freezing point depression (Heimburg and Jackson, 2007c). So it is difficult to force the membrane through the fluid-gel transition, thereby the action potential generation is suppressed. It is noted that freezing point depression is a general thermodynamic phenomenon and not chemical in nature (Anderson et al., 2009). Adv. Radio Sci., 8, 75–79, 2010 S The nerve changes dimensions and gets thicker during nerve pulse. The voltage pulse accompanies the propagating signal, as a result of these transient geometric changes of the membrane. Both the theories explain the existence of voltage pulse in nerve pulse propagation. However, as an information transmitting pulse in H-H model and as a by-product signal in soliton theory. 5 Discussion Hodgkin-Huxley model of nerve pulse propagation is undoubtedly a remarkable achievement in the field of physiology. However, the theory could not explain the physical phenomena such as reversible heat changes, density changes and geometrical changes observed in the experiments (Iwasa et al., 1980; Tasaki, 1982, 1999; Tasaki et al., 1989, Tasaki and Byrne, 1992). Soliton model is an alternative model, convincingly explaining some of the unexplained observations in the experiments. However, there are several other questions that this www.adv-radio-sci.net/8/75/2010/ R. Appali et al.: A comparision of Hodgkin-Huxley and soliton neural theories has to answer like ion flow involvement in nerve signal propagation as stated by H-H model and also the faster propagation in myelinated nerves than in unmyelinated. 6 Context This is a theoretical comparison of both the theories. This has been carried out to analyze and choose a detailed neuron model which fits in for the goals of our project. We are examining the soliton theory to model and simulate the action potential and the coupling of electrode for a detailed study of the neuron. Acknowledgements. We are grateful to DFG (German Science Foundation) for funding our project in the Research Training Group 1505/1. References Andersen, S., Jackson, A. D., and Heimburg, T.: Towards a thermodynamic theory of nerve pulse propagation, Progr. Neurobiol., 88, 104–113, 2009. Bernstein, J.: Studies on the thermodynamics of bioelectric currents, Pflügers Arch. Ges. Physiol., 92, 521–562, 1902. Cole, K. S. and Curtis, H. J.: Electric impedance of the squid giant axon during activity, J. Gen. Physiol., 22, 649–670, 1939. Gerstner, W. and Kistler, M. W.: Spiking neuron models. Single neurons, populations, plasticity, Cambridge Univ. Press, p. 34, 2002. Häusser, M.: The Hodgkin-Huxley theory of the action potential, Nature neuroscience, 3, 1165, doi:10.1038/81426, 2000. Heimburg, T. and Jackson, A. D.: On soliton propagation in biomembranes and nerves, Proc. Natl. Acad. Sci. USA., 102, 9790–9795, 2005. www.adv-radio-sci.net/8/75/2010/ 79 Heimburg, T.: Chapter 6: lipid melting, in: Thermal Biophysics of Membranes, Wiley–VCH, pp. 75–97, 2007a. Heimburg, T. and Jackson, A. D.: On the action potential as a propagating density pulse and the role of anesthetics, Biophys. Rev. Lett., 2, 57–78, 2007a. Heimburg, T.: Chapter 7: phase diagrams, in: Thermal Biophysics of Membranes, Wiley–VCH, pp. 99–122, 2007b. Heimburg, T. and Jackson, A. D.: Thermodynamics of the nervous impulse. In: Nag, K. 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Tasaki, I.: Physiology and Electrochemistry of Nerve Fibers, Academic Press, New York, 1982. Tasaki, I.: Evidence for phase transitions in nerve fibers, cells and synapses, Ferroelectrics, 220, 305–316, 1999. Tasaki, I. and Byrne, P. M.: Heat production associated with a propagated impulse in bullfrog myelinated nerve fibers, Jpn. J. Physiol., 42, 805–813, 1992. Tasaki, I., Kusano, K., and Byrne, P. M.: Rapid mechanical and thermal changes in the garfish olfactory nerve associated with a propagated impulse, Biophys. J., 55, 1033–1040, 1989. Adv. Radio Sci., 8, 75–79, 2010