### Quantum model of nerve pulse III: Relation to Hodgkin-Huxley model

Before the replacement of Hodgkin-Huxley model with a genuinely quantal model can be taken seriously, one must answer many difficult questions which also Hodgkin and Huxley must have faced as they developed their own model. In the following I will go through the basic questions and quantum answers to them.

** 1. Questions and answers**

* Q: In the resting state membrane potential is negative and cell has a negative net charge. What stabilizes the cell against the leakage of the negative charge if pumps and channels are not responsible for this?*

A: The findings about the strange behavior of cell membrane inspire TGD based answer. Cell membrane space-time sheet is its own quantum world and the flow of ions occurs only in the presence of magnetic flux tubes connecting it to the external world. These currents a however oscillatory Josephson currents if dissipation is absent. Hence there is no need to cut completely the connections to the external world.

* Q: How the resting state can result spontaneously if pumps are absent?*

A: If ionic currents are Josephson currents, they are automatically oscillating and the return to the original state is guaranteed. The flux tubes carrying the ionic currents will be assumed to connect axonal microtubules to the space-time sheet of the cell interior. Consider first the most obvious objections.

- Dark ions could not transform to ordinary ones in the exterior of the cell membrane. This might indeed kill the model.
- If ionic currents are Josephson currents, they are automatically oscillating and the return to the original state is guaranteed. The objection is that all biologically important ions are not bosons and the model for high T
_{c}super-conductor in its recent form allows only electronic and protonic Cooper pairs at room temperature [8]. TGD based nuclear physics however predicts the possibility of exotic nuclei for which one or more color bonds connecting nucleons to the nuclear string are charged. These exotic nuclei with electronic states identical to those of genuine ions could save the situation. The table below describes how cyclotron frequencies for B=.2 Gauss of the most important ions are modified in the simplest replacements with exotic ions. For instance, the notation Mg^{++}_{-}tells that there is double electronic ionization and electron shell of Argon as usual but that one color bond is negatively charged.^{23}Na^{+}→^{19}Ne_{+}: 13.1 Hz → 15.7 Hz^{23}Na^{+}^{24}→ Mg^{++}_{-}: 13.1 Hz→ 12.5 Hz^{39}K^{+}^{40}→ A_{+}: 7.7 Hz→ 7.5 Hz^{39}K^{+}→^{40}Ca^{++}_{-}: 7.7 Hz→ 7.5 Hz^{35}Cl^{-}→^{40}A_{-}: 8.6 Hz →7.5 Hzf

_{c}(K^{+}) and f_{c}(Cl^{-}) are replaced with the frequency 7.5 Hz and one can do only using the cyclotron frequencies f_{c}(Ca^{++})/2=7.5 Hz, f_{c}(Mg^{++})=12.5 Hz, and f_{c}(Ca^{++})=15 Hz. The nominal values of the lowest Schumann frequencies are 7.8 Hz and 14.3 Hz. All ions with relevance for nerve pulse and EEG could be bosonic ions or bosonic pseudo-ions. I do not know how well the needed ionization mechanisms are understood in the standard framework.

_{J}propto hbar

^{2}and hbar /Ω propto hbar for solitonic situation. ΔQ is very small for the ordinary value of hbar : also the oscillation period is very small. For large values of hbar situation changes and large maximal ion transfers are possible.

An hbar increasing phase transition could be involved with the generation of the nerve pulse. Quantum criticality during nerve pulse generation indeed suggest the presence of flux tubes with varying values of hbar . The lifetimes of the connected flux tubes could be proportional to hbar at criticality. A fractal hierarchy of pulses and EEG like oscillations of the membrane potential corresponding to various values of hbar is suggestive.

* Q: Can one make this more quantitative?*

A: One can construct a model based on Sine-Gordon wave equation [7] for the phase different Φ between the superconductors connected by Josephson junction sequences defined by magnetic flux tubes and idealizable as a continuous Josephson junction.

- For a Josephson junction idealizable as a hollow cylinder with radius R and thickness d the expression of the Josephson current reads as
J= J

_{0}sin(Ze∫ Vdt/hbar)J

_{0}is in case of cell membrane given byJ

_{0}= (Ze2π dR/Λ^{2}) ×(hbar/m) ,where R and d would be now the radius and thickness of the axon, Λ is the magnetic peneration length, and m is the mass of the charge carrier. Although this expression does not hold true as such when Josephson junctions are replaced by magnetic flux tubes connecting microtubuli and axon, one can can safely make some qualitative conclusions. The amplitude of the Josephson current increases with hbar . For electron the value of the amplitude is by a factor x≈ Am

_{p}/m_{e}≈ 2^{11}A larger than for ion with a mass number A. This gives for electron Cooper pairs a unique role as an initiator of the nerve pulse. Note that the amplitudes of the Josephson currents of electron and ions are quite near to each other if one has hbar (ion)= 2^{11}A×hbar: this might explain why the powers of 2^{11}for hbar seem to be favored. - Electronic Josephson current dominates and makes it ideal for the generation of nerve pulse (kick to gravitational pendulum). This is possible if the net amount of electronic charge is so small that it flows out during the generation of flux tubes. For ions this need not occur even if ion densities are of same order of magnitude. Constant voltage V creates an oscillating current and no catastrophic leakage takes place and the resting state results automatically. The ionic Josephson currents assignable to the magnetic flux tubes connecting microtubules through the cell membrane to the external world could be responsible for the nerve pulse.
- The mechanical analog for Sine-Gordon system [8] assignable to Josephson junction is rotating pendulum but one must be cautious in applying this analogy. There are two options concerning the modeling of the situation.
- Membrane potential represents an external voltage V(t) and one has Φ
_{i}= Z_{i}e∫ Vdt/hbar, where Φ_{i}is the phase difference between Bose-Einstein condensates. - System is autonomous and membrane potential V(t)=hbar (dΦ
_{i}/dt)/Z_{i}e is completely determined by the dynamics of any phase Φ_{i}. This option is highly predictive and discussed in the sequel.

- Membrane potential represents an external voltage V(t) and one has Φ
- The analogy with gravitational pendulum allows to identify the phase angle Φ as the counterpart of angle Θ characterizing angular position of mathematical pendulum (note that this analogy can be misleading since it implicitly brings in 3-D thinking).
- In this picture rotating pendulum corresponds to a soliton sequence containing infinite number of solitons: both stationary and moving soliton sequences are obtained. The sign of Ω=dΦ/dt is fixed and approximately constant for large values of Ω. Resting potential could correspond to this kind of situation and Ω ≈ 2π kHz is suggested by kHz synchrony. A mechanism of this synchrony will be discussed below. For large values of hbar even values of Ω in EEG range could correspond to membrane potential. For large values of Ω one as V≈ hbarΩ
_{i}/Z_{i}e. If also EEG rhythms correspond to Ω they must correspond to different values of hbar and f propto 1/hbar would hold true. Changes in the dominating EEG rhythm (40 Hz, 10 Hz, 5 Hz,..) could correspond to phase transitions changing hbar to given value for a large number of axons. The maximal charge transfer during single period is proportional to Δ Q propto 1/Ω. - Hyperpolarization/polarization would mean fastening/slowing down of the pendulum rotation and slowing down would make the system unstable. Near criticality against the generation of nerve pulse would mean that pendulum is rotating rather slowly (Ω<< f
_{J}) so that a small kick can transform rotation to oscillation. The sign of V propto dΦ/dt would change and large amplitude oscillatory motion would result for single period only after which a kick in opposite direction would lead back to the resting state. Membrane potential varies between the resting potential V_{0}=-75 V and V_{1}=+40 V during nerve pulse: V_{1}>|V_{0}| would have killed the model. Note that V_{1}=40 V is rather near to the critical potential about V_{1}=50 V: ideally these potentials should be identical. - The so called breathers -both stationary and moving- correspond to soliton-antisoliton bound state (see the visualization here). Breathers could be identified as large amplitude oscillations around Φ=0 ground state. Physical intuition suggests that breathers are possible also for a ground state corresponding to a rotating pendulum (representing moving or stationary waves). They would correspond to kicking of one pendulum in a sequence of penduli along z-axis rotating in phase at the initial moment. The kick could correspond to a genuine external perturbation generated by a pair electronic supra current pulses of opposite sign giving constant velocity increments ΔΩ initiating and halting the nerve pulse just like they would do in the case of tqc but in opposite time order. If the background corresponds to a propagating EEG wave, also nerve pulse is expected to propagate with same velocity. The propagation direction of EEG wave would also explain why nerve pulses propagate only in single direction.

- In this picture rotating pendulum corresponds to a soliton sequence containing infinite number of solitons: both stationary and moving soliton sequences are obtained. The sign of Ω=dΦ/dt is fixed and approximately constant for large values of Ω. Resting potential could correspond to this kind of situation and Ω ≈ 2π kHz is suggested by kHz synchrony. A mechanism of this synchrony will be discussed below. For large values of hbar even values of Ω in EEG range could correspond to membrane potential. For large values of Ω one as V≈ hbarΩ
- For the ordinary value of hbar , the frequency of the Josephson current corresponds to that assignable to energy .07 eV being around 1.6×10
^{13}Hz and quite high. For x==hbar /hbar_{0}=2^{44}the frequency would be near to cyclotron frequency of about 1 Hz assignable to DNA strands. For x=3× 2^{3× 13}the frequency would be near to the fundamental 10 Hz frequency which is secondary p-adic time scale associated with electron and correspond to the temporal duration of negative energy space-time sheet assignable to electron. For x=3× 2^{3× 11}one would obtain a 640 Hz frequency which corresponds to the time scale of nerve pulse. It seems clear that the original hypothesis that only powers of 2^{11}define the spectrum of Planck constant is too restrictive. The requirement that cyclotron frequencies and Josephson frequencies are proportional to each other for small oscillations would guarantee resonant behavior for common strength of the magnetic field would give hbar propto A. This would require that each ion species lives at its own flux tubes.

*Q: What instabilizes the axon? Why the reduction rather than increase of the magnitude of the membrane potential induces the instability? Why the reduction of the resting potential below the critical value induces nerve pulse?*

A: Large enough voltage pulse between microtubules and membrane could generate electronic DC supra current. The introduction of a small amount of positive charge to the inner lipid layer and staying there for some time would generate the voltage pulse between microtubules and lipid layer so that DC electronic supra current would be induced, and induce the reduction Δ V≈ .02 eV of the magnitude of the membrane potential. A similar introduction of negative charge would induce hyperpolarization and the direction of the current would be opposite if it is generated at all. The mechanism generating the small positive charge to the inner lipid layer could be based on the exchange of exotic W bosons between pairs of exotic nuclei at opposite sides of the cell membrane so that the negative charge of the inner lipid layer would be reduced.

* Q: Can one understand the observed radial force, the increase of the radius of axons and the reduction of its thickness, and heating followed by cooling?*

A: The observed outward force acting on a test system might be due to the ionic Josephson currents to which the test system responds. During the second half of the pulse the sign of the ionic force is predicted to change. The pressure caused by the electronic Josephson current pulse before the connection of flux tubes to single flux tube might relate to the increase of the radius of the axonal membrane and with the reduction of its thickness as well as the slight increase of its temperature as being due to the electrons which heat the lipid layer as they collide with it. The ions return at the second half of the pulse and could transfer the heat away by convection.

- This hypothesis gives the estimate for the force f per unit area as
f

_{2e}(t)= (dn(lipid)/da) ×(J(t)/2e)× hbar k= (dn(lipid)/da) × U× (hbar

^{2}k/2m_{e})× sin(ω_{J}(2t)) ,U= (2π A/Λ

^{2}) .The parameter A corresponds to the parameter dR in the case that Josephson junctions have the thickness of axonal membrane, and is not relevant for order of magnitude estimate. R corresponds to the distance between microtubules and cell exterior space-time sheet to which flux tubes end. dn(lipid)/da is the 2-D density Josephson junctions equal to the density of lipids.

k≈ 1/R is the wave vector of electron Cooper pair at the magnetic flux tube. The 3-momentum of electron is enormous for the proposed value of hbar , and the only possible interpretation is that the four-momentum of electron is virtual and space-like and corresponds to exchange of space-like virtual photon describing Coulomb interaction with lipid layer.

Λ

^{2}satisfies in the first approximation the formulaΛ

^{-2}= (4π e^{2}n_{e}/m_{e})+ ∑_{I}(4π e^{2}n_{I}/Am_{I})= α_{em}16π^{2}×hbar_{0}[(n_{e}/m_{e})+ ∑_{I}(n_{I}/A_{I}m_{I})].Note that there is no real dependence on hbar . Above critical voltage electrons could dominate in the expression but during nerve pulse ions should give the dominating contributions. U cannot be too far from unity.

- From this one can integrate the total momentum of Cooper pairs transferred to the lipid layer before the flux tubes fuse together if one knows the value of time t when this happens. F propto hbar
^{2}/m_{e}^{2}proportionality implies that for the required large value of hbar /hbar_{0}≈ 3× 2^{33}the force is by a factor 6× 10^{20}stronger than for hbar_{0}. - The force caused by ionic Josephson currents on piston is given by
f(t)= ∑

_{I}(2me/m_{I}) (2/Z_{I}) × f_{2e}(τ)τ=(Z

_{I}/2)×(Ω/ω_{J})× t . - The comparison with the observed force gives estimate for the value of magnetic penetration length and thus density of electrons at the flux tube.
According to [3] in one particular experiment the force on piston of area S= .01 cm

^{2}at the maximum of voltage (t= 2π/Ω) is F= 2 nN. This gives a killer test for the model. One obtains an estimate for the parameter U=(Λ^{2}/2π A) asU=Λ

^{2}/2πA= (dn(lipid)/da) × (hbar^{2}k/m_{p}cF)× ∑_{I}(2/A_{I}Z_{I}) .The value of U should not deviate too much from unity. One can use the estimates

hbar/hbar

_{0}=3×2^{33}, k=2π/R.Note that the experimental arrangement forces to use this value of k. The actual value in normal situation would be smaller and depends on the distance of the boundary of the cell exterior space-time sheet on microtubules. Using the values d=10 nm and R=5 μm this gives

U≈ ∑

_{I}(2/A_{I}Z_{I})× X ,X= 9× 2

^{66}× (hbar_{0}^{2}2π/m_{p}cFR)×(S/S(lipid)).The factor X=.9267 is of order unity! Taking into account that hbar/hbar

_{0}is enormously large number it is difficult to believe that the result could be mere accident. Hence U does not deviate too much from unity and there are good hopes that the model works.For n

_{I}= x_{I}/a^{3}, a=10^{-10}m, and A= dR one obtains a direct estimate which combined with above estimate gives two conditions which should be consistent with each other:U= 76.1×∑

_{I}(x_{I}/A_{I}) ,U= .93×∑

_{I}(2/A_{I}Z_{I}) .These estimates are consistent for x

_{I}≈ 10^{-2}, which makes sense.

* Q: Where the primary wave propagates: along axon or along microtubules?*

A: This question need not make sense if microtubules and axon are connected by magnetic flux tubes to form single quantum coherent system. That axonal microtubules have constant electric field which is always in same direction could explain why the background soliton sequences and nerve pulses propagate always in the same direction and suggests that the primary wave propagates along microtubules. On the other hand, if W exchange between cell exterior and exterior reduces the negative charge of the inner lipid layer then axon could be seen as initiator. This could induce conformational or gel-sol phase transition propagating along microtubule and inducing the pair of voltage pulses in turn inducing the fusion of flux tubes at cell membrane which in turn would induce criticality of the axonal membrane. For this option axonal soliton would be a shadow of the microtubular soliton rather than completely independent dynamical process.

* Q: How nerve pulse velocities are determined?*

A: At first glance it seems v could be determined by boundary conditions guaranteing synchronization of neuronal activity rather than by dissipation as in Hodkin-Huxley model. As a matter fact, dissipation turns out to affect also v just because it is determined by boundary conditions!

- Hodkin-Huxley model would suggest that nerve pulse velocity v is dictated by frictional effects as an analog of a drift velocity. The rough order of magnitude estimates for the velocities of conformational waves along micro-tubuli are consistent with the velocities of nerve pulses. The proportionality v propto d of nerve pulse velocity to nerve axonal radius might be understood as resulting on the dependence on the length of flux tubes connecting axon and microtubules and mediating a frictional feedback interaction from axon. Feedback would be naturally reduced as d increases. Feedback interaction could explain also the sensitivity of the thermal parameters of the axonal membrane to the proteins in its vicinity. If the frictional feedback is due to the environmental noise at the axon amplified at quantum criticality this is what one expects. Quite generally, quantum criticality would explain the high sensitivity of the thermal parameters on noise. Saltation cannot be responsible for the higher conduction velocity in myelin sheathed portions of axon. The insulation would reduce the environmental noise at the level of axons and thus reduce the frictional feedback from axon to the microtubules.
- The introduction of friction is however problematic in the recent situation. In absence of boundary conditions Sine-Gordon equation predicts for the propagating soliton sequences a continuous velocity spectrum and friction should affect Ω and V but it is not clear whether it can affect v.
- In this framework the boundary boundary conditions at the ends of the axon or some subunit of axon would dictate the values of v: γΩ L/v=n2π corresponds to periodic boundary conditions (note that γ=(1-(v/c)
^{2})^{1/2}≈ 1 holds true). v=ΩL/n2π implies that friction indeed affects also v. - This relationship states that the time taken by the nerve pulse propagate through the axon is always T= L/v =n2π/Ω: this would synchronize neurons and Ω≈ 2π kHz is suggested by the well-known 1 kHz synchrony difficult to understand in the standard framework where v would be determined by chemistry rather than geometry. Myelin shielding could in this picture guarantee that coherent wave propagation is possible over the entire axon so that boundary conditions can be applied.
- This would give v≈ ΩL/n2π≤ ΩL/2π. Ω= 2π kHz and n=1 would give for L in the range 1 cm -10 cm v in the range 10 m/s-100 m/s corresponding roughly to the observed range of values. For short axons velocity would be lower: for L=10 μm one would have v= .01 m/s. For longer axons the value of n could be higher or the axon would decompose into structural units for which periodic boundary conditions are satisfied. The sections between Ranvier nodes have length measured in millimeters as are also the lengths of axonal microtubules and 1 mm would correspond to a velocity of 1 m/s. The actual velocity for the myelinated sections varies between 18-100 m/s so that basic structural units should be longer.
- The proportionality of v to the radius of axon would follow from the proportionality of the length of the axon or its basic sub-unit (not longer than ≈ 10 cm) to its radius: the simplest geometric explanation for this would be in terms of scaling invariance of the axonal geometry consistent with fractality of TGD Universe. In the standard framework this proportionality would be explained by the minimization of dissipative losses in the case of long axons: one cannot exclude some variant of this explanation also now since friction indeed reduces v.
- There is an electric field associated with microtubules (always in same direction). Could this electric field play the role of external force feeding energy and momentum to the moving soliton sequence to compensate dissipation so that v would have interpretation as a drift velocity?

- In this framework the boundary boundary conditions at the ends of the axon or some subunit of axon would dictate the values of v: γΩ L/v=n2π corresponds to periodic boundary conditions (note that γ=(1-(v/c)

*Q: Can one understand EEG in this framework?*

A: Just like kHz waves also EEG generating waves could correspond to propagating soliton sequences. Since V is not affected, the value of hbar must be much larger and one must have hbar propto f, where f defines the EEG rhythm. It is known that EEG amplitudes associated with EEG rhythms behave roughly like 1/f. This can be understood. By Maxwell's equation the divergence of electromagnetic field tensor is proportional to 4-current implying the amplitude of EEG identified as Josephson radiation is proportional J_{0}/Ω and therefore to hbar. The propagation velocity v= ΩL/2πn of EEG generating waves is rather slow as compared to kHz waves: for f=10 Hz one would have 10 cm long axon v=1 m /s. Synchronization results automatically from periodic boundary conditions at the ends of the axons.

Nerve pulses during EEG rhythms would have much slower velocity of propagation and the duration of nerve pulse would be much longer. The maximal charge transfer would be proportional to 1/hbar. It would thus seem that EEG and nerve pulse activity should exclude each other for a given axon. Ω is however smaller so that the generation of nerve pulse is easier unless also ion densities are lower so that J_{0} (analogous to gravitational acceleration g in pendulum analogy) is reduced. Perhaps this takes place. The consistency with the propagation velocity of microtubular conformational (or even gel-sol-gel) waves might pose additional constraints on v and thus on frequencies Ω for which nerve pulses are possible. That ordinary EEG is not associated with ordinary cells might be due to the fact that hbar is much smaller: the fractal analog of EEG generating waves could be present but these EEG waves would correspond to faster oscillations in accordance with the view about evolution as an increase of hbar.

For background see that chapter Quantum Model of Nerve Pulse of "TGD and EEG".

** References**

[1] * Soliton model*.

[2] T. Heimburg and A. D. Jackson (2005),* On soliton propagation in biomembranes and nerves*, PNAS vol. 102, no. 28, p.9790-9795.

[3] T. Heimburg and A. D. Jackson (2005), * On the action potential as a propagating density pulse and the role of anesthetics*, arXiv : physics/0610117 [physics.bio-ph].

[4] K. Graesboll (2006), * Function of Nerves-Action of Anesthetics*, Gamma 143, An elementary Introduction.

[5] * Physicists challenge notion of electric nerve impulses; say sound more likely*.

[6]* Saltation*.

[7] * Sine-Gordon*

[8] The chapter * Bio-Systems as Super-Conductors: part I* of "The Quantum Hardware of Living Matter".

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