Two-wave model of the muscle contraction

BioSystems 96 (2009) 136–140

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Two-wave model of the muscle contraction Marcin Molski ∗ Department of Theoretical Chemistry, Faculty of Chemistry, A. Mickiewicz University of Pozna´ n, ul. Grunwaldzka 6, PL 60-780 Pozna´ n, Poland

a r t i c l e

i n f o

Article history: Received 14 November 2008 Received in revised form 6 January 2009 Accepted 9 January 2009 Keywords: Muscle contraction Space-like fields Feinberg equation Corben theory Quantum potential

a b s t r a c t The Matsuno model of the muscle contraction is considered in the framework of the two-wave Corben’s theory of composite objects built up of both time- and space-like components. It has been proved that during muscle contraction the locally coherent aggregates distributed along the actin filament interact by means of space-like fields, which are solutions of the relativistic Feinberg equation. The existence of such interactions and lack of decoherence are conditions sine qua non for appearance of the quantum entanglement between actin monomers in an ATP-activated filament. A possible role of a quantum potential in the muscle contraction is discussed and the mass of the carrier of space-like interactions is estimated m0 = 7.3 × 10−32 g (46 eV).

1. Introduction In the process of evolution living systems have developed various intricate communication capabilities allowing effective transfer of information on and between all levels of biological existence (organism, organ, tissue, cell, subcellular organelles, biomolecules). The effective transfer of information enables an organism to coordinate numerous biological functions and processes leading to self-organization and cooperation, whose spatial scale spans nine orders of magnitude: from 10−9 m for approximate intermolecular spacing to about 1 m—the unit of height for animals and human beings. Besides the electromagnetic fields, the remaining carriers of information (molecules, ions, electric currents) seem too inert to play a coordinating role on a such wide scale. One of the most intriguing biological collective phenomenon is cell motility in general, and muscle contraction in particular. On the molecular level this process is generated by an actin filament sliding on myosin molecules in the presence of hydrolized ATP (adenosine triphosphate). In the experimental research performed by Hatori et al. (2001) it has been demonstrated that during muscle contraction an actin in the presence of hydrolyzed ATP, induces coherent magnetic dipoles over the entire filament. Since the magnetic dipole–dipole interaction energy is much less than the thermal energy of the environment (at the temperature of the organism), thermal influences would easily destroy the magnetic alignment and coherence. The observed alignment along the ATPactivated actin filament, indicates that there exists an ordering

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factor another type than the magnetic dipole–dipole interaction. Acccording to Matsuno (1999), the magnetic ordering, which generates the coherent state of the actin filament may appear due to the quantum entanglement (Schrödinger, 1935) linking actin monomers in the ATP-activated phase. The Matsuno model cleary demonstrates that functioning of the living systems in general and muscles in particular requires a coherent cooperation and space-like (non-local) correlation (via quantum entanglement) of biological entities on both the microlevel (actin, myosin, ATP) and the macro-level (muscle). A question emerges: has the nature incorporated the space-like quantal effects (entaglement, coherence, interaction at-the-distance, non-local correlations) in functioning of living systems? Much effort has been undertaken to answer this question (Penrose, 1994; Josephson and Pallikari-Virast, 1991; Matsuno, 1999; Thaheld, 2003; Pizzi et al., 2004a, b; Molski and Konarski, 2003; Molski, 2006). According to Penrose (1994, p. 373): “Despite the technical difficulty of performing experiments which can detect such largedistance quantum effects, we should not rule out the possibility of Nature having found biological ways of doing a great deal more. The “ingenuity” that is to be found in biology should never be underestimated.” Josephson and Pallikari-Virast (1991) has established the thesis that “. . . life may exist from the beginning as a cooperative whole directly interconnected at-the-distance by Bell type non-local interactions, following which modifications through the course of evolution cause organisms to be interconnected directly with each other and with object to an extent that is adapted to circumstances.” According to the physical interpretation (Einstein et al., 1935; Freedman and Clauser, 1972)a system is space-like, when interactions or mutual influences (e.g. correlations) between its

M. Molski / BioSystems 96 (2009) 136–140

constituents are not subjected to limitation involving the velocity of light (in vacuum) as the upper limit. In such non-local systems the interactions at-a-distance including instantaneous non-local correlations between long range separated objects are possible (Tittel et al., 1998). Since the pioneer work by Einstein et al. (1935) was published the non-local systems have become the subject of intensive studies. The experiments by Freedman and Clauser (1972), Aspect et al. (1982a, b) proved that the statistical correlations between measured polarization of the pair of photons are inconsistent with the Bell’s (1964, 1993) inequality. It means, that separable quantal sytems can be interconnected via non-local EPR correlations. It should be pointed out, however, that according to the standard formulation of quantum mechanics, the EPR effect and entanglement cannot be used to send superluminal signals from one object to another—there is no means to transfer messages via this procedure (Eberhard, 1977a, b; Ghirardi et al., 1980, 1992). However, this prohibition is broken if quantum mechanics is allowed to be “slightly” non-linear (Polchinski, 1991) or is interpreted in transactional(Cramer, 1977) or tachyonic (Molski, 1998) terms. In view of this, we cannot exclude a space-like transfer of influences or correlations between actin monomers in ATP-activated filament, even though no signal or energy is passed between them (Stapp, 1977). The main objective of the present study is to reconsider the Matsuno (1999) model in the framework of the Corben’s (1977a,b, 1978a,b) theory, which includes in description of quantal systems both time- and space-like1 states. In particular, it will be proved that during muscle contraction the locally coherent aggregates distributed along the actin filament interact by means of space-like fields, which are solutions of the relativistic Feinberg (1967) equation. The possible role of the quantum potential in the muscle contraction will be also discussed.

137

Its solution takes the form ˛(t, x) = f (x − vactin t)

(5)

in which f is an arbitrary function varying smoothly in time and space. Because of the spatiotemporal smoothness of the phase factor ˛(t, x), each local aggregate of mass Mc , is coherent and entangled in the quantum manner (Matsuno, 1999). 3. The Space-Like Phase Function The solution (5) represents phase propagating unidirectionally with the phase velocity vp identical to the translational velocity vactin of the actin filament sliding on myosin molecules.2 This fact is worth pointing out as the phase vp and group vg velcities are interrelated

vg =

c2

vp

,

(6)

where c is the velocity of light in vacum. The relation (6) indicates that for vp = vactin < c, the associated group velocity is vg > c, and this is the main criterion (Recami, 1986; Horodecki, 1988; Horodecki, 1991; Molski, 1998) of the space-like characteristic of the phase function (t, x) = ei˛(t,x) .

(7)

If Matsuno’s model is correct within the framework of de Broglie matter wave theory, the group velocity specific to the waves inducing condensation into a single quantum state must be greater than c because of the phase-group velocity relation (6). The de Broglie waves whose group velocity is greater than c must be space-like (tachyonic). To clarify this point, we notice that the phase function (7) can be given in the equivalent form of a plane wave

2. The Matsuno Model

(t, x)+ = ei(kx−ωt) ,

In the Matsuno (1999) model of the muscle contraction an actin filament slides on myosin molecules unidirectionally along the x-axis with translational velocity vactin . In the quantal picture, a locally coherent aggregate with the rest mass Mc distributed along the actin filament is described by the interaction-free time-like Schrödinger equation

which is also solution of Eq. (4) specified in the form

i¯h

∂free (t, x) ∂t

=−

2 h ¯ ∂2 free (t, x) . 2Mc ∂x2

(1)

When the local aggregate interacts with other neighboring aggregates, the wavefuction free is slightly modified to the form  (t, x) = free (t, x) ei˛(t,x)

(2)

in which the phase factor ˛(t, x) includes the contribution from the interactions. According to Matsuno (1999) model the wavefunction  (t, x) satisfies the time-like Schrödinger equation 2

i¯h

h ¯ ∂2  (t, x) ∂ (t, x) , =− 2Mc ∂t ∂x2

(3)

Â+ (t, x)+ = 0,

vp =

ω = vactin . k

(9)

(10)

Such wave propagates in the +x-direction in contradistinction to the wave (t, x)− = ei(kx+ωt)

(11)

propagating in the −x-direction and satisfying the equation Â− (t, x)− = 0,

Â− =

∂ ∂ − vactin . ∂t ∂x

(12)

Multiplying operators Â− and Â+ , we arrive at the second-order differential equation



Â− Â+ (t, x) =

∂˛(t, x) ∂˛(t, x) + vactin = 0. ∂t ∂x

in which

∂2 ∂2 − v2actin 2 2 ∂t ∂x

(t, x) = ei(kx±ωt) . 1 In the particle picture, the notion time-like refers to the objects, which move with subluminal velocity v ∈ (0, c), whereas the notion space-like characterizes the objects, which move with superluminal velocity v ∈ (c, ∞). In the wave picture the time(space)-like waves propagate with group velocity smaller (greater) then that of the light in vacuum.

∂ ∂ + vactin . ∂t ∂x

This plane wave is characterized by the wavevector k = 2/, frequency ω = 2 and the phase velocity

which, under substitution (2) into (3), produces in the slowly varying limit of the phase factor, the phase equation (Matsuno, 1999) (4)

Â+ =

(8)



(t, x) = 0

(13)

(14)

2 It should be pointed out that an actin filament with heat acceptors attached to its Cys374 residue in each actin monomer could move unidirectionally under heat pulsation in the total absence of ATP and myosin (Kawaguchi and Honda, 2007).

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M. Molski / BioSystems 96 (2009) 136–140

Introducing into (13) the useful relation (Molski, 1998)

v2p c2

2

=1−

m0 c 2 2 h ¯ k2

v2actin c2

=

with respect to an invariant interaction condition (15)



m 2 c2 ω2 − k2 = − 02 2 c h ¯ one gets ∂2 − 2 c ∂t 2



(16)



1−

m02 c 2



2 h ¯ k2

∂2 ∂x2

 (t, x) = 0.



∂2 c 2 ∂t 2

−

m 2c ∂2 − 02 2 ∂x h ¯

(18)

Here, m0 stands for the mass associated with the space-like field (t, x). The derived above equation is well-known in the theory of space-like objects. It represents the relativistic Feinberg (1967, 1970) equation describing propagation of the space-like field (t, x) in two-dimensional space-time continnum (ct, x). If, according to Matsuno (1999), the function (t, x) represents the contribution from interactions between the neighboring aggregates along the actin filament, those interactions are of space-like type. In such circumstances, m0 can be interpreted as the mass of the carrier of space-like interactions. 4. The Corben Model In the wave physics the phase velocity refers to the extent of temporal difference of the phase belonging to a single quantum state, while group velocity refers to the temporal difference of the resultant phase belonging to a group of quantum states. In the Matsuno’s (1999) model the phase velocity intrinsic to the quantum state of condensation is equal to the actin’s sliding velocity, while the phase velocity of the associated de Broglie wave is always greater than light velocity c though its group velocity is less than c. Bearing in mind that the actin velocity vactin should be equal to the group velocity of the associated time-like field free (t, x), we arrive at an interesting interpretative inconsistency. If vg = / vactin then two different group velocities are associated with the field  (t, x): one superluminal vg > c connected with (t, x) and another subluminal vactin < c associated with free (t, x). This surprising situation is familiar for the composite systems built up of both time- and space-like components. The theory of such systems was developed by Corben (1977); Cramer (1977); Corben (1978a, b). He showed that a free time-like object of rest mass3 Mc and a free space-like object of rest mass m0 can trap each other in a relativistically invariant way. If Mc > m0 the compound object is invariably of time-like type with rest mass



M0 =



Mc2 − m02

(19)

described by a wave function  = free 

(20)

satisfying a time-like Klein–Gordon wave equation



∂ ∂ + (M0 c/¯h)

3

2



 =0

Notation adopted to that applied in the Section 2.

∂2 c 2 ∂t 2

∂2 ∂2 ∂2 − 2 − 2 2 ∂x ∂y ∂z

−

(23)

is the d’Alembert operator. Such a system of particles is built up of two coupled objects associated with a time-like free wave and a space-like  one being solutions of time-like Klein–Gordon equation ∂ ∂ + (Mc c/¯h)

2



free = 0,

(24)

and space-like Feinberg (1967) equation

 2

(t, x) = 0.

∂ ∂ =

 (17)

After differentiation of the wave function (t, x) twice with respect to space coordinate, the above wave equation yields



(22)

In the above equations

derived from the dispersion formula (Molski, 1998) valid for spacelike matter waves associated with a particle of mass m0



∂ free ∂  = 0.

(21)



∂ ∂ − (m0 c/¯h)

2



 = 0.

(25)

The free and  fields interact and lock to form a plane wave  , which is time-like when Mc > m0 . One cannot combine two time(space)like states in this way, because application of condition (22) to such states leads to imaginary momenta and exponentially increasing (not normalizable) wave functions. Hence, two time-like or two space-like objects cannot thus trap each other to yield a stable bound system (Corben, 1977; Cramer, 1977; Corben, 1978a, b). In non-relativistic regime in two-dimensional space-time (ct, x), the time-like Eqs. (21) and (24) reduce to 2

i¯h

∂ (t, x) h ¯ ∂2  (t, x) =− , 2 2 ∂t ∂x2 2 M −m c

i¯h

∂free (t, x) ∂t

=−

(26)

0

2 h ¯ ∂2 free (t, x) . 2Mc ∂x2

(27)

A comparison of (3), (1), (18) with (26), (27), (25) reveals, that fundamental equations of the Matsuno model are consistent with the Corben’s theory. Only the one exception is the effective mass  Mc2 − m02 appearing in Eq. (26), which in Eq. (3) is approxM0 = iated by Mc . This result indicates that the mass Mc of the local aggregate diminishes due to the space-like interactions between aggregates. 5. The Tachyokinematic Effect Taking into account the disspersion formula (16) in the fourmomentum form 

E2   − P 2 = −m02 c 2 c2

(28)

one may calculate the momentum P  = m0 c of infinite speed tachyon (E  = 0). If local aggregate of mass Mc interacts with such particle the aggregate attains the velocity vactin and momentum Mc vactin . In such circumstances the momentum conservation principle Pc = Mc vactin = P  = m0 c ⇒ m0 =

Mc vactin c

(29)

permits calculating the mass of the carrier of space-like interactions between aggregates. The relation (29) represents the so-called tachyokinematic (Molski, 1998) or tachyoelectric (Steyaert, 2000) effect. If aggregate endowed with the mass Mc carries the momentum Pc = Mc vactin = 2.2 × 10−21 erg s/cm (Matsuno, 1999), then for c = 29, 979, 245, 800 cm/s one gets m0 = 7.3 × 10−32 g (46 eV). This mass is greater than the electron neutrino mass 0.7 × 10−34 g (0.04 eV) obtained in the Kamiokande experiment. On the other hand, it conforms acceptably with the mass 44 eV of the superluminal cosmic electron neutrino, estimated by Ni and Chang (2001) from the data obtained after the supernova explosion (SN 1987A)

M. Molski / BioSystems 96 (2009) 136–140

(Bionta et al., 1987; Hirata, 1987).It is worth pointing out also that the experimental investigation of the antineutrino electron mass in the Troitsk experiment (Belesev et al., 2008) provided the range of the squared mass −0.8 ≤ m2 ≤ 0 eV2 . It indicates that electron antineutrino emitted in the tritium ␤-decay is either massless or is a space-like particle endowed with imaginary mass and superluminal velocity (Feinberg, 1967). Theory of the space-like neutrino has been developed among other by Chodos et al. (1985); Ciborowski ´ and Rembielinski (1999); Ni and Chang (2001); Ehrlich (2003).

139

of the actin filament vactin . It is interesting to note that the identical formula as (38) has been employed by Steyaert (2000) in astrophysics to describe the tachyoelectric effect in which bradyon with rest mass Mc recoils with the kinetic energy (m0 c)2 /(2Mc ) due to absorption of tachyon of mass m0 . 7. Results and Discussion

in which Pc = Mc vactin is x-component of the momentum of the aggregate of mass Mc along the actin filament whereas E is its translational energy. Hence, the factorized function (2) turns out to be

The quantum model of the muscle contraction proposed by Matsuno (1999) is consistent both with the Corben and de Broglie–Bohm quantum theory,4 provided that we take into account the change of the mass Mc of the local aggregate due to influence of the space-like interactions between neighboring aggregates. The proposed interpretation of the Matsuno (1999) model predicts that those interactions slightly decrease the mass Mc of the locally coherent aggregation according to the formula (19). If we assume lack of interactions then a local aggregate with its mass Mc along the actin filament, carries the momentum Pc = Mc vactin = 2.2 × 10−21 erg s/cm and the associated de Broglie wavelength is (Matsuno, 1999)

 (t, x) = (t, x)e(i/¯h)S(t,x) .

B =

6. The Quantum Potential In the Matsuno (1999) model the solution of the interaction-free Schrödinger Eq. (1) can be given in the form free (t, x) = e(i/¯h)S(t,x) ,

S(t, x) = Pc x − Et

(30)

(31)

It is interesting to note that identical form of the wave function appears in the de Broglie–Bohm quantum theory (de Broglie, 1960; Bohm, 1952a, b). In this approach, the wave function (31) is solution of the time-like Klein–Gordon Eq. (21) for a massive particle of rest mass M0 . Introducing (31) into (21) and then separating the latter into real and imaginary parts one obtains two equations (de Broglie, 1960, p. 112) ¯ ∂ ∂ (t, x)/(t, x), [∂0 S(t, x)] − [∂x S(t, x)] = M02 c 2 + h 2

2

2

∂0 (t, x)∂0 S(t, x) − ∂x (t, x)∂x S(t, x) =

− 12 (t, x)∂ ∂ S(t, x).

(32) (33)

Introducing S(t, x) into (32) one gets 2

(E/c) − Pc2 = M02 c 2 +

2 h ¯ ∂ ∂ (t, x)  = M02 c 2 + m02 c 2 = Mc2 c 2 . (t, x) c2

(34) Hence, (32) represents Jacobi’s relativistic equation for a particle endowed with an effective mass (de Broglie, 1960, p. 116)



Mc =

M02 +

2 h ¯ ∂ ∂ (t, x) . (t, x) c2

(35)

In the non-relativistic regime the following approximation hold (de Broglie, 1960, p. 121) Mc  M0 + Q (t, x)/c 2 ,

(36)

∂ ∂ (t, x) h ¯ = 2M0 (t, x) 2

Q (t, x) =

(m0 c)2 2M0

(37)

in which the quantity Q (t, x) is generally interpreted as a q uantum potential related to a quantum field existing even in the absence of a field of classical type, e.g. gravitational or electromagnetic. If (t, x) represents the contribution from interactions between the neighboring aggregates along the actin filament (Matsuno, 1999), an origin of those interactions should be quantum potential (37). Its source is the space-like field (t, x) associated with a carrier of space-like interactions, endowed with the mass m0 . Employing the tachyokinematic effect, represented by Eq. (29), and approximation M0 ≈ Mc one may rewrite Eq. (37) to the form Q (t, x) ≈

(m0 c)2 2Mc

=

Mc v2actin 2

,

h

vactin Mc

= 4.5 nm,

(39)

where h stands for the Planck’s constant. When the locally coherent aggregates interact each other and condense in a single quantum state, they carry the momentum Pc = vactin





Mc2 − m02 ,

(40)

hence, the associated de Broglie wavelength B =

vactin

h





Mc2 − m02

(41)

slightly differs from that predicted by the Matsuno (1999) model. Since the mass m0 = 7.3 × 10−32 g is extremely small, the change in B has no measurable effect on the range of quantum coherence of the actin monomers endowed with the diameter 2.5 nm (Matsuno, 1999). When myosin molecules are sufficiently available, the sliding velocity of the actin filament is equal vactin ≈ 9 × 10−4 cm/s (Matsuno, 1999). Taking advantage of the relation (6) one can estimate the group velocity of the superluminal phase wave (7). For c ≈ 3 × 1010 cm/s one gets vg = 1024 cm/s. Interactions, which propagate with such group velocity are interpreted in our subluminal world as “instantaneous” as they are 1014 times faster then the velocity of light. In the Matsuno model the Schrödinger Eq. (3) does not contain a potential energy term representing interactions between locally coherent aggregates. The contributions from those interactions are included in the phase factor of the wave function (2). In such circumstances a source of the interactions can be a quantum potential related to a quantum field existing even in the absence of a field of classical type, e.g. gravitational or electromagnetic. The quantum potential plays fundamental role in the de Broglie–Bohm theory because of its non-local character (Bohm and Hiley, 1975). If the Matsuno model is consistent with the de Broglie–Bohm theory the quantum potential and tachyokinematic effect can be employed in the non-local interpretation of the muscle contraction. In particular, the quantum potential can be responsible for the long-range coordination, coherence and entanglement of the actin monomers. In such circumstances the muscle contraction should belong to the class of biological space-like phenomena, which take place over the

(38)

which indicates that the quantum potential is approximately equal to the kinetic energy of aggregates moving with the sliding velocity

4 The Corben and de Broglie–Bohm theory can be unified to the one general quantum theory (Molski, 1998).

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M. Molski / BioSystems 96 (2009) 136–140

space-like separation of the actin monomers aligned along the filament. Such space-like interactions and lack of decoherence are conditions sine qua non for appearance of the quantum entanglement between actin monomers in an ATP-activated filament. It is worth noticing that a similar space-like correlations between shielded basins of human neurons have been recently discovered by Pizzi et al. (2004a, b). The authors proved that this process utilizes a sort of spontaneously generated entanglement between separated neural networks: after an initial stage where the system interacted by direct contact, also in the following stage where the system has been separated into two sections, a sort of space-like correlations persisted between sections. It should be pointed out, however, that although both phenomena are similar each to other (i.e. non-local), their origin is different. In the first case we have entanglement within the molecular aggregates, whereas in the second one—between specialized cells. In such circumstances the non-local interactions observed by Pizzi et al. (2004a, b) belong to the class of macroscopic quantum effects (Leggett, 1987) operating on mesoscopic biological scale,5 in contradistinction to those in the Matsuno model, which are typical microscopic (quantal) interactions. References Aspect, A., Grangier, P., Roger, G., 1982a. Experimental realization of Einstein– Podolsky–Rosen gedanken experiment: a new violation of Bell’s inequalities. Phys. Rev. Lett. 48, 91–94. Aspect, A., Dalibard, J., Roger, G., 1982b. Experimental test of Bell’s inequalities using time-varying analyzers. Phys. Rev. Lett. 49, 1804–1807. Bionta, R.M., et al., 1987. Observation of a neutrino burst in coincidence with supernova 1987A in the large magellanic cloud. Phys. Rev. Lett. 58, 1494–1496. Belesev, A.I., et al., 2008. Investigation of space-charge effects in gaseous tritium as a source of distortions of the beta spectrum observed in the Troitsk neutrino-mass experiment. Phys. At. Nucl. 71, 427–436. Bell, J.S., 1964. On the Einstein–Podolsky–Rosen paradox. Physics 1, 195–200. Bell, J.S., 1993. Speakable and Unspeakable in Quantum Mechanics. Cambridge University Press, Cambridge. Bohm, D., 1952a. A suggested interpretation of the quantum theory in terms of “hidden” variables. Phys. Rev. I 85, 166–179. Bohm, D., 1952b. A suggested interpretation of the quantum theory in terms of “hidden” variables. Phys. Rev. II 85, 180–193. Bohm, D., Hiley, B.J., 1975. On the intuitive understanding of nonlocality as implied by quantum theory. Found. Phys. 5, 93–109. de Broglie, L., 1960. Non-Linear Wave Mechanics. A Causal Interpretation. Elsevier, Amsterdam. ´ Ciborowski, J., Rembielinski, J., 1999. Tritium decay and the hypothesis of tachyonic neutrinos. Eur. Phys. J. C 8, 157–161. Chodos, A., Hauser, A.I., Kosteleck’y, V.A., 1985. The neutrino as a tachyon. Phys. Lett. B 150, 431–435. Corben, H.C., 1977. Relativistic selftrapping for hadrons. Lett. Nuovo Cimento 20, 645–648. Corben, H.C., 1978a. Electromagnetic and hadronic properties of tachyons. In: Recami, E. (Ed.), Tachyons, Monopoles and Related Topics. North-Holland, Amsterdam, Holland, pp. 31–41. Corben, H.C., 1978b. The , f  , F1 KSpectrum. Lett. Nuovo Cimento 22, 116–120. Cramer, J.G. Quantum nonlocality and the possibility of superluminal effects. In: Proceedings of the NASA Breakthrough Propulsion. Physics Workshop, Cleveland, OH, 1977. Eberhard, P.H., 1977a. Bell’s theorem without hidden variables. Il Nuovo Cimento B 38, 75–80.

5 Neurons are eucariotic cells, which have dimension of 10–100 ␮m; some axons (e.g. in sciatic nerve) can attain even 1 m.

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