ECCLES’S PSYCHONS COULD BE ZERO-ENERGY TACHYONS
Syamala Hari

ABSTRACT
This paper suggests that mental units called psychons by Eccles could be tachyons defined theoretically by physicists sometime ago. Although experiments to detect faster-than-light particles have not been successful so far, recently, there has been renewed interest in tachyon theories in various branches of physics. We suggest that tachyon theories may be applicable to brain physics. Eccles proposed an association between psychons and what he called dendrons which are dendrite bundles and basic anatomical units of the neocortex for reception. We show that a zero-energy tachyon could act as a trigger for exocytosis (modeled by Beck and Eccles as a quantum tunneling process), not merely at a single presynaptic terminal but at all selected terminals in the interacting dendron by momentarily transferring momentum to vesicles, thereby decreasing the effective barrier potential and increasing the probability of exocytosis at all those selected terminals at the same time. This is consistent with the tachyon characteristic that it is a nonlocal phenomenon produced and absorbed instantaneously and nonlocally by detectors acting in a coherent and cooperative way.

1. INTRODUCTION

Unlike many other prominent neuroscientists, Sir John Eccles rejected the notion of mind brain identity and developed a theory of the mind, known as dualist-interactionism. In his book, How the Self Controls Its Brain, Eccles (1994), shows that mind-brain action can be explained without violating the conservation of energy if account is taken of quantum physics and the available knowledge concerning the microstructure of the neocortex. Eccles calls some fundamental neural units of the cerebral cortex dendrons, and proposes that each of the 40 million dendrons is linked with a mental unit, or psychon, representing a unitary conscious experience. According to Eccles, in willed actions and thought, psychons act on dendrons and momentarily increase the probability of the firing of selected neurons. Based on physicist Friedrich Beck’s (Beck, Eccles 1992) quantum mechanical analysis of bouton exocytosis, he proposes the hypothesis that mental intention (the volition) becomes neurally effective by momentarily increasing the probability of exocytosis in selected cortical areas.

A nerve impulse propagating into a presynaptic terminal triggers a process called exocytosis that causes opening of a channel and releasing transmitter molecules into the synaptic cleft. Exocytosis is the basic activity that initiates information flow between neurons in chemical synapses. It is triggered with some small probability by an arriving nerve impulse. The detailed model proposed by Beck and Eccles (1992) is based on the quantum concept of quasiparticles, reflecting the particle aspect of a collective mode. Their proposed model refers to tunneling processes of two-state quasiparticles, resulting in state collapses. It yields a probability of exocytosis in agreement with empirical observations. The quantum treatment of exocytosis links the neocortical activity with the existence of a large number of quantum probability amplitudes, since there are more than 100,000 boutons in a bundle of dendrites called a dendron. Eccles’s rationale for the hypothesis of mental interaction includes the argument that mental intention must be neurally effective by momentarily increasing the probabilities for exocytosis in a whole dendron in selected areas and coupling the large number of probability amplitudes to produce coherent action because in the absence of mental activity these probability amplitudes would act independently, causing fluctuating EPSPs in the pyramidal cell.

In this paper, we suggest that the so called psychons could be tachyons defined by physicists sometime ago (Bilaniuk, Deshpande, & Sudarshan 1962; Feinberg 1967). Theoretically, the existence of tachyons is not contrary to the principle of Relativity. Although experiments to detect faster-than-light particles have not been successful so far, recently, there has been renewed and keen interest in tachyon theories across a spectrum of research areas as diverse as particle physics to cosmology and string theory. In this paper, we suggest that tachyon theories may be applicable to brain physics as well. We show how a zero-energy tachyon could act as a trigger for exocytosis in a whole dendron by momentarily decreasing the effective potential (external potential plus the quantum potential also called the Bohmian potential) of a multitude of quasiparticles and thereby enhancing the tunneling process across the whole dendron.

A tachyon is a wave packet associated with an imaginary mass value. The author has held the view since a long time (Hari, 2002, 2007; Vishnubhatla, 1985, 1986, 1987) that thought processes in a brain involve tachyons (faster than light objects) defined by physicists sometime ago. Some intuitive rationale for this hypothesis may be found in the stated references. Tachyons are not material particles but nonlocal objects that can be associated with waves which are nonlocal in space but localized in time. Tachyons cannot be created at a single point of space but can only be created or absorbed by systems of ordinary particles distributed over a region. Thus, tachyons do not travel from one point of space to another with a speed faster than light but they create that impression because their influence can be simultaneously detected by spatially separated detectors (Shay & Miller 1978).

2. BECK-ECCLES QUANTUM MECHANCAL MODEL OF EXOCYTOSIS

In the Beck-Eccles (1992) quantum mechanical model of exocytosis, they adopt the following concept: preparation for exocytosis means bringing the presynaptic vesicular grid into a metastable state from which exocytosis can occur. The process of exocytosis is then modeled by the motion of a quasiparticle with one degree of freedom along a collective coordinate, and over an activation barrier. The motion is characterized by a potential energy V(q), which may take on a positive value before exocytosis according to the metastable situation, then rises toward a maximum and acts as the barrier, which the quasiparticle has to tunnel through, and finally drops to zero. This quantum mechanical tunneling process consists of two states: one in which the particle has not crossed the barrier and exocytosis does not take place, and the other state in which the particle crosses the barrier and exocytosis takes place. Eccles hypothesizes that mental intention is responsible for the action of selecting the state, where exocytosis occurs, by momentarily increasing the probability of exocytosis. The time-dependent process of exocytosis is described by Beck and Eccles (1992: p3) by a one-dimensional Schrödinger equation for the wave function y(q; t):
iћ∂_ty(q; t) = − (ћ²/2M)(∂_q ² + V(q))y(q, t) (SE1)
where ∂ denotes differentiation with respect to its suffix, M is the mass of the particle and ћ is the Plank’s.

In the following sections, we show that a zero-energy tachyon can trigger the crossing of the potential barrier (in other words, collapse of the quasiparticle wave-function to the state, where exocytosis occurs) by momentarily transferring momentum to the quasiparticle (somewhat like giving it a push!) without any exchange of energy. Thereby, the particle’s effective potential, that is, the external potential plus the quantum potential is decreased to a value below the particle’s total energy and allows the particle to tunnel through the barrier. Being a field, the tachyon pushes all the boutons that it interacts with, at the same time!

3. ZER-ENERGY TACHYON’S ELECTROMAGNETIC FIELD

The Klein-Gordon equation for a free tachyon having negative squared-mass − µ² (where µ is a positive real number) is written as

(∂_t² /c²− D − m²)ψ(x, t) = 0, (1)

where x is the vector (x, y, z), D = ∂_x²+ ∂_y²+ ∂_z², again ∂ denotes differentiation with respect to its suffix, and c, the speed of light in free space, and m = µc/ћ. Separating the time dependence of ψ by writing ψ(x, t) = ψ(x)ψ′(t), we get solutions e^dit ψ(x) of equation (1), where ψ(x) satisfies

[−D − k²] ψ(x) = 0, and ω² /c² = k² –m². (2)

The frequency ω is real only for k ≥ m; plane wave solutions e^{i(ωt – k.x)} are not a complete set in space because of this condition and a superposition of them cannot be localized in space (Feinberg 1967). The phase velocity of such a wave e^{i(ωt – k.x)} is < c and group velocity of the associated wave packet is > c. On the other hand, a superposition of solutions of (1) can be localized in time because solutions e^±iωt ψ(x, t) exist for all real ω. A point of view shared by physicists is that tachyons are essentially nonlocal in space but localized in time. A tachyon’s interaction with matter is non-local (Feinberg 1967; Sudarshan 1970; Shay & Miller, 1978) resulting in instantaneous action at a distance. A tachyon wave is a strictly nonlocal phenomenon in a dispersive medium produced and absorbed instantaneously and nonlocally by detectors acting in a coherent and cooperative way (Shay & Miller, 1978).

In the frame of reference in which the energy of a tachyon vanishes, the magnitude of the momentum is equal to mc and rather than being at rest, the tachyon travels with infinite speed. The interaction of such a tachyon with ordinary matter would be to transfer no energy but all its momentum instantaneously in a manner analogous to a rigid body’s transferring impulses instantaneously in a collision without exchanging energy (Sudarshan 1970). In the following sections, we will show that a zero-energy tachyon’s interaction with a dendron would result in transferring momentum to various boutons in a dendron and thereby decrease the effective barrier potential and increase the probability of exocytosis in all its boutons. A zero-energy solution of (1) corresponds to frequency ω = 0 and k² = m² and satisfies
D F(x) = − m² F(x). (3)
Equation (3) has multiple linearly independent solutions F(x) corresponding to a given value of m. Each solution represents a field with zero energy and capable of exchanging momentum with a particle of matter. We take F(x) to be real.

To describe the interaction of a field satisfying equation (3) with a particle whose motion is governed by a Schrödinger equation, we associate an electromagnetic field with a solution of (3) as follows. Consider the field φ(x, t) = e^imct F(x), and the four vector à = (−grad φ(x, t), ∂_τφ(x, t)), where τ = ct. Writing Ā = −gradφ and U = ∂_τφ = imφ, we find that equation (3) implies that Ā and U satisfy the following Poisson equations of the vector and scalar potentials of an electromagnetic field whose current density and charge density are both zero.

(D − ∂_t²/c²)Ā(x, t) = 0, (D − ∂_t²/c²)U(x, t) =0.

Moreover, Ā and U satisfy the Lorentz gauge condition: divĀ+(1/c)∂_tU= 0. Hence Ā and U = imφ can be the vector and scalar potentials of an electromagnetic field. Note that the potentials Ā and U give rise to zero fields E and B because E = − gradU − (1/c)∂_t Ā = 0 and B = curlĀ = 0. Nonetheless, we will use U and Ā to describe the interaction of a solution of equation (3) with a particle of matter since these potentials may produce observable effects other than and independent of the electric and magnetic fields also produced by them (Aharonov, Bohm, 1959). Like Beck & Eccles (1992), we assume that the interaction of the tachyon (psychon) with the dendron is momentary and take t=0 as the time of interaction. We find that at t = 0 the scalar potential imF(x) is purely imaginary whereas the vector potential − gradF(x) is real and therefore, a zero-energy tachyon would only transfer momentum to a charged particle but no energy. This is consistent with the well known roles of scalar and vector potentials: the scalar potential is a store of field energy, and the vector potential is a momentum potential (a store of field momentum available to charge motion) (Konopinski, 1978).

4. TACHYON DENDRON INTERACTION

In the Beck-Eccles Beck (1992) model, the various boutons in a dendron, have probabilities of exocytosis that are independent of one another. Hence, the wave- function of all boutons together, is the product of their individual wave-functions, and the Hamiltonian of the total system is the sum of the individual Hamiltonians; in other words, each bouton has its motion describes by an equation of the form (SE1) with no term of interaction with any other bouton. Therefore, to describe the effect of the electromagnetic potentials described above on each bouton, we will introduce the necessary electromagnetic interaction terms into the equation (SE1) and the discussion that follows will be applicable to any bouton in the dendron.

In terms of scalar and vector potentials V and a, the following is the Schrodinger equation in three dimensions of a particle with charge e and mass M interacting with the EM field defined by V and a:
iћ∂_ty = (1/2M)[(ћ/i)grad − (e/c)a(x, t)]²y+ {eV + V}y

In the present case of a bouton, the quasiparticle’s motion depends on a single degree of freedom q given in equation (SE1). The coordinate q is a collective coordinate and may not necessarily be identical with one of its Cartesian coordinates x, y, z. Still, its position vector r = r(q, t) = r(q) is a function of q; we assume that r does not depend explicitly on time because the external potential V of the bouton in equation (SE1) depends on q alone. Hence, the terms a = − e^imct grad_jF(r_j) and V=e^imct imF(r_j) that enter into (SE1) have their explicit time dependence only in the factor e^imct because Ф(r_j) and grad_jФ(r_j) do not depend explicitly on time. Thus, after the electromagnetic interaction with the tachyon field, the equation (SE1) of the jth bouton in the dendron changes to the equation (SE IN) below, where q_j is its degree of freedom:

iћ∂_ty_j(q_j, t)=(1/2M_j)[Î_j(ћ/i)∂_qj−(e_j/c)e^imctA(r_j)]²y(q_j, t) + [e^imct ie_jmФ(r_j) + V(q_j)]y_j(q_j, t), j = 1, 2,…., N. (SE IN)

In equation (SE IN), e_j is the quasiparticle’s charge, M_j its mass, Î_j is the unit vector along its velocity v_j, V(q_j) is the value of an external potential of the dendron, A(r_j) = − grad_jФ(r_j) and grad_j denotes differentiation with respect to the components of r_j, and N is the number of boutons that would undergo exocytosis. We note that on the right side of (SE IN), Î_j(ћ/i)∂_qj corresponds to the particle’s canonical momentum and M_jv_j −(e_j/c)e^imctA(r_j) is its kinetic momentum. Hereafter, to simplify notation, we will drop the suffix j while describing the motion of a single bouton.

Now, considering that v = Îdq/dt = dr/dt = ∂_qx dq/dt + ∂_qy q/dt + ∂_qz dq/dt, it follows that

Î=(∂_qx, ∂_qy, ∂_qz), A(r).Î=gradФ(r).Î = dФ/dq, and A(r)=dФ/dq Î + Â, (4)

where  is the component of A(r) perpendicular to Î. Substituting (4) in equation (SE IN) the bouton’s Schrodinger equation after interaction is

iћ∂_ty=1/2M[((ћ/i)∂−(e/c)e^imct dФ/dq)²+(eÂ/c)²/2M]y
+ [iemФ(r)e^imct + V(q)]y. (SE2)

Writing y′(q, t) = y′(q, t)e^{i(eФ/cћ)(cosmct +isinmct)} in (SE2) the following is the Schrodinger equation of the bouton with the wave-function y′.
iћ∂_t y′ = − (ћ²/2M)[∂_q² + (eÂ/c)²]y′ + V(q)y′. (SE3)

Equation (SE3) shows that the particle’s interaction with the potentials Ā and V adds the factor (eФ/cћ)cosmct to the phase of the wave-function of (SE1).

To describe the tunneling process in the language of Bohmian mechanics, we will write the wave-function y(x, t) in equation (SE1) as
y(q, t) = R(q, t)e^{iS(q, t)/ћ}, (5)

where R and S are real valued functions. Equating the real and imaginary parts on both sides of (SE1), we obtain the following two equations:

∂_tS + (∂_qS)² /2M + Q + V(q) = 0 (B1)

∂_tR² + ∂_q(R² ∂_qS)/M = 0, (B2)

where Q = −ћ²(∂_q²R)/2MR is called the quantum potential. (B1) is similar to the Hamilton-Jacobi equation of classical mechanics; we have: − ∂_tS = E, the total energy of the particle, and ∂_qS = the particle’s momentum. (B2) is the equation of continuity relating the probability density R² of the particle being at the position q at time t, to the probability current R²(∂_qS)/M. Once (SE1) is solved for the wave-function y(q, t), the particle’s trajectories can be computed classically from

Mdq/dt = ∂_qS or Md²q/dt² = − ∂_q(Q + V) (B3)

by prescribing initial conditions. Consider the equation (B1) at the first classical turning point where E = V, Q = 0; the particle’s kinetic energy = (∂_qS)²/2M = 0 and the tunneling process begins here. As the potential V increases and becomes > E, motion is classically forbidden. As long as the particle remains in the state of no exocytosis, in other words, it has not crossed the barrier V> E, the particle’s momentum ∂_qS remains 0 and the quantum potential Q adjusts itself so that Q+V = E; Q+V cannot be > E because (∂_qS)² cannot be negative. On the other hand, Q+V can be < E although V > E; if so, the second equation of (B3) gives trajectories penetrating the barrier and (B1) gives a nonzero kinetic energy (∂_qS)²/2M. Therefore, V > E and Q+V < E corresponds to the state where exocytosis has taken place.

If the tachyon interaction takes place when the particle is at or within the barrier V > E, its effect on equations (B1) and (B2) can be obtained by substituting

y′(q, t) = R′(q, t)e^{i(S(q, t)+ (eФ/c)cosmct)/ћ}

in (SE3), and equating real parts on both sides of equation (SE2). After interaction, (B1) and (B2) change respectively to (B1′) and (B2′) below.

∂_tS − eФm sinmct + [(∂_qS + (e/c)cosmct dФ/dq)²+(eÂ/c)²]/2M +Q+V= 0, ( B1')
∂_tR′² + ∂_q(R′² [∂_qS +(e/c)cosmctdФ/dq]/M = 0, (B2′)
where Q′ = −ћ²(∂_q²R′)/2MR′. At t=0, the equation (B1′) is
−E + [(∂_qS + (e/c)dФ/dq)² +(eÂ/c)²]/2M + Q′ +V = 0 (B1′ t=0)

The first term ∂_tS = −E is total energy of the particle and same as in (B1) because the interaction does not involve exchange of energy. The second term in brackets is the particle’s kinetic energy. At t=0, from (B1) we have ∂_qS = 0. Hence, we find (using (4)) that the kinetic energy is ((e/c)gradF(r))²/2M and momentum is egradF(r)/c which is acquired by the particle as a result of the interaction. Therefore, if gradF(r)≠ 0 at the position r, then from (B1′ t=0), we have
Q′ + V = E − ((e/c)gradF(r))² /2M < E (6)

If the effective potential Q′ + V < E, the second equation below in (B3′)
Mdq/dt = ∂_qS Md²q/dt² = − ∂_q(Q′ + V) (B3′)
can be used to determine the particle’s trajectory classically and permits trajectories to penetrate through the barrier. Thus, except at those points where gradF vanishes, the quantum potential acts to lower the barrier momentarily to permit trajectories to pass through the barrier. Indeed, it turns out that for sufficiently small m, the whole dendron will be within a region where gradF does not vanish. This can be seen as follows. Assuming that gradF does not vanish at the origin (if not we can change the origin to a point where gradF does not vanish by a translation of the origin of co-ordinates), consider the coordinates x′ = mx. Equation (4) when expressed in terms of x′ becomes
D′ F′(x′) = − F′(x′). (7)
Since gradF(x) = m grad′ F′(x′), grad′ F′(x′) does not vanish at x′ = 0. Hence there is an r such that grad′ F′(x′) ≠ 0 in the region x′ < r. This implies that gradF(x) ≠ 0 when mx < r or x < r/m; r/m would include the entire dendron if m is sufficiently small.

The solution F(r) may be assumed to have been normalized to satisfy the condition that the total momentum acquired by all the boutons in the interacting dendron equals m, the momentum of the tachyon because the solution F_N(r) is given by
F_N(r) = mF(x) /(∑^N _{j = 1}gradF(r_j)) (8)does satisfy the condition: ∑^N _{j = 1}gradF_N(r_j) = m.

5. HOW IS THIS THEORY VERIFIED EXPERIMENTALLY?

So far, in the above sections, we have been describing only the theoretical possibility of a tachyon’s interaction with a dendron resulting in exocytosis at its multiple boutons to produce a strong enough EPSP by a pyramidal cell. Clearly, unless there is a way to verify the theory by making measurements on the brain, the theory will be no more than a speculation or hypothesis. Leaving a detailed analysis of such verification to a future paper, here, we suggest a possible approach to such verification.

After the momentary tachyon interaction is turned off, the particle’s motion reverts back to that described by equations (B1), (B2), and (B3) but with a different initial condition: at time t+δt the particle’s momentum is ∂_qS = eA/c where A=A(r), r being the particle position at time t+δt. Therefore, after time t+δt, the second equation B3 is

Md²q/dt² = − ∂_q(Q + V −(∂_qS)²) ≤ 0. (9)
Assuming that the interaction provides kinetic energy which is very small relative to E, and therefore neglecting (∂_qS)² on the right hand side of (9), from equation (9) we obtain a constant momentum eA/c for the particle for its journey through the barrier. Hence the time taken by the particle to cross the barrier is McW/eA where W is the barrier width. This should be the time t of metastable instability mentioned in (Beck & Eccles 1992: p 4). Thus an estimate of A can be obtained from

eA = WMc/t (10)

if the barrier width W and the position r(q) as a function of the coordinate q are known. Obtaining an estimate of m is really not important but the existence of a value of m which satisfies equation (10) can be considered to validate the proposed hypothesis of tachyon interaction with the brain. Assuming that the barrier width is of the order of angstroms, from the numerical values given by Beck & Eccles (1992), eA is found to be in the order of 10⁻¹³ to 10⁻¹²eV.

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