## 1. Introduction

This project calculates quantum ${\mathrm{Ca}}^{2+}$ interactions with EEG. In this paper, EEG is synonymous with large-scale neocortical firings during attentional tasks as measured by large-amplitude electroencephalographic (EEG) recordings. In this paper, only very specific calcium ions, ${\mathrm{Ca}}^{2+}$, are considered, those arising from regenerative calcium waves generated at tripartite neuron-astrocyte-neuron synapses. Indeed, it is important to note that ${\mathrm{Ca}}^{2+}$ ions, and specifically ${\mathrm{Ca}}^{2+}$ waves, influence many processes in the brain, but this study focuses on free waves generated at tripartite synapses because of their calculated direct interactions with large synchronous neuronal firings.

Section 2 reviews the background of the main model used, Statistical Mechanics of Neocortical Interactions (SMNI).

Section 3 reviews the code Adaptive Simulated Annealing (ASA), used for optimization of many systems—fitting models to real data, e.g., fits to EEG data reported here.

Section 4 reviews the development of path-integral codes, PATHINT and qPATHINT, used for propagation of conditional probabilities and quantum-mechanical wave-functions, as reported here.

Section 5 gives new results of inclusion of quantum-mechanical interactions of

${\mathrm{Ca}}^{2+}$ wave-packets with EEG.

Section 6 reviews some applications of this project.

The theory and codes for ASA and [q]PATHINT have been well tested across many disciplines by multiple users. This particular project most certainly is speculative, but it is testable. As reported here, fitting such models to EEG tests some aspects of this project. This is a somewhat indirect path, but not novel to many physics paradigms that are tested by experiment or computation. A detailed future path is described in the [q]PATHINT review Section.

While SMNI has been developed since 1981, and been confirmed by many tests, this evolving model including ionic scales has been part of multiple papers relatively recently, since 2012. Classical physics calculations support these extended SMNI models and are consistent with experimental data. Quantum physics calculations also support these extended SMNI models and, while they too are consistent with experimental data, it is quite speculative that they can persist in neocortex. Admittedly, it is surprising that detailed calculations continue to support this model, and so it is worth continued examination it until it is theoretically or experimentally proven to be false.

## 2. Statistical Mechanics of Neocortical Interactions (SMNI)

SMNI has been developed since 1981, scaling aggregate synaptic interactions to neuronal firings, up to minicolumnar-macrocolumnar columns of neurons to mesocolumnar dynamics, up to columns of neuronal firings, up to regional macroscopic sites [

1,

2,

3,

4,

5,

6].

SMNI has calculated agreement/fits with experimental data from various aspects of neocortical interactions, e.g., properties of short-term memory (STM) [

7], including its capacity (auditory

$7\pm 2$ and visual

$4\pm 2$) [

8,

9], duration, stability, primacy versus recency rule, as well other phenomenon, e.g., Hick’s law [

10,

11,

12], interactions within macrocolumns calculating mental rotation of images, etc. [

2,

3,

4,

5,

6]. SMNI scaled mesocolumns across neocortical regions to fit EEG data [

7,

13,

14].

Figure 1 depicts this model [

3].

#### 2.1. Synaptic Interactions

The short-time conditional probability distribution of firing of a given neuron firing given just-previous firings of other neurons is calculated from chemical and electrical intra-neuronal interactions [

2,

3]. Given its previous interactions with

k neurons within

${\tau}_{j}$ of 5–10 ms, the conditional probability that neuron

j fires

$({\sigma}_{j}=+1)$ or does not fire

$({\sigma}_{j}=-1)$ is

The contribution to polarization achieved at an axon given activity at a synapse, taking into account averaging over different neurons, geometries, etc., is given by $\mathsf{\Gamma}$, the “intra-neuronal” probability distribution. $\mathsf{\Psi}$ is the “inter-neuronal” probability distribution, of thousands of quanta of neurotransmitters released at one neuron’s presynaptic site effecting a (hyper-)polarization at another neuron’s postynaptic site, taking into account interactions with neuromodulators, etc. This development holds for $\mathsf{\Gamma}$ Poisson, and for $\mathsf{\Psi}$ Poisson or Gaussian.

${V}_{j}$ is the depolarization threshold in the somatic-axonal region. ${v}_{jk}$ is the induced synaptic polarization of E or I type at the axon, and ${\varphi}_{jk}$ is its variance. The efficacy ${a}_{jk}$ is a sum of ${A}_{jk}$ from the connectivity between neurons, activated if the impinging k-neuron fires, and ${B}_{jk}$ from spontaneous background noise. The efficacy is related to the impedance across synaptic gaps.

#### 2.2. Neuronal Interactions

Aggregation up to the mesoscopic scale from the microscopic synaptic scale uses mesoscopic probability

PM represents a mesoscopic scale of columns of

N neurons, with subsets

E and

I, represented by

${p}_{{q}_{i}}$. The “delta”-functions

$\delta $-constraint represents an aggregate of many neurons in a column.

G is used to represent excitatory (

E) and inhibitory (

I) contributions.

$\overline{G}$ designates contributions from both

E and

I.

The path integral is derived in terms of mesoscopic Lagrangian L. The short-time distribution of firings in a minicolumn, given its just previous interactions with all other neurons in its macrocolumn, is thereby defined.

#### 2.3. Columnar Interactions

In the prepoint (Ito) representation the SMNI Lagrangian

L is

The threshold factor

${F}^{G}$ is derived as

where

${A}_{{G}^{\prime}}^{G}$ is the columnar-averaged direct synaptic efficacy,

${B}_{{G}^{\prime}}^{G}$ is the columnar-averaged background-noise contribution to synaptic efficacy. The “

${}^{\u2021}$” parameters arise from regional interactions across many macrocolumns.

#### 2.4. SMNI Parameters From Experiments

All values of parameters and their bounds are taken from experimental data, not arbitrarily fit to specific phenomena.

${N}^{G}$ = {${N}^{E}=160$, ${N}^{I}=60$} was set for for visual neocortex, {${N}^{E}=80$, ${N}^{I}=30$} was set for all other neocortical regions, ${M}^{{G}^{\prime}}$ and ${N}^{{G}^{\prime}}$ in ${F}^{G}$ are afferent macrocolumnar firings scaled to efferent minicolumnar firings by $N/{N}^{*}\approx {10}^{-3}$. ${N}^{*}$ is the number of neurons in a macrocolumn, about ${10}^{5}$. ${V}^{\prime}$ includes nearest-neighbor mesocolumnar interactions. $\tau $ is usually considered to be on the order of 5–10 ms.

Other values also are consistent with experimental data, e.g., ${V}^{G}=10$ mV, ${v}_{{G}^{\prime}}^{G}=0.1$ mV, ${\varphi}_{{G}^{\prime}}^{G}={0.03}^{1/2}$ mV.

Nearest-neighbor interactions among columns give dispersion relations that were used to calculate speeds of mental visual rotation [

2,

3].

The wave equation cited by EEG theorists, permitting fits of SMNI to EEG data [

15], was derived using the variational principle applied to the SMNI Lagrangian.

This creates an audit trail from synaptic parameters to the statistically averaged regional Lagrangian.

#### 2.5. Previous Applications

#### 2.5.1. Verification of basic SMNI Hypothesis

The core SMNI hypothesis first developed circa 1980 [

1,

2,

3] is that highly synchronous patterns of neuronal firings in fact process high-level information. Only since 2012 has this hypothesis been verified experimentally [

16,

17].

#### 2.5.2. SMNI Calculations of Short-Term Memory (STM)

SMNI calculations agree with observations [

2,

3,

4,

5,

6,

7,

11,

13,

18,

19,

20,

21,

22,

23,

24,

25,

26]: This list includes:

capacity (auditory

$7\pm 2$ and visual

$4\pm 2$) [

4]

primacy versus recency rule [

5,

27]

Hick’s law (reaction time and

g factor) [

11]

nearest-neighbor minicolumnar interactions

$=>$ mental rotation of images [

2,

3]

derivation of basis for EEG [

15,

28]

#### 2.5.3. Three Basic SMNI Models

Three basic models were developed by slightly changing the background firing component of the columnar-averaged efficacies ${B}_{{G}^{\prime}}^{G}$ within experimental ranges, which modify ${F}^{G}$ threshold factors to yield in the conditional probability:

- (a)
case EC, dominant excitation subsequent firings

- (b)
case IC, inhibitory subsequent firings

- (c)
case BC, balanced between EC and IC

This is consistent with experimental evidence of shifts in background synaptic activity under conditions of selective attention [

29,

30], This enables a Centering Mechanism (CM) on case BC, giving

${\mathrm{BC}}^{\prime}$, wherein the numerator of

${F}^{G}$ only has terms proportional to

${M}^{{E}^{\prime}}$,

${M}^{{I}^{\prime}}$ and

${M}^{\u2021{E}^{\prime}}$, i.e., zeroing other constant terms by resetting the background parameters

${B}_{{G}^{\prime}}^{G}$, still within experimental ranges. This brings in a maximum number of minima into the physical firing

${M}^{G}$-space, due to the minima of the new numerator in being in a parabolic trough defined by

about which nonlinearities develop multiple minima identified with STM phenomena.

In current projects a Dynamic CM (DCM) model is used, resetting

${B}_{{G}^{\prime}}^{G}$ every few epochs of

$\tau $. Such changes in background synaptic activity on such time scales are seen during attentional tasks [

29].

#### 2.6. Comparing EEG Testing Data with Training Data

Using EEG data from

http://physionet.nlm.nih.gov/pn4/erpbci [

31,

32], SMNI was fit to highly synchronous waves (P300) during attentional tasks, for each of 12 subjects, it was possible to find 10 Training runs and 10 Testing runs [

22].

Spline-Laplacian transformations on the EEG potential $\mathsf{\Phi}$ are proportional to the SMNI ${M}^{G}$ firing variables at each electrode site. The electric potential $\mathsf{\Phi}$ is experimentally measured by EEG, not $\mathbf{A}$, but both are due to the same currents $\mathbf{I}$. Therefore, $\mathbf{A}$ is linearly proportional to $\mathsf{\Phi}$ with a simple scaling factor included as a parameter in fits to data. Additional parameterization of background synaptic parameters, ${B}_{{G}^{\prime}}^{G}$ and ${B}_{{E}^{\prime}}^{\u2021E}$, modify previous work.

The $\mathbf{A}$ model outperformed the no-$\mathbf{A}$ model, where the no-$\mathbf{A}$ model simply has used $\mathbf{A}$-non-dependent synaptic parameters. Cost functions with an $\left|\mathbf{A}\right|$ model were much worse than either the $\mathbf{A}$ model or the no-$\mathbf{A}$ model. Runs with different signs on the drift and on the absolute value of the drift also gave much higher cost functions than the $\mathbf{A}$ model.

#### 2.7. STM PATHINT Calculations

#### 2.7.1. PATHINT STM

The evolution of a Balanced Centered model (BC) after 500 foldings of $\mathsf{\Delta}t=0.01$, 5 unit of relaxation time $\tau $, exhibits the existence of ten well developed peaks. These peaks are identified with possible trappings of firing patterns.

This describes the “

$7\pm 2$” rule, as calculated by SMNI PATHINT in

Figure 2 [

33].

#### 2.7.2. PATHINT STM Visual

The evolution of a Balanced Centered Visual model (BCV) after 1000 foldings of $\mathsf{\Delta}t=0.01$, 10 unit of relaxation time $\tau $, exhibits the existence of four well developed peaks. These peaks are identified with possible trappings of firing patterns. Other peaks at lower scales are clearly present, numbering on the same order as in the BC’ model, as the strength in the original peaks dissipates throughout firing space, but these are much smaller and therefore much less probable to be accessed.

This describes the “

$4\pm 2$” rule for visual STM, as calculated by SMNI PATHINT in

Figure 3 [

33].

#### 2.8. Tripartite Synaptic Interactions

The human brain contains over

${10}^{11}$ cells, about half of which are neurons. The other half are glial cells. Astrocytes comprise a good fraction of glial cells, possibly the majority. Many papers examine the influence of astrocytes on synaptic processes [

34,

35,

36,

37,

38,

39,

40,

41,

42].

http://www.astrocyte.info claims they are the most numerous cells in the human brain. Unlike the previous ideology of astrocytes being “filler” cells, they are very active in the central nervous system and greatly outnumber neurons,

Glutamate release from astrocytes through a

${\mathrm{Ca}}^{2+}$-dependent mechanism can activate receptors at the presynaptic terminals. Intercellular calcium waves (ICWs) may travel over hundreds of astrocytes propagating over many neuronal synapses. ICWs contribute to control synaptic activity. Glutamate is released in a regenerative manner, with subsequent cells that are involved in the calcium wave releasing additional glutamate [

43].

$\left[{\mathrm{Ca}}^{2+}\right]$ (concentrations of ${\mathrm{Ca}}^{2+}$) affect increased release probabilities at synaptic sites, by enhancing the release of gliotransmitters. (Free ${\mathrm{Ca}}^{2+}$ waves are considered here, not intracellular astrocyte calcium waves in situ which also increase neuronal firings.)

These free regenerative ${\mathrm{Ca}}^{2+}$ waves, arising from astrocyte-neuron interactions, couple to the magnetic vector potential $\mathbf{A}$ produced by highly synchronous collective firings, e.g., during selective attention tasks, as measured by EEG.

#### 2.8.1. Canonical Momentum $\mathbf{\Pi}=\mathbf{p}+q\mathbf{A}$

As derived in the Feynman (midpoint) representation of the path integral, the canonical momentum,

$\mathbf{\Pi}$, defines the dynamics of a moving particle with momentum

$\mathbf{p}$ in an electromagnetic field. In SI units,

where

$q=-2e$ for

${\mathrm{Ca}}^{2+}$,

e is the magnitude of the charge of an electron

$=1.6\times {10}^{-19}$ C (Coulomb), and

$\mathbf{A}$ is the electromagnetic vector potential. (In Gaussian units

$\mathbf{\Pi}=\mathbf{p}+q\mathbf{A}/c$, where

c is the speed of light.)

$\mathbf{A}$ represents three components of a 4-vector.

#### 2.8.2. Vector Potential of Wire

A columnar firing state is modeled as a wire/neuron with current

$\mathbf{I}$ measured in A = Amperes = C/s,

along a length

z observed from a perpendicular distance

r from a line of thickness

${r}_{0}$. If far-field retardation effects are neglected, this yields

where

$\mu $ is the magnetic permeability in vacuum

$=4\pi {10}^{-7}$ H/m (Henry/meter). Note the insensitive log dependence on distance.

The contribution to $\mathbf{A}$ includes minicolumnar lines of current from hundreds to thousands of macrocolumns, within a region not so large to include many convolutions, but still contributing to large synchronous bursts of EEG.

Electric $\mathbf{E}$ and magnetic $\mathbf{B}$ fields, derivatives of $\mathbf{A}$ with respect to r, do not possess this logarithmic insensitivity to distance, and therefore they do not linearly accumulate strength within and across macrocolumns.

Estimates of contributions from synchronous firings to P300 measured on the scalp are tens of thousands of macrocolumns spanning 100 to 100’s of cm${}^{2}$. Electric fields generated from a minicolumn may fall by half within 5–10 mm, the range of several macrocolumns.

There are other possible sources of magnetic vector potentials not described as wires with currents [

44]. Their net effects plausibly would be included the vector magnetic potential of net synchrous firings, but not their functional forms as derived here.

#### 2.8.3. Effects of Vector Potential on Momenta

The momentum $\mathbf{p}$ for a ${\mathrm{Ca}}^{2+}$ ion with mass $m=6.6\times {10}^{-26}$ kg, speed on the order of 50 $\mathsf{\mu}$m/s to 100 $\mathsf{\mu}$m/s, is on the order of ${10}^{-30}$ kg-m/s. Molar concentrations of ${\mathrm{Ca}}^{2+}$ waves, comprised of tens of thousands of free ions representing about 1% of a released set, most being buffered, are within a range of about 100 $\mathsf{\mu}$m to as much as 250 $\mathsf{\mu}$m, with a duration of more than 500 ms, and with [${\mathrm{Ca}}^{2+}$] ranging from 0.1–5 $\mathsf{\mu}$M ($\mathsf{\mu}$M = ${10}^{-3}$ mol/m${}^{3}$).

The magnitude of the current is taken from experimental data on dipole moments

$\mathbf{Q}=\left|\mathbf{I}\right|\widehat{\mathbf{z}}$ where

$\widehat{\mathbf{z}}$ is the direction of the current

$\mathbf{I}$ with the dipole spread over

z.

$\mathbf{Q}$ ranges from 1 pA-m =

${10}^{-12}$ A-m for a pyramidal neuron [

45], to

${10}^{-9}$ A-m for larger neocortical mass [

46]. These currents give rise to

$q\mathbf{A}\approx {10}^{-28}$ kg-m/s. The velocity of a

${\mathrm{Ca}}^{2+}$ wave can be ≈20–50

$\mathsf{\mu}$m/s. In neocortex, a typical

${\mathrm{Ca}}^{2+}$ wave of 1000 ions, with total mass

$m=6.655\times {10}^{-23}$ kg times a speed of ≈20–50

$\mathsf{\mu}$m/s, gives

$\mathbf{p}\approx {10}^{-27}$ kg-m/s.

Taking ${10}^{4}$ synchronous firings in a macrocolumn, leads to a dipole moment $\left|\mathbf{Q}\right|={10}^{-8}$ A-m. Taking z to be ${10}^{2}\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}$m $={10}^{-4}$ m, a couple of neocortical layers, gives $\left|q\mathbf{A}\right|\approx 2\times {10}^{-19}\times {10}^{-7}\times {10}^{-8}/{10}^{-4}$ = ${10}^{-28}$ kg-m/s,

#### 2.8.4. Reasonable Estimates

Estimates used here for $\mathbf{Q}$ come from experimental data. These include shielding and material effects. When coherent activity among many macrocolumns associated with STM is considered, $\left|\mathbf{A}\right|$ may be much larger. Since ${\mathrm{Ca}}^{2+}$ waves influence synaptic activity, there is direct coherence between these waves and the activity of $\mathbf{A}$.

Classical physics calculates $q\mathbf{A}$ from macroscopic EEG to be on the order of ${10}^{-28}$ kg-m/s, while the momentum $\mathbf{p}$ of a ${\mathrm{Ca}}^{2+}$ ion is on the order of ${10}^{-30}$ kg-m/s. This numerical comparison demonstrates the possible importance of the influence of $\mathbf{A}$ on $\mathbf{p}$ at classical scales.

This project fits the SMNI model to EEG data. Direct calculations in classical and quantum physics support the concepts presented here, e.g., that ionic calcium momentum-wave effects among neuron-astrocyte-neuron tripartite synapses modify background SMNI parameters and create feedback between ionic/quantum and macroscopic scales [

7,

20,

21,

22,

23,

25,

26].

#### 2.9. Model of Models (MOM)

Deep Learning (DL) has invigorated AI approaches to parsing data in complex systems, often to develop control processes of these systems. A couple of decades ago, Neural Net AI approaches fell out of favor when concerns were apparent that such approaches offered little guidance to explain the “why” or “how” such algorithms worked to process data, e.g., contexts which were deemed important to deal with future events and outliers, etc.

The success of DL has overshadowed these concerns. However, that should not diminish their importance, especially if such systems are placed in positions to affect lives and human concerns; humans are ultimately responsible for structures they build.

An approach to dealing with these concerns can be called Model of Models (MOM). An argument in favor of MOM is that humans over thousands of years have developed models of reality across many disciplines, e.g., ranging over Physics, Biology, Mathematics, Economics, etc.

A good use of DL might be to process data for a given system in terms of a collection of models, then again use DL to process the models over the same data to determine a superior model of models (MOM). Eventually, large DL (quantum) machines could possess a database of hundreds or thousands of models across many disciplines, and directly find the best (hybrid) MOM for a given system.

In particular, SMNI offers a reasonable model upon which to further develop MOM, wherein multiple scales of observed interactions are developed. This is just one example of how physics modeling and computational physics can be used to better understand complex systems.

#### Ideas by Statistical Mechanics

A project sympathetic to this MOM context was proposed as Ideas by Statistical Mechanics (ISM) [

47,

48,

49]. using ASA [

50,

51,

52] to fit parameters of a generic nonlinear multivariate colored-noise Gaussian-Markovian short-time conditional probability distribution to data, useful for many systems.

Models developed using ASA have been applied in many contexts across many systems [

53], including applications to neural networks [

54].

Many of these ASA applications have used Ordinal representations of features, to permit parameterization of their inclusion into models, quite similar in spirit to DL.

ASA can be used again in the expanded context of MOM. This is suggested as a first step in a new discipline to which MOM is to be applied, to help develop a range of parameters useful for DL, as DL by itself may get stuck in non-ideal local minima of the importance-sampled space. Then, after a reasonable range of models is found, DL can take over to permit much more efficient and accurate development of MOM for a given discipline/system.

## 5. Results Including Quantum Scales

The wave function

${\psi}_{\mathrm{e}}$ describing the interaction of

$\mathbf{A}$ with

$\mathbf{p}$ of

${\mathrm{Ca}}^{2+}$ wave packets was derived in closed form from the Feynman representation of the path integral using path-integral techniques [

87], modified here to include

$\mathbf{A}$.

where

${\psi}_{0}$ is the initial Gaussian packet,

${\psi}_{F}$ is the free-wave evolution operator,

ℏ is the Planck constant,

q is the electronic charge of

${\mathrm{Ca}}^{2+}$ ions,

m is the mass of a wave-packet of 1000

${\mathrm{Ca}}^{2+}$ ions,

$\mathsf{\Delta}{\mathbf{r}}^{2}$ is the spatial variance of the wave-packet, the initial momentum is

${\mathbf{p}}_{0}$, and the evolving canonical momentum is

$\mathbf{\Pi}=\mathbf{p}+q\mathbf{A}$. Detailed calculations show that

$\mathbf{p}$ of the

${\mathrm{Ca}}^{2+}$ wave packet and

$q\mathbf{A}$ of the EEG field make about equal contributions to

$\mathbf{\Pi}$.

#### 5.1. SMNI + ${\mathrm{Ca}}^{2+}$ Wave-Packet

Tripartite influence on synaptic

${B}_{{G}^{\prime}}^{G}$ is measured by the ratio of packet’s

$<\mathbf{p}\left(t\right){>}_{\psi \ast \psi}$ to

$<{\mathbf{p}}_{0}\left({t}_{0}\right){>}_{\psi \ast \psi}$ at the onset of each attentional task. Here

$<{>}_{\psi \ast \psi}$ is taken over

${\psi}_{\mathrm{e}}^{*}\phantom{\rule{0.166667em}{0ex}}{\psi}_{\mathrm{e}}$.

$\mathbf{A}$ changes slower than

$\mathbf{p}$, so static approximation of

$\mathbf{A}$ used to derive

${\psi}_{\mathrm{e}}$ and

$<\mathbf{p}{>}_{\psi \ast \psi}$ is reasonable to use within P300 EEG epochs, resetting

$t=0$ at the onset of each classical EEG measurement (1.953 ms apart), using the current

$\mathbf{A}$. This permits tests of interactions across scales in a classical context.

#### 5.2. Supercomputer Resources

About 1000 h of supercomputer CPUs are required for an ASA fit of SMNI to the same EEG data used previously, i.e., from

http://physionet.nlm.nih.gov/pn4/erpbci [

31,

32], using mostly the same codes used previously [

22]. Many such sets of runs are required. Including quantum processes will take even longer.

#### 5.3. Results Using $<\mathbf{p}{>}_{\psi \ast \psi}$

$<\mathbf{p}{>}_{\psi \ast \psi}$ was used in classical-physics SMNI fits to EEG data using ASA. Runs using 1M or 100K generated states gave results not much different. Training with ASA used 100K generated states over 12 subjects with and without

$\mathbf{A}$, followed by 1000 generated states with the simplex local code contained with ASA. Training and Testing runs on XSEDE.org for this project has taken an equivalent of several months of CPU on the XSEDE.org UCSD platform Comet. These calculations use one additional parameter across all EEG regions to weight the contribution to synaptic background

${B}_{{G}^{\prime}}^{G}$.

$\mathbf{A}$ is taken to be proportional to the currents measured by EEG, i.e., firings

${M}^{G}$. Otherwise, the “zero-fit-parameter” SMNI philosophy was enforced, wherein parameters are picked from experimentally determined values or within experimentally determined ranges [

4].

As with previous studies using this data, results sometimes give Testing cost functions less than the Training cost functions. This reflects on great differences in data, likely from great differences in subjects’ contexts, e.g., possibly due to subjects’ STM strategies only sometimes including effects calculated here. Further tests of these multiple-scale models with more EEG data are required, and with the PATHINT-qPATHINT coupled algorithm described previously.

Table 1 gives recent results on such tests. Cost functions are the effective Action,

${A}_{eff}$, which is

$L\phantom{\rule{0.166667em}{0ex}}\mathsf{\Delta}t-log\left(\mathrm{prefactor}\right)$, where the prefactor multiplier of the exponential arises from the normalization of the short-time conditional probability distribution and

$L\phantom{\rule{0.166667em}{0ex}}\mathsf{\Delta}t$ is the argument of the exponential factor. Equation (3) defines the Lagrangian

L, and the normalization is defined in

$DM$ in Equation (11).

#### 5.4. Quantum Zeno Effects

The quantum-mechanical wave function of the wave packet was shown to “survive” overlaps after multiple collisions, due to their regenerative processes during the observed long durations of hundreds of ms. Thus,

${\mathrm{Ca}}^{2+}$ waves may support a Zeno or “bang-bang” effect which may promote long coherence times [

88,

89,

90,

91,

92,

93,

94,

95,

96].

Of course, the Zeno/“bang-bang” effect may exist only in special contexts, since decoherence among particles is known to be very fast, e.g., faster than phase-damping of macroscopic classical particles colliding with quantum particles [

97].

The wave may be perpetuated by the constant collisions of ions as they enter and leave the wave packet due to the regenerative collisions by the Zeno/“bang-bang” effect. qPATHINT can calculate the coherence stability of the wave due to serial shocks.

#### Survival of Wave Packet

In momentum space, the wave packet

$\varphi (\mathbf{p},t)$ is considered as being “kicked” from

$\mathbf{p}$ to

$\mathbf{p}+\delta \mathbf{p}$. Assume that random repeated kicks of

$\delta \mathbf{p}$ result in

$<\delta \mathbf{p}>\approx 0$, and that each kick keeps the variance

$\mathsf{\Delta}{(\mathbf{p}+\delta \mathbf{p})}^{2}\approx \mathsf{\Delta}{\left(\mathbf{p}\right)}^{2}$. Then, the overlap integral at the moment

t of a typical kick between the new and old state is

where

$\varphi (\mathbf{p}+\delta \mathbf{p},t)$ is the normalized wave function in

$\mathbf{p}+\delta \mathbf{p}$ momentum space. A crude estimate is obtained of the survival time amplitude

$A\left(t\right)$ and survival probability

$p\left(t\right)$ [

90],

Even many small repeated kicks do not appreciably affect the real part of $\varphi $, and these projections do not appreciably destroy the original wave packet, giving a survival probability per kick as $p\left(t\right)\approx exp(-2.5\times {10}^{-7})\approx 1-2.5\times {10}^{-7}$.

The time-dependent phase terms are sensitive to times of tenths of a sec. These times are prominent in STM and in synchronous neural firings. Therefore, $\mathbf{A}$ effects on ${\mathrm{Ca}}^{2+}$ wave functions may maximize their influence on STM at frequencies consistent with synchronous EEG during STM.

All these calculations support this model, in contrast to other models of quantum brain processes without such specific calculations and support [

98,

99,

100].