This project calculates quantum

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.

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 [

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

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 [

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

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

The path integral is derived in terms of mesoscopic Lagrangian

In the prepoint (Ito) representation the SMNI Lagrangian

The threshold factor

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

Other values also are consistent with experimental data, e.g.,

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

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

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

The core SMNI hypothesis first developed circa 1980 [

SMNI calculations agree with observations [

capacity (auditory

duration [

stability [

primacy versus recency rule [

Hick’s law (reaction time and

nearest-neighbor minicolumnar interactions

derivation of basis for EEG [

Three basic models were developed by slightly changing the background firing component of the columnar-averaged efficacies

case EC, dominant excitation subsequent firings

case IC, inhibitory subsequent firings

case BC, balanced between EC and IC

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

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

Using EEG data from

Spline-Laplacian transformations on the EEG potential

The

The evolution of a Balanced Centered model (BC) after 500 foldings of

This describes the “

The evolution of a Balanced Centered Visual model (BCV) after 1000 foldings of

This describes the “

The human brain contains over

Glutamate release from astrocytes through a

These free regenerative

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

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

The contribution to

Electric

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

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

The momentum

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

Taking

Estimates used here for

Classical physics calculates

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 [

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.

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

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

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.

Nonlinear and/or stochastic systems often require importance-sampling algorithms to scan or to fit parameters. Methods of simulated annealing (SA) are often used. Proper annealing (not “quenching”) possesses a proof of finding the deepest minimum in searches.

The ASA code is open-source software, and can be downloaded and used without any cost or registration at

This algorithm fits empirical data to a cost function over a

This ASA algorithm is faster than fast Cauchy annealing, which has schedule

For parameters

The default ASA uses the same type of annealing schedule for the acceptance function

All default functions in ASA can be overridden with user-defined functions.

The ASA code [

chaotic systems [

combat simulations [

financial systems: bonds, equities, futures, options [

neuroscience [

optimization

The path integral in the Feynman (midpoint) representation is used to examine discretization issues in time-dependent nonlinear systems [

Non-constant diffusions add terms to drifts, and a Riemannian-curvature potential

In the Ito (prepoint) representation:

Here the diagonal diffusions are

The midpoint derivation derives a Riemannian geometry with metric defined by the inverse of the covariance matrix

An Ito prepoint discretization for the same probability distribution

Three basic different approaches are mathematically equivalent:

Fokker-Planck/Chapman-Kolmogorov partial-differential equations

Langevin coupled stochastic-differential equations

Lagrangian or Hamiltonian path-integrals

All three are described here as many researchers are familiar with at least one of these approaches to complex systems.

The path-integral approach is useful to define intuitive physical variables from the Lagrangian

Differentiation especially of noisy systems often introduces more noise. The path-integral often gives superior numerical performance because integration is a smoothing process.

The Stratonovich (midpoint discretized) Langevin equations can be analyzed in terms of the Wiener process

The Fokker-Planck, sometimes defines as Chapman-Kolmogorov, partial differential equation is:

Path integrals and PATHINT have been applied across several disciplines, including combat simulations [

qPATHINT is an N-dimensional code developed to calculate the propagation of quantum variables in the presence of shocks. Many real systems propagate in the presence of sudden changes of state dependent on time. qPATHINT is based on the classical-physics code, PATHINT, which has been useful in several systems across several disciplines. Applications have been made to SMNI and Statistical Mechanics of Financal Markets (SMFM) [

To numerically calculate the path integral for serial changes in time, standard Monte Carlo techniques generally are not useful. PATHINT was originally developed for this purpose. The PATHINT C code of about 7500 lines of code using the GCC C-compiler was rewritten to use double complex variables instead of double variables, and further developed for arbitrary N dimensions, creating qPATHINT. The outline of the code is described here for classical or quantum systems, using generic coordinates

The distribution (probabilities for classical systems, wave-functions for quantum systems) can be numerically approximated to a high degree of accuracy using a histogram procedure, developing sums of rectangles of height

Many real-world systems propagate in the presence of continual “shocks”.

In SMNI, collisions occur via regenerative

In SMFM applications, shocks occur due to future dividends, changes in interest rates, changes in asset distributions, etc.

A one-dimensional path-integral in variable

This yields

Several projects have used this algorithm [

Explicit dependence of

This constrains the dependence of the covariance of each variable to be a (nonlinear) function of that variable to present a rectangular underlying mesh. Since integration is inherently a smoothing process [

By considering the contributions to the first and second moments, conditions on the time and variable meshes can be derived. Thus

SMNI qPATHINT has emphasized the requirement of broad-banded kernels for oscillatory quantum states.

SMFM PATHTREE, and its derived qPATHTREE, is a different options code, based on path-integral error analyses, permitting a new very fast binary calculation, also applied to nonlinear time-dependent systems [

SMFM [q]PATHINT for (American) financial options: Calculate at each node of each time slice—back in time.

SMNI [q]PATHINT: Calculate at each node of each time slice—forward in time.

At each node of each time slice, a proposed algorithm is to calculate quantum-scale

PATHINT using the Classical SMNI Lagrangian

qPATHINT using the Quantum

Sync in time during P300 attentional tasks.

Time/phase relations between classical and quantum systems may be important.

ASA-fit synchronized classical-quantum PATHINT-qPATHINT model to EEG data.

The wave function

Tripartite influence on synaptic

The XSEDE.org University of California San Diego (UCSD) supercomputer resource is Comet, described at

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

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.

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,

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 [

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.

In momentum space, the wave packet

These numbers yield:

Even many small repeated kicks do not appreciably affect the real part of

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,

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

There is the possibility of carrying pharmaceutical products in nanosystems that could affect unbuffered

At the onset of a

An area of the receptor of the nanosystem of 1 nm

The nano-roboot could be switched on/off at a regional/columnar level by sensitivity to local electric/magnetic fields. Highly synchronous firings during STM processes can be affected by these piezoelectric nanosystems which affect background/noise efficacies via control of

There is interest in researching possible quantum influences on highly synchronous neuronal firings relevant to STM to understand connections to consciousness and “Free Will” (FW) [

If experimental evidence is gained of quantum-level processes of tripartite synaptic interactions with large-scale synchronous neuronal firings, then FW may be established using the Conway-Kochen quantum no-clone “Free Will Theorem” (FWT) [

The essence of FWT is that, since quantum states cannot be cloned, a

The SMNI model has demonstrated it is faithful to experimental data, for EEG recordings under STM experimental paradigms. qPATHINT permits an inclusion of quantum scales in the multiple-scale SMNI model, by evolving

This quantum path-integral algorithm with serial random shocks will be further studied as it can be used for many quantum systems.

The author thanks the Extreme Science and Engineering Discovery Environment (XSEDE.org), for supercomputer grants since February 2013, starting with Electroencephalographic field influence on calcium momentum waves, one under PHY130022 and two under TG-MCB140110. The current grant under TG-MCB140110, Quantum path-integral qPATHTREE and qPATHINT algorithms, was started in 2017, and renewed through December 2018.

The author thanks the Extreme Science and Engineering Discovery Environment (XSEDE.org), for supercomputer grants since February 2013, starting with “Electroencephalographic field influence on calcium momentum waves”, one under PHY130022 and two under TG-MCB140110. The current grant under TG-MCB140110, “Quantum path-integral qPATHTREE and qPATHINT algorithms”, was started in 2017, and renewed through December 2018. XSEDE grants have spanned several projects described in

The author declares no conflict of interest.

Illustrates three SMNI biophysical scales [

Illustrates SMNI STM Model BC at the evolution at 5

Illustrates SMNI STM Model BCV at the evolution at 10

Column 1 is the subject number; the other columns are cost functions. Columns 2 and 3 are no-

Sub | TR0 | TE0 | TRA | TEA | sTR0 | sTE0 | sTRA | sTEA |
---|---|---|---|---|---|---|---|---|

s01 | 85.75 | 121.23 | 84.76 | 121.47 | 120.48 | 86.59 | 119.23 | 87.06 |

s02 | 70.80 | 51.21 | 68.63 | 56.51 | 51.10 | 70.79 | 49.36 | 74.53 |

s03 | 61.37 | 79.81 | 59.83 | 78.79 | 79.20 | 61.50 | 75.22 | 79.17 |

s04 | 52.25 | 64.20 | 50.09 | 66.99 | 63.55 | 52.83 | 63.27 | 64.60 |

s05 | 67.28 | 72.04 | 66.53 | 72.78 | 71.38 | 67.83 | 69.60 | 68.13 |

s06 | 84.57 | 69.72 | 80.22 | 64.13 | 69.09 | 84.67 | 61.74 | 114.21 |

s07 | 68.66 | 78.65 | 68.28 | 86.13 | 78.48 | 68.73 | 75.57 | 69.58 |

s08 | 46.58 | 43.81 | 44.24 | 49.38 | 43.28 | 47.27 | 42.89 | 63.09 |

s09 | 47.22 | 24.88 | 46.90 | 25.77 | 24.68 | 47.49 | 24.32 | 49.94 |

s10 | 53.18 | 33.33 | 53.33 | 36.97 | 33.14 | 53.85 | 30.32 | 55.78 |

s11 | 43.98 | 51.10 | 43.29 | 52.76 | 50.95 | 44.47 | 50.25 | 45.85 |

s12 | 45.78 | 45.14 | 44.38 | 46.08 | 44.92 | 46.00 | 44.45 | 46.56 |