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\def\makelabel##1{\upshape ##1}}% } {\end{list}\end{raggedright}} \newenvironment{specHead}[2]% {\vspace{20pt}\hrule\vspace{10pt}% \phantomsection\label{#1}\markright{#2}% \pdfbookmark[2]{#2}{#1}% \hspace{-0.75in}{\bfseries\fontsize{16pt}{18pt}\selectfont#2}% }{} \def\TheFullDate{2021-05-15 (revised: 15 May 2021)} \def\TheID{\makeatother } \def\TheDate{2021-05-15} \title{Hybrid Classical-Quantum Computing: Applications to Statistical Mechanics of Neocortical Interactions} \author{}\makeatletter \makeatletter \newcommand*{\cleartoleftpage}{% \clearpage \if@twoside \ifodd\c@page \hbox{}\newpage \if@twocolumn \hbox{}\newpage \fi \fi \fi } \makeatother \makeatletter \thispagestyle{empty} \markright{\@title}\markboth{\@title}{\@author} \renewcommand\small{\@setfontsize\small{9pt}{11pt}\abovedisplayskip 8.5\p@ plus3\p@ minus4\p@ \belowdisplayskip \abovedisplayskip \abovedisplayshortskip \z@ plus2\p@ \belowdisplayshortskip 4\p@ plus2\p@ minus2\p@ \def\@listi{\leftmargin\leftmargini \topsep 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\textcolor{GJMediumGrey}{\rule{\textwidth}{2pt}} \vskip16pt \textcolor{GJBlue}{\large\bfseries\abstractname\space} }{% \vskip8pt \textcolor{GJMediumGrey}{\rule{\textwidth}{2pt}} \vskip16pt } \usepackage[absolute,overlay]{textpos} \makeatother \usepackage{lineno} \linenumbers \begin{document} \author[1]{Lester Ingber} \affil[1]{ Physical Studies Institute LLC} \renewcommand\Authands{ and } \date{\small \em Received: 10 April 2021 Accepted: 30 April 2021 Published: 15 May 2021} \maketitle \begin{abstract} Several commerical quantum computers are now available that offer Hybrid Classical-Quantum computing. Application is made to a classical-quantum model of human neocortex, Statistical Mechanics of Neocortical Interactions (SMNI), which has had its applications published in many papers since 1981. However, this project only uses Classical (super-) computers.Since 2015, PATHINT, has been used as a numerical algorithm for folding path-integrals. Applications in several systems in several disciplines has generalized been from 1 dimension to N dimensions, and from classical to quantum systems, qPATHINT. Papers have applied qPATHINT to neocortical interactions and financial options.The classical space described by SMNI applies nonlinear nonequilibrium multivariate statistical mechanics to synaptic neuronal interactions, while the quantum space described by qPATHINT applies synaptic contributions from Ca2+ waves generated by astrocytes at tripartite neuron-astrocyte-neuron sites. \end{abstract} \keywords{path integral, quantum systems, neocortical interactions.} \begin{textblock*}{18cm}(1cm,1cm) % {block width} (coords) \textcolor{GJBlue}{\LARGE Global Journals \LaTeX\ JournalKaleidoscope\texttrademark} \end{textblock*} \begin{textblock*}{18cm}(1.4cm,1.5cm) % {block width} (coords) \textcolor{GJBlue}{\footnotesize \\ Artificial Intelligence formulated this projection for compatibility purposes from the original article published at Global Journals. However, this technology is currently in beta. \emph{Therefore, kindly ignore odd layouts, missed formulae, text, tables, or figures.}} \end{textblock*} \let\tabcellsep& \par https://quantumzeitgeist.com/tensorflow-for-quantumhits-first-birthday/ https://www.tensorflow.org/quantum Two previous XSEDE project-codes are merged for this project, "Electroencephalographic field influence on calcium momentum waves" and "Quantum pathintegral qPATHTREE and qPATHINT algorithms". These Classical codes on Classical computers define a Hybrid Classical-Quantum system.\par Previous fits to electroencephalographic (EEG) data have been published using quantum wave-packets defining a specific class of (re-)generated Ca2+ ions at tripartite neuron-astrocyte-neuron sites, which influence synaptic interactions \hyperref[b28]{[Ingber(2018)}]. Since 2011 {\ref [Ingber(2011)}, {\ref Ingber(2012a)}], During tasks requiring short-term memory (STM), classical as well as quantum calculations are consistent with interactions between momenta p of these wave-packets with a magnetic vector potential A generated by highly synchronous firings of many thousands of neocortical neurons. An anayltic (closed-form) path-integral calculation of the quantum process, in terms of a wavefunction with expectation value p interacting with A, defines quantum interactions coupled to a macroscopic system \hyperref[b26]{[Ingber (2017a)}].\par It is important to further address this system using these to enable realistic inclusion of shocks to the wave-packet. Due to the regenerative process of the wave-packet, e.g., due to collisions between Ca2+ ions in the wave-packet causing some ions to leave the wave-packet during its hundreds of msec lifetime, or as new ions enter from the astrocyte processes, the wavefunction is repeatedly projected onto quantum subspaces. These may be considered as random processes \hyperref[b40]{[Ross (2012)}]. PATHTREE/qPATHTREE and PATHINT/qPATHINT can include shocks in the evolution of a short-time probability distribution over thousands of foldings \hyperref[b25]{[Ingber(2016a)}, Ingber(2017a), \hyperref[b27]{Ingber (2017b)}]. \section[{b) SMNI}]{b) SMNI}\par Statistical Mechanics of Neocortical Interactions (SMNI) was developed in the late 1970's {\ref [Ingber(1981)}, {\ref Ingber(1982)}, {\ref Ingber(1983)}] and enhanced since by fitting experimental data from short-term memory (STM) and electroencephalography (EEG), including papers on fits to attention data \hyperref[b28]{[Ingber(2018)}] and affective data [Alakus \textasciitilde al.(2020) Alakus, Gonen, Turkoglu, Ingber(2021b)]. \section[{c) PATHINT}]{c) PATHINT}\par The path-integral defines code that numerically propagates short-time conditional probability distributions \hyperref[b17]{[Ingber \textasciitilde al.(1991)}Ingber, Fujio, Wehner, Wehner Wolfer(1983a), Wehner \hyperref[b47]{Wolfer(1983b)}]. This was generalized to PATHINT in N dimensions, and applied in several disciplines {\ref [Ingber(2000a)} \section[{d) qPATHINT}]{d) qPATHINT}\par PATHTREE and PATHINT were generalized to quantum systems, qPATHTREE and qPATHINT \hyperref[b25]{[Ingber(2016a)}, Ingber(2017b), \hyperref[b28]{Ingber(2017c)}].\par A companion paper treats "Hybrid classicalquantum computing: Applications to statistical mechanics of financial markets" \hyperref[b31]{[Ingber(2021c)}]. \section[{e) Organization of Paper}]{e) Organization of Paper}\par Section 2 further describes SMNI in the context of this project.\par Section 3 further describes Adaptive Simulated Annealing (ASA) in the context of this project.\par Section 4 further describes qPATHINT in the context of this project.\par Section 5 describes how the calculation proceeds between SMNI and qPATHINT. Section 6 describes performance and scaling issues.\par Section 7 gives motivation to this project. Calculations performed on the Ookami supercomputer at StonyBrook.edu test a closed-form derived solution with and without shocks. Shocks clearly lower cost functions. Section 8 is the Conclusion. \section[{f) Caveat}]{f) Caveat}\par As stated previously in these projects \hyperref[b28]{[Ingber(2018)}], "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." \section[{II. SMNI}]{II. SMNI}\par Statistical Mechanics of Neocortical Interactions (SMNI) has been developed since 1981, in over 30+ papers, scaling aggregate synaptic interactions describing neuronal firings, scaling minicolumnarmacrocolumnar columns of neurons to mesocolumnar dynamics, and scaling columns of neuronal firings to regional (sensory) macroscopic sites identified in EEG studies {\ref [Ingber(1981)}, {\ref Ingber(1982)}, {\ref Ingber(1983)}, {\ref Ingber(1984)}, {\ref Ingber(1985a)}, {\ref Ingber(1994)}].\par Success of SMNI has been to discover agreement/fits with experimental data from modeled aspects of neocortical interactions, e.g., properties of short-term memory (STM) {\ref [Ingber(2012a)}], including its capacity (auditory 72 and visual 42), duration, stability, primacy versus recency rule {\ref [Ingber(1984)}, {\ref Ingber (1985a)}, {\ref Ingber(1994)}, \hyperref[b19]{Ingber(1995)}, {\ref Ingber(2000b)}, Ingber(2012a), Ingber \hyperref[b19]{Nunez(1995)}], EEG dispersion relations {\ref [Ingber(1985b)}], as well other phenomenon, e.g., Hick's law \hyperref[b16]{[Hick(1952)}, {\ref Ingber(1999)}, \hyperref[b33]{Jensen (1987)}], nearest-neighbor minicolumnar interactions within macrocolumns calculating rotation of images, etc {\ref [Ingber(1982)}, {\ref Ingber(1983)}, {\ref Ingber(1984)}, {\ref Ingber (1985a)}, {\ref Ingber(1994)}]. SMNI was also scaled to include mesocolumns across neocortical regions to fit EEG data \hyperref[b4]{[Ingber(1997)}, Ingber(2012a), Ingber(2012b), Ingber (2018), Ingber \hyperref[b20]{Nunez(2010)}]. \section[{a) XSEDE EEG Project}]{a) XSEDE EEG Project}\par The Extreme Science and Engineering Discovery Environment (XSEDE.org) project since February 2013, "Electroencephalographic field influence on calcium momentum waves," has fit SMNI to EEG data, developing ionic Ca2+ momentum-wave effects among neuron-astrocyte-neuron tripartite synapses modified parameterization of background SMNI parameters. Both classical and quantum physics support the development of the vector magnetic potential of EEG from highly synchronous firings, e.g., as measured during selective attention, as directly interacting with these momentum-waves, creating feedback between these ionic/quantum and macroscopic scales {\ref [Ingber(2012a)} \section[{i. qPATHINT for SMNI}]{i. qPATHINT for SMNI}\par qPATHINT includes quantum regenerative process to Ca2+ wave-packets as reasonable shocks to the waves that typically do not damage its coherence properties. A proof of principal has been published \hyperref[b27]{[Ingber(2017b)}]. \section[{b) SMNI With A}]{b) SMNI With A}\par A model of minicolumns as wires supporting neuronal firings, largely from large neocortical excitatory pyramidal cells in layer V (of I-VI), gives rise to currents, in turn giving rise to electric potentials measured as scalp EEG {\ref [Ingber(2011)}, Ingber(2012a), Nunez \textasciitilde al.(2013)Nunez, Srinivasan, Ingber]. This gives rise to a magnetic vector potential.?? = ?? 4?? ??log ? ?? ?? 0 ? (1)\par Note the log-insensitive dependence on distance. In neocortex, ?? ? ?? 0 , where 0 is the magnetic permeability in vacuum = 4??10 ?7 H/m (Henry/meter), where Henry has units of kg-m-C ?2 , the conversion factor from electrical to mechanical variables. For oscillatory waves, the magnetic field B = ? × A and the electric field E = (????/ð??"ð??")? × ? × A do not have this log dependence on distance. Thus, A fields can contribute collectively over large regions of neocortex {\ref [Ingber(2012a)} The magnitude current is deermined by experimental data on dipole moments Q = |I|z ? where z ? is the direction of the current I with the dipole spread over z. Q ranges from 1 pA-m = 10 -12 A-m for a pyramidal neuron \hyperref[b35]{[Murakami Okada(2006)}], to 10 ?9 A-m for larger neocortical mass \hyperref[b37]{[Nunez Srinivasan(2006)}]. Currents give rise to ??A ? 10 ?28 kg-m/s. The velocity of a Ca 2+ wave can be ?20-50 ??m/s. In neocortex, a typical Ca 2+ wave of 1000 ions, with total mass ?? = 6.655 × 10 ?23 kg times a speed of ?20-50 ??m/s, gives p ? 10 ?27 kgm/s. This yields p to be on the same order as ??A. \section[{i. Results Including Quantum Scales}]{i. Results Including Quantum Scales}\par The wave-function ð??"ð??" e of the interaction of A with p of Ca 2+ wave-packets was derived in closed form from the Feynman representation of the path integral using path-integral techniques, not including shocks \hyperref[b44]{[Schulten(1999)}], modified here to include A.ð??"ð??" e (??) = ? ???? 0 ð??"ð??" 0 ð??"ð??" ?? = ? 1 ? ?????/(??Î?"?? 2 ) 1 + ?????/(??Î?"?? 2 ) ? 1/4 [??Î?"?? 2 \{1 + [???/(??Î?"?? 2 )] 2 \}] ?1/4 × exp ?? [???(?? 0 +????)??/?? ] 2 2Î?"?? 2 1??????/(??Î?"?? 2 ) 1+[???/(??Î?"?? 2 )] 2 + ?? ?? 0 ??? ? ? ?? (?? 0 +????) 2 ?? 2??? ?\textbf{(2)}\par PATHINT also has been used with the SMNI Lagrangian ?? for STM for both auditory and visual memory \hyperref[b11]{[Ericsson Chase(1982)}, Zhang \hyperref[b50]{Simon(1985)}], calculating the stability and duration of STM, the observed 7 ± 2 capacity rule of auditory memory, and the observed 4 ± 2 capacity rule of visual memory {\ref [Ingber(2000a)}, Ingber \hyperref[b19]{Nunez(1995)}]. \section[{ii. Results using < ?? > ð??"ð??" * ð??"ð??"}]{ii. Results using < ?? > ð??"ð??" * ð??"ð??"}\par The author previously used his derived analytic expression for < ?? > ð??"ð??" * ð??"ð??" in classical-physics SMNI fits to EEG data using ASA \hyperref[b26]{[Ingber(2016b)} Calculations used one additional parameter across all EEG regions to weight the contribution to synaptic background ?? ?? ? ?? . A was proportional to the currents measured by EEG, i.e., firings ?? ?? . The "zero-fit-parameter" SMNI philosophy was enforced, wherein parameters are selected and enforced between experimentally determined ranges {\ref [Ingber(1984)}].\par Sopmetimes Testing cost functions were less than their Training cost functions, a result sometimes found in previous studies using this data. This likely is due to great differences in data, likely from great differences in subjects' contexts, e.g., possibly due to subjects' STM strategies including effects calculated here. ASA optimizations in this project always included "finishing" ASA importance-sampling with the modified Nelder-Mead simplex code included in the ASA code to ensure best precision.\par iii. Assumptions for quantum SMNI Some assumptions made for this quantum enhancement of SMNI can be determined by future experiments.\par The quantum wave-function of the Ca 2+ wavepacket was calculated, adding multiple collisions due to their regenerative processes, and it was demonstrated that overlaps with just-previous wave-functions during the observed long durations of hundreds of ms typical of Inclusion of repeated random shocks to the above wave-function ð??"ð??" ?? (??) demonstrated only small effects on the projections of the wave-packet after these shocks, i.e., the survival time was calculated \hyperref[b13]{[Facchi Pascazio(2008)}, \hyperref[b28]{Ingber(2018)}].\par Of course, the Zeno/"bang-bang" effect may exist only in special contexts, given that decoherence among particles is known to be very fast \hyperref[b39]{[Preskill(2015)}]. Therefore, the constant collisions of Ca 2+ ions as they enter and leave the Ca 2+ wave-packet due to the regenerative process that maintains the wave may perpetuate at least part of the wave, permitting the Zeno/"bang-bang" effect. qPATHINT as used here provides an opportunity to explore the coherence stability of the wave due to serial shocks of this process. \section[{iv. Nano-Robotic Applications}]{iv. Nano-Robotic Applications}\par It is possible that the above considerations could lead to pharmaceutical products contained in nanosystems that could affect unbuffered Ca 2+ waves in neocortex \hyperref[b24]{[Ingber(2015)}]. A Ca 2+ -wave momentumsensor could act like a piezoelectric device \hyperref[b28]{[Ingber (2018)}].\par The nano-robot would be sensitive to local electric/magnetic fields. Highly synchronous firings during STM processes could be directed by piezoelectric nanosystems to affect background/noise efficacies via control of Ca 2+ waves. This could affect the influence of Ca 2+ waves via the vector potential A, etc.\par v. Free will Further qPATHINT calculations could give additional support to researching possible quantum influences on highly synchronous neuronal firings relevant to STM, yielding connections to consciousness and "Free Will" (FW).\par As described previously [Ingber(2016a), Ingber(2016b)], experimental feedback from quantumlevel processes of tripartite synaptic interactions with large-scale synchronous neuronal firings, recognized as being highly correlated with STM and states of attention, may be established using the quantum no-clone "Free Will Theorem" (FWT) \hyperref[b9]{[Conway Kochen(2006)}, Conway \hyperref[b10]{Kochen(2009)}].\par Ca 2+ quantum wave-packets may generate states proven to have not previously existed, since quantum states cannot be cloned. In this context, these quantum states may be influential in large-scale patterns of synchronous neuronal firings, rendering these patterns as truly new. The FWT may consider these patterns as new decisions not solely based on previous decisions. These considerations are quite different and independent of other philosophical considerations, e.g., as in https://plato.stanford.edu/ entries/qt-consciousness/.\par Note that only recently has the core SMNI hypothesis since circa 1980 {\ref [Ingber(1981)}, {\ref Ingber(1982)}, {\ref Ingber(1983)}], that highly synchronous patterns of neuronal firings process high-level information, been verified experimentally \hyperref[b2]{[Asher(2012)} \section[{c) qPATHINT for SMNI}]{c) qPATHINT for SMNI}\par A previous project tested applications of qPATHTREE and qPATHINT. The wave-function ð??"ð??" is numerically propagated from its initial state, growing into a tree of wave-function nodes. At each node, interaction of the of Ca 2+ wave-packet, via its momentum p, with highly synchronous EEG, via its collective magnetic vector potential A, determines changes of timedependent phenomena. Changes occur at microscopic scales, e.g., due to modifications of the regenerative wave-packet as ions leave and contribute to the wavepacket, determining the effect on tripartite contributions to neuron-astrocyte-neuron synaptic activity, affecting both p and A. Such changes also influence macroscopic scales, e.g., changes due to external and internal stimuli affecting synchronous firings, and thereby A. At every time slice, quantum effects on synaptic interactions are determined by expected values of the interactions over probabilities (ð??"ð??" * ð??"ð??") determined by the wave-functions at their nodes.\par Due to the form of the quantum Lagrangian/Hamiltonian, a multiplicative Gaussian form (with nonlinear drifts and diffusions) is propagated. This permits a straight-forward use of Gaussian quadratures for numerical integration of the expectation of the momenta of the wave-packet, i.e., of < ??(??) > ð??"ð??" * ð??"ð??" . E.g., see https://en.wikipedia.org/wiki/Gaussian\textunderscore quadrature \section[{d) Comparing EEG Testing Data with Training Data}]{d) Comparing EEG Testing Data with Training Data}\par Using EEG data from {\ref [} due to the same currents I. Therefore, A is linearly proportional to ? with a simple scaling factor as a parameter in fits. Additional parameterization of background synaptic parameters also are included, ?? ?? ? ?? and ?? ?? ? ??? . \section[{e) Investigation into Spline-Laplacian Transformation}]{e) Investigation into Spline-Laplacian Transformation}\par As is common practice, codes for the Spline-Laplacian transformations were applied to all electrodes measured on the scalp. However, the author thinks that the transformation should be applied to each Region of neocortex separately (e.g., visual, auditory, somatic, abstract, etc.), since each region typically participates in attention differently. This process is further tested in this project. \section[{III.}]{III.} \section[{ASA Algorithm}]{ASA Algorithm}\par For parameters?? ?? ?? ? [?? ?? , ?? ?? ]\textbf{(3)}\par sampling with the random variable ?? ???? ?? ? [?1,1] ?? ??+1 ?? = ?? ?? ?? + ?? ?? (?? ?? ? ?? ?? )\textbf{(4)}\par the default generating function isð??"ð??" ?? (??) = ? ?? ??=1 1 2ln (1+1/?? ?? )(|?? ?? |+?? ?? ) ? ? ?? ??=1 ð??"ð??" ?? ?? (?? ?? )\textbf{(5)}\par in terms of "temperatures" for parameters {\ref [Ingber(1989)}]?? ?? = ?? ??0 exp(??? ?? ?? 1/?? )\textbf{(6)}\par The default ASA uses the same distribution for the annealing schedule for the acceptance function ? used for the generating function ð??"ð??".\par The ASA default functions can be substituted with user-defined functions {\ref [Ingber(1993)}, \hyperref[b23]{Ingber (2012c)}].\par ASA has been applied to studies of COVID-19, fitting forms like ???? ?? , for variables ?? and parameters ?? and ??, in the drifts and covariances of conditional probability distributions \hyperref[b32]{[Ingber(2021d)}].\par With over 150 OPTIONS, ASA permits robust tuning over many classes of nonlinear stochastic systems. These many OPTIONS help ensure that ASA can be used robustly across many classes of systems. \section[{IV.}]{IV.} \section[{Path-Integral Methodology a) Generic Applications}]{Path-Integral Methodology a) Generic Applications}\par Many systems are defined by (a) Fokker-Planck/Chapman -Kolmogorov partial -differential equations, (b) Langevin coupled stochastic-differential equations, and (c) Lagrangian or Hamiltonian pathintegrals. All three such systems of equations are equivalent mathematically, when limits of discretized variables are taken in the defined induced Riemannian geometry of the system due to nonlinear and time-dependent diffusions {\ref [Ingber(1982)}, {\ref Ingber(1983)} i \section[{. Path-Integral Algorithm}]{. Path-Integral Algorithm}\par In classical physics, the path integral of N variables indexed by ??,, at multiple times indexed by ??, is defined in terms of its Lagrangian ??:??[?? ?? |?? ?? 0 ]????(??) = ? ? ? ????exp ??min ? ?? ?? 0 ???? ? ??? ??(??(?? 0 ) = ?? 0 )??(??(??) = ?? ?? ) ???? = lim ?? ?? ? ?? +1 ??=1 ð??"ð??" 1/2 ? ?? (2??Î?"??) ?1/2 ???? ?? ?? ??(?????, ?? ?? , ??) = 1 2 (????? ? ð??"ð??" ?? )ð??"ð??" ???? ? (????? ? ? ð??"ð??" ?? ? ) + ??/6 ð??"ð??" ???? ? = (ð??"ð??" ???? ? ) ?1 , ð??"ð??" = det(ð??"ð??" ???? ? )\textbf{(7)}\par The diagonal diffusion terms are ð??"ð??" ???? and the drift terms are ð??"ð??" ?? . If the diffusions terms are non-constant, there are additional terms in the drift, and in a Riemannian-curvature potential R/6 for dimension > 1 in The path-integral approach is useful to give mathematical support to physically intuitive variables in the Lagrangian ??, \par Canonical Momenta Indicators (CMI = ? ?? ) were used successfully in neuroscience \hyperref[b22]{[Ingber (1996)}, \hyperref[b4]{Ingber(1997)}, {\ref Ingber(1998)}], combat analyses \hyperref[b4]{[Bowman Ingber(1997)}], and financial markets \hyperref[b22]{[Ingber(1996)}, Ingber ].\par The histogram algorithm numerically approximates sums of rectangles of height ?? ?? and width Î?"?? ?? at points ?? ?? . E.g., consider a one-dimensional system. In the prepoint Ito discretization, the pathintegral representation is given by the kernel ?? for each of its intermediate integrals, as??(??; ?? + Î?"??) = ? ???? ? [ð??"ð??" 1/2 (2??Î?"??) ?1/2 exp(??? Î?"??)]??(?? ? ; ??) = ? ???? ? ??(??, ?? ? ; Î?"??)??(?? ? ; ??) ??(??; ??) = ? ?? ??=1 ??(?? ? ?? ?? )?? ?? (??) (t) ??(?? ? ?? ?? ) = 1, (?? ?? ? 1 2 Î?"?? ???1 ) ? ?? ? (?? ?? + 1 2 Î?"?? ?? ); 0, otherwise\textbf{(9)}\par This yields?? ?? (?? + Î?"??) = ?? ???? (Î?"??)?? ?? (??)\textbf{(10)}\par ?? ???? is a banded matrix of the Gaussian short-time probability centered about the (possibly timedependent) drift. Explicit dependence of ?? on time ?? can be included. The mesh Î?"?? ?? is strongly dependent on diagonal elements of the diffusion matrix, e.g.,Î?"?? ?? ? (Î?"??ð??"ð??" ???? ) 1/2\textbf{(11)}\par The covariance of each variable is a (nonlinear) function all variables, presenting a rectangular mesh. Given that integration is a smoothing process [ {\ref Ingber(1990)}], fitting the data with integrals over the short-time probability distribution permits coarser meshes than the corresponding stochastic differential equation(s). The coarser resolution is appropriate for a numerical solution of the time-dependent path integral. Consideration of first and second moments yields conditions on the time and variable meshes \hyperref[b46]{[Wehner Wolfer(1983a)}]. A scan of the time slice, Î?"?? ? ?? ? ?1 where ?? ? is the uniform/static Lagrangian, gives most important contributions to the probability distribution ??. \section[{ii. Direct Kernel Evaluation}]{ii. Direct Kernel Evaluation}\par Several projects have used this algorithm \hyperref[b19]{[Ingber Nunez(1995)} \hyperref[b19]{[Ingber Nunez(1995)}], and Statistical Mechanics of Financial Markets (SMFM) {\ref [Ingber(2000a)}].\par The 1-dimensional PATHINT code was generalized by the author to N dimensions. Also, a quantum generalization was made, changing all variables and functions to complex variables, encompassing about 7500 lines of PATHINT code. The N-dimensional code was developed for classical and quantum systems \hyperref[b25]{[Ingber(2016a)}, Ingber(2017a), \hyperref[b27]{Ingber(2017b)}]. \section[{iii. Monte Carlo vs Kernels}]{iii. Monte Carlo vs Kernels}\par Path-integral numerical applications often use Monte Carlo techniques [O'Callaghan \hyperref[b38]{Miller(2014)}]. This includes the author's ASA code using ASA\textunderscore SAMPLE OPTIONS {\ref [Ingber(1993)}]. However, this project is concerned with time-sequential serial random shocks, which is not conveniently treated with Monte-Carlo/importance-sampling algorithms. T ij (?t) = 2 ?x i?1 + ?x i x i +?x i /2 x i ??x i?1 /2 dx x j +?x j /2 x j ??x j?1 /2 dx G(x, x ; ?t b) Quantum Path Integral Algorithms i.\par Scaling Issues qPATHINT has been tested with shocks to Ca 2+ waves [Ingber(2017b)], using the basic code also used for for quantum options on quantum money \hyperref[b26]{[Ingber(2017a)}]. This has illustrated computational scaling issues, further described in the Performance and Scaling Section. \section[{ii. Imaginary Time}]{ii. Imaginary Time}\par Imaginary-time Wick rotations permit imaginarytime to be transformed into real-time. Unfortunately, numerical calculations, after multiple foldings of the path integral, leaves no audit trail back to imaginary time to extract phase information (private communication with several authors of path-integral papers, including Larry Schulman on 18 Nov 2015) \hyperref[b43]{[Schulman(1981)}].\par V. \section[{SMNI with Qpathint}]{SMNI with Qpathint}\par This defines a process fitting EEG using SMNI with qPATHINT numerically calculating the Quantum path-integral between EEG epochs. At the beginning of each EEG epoch time is reset (t=0); the wave-function is decohered ("collapsed") by any EEG measurement. Until the end of any EEG epoch, there are multiple calls to SMNI functions to calculate the evolution of the Classical distribution. This replaces the author's Quantum path-integral closed-form time-dependent analytic solution. \section[{VI.}]{VI.} \section[{Performance and Scaling}]{Performance and Scaling}\par Code from a previous XSEDE grant "Electroencephalographic field influence on calcium momentum waves", is used for SMNI EEG fits. Code from a XSEDE previous XSEDE grant "Quantum pathintegral qPATHTREE and qPATHINT algorithm", is used for qPATHINT runs. \section[{a) Scaling Estimates}]{a) Scaling Estimates}\par Estimates XSEDE.org's Expanse using 'gcc -O3'. Expanse is described in https://www.sdsc.edu/support/user\textunderscore guides/expanse i. SMNI 100 ASA-iterations taking 7.12676s yields 0.0712676 sec/ASA-iteration over 2561 EEG epochs. With '-g' the total time is 29.9934s. \section[{ii. qPATHINT}]{ii. qPATHINT}\par The qPATHINT code uses a variable mesh covering 1121 points along the diagonal, with a maximum off-diagonal spread of 27. Corners require considerable CPU time to take care of boundaries. Oscillatory wave functions require a large off-diagonal spread \hyperref[b27]{[Ingber(2017b)}]. dt=0.0002 secs requires 10 foldings of the distribution. This takes the code 0.0002 secs/qIteraction. With '-g' the code takes 0.004s to run.\par iii. Projected Hours/Service Units (SUs) for this Project nSubjects x 2 (switch Train/Test) yields a 24array set of 1-node jobs.\par ASA-iterations x (SMNI\textunderscore time/ASA-iteration + nEpochs x qIterations x qPATHINT\textunderscore time/qIteration) yields 100,000 x (0.07 + 2500 x 10 x 0.0002) = 507,000 sec = 140 hr/run = 6 day/run. Time for Gaussian quadratures calculations is not appreciable: https://en.wikipedia.org/wiki/Gaussian\textunderscore quadrature Maximum duration of a normal job is 2 days. ASA has built in a simple way of ending jobs with printout required to restart, including sets of random numbers generated. \section[{VII.}]{VII.} \section[{Closed-Form Calculations}]{Closed-Form Calculations}\par Calculations on the Ookami supercomputer at StonyBrook.edu tested the path-integral derived analytic (closed-form) expression \hyperref[b28]{[Ingber(2018)}] for astrocyte Ca 2+ wave synaptic interactions< ?? > ð??"ð??" * ð??"ð??" = ?? ð??"ð??" * ð??"ð??" ?? ??? 0 = ??A +p 0 ?? 1/2 |Î?"r | ? (???) 2 +(??Î?"r 2 ) 2 ???+??Î?"r 2 ? 1/2 (12)\par Shocks were inserted into the mass ?? of the wave packet of 1000 ions, using a random number generator that contributed up to 1\% of the synaptic contribution due to Ca 2+ wave contributions. I.e., the mass was perturbed as ?? = ??(1 ? ??) + ????, where ?? = 0.1 and ?? is a random number between 0 and 1. The results are given in Table {\ref 1}.\par ASA was used with 200000 valid generated states for optimizations, then the modified Nelder-Mead code was used to sometimes gain extra precision with 5000 valid generated states. Training and Testing sets of data were used for 12 subjects, then the Training and Testing sets were exchanged. Runs were done with the added shocks and without these shocks. Each of the 48 runs took about 2 days on the Ookami supercomputer at StonyBrook.edu .\par As commented previously \hyperref[b28]{[Ingber(2018)}], "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."\par Volume XXI Issue VI Version I The number 2561 of EEG epochs is a region of high amplitude of times from 17 to 22 secs after the tasks began, defining epochs to be about 0.002 sec.\par Table 1: Comparison using analytic derivation without and with shocks. The subject numbers are given as sNN, and a "-X" represents exchanging Training and Testing sets of EEG data. Under Results an "I" represents an improvement of a lower cost functions with shocks versus no-shocks. An "N" represents better results with no-shocks versus shocks. These results show 16 "I"s and 8 "N"s, clearly in favor of shocks. \section[{Conclusion}]{Conclusion}\par A numerical path-integral methodology is used by SMNI to fit EEG to test quantum evolution of astrocyte-(re-)generated wave-packets of Ca 2+ ions that suffer shocks due collisions and regeneration of free ions. SMNI is generalized with quantum variables using qPATHINT.\par SMNI has fit experimental data, e.g., STM and EEG recordings under STM experimental paradigms. qPATHINT includes quantum scales in the SMNI model, evolving Ca 2+ wave-packets with momentum p serial shocks, interacting with the magnetic vector potential A due to EEG, via the (p + ??A) interaction at each node at each time slice ??, in time with experimental EEG data.\par Published pilot studies have given rationales for developing this particular quantum path-integral algorithm, to study serial random shocks that occur in many systems. This quantum version can be used for many quantum systems, which is increasingly important as experimental data is increasing at a rapid pace for many quantum systems. \begin{figure}[htbp] \noindent\textbf{} \par \begin{longtable}{P{0.8115691489361702\textwidth}P{0.020345744680851062\textwidth}P{0.011303191489361701\textwidth}P{0.006781914893617021\textwidth}} \multicolumn{4}{l}{https://docs.ocean.dwavesys.com/projects/hybrid/en/l}\\ atest/index.html\tabcellsep \tabcellsep \tabcellsep \\ \multicolumn{2}{l}{https://www.rigetti.com/what}\tabcellsep \tabcellsep \\ \multicolumn{3}{l}{https://azure.microsoft.com/en-us/solutions/hybrid-}\tabcellsep \\ cloud-app/\#overview\tabcellsep \tabcellsep \tabcellsep \\ \multicolumn{4}{l}{https://www.ibm.com/it-infrastructure/z/capabilities/}\\ hybrid-cloud\tabcellsep \tabcellsep \tabcellsep \\ These Classical\tabcellsep computers\tabcellsep often\tabcellsep run\\ \multicolumn{4}{l}{optimization program in systems that are described by}\\ \multicolumn{4}{l}{quantum variables using these companies' Quantum}\\ computers [\tabcellsep \tabcellsep \tabcellsep \end{longtable} \par {\small\itshape [Note: Introduction a) Hybrid Computing everal commercial Classical-Quantum computers now can be accessed via the Cloud, e.g., Rigetti, D-Wave, Microsoft, and IBM [Ingber(2021a)]; see]} \caption{\label{tab_0}}\end{figure} \begin{figure}[htbp] \noindent\textbf{} \par \begin{longtable}{P{0.2867469879518072\textwidth}P{0.2867469879518072\textwidth}P{0.2765060240963855\textwidth}} ,\tabcellsep Ingber(2012b),\tabcellsep Ingber(2015),\\ Ingber(2016b),\tabcellsep Ingber(2017b),\tabcellsep Ingber(2017c),\\ Ingber(2018),\tabcellsep \tabcellsep \end{longtable} \par \caption{\label{tab_2}}\end{figure} \begin{figure}[htbp] \noindent\textbf{} \par \begin{longtable}{} \end{longtable} \par \caption{\label{tab_3}}\end{figure} \begin{figure}[htbp] \noindent\textbf{} \par \begin{longtable}{P{0.85\textwidth}} 4\\ Citi \textasciitilde al. (2010)Citi, Poli, Cinel, Goldberger \textasciitilde al. (2000)Goldberger, Amaral, Glass, Hausdorff, Ivanov, Mark, Mietus, Moody, Peng, Stanley] http://physionet.nlm.nih.gov/pn4/erpbci As was done previously, fitting SMNI to highly synchronous waves (P300) during attention tasks, for each of 12 subjects, it is possible to find 10 Training runs and 10 Testing runs [Ingber(2016b)]. A region of continuous high amplitude of 2561 lines represents times from 17 to 22 secs after the tasks began. Spline-Volume XXI Issue VI Version I ( E )\end{longtable} \par \caption{\label{tab_4}}\end{figure} \footnote{( E )} \footnote{Year 2021 © 2021 Global Journals Hybrid Classical-Quantum Computing: Applications to Statistical Mechanics of Neocortical Interactions} \footnote{© 2021 Global Journals Hybrid Classical-Quantum Computing: Applications to Statistical Mechanics of Neocortical Interactions} \backmatter \subsection[{Acknowledgments}]{Acknowledgments}\par The author thanks the Extreme Science and Engineering Discovery Environment (XSEDE.org), for yearly supercomputer grants since February 2013. The author also thanks the Ookami supercomputer at StonyBrook.edu for its supercomputer resources for calculations.\par Note that in accordance with common practice, standalone website URLs are not necessarily referenced at the end, but often appear within the body of text. 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