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\title{Hybrid Classical-Quantum Computing: Applications to Statistical Mechanics of Neocortical Interactions}
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             \author[1]{Lester  Ingber}

             \affil[1]{  Physical Studies Institute LLC}

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\date{\small \em Received: 10 April 2021 Accepted: 30 April 2021 Published: 15 May 2021}

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\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.}

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\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 \\
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\multicolumn{4}{l}{optimization program in systems that are described by}\\
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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. E.g., see https://libanswers.snhu.edu/faq/8627 https://www.scribbr.com/apa-examples/cite-a-website/ https://libguides.css.edu/APA7thEd/Webpages 			  			 \par
Computer Modelling 23 \hyperref[b3]{(3)}, 43-53. https:// www. ingber.com/path96\textunderscore duffing.pdf. 23.  {\ref [Ingber Wilson (1999)}] Ingber, L., Wilson, J.\par
(1999). Volatility of volatility of financial markets. Mathematical Computer Modelling 29 \hyperref[b5]{(5)}, 39-57. https://www.ingber.com/markets99\textunderscore vol.pdf. 24.  {\ref [Ingber Wilson(2000)}] Ingber, L., Wilson, J.  {\ref (2000)}.\par
Statistical mechanics of financial markets: Exponential modifications to black-scholes. 			 			  				\begin{bibitemlist}{1}
\bibitem[Illinois]{b45}\label{b45} 	 		\textit{},  		 			Illinois 		.  		 \url{http://www.ks.uiuc.edu/Services/Class/PHYS480/}  		Urbana, IL.  	 
\bibitem[Simon ()]{b50}\label{b50} 	 		\textit{},  		 			Zhang Simon 		.  		1985.  	 
\bibitem[Citi et al. ()]{b7}\label{b7} 	 		\textit{},  		 			Poli Citi 		,  		 			L Citi 		,  		 			R Poli 		,  		 			C Cinel 		.  		2010. 2010.  	 	 (Citi \textasciitilde al.) 
\bibitem[Salazar et al. ()]{b41}\label{b41} 	 		\textit{},  		 			Dotson Salazar 		,  		 			Gray ] Bressler 		,  		 			R Salazar 		,  		 			N Dotson 		,  		 			S Bressler 		,  		 			C Gray 		.  		2012. 2012.  	 	 (Salazar \textasciitilde al.) 
\bibitem[Alakus et al. ()]{b0}\label{b0} 	 		\textit{},  		 			Gonen Alakus 		,  		 			T Turkoglu] Alakus 		,  		 			M Gonen 		,  		 			I Turkoglu 		.  	 	 		\textit{References Références Referencias}  		2020. 2020. 1.  	 	 (Alakus \textasciitilde al.) 
\bibitem[Chakrabarti et al. ()]{b6}\label{b6} 	 		\textit{A threshold for quantum advantage in derivative pricing},  		 			S Chakrabarti 		,  		 			R Krishnakumar 		,  		 			G Mazzola 		,  		 			N Stamatopoulos 		,  		 			S Woerner 		,  		 			W Zeng 		.  		 arXiv:2012.03819v2.  		2020. New York, NY: Goldman Sachs.  	 	 (Technical Report) 	 (quant-ph) 
\bibitem[Ingber et al. ()]{b23}\label{b23} 	 		‘Adaptive simulated annealing’.  		 			L Ingber 		,  		 			H Oliveira 		,  		 			A Petraglia 		,  		 			L Ingber 		,  		 			M Machado 		.  		 \url{https://www.ingber.com/asa11\textunderscore options.pdf}  	 	 		\textit{Stochastic global optimization and its applications with fuzzy adaptive simulated annealing},  				 			M Petraglia 		 (ed.)  		 (New York)  		2012c. Springer. p. .  	 	 (Ingber(2012c). Invited Paper) 
\bibitem[Asher ()]{b2}\label{b2} 	 		\textit{Brain's code for visual working memory deciphered in monkeys NIHfunded study},  		 			J Asher 		.  		 NIH Press Release.  		 \url{http://www.nimh.nih.gov/news/science-news/2012/in-sync-brain-waves-hold-memory-of-objects-just-seen.shtml}  		2012. NIH, Bethesda, MD.  	 	 (Technical Report) 
\bibitem[Ingber ()]{b24}\label{b24} 	 		‘Calculating consciousness correlates at multiple scales of neocortical interactions’.  		 			L Ingber 		.  		 \url{https://www.ingber.com/smni15\textunderscore calc\textunderscore conscious.pdf}  	 	 		\textit{Horizons in Neuroscience Research},  				 			A Costa,  			E Villalba 		 (ed.)  		 (Nova, Hauppauge, NY)  		2015. p. .  	 	 (Ingber(2015). Invited paper.) 
\bibitem[Bowman and Ingber (1997)]{b4}\label{b4} 	 		‘Canonical momenta of nonlinear combat’.  		 			M Bowman 		,  		 			L Ingber 		.  		 \url{https://www.ingber.com/combat97\textunderscore cmi.pdf}  	 	 		\textit{Proceedings of the 1997 Simulation Multi-Conference},  				 (the 1997 Simulation Multi-ConferenceAtlanta, GA; San Diego, CA)  		1997. 1997. 6-10 April 1997. Bowman Ingber. p. .  	 	 (Society for Computer Simulation) 
\bibitem[Content-specific fronto-parietal synchronization during visual working memory Science]{b42}\label{b42} 	 		‘Content-specific fronto-parietal synchronization during visual working memory’.  		 \xref{http://dx.doi.org/10.1126/science}{10.1126/science}.  		 \url{https://dx.doi.org/10.1126/science.1224000}  	 	 		\textit{Science}  		338  (6110)  p. .  	 
\bibitem[[murakami Okad ; Murakami and Okada ()]{b35}\label{b35} 	 		‘Contributions of principal neocortical neurons to magnetoencephalography and electroencephalography signals’.  		 			S [murakami Okad ; Murakami 		,  		 			Y Okada 		.  	 	 		\textit{Journal of Physiology}  		2006. 2006. 575  (3)  p. .  	 
\bibitem[Database for an emotion recognition system based on EEG signals andvarious computer games -GAMEEMO Biomedical Signal Processing and Control (101951)]{b1}\label{b1} 	 		‘Database for an emotion recognition system based on EEG signals andvarious computer games -GAMEEMO’.  		 \xref{http://dx.doi.org/10.1016/j.bspc.2020.101951}{10.1016/j.bspc.2020.101951}.  		 \url{https://doi.org/10.1016/j.bspc.2020.101951}  	 	 		\textit{Biomedical Signal Processing and Control}  		101951. 60 p. .  	 
\bibitem[Documenting, modelling and exploiting P300 amplitude changes due to variable target delays in Donchin's speller Journal of Neural Engineering]{b8}\label{b8} 	 		‘Documenting, modelling and exploiting P300 amplitude changes due to variable target delays in Donchin's speller’.  		 \xref{http://dx.doi.org/10.1088/1741-2560/7/5/056006}{10.1088/1741-2560/7/5/056006}.  		 \url{https://dx.doi.org/10.1088/1741-2560/7/5/056006}  	 	 		\textit{Journal of Neural Engineering}  		7  (056006)  p. .  	 
\bibitem[[nunez Srinivasan ; Nunez and Srinivasan ()]{b37}\label{b37} 	 		\textit{Electric Fields of the Brain: The Neurophysics of EEG},  		 			P [nunez Srinivasan ; Nunez 		,  		 			R Srinivasan 		.  		2006. 2006. London: Oxford University Press.  	 	 (2nd Ed) 
\bibitem[Electroencephalographic field influence on calcium momentum waves Journal of Theoretical Biology]{b21}\label{b21} 	 		‘Electroencephalographic field influence on calcium momentum waves’.  		 \xref{http://dx.doi.org/10.1016/j.jtbi.2013.11.002}{10.1016/j.jtbi.2013.11.002}.  		 \url{https://doi.org/10.1016/j.jtbi.2013.11.002}  	 	 		\textit{Journal of Theoretical Biology}  		343 p. .  	 
\bibitem[Ingber ()]{b27}\label{b27} 	 		‘Evolution of regenerative ca-ion wave-packet in neuronal-firing fields: Quantum path-integral with serial shocks’.  		 			L Ingber 		.  		 \url{https://www.ingber.com/path17\textunderscore quantum\textunderscore pathint\textunderscore shocks.pdf.51}  	 	 		\textit{International Journal of Innovative Research in Information Security}  		2017b. 4  (2)  p. .  	 	 (Ingber(2017b). Ingber(2017c)) 
\bibitem[Ericsson and Chase ()]{b11}\label{b11} 	 		‘Exceptional memory’.  		 			K Ericsson 		,  		 			W Chase 		.  	 	 		\textit{American Scientist}  		1982. 1982. 70 p. .  	 
\bibitem[Ingber ()]{b32}\label{b32} 	 		‘Forecasting COVID-19 with importance-sampling and pathintegrals’.  		 			L Ingber 		.  		 \url{https://www.ingber.com/asa20\textunderscore covid.pdf.57}  	 	 		\textit{WSEAS Transactions on Biology and Biomedicine}  		2021d. 1987.  	 	 (Ingber(2021d)) 
\bibitem[Langouche et al. ()]{b34}\label{b34} 	 		‘Functional Integration and Semiclassical Expansions’.  		 			Roekaerts Langouche 		,  		 			F Tirapegui] Langouche 		,  		 			D Roekaerts 		,  		 			E Tirapegui 		,  		 			Muller 		,  		 			Gherardini 		,  		 			M Caruso] Muller 		,  		 			S Gherardini 		,  		 			F Caruso 		.  		 arXiv:1607.08871v1.  	 	 		\textit{Reidel}  		1982. 1982. 2016. 2016. 60.  	 	 (Technical Report) 	 (Quantum zeno dynamics through stochastic protocols. quant-ph) 
\bibitem[Ingber ()]{b31}\label{b31} 	 		‘Hybrid classicalquantum computing: Applications to statistical mechanics of financial markets’.  		 			L Ingber 		.  		 2021: CQCA.  		 \url{https://www.ingber.com/markets21\textunderscore hybrid\textunderscore smfm.pdf}  	 	 		\textit{Physical Studies Institute}  		2021c.  	 	 (Technical Report) 	 (Ingber(2021c)) 
\bibitem[Ingber et al. ()]{b17}\label{b17} 	 		‘Mathematical comparison of combat computer models to exercise data’.  		 			Chen Ingber 		,  		 			Mondescu 		,  		 			Muzzall 		,  		 			L Ingber 		,  		 			C Chen 		,  		 			R Mondescu 		,  		 			D Muzzall 		,  		 			M Renedo 		,  		 			Ingber 		,  		 			Fujio 		,  		 			L Wehner] Ingber 		,  		 			H Fujio 		,  		 			M Wehner 		.  		 \url{https://www.ingber.com/combat91\textunderscore data.pdf}  	 	 		\textit{Mathematical Computer Modelling}  		2001. 2001. 1991. 1991. 64  (5)  p. .  	 	 (Physical Review E) 
\bibitem[Miller et al. ()]{b38}\label{b38} 	 		‘Measurement-induced localization of an ultracold lattice gas’.  		 			[o'callaghan Miller 		,  		 			; O'callaghan 		,  		 			M Miller 		,  		 			B Patil 		,  		 			Chakram 		,  		 			Y Vengalattore] Patil 		,  		 			S Chakram 		,  		 			M Vengalattore 		.  		 \xref{http://dx.doi.org/10.1103/PhysRevE.89.0421265}{10.1103/PhysRevE.89.0421265}.  		 \url{http://link.aps.org/doi/10.1103/PhysRevLett.115.140402}  	 	 		\textit{Physical Review Letters}  		2014. 2014. 042124. 2015. 2015. 140402. 89 p. .  	 	 (Physical Review E) 
\bibitem[Ingber et al. ()]{b20}\label{b20} 	 		‘Neocortical dynamics at multiple scales: EEG standing waves, statistical mechanics, and physical analogs’.  		 			L Ingber 		,  		 			P Nunez 		,  		 			Ingber 		,  		 			Pappalepore 		,  		 			L Stesiak] Ingber 		,  		 			M Pappalepore 		,  		 			R Stesiak 		.  		 \url{https://www.ingber.com/smni10\textunderscore multiple\textunderscore scales.pdf21}  	 	 		\textit{Mathematical Biosciences}  		2010. 2014. 2014. 229 p. .  	 	 (Ingber \textasciitilde al.) 
\bibitem[Jensen et al. ()]{b33}\label{b33} 	 		‘Non-hermitian dynamics in the quantum zeno limit’.  		 			A Jensen 		,  		 			Caballero Kozlowski 		,  		 			Benitez 		,  		 			] Mekhov 		,  		 			W Kozlowski 		,  		 			S Caballero-Benitez 		,  		 			I Mekhov 		.  		 arXiv1510.04857.  	 	 		\textit{Speed of Information-Processing and Intelligence},  				 			P Vernon 		 (ed.)  		 (Ablex, Norwood, NJ; Oxford, Oxford, UK)  		1987. 2015. 2015. 58 p. .  	 	 (Technical Report) 	 (Individual differences in the hick paradigm. quant-ph) 
\bibitem[[wehner Wolfer ; Wehner and Wolfer ()]{b48}\label{b48} 	 		‘Numerical evaluation of path integral solutions to fokker-planck equations. III. time and functionally dependent coefficients’.  		 			M [wehner Wolfer ; Wehner 		,  		 			W Wolfer 		.  	 	 		\textit{Physical Review A}  		1987. 1987. 35 p. .  	 
\bibitem[[wehner Wolfer ; Wehner and Wolfer ()]{b46}\label{b46} 	 		‘Numerical evaluation of path-integral solutions to fokker-planck equations. I’.  		 			M [wehner Wolfer ; Wehner 		,  		 			W Wolfer 		.  	 	 		\textit{Physical Review A}  		1983a. 1983a. 27 p. .  	 
\bibitem[[wehner Wolfer ; Wehner and Wolfer ()]{b47}\label{b47} 	 		‘Numerical evaluation of path-integral solutions to fokker-planck equations. II. restricted stochastic processes’.  		 			M [wehner Wolfer ; Wehner 		,  		 			W Wolfer 		.  	 	 		\textit{Physical Review A}  		1983b. 1983b. 28 p. .  	 
\bibitem[Hick ()]{b16}\label{b16} 	 		‘On the rate of gains of information’.  		 			W Hick 		.  	 	 		\textit{Quarterly Journal Experimental Psychology}  		1952. 34  (4)  p. .  	 	 (Hick (1952)) 
\bibitem[Ingber and Mondescu ()]{b18}\label{b18} 	 		‘Optimization of trading physics models of markets’.  		 			L Ingber 		,  		 			R Mondescu 		.  		 \url{https://www.ingber.com/markets01\textunderscore optim\textunderscore trading.pdf}  	 	 		\textit{Neural Networks in Financial Engineering}  		2001. 12  (4)  p. .  	 	 (IEEE Transactions Neural Networks) 
\bibitem[Benedetti et al. ()]{b3}\label{b3} 	 		‘Parameterized quantum circuits as machine learning models’.  		 			Lloyd Benedetti 		,  		 			Sack 		,  		 			M Benedetti 		,  		 			E Lloyd 		,  		 			S Sack 		,  		 			M Fiorentini 		.  		 \xref{http://dx.doi.org/10.1088/2058-9565/ab5944}{10.1088/2058-9565/ab5944}.  		 \url{https://doi.org/10.1088/2058-9565/ab5944}  	 	 		\textit{Quantum Science and Technology}  		2019. 2019. 4  (4) .  	 	 (Benedetti \textasciitilde al.) 
\bibitem[Ingber ()]{b25}\label{b25} 	 		‘Path-integral quantum PATHTREE and PATHINT algorithms’.  		 			L Ingber 		.  		 \url{https://www.ingber.com/path16\textunderscore quantum\textunderscore path.pdf}  	 	 		\textit{International Journal of Innovative Research in Information Security}  		2016a. 3  (5)  p. .  	 	 (Ingber(2016a)) 
\bibitem[Ingber et al. ()]{b22}\label{b22} 	 		‘Pathintegral evolution of chaos embedded in noise: Duffing neocortical analog. Mathematical 44’.  		 			Ingber 		,  		 			Srinivasan 		,  		 			L Nunez] Ingber 		,  		 			R Srinivasan 		,  		 			P Nunez 		,  		 			L Ingber 		.  		 \url{https://www.ingber.com/smni12\textunderscore vectpot.pdf}  	 	 		\textit{Current Progress Journal}  		1996. 1996. 2012b. 1  (1)  p. .  	 	 (Influence of macrocolumnar EEG on ca waves) 
\bibitem[Goldberger et al. ()]{b15}\label{b15} 	 		‘Physio Bank, PhysioToolkit, and Physio Net: components of a new research resource for complex physiologic signals’.  		 			Amaral Goldberger 		,  		 			Glass 		,  		 			Hausdorff 		,  		 			Ivanov 		,  		 			Mark 		,  		 			Mietus 		,  		 			Moody 		,  		 			Stanley ] Peng 		,  		 			A Goldberger 		,  		 			L Amaral 		,  		 			L Glass 		,  		 			J Hausdorff 		,  		 			P Ivanov 		,  		 			R Mark 		,  		 			J Mietus 		,  		 			G Moody 		,  		 			C K Peng 		,  		 			H Stanley 		.  	 	 		\textit{Circulation}  		2000. 2000. 101  (23)  p. .  	 	 (Goldberger \textasciitilde al.) 
\bibitem[Ingber ()]{b30}\label{b30} 	 		‘Quantum calcium-ion affective influences measured by EEG’.  		 			L Ingber 		.  		 \url{https://www.ingber.com/quantum20\textunderscore affective.pdf}  	 	 		\textit{International Journal Physics and Space Science}  		2021b.  	 	 (Ingber(2021b)) 
\bibitem[Schulten ()]{b44}\label{b44} 	 		‘Quantum mechanics’.  		 			K Schulten 		.  	 	 		Lecture Notes  		1999. 1999.  	 	 (Technical Report) 
\bibitem[Preskill ()]{b39}\label{b39} 	 		\textit{Quantum mechanics},  		 			J Preskill 		.  		 Lecture Notes.  		 \url{http://www.theory.caltech.edu/people/preskill/ph219/}  		2015. 2015. Caltech, Pasadena, CA.  	 	 (Technical Report) 
\bibitem[Ingber and Ingber ()]{b28}\label{b28} 	 		‘Quantum path-integral qPATHINT algorithm’.  		 			L Ingber 		,  		 			L Ingber 		.  		 \xref{http://dx.doi.org/10.2174/1874110X01711010119.52}{10.2174/1874110X01711010119.52}.  		 \url{https://www.ingber.com/smni18\textunderscore quantumCaEEG.pdfandhttps://doi.org/10.3390/sci1010020}  	 	 		\textit{The Open Cybernetics Systemics Journal}  		2017c. 2018. 2018. 11  (7)  p. .  	 	 (EEG. Sci) 
\bibitem[Burgarth et al. ()]{b5}\label{b5} 	 		\textit{Quantum zeno dynamics from general quantum operations},  		 			Facchi Burgarth 		,  		 			Nakazato 		,  		 			Pascazio 		,  		 			D Burgarth 		,  		 			P Facchi 		,  		 			H Nakazato 		,  		 			S Pascazio 		,  		 			K Yuasa 		.  		 arXiv:1809.09570v1.  		2018. 2018. Aberystwyth, UK: Aberystwyth U.  	 	 (Technical Report) 	 (Burgarth \textasciitilde al.. quant-ph) 
\bibitem[[facchi Pascazio ; Facchi and Pascazio ()]{b13}\label{b13} 	 		‘Quantum zeno dynamics: mathematical and physical aspects’.  		 			P [facchi Pascazio ; Facchi 		,  		 			S Pascazio 		.  	 	 		\textit{Journal of Physics A}  		2008. 2008. 41  (493001)  p. .  	 
\bibitem[Giacosa and Pagliara ()]{b14}\label{b14} 	 		‘Quantum zeno effect by general measurements’.  		 			G Giacosa 		,  		 			G Pagliara 		.  	 	 		\textit{Giacosa Pagliara},  				2014. 2014. 052107 p. .  	 
\bibitem[Ingber and Nunez ()]{b19}\label{b19} 	 		‘Statistical mechanics of neocortical interactions: High resolution path-integral calculation of short-term memory’.  		 			L Ingber 		,  		 			P Nunez 		.  		 \url{https://www.ingber.com/smni95\textunderscore stm.pdf.20}  	 	 		\textit{Physical Review E}  		1995. 51  (5)  p. .  	 	 (Ingber Nunez (1995). Ingber Nunez (2010)) 
\bibitem[Ingber and Ingber ()]{b26}\label{b26} 	 		‘Statistical mechanics of neocortical interactions: Large-scale EEG influences on molecular processes’.  		 			L Ingber 		,  		 			L Ingber 		.  		 \xref{http://dx.doi.org/10.1016/j.jtbi.2016.02.003.49}{10.1016/j.jtbi.2016.02.003.49}.  		 \url{https://www.ingber.com/path17\textunderscore quantum\textunderscore options\textunderscore shocks.pdf}  	 	 		\textit{International Journal of Innovative Research in Information Security}  		2016b. 2017a. 395  (2)  p. .  	 	 (Journal of Theoretical Biology) 
\bibitem[Zhang and Simon ()]{b51}\label{b51} 	 		‘STM capacity for chinese words and idioms: Chunking and acoustical loop hypotheses’.  		 			G Zhang 		,  		 			H Simon 		.  	 	 		\textit{Memory \& Cognition}  		1985. 13 p. .  	 
\bibitem[Schulman ()]{b43}\label{b43} 	 		\textit{Techniques and Applications of Path Integration},  		 			L Schulman 		.  		1981. 1981. New York: J. Wiley \& Sons.  	 
\bibitem[Ingber (ed.) ()]{b29}\label{b29} 	 		\textit{Technological Breakthroughs and Future Business Opportunities in Education, Health, and Outer Space},  		 			L Ingber 		.  		 \url{https://www.ingber.com/quantum20\textunderscore techcast.pdf}  		Hooke, A. (ed.)  		2021a. Hershey, PA: IGI Global. p. .  	 	 (Revisiting our quantum world) 
\bibitem[Conway Kochen ; Conway and Kochen ()]{b9}\label{b9} 	 		\textit{The free will theorem},  		 			[ Conway Kochen ; Conway 		,  		 			J Kochen 		,  		 			S 		.  		 arXiv:quant-ph/0604079.  		2006. 2006. Princeton U, Princeton, NJ.  	 	 (Technical Report) 	 (quant-ph) 
\bibitem[Conway Kochen ; Conway and Kochen ()]{b10}\label{b10} 	 		‘The strong free will theorem’.  		 			[ Conway Kochen ; Conway 		,  		 			J Kochen 		,  		 			S 		.  	 	 		\textit{Notices of the American Mathematical Society}  		2009. 2009. 56  (2)  p. .  	 
\bibitem[Nunez et al. ()]{b36}\label{b36} 	 		‘Theoretical and experimental electrophysiology in human neocortex: Multiscale correlates of conscious experience’.  		 			Nunez 		,  		 			Srinivasan 		,  		 			P Ingber] Nunez 		,  		 			R Srinivasan 		,  		 			L Ingber 		.  		 \xref{http://dx.doi.org/10.1002/9783527671632.ch06}{10.1002/9783527671632.ch06}.  		 \url{https://doi.org/10.1002/9783527671632.ch06}  	 	 		\textit{Multiscale Analysis and Nonlinear Dynamics: From genes to the brain},  				 			M Pesenson 		 (ed.)  		 (New York)  		2013. 2013. Wiley. p. .  	 	 (Nunez \textasciitilde al.) 
\bibitem[Wu et al. ()]{b49}\label{b49} 	 		‘Time-dependent decoherence-free subspace’.  		 			Wang Wu 		,  		 			Yi ] Wu 		,  		 			S Wang 		,  		 			L Yi 		,  		 			X 		.  	 	 		\textit{Journal of Physics A}  		2012. 2012. 405305 p. .  	 	 (Wu \textasciitilde al.) 
\bibitem[Ross ()]{b40}\label{b40} 	 		‘Understanding calcium waves and sparks in central neurons’.  		 			W Ross 		.  	 	 		\textit{Nature Reviews Neuroscience}  		2012. 13 p. .  	 	 (Ross (2012) 
\bibitem[Facchi et al. ()]{b12}\label{b12} 	 		‘Unification of dynamical decoupling and the quantum zeno effect’.  		 			Lidar Facchi 		,  		 			P Pascazio] Facchi 		,  		 			D Lidar 		,  		 			S Pascazio 		.  	 	 		\textit{Physical Review A}  		2004. 2004. 032314. 69 p. .  	 	 (Facchi \textasciitilde al.) 
\end{bibitemlist}
 			 		 	 
\end{document}
