Hybrid classical-quantum fitting attention states to statistical mechanics of neocortical interactions

Hybrid Classical-Quantum computing has already arrived at sev eral commercial quantum computers, offered to researchers and businesses. Here, 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, a path-integral algorithm, PATHINT, used previously to accurately describe several systems in several disciplines, has been generalized from 1 dimension to N dimensions, and from classical to quantum systems, qPATHINT. Published papers have described the use of 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+ wav es generated by astrocytes at tripartite neuron-astrocyte-neuron sites. leaving wav e packets should included using between electroencephalographic extends to multiple scales of interaction between classical ev ents quantum the quantum processes by qPATHINT. PI’s Adaptive Simulated Annealing importance-sampling optimization code fitting combined classical-quantum system. Gaussian is for numerical calculation of momenta expectations of the astrocyte processes that contribute to SMNI synaptic interactions. This project thereby demonstrates how some hybrid classical-quantum systems may be calculated quite well using only classical (super-)computers.


Organization of paper
Section 2 further describes SMNI in the context of this project. Section 3 further describes qPAT HINT in the context of this project. Section 4 describes howthe calculation proceeds between SMNI and qPAT HINT. Section 5 describes performance and scaling issues. Section 6 is the Conclusion.

Caveat
As stated previously in these projects (Ingber,2018), "The theory and codes for ASA and [q]PAT HINT have been well tested across manyd isciplines by multiple users. This particular project most certainly is speculative,b ut 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 noveltomanyphysics paradigms that are tested by experiment or computation."

qPATHINT For SMNI
The present qPAT HINT code determines howt he quantum regenerative process that defines Ca 2+ wavepackets also may produce reasonable shocks to the wav esw ithout seriously damaging its coherence properties. A proof of principal of its use has been published (Ingber,2017a).

Results Including Quantum Scales
Without random shocks, the wav e-function ψ e describing the interaction of A with p of Ca 2+ wavepackets was derivedi nc losed form from the Feynman representation of the path integral using pathintegral techniques (Schulten, 1999), modified here to include A.
where ψ 0 is the initial Gaussian wav e-packet, ψ F is the free-wav e ev olution operator, h is the Planck constant, q is the electronic charge of Ca 2+ ions, m is the mass of a wav e-packet of 1000 Ca 2+ ions, ∆r 2 is the spatial variance of the wav e-packet, the initial momentum is p 0 ,a nd the evolving canonical momentum is Π=p + qA.D etailed calculations showthat p of the Ca 2+ wave-packet and qA of the EEG field makeabout equal contributions to Π.

SMNI + Ca 2+ wave-packet
Tripartite influence on synaptic B G G′ is measured by the ratio of the wav e-packet's<p(t)> ψ *ψ to < p 0 (t 0 )> ψ *ψ at the onset of each attention task, measured just after each EEG measurement. Here <> ψ *ψ is taken over ψ * e ψ e .
A changes slower than p,s os tatic approximation of A used to derive ψ e and < p > ψ *ψ wasd eemed reasonable to use within P300 EEG epochs, resetting t = 0a tt he onset of each classical EEG measurement (1.953 ms apart), using the current-time A.T his permits tests of interactions across scales in a classical context. This result contains ht,thereby explicitly containing quantum h as well as explicit time-dependence t,and is therefore directly testable. Note that this analytic (closed-form) solution is replaced in this project by L.Ingber qPAT HINT,p ermitting shocks to astrocyte-(re-)generated Ca 2+ wave-packets to be included in the fits to EEG data.

SMNI Context
As hort description of the SMNI structure is required to understand howt he qPAT HINT development is used in this multiple-scale model (Ingber,2016b). After a statistical-mechanical aggregation of synaptic, neuronal and columnar scales (Ingber,1 982;Ingber,1983), the SMNI Lagrangian L in the prepoint (Ito) representation is where G ={ E, I }f or chemically independent excitatory and inhibitory synaptic interactions, N G = {N E = 160, N I = 60} was chosen for visual neocortex, {N E = 80, N I = 30} was chosen for all other neocortical regions, M G′ and N G′ in F G are afferent macrocolumnar firings scaled to efferent minicolumnar firings by N /N * ≈ 10 −3 ,and N *isthe number of neurons in a macrocolumn, about 10 5 . τ is usually considered to be on the order of 5−10 ms. V ′ includes nearest-neighbor mesocolumnar interactions (Ingber,1982;Ingber,1983). The threshold factor F G is derivedas where A G G′ is the columnar-averaged direct synaptic efficacy, B G G′ is the columnar-averaged backgroundnoise contribution to synaptic efficacy. A G G′ and B G G′ have been scaled by N */N ≈ 10 3 keeping F G invariant. Other values are consistent with experimental data, e.g., V G = 10 mV, v G G′ = 0. 1 mV, φ G G′ = 0. 03 1/2 mV.T he " ‡ "parameters arise from regional interactions across manymacrocolumns. ADynamic Centering Mechanism (DCM) model is used (Ingber,2016b), wherein the B G G′ are reset every fewe pochs of τ ,p arameterized to include contributions from tripartite neuron-astrocyte-neuron contributions. The DCM is consistent with experimental observations of shifts in synaptic activity during attention tasks (Mountcastle et al,1 981;Briggs et al,2 013). PAT HINT also has been successfully used with the SMNI Lagrangian L to calculate properties of STM for both auditory and visual memory (Ericsson & Chase, 1982;G. Zhang & Simon, 1985) calculating the stability and duration of STM, the observed 7 ± 2c apacity rule of auditory memory and the observed 4 ± 2c apacity rule of visual memory (Ingber,2000a;Ingber & Nunez, 1995).

Results
Using < p > ψ *ψ < p > ψ *ψ wasu sed in classical-physics SMNI fits to EEG data using ASA (Ingber,P appalepore & Stesiak, 2014; Ingber,2 016b). Runs using 1M or 100K generated states gav e results not much different. Training with ASA used 100K generated states over1 2s ubjects with and without A,f ollowed by 1000 generated states with the simplexl ocal code contained with ASA to check precision. Training and Testing runs on XSEDE.orgf or this project took an equivalent of several months of CPU on the XSEDE.orgU CSD San Diego Supercomputer (SDSC) platform Comet. These calculations used one additional parameter across all EEG regions to weight the contribution to synaptic background B G G′ . A is taken to be proportional to the currents measured by EEG, i.e., firings M G .O therwise, the "zero-fitparameter" SMNI philosophyw as enforced, wherein parameters are picked from experimentally determined values or within experimentally determined ranges (Ingber,1984). As with previous studies using this data, results sometimes gav e 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 qPAT HINT coupled algorithm described previously.N ote that ASA optimizations in this project always include "finishing off" ASA importance-sampling with a modified Nelder-Mead simplexc ode (included in the ASA code) to ensure best precision.

Assumptions forquantum SMNI
There are assumptions made for this quantum enhancement of SMNI that can only be determined by future experiments. In the context of quantum mechanics, the wav e-function of the Ca 2+ wave-packet was calculated, and it wasd emonstrated that overlap with multiple collisions, due to their regenerative processes, during the observed long durations of hundreds of ms of typical Ca 2+ waves ( Ingber,P appalepore & Stesiak, 2014;Ingber,2015;Ingber,2016b;Ingber,2017a;Ingber,2017c;Ingber,2018) support a Zeno or "bang-bang" effect (Facchi, Lidar & Pascazio, 2004;Facchi & Pascazio, 2008 Explicit calculations of reasonable repeated random shocks to the above wav e-function ψ F (t) demonstrated only small effects on the repeated projections of this wav e-packet after these shocks, i.e., the survivaltime was calculated (Facchi & Pascazio, 2008;Ingber,2018). 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 (Preskill, 2015). Here, the constant collisions of Ca 2+ ions as theyenter and leave the Ca 2+ wave-packet due to the regenerative process that maintains the wav e,may perpetuate at least part of the wav e,p ermitting the Zeno/"bang-bang" effect. In anyc ase, qPAT HINT as used here provides an opportunity to explore the coherence stability of the wav e due to serial shocks of this process.

Free Will
In addition to the intrinsic interest of researching STM and multiple scales of neocortical interactions via EEG data, there is interest in researching possible quantum influences on highly synchronous neuronal firings relevant to STM to understand possible connections to consciousness and "Free Will" (FW). As pointed out in some papers (Ingber,2 016a; Ingber,2 016b), if neuroscience evere stablishes experimental feedback from quantum-levelp rocesses of tripartite synaptic interactions with large-scale synchronous neuronal firings, that are nowrecognized as being highly correlated with STM and states of attention, then FW may yet be established using the quantum no-clone "Free Will Theorem" (FWT) (Conway & Kochen, 2006;Conway & Kochen, 2009). Basically,this means that a Ca 2+ quantum wav e-packet may generate a state proventohav e not previously existed; quantum states cannot be cloned. In the context of this basic premise, this state may be influential in a large-scale pattern of synchronous neuronal firings, thereby rendering this pattern as a truly newp attern not having previously existed. The FWT shows that this pattern, considered as a measurement of the Ca 2+ quantum wav e-packet, is correctly identified as itself being a newd ecision not solely based on previous decisions, evenu nder reasonably stochastic experimental and real-life conditions. These considerations require detailed calculations, which are quite different and independent of other philosophical considerations, e.g., as in https://plato.stanford.edu/entries/qt-consciousness/ .

qPATHINT For SMNI
An XSEDE.orggrant, "Quantum path-integral qPAT HTREE and qPAT HINT algorithms", was used to test and enhance applications of qPAT HTREE and qPAT HINT.T he wav e-function ψ is propagated for its initial state, numerically growing into a tree of wav e-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,iscalculated to determine changes due to time-dependent phenomena. Such changes occur at microscopic scales, e.g., due to modifications of the regenerative wav e-packet as ions leave and contribute to the wav e-packet, thereby determining the effect on tripartite contributions to neuron-astrocyte-neuron synaptic activity,a ffecting both p and A.S uch changes also may occur at macroscopic scales, e.g., changes due to external and internal stimuli affecting synchronous firings and thereby A.A te very time slice, quantum effects on synaptic interactions are determined by expected values of the interactions over probabilities (ψ * ψ )determined by the wav e-functions at their nodes. 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 expection of the momenta of the wav e-packet, i.e., of < p(t)> ψ *ψ .E .g., see https://en.wikipedia.org/wiki/Gaussian_quadrature

Comparing EEG Testing Data with Training Data
Using EEG data from (Goldberger et al,2000;Citi et al,2010) http://physionet.nlm.nih.gov/pn4/erpbci SMNI was fit to highly synchronous wav es( P300) during attention tasks, for each of 12 subjects, it was possible to find 10 Training runs and 10 Testing runs (Ingber,2 016b). A region of continuous high amplitude of 2561 lines represents times from 17 to 22 secs after the tasks began. Spline-Laplacian transformations on the EEG potential Φ are proportional to the SMNI M G firing variables at each electrode site. The electric potential Φ is experimentally measured by EEG, not A,b ut both are due to the same currents I.T herefore, A is linearly proportional to Φ with a simple scaling factor included as a parameter in fits to data. Additional parameterization of background synaptic parameters, B G G′ and B ‡E E′ ,modify previous work. The A model outperformed the no-A model, where the no-A model simply has used A-non-dependent synaptic parameters. Cost functions with an |A|m odel were much worse than either the A model or the no-A model. Runs with different signs on the drift and on the absolute value of the drift also gav e much higher cost functions than the A model.

Investigation into Spline-Laplacian Transformation
As is common practice, codes for the Spline-Laplacian transformations were applied to all electrodes measured on the scalp. However, the PI thinks that the transformation should be applied to each Region of neocortexs eparately (e.g., visual, auditory,s omatic, abstract, etc.), since each region typically participates in attention differently.T his process is tested in this project.

Summary
Previous publications have shown howE EG data, under experimental paradigms measuring attention states with varying conditions ranging from studies of alcoholism to different stimuli presented to subjects, can be used to fit the SMNI description of multiple scales of neocortical activity,e.g., including synaptic variables and parameters, collective columnar firings within large regions of neocortex, etc. Recent papers have included models of Ca 2+ -wav esm odifying parameterization of background SMNI synaptic parameters. Data was fit to this model using the PI'sA SA code (Ingber,1 993;Ingber,2 012a). These projects demonstrated that fits with this inclusion were better than fits without. Using qPAT HINT,q uantum scales of interaction can nowb ei ncluded. A reasonable measure of the influence of these scales is to fit real EEG data under other published experimental paradigms. This is accomplished using the above calculation of ψ (t)w ith realistic serial shocks, calculating updates to p(t) and A(t)a te ach time slice t corresponding to the EEG data for both training and testing sets of data. While PAT HINT could synchronously also be evolved using the SMNI Lagrangian, it likely is convenient and accurate to use the short-term SMNI Lagrangian as in previous studies to fit EEG and at each epoch.

Generic Applications
There are manys ystems that are well defined by (a) Fokker-Planck/Chapman-Kolmogorovp artialdifferential equations, (b) Langevin coupled stochastic-differential equations, and (c) Lagrangian or Hamiltonian path-integrals. All three such systems of equations are mathematically equivalent, when care is taken to properly takelimits of discretized variables in the well-defined induced Riemannian geometry of the system due to nonlinear and time-dependent diffusions (Schulman, 1981;Langouche et al,1 982;Ingber,1982;Ingber,1983).

Path-Integral Algorithm
The path integral of a classical system of N variables indexedb y i,a tm ultiple times indexedb y ρ,i s defined in terms of its Lagrangian L: Here the diagonal diffusion terms are g ii and the drift terms are g i .I ft he diffusions terms are not constant, then there are additional terms in the drift, and in a Riemannian-curvature potential R/6 for dimension > 1 in the midpoint Stratonovich/Feynman discretization (Langouche et al,1982). The path-integral approach is particularly useful to precisely define intuitive physical variables from the Lagrangian L in terms of its underlying variables q i : E.g., Canonical Momenta Indicators (CMI = Π i )w ere used successfully in neuroscience (Ingber, Ingber 1996;Ingber & Mondescu, 2001). The histogram procedure recognizes that the distribution can be numerically approximated to a high degree of accuracybysums of rectangles of height P i and width ∆q i at points q i .F or convenience only, just consider a one-dimensional system. In the prepoint Ito discretization, the path-integral representation can be written in terms of the kernel G,for each of its intermediate integrals, as P(x; t +∆t) = ∫ dx′[g 1/2 (2π ∆t) −1/2 exp(−L ∆t)]P(x′; t) = ∫ dx′G(x, x′; ∆t)P(x′; t) This yields T ij is a banded matrix representing the Gaussian nature of the short-time probability centered about the (possibly time-dependent) drift. Explicit dependence of L on time t also has been included without complications. Care must be used in developing the mesh ∆q i ,which is strongly dependent on diagonal elements of the diffusion matrix, e.g., This constrains the dependence of the covariance of each variable to be a (nonlinear) function of that variable to present an approximately rectangular underlying mesh. Since integration is inherently a smoothing process (Ingber,1 990), fitting the data with integrals overt he short-time probability distribution, this permits the use of coarser meshes than the corresponding stochastic differential equation(s). For example, the coarser resolution is appropriate, typically required, for a numerical solution of the time-dependent path integral. By considering the contributions to the first and second moments, conditions on the time and variable meshes can be derived( Wehner & Wolfer,1 983a). For non-zero drift, the time slice may be determined by a scan of ∆t ≤ L −1 ,w here L is the uniform/static Lagrangian, respecting ranges giving the most important contributions to the probability distribution P.

Scaling Issues
The use of qPAT HINT has been tested with shocks to Ca 2+ waves( Ingber,2 017a), using the same basic code used for "contrived" developed for quantum options on quantum money (Ingber,2017b), serving to illustrate some computational scaling issues, further described in the Performance and Scaling Section.

Imaginary Time
Imaginary-time Wick rotations transform imaginary time (the primary source of imaginary dependencies) into a real-variable time. However, when used with numerical calculations, after multiple foldings of the path integral, usually there is 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 Nov2 015) (Schulman, 1981).

SMNI With qPATHINT
The above considerations define a clear process of application of fitting EEG using SMNI, with qPAT HINT numerically calculating the path-integral at each time between EEG epochs. At the beginning of each EEG epoch, time is reset (t = 0) since the wav e-function is considered have been decohered ("collapsed") by a prior EEG measurement; until the end of that EEG epoch there are multiple calls to SMNI functions to calculate the evolution of the conditional short-time probability distribution, and each call also calls qPAT HINT for numerical calculation of the path-integral, instead of as in previous work using the PI'sderivedlong-time path-integral that has an analytic time-dependent closed-form solution.

Performance and Scaling
Code is used from XSEDE previous XSEDE grant "Electroencephalographic field influence on calcium momentum wav es", for SMNI EEG fits. Code is used from XSEDE previous XSEDE grant "Quantum path-integral qPAT HTREE and qPAT HINT algorithm", for qPAT HINT runs.

Scaling Estimates
Estimates belowwere made on Expanse using 'gcc -O3'. In this context, all debugging flags are not used (e.g., '-g') unless specifically noted otherwise, as recommended in https://www.sdsc.edu/support/user_guides/expanse 5.1.1. SMNI 100 ASA-iterations taking 7.12676s yields 0.0712676 sec/ASA-iteration over2 561 EEG epochs. Note that with '-g' the total time is 29.9934s. The number 2561 of EEG epochs, represents a region of continuous high amplitude of 2561 lines representing times from 17 to 22 secs after the tasks began. This sets time between epochs to be about 0.002 sec.

qPATHINT
The current code uses a variable mesh covering 1121 points along the diagonal, with a maximum offdiagonal spread of 27; corners require considerable CPU time to takecare of boundaries, etc. Oscillatory wave functions require a large off-diagonal spread (Ingber,2017a).
If dt = 0. 0002, this requires 10 foldings of the distribution. This takes the code 0.002s to run, giving 0.0002/qIteraction. Note that with '-g' the code takes 0.004s to run. L.Ingber

Projected SUs for This
Project nSubjects x 2 (switch Train/Test) yields a 24-array set of 1-node jobs. ASA-iterations x (SMNI_time/ASA-iteration + nEpochs x qIterations x qPAT HINT_time/qIteration) yields 100,000 x (0.07 + 2500 x 10 x 0.0002) = 507,000 sec = 140 hr/run = 6 day/run where time for the Gaussian quadratures calculations is not appreciable: https://en.wikipedia.org/wiki/Gaussian_quadrature Note the 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, so this is quite feasible.

XSEDE Ticket with Mahidhar Tatineni
In tickets.xsede.org#148054, Mahidhar Tatineni replied: "Thanks, Lester.T hat makes it clear.Ithink in this case using the "shared" partition with array jobs will be perfect and you can restart every 48 hours (makesure you say this in the proposal so that reviewers are aw are you can restart).
So you need a total of 24 x 140 = 3360 SUs for each set which is completely reasonable. If you do 100 such sets you will need ˜350K SUs which is completely fine from a request point of view(as long as the runs are justified for the science being done and there is a clear computational plan associated with it). Mahidhar"

Summary of ECSS outcome from 22 July 2020
Belowa re quotes from email correspondence concerning my Extended Collaborative Support Services (ECSS) grant to establish whether I have been using XSEDE resources optimally. Gujral, Madhusudan to Lester Dear Lester, Ic onveyed your your thanks to Mahi. He is great guy always ready to help evena to dd hours. He just communicated to me that node sharing will be available on Expanse as well. You should explain on Scaling and parallelization efficiencys ection that your application is not multi-threaded and you use single core on comet to run your jobs. This givese ff iciencyo f1 ,w hich is maximum value achievable. However, you run array of jobs in one submission and each job uses a single core. This is most efficient use of resources because node sharing is allowed on Comet. It won'thurt to write that you have consulted XSEDE staffonthis matter. There might be another unpleasant reason whyyour proposal gets criticism. People are given3-weeks to reviewp roposal and manyat imes people do not get to it till the last day.I ns uch cases, it get cursorily reviewed without attention to details and there is column in the section "is parallel efficiencyofthe code provided?" which has to be filled. There are situations when code runs on a single core and check column is redundant. Is hall bring up this matter with my manager,B ob Sinkovits, at SDSC when he returns from vacation next week.

Conclusion
Ak ernel numerical path-integral methodology is presented, to be used with SMNI to fit EEG as a test of the influence of quantum evolution of astrocyte-(re-)generated wav e-packets of Ca 2+ ions, that suffer shocks due collisions and regeneration of free ions. SMNI is generalized by including quantum variables using qPAT HINT.I tshould be noted that qPAT HINT requires large kernel bands for oscillatory states. The SMNI model has demonstrated it is faithful to experimental data, e.g., STM and EEG recordings under STM experimental paradigms. qPAT HINT nowp ermits a newi nclusion of quantum scales in the L.Ingber multiple-scale SMNI model, by evolving Ca 2+ wave-packets with momentum p,w ith serial shocks, interacting with the magnetic vector potential A derivedf rom EEG data, via a (p + qA)i nteraction, calculated at each node at each time slice t,m arching forward in time lock-step with experimental EEG data. Published pilot studies give a rationale for further developing this particular quantum path-integral algorithm based on folding kernels, as this can be used to study serial random shocks that occur in many real systems. Furthermore, this quantum version can be used for manyq uantum systems, which are becoming increasingly important as experimental data is increasing at a rapid pace for manyq uantum systems.