The project Statistical Mechanics of Neocortical Interactions (SMNI) has been developed in over 30+ papers since 1981, scaling aggregate synaptic interactions to describe neuronal firings, then scaling minicolumnar-macrocolumnar columns of neurons to mesocolumnar dynamics, and then scaling columns of neuronal firings to regional (sensory) macroscopic sites identified in electroencephalographic (EEG) studies [1L. Ingber, "Statistical mechanics of neocortical interactions. I. basic formulation", Physica D, vol. 5, pp. 83-107. Available from: https://www.ingber.com/ smni82_basic.pdf
[http://dx.doi.org/10.1016/0167-2789(82)90052-5] -5L. Ingber, "Statistical mechanics of neocortical interactions: Path-integral evolution of short-term memory", Physical Review E, vol. 49, no. 5, pp. 4652-4664. Available from: https://www.ingber.com/smni94_stm.pdf
[http://dx.doi.org/10.1103/PhysRevE.49.4652] [PMID: 9961760] ].

The measure of the success of SMNI has been to discover agreement/fits with experimental data from various modeled aspects of neocortical interactions, e.g., properties of short-term memory (STM) [6L. Ingber, “Columnar EEG magnetic influences on molecular development of short-term memory,” In: Short-Term Memory: New Research, G. Kalivas and S. Petralia, Eds., NY: Nova, 2012, pp. 37-72. Available from: https://www.ingber.com/smni11_stm_scales.pdf], including its capacity (auditory 7±2 and visual 4±2), duration, stability, primacy versus recency rule, as well other phenomenon, e.g., Hick’s law [7W. Hick, "On the rate of gains of information", Quarterly Journal Experimental Psychology, vol. 34, no. 4, pp. 1-33.-9A. Jensen, Individual differences in the hick paradigm.Speed of Information-Processing and Intelligence., Ablex: Norwood, NJ, pp. 101-175.], nearest-neighbor minicolumnar interactions within macrocolumns calculating rotation of images, etc [1L. Ingber, "Statistical mechanics of neocortical interactions. I. basic formulation", Physica D, vol. 5, pp. 83-107. Available from: https://www.ingber.com/ smni82_basic.pdf
[http://dx.doi.org/10.1016/0167-2789(82)90052-5] -5L. Ingber, "Statistical mechanics of neocortical interactions: Path-integral evolution of short-term memory", Physical Review E, vol. 49, no. 5, pp. 4652-4664. Available from: https://www.ingber.com/smni94_stm.pdf
[http://dx.doi.org/10.1103/PhysRevE.49.4652] [PMID: 9961760] ]. SMNI was also scaled to include mesocolumns across neocortical regions to fit EEG data [10L. Ingber, "Statistical mechanics of neocortical interactions: Applications of canonical momenta indicators to electroencephalography", Physical Review E, vol. 55, no. 4, pp. 4578-4593. Available from: https://www.ingber.com/smni97_cmi.pdf
[http://dx.doi.org/10.1103/PhysRevE.55.4578] , 6L. Ingber, “Columnar EEG magnetic influences on molecular development of short-term memory,” In: Short-Term Memory: New Research, G. Kalivas and S. Petralia, Eds., NY: Nova, 2012, pp. 37-72. Available from: https://www.ingber.com/smni11_stm_scales.pdf].

The project Statistical Mechanics of Financial Markets (SMFM) has been developed to model financial systems [17L. Ingber, "Statistical mechanics of nonlinear nonequilibrium financial markets", Mathematical Modelling, vol. 5, no. 6, pp. 343-361. Available from: https://www.ingber.com/markets84_statmech.pdf
[http://dx.doi.org/10.1016/0270-0255(84)90022-8] -21L. Ingber, "Automated internet trading based on optimized physics models of markets", Intelligent Internet-Based Information Processing Systems, .https://www.ingber.com/markets03_automated.pdf
[http://dx.doi.org/10.1142/9789812795342_0009] ].

There are many systems that are well defined by (a) Fokker-Planck/Chapman-Kolmogorov partial-differential 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 take limits of discretized variables in the well-defined induced Riemannian geometry of the system due to nonlinear and time-dependent diffusions [32F. Langouche, D. Roekaerts, and E. Tirapegui, Functional Integration and Semiclassical Expansions., D. Reidel: Dordrecht, The Netherlands, .
[http://dx.doi.org/10.1007/978-94-017-1634-5] , 33L. Schulman, Techniques and Applications of Path Integration., J. Wiley and Sons: New York, .].

The path integral of a classical system of N variables indexed by i at multiple times indexed by ρ is defined in terms of its Lagrangian L:

(1)

Here the diagonal diffusion terms are gⁱⁱ and the drift terms are g_i = -∂Φ/∂qⁱ. If the 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 [32F. Langouche, D. Roekaerts, and E. Tirapegui, Functional Integration and Semiclassical Expansions., D. Reidel: Dordrecht, The Netherlands, .
[http://dx.doi.org/10.1007/978-94-017-1634-5] ].

The path-integral approach is particularly useful to precisely define intuitive physical variables from the Lagrangian L in terms of its underlying variables qⁱ:

(2)

The histogram procedure recognizes that the distribution can be numerically approximated to a hight degree of accuracy by sums of rectangles of height Pⁱ and width Δqⁱ at points qⁱ . For convenience 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

(3)

This yields

(4)

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 can be included without complications. Care must be used in developing the mesh Δqⁱ, which is strongly dependent on diagonal elements of the diffusion matrix, e.g.,

(5)

This constrains the dependence of the covariance of each variable to be a (nonlinear) function of that variable to present a rectangular-ish underlying mesh. Since integration is inherently a smoothing process [34L. Ingber, "Statistical-mechanical aids to calculating term-structure models", Physical Review A, vol. 42, no. 12, pp. 7057-7064. Available from: https://www.ingber.com/markets90_interest.pdf
[http://dx.doi.org/10.1103/PhysRevA.42.7057] [PMID: 9904019] ], fitting the data with integrals over the 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 [35M. Wehner, and W. Wolfer, "Numerical evaluation of path-integral solutions to Fokker-Planck equations. I", Physical Review A, vol. 27, pp. 2663-2670.
[http://dx.doi.org/10.1103/PhysRevA.27.2663] ]. For non-zero drift, the time slice may be determined by a scan of , where is the uniform/static Lagrangian, respecting ranges giving the most important contributions to the probability distribution P. Thus Δt can be measured by the diffusion divided by the square of the drift.

Many path-integral numerical applications use Monte Carlo techniques [42M. O’Callaghan, and B.N. Miller, "Path integral Monte Carlo on a lattice: extended states", Physical Review E, vol. 89, no. 4, p. 042124.
[http://dx.doi.org/10.1103/PhysRevE.89.042124] [PMID: 24827210] ]. This approach includes the author’s Adaptive Simulated Annealing (ASA) code using its ASA_SAMPLE OPTIONS [43L. Ingber, “Adaptive simulated annealing (ASA),” Caltech Alumni Association, Pasadena, CA, Technical Report, Global optimization C-code, 1993. Available from: https://www.ingber.com/#ASA-CODE]. This project is concerned with serial (time-sequential) random shocks, not conveniently treated with Monte-Carlo/importance-sampling algorithms.

A “contrived” qPATHINT code was developed for quantum options, which serves to illustrate some computational scaling issues, using data and parameters from dimension 1 for all N dimensions.

The volume of the kernel required for computation is measured by [IJ]^D, where D = dimension, I is the number of elements in the diagonal (or the drift-shifted diagonal in some cases), J is the width of the off-diagonal. Many loops in the code also have D-dependent inner loops. J has been as high as 13, requiring a span of 13+13+1 = 27 elements, needed for some oscillatory systems in previous studies [14L. Ingber, "Evolution of regenerative ca-ion wave-packet in neuronal-firing fields: Quantum path-integral with serial shocks", International Journal of Innovative Research in Information Security, vol. 4, no. 2, pp. 14-22. Available from: https://www.ingber.com/ path17_quantum_pathint_shocks.pdf]; in contrast, currently [q]PATHTREE is only a binary tree with J. Note that scales as sqrt(E), where E = number epochs in [t₀ , T] (the diffusion is proportional to dt^1/2). I is determined in each dimension, by dqⁱ that must span [q_min, q_max], where dq_i = (dtgⁱⁱ)^1/2,dt = (T -t₀ )/E.

Examples for this paper were done with with J = 3, T = 1.5, t₀ = 0, q_min = 1, q_max = 12, I = imxall J = jmxall Prob_Mesh_Size=ijkcnt=number-nonzero-elements. For larger J and/or larger E, it is possible to decrease E and T, such that fewer meshes can be developed, and qPATHINT would propagate from t₀ to T_1, from T_1 to T_2, ..., from T_s-1 to T_s=T, with write/read of probabilities/wave-functions developed at each intermediate T_[].

Runs were done for 16 Strikes for an American option [44J. Hull, Options, Futures, and Other Derivatives., 4th edPrentice Hall: Upper Saddle River, NJ, .] (each Strike require independent propagation of the distribution), requiring a total of 1364 foldings.

To appreciate requirements of kernel memory as a function of dimension, for the mesh considered in this pilot study,

D=1:imxall: 27, jmxall: 7, ijkcnt: 189

D=2:imxall: 729, jmxall: 49, ijkcnt: 35721

D=3:imxall: 19683, jmxall: 343, ijkcnt: 6751269

D=4:imxall: 531441, jmxall: 2401, ijkcnt: 1275989841

D=5:imxall: 14348907, jmxall: 16807, ijkcnt: 241162079949

D=6:imxall: 387420489, jmxall: 117649, ijkcnt: 45579633110361

D=7:imxall: 10460353203, jmxall: 823543, ijkcnt: 8614550657858229

where =imxall, =jmxall, and kernel size = ijkcnt.

The results to date show that D=3 is reasonable with current systems constraints. D=4 is only possible with very long run times exceeding 47 CPU hours on XSEDE San Diego Supercomputer (SDSC.edu) Comet. D=5,6 are possible, but not one epoch evolution completed within 47 CPU hours on Comet. Comet is a 2.0 Petaflop (PF) Dell integrated compute cluster, with next-generation Intel Haswell processors (with AVX2), interconnected with Mellanox FDR InfiniBand in a hybrid fat-tree topology. Full bisection bandwidth is available at rack level (72 nodes) and there is a 4:1 oversubscription cross-rack. Compute nodes feature 320 GB of SSD storage and 128GB of DRAM per node.

Attempting D=7 leads directly to ’*** laginit = 9 ***’, where the code crashes with too large ’(long int) imxall’:

if ((pnew = (double complex *) calloc (imxall, sizeof (double complex))) == NULL) return (9);

The same results are obtained with ’(long long int) imxall’, which reverts to ’(long int) imxall’ on most platforms.

It is possible to run the code saving all elements of the kernel is a large matrix, instead of calculating the Lagrangian and its kernel each evolution. However, this is not useful for time-dependent problems as considered here, and the ijkcnt-size of memory required prevents any runs with dimension > 3.

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 Nov 2015) [33L. Schulman, Techniques and Applications of Path Integration., J. Wiley and Sons: New York, .].

Previous papers have modeled minicolumns as wires which support neuronal firings, mainly from large neocortical excitatory pyramidal cells in layer V (of I-VI), giving rise to currents which in turn gives rise to electric potentials measured as scalp EEG [45L. Ingber, “Computational algorithms derived from multiple scales of neocortical processing,” In: Pointing at Boundaries: Integrating Computation and Cognition on Biological Grounds, A. Pereira, E. Massad, and N. Bobbitt, Eds., Springer: New York, 2011, pp. 1-13. Invited Paper. Available from: https://www.ingber.com/smni11_cog_comp.pdf, 6L. Ingber, “Columnar EEG magnetic influences on molecular development of short-term memory,” In: Short-Term Memory: New Research, G. Kalivas and S. Petralia, Eds., NY: Nova, 2012, pp. 37-72. Available from: https://www.ingber.com/smni11_stm_scales.pdf, 16P. Nunez, R. Srinivasan, and L. Ingber, Theoretical and experimental electrophysiology in human neocortex: Multiscale correlates of conscious experience.Multiscale Analysis and Nonlinear Dynamics: From genes to the brain., Wiley: New York, pp. 149-178.
[http://dx.doi.org/10.1002/9783527671632.ch06] ]. This gives rise to a magnetic vector potential

(6)

which has a log-insensitive dependence on distance. In the brain, µ ≈ µ₀ , where is the magnetic permeability in vacuum = 4π10^-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 do not have this log dependence on distance. Thus, A fields can contribute collectively over large regions of neocortex [6L. Ingber, “Columnar EEG magnetic influences on molecular development of short-term memory,” In: Short-Term Memory: New Research, G. Kalivas and S. Petralia, Eds., NY: Nova, 2012, pp. 37-72. Available from: https://www.ingber.com/smni11_stm_scales.pdf, 11L. Ingber, "Influence of macrocolumnar EEG on ca waves", Current Progress Journal, vol. 1, no. 1, pp. 4-8. Available from: https://www.ingber.com/smni12_vectpot.pdf
[http://dx.doi.org/10.2139/ssrn.2091676] -16P. Nunez, R. Srinivasan, and L. Ingber, Theoretical and experimental electrophysiology in human neocortex: Multiscale correlates of conscious experience.Multiscale Analysis and Nonlinear Dynamics: From genes to the brain., Wiley: New York, pp. 149-178.
[http://dx.doi.org/10.1002/9783527671632.ch06] ]. The magnitude of the current is taken from experimental data on dipole moments Q =| I | where 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 [46S. Murakami, and Y. Okada, "Contributions of principal neocortical neurons to magnetoencephalography and electroencephalography signals", Journal of Physiology, vol. 575, no. Pt 3, pp. 925-936.
[http://dx.doi.org/10.1113/jphysiol.2006.105379] [PMID: 16613883] ], to 10^-9 A-m for larger neocortical mass [47P. Nunez, and R. Srinivasan, Electric Fields of the Brain: The Neurophysics of EEG., 2nd edOxford University Press: London, .
[http://dx.doi.org/10.1093/acprof:oso/9780195050387.001.0001] ]. These currents give rise to qA≈10^-28 kg-m/s. The velocity of a Ca²⁺ wave can be ≈20-50 µm/s. In neocortex, a typical Ca⁺² wave of 1000 ions, with total mass m=6.655×10^-23 kg times a speed of ≈20-50 µm/s, gives p≈10^-27 kg-m/s.

Without random shocks, the wave function ψ_e representing the interaction of the EEG magnetic vector potential A with the momenta p of Ca²⁺ wave packets was derived to be [14L. Ingber, "Evolution of regenerative ca-ion wave-packet in neuronal-firing fields: Quantum path-integral with serial shocks", International Journal of Innovative Research in Information Security, vol. 4, no. 2, pp. 14-22. Available from: https://www.ingber.com/ path17_quantum_pathint_shocks.pdf]

(7)

where ψ₀ is the initial Gaussian packet, ψ_F is the free-wave evolution operator, is the Planck constant, q is the electronic charge of Ca²⁺ ions, m is the mass of a wave-packet of 1000 Ca²⁺ ions, Δr² is the spatial variance of the wave-packet, the initial canonical momentum is II₀ = p₀ + qA₀ , and the evolving canonical momentum is II = P+qA. Detailed classical and quantum calculations have shown that P of the Ca²⁺ wave packet and qA of the large-scale EEG field make about equal contributions to II.

This calculation is for any time t, so the path integral was folded over many times numerically to give agreement with this results for long times [14L. Ingber, "Evolution of regenerative ca-ion wave-packet in neuronal-firing fields: Quantum path-integral with serial shocks", International Journal of Innovative Research in Information Security, vol. 4, no. 2, pp. 14-22. Available from: https://www.ingber.com/ path17_quantum_pathint_shocks.pdf]. Then, the path integral was calculated with random shocks to find a reasonable threshold that still possessed the character of the original wave function.

All this was done for a one-dimensional system. EEG is produced primarily by highly synchronous firings of pyramidal neurons, also the source of A fields, relatively orthogonal to the 6 layers of neocortex. A realistic calculation would require a 3-dimensional calculation over sets of input parameters. After this, contact with real EEG must be made to see if better fits are obtained including this more realistic modeling of Ca²⁺ -wave interaction with EEG than in previous XSEDE studies [13L. Ingber, "Statistical mechanics of neocortical interactions: Large-scale EEG influences on molecular processes", Journal of Theoretical Biology, vol. 395, pp. 144-152. Available from: https://www.ingber.com/smni16_large-scale_molecular.pdf
[http://dx.doi.org/10.1016/j.jtbi.2016.02.003] [PMID: 26874226] -15L. Ingber, M. Pappalepore, and R.R. Stesiak, "Electroencephalographic field influence on calcium momentum waves", Journal of Theoretical Biology, vol. 343, pp. 138-153. Available from: https://www.ingber.com/smni14_EEG_ca.pdf
[http://dx.doi.org/10.1016/j.jtbi.2013.11.002] [PMID: 24239957] ]. This will require coupling models of ψ with the current SMNI model as discussed in the next section.

4.2.3. SMNI Context

A short description of the SMNI structure is required to understand how the qPATHINT development is used in this multiple-scale model [13L. Ingber, "Statistical mechanics of neocortical interactions: Large-scale EEG influences on molecular processes", Journal of Theoretical Biology, vol. 395, pp. 144-152. Available from: https://www.ingber.com/smni16_large-scale_molecular.pdf
[http://dx.doi.org/10.1016/j.jtbi.2016.02.003] [PMID: 26874226] ].

After a statistical-mechanical aggregation of synaptic, neuronal and columnar scales [1L. Ingber, "Statistical mechanics of neocortical interactions. I. basic formulation", Physica D, vol. 5, pp. 83-107. Available from: https://www.ingber.com/ smni82_basic.pdf
[http://dx.doi.org/10.1016/0167-2789(82)90052-5] , 2L. Ingber, "Statistical mechanics of neocortical interactions. dynamics of synaptic modification", Physical Review A, vol. 28, pp. 395-416. Available from: https://www.ingber.com/smni83_dynamics.pdf
[http://dx.doi.org/10.1103/PhysRevA.28.395] ], the SMNI Lagrangian L in the prepoint (Ito) representation is

(8)

where G = {E,I} for chemically independent excitatory and inhibitory synaptic interactions. All values of parameters were taken within ranges of experimental data: 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^* is the number of neurons in a macrocolumn, about 10⁵.V^' includes nearest-neighbor mesocolumnar interactions [1L. Ingber, "Statistical mechanics of neocortical interactions. I. basic formulation", Physica D, vol. 5, pp. 83-107. Available from: https://www.ingber.com/ smni82_basic.pdf
[http://dx.doi.org/10.1016/0167-2789(82)90052-5] , 2L. Ingber, "Statistical mechanics of neocortical interactions. dynamics of synaptic modification", Physical Review A, vol. 28, pp. 395-416. Available from: https://www.ingber.com/smni83_dynamics.pdf
[http://dx.doi.org/10.1103/PhysRevA.28.395] ]. τ is usually considered to be on the order of 5-10 ms. The threshold factor F^G is derived as

(9)

where A^G_G^' is the columnar-averaged direct synaptic efficacy, B^G_G^' is the columnar-averaged background-noise 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. The parameters arise from regional interactions across many macrocolumns.

Three basic models were developed with slight adjustments of the parameters [3L. Ingber, "Statistical mechanics of neocortical interactions. derivation of short-term-memory capacity", Physical Review A, vol. 29, pp. 3346-3358. Available from: https://www.ingber.com/smni84_stm.pdf
[http://dx.doi.org/10.1103/PhysRevA.29.3346] ], changing the firing component of the columnar-averaged efficacies within experimental ranges, which modify F^G threshold factors to yield (a) case EC, dominant excitation subsequent firings in the conditional probability, or (b) case IC, inhibitory subsequent firings, or (c) case BC, balanced between EC and IC. A Centering Mechanism (CM) on case BC yields case wherein the numerator of F^G only has terms proportional to M^E', M^I' and , i.e., zeroing other constant terms by resetting the background parameters , still within experimental ranges. This has the net effect of bringing in a maximum number of minima into the physical firing M^G -space. The minima of the numerator then defines a major parabolic trough,

(10)

about which other SMNI nonlinearities bring in multiple minima calculated to be consistent with STM phenomena. Here, A dynamic CM (DCM) model is used [13L. Ingber, "Statistical mechanics of neocortical interactions: Large-scale EEG influences on molecular processes", Journal of Theoretical Biology, vol. 395, pp. 144-152. Available from: https://www.ingber.com/smni16_large-scale_molecular.pdf
[http://dx.doi.org/10.1016/j.jtbi.2016.02.003] [PMID: 26874226] ], wherein the are reset every few epochs of τ, parameterized to include contributions from tripartite neuron-astrocyte-neuron contributions. PATHINT also has been successfully used with the SMNI Lagrangian L to calculate properties of STM for both auditory and visual memory [64K.A. Ericsson, and W.G. Chase, "Exceptional memory", American Scientist, vol. 70, no. 6, pp. 607-615.
[PMID: 7181217] , 65G.J. Zhang, and H.A. Simon, "STM capacity for Chinese words and idioms: chunking and acoustical loop hypotheses", Memory and Cognition, vol. 13, no. 3, pp. 193-201.
[http://dx.doi.org/10.3758/BF03197681] [PMID: 4046819] ] 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 [23L. Ingber, "High-resolution path-integral development of financial options", Physica A, vol. 283, no. 3-4, pp. 529-558. Available from: https://www.ingber.com/markets00_highres.pdf
[http://dx.doi.org/10.1016/S0378-4371(00)00229-6] , 36L. Ingber, and P.L. Nunez, "Statistical mechanics of neocortical interactions: High-resolution path-integral calculation of short-term memory", Physical Review E, vol. 51, no. 5, pp. 5074-5083. Available from: https://www.ingber.com/smni95_stm.pdf
[http://dx.doi.org/10.1103/PhysRevE.51.5074] [PMID: 9963220] ].

Options models describe the market value of an option, V as

(11)

where S is the asset price, and σ is the standard deviation, or volatility of S, and r is the short-term interest rate.

The basic Black-Scholes (BS) options model [66F. Black, and M. Scholes, "The pricing of options and corporate liabilities", The Journal of Political Economy, vol. 81, pp. 637-659.
[http://dx.doi.org/10.1086/260062] ] considers a portfolio Π in terms of Δ,

(12)

in a market with Gaussian-Markovian (“white”) noise X and drift µ ,

(13)

where V(S,t) inherits a random process from S,

(14)

This yields

(15)

Financial options are generally described and traded using “Greeks”:

(16)

The portfolio to be hedged is often considered to be “risk-neutral,” if Δ is chosen such that

(17)

The expected risk-neutral return of Π is

(18)

where S is the asset price σ, and is the standard deviation, or volatility of S, and r is the short-term interest rate. For example, the basic equation can apply to many models of interpretations of prices given to V, e.g., puts or calls, and to S, e.g., stocks or futures, dividends, etc.

Path-integral algorithms are very useful for the calculation of the evolving probability distribution defined by this stochastic system, especially for realistic non-Black-Scholes financial options requiring time-dependent nonlinear drifts and diffusions, e.g., as developed by the author [23L. Ingber, "High-resolution path-integral development of financial options", Physica A, vol. 283, no. 3-4, pp. 529-558. Available from: https://www.ingber.com/markets00_highres.pdf
[http://dx.doi.org/10.1016/S0378-4371(00)00229-6] , 24L. Ingber, and J. Wilson, "Volatility of volatility of financial markets", Mathematical Computer Modelling, vol. 29, no. 5, pp. 39-57. Available from: https://www.ingber.com/markets99_vol.pdf
[http://dx.doi.org/10.1016/S0895-7177(99)00048-5] ]. In particular, American options (with possible early exercise), the most popular options and calculated here, have no closed solutions and must be done numerically. Other authors also have applied classical path-integral techniques to options [67B. Balaji, "Option pricing formulas and nonlinear filtering: a Feynman path integral perspective", Signal Processing, Sensor Fusion, and Target Recognition XXII, vol. 8745, pp. 1-10.
[http://dx.doi.org/10.1117/12.2017901] ].

Dividends and stress-testing for possible future events require including “shocks” to the system. A 1-dimensional calculation of quantum options using qPATHINT with shocks was published [31L. Ingber, "Options on quantum money: Quantum path-integral with serial shocks", International Journal of Innovative Research in Information Security, vol. 4, no. 2, pp. 7-13. Available from: https://www.ingber.com/path17_quantum_options_shocks.pdf]. Future studies including volatility of volatility [23L. Ingber, "High-resolution path-integral development of financial options", Physica A, vol. 283, no. 3-4, pp. 529-558. Available from: https://www.ingber.com/markets00_highres.pdf
[http://dx.doi.org/10.1016/S0378-4371(00)00229-6] ] are required to better understand quantum financial options.

For SMFM, the use of qPATHINT for quantum options is similar to the use of PATHINT for classical options. For PATHINT, the probability distribution of the underlying (e.g., “price”) is propagated from its initial state, numerically growing into a tree of probability nodes. Then, starting backward from the final time at the maturity date of the option, at each node a calculation is performed, e.g., for American options comparing the strike price to the price at that node, and a decision is made, e.g., whether to exercise the option at that node, etc. Other calculations also can be made at each node or time slices of nodes, e.g., of Greeks that might enter decisions that can be applied to classical events, inclusion of dividends, changes of interest rates, etc.

For qPATHINT, the evolving distribution might itself represent monetary value, a link in a quantum blockchain, a “wave-function” itself whose associated probability (complex square) is the entity being traded, etc. For quantum options, the nature of the evolving distribution likely would be determined by the market. Experience with the pilot SMNI test demonstrates the need for wide kernel bands for oscillatory systems.

For classical PATHINT, previous publications have shown how real-life strike data is successfully fit/predicted for use in trading financial options [23L. Ingber, "High-resolution path-integral development of financial options", Physica A, vol. 283, no. 3-4, pp. 529-558. Available from: https://www.ingber.com/markets00_highres.pdf
[http://dx.doi.org/10.1016/S0378-4371(00)00229-6] , 22L. Ingber, C. Chen, R.P. Mondescu, D. Muzzall, and M. Renedo, "Probability tree algorithm for general diffusion processes", Physical Review E, vol. 64, no. 5 Pt 2, p. 056702. Available from: https://www.ingber.com/path01_pathtree.pdf
[http://dx.doi.org/10.1103/PhysRevE.64.056702] [PMID: 11736136] , 24L. Ingber, and J. Wilson, "Volatility of volatility of financial markets", Mathematical Computer Modelling, vol. 29, no. 5, pp. 39-57. Available from: https://www.ingber.com/markets99_vol.pdf
[http://dx.doi.org/10.1016/S0895-7177(99)00048-5] ]. Similarly, qPATHINT can be tested when quantum strikes or other traded entities become available.

For SMNI, the use of qPATHINT is envisioned to be similar to its use for SMFM. The wave function ψ is propagated for its initial state, numerically growing into a tree of wave-function nodes. At each node, going forward instead of back in time as for SMFM, interaction of the Ca²⁺ of wave-packet, via its momentum P, with highly synchronous EEG, via its collective magnetic vector potential A, is calculated to determine changes due to time-dependent phenomena. Such changes occur at microscopic scales, e.g., due to modifications of the regenerative wave-packet as ions leave and contribute to the wave packet, thereby determining the effect on tripartite contributions to neuron-astrocyte-neuron synaptic activity, affecting both P and A. Such changes also may occur at 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(ψ^*ψ) determined by the wave-functions at their nodes.

Previous publications have shown how real-life EEG data, under experimental paradigms measuring attentional 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 Ca²⁺ -waves modifying parameterization of background SMNI synaptic parameters. Data was fit to this model using the author’s Adaptive Simulated Annealing code [43L. Ingber, “Adaptive simulated annealing (ASA),” Caltech Alumni Association, Pasadena, CA, Technical Report, Global optimization C-code, 1993. Available from: https://www.ingber.com/#ASA-CODE, 68L. Ingber, Adaptive simulated annealing.Stochastic global optimization and its applications with fuzzy adaptive simulated annealing., Springer: New York, pp. 33-61. Available from: https://www.ingber.com/asa11_options.pdf.. ]. These projects demonstrated that fits with this inclusion were better than fits without.

Using qPATHINT, quantum scales of interaction can now be included. Again, a reasonable measure of the influence of these scales will be to fit real EEG data under other published experimental paradigms. This will be accomplished using the above calculation of ψ(t) with realistic serial shocks, calculating updates to P(t) and A(t) at each time slice corresponding to the EEG data for both training and testing sets of data. While PATHINT could synchronously also be evolved using the SMNI Lagrangian, i.e., considering a joint quantum-classical probability distribution, it likely is convenient and accurate to use the short-term SMNI Lagrangian as in previous studies to fit EEG and at each epoch; note that the columnar neocortical system propagates slower than the faster Ca²⁺ wave-packet system. This permits a robust inclusion of quantum-scale interactions into the SMNI multiple-scale framework.

An additional burden of reality is of course placed on the specific (P+qA) interaction developed since 2012, between P of Ca²⁺ wave packets and qA from EEG fields. However, the methodology developed here still will be useful for other interactions that might be relevant to include these expanded multiple scales of neocortical interactions.