Parameterization of quantum interactions

Background: Previous papers have dev eloped a statistical mechanics of neocortical interactions (SMNI) fit to short-term memory and EEG data (Ingber, 2018). Adaptive Simulated Annealing (ASA) was used for all fits to data. A numerical path-integral for quantum systems, qPATHINT, was used. Objective: The quantum path-integral for Calcium ions was used to derive a closed-form analytic solution at arbitrary time. The quantum effects is parameterized here, whereas the previous 2018 paper applied a nominal ratio of 1/2 to these effects. Method: Methods of mathematical-physics for optimization and for path integrals in classical and quantum spaces are used. The quantum path-integral is used to derive a closed-form analytic solution at arbitrary time, and is used to calculate interactions with classical-physics SMNI interactions among scales. Results: The mathematical-physics and computer parts of the study are successful, in that three cases with Subjects (blind to this author) after 1,000,000 visits to the cost function gav e: Subject-07 = 0.04, Subject-08 = 0.55, and Subject-09 = 1.00. All other 9 Subjects gav e 0.


Introduction
This project calculates quantum Ca 2+ using EEG for data.Only specific calcium ions Ca 2+ are considered, those arising from regenerative calcium wav esgenerated at tripartite neuron-astrocyte-neuron synapses.This project is speculative,but it is testable, e.g., by fitting EEG.SMNI has been developed since 1981.This evolving model including ionic scales have been published since 2012.Quantum physics calculations also support these extended SMNI models.

Synaptic Interactions
The short-time conditional probability distribution of firing of a givenn euron firing givenj ust-previous firings of other neurons is calculated from chemical and electrical intra-neuronal interactions (Ingber, 1982;Ingber,1 983).With previous interactions with k neurons within τ j of 5−10 msec, the conditional probability that neuron j fires (σ j =+1) or does not fire (σ j =−1) is V j is the depolarization threshold in the somatic-axonal region.v jk is the induced synaptic polarization of E or I type at the axon, and φ jk is its variance.The efficacy a jk is a sum of A jk from the connectivity between neurons, activated if the impinging k-neuron fires, and B jk from spontaneous background noise.

Neuronal Interactions
Aggregation up to the mesoscopic scale from the microscopic synaptic scale uses mesoscopic probability M represents a mesoscopic scale of columns of N neurons, with subsets E and I ,represented by p q i .T he "delta"-functions δ -constraint represents an aggregate of manyn eurons in a column.G is used to represent excitatory (E)and inhibitory (I )contributions.G designates contributions from both E and I .The path integral is derivedi nt erms of mesoscopic Lagrangian L.T he short-time distribution of firings in a minicolumn, givenits just previous interactions with all other neurons in its macrocolumn is thereby defined.

Columnar Interactions
In the prepoint (Ito) representation the SMNI Lagrangian L is

SMNI Parameters From Experiments
Va lues of parameters and their bounds are taken from experimental data, not fit to specific phenomena.
N G ={N E = 160, N I = 60} was set for for visual neocortex, {N E = 80, N I = 30} was set 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 .N * is the number of neurons in a macrocolumn, about 10 5 .V ′ includes nearest-neighbor mesocolumnar interactions.τ is usually considered to be on the order of 5−10 ms.V G = 10 mV, v G G′ = 0. 1 mV, φ G G′ = 0. 03 1/2 mV.Nearest-neighbor interactions among columns give dispersion relations consistent with speeds of mental visual rotation (Ingber,1982;Ingber,1983).
The wav e equation cited by EEG theorists, permitting fits of SMNI to EEG data (Ingber,1995), is derived using the variational principle applied to the SMNI Lagrangian.This creates an audit trail from synaptic parameters to the averaged regional Lagrangian.

Basic SMNI Model
Consistent with experimental evidence of shifts in background synaptic activity under conditions of selective attention (Mountcastle et al,1 981; Briggs et al,2 013), a Centering Mechanism (CM) on case L.Ingber BC, giving BC′,w here the numerator of F G only has terms proportional to M E′ , M I ′ and M ‡E′ ,i .e., zeroing other constant terms by resetting the background parameters B G G′ ,still within experimental ranges.This brings in a maximum number of minima into the physical firing M G -space, due to the minima of the newnumerator in being in a parabolic trough defined by

A E
E M E − A E I M I = 0( 4) about which nonlinearities develop multiple minima identified with STM phenomena.ADynamic CM (DCM) model is used, resetting B G G′ ev ery fewepochs of τ .S uch changes in background synaptic activity on such time scales are seen during attentional tasks (Briggs et al,2013).

Comparing EEG Testing Data with Training Data
EEG data was used from http://physionet.nlm.nih.gov/pn4/erpbci(Goldberger et al,2 000; Citi et al, 2010), SMNI was again fit to highly synchronous wav es( P300) during attentional tasks, for each of 12 subjects (Ingber,2 016b).The electric potential Φ is experimentally measured by EEG, but both are due to the same currents I.A is linearly proportional to Φ with a scaling factor included as a parameter in fits to data.Additional parameterization of background synaptic parameters, B G G′ and B ‡E E′ ,m odify previous work.

Canonical Momentum Π=p + qA
In the Feynman (midpoint) representation of the path integral, the canonical momentum, Π,d efines the dynamics of a moving particle with momentum p in an electromagnetic field.In SI units, Π=p + qA (5) where q =−2e for Ca 2+ , e is the magnitude of the charge of an electron = 1.6 × 10 −19 C( Coulomb), and A is the electromagnetic vector potential.A represents three components of a 4-vector.

Vector Potential of Wire
Acolumnar firing state is modeled as a wire/neuron with current I measured in A = Amperes = C/s, along a length z observed from a perpendicular distance r from a line of thickness r 0 .I ff ar-field retardation effects are neglected, this yields where µ is the magnetic permeability in vacuum = 4π 10 −7 H/m (Henry/meter).
A includes minicolumnar lines of current from hundreds to thousands of macrocolumns, within a region not so large to include manyconvolutions, but still contributing to large synchronous bursts of EEG.Electric E and magnetic B fields, derivativeso f A with respect to r,d on ot possess this logarithmic insensitivity to distance, and theydonot linearly accumulate strength within and across macrocolumns.
Estimates of contributions from synchronous firings to P300 measured on the scalp are tens of thousands of macrocolumns spanning 100 to 100'sofcm 2 .E lectric fields generated from a minicolumn may fall by half within 5−10 mm, the range of several macrocolumns.

Reasonable Estimates
Classical physics calculates qA from macroscopic EEG to be on the order of 10 −28 kg-m/s, while the momentum p of a Ca 2+ ion is on the order of 10 −30 kg-m/s.This numerical comparison includes the influence of A on p at classical scales.Direct calculations in both classical and quantum physics showi onic calcium momentum-wav e effects neuron-astrocyte-neuron tripartite synapses modify background SMNI parameters and create feedback between ionic/quantum and macroscopic scales (Ingber,2012a;Ingber,2012b;Nunez et al,2013;Ingber L.Ingber et al,2014;Ingber,2015;Ingber,2016b;Ingber,2017a).

PATHINT/qPATHINT Code
qPAT HINT is an N-dimensional code which calculates the propagation of quantum variables in the presence of shocks.Applications have been made to SMNI and Statistical Mechanics of Financal Markets (SMFM) (Ingber,2017a;Ingber,2017b;Ingber,2017c).
The PAT HINT C code of 7500 lines of code using the GCC C-compiler was rewritten to use double complexv ariables instead of double variables, developed for arbitrary N dimensions, creating qPAT HINT (Ingber,2016a;Ingber,2017a;Ingber,2017b).

Results Including Quantum Scales
The wav e function ψ e describing the interaction of A with p of Ca 2+ wave packets was derivedi nc losed form from the Feynman representation of the path integral using path-integral techniques (Schulten, 1999), modified to include A.
p of the Ca 2+ wave packet and qA of the EEG field makeabout equal contributions to Π (Ingber,2015).

SMNI + Ca 2+ wave-packet
Tripartite influences on synaptic B G G′ ,ismeasured by the ratio of packet's<p(t)> ψ *ψ to < p 0 (t 0 )> ψ *ψ ,at the onset of each attentional task.Here <> ψ *ψ is taken over ψ * e ψ e .A changes slower than p,a nd the static approximation of A used to derive ψ e and < p > ψ *ψ is use within P300 EEG epochs, resetting t = 0 at the onset of each classical EEG measurement (1.953 ms apart), using the current A.

Results
Using < p > ψ *ψ < p > ψ *ψ wasu sed in classical-physics SMNI fits to EEG data using ASA.Runs using 1M or 100K generated states gav e results not much different.The current calculations use 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-fit-parameter" SMNI philosophywas 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 give Testing cost functions less than the Training cost functions.This is due to differences in data, likely from differences in subjects' contexts, e.g., possibly due to subjects' STM strategies.Further tests of these multiple-scale models with more EEG data are required, and with the PAT HINT-qPAT HINT coupled algorithm described previously.

Results
The mathematical-physics and computer parts of the study are successful, in that three cases with Subjects (blind to this author) after 1,000,000 visits to the cost function gav e:S ubject-07 = 0.04, Subject-08 = 0.55, and Subject-09 = 1.00.All other 9 Subjects gav e 0.

Conclusion
The SMNI model demonstrates can be very well fit to experimental data, e.g., EEG recordings under STM experimental paradigms.qPAT HINT permits an inclusion of quantum scales in the multiple-scale SMNI model, by evolving Ca 2+ wave-packets with momentum p,including serial shocks, interacting with the magnetic vector potential A derivedfrom EEG data, marching forward in time with experimental EEG data.