# Quantum Calcium-Ion Interactions with EEG

## Abstract

**:**

**Background**: Previous papers have developed a statistical mechanics of neocortical interactions (SMNI) fit to short-term memory and EEG data. Adaptive Simulated Annealing (ASA) has been developed to perform fits to such nonlinear stochastic systems. An N-dimensional path-integral algorithm for quantum systems, qPATHINT, has been developed from classical PATHINT. Both fold short-time propagators (distributions or wave functions) over long times. Previous papers applied qPATHINT to two systems, in neocortical interactions and financial options.

**Objective**: In this paper the quantum path-integral for Calcium ions is used to derive a closed-form analytic solution at arbitrary time that is used to calculate interactions with classical-physics SMNI interactions among scales. Using fits of this SMNI model to EEG data, including these effects, will help determine if this is a reasonable approach.

**Method**: Methods of mathematical-physics for optimization and for path integrals in classical and quantum spaces are used for this project. Studies using supercomputer resources tested various dimensions for their scaling limits. In this paper the quantum path-integral is used to derive a closed-form analytic solution at arbitrary time that 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 there is modest improvement of cost/objective functions used to fit EEG data using these models.

**Conclusions**: This project points to directions for more detailed calculations using more EEG data and qPATHINT at each time slice to propagate quantum calcium waves, synchronized with PATHINT propagation of classical SMNI.

## 1. Introduction

## 2. Statistical Mechanics of Neocortical Interactions (SMNI)

#### 2.1. Synaptic Interactions

#### 2.2. Neuronal Interactions

#### 2.3. Columnar Interactions

#### 2.4. SMNI Parameters From Experiments

#### 2.5. Previous Applications

#### 2.5.1. Verification of basic SMNI Hypothesis

#### 2.5.2. SMNI Calculations of Short-Term Memory (STM)

#### 2.5.3. Three Basic SMNI Models

- (a)
- case EC, dominant excitation subsequent firings
- (b)
- case IC, inhibitory subsequent firings
- (c)
- case BC, balanced between EC and IC

#### 2.6. Comparing EEG Testing Data with Training Data

#### 2.7. STM PATHINT Calculations

#### 2.7.1. PATHINT STM

#### 2.7.2. PATHINT STM Visual

#### 2.8. Tripartite Synaptic Interactions

#### 2.8.1. Canonical Momentum $\mathbf{\Pi}=\mathbf{p}+q\mathbf{A}$

#### 2.8.2. Vector Potential of Wire

#### 2.8.3. Effects of Vector Potential on Momenta

#### 2.8.4. Reasonable Estimates

#### 2.9. Model of Models (MOM)

#### Ideas by Statistical Mechanics

## 3. Adaptive Simulated Annealing (ASA) Algorithm

#### 3.1. Importance Sampling

#### 3.2. Outline of ASA Algorithm

#### 3.3. ASA Applications

## 4. Path-Integral Algorithms PATHINT and qPATHINT

#### 4.1. Path Integral in Stratonovich (Midpoint) Representation

#### 4.2. Path Integral in Ito (Prepoint) Representation

#### 4.3. Path-Integral Riemannian Geometry

#### 4.4. Three Approaches Are Mathematically Equivalent

- (a)
- Fokker-Planck/Chapman-Kolmogorov partial-differential equations
- (b)
- Langevin coupled stochastic-differential equations
- (c)
- Lagrangian or Hamiltonian path-integrals

#### 4.4.1. Stochastic Differential Equation (SDE)

#### 4.4.2. Partial Differential Equation (PDE)

#### 4.5. PATHINT Applications

#### 4.6. PATHINT/qPATHINT Code

#### 4.6.1. Shocks

#### 4.6.2. PATHINT/qPATHINT Histograms

#### 4.6.3. Meshes For [q]PATHINT

#### 4.7. Lessons Learned From SMFM and SMNI

#### Calculations At Each Node At Each Time Slice

- PATHINT using the Classical SMNI Lagrangian
- qPATHINT using the Quantum ${\mathrm{Ca}}^{2+}$ wave-packet Lagrangian
- Sync in time during P300 attentional tasks.
- Time/phase relations between classical and quantum systems may be important.
- ASA-fit synchronized classical-quantum PATHINT-qPATHINT model to EEG data.
- $\mathbf{A}$ is determined experimentally from EEG, and includes all synaptic background ${B}_{{G}^{\prime}}^{G}$ effects.

## 5. Results Including Quantum Scales

#### 5.1. SMNI + ${\mathrm{Ca}}^{2+}$ Wave-Packet

#### 5.2. Supercomputer Resources

#### 5.3. Results Using $<\mathbf{p}{>}_{\psi \ast \psi}$

#### 5.4. Quantum Zeno Effects

#### Survival of Wave Packet

## 6. Quantum Applications

#### 6.1. Nano-Robotic Applications

#### 6.2. Free Will

## 7. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Illustrates three SMNI biophysical scales [2,3]: (a)-(a${}^{*}$)-(a’) microscopic neurons; (b)-(b’) mesocolumnar domains; (c)-(c’) macroscopic regions; (a${}^{*}$): synaptic inter-neuronal interactions, scaled up to mesocolumns, phenomenologically described by the mean and variance of a distribution $\mathsf{\Psi}$. (

**a**): intraneuronal transmissions phenomenologically described by the mean and variance of $\mathsf{\Gamma}$. (

**a’**): collective mesocolumnar-averaged inhibitory (I) and excitatory (E) neuronal firings M; (

**b**): vertical organization of minicolumns including their horizontal layers, yielding a physiological entity, the mesocolumn (

**b’**): overlapping mesocolumns at locations r and ${r}^{\prime}$ from times t and $t+\tau $, $\tau $ on the order of 10 ms; (

**c**): macroscopic regions of neocortex arising from many mesocolumnar domains (

**c’**): regions coupled by long-ranged interactions.

**Table 1.**Column 1 is the subject number; the other columns are cost functions. Columns 2 and 3 are no-

**A**model’s Training (TR0) and Testing (TE0). Columns 4 and 5 are

**A**model’s Training (TR

**A**) and Testing (TE

**A**). Columns 6 and 7 are switched no-

**A**model’s Training (sTR0) and Testing (sTE0). Columns 8 and 9 are switched

**A**model’s Training (sTR

**A**) and Testing (sTE

**A**).

Sub | TR0 | TE0 | TRA | TEA | sTR0 | sTE0 | sTRA | sTEA |
---|---|---|---|---|---|---|---|---|

s01 | 85.75 | 121.23 | 84.76 | 121.47 | 120.48 | 86.59 | 119.23 | 87.06 |

s02 | 70.80 | 51.21 | 68.63 | 56.51 | 51.10 | 70.79 | 49.36 | 74.53 |

s03 | 61.37 | 79.81 | 59.83 | 78.79 | 79.20 | 61.50 | 75.22 | 79.17 |

s04 | 52.25 | 64.20 | 50.09 | 66.99 | 63.55 | 52.83 | 63.27 | 64.60 |

s05 | 67.28 | 72.04 | 66.53 | 72.78 | 71.38 | 67.83 | 69.60 | 68.13 |

s06 | 84.57 | 69.72 | 80.22 | 64.13 | 69.09 | 84.67 | 61.74 | 114.21 |

s07 | 68.66 | 78.65 | 68.28 | 86.13 | 78.48 | 68.73 | 75.57 | 69.58 |

s08 | 46.58 | 43.81 | 44.24 | 49.38 | 43.28 | 47.27 | 42.89 | 63.09 |

s09 | 47.22 | 24.88 | 46.90 | 25.77 | 24.68 | 47.49 | 24.32 | 49.94 |

s10 | 53.18 | 33.33 | 53.33 | 36.97 | 33.14 | 53.85 | 30.32 | 55.78 |

s11 | 43.98 | 51.10 | 43.29 | 52.76 | 50.95 | 44.47 | 50.25 | 45.85 |

s12 | 45.78 | 45.14 | 44.38 | 46.08 | 44.92 | 46.00 | 44.45 | 46.56 |

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Ingber, L.
Quantum Calcium-Ion Interactions with EEG. *Sci* **2019**, *1*, 20.
https://doi.org/10.3390/sci1010020

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Ingber L.
Quantum Calcium-Ion Interactions with EEG. *Sci*. 2019; 1(1):20.
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Ingber, Lester.
2019. "Quantum Calcium-Ion Interactions with EEG" *Sci* 1, no. 1: 20.
https://doi.org/10.3390/sci1010020