Applications of Quantum Mechanics, Laws of Classical Physics, and Differential Calculus to Evaluate Source Localization According to the Electroencephalogram

Abstract

Applications of quantum mechanics and physics-based calculus allow for advanced mathematical modeling of source localization estimations. Because EEG waveforms can be modeled as continuous or discrete quantum matter, algorithmic models that estimate current source density must also consider the statistical properties of the dipole fields that are etiologically relevant to the reflected waveforms. Further applications of quantum physics to the electroencephalogram (EEG) suggest that neurodynamic behavior that originates in thalamo-cortical regions compared to cortico-cortical regions yield divergent 3-dimensional dispersions of wave forms and quantum energy. Evaluations of the dispersion of quantum energy and dipole magnetic fields according to classical physics and electromagnetism indicate that the area of tissue by which the oscillatory mechanisms are thought to originate inherently influences the algorithmic modeling and estimations current source density. Principles and algorithms related to classical physics are included within this paper to evaluate limitations of algorithmic models of source localization and the inverse problem.

1. Introduction

The EEG is a neuroimaging device that reflects the electrophysiological activity of the cerebrum that is generated by the synchronistic firing of nerve cells [1, 2]. Because the EEG reflects dynamic neurocognitive processes according to millisecond variations, the statistical properties of neural activation and processes fluctuate according to time, depth, and orientation of the neural generators. Measurement of the electrical activity reflected by the EEG is best conceptualized as a nonlinear stochastic process that is inherently spatio-temporally dependent [1, 2].

Extracellular currents are derived from excitatory and inhibitory postsynaptic potentials allow for dipole moment per unit volume approximations [1, 3]. The dipole moment per unit volume approximations allow for the derivation of weighted estimations of current source density according to neural generators across and between local tissues and the orthogonal orientation of nerve cells within the neocortical layers [1].

While measurements of extracellular currents can be measured using divergent mathematical algorithms (i.e., phase shift, phase lock, coherence, and lagged coherence), the interaction between neurons that generate excitatory and inhibitory postsynaptic potentials must be evaluated according to time, frequency, and the location of the neural generators [4, 5]. Utilization of the Fast Fourier Transformation (FFT) allows for the derivation of EEG frequency bands from the raw signal according to time. FFT analyses elucidate phase angle and phase differences from the EEG frequency domains. When the phase angle is stable or constant, this indicates that coherence = 1.0 whereas coherence = 0 when there are phase differences due to moment to moment variations in the phase angle. Thus, coherence elucidates communication between distal and contiguous neuroanatomical regions and therefore, across and between functional networks via coupled neural oscillators [4, 5].

Mathematical modeling of current sources can be reflected as P (r′, t) when the estimation is derived according to dipole moment per unit volumetric approximations [1]. This algorithmic model asserts that r′ is the estimated location of the neural source according to volume where t indicates time. Limitations of source localization methods are attributable to the inverse problem and assumptions related to the head volume conduction model [1, 2, 3, 4, 5, 6]. The head volume conduction model is significantly flawed because there are multiple layers of biological material that the electrical current must surpass before it can be reflected on the scalp [1, 7]. Furthermore, because of the anisotropic conductivity of the brain, which is attributable to the inhomogeneous conductivity of the cerebrospinal fluid, dura and cerebral tissue, there are an infinite number of solutions to the inverse solution [1, 8, 9, 10].

2. Applications of Maxwell’s law

Applications of Maxwell’s equations to the EEG incorporate fundamental principles related to thermodynamics as derived by Faraday’s law of electromagnetic induction and Gauss’ law of electromagnetism [11, 12, 13, 14]. Maxwell’s equations establish that the electrical displacement of neural generators and the field produced by these neurophysiological generators inherently influence accurate estimations of current source density [14].

There are four algorithms that comprise Maxwell’s equations [15, 16, 17]. Eq. (1) defines 𝛁 as a nabla operator [14]. The nabla must act upon the other variables denoted within the equation in order to derive estimations of current source density. E is defined as the electrical current. When E is considered as a cross product with the nabla, it is a mathematical constant that is derived as a function of the partial derivation, 𝛛, of the magnetic field, B, which varies according to temporal changes, t [14].

$$\nabla \times E=-\frac{\partial }{\partial t}B\text{,}$$E1

Considerations of Eq. (1) indicate that the nabla operator denotes a gradient of a scalar field when this operator is utilized to derive estimations of neural generators according to a Cartesian coordinate system [15]. Thus, this equation attempts to utilize statistical weighting to estimate Laplacian operators that are derived from electromagnetic fields at different points within the head volume conduction model. Because the activity reflected in an EEG recording is derived from the activity of orthogonally oriented neurons, estimations of current source density must account for the errors in estimation that are a result of the inability to determine the precise location of the neurobiological generators that produce dipole magnetic fields [15]. It is important to denote that 𝛁 x E represents a cross-product between these two factors and does not represent an order of multiplication [14].

Eq. (2) indicates that the cross product between the magnetic field strength, H, and the gradient of a scalar field, 𝛁, is equal to current source density, J, as it relates to a partial derivation, 𝛛, of time, t. The field of electric displacement is denoted as D [15].

$$\nabla xH=J+\frac{\partial }{\partial t}D\text{,}$$E2

Eq. (3) indicates that the gradient of a scalar field as a cross product between the electrical displacement, D, allows for the derivation of the unbounded charge density, p [15]. Eq. (3) is derived from Gauss’ law of electromagnetism [15]. Gauss’ law of electromagnetism indicates that the electrical flux that is reflected by a closed surface must be proportional to the total electrical charge within that surface [14, 16]

$$\nabla \cdot D=p$$E3

Applications of Maxwell’s law and Gauss’ law of electromagnetism incorporate the divergence theorem [17]. The divergence theorem states that the properties of a continuous entity, such as electrical flux, within a closed circuit is equal to the spread of that continuous entity over the volume of the closed-circuit system. Thus, because the flow of a continuous entity must be calculated as a 3-dimensional dispersion, the calculation must consider the magnitude, directionality, and the time by which the divergence occurs. This theorem indicates that a geometric interpretation of the dispersion of a waveform that uses a 3-coordinate configuration system is equivalent to the derivation of current source density [17].

Eq. (4) indicates that the sum of the voltages within a closed-circuit loop must inherently sum to zero [15]. This assertion is consistent with Laplacian physics and Kirchhoff’s Circuit Theory [18].

$$\nabla \cdot B=0$$E4

3. Laplacian physics

Geometric models and calculus-based physics in EEG source localization include applications of Coulomb’s inverse-square law [19]. Coulomb’s law indicates that the magnitude of the electrical force between two particles is inversely related to the distance between the two charges. Thus, as distance increases, the magnitude of the two charges decrease. This law also indicates that electrical force increases as the charge of two particles increase. Coulomb’s law can be applied to estimations of source localization if the difference in magnitude between divergent neural generators is considered similarly to the difference in charge of two particles. Specifically, as the distance between electrophysiological generators from the scalp increases, there is a decreased probability that this activity will be reflected on the surface. Coulomb’s inverse square law is mathematically defined in Eq. (5) [19].

$$\left|FE\right|=k\frac{q1q2}{r^2}$$E5

Because the depth of a signal inherently influences the electrical field within a volume conductor, applications of Ohm’s law to the EEG are also related to source localization [19]. Algorithmic modeling of Ohm’s law can be reflected according to vector computations. Eq. (6) denotes that E(r) is the electrical field represented at a specific location, p is resistance, and j is current source density [19].

$$E\left(r\right)=pj\left(r\right)$$E6

Gradient derivations are also employed to evaluate the Laplacian operators [19, 20]. Estimations of Cartesian coordinates to evaluate locations of electrical generators are determined by computations that derive the statistical term weights in vector space models. Eq. (7) indicates that the electrical field is computed as a vector function that fluctuates according to time. The derivation of the estimated coordinates account for the time and direction of the magnetic field which inherently affects the magnitude of power reflected by electrical generators. Eq. (7) defines 𝛛 as a partial derivative of V(r) according to its location and i, j, and k as individual Cartesian coordinates according to their orthogonal directionality. The orientation of the generator inherently affects the weighting or probabilistic influence of each coordinate in relation to the electrical signal. Grad indicates a gradient of V(r) which is a measurement of the rate of change of electrical flux within a physical construct according to its radial direction [19, 20].

$$\mathit{Grad}\left(V\left(r\right)\right)=\frac{\partial V\left(r\right)}{\partial x}\hat{i}+\frac{\partial V\left(r\right)}{\partial y}\hat{j}+\frac{\partial V\left(r\right)}{\partial z}\hat{k}$$E7

Computations of Laplacian operators require considerations of resistivity [1, 10, 21]. Because the resistivity of biologic materials within the EEG are anisotropic, estimations of Laplacian operators according to the head volume conduction model are inherently flawed. The head volume conduction model assumes isotropy or uniform conductivity. Weighted approximations of the resistivity within a biologic medium must account for the time, region, directionality and the type of tissue by which the electrical currents propagate. Because electrical currents indicate a non-uniform distribution, a cross-section within the brain may present divergent resistive properties dependent upon the Laplacian operator being cortical or subcortical in origin [21].

Further applications of Maxwell’s equations include the Maxwell-Rayleigh algorithm [21]. This algorithm is utilized to calculate the resistive properties that occur within a spherical conductive model. Thus, the Maxwell-Rayleigh equation incorporates geometric configurations to estimate of gradations of resistivity that are attributable to a homogenous suspension that exists within a volume conductor. This application is directly related to the estimation of current source density as the brain is surrounded by cerebrospinal fluid. Eq. (8) defines η is resistance according to a specific medium, ηs is fluid resistivity, h is configured according to the geometrical shape of the volume conductor, and 𝛂 is the partial derivative of the volume by which the suspension occupies [21].

$$\frac{\eta }{\eta s}=\frac{1+h\alpha }{1-\alpha }$$E8

Considerations of algorithms that attempt to model estimations of current source density according to the EEG must consider physiological constraints and mathematical limitations [1]. Derivations of the cumulative density function according to Eq. (9) does not consider the resistivity of the biological material by which the electrical signals must bypass. The resistivity of these biological materials alters the conductivity of the electrical signals that are reflected on the scalp. Eq. (9) defines 𝚽 as the cumulative magnetic flux, for a given region, r, according to time, t. Determination of the cumulative magnetic flux is a function of the interaction between independent variables that vary according to time. This equation defines ∫ as the integral of B, the imaginary component, GH as the weighted term of volume conductance, P (r′, t) as the three-dimensional electrical field, and dV as the electrical flux according to density. V is defined as volume [1].

$$\varphi S\left(r,t\right)={\displaystyle \int }GH\left(r,r\text{'}\right)P\left(r\text{'},t\right)dV\left(r\text{'}\right)$$E9

GH is a direct application of Coulomb’s law of electrostatic force with regard to source localization. This equation indicates that GH is always larger for cortical generators compared to subcortical generators [1]. Thus, this is a direct application of Coulomb’s law because as the distance between the electrophysiological generators and the scalp increases, the probability that the activity will be shown on the surface decreases. The reduced probability is attributable to Coulomb’s law that indicates that the magnitude of electric force decreases as a factor of 1/r² where r is defined as the distance between the two charges or generators that produce the dipole field [1, 19].

4. Quantum mechanics and Schrodinger equation applications

Applications of quantum mechanics indicate that linear models that are utilized to conceptualize EEG data are inherently flawed [22]. Linear models that attempt to model EEG signals according to volume conduction include the linear instantaneous time-invariant mixing algorithm [23]. The linear instantaneous time-invariant mixing model indicates that the generation of electrophysiological signals can be modeled as a linear function. This is attributable to the assertion that the cumulative electrical activity reflected on the scalp is a result of the mixing of underlying sources within the head volume conduction model. Congedo and Sherlin [23] indicate that the mixing coefficients or the individual sources do not vary according to time and are therefore fixed. Eq. (10) defines xi as the voltage, ai and aiM as the mixing coefficients, s as the source, and t as time [23].

$$\mathit{xi}\left(t\right)=\mathit{ai}1\cdot s1\left(t\right)+\mathit{ai}2\cdot s2\left(t\right)+\dots +\mathit{aiM}\cdot sM\left(t\right)$$E10

Eq. (10) negates the inclusion of factors related to the resistivity of the biologic medium by which the electrical signals are derived [23]. Furthermore, this equation is inherently flawed in that the mixing coefficients are deemed as time invariant factors. Considerations of the inverse problem and Coulomb’s law of electrical force indicate that the estimation of the electrical sources is dependent upon the depth, orientation, and number of neurons that generate the EEG rhythms that are reflected on the scalp [1, 7].

Applications of Heinsenberg’s Uncertainty Principle and Schrodinger’s equation further exemplify the error associated with conceptualizing the EEG as a linear function [22, 24]. These principles and algorithms indicate that EEG waveforms can be modeled as a summation of individual waves and quantum particles. Heinsenberg’s Uncertainty Principle indicates that statistical modeling can be utilized in a priori estimations of the location and position of quantum objects as the precise location and momentum of particles cannot be simultaneously determined. Eq. (11) is a representation of the time independent derivative of the Schrodinger equation [22, 24].

$$i\hslash \frac{\partial \psi \left(x,t\right)}{\partial t}=-\frac{{\hslash }^2}{2m}{\nabla }^2\psi \left(x,t\right)+V\left(x,t\right)+\psi \left(x,t\right)$$E11

Eq. (11) defines iħ as energy, 𝛛 as the partial derivative of the waveform according to time [24]. ħ is Plank’s constant, 𝚿 indicates the wave function, and 𝛁 indicates the degree to which the wave function changes according to space relative to the conductor by which it is modeled. V is defined as the magnitude or force of the particle according to location and time where m represents the mass of the particle. Because the Schrodinger equation indicates that waveforms within the EEG can be conceptualized as quantum particles, this suggests that the energy or magnitude of the quanta that produce neuro-oscillatory behavior and EEG rhythms can be defined as a function of kinetic energy [24].

4.1 EEG application of Schrodinger equation

The Schrodinger equation exemplifies the inherent limitations of the head volume conduction model and, therefore, the inverse problem. Albeit Nunez and Srinivasan [21] have derived estimations of resistivity for divergent biological materials such as the skull, cerebrospinal fluid, dura, blood and the cortex, the Schrodinger equation indicates that the precise location of individual quantum particles and waveforms cannot be determined unless invasive procedures are utilized to isolate these particles [21, 22]. Postulates of the Schrodinger equation indicate that probability distributions are utilized to estimate the dispersion of each wave form [24]. This postulate can be applied to the EEG in relation to the derivation of dipole magnetic fields.

Because multiple ‘neurorhythmicities’ are generated at any given time, where each waveform will have its own probabilities and estimated dispersions of the dipole fields related to that wave, there is not a unique solution to the inverse problem. Derivation of the individual statistical weights for each waveform that is cortical or subcortical in origin is mathematically infeasible. These mathematical and physiological challenges that are specific to the head volume conduction model and inverse problem indicate the necessity to incorporate integrals and imaginary components to reduce statistical error in estimations of source localization.

5. Conclusion

The classification of EEG waveforms as continuous or discrete quantum matter and the output of the EEG as a linear or nonlinear function inherently affects algorithmic modeling for estimations of source localization [23, 24]. While Nunez and Srinivasan [21] have estimated the degree to which specific biologic mediums and materials possess resistive properties, these parametric configurations do not provide a solution to the inverse problem. Applications of quantum mechanics indicate that neurodynamic behavior that originates in thalamo-cortical compared to cortico-cortical regions yield divergent 3-dimensional dispersions of wave forms and quantum energy [21].

The area of tissue by which the oscillatory mechanisms are thought to originate inherently determines the validity and applicability of Ohm’s law to the EEG [1]. Specifically, the amount of tissue by which the neuro-oscillatory generators are thought to be dispersed distorts mathematical estimations of source localization. As the magnitude of space where neural generators are thought to originate increases, Ohm’s law may only be applicable when estimates of current source density are calculated using matrices to account for the macroscopic and anisotropic heterogeneities that permeate calculations of resistivity in subcortical and cortical regions [21].

Applications of the Schrodinger equation and Heisenberg’s Uncertainty Principle indicate that if EEG waveforms are considered as a summation of discrete particles and individual waveforms, the precise cytoarchitectural boundaries and power of the specific quantum particles that comprise these waveforms cannot be derived simultaneously. Estimations of current source density must obey laws of physics related to thermodynamics and electromagnetism. Thus, laws of classical physics and quantum mechanics such as Coulomb’s inverse square law, Gauss’s law of electromagnetism, and Maxwell’s equations can be applied to derive estimations of the Cartesian coordinates that are utilized in 3-dimensional dipole electrical field per unit volumetric estimations of current source density [1, 19].