1. Overview: The Electrophysiological Paradigm and Diagnostic Boundaries
Electroencephalography (EEG), pioneered by Hans Berger in 1924, remains the unrivaled gold standard for interrogating macroscopic neural population dynamics with sub-millisecond temporal resolution ($\Delta t < 1\text{ ms}$). By directly registering the synchronous post-synaptic potentials (PSPs) generated by vertically oriented cortical pyramidal neurons, scalp EEG captures the instantaneous firing patterns of cerebral cortex assemblies. However, clinical enthusiasm often obscures a profound physical reality: scalp EEG is inherently constrained by the bioelectromagnetic forward and inverse problems, severe skull volume conduction smearing, and negligible spatial specificity for deep subcortical structures.
Primary Clinical Indications (Gold Standard)
Epilepsy & Paroxysmal Events: Essential for distinguishing focal vs. generalized epileptic seizures [1], mapping epileptogenic zones via spike-and-wave discharges, diagnosing non-convulsive status epilepticus (NCSE), and titrating anti-seizure medications.
Sleep Medicine (Polysomnography): Distinguishing NREM stages (sleep spindles, K-complexes, delta slow-wave activity) from REM sleep atonia [2], narcolepsy, and REM sleep behavior disorder (RBD).
Encephalopathy & Brain Death: Quantifying diffuse cerebral metabolic distress (triphasic waves in hepatic failure, burst suppression under anesthesia, and electrocerebral silence for brain death confirmation).
Clinical Pitfalls & Misapplications
Structural Space-Occupying Lesions: Glioblastomas, subdural hematomas, and acute ischemic strokes generate non-specific focal polymorphic delta activity (PDA) [3]. Attempting to characterize tumor margins or tissue viability using EEG alone is clinically hazardous; high-resolution neuroimaging (MRI/CT) is mandatory.
Psychiatric & Affective Disorders: Major Depressive Disorder, Generalized Anxiety, and Schizophrenia cannot be reliably diagnosed using surface EEG or commercial "qEEG brain maps". Frontal alpha asymmetry and spectral shifts exhibit high inter-individual variance, poor test-retest reliability, and lack diagnostic specificity.
3. Mathematical Formulations & Bioelectromagnetic Principles
Scalp electroencephalography measures the macroscopic electric potential $\Phi(\mathbf{r})$ generated by the spatial summation of microscopic current dipoles within cortical pyramidal cell columns.
Quasi-Static Maxwell & The Forward Volume Conduction Equation
Because neural frequencies of clinical interest ($\le 100\text{ Hz}$) have wavelengths $\lambda_{\text{EM}} \gg \text{head radius}$, capacitive and inductive tissue effects are negligible. The bioelectric forward problem is governed by Poisson's equation for inhomogeneous, anisotropic conductive media:
$$\nabla \cdot (\sigma(\mathbf{r}) \nabla \Phi(\mathbf{r})) = -\sum_{i=1}^{P} I_i \delta(\mathbf{r} - \mathbf{r}_i) = -\nabla \cdot \mathbf{J}^p(\mathbf{r})$$
where $\sigma(\mathbf{r})$ is the position-dependent conductivity tensor ($\sigma_{\text{brain}} \approx 0.33\text{ S/m}$, $\sigma_{\text{CSF}} \approx 1.79\text{ S/m}$, $\sigma_{\text{skull}} \approx 0.0042\text{ S/m}$, $\sigma_{\text{scalp}} \approx 0.33\text{ S/m}$), $\Phi(\mathbf{r})$ is the scalar potential, and $\mathbf{J}^p(\mathbf{r})$ represents the primary impressed neural current source density.
The Ill-Posed Inverse Problem & Tikhonov Regularization
Discretizing $M$ scalp electrodes and $N$ cortical dipole generators ($N \gg M$) yields the linear lead field system $\mathbf{v} = \mathbf{L} \mathbf{j} + \mathbf{\epsilon}$, where $\mathbf{L} \in \mathbb{R}^{M \times 3N}$ is the Lead Field Matrix. Because infinitely many distinct 3D source distributions can produce identical 2D scalp voltage topographies (Helmholtz ill-posedness), the inverse solution requires mathematical constraints:
$$\hat{\mathbf{j}}_{\text{Tikhonov}} = \arg\min_{\mathbf{j}} \left\{ \|\mathbf{v} - \mathbf{L}\mathbf{j}\|_2^2 + \lambda^2 \|\mathbf{W} \mathbf{j}\|_2^2 \right\} = (\mathbf{L}^T \mathbf{L} + \lambda^2 \mathbf{W}^T \mathbf{W})^{-1} \mathbf{L}^T \mathbf{v}$$
where $\lambda$ is the regularization parameter determined via the L-curve criterion and $\mathbf{W}$ is a depth-weighting matrix penalizing superficial cortical bias.
Hemodynamic & Metabolic Multimodal Coupling
Simultaneous EEG-fMRI bridges neural electrophysiology and hemodynamics via the canonical Hemodynamic Response Function (HRF) $h(t)$:
$$y_{\text{BOLD}}(t) = \left[ \sum_{k} \delta(t - t_k) \ast h(t) \right] + \mathbf{X}\beta + \varepsilon, \quad h(t) = \left(\frac{t}{d_1}\right)^{a_1} e^{-\frac{t - d_1}{b_1}} - c\left(\frac{t}{d_2}\right)^{a_2} e^{-\frac{t - d_2}{b_2}}$$
where epileptiform spikes detected on EEG at times $t_k$ serve as discrete regressors convolved with $h(t)$ to generate voxel-wise general linear models (GLM) in fMRI space [4, 5].