Gerald Cooray

This chapter examines the scaling properties of electromagnetic radiation from finite oscillating current systems within classical electrodynamics. Two complementary formulations are considered: a frequency-domain analysis of standing-wave current distributions along thin wires and a time-domain analysis of transient current pulses propagating on thin wires. In both cases, physically motivated constraints on spatial coherence and microscopic applicability lead to finite maximum values of radiated energy and action for a given source charge. The frequency-domain analysis incorporates a cosmological coherence bound and a microscopic applicability bound associated with the breakdown of continuum classical electrodynamics at very small wavelengths. Within this constrained parameter space, the radiated energy per period reaches a finite maximum. When the source charge is identified with the elementary charge, this maximum approaches the product of Planck’s constant and frequency, while the corresponding wavelength approaches the reduced Bohr radius.The time-domain analysis introduces electromagnetic action as radiated energy multiplied by effective pulse duration. This quantity is remarkably insensitive to detailed waveform shape when expressed in terms of total charge. The analysis likewise incorporates cosmological and microscopic bounds associated with the limits of continuum classical electrodynamics. Within the constrained parameter space, the action associated with the elementary charge equals the reduced Planck constant when the characteristic pulse propagation length coincides with the reduced Bohr radius. These results are interpreted not as derivations of quantum mechanics from classical theory, but as constrained scaling relations within classical electrodynamics.

Background: Analysis of SPES responses often relies on averaging repeated stimulation trials to improve signal quality. However, this may obscure clinically relevant trial-to-trial variation. We tested whether explicitly modelling cross-trial dependencies improves localisation of the epileptogenic zone, using concordance with the clinical SOZ as a proxy endpoint, and explored whether resection of model-positive channels is associated with postsurgical seizure freedom. Methods: We developed an interleaved Hierarchical Attention Transformer (HAT), a deep learning architecture that models cross-trial and cross-channel dependencies in SPES responses without averaging. We compared it with two baselines averaging responses or trial embeddings. Models were evaluated with patient-held-out, repeated five-fold cross-validation on SPES data from 35 patients. Robustness to reduced trial availability at inference was assessed by restricting test inputs to 1 or 5 trials. Associations with surgical outcome were assessed using AUROC and patient-level tests on the proportion of model-positive channels resected. Results: The HAT had higher SOZ concordance than the trial-averaged baseline (AUROC 0.762 vs 0.721; mean paired difference 0.041; one-sided 95% lower confidence bound 0.009; Holm-adjusted p = 0.0197). Performance changed little when inference was restricted to 1 trial. Outcome analyses did not provide statistical evidence that seizure-free patients had a higher proportion of model-positive channels resected (AUROC 0.634; p = 0.102). Conclusions: Modelling cross-trial dependencies improved concordance with the SOZ compared with trial-averaged approaches, while remaining robust to reduced trial availability at inference. Associations with postsurgical outcome were inconclusive, consistent with limited sample size and training on SOZ labels rather than outcome-aligned labels. Key points: Cross-trial modelling of SPES responses improved concordance with the clinical SOZ compared with trial-averaged baselines. In a 35-patient cohort, SOZ AUROC was higher than a trial-averaged baseline (0.762 vs 0.721). Performance was stable when inference used as few as 1 trial, with the largest drop in the trial-averaged baseline. Resection of model-positive channels did not provide statistical evidence of an association with seizure freedom (AUROC 0.634; one-sided Mann–Whitney p = 0.102).

Cortical activity recorded through EEG and MEG reflects complex dynamics that span multiple temporal and spatial scales. Spectral analyses of these signals consistently reveal power-law behavior, a hallmark of turbulent systems. In this paper, we derive a kinetic equation for neural field activity based on wave turbulence theory, highlighting how quantities such as energy and pseudo-particle density flow through wave-space ( k -space) via direct and inverse cascades. We explore how different forms of nonlinearity–particularly 3-wave and 4-wave interactions–shape spectral features, including harmonic generation, spectral dispersion, and transient dynamics. While the observed power-law decays in empirical data are broadly consistent with turbulent cascades, variations across studies—such as the presence of dual decay rates or harmonic structures—point to a diversity of underlying mechanisms. We argue that although no single model fully explains all spectral observations, key constraints emerge: namely, that cortical dynamics exhibit features consistent with turbulent wave systems involving both single and dual cascades and a mixture of 3- and 4-wave interactions. This turbulence-based framework offers a principled and unifying approach to interpreting large-scale brain activity, including state transitions and seizure dynamics.

Understanding how the path of streamers is influenced by background electric fields is crucial in leader progression models and electrical breakdown models in long gaps. While numerous advanced models of streamers exist, applying them to leader progression models to track streamer movement remains computationally intensive and impractical. In this study, we employ one of the simplest streamer models available in the literature to investigate how streamers are deflected in the presence of background electric fields. Our analysis identifies the key parameters that govern this interaction. Additionally, we estimate the time and length scales over which streamers are diverted by a background electric field of a specified strength.

Cortical activity recorded through EEG and MEG reflects complex dynamics that span multiple temporal and spatial scales. Spectral analyses of these signals consistently reveal power-law behaviour, a hallmark of turbulent systems. In this paper, we derive a kinetic equation for neural field activity based on wave turbulence theory, highlighting how quantities such as energy and pseudo-particle density flow through wave-space (k-space) via direct and inverse cascades. We explore how different forms of nonlinearity, particularly 3-wave and 4-wave interactions, shape spectral features, including harmonic generation, spectral dispersion, and transient dynamics. While the observed power-law decays in empirical data are broadly consistent with turbulent cascades, variations across studies, such as the presence of dual decay rates or harmonic structures, point to a diversity of underlying mechanisms. We argue that although no single model fully explains all spectral observations, key constraints emerge: namely, that cortical dynamics exhibit features consistent with turbulent wave systems involving both single and dual cascades and a mixture of 3- and 4-wave interactions. This turbulence-based framework offers a principled and unifying approach to interpreting large-scale brain activity, including state transitions and seizure dynamics.

Cortico‐cortical evoked potentials (CCEPs) are an active electrophysiological technique used during intracranial electroencephalography to evaluate the effective connectivity and influence of therapeutic stimulation between distinct cortical regions and pinpoint epileptogenic zones (EZs) in patients with epilepsy. Various methodologies have been implemented to analyze CCEPs and characterize the epileptogenic networks for EZ localization. Despite its promise, their interpretation remains challenging due to the large volumes of spatially and temporally complex data generated. Early studies focused largely on qualitative descriptors and predefined, semi‐quantitative features such as waveform morphology and peak latencies. However, these methods are limited by the significant heterogeneity in CCEP waveform conformations across patients and cortical regions. The specific technique used for extraction of features, such as the spectral band power and root mean squared values, remains open to empirical refinement, as does choice of appropriate latency windows, with no consensus reached regarding the optimal approach. Graph theoretical metrics such as degree centrality, betweenness centrality, and clustering coefficients can provide a rich representation of epileptogenic network connectivity. However, these metrics are often abstract and difficult to interpret in a clinical setting or to the non‐expert, and their neuroscientific substrates remains poorly understood. The lack of standardization in stimulation protocol and data‐processing pipelines has further contributed to inconsistency in reported findings. Emerging machine learning approaches have been increasingly applied to CCEP data, offering a more data‐driven and potentially generalizable way to identify electrophysiological biomarkers of the epileptogenic effective connectivity. In this article, we discuss qualitative, quantitative, and spectral features; network‐analytical metrics; and more recently, data driven methodologies aimed at improving the interpretability and clinical utility of CCEP data.

Macroscopic studies of cortical tissue reveal a prevalence of oscillatory activity, that reflect a fine tuning of neural interactions. This research extends neural field theories by incorporating generalized oscillatory dynamics into previous work on conservative or semi-conservative neural field dynamics. Prior studies have largely assumed isotropic connections among neural units; however, this study demonstrates that a broad range of anisotropic and fluctuating connections can still sustain oscillations. Using Lagrangian field methods, we examine different types of connectivity, their dynamics, and potential interactions with neural fields. From this theoretical foundation, we derive a framework that incorporates Hebbian and non-Hebbian learning – i.e., plasticity – into the study of neural fields via the concept of a connectivity field.