Mathematical Formulation of the HULYAS Field Dynamics for Biological Systems

The HULYAS framework introduces a pioneering mathematical model for oscillatory dynamics in biological systems, centered on a scalar field phi(x,t) governed by a fundamental frequency of 1.287 Hz. This framework is the first to mathematically integrate cardiac, neural, and mitochondrial dynamics into a unified model, producing precise harmonic solutions at 1.000 Hz for cardiac rhythms and 39.897 Hz for neural oscillations, alongside energy scales of approximately 10^-19 J (cardiac, mitochondrial) and 10^-20 J (neural). The model is defined by the master partial differential equation Box phi - mu^2 phi + sum C_k(phi) = J, where mu^2 = 10^-20 J^-1 m^2 and J(x,t) is a source term in J m^-3. Coupling terms are rigorously derived: C_k(phi) = 10^-20/k! phi^k, with k=6 for cardiac dynamics (based on hexagonal lattice symmetry to ensure stable limit cycles), k=5 for neural dynamics (derived from odd-powered polynomial potentials to model spike-like solutions), and k=1 for mitochondrial dynamics (reflecting linear response for energy transfer). The base solution phi(t) = phi_0 exp(1.287 t) generates harmonics, with the cardiac mode achieving 1.000 Hz via subharmonic division (1.287/1.287) and the neural mode yielding 39.897 Hz through harmonic multiplication (31 * 1.287). A 0.103 Hz discrepancy in the neural frequency (0.26% error relative to 40 Hz) is addressed by proposing a cubic term g phi^3 to account for higher-order nonlinear effects. The framework provides a verification protocol using finite-difference methods with a timestep <= 0.001 s, applied to systems with masses 10^-12 kg (cardiac) and 10^-15 kg (neural), and initial conditions phi = 10^-19 J (cardiac) or 10^-20 J (neural). An experimental protocol is included, involving application of a 1.287 Hz electromagnetic field (10^-6 T, 30 minutes) to bovine heart mitochondria, with ATP production measured via luciferase assay in triplicate, compared to no-field and 2 Hz controls. This model aligns with biological frequency ranges (0.8–1.2 Hz for cardiac, 30–100 Hz for neural) and energy scales, offering a robust, self-consistent dynamical system for mathematical biophysics. Shared under a Creative Commons license, the HULYAS framework invites researchers to test and extend its applications, advancing the quantitative understanding of complex biological oscillations.