A real-time path-integral framework for stochastic neural fields via the Henstock–Kurzweil integral and Külbs–Steadman spaces
Abstract
We develop a real-time functional-integral framework for stochastic neural fields based on the Henstock–Kurzweil gauge integral and the Kuelbs–Steadman (KS2) spaces. Unlike classical approaches that rely on a Wick rotation to convert an ill-defined oscillatory path integral into a Wiener integral, the gauge integral works directly in real time: a point-dependent gauge replaces the translation-invariant measure, and conditionally convergent sums become genuine gauge limits. This is essential for biological systems whose dynamics are intrinsically real-time and punctuated by discontinuous spikes. As a concrete test we treat the subjective-centered model of Titov, in which conscious cognitive activity arises from the interaction of a noisy «intentiating» neural field with an «intential» repertoire of stable patterns (exitoms) coupled through the connectome. Discretising the cortex into a finite grid and encoding the linearised Wilson–Cowan dynamics by a symmetric positive- definite matrix A = αI – βW, we obtain a closed-form partition function Z = (2π)^(n/2) (det A)^(-1/2) and the two-point correlation ⟨φ_k φ_l⟩ = Z (A⁻¹)_kl. The cognitive transition probability into an exitom reduces to a one-dimensional Gaussian tail, yielding the formula P = N((m–θ)/√a) in terms of the standard normal CDF. The Wilson–Cowan evolution operator is split via the Chernoff/Lie–Trotter product formula, and we prove that time-sliced approximants converge to the exact propagator exp(t(D+C)), which satisfies the associated Fokker–Planck equation. Every statement is formalised in the Lean 4 / Mathlib file HkNeuralField.lean; all proofs are sorry-free and depend only on propext, Classical.choice, and Quot.sound. The paper also discusses parametric sensitivity of cognitive transitions, the extension to spike-driven activity on the Skorokhod space, an algorithmic bridge from connectome to real-time brain–computer interfaces, and a philosophical interpretation of the gauge integral as a mathematical model of subjectivity.
Related
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BibTeX
@misc{титов2026realtime,
title = {{A real-time path-integral framework for stochastic neural fields via the Henstock–Kurzweil integral and Külbs–Steadman spaces}},
author = {К В Титов and Y. N. Berdinsky and A. S. Ushakov},
howpublished = {HAL},
year = {2026},
month = oct,
doi = {10.5281/zenodo.21726881},
url = {https://hal.science/hal-05711261v3},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A real-time path-integral framework for stochastic neural fields via the Henstock–Kurzweil integral and Külbs–Steadman spaces
AU - К В Титов
AU - Y. N. Berdinsky
AU - A. S. Ushakov
PY - 2026
DA - 2026/10/05
PB - HAL
DO - 10.5281/zenodo.21726881
UR - https://hal.science/hal-05711261v3
AB - We develop a real-time functional-integral framework for stochastic neural fields based on the Henstock–Kurzweil gauge integral and the Kuelbs–Steadman (KS2) spaces. Unlike classical approaches that rely on a Wick rotation to convert an ill-defined oscillatory path integral into a Wiener integral, the gauge integral works directly in real time: a point-dependent gauge replaces the translation-invariant measure, and conditionally convergent sums become genuine gauge limits. This is essential for biological systems whose dynamics are intrinsically real-time and punctuated by discontinuous spikes. As a concrete test we treat the subjective-centered model of Titov, in which conscious cognitive activity arises from the interaction of a noisy «intentiating» neural field with an «intential» repertoire of stable patterns (exitoms) coupled through the connectome. Discretising the cortex into a finite grid and encoding the linearised Wilson–Cowan dynamics by a symmetric positive- definite matrix A = αI – βW, we obtain a closed-form partition function Z = (2π)^(n/2) (det A)^(-1/2) and the two-point correlation ⟨φ_k φ_l⟩ = Z (A⁻¹)_kl. The cognitive transition probability into an exitom reduces to a one-dimensional Gaussian tail, yielding the formula P = N((m–θ)/√a) in terms of the standard normal CDF. The Wilson–Cowan evolution operator is split via the Chernoff/Lie–Trotter product formula, and we prove that time-sliced approximants converge to the exact propagator exp(t(D+C)), which satisfies the associated Fokker–Planck equation. Every statement is formalised in the Lean 4 / Mathlib file HkNeuralField.lean; all proofs are sorry-free and depend only on propext, Classical.choice, and Quot.sound. The paper also discusses parametric sensitivity of cognitive transitions, the extension to spike-driven activity on the Skorokhod space, an algorithmic bridge from connectome to real-time brain–computer interfaces, and a philosophical interpretation of the gauge integral as a mathematical model of subjectivity.
ER -