Neuroeconometrics and Neuroconnectometrics: Mathematical Foundations, Causal Identification, and Topological Manifolds
Abstract
Neuroeconometrics and Neuroconnectometrics: Mathematical Foundations, Causal Identification, and Topological Manifolds presents an axiomatic, measure-theoretic, and differential-topological foundation unifying decision neurobiology, high-dimensional econometrics, and structural connectomics. The monograph resolves the fundamental inverse and measurement problems inherent in neuroimaging modalities (fMRI, MEG, EEG, two-photon calcium imaging) by formulating neural latent utility inversion through regularized Fredholm and weakly singular Volterra integral operators, stochastic jump-diffusion processes, and Hamilton-Jacobi-Bellman viscosity solutions. The treatise introduces rigorous causal identification regimes for transcranial magnetic stimulation (TMS) and optogenetic interventions via non-local Maxwell resolvents and Rubin-Angrist local average treatment effect (LATE) boundaries. On structural and dynamic network architectures, the work establishes exact spectral Hodge Laplacians on simplicial complexes, geodesic regression on the Riemannian manifold of positive-definite covariance matrices, and network control theory under port-Hamiltonian dissipative energy bounds. Subgraph communicability and matrix exponentials are proven to govern risk-preference curvature, while graphon mean field games (G-MFG) model multi-agent strategic equilibrium. High-dimensional graph Elastic Net estimators and continuum spatial autoregressive (C-SAR) field equations on 2D Riemannian manifolds are accompanied by non-asymptotic oracle inequalities, minimax rate bounds, and dynamic Schrödinger bridge optimal transport solvers. Package Contents: Complete 12-chapter research monograph, 10-part scholarly front matter, 8-part back matter (chronological timeline 1850–2026, technical glossary, substantive endnotes, comprehensive bibliography, conceptual index), open-science benchmarking protocols (HCP S1200 / UK Biobank), and verified reference algorithms implemented in pyneuroconnect.
