Dynamics and learning in cortical neuronal networks

The brain is a large many-body system whose function emerges from collective neuronal activity, yet current theoretical approaches often split between bottom-up statistical mechanics (biophysically grounded but analytically hard at the behavioral scale) and top-down machine learning (algorithmically powerful but weakly constrained by biological implementation). This thesis develops mathematical tools that operate between these perspectives, advancing the theory of stochastic neural populations while proposing dynamical mechanisms for rapid, biologically plausible adaptation. First, we establish a rigorous spectral theory for the Fokker–Planck operators arising in population density approaches to integrate-and-fire neurons with fire-and-reset boundary conditions. Using the framework of boundary eigenvalue operator functions, we provide a functional-analytic foundation for spectral decompositions. A central result is the resolution of the completeness problem: by proving Birkhoff regularity we establish completeness of the biorthogonal system of root functions. This rigorous setting also enables a systematic characterization of exceptional points of the spectrum as defective eigenvalues and motivates a smooth regularization of mode projections near spectral singularities. Second, we develop a theory of non-stationary inter-spike-interval (ISI) statistics beyond renewal and quasi-renewal approximations. We formulate the dynamics on an augmented state space (v, τ) (membrane potential and time since last spike), deriving a two-dimensional Fokker–Planck equation that unifies voltage-based and age-structured descriptions. From this framework we obtain exact ISI statistics far from stationarity, derive a tractable hierarchy of moment equations and develop an analytical linear response theory for the inter-spike intervals. Finally, we shift to a computational perspective and study how neuronal circuits may adapt rapidly without synaptic weight updates. Building on reservoir computing and latent-variable readout parametrizations, we propose gain-modulated recurrent architectures that implement gradient-descent-like learning dynamics in real time through neural activity modulation. This provides a concrete biophysical hypothesis for in-context adaptation in cortical circuits and a bridge between modern machine-learning principles and biological constraints.