We study Fokker-Planck operators arising in the population density approach to networks of spiking neuron models, where a neuronal ensemble is described by a probability density evolving under drift–diffusion in membrane potential subject to a fire-and-reset boundary condition. These operators are intrinsically non-self-adjoint as they model an out-of-equilibrium system where detailed balance is broken, yet spectral decompositions are routinely invoked in applications.
Using the framework of boundary eigenvalue operator functions, we provide a functional-analytic foundation for spectral decompositions. A central result is the proof of Birkhoff regularity, which establishes the completeness of the biorthogonal system of root functions. This 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 introduce a theory of non-stationary inter-spike-interval statistics beyond renewal and quasi-renewal approximations. We formulate the dynamics on an augmented state space, simultaneously tracking membrane potential and the time since the last spike, and derive a two-dimensional Fokker–Planck equation that unifies voltage-based and age-structured descriptions. From this representation we obtain exact inter-spike-interval statistics far from stationarity, derive a tractable hierarchy of moment equations, and develop an analytical linear response theory for ISIs in time-dependent regimes.