New spectral theory reveals how neuron refractoriness drives oscillations

Researchers Luca Falorsi, Gianni Valerio Vinci, and Maurizio Mattia have solved a long-standing open problem in computational neuroscience: incorporating an absolute refractory period into the population density approach for spiking neurons. Their paper, posted on arXiv, builds an operator-theoretic framework that augments the state space to include refractory history. The problem is recast as a non-self-adjoint boundary eigenvalue problem for the Fokker-Planck operator, allowing a full spectral characterization of the generator. This proves dissipativity and the existence of a contraction semigroup, and identifies defective eigenvalues as exceptional points where oscillatory modes emerge from coalescing relaxational modes.

Within linear response theory, the team derives an exact transfer function that correctly accounts for boundary conditions modulated by external input—correcting earlier heuristic derivations and revealing additional threshold-noise contributions. Applying this transfer function under a mean-field approximation, they show that refractoriness in populations of interacting neurons can facilitate the onset of limit cycles, i.e., stable oscillations in firing rate. This provides a rigorous foundation for spectral decomposition methods in computational neuroscience and opens the door to further mathematical analysis of neural network dynamics.