AISTROSIGHT researchers uncover geometric basis for neural population models

A new paper from AISTROSIGHT researchers Hugues Berry and Leonardo Trujillo, posted on arXiv in May 2026, tackles a long-standing mystery in theoretical neuroscience: why the Lorentzian (Cauchy) distribution keeps appearing in mean-field reductions of coupled oscillators and spiking neurons. The Lorentzian Ansatz has been used for nearly two decades as a heuristic trick to produce low-dimensional descriptions of large neural populations, but its deep mathematical origin remained unclear.

Berry and Trujillo now show that the Ansatz is not arbitrary — it is geometrically necessary. By mapping the dynamics onto the circle via stereographic projection, they demonstrate that the Cauchy-Lorentz family is the unique connected two-dimensional family of continuous probability densities invariant under the induced projective transport from Riccati dynamics. Under the full projective action, only the Lorentzian family survives. This elegantly explains why the Ott-Antonsen reduction (2008) and the Montbrió-Pazó-Roxin reduction (2015) work so well, and why attempts to use Gaussian distributions for similar closures inevitably break down.