A Fokker-Planck formalism for diffusion with finite increments and absorbing boundaries

Gaussian white noise is frequently used to model fluctuations in physical systems. In Fokker-Planck theory, this leads to a vanishing probability density near the absorbing boundary of threshold models. Here we derive the boundary condition for the stationary density of a first-order stochastic differential equation for additive finite-grained Poisson noise and show that the response properties of threshold units are qualitatively altered. Applied to the integrate-and-fire neuron model, the response turns out to be instantaneous rather than exhibiting low-pass characteristics, highly non-linear, and asymmetric for excitation and inhibition. The novel mechanism is exhibited on the network level and is a generic property of pulse-coupled systems of threshold units.

💡 Research Summary

The paper addresses a fundamental limitation of traditional Fokker‑Planck (FP) approaches that model stochastic dynamics with Gaussian white noise. In the classic formulation the diffusion term is continuous, and for systems with an absorbing boundary (e.g., a threshold in integrate‑and‑fire neurons) the stationary probability density necessarily vanishes at the boundary. Consequently, the response of such threshold units behaves like a low‑pass filter: a gradual accumulation of fluctuations is required before the state can cross the threshold.

Real biological and engineered pulse‑coupled systems, however, receive inputs that are discrete events of finite size – synaptic postsynaptic potentials, spikes in digital circuits, or price jumps in finance. These inputs are well described by a Poisson point process with a fixed jump amplitude. The authors therefore consider a first‑order stochastic differential equation (SDE) of the form

dx = f(x) dt + dJ(t),

where dJ(t) is an additive Poisson noise term with rate λ and jump size a. By applying the master‑equation formalism they derive the corresponding FP equation that contains a non‑local jump operator

L_J p(x) = λ