Stochastic Resonance in Continuous and Spiking Neuron Models With Levy Noise

Abstract

Levy noise can help neurons detect faint or subthreshold signals. Levy noise extends standard Brownian noise to many types of impulsive jump-noise processes found in real and model neurons as well as in models of finance and other random phenomena. Two new theorems and the ItO calculus show that white Levy noise will benefit subthreshold neuronal signal detection if the noise process's scaled drift velocity falls inside an interval that depends on the threshold values. These results generalize earlier "forbidden interval" theorems of neuronal "stochastic resonance" (SR) or noise-injection benefits. Global and local Lipschitz conditions imply that additive white Levy noise can increase the mutual information or bit count of several feedback neuron models that obey a general stochastic differential equation (SDE). Simulation results show that the same noise benefits still occur for some infinite-variance stable Levy noise processes even though the theorems themselves apply only to finite-variance Levy noise. The proves the two ItO-theoretic lemmas that underlie the new Levy noise-benefit theorems.

Extending Stochastic Resonance for Neuron Models to General LÉvy Noise

“…This has found a wide range of applications in physics, biology, and medicine (see, e.g., [3] and the extensive bibliography in [5]). However, almost all applications up to now have employed Gaussian noise which has continuous sample paths.…”

“…Patel and Kosko [5] have recently published the first paper dealing with SR where the noise is a general Lévy process, however they restricted to the case where the noise has finite variance. …”

The Kalman-Bucy filter for integrable Lévy processes with infinite second moment

“…There have been many interesting applications of these processes, in e.g. mathematical finance (see Chapter 5 of [2] and [9]), evolutionary biology [5] and signal processing [16,3].…”

Non-Gaussian, non-dynamical stochastic resonance

“…Despite some controversies regarding observability of Lévy flights [26,27] and theoretical issues, e.g. unbounded variance the area of applicability of Lévy stable noises and distributions is steadily growing over time including noise induced effects [10,[28][29][30][31][32][33][34][35][36][37][38], neuron systems [39,40], epidemiology [41,42], ecology [43,44], light scattering [45] and many others.…”