AbstractIn this research we have analyzed functional magnetic resonance imaging (fMRI) signals of different networks in the brain under resting state condition.To such end, the dynamics of signal variation, have been conceived as a stochastic motion, namely it has been modelled through a generalized Langevin stochastic differential equation, which combines a deterministic drift component with a stochastic component where the Gaussian noise source has been replaced with α-stable noise.The parameters of the deterministic and stochastic parts of the model have been fitted from fluctuating data. Results show that the deterministic part is characterized by a simple, linear decreasing trend, and, most important, the α-stable noise, at varying characteristic index α, is the source of a spectrum of activity modes across the networks, from those originated by classic Gaussian noise (α = 2), to longer tailed behaviors generated by the more general Lévy noise (1 ≤ α < 2).Lévy motion is a specific instance of scale-free behavior, it is a source of anomalous diffusion and it has been related to many aspects of human cognition, such as information foraging through memory retrieval or visual exploration.Finally, some conclusions have been drawn on the functional significance of the dynamics corresponding to different α values.Author SummaryIt has been argued, in the literature, that to gain intuition of brain fluctuations one can conceive brain activity as the motion of a random walker or, in the continuous limit, of a diffusing macroscopic particle.In this work we have substantiated such metaphor by modelling the dynamics of the fMRI signal of different brain regions, gathered under resting state condition, via a Langevin-like stochastic equation of motion where we have replaced the white Gaussian noise source with the more general α-stable noise.This way we have been able to show the existence of a spectrum of modes of activity in brain areas. Such modes can be related to the kind of “noise” driving the Langevin equation in a specific region. Further, such modes can be parsimoniously distinguished through the stable characteristic index α, from Gaussian noise (α = 2) to a range of sharply peaked, long tailed behaviors generated by Lévy noise (1 ≤ α < 2).Interestingly enough, random walkers undergoing Lévy motion have been widely used to model the foraging behaviour of a range of animal species and, remarkably, Lévy motion patterns have been related to many aspects of human cognition.
A Stochastic Pitchfork Bifurcation in Most Probable Phase Portraits
