The Foraging Brain: Evidence of Lévy Dynamics in Brain Networks

Intrinsic bursts facilitate learning of Lévy flight movements in recurrent neural network models

Isolated spikes and bursts of spikes are thought to provide the two major modes of information coding by neurons. Bursts are known to be crucial for fundamental processes between neuron pairs, such as neuronal communications and synaptic plasticity. Neuronal bursting also has implications in neurodegenerative diseases and mental disorders. Despite these findings on the roles of bursts, whether and how bursts have an advantage over isolated spikes in the network-level computation remains elusive. Here, we demonstrate in a computational model that not isolated spikes, but intrinsic bursts can greatly facilitate learning of Lévy flight random walk trajectories by synchronizing burst onsets across a neural population. Lévy flight is a hallmark of optimal search strategies and appears in cognitive behaviors such as saccadic eye movements and memory retrieval. Our results suggest that bursting is crucial for sequence learning by recurrent neural networks when sequences comprise long-tailed distributed discrete jumps.

Thermodynamics of Superdiffusion Generated by Lévy-Wiener Fluctuating Forces

Scale free Lévy motion is a generalized analogue of the Wiener process. Its time derivative extends the notion of “white noise” to non-Gaussian noise sources, and as such, it has been widely used to model natural signal variations described by an overdamped Langevin stochastic differential equation. Here, we consider the dynamics of an archetypal model: a Brownian-like particle is driven by external forces, and noise is represented by uncorrelated Lévy fluctuations. An unperturbed system of that form eventually attains a steady state which is uniquely determined by the set of parameter values. We show that the analyzed Markov process with the stability index α<2 violates the detailed balance, i.e., its stationary state is quantified by a stationary probability density and nonvanishing current. We discuss consequences of the non-Gibbsian character of the stationary state of the system and its impact on the general form of the fluctuation–dissipation theorem derived for weak external forcing.

Lévy flights in the presence of a point sink of finite strength

In this paper, the absorption of a particle undergoing Lévy flight in the presence of a point sink of arbitrary strength and position is studied. The motion of such a particle is given by a modified Fokker-Planck equation whose exact solution in the Laplace domain can be described in terms of the Laplace transform of the unperturbed (absence of the sink) Green's function. This solution for the Green's function is a well-studied, generic result which applies to both fractional and usual Fokker-Planck equations alike. Using this result, the propagator and the absorption-time distribution are obtained for free Lévy flight and Lévy flight in linear and harmonic potentials in the presence of a delta function sink, and their dependence on the sink strength is analyzed. Analytical results are presented for the long-time behavior of the absorption-time distribution in all three above-mentioned potentials. Simulation results are found to corroborate closely with analytical results.