An important open problem in the theory of Lévy flights concerns the analytically tractable formulation of absorbing boundary conditions. Although numerical studies using the correctly defined nonlocal approach have yielded substantial insights regarding the statistics of first passage, the resultant modifications to the dynamical equations hinder the detailed analysis possible in the absence of these conditions. In this study it is demonstrated that using the first-hit distribution, related to the first passage leapover, as the absorbing sink preserves the tractability of the dynamical equations for a particle undergoing Lévy flight. In particular, knowledge of the first-hit distribution is sufficient to fully determine the first passage time and position density of the particle, without requiring integral truncation or numerical simulations. In addition, we report on the first-hit and leapover properties of first passages and arrivals for Lévy flights of arbitrary skew parameter, and extend these results to Lévy flights in a certain ubiquitous class of potentials satisfying an integral condition.

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Interactions of large numbers of spiking neurons give rise to complex neural dynamics with fluctuations occurring at multiple scales. Understanding the dynamical mechanisms underlying such complex neural dynamics is a long-standing topic of interest in neuroscience, statistical physics and nonlinear dynamics. Conventionally, fluctuating neural dynamics are formulated as balanced, uncorrelated excitatory and inhibitory inputs with Gaussian properties. However, heterogeneous, non-Gaussian properties have been widely observed in both neural connections and neural dynamics. Here, based on balanced neural networks with heterogeneous, non-Gaussian features, our analysis reveals that in the limit of large network size, synaptic inputs possess power-law fluctuations, leading to a remarkable relation of complex neural dynamics to the fractional diffusion formalisms of non-equilibrium physical systems. By uniquely accounting for the leapovers caused by the fluctuations of spiking activity, we further develop a fractional Fokker-Planck equation with absorbing boundary conditions. This body of formalisms represents a novel fractional diffusion theory of heterogeneous neural networks and results in an exact description of the network activity states. This theory is further implemented in a biologically plausible, balanced neural network and identifies a novel type of network state with rich, nonlinear response properties, providing a unified account of a variety of experimental findings on neural dynamics at the individual neuron and the network levels, including fluctuations of membrane potentials and population firing rates. We illustrate that this novel state endows neural networks with a fundamental computational advantage; that is, the neural response is maximised as a function of structural connectivity. Our theory and its network implementations provide a framework for investigating complex neural dynamics emerging from large networks of spiking neurons and their functional roles in neural processing.

Purkinje cell (PC) synapses onto cerebellar nuclei (CbN) neurons convey signals from the cerebellar cortex to the rest of the brain. PCs are inhibitory neurons that spontaneously fire at high rates, and many uniform sized PC inputs are thought to converge onto each CbN neuron to suppress or eliminate firing. Leading theories maintain that PCs encode information using either a rate code, or by synchrony and precise timing. Individual PCs are thought to have limited influence on CbN neuron firing. Here, we find that single PC to CbN synapses are highly variable in size, and using dynamic clamp and modelling we reveal that this has important implications for PC-CbN transmission. Individual PC inputs regulate both the rate and timing of CbN firing. Large PC inputs strongly influence CbN firing rates and transiently eliminate CbN firing for several milliseconds. Remarkably, the refractory period of PCs leads to a briefly elevation of CbN firing prior to suppression. Thus, PC-CbN synapses are suited to concurrently convey rate codes, and generate precisely-timed responses in CbN neurons. Variable input sizes also elevate the baseline firing rates of CbN neurons by increasing the variability of the inhibitory conductance. Although this reduces the relative influence of PC synchrony on the firing rate of CbN neurons, synchrony can still have important consequences, because synchronizing even two large inputs can significantly increase CbN neuron firing. These findings may be generalized to other brain regions with highly variable sized synapses.