Average trajectory of returning walks

Abstract

We compute the average shape of trajectories of some one-dimensional stochastic processes x(t) in the (t, x) plane during an excursion, i.e. between two successive returns to a reference value, finding that it obeys a scaling form. For uncorrelated random walks the average shape is semicircular, independently from the single increments distribution, as long as it is symmetric. Such universality extends to biased random walks and Levy flights, with the exception of a particular class of biased Levy flights. Adding a linear damping term destroys scaling and leads asymptotically to flat excursions. The introduction of short and long ranged noise correlations induces non trivial asymmetric shapes, which are studied numerically.

Power-law statistics and universal scaling in the absence of criticality

“…We add that beyond the collapse of the avalanche trajectories, the shapes onto these avalanche collapse may convey important information, as noted and investigated in the context of one-dimensional random walks [22,23]. We observe indeed that the shape of the network-generated avalanches are not similar to the shapes obtained in the Brownian or Ornstein-Uhlenbeck case, and may similarly contain an information that goes beyond pure shape collapse reported in neural data [25].…”

“…However, the method of analyzing the amplitude of negative LFP peaks was shown to produce spurious power laws scalings [21] regardless of the spike activity of cells. Indeed, identical scalings were found in surrogate data, positive LFP peaks (that are independent of spiking activity), and also arise in elementary purely stochastic signals, such as excursions of OrnsteinUhlenbeck processes through thresholds away from the mean, or in one-dimensional random walks [22,23]: both duration and time of excursions show power-law statistics, and display shape invariance. It was further shown in that both in data and surrogate models, statistical significance of these power-laws of LFP peak was poor, and depended on the threshold chosen.…”

Scale-Free Bursting in Human Cortex following Hypoxia at Birth

“…Burst shape analysis is a sharper probe than analysis of size distribution alone, because systems with the same power laws can exhibit different average shapes (Sethna et al, 2001). Scale-dependent deviations in average shapes, such as flattening and asymmetrical skewing (Spasojević et al, 1996;Baldassarri et al, 2003;Colaiori et al, 2004;Zapperi et al, 2005), have been observed in both empirical data and theoretical models, yielding new insights into the underlying mechanisms. To improve the signal-to-noise ratio for the average shape analysis, we pooled bursts over narrow ranges of durations and averaged within these bins (Papanikolaou et al, 2011), rather than averaging only over bursts of exactly the same duration.…”

“…1E). Next we rescaled the bursts to have unit area and unit duration (Colaiori et al, 2004;Fig. 1F ).…”

“…Crackling noise occurs when interactions in a system are in a critical balance between amplification and dissipation (Colaiori et al, 2004). Given the critical balance of excitatory and inhibi-tory activities in the cortex (Shu et al, 2003), crackling noise might be expected to arise in the brain.…”