Fig 1.
(Color) Avalanche size and duration distributions from an inhomogeneous Poisson process (IPP, blue, see methods), which approximates power laws that resemble those observed in typical experiments.
Shuffling the events of the IPP results in a homogeneous Poisson process (HPP, gray). For HPPs, the size and duration distributions are (approximately) exponential rather than following a power law. Dashed lines indicate power laws with exponents -1.6 and -2.0 in the left and right panel, respectively.
Fig 2.
(Color) Avalanche size and duration distributions for three example processes, as exemplified in the raster plots above, all with the same mean rate: A&B. homogeneous Poisson process, C&D. inhomogeneous Poisson process, E&F. critical branching process (BP). Different colors represent different bin sizes, Δt, at r = 1 (or equivalently different rates r at Δt = 1). Colored lines or dots are numerical results; black lines are analytical results. A-D. For both the homogeneous and the inhomogeneous Poisson processes, an increase in Δt (or r) makes the size distribution PS(s) and the duration distribution PD(d) flatter. E&F. For the critical system, a change in Δt (or r) hardly changes PS(s), which shows a power law with exponent -1.5 (dashed). The slope of PD(d) changes systematically, because d is in units of bins. In units of time steps, PD(d) would also change very little and show the exponent -2 (dashed).
Fig 3.
(Color) Avalanche size and duration distributions and additional time-series measures for: A. a homogeneous Poisson process (HPP), B. an inhomogeneous Poisson process (IPP, same as Fig 1), and C. a near-critical branching process (BP) with branching parameter σ = 0.999 = σc − 10−3. Circles represent numerical results; black lines represent analytical results; and dashed line are reference power laws. The mean rate of all processes is r = 1, and the bin size (if relevant) is Δt = 1. The top row shows representative examples of raster plots for each process. While HPPs do not follow power laws, the avalanche size distribution of the example IPP does approximate a power law with cutoff, comparable to distributions obtained in experiments and in simulations of critical branching processes.
Fig 4.
(Color) Superposition of exponential distributions that arise from windows with a fixed Poisson firing rate can combine to power law distributions with a cutoff.
A. The weighting function w(r) that leads to a power law with an exponent of -2 in the duration distribution; the analytical result is shown as a dashed black line, and a specific stochastic realization that we used for panels B-D is shown as a non-broken red line. B. Avalanche duration distributions PD(d) arising from the weighting function in A, using either the red weighting function (solid line) or the analytical expression with the same integration limits, i.e., 0.01 to 5 (dashed lines). Colors indicate different bin sizes Δt (in units of 1/r), and the dashed black line is a reference power law with exponent -2. C. Effect of bin size on the cutoff. The different functions depict the theoretical cutoff function Δγ imposed on the target duration distribution PD(d) ~ d−2 for different bin sizes (see text for details; same color code as in B). D. Avalanche size distributions PS(s) from the same red weighting function as in A also show an approximate power law with an exponent characteristic of branching processes (exponent -1.5 is indicated by the dashed black line).
Fig 5.
(Color) Avalanche size and duration distributions obtained for a continuously varying IPP are well approximated by power-law distributions with an exponent of -1.5 (A), and are well approximated by the analytical results, shown for bin size 1 (B). The IPP was realized as a sinusoidal with period T=250s and offset 1 (i.e. sin(t/T)+1), as sketched in the inset. The resulting mean rate is unity. Colored lines correspond to different bin sizes, circles depict analytical results, and the dashed black line depicts a reference power law with an exponent of -1.5.
Fig 6.
(Color) This graph illustrates that combining a number of size distributions recorded in different sessions or in different animals can easily yield approximate power laws.
Depicted are the avalanche size distributions PS(s) from 12 spike recording sessions in Macaque monkeys used in [35] (gray; Δt = 4 ms), and from summing over the 12 individual PS(s) (red; plotted with offset). Dashed line: power law with slope -1.6.
Fig 7.
Avalanche Definition. A. For the avalanche analyses, events from all channels or units are combined into a single vector of activity per bin, A(t|Δt), which is a function of the bin size (Δt). An avalanche is defined as the set of events in a sequence of non-empty time bins. Empty time bins are denoted in blue, events in red. The avalanche size s is defined as the total number of events in an avalanche, the avalanche duration d is defined as its length in bins (depicted above the raster plot). B. With changing the bin size, avalanche measures can also change (modified from [30]).
Fig 8.
The function B(s) approaches its limit lims→∞ B(s) = −∞ very slowly, and thus can be approximated by a constant for large intervals of s.