Figure 1.
Observation window limit introduces cut-off in avalanche size distributions.
(A) Sketch of macaque brain with microelectrode array locations (squares; black: premotor cortex, monkey 1; gray: prefrontal cortex, monkey 2). CS: central sulcus; PS: principal sulcus. (B) An example trace of an LFP signal, showing the detection of nLFPs (marked by asterisks) using a threshold of −2.5 SD (dashed line). (C) Raster plot of nLFPs detected from all 91 electrodes (monkey 1) in a period of 2 seconds. nLFPs are represented by individual dots in the plot. (D) nLFPs occurring during either the same or consecutive time bins are detected as a spatiotemporal cluster with the size, s, defined as the number of nLFPs involved. (E) Avalanche size distributions are plotted in double-logarithmic coordinates for four observation windows, i.e., groups of electrodes in the recording array. The size of the observation window, N, is defined as the number of electrodes within the window (see inset for spatial coverage of the windows). The positions of arrows indicate the values of the corresponding N. (F) Continuous avalanche size distributions are plotted for the same observation windows with the size of an avalanche, sAmp, defined as the summated absolute amplitudes of all nLFPs involved. The positions of arrows indicate the values of N× mean absolute nLFP amplitude across all electrodes. For visual comparison, a power law with exponent −1.5 is shown in E and F (dashed lines). E is re-plotted from [17], [32].
Figure 2.
Local activity propagation leads to avalanche dynamics that can be observed by windows with varying sizes.
(A) Small avalanches identified within small windows are parts of larger avalanches identified in large windows. Examples of the spatiotemporal pattern of an avalanche as observed through windows of increasing size. Note that for the smallest window, the avalanche was separated into two smaller avalanches. (B) Probability map of nLFP propagations, showing the probability, p, of detecting a decedent nLFP at certain location in the next time bin (2 ms, upper row; 4 ms, lower row) after a single nLFP has been detected. The initial nLFP is always positioned at the center of the map (0, 0) and the unit of distance, Δd, is the inter-electrode distance of the recording array (0.4 mm). (C) Estimation of balanced propagation depends on window size. The estimated branching parameter, σ, increases with window N, approaching the critical value of σ = 1. (D) Branching parameter as a function of avalanche size, σ(s), is plotted for the four observation windows used in A (color coded). Individual dots represent average σ for avalanches with different sizes, s = 1, …, N for monkey 1 (monkey 2 gave similar results; not shown).
Figure 3.
Characteristics of spatial patterns for avalanches observed before and after the cut-off.
(A) Avalanches observed with a window of N = 24 (see inset). Top panel: the probability distribution is redrawn from Fig. 1E (Monkey 1). Bottom panel: five randomly chosen spatial avalanche patterns each are shown for s = 3, 6, …, 24. In addition, all 19 spatial patterns for avalanches larger than the observation window size (i.e., s>24) are depicted. The number of times that any specific electrode participated in a given avalanche is color-coded. (B) Same as A for the largest observation window with N = 91 electrodes. Only 15 example patterns with s>91 are depicted. (C) The average spatial extent of avalanches, quantified by the number of unique electrodes involved in an avalanche, is plotted as a function of avalanche size for different observation windows. Horizontal dashed lines indicate window size N. The diagonal red line indicates equality. (D) Average percentage of electrodes that do not exhibit repeated activation in an avalanche is plotted as a function of the avalanche size for different observation windows. Vertical dashed lines correspond to the different observation window sizes. The observation windows used are the same as those in Figure 1E.
Figure 4.
The two-layer model exhibits dynamics similar to LFP-based cortical neuronal avalanches.
(A) The diagram of the model, showing a part of the two-dimensional network of binary neurons and the generation of signals at the “electrode level”, i.e., the local spiking activity (LSA). The LSA sampled by simulated electrodes is produced by summation of spiking activities from spatially compact, non-overlapping 10 by 10 neuronal groups (dark gray and blue nodes) and subsequent temporal smoothing. (B) The size distribution of spike avalanches (n = 150,000; red) in the critically tuned network follows a power law with exponent −1.5 (dashed line). (C) Example trace of raw (blue) and temporally smoothed (black) LSA activities (half-width of the Gaussian smoothing window: 30 time steps). LSA peaks (red dots) were detected by applying a threshold of LSA = 0.1. (D) Raster of LSA peaks detected at the electrode level (individual dots represent LSA peaks). (E) Avalanche size distributions observed at the electrode level of the model with local connectivity are plotted for four different observation windows (n = 50,105 avalanches for N = 100). Inset: probability of LSA propagation across the two-dimensional array of simulated electrodes. The positions of arrows indicate the corresponding window sizes. The dotted line is a power law with exponent of −1.5. (F) The estimated branching parameter, σ, is plotted against the observation window size N. (G) The average spatial extent, quantified by the number of unique electrodes involved in an avalanche, is plotted against avalanche size for different observation windows. The horizontal dotted lines indicate window sizes (same as E). The diagonal dotted line indicates equality. (H) Percentage of electrodes without repeated activation during an avalanche is plotted as a function of avalanche size. (I) The same as in E for all-to-all connectivity (inset shows the probability of LSA propagation across the electrodes).
Figure 5.
Quantification of the cut-off.
(A) Rescaled cluster size distributions for monkey 1 show the collapse of the distributions before s/N = 1 (vertical arrow) and different cut-off behaviour for s/N>1 for four different array sizes. (B) The same as A for the model. (C) Cut-off behaviour of cluster size distributions that were obtained for different temporal filter settings. Data was filtered with the same lower cut-off frequency (1 Hz) but different upper cut-off frequencies, fhigh = 100, 150, and 250 Hz (monkey 1; vertical arrow indicates the array size: N = 24 electrodes). (D) The same for the model. Raw LSA traces were smoothed with Gaussian filters of various half-widths (wh = 10, 15, and 30 time steps; array size: N = 30 electrodes). (E) Cut-off index CI (Eq. 7) for size distributions in monkey 1 for all combinations of fhigh and N. (F) CI for distributions obtained from the model with different values of temporal smoothing (wh) and N. To estimate α for the calculation of CI for the model, smin = 4 was used.
Figure 6.
Size relationship of avalanches is only preserved for avalanches smaller than the observation window size.
(A) Observing an avalanche of size sN1 ≤ N1 predicts the size sN2 of the corresponding avalanche observed in window N2 < N1. This prediction power is lost for sN1>N1. The sizes of nLFP clusters were measured for a window of size N1 and plotted against the corresponding cluster sizes that were obtained for a window half as large, i.e., N2 = 0.5×N1 (monkey 1). Vertical arrows indicate the sizes of the larger window. Shown are averages for each size sN1 (gray symbols) and smoothed lines for better visualization (×: N1 = 20, +: N1 = 40, o: N1 = 80). The smaller window with N2 electrodes was completely contained within the larger window with N1 electrodes. (B) The same as A for the model. (C) The same analysis for various values of the upper cut-off frequency, fhigh (N1 = 40, N2 = 20; monkey 1). (D) The same as C for the model with various settings for temporal smoothing of the raw LSA signal.
Figure 7.
The impact of a cut-off on visualizing a power-law distribution and estimating the exponent.
(A) Probability mass function (PMF, inset; calculated as , where
is the Riemann zeta function) and the corresponding complementary cumulative distribution [CCDF, defined as
] for a power-law distribution without cut-off, i.e., the power law holds for arbitrarily large k. The exponents are −1.5 and −0.5 for the PMF and CCDF, respectively. (B) PMFs (inset; defined as
if k ≤ kmax and
if k>kmax, where kmax is the cut-off size) and corresponding CCDFs [defined as
] for power-law distributions with cut-off sizes, kmax = 102, 103, 104, and 105 [dashed lines: power law with exponent α = −1.5 (inset) and −0.5 shown for comparison]. (C) CCDFs for cluster sizes in monkey 1 (see Fig. 1E for corresponding PMFs). (D) Power-law exponents were estimated for synthetic data with varying cut-off size, N, ranging from 8 to 104, assuming the correct model with upper bound (smax = N, red) or an incorrect model without cut-off (smax = ∞, black). Exponents were estimated using a maximum-likelihood approach (shown are the means with error bars indicating the standard deviation across n = 10 synthetic distributions). (E) Power-law exponents were estimated for size distribution of monkey 1 with varying cut-off size, N, ranging from 10 to 91, assuming the correct model with upper bound (smax = N, red) or an incorrect model without cut-off (smax = ∞, black).