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. 2009 Apr 29:10:40.
doi: 10.1186/1471-2202-10-40.

Subsampling effects in neuronal avalanche distributions recorded in vivo

Affiliations

Subsampling effects in neuronal avalanche distributions recorded in vivo

Viola Priesemann et al. BMC Neurosci. .

Abstract

Background: Many systems in nature are characterized by complex behaviour where large cascades of events, or avalanches, unpredictably alternate with periods of little activity. Snow avalanches are an example. Often the size distribution f(s) of a system's avalanches follows a power law, and the branching parameter sigma, the average number of events triggered by a single preceding event, is unity. A power law for f(s), and sigma = 1, are hallmark features of self-organized critical (SOC) systems, and both have been found for neuronal activity in vitro. Therefore, and since SOC systems and neuronal activity both show large variability, long-term stability and memory capabilities, SOC has been proposed to govern neuronal dynamics in vivo. Testing this hypothesis is difficult because neuronal activity is spatially or temporally subsampled, while theories of SOC systems assume full sampling. To close this gap, we investigated how subsampling affects f(s) and sigma by imposing subsampling on three different SOC models. We then compared f(s) and sigma of the subsampled models with those of multielectrode local field potential (LFP) activity recorded in three macaque monkeys performing a short term memory task.

Results: Neither the LFP nor the subsampled SOC models showed a power law for f(s). Both, f(s) and sigma, depended sensitively on the subsampling geometry and the dynamics of the model. Only one of the SOC models, the Abelian Sandpile Model, exhibited f(s) and sigma similar to those calculated from LFP activity.

Conclusion: Since subsampling can prevent the observation of the characteristic power law and sigma in SOC systems, misclassifications of critical systems as sub- or supercritical are possible. Nevertheless, the system specific scaling of f(s) and sigma under subsampling conditions may prove useful to select physiologically motivated models of brain function. Models that better reproduce f(s) and sigma calculated from the physiological recordings may be selected over alternatives.

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Figures

Figure 1
Figure 1
Comparison of subsampling effects in avalanche distributions and branching parameters from model systems and in vivo LFP data. (A) Avalanche distributions calculated from events sampled on a small fraction of the model sites (as indicated in B). None of the models shows a power law for f(s). Note that the characteristic peaks in f(s) are only expressed in the ASM and only when the distance between the sampling sites is small (left column). Avalanche distributions of the ASM (full lines), the RNM (dashed lines) and the FFM (dotted lines) are plotted. The colours indicate the different bin sizes (blue 2 steps; green 4 steps; red 8 steps). (B) Recording electrode configurations and corresponding sampling sites used in the simulations. The circles indicate the position of the electrodes, full circles indicate the electrodes that provided data for the evaluation of the LFPs. The inter electrode distance is given at the bottom of each figure. The full circles at the same time indicate the configuration of the subset of sampling sites sampled in the models. The left part of each figure indicates the position of the subsets of sampling sites with respect to the grid the model was simulated on. The left figure indicates the subset of sampling sites s2: 4×4 sites with distance 2 grid units (g.u.) between the sites, located in the center of the grid. The middle figure 4 enotes the subset of sampling sites s5, and the right the subset of sampling sites c2. (C) Avalanche size distributions f(s) for the binary events calculated from the LFPs. The colours indicate the bin size (blue 2 ms; green 4 ms; red 8 ms): left figure – Monkey1 (LFP M1), middle – Monkey2, LFP2 M2, right – Monkey3, LFP M3. The corresponding electrode configurations are plotted in part (B). (D) Branching parameter σ over bin size (in ms for LFP data and steps for simulated data) for the subsampled models and the LFP data. (Left) σ for sampling on subset s2 (related to the LFP recording geometry in M1). Green, solid line: σ for LFP of M1; dashed lines: σ for the subsampled ASM, FFM, and RNM; (Middle) σ for sampling on subset s5 (LFP of M2), same colour codes; (Right) σ for sampling on subset c2 (LFP of M3), same colour codes.
Figure 2
Figure 2
Definition of binary events from raw LFP data and detection of avalanches. (A) Sample LFP traces recorded simultaneously on 14 electrodes in monkey 1. LFPs were z-transformed with respect to pre-trial baseline. The red dots indicate binary events calculated from the LFP traces (see figure part B). (B) Algorithm for calculating binary events from the LFPs. Deflection lobes under the LFP trace are coloured in grey. Blue bars indicate the value for the area under a deflection lobe between two zero crossings. A binary event (red dot) is generated, if the absolute value of the area exceeds a threshold of 5SD of the absolute areas of deflection lobes in the baseline. (C) Avalanche definition. Binary events are concatenated in temporal bins (here: 4 ms). The avalanche size s is the total number of events in subsequent nonempty time bins. The single-step branching ratio σ' for the transition from one time bin to the next is calculated as described in the methods. The branching parameter σ is defined as the average of all single-step branching parameters. (D) Definition of the drop parameter δ for avalanche distributions with peaks at the number of sampled sites/electrodes and its multiples (N, 2N, ..). The drop delta is the difference of the value of f(s) at N and the value at this point obtained by linear extrapolation from the right (see methods).
Figure 3
Figure 3
Avalanche distributions and branching parameters for fully sampled model sytems. (A) Avalanche Distributions for the fully sampled SOC models evaluated in logarithmic binning. The avalanche distributions of the fully sampled models do not change with the bin size, due to the infinite separation between subsequent avalanches in the models. All f(s) follow a power law for s < 500. The steeper decay for large s is caused by the finite size of the models. Solid line – Abelian sandpile model (ASM). Dashed line – random neighbour model (RNM). Dotted line – forest fire model (FFM). Avalanche distributions have been set apart for better visibility by multiplication with a constant factor per curve. (B) Dependence of the branching parameter σ on the bin size. Sigma is near unity for small bin sizes. For larger bin sizes sigma decays due to the finite size of the models. Solid line – Abelian sandpile model (ASM). Dashed line – random neighbour model (RNM). Dotted line – forest fire model (FFM). Bin size given in simulation steps.
Figure 4
Figure 4
Approximation of a power law distribution with increasing coverage of the system when subsampling. Avalanche size distributions f(s) from models of grid size 50×50, sampled on centred, compact subareas of size 4×4 (purple), 10×10 (light blue), 20×20 (red), 25×25 (green) and fully sampled (blue). (A) ASM. (B) FFM. Note, how the characteristic subsampling effects vanish for the ASM (peaks) and the FFM (peak, steep drop off) with increasing coverage of the system.
Figure 5
Figure 5
Avalanche size distributions f(s) for shuffled data. Avalanche size distribution f(s) obtained from trial shuffled data plotted in semi-logarithmic coordinates. Here, straight lines indicate exponential distributions and not a power law. Coloured lines are averages over 1000 different shufflings. Grey lines around the coloured lines are results obtained for single shufflings. (A) ASM; bin sizes: 2 (purple), 4 (green), 8 (red). (B) LFP data from M1; bin sizes: 2 (purple), 4 (green), 8 (red).
Figure 6
Figure 6
Distribution of inter event intervals (IEIs). (A) IEIs evaluated for data obtained from a single site in M1 (olive), M2 (dark green), M3 (light green), and of the ASM (black). (B) IEIs evaluated for the data obtained from all electrodes in M1 (olive), M2 (dark green), and M3 (light green). (C) IEIs evaluated for data obtained by subsampling the ASM on s2 (black), s5 (dark grey), and c2 (light grey).

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