Results empirical data sets.

<p>(A) Running time of the complete algorithm by number of nodes plus number of edges ∣<b>V</b>∣ + ∣<b>E</b>∣; (B) Mean percentage of tagged, potentially spurious edges by chosen threshold <i>θ</i> after application of the algorithm, error bars indicate 1 standard deviation (SD); the value for <i>θ</i> obtained from bootstrapping in two example data sets is marked in red; (C) Mooney Stimulus [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0140530#pone.0140530.ref042" target="_blank">42</a>]; (D) Cortical sources after beamforming of MEG data (l.,left; r., right: l. orbitofrontal cortex (OFC); r. middle frontal gyrus (MiFG); l. inferior frontal gyrus (IFG left); r. inferior frontal gyrus (IFG right); l. anterior inferotemporal cortex (aTL left); l. cingulate gyrus (cing); r. premotor cortex (premotor); r. superior temporal gyrus (STG); r. anterior inferotemporal cortex (aTL right); l. fusiform gyrus (FFA); l. angular/supramarginal gyrus (SMG); r. superior parietal lobule/precuneus (SPL); l. caudal ITG/LOC (cITG); r. primary visual cortex (V1)), see also [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0140530#pone.0140530.ref040" target="_blank">40</a>]; (E) Example of removal of tagged edges: MEG data of a face detection task in two subjects. First column shows transfer entropy values prior to detection of potentially spurious edges (<b>Pre</b>). The second column shows color-coded tagged edges (red: Potential cascade effects, blue: potential common drive effects; <i>θ</i> = 3<i>ms</i>). The third column shows the network of directed interactions after removal of all tagged edges (<b>Post</b>).</p

Schematic example of a subproblem of the proposed algorithm.

<p>(A) Example subproblem <math><msubsup><mi>L</mi><mrow><msub><mi>w</mi><mi>i</mi></msub><mo>,</mo><msub><mi>v</mi><mi>j</mi></msub></mrow><mi>n</mi></msubsup></math>: At the <i>n</i>^<i>th</i> algorithmic step, we search for all paths of weight <i>w</i>_<i>i</i> leading to node <i>v</i>_<i>j</i>; (B) Finding a solution for the current subproblem by investigating solutions to prior subproblems: We investigate all predecessors <i>v</i>_<i>p</i> of the current node <i>v</i>_<i>j</i>; if there exists a solution to <math><msubsup><mi>L</mi><mrow><msub><mi>w</mi><mi>p</mi></msub><mo>,</mo><msub><mi>v</mi><mi>p</mi></msub></mrow><mi>M</mi></msubsup></math>, i.e., there is a solution to the prior subproblem <math><mrow><msub><mi>v</mi><mi>s</mi></msub><mo>⇝</mo><msub><mi>w</mi><mi>p</mi></msub><msub><mi>v</mi><mi>p</mi></msub></mrow></math> of finding a path of weight <i>w</i>_<i>p</i> leading from <i>v</i>_<i>s</i> to <i>v</i>_<i>p</i>, and <i>w</i>_<i>p</i> + <i>w</i>_{(<i>v</i>_<i>p</i>, <i>v</i>_<i>j</i>)} = <i>w</i>_<i>i</i>, we find a solution to the current subproblem <math><msubsup><mi>L</mi><mrow><msub><mi>w</mi><mi>i</mi></msub><mo>,</mo><msub><mi>v</mi><mi>j</mi></msub></mrow><mi>n</mi></msubsup></math>.</p

Results running time.

<p>Running times [log(s)] for dynamic programming (A, B) and backtracking (C, D) by number of vertices ∣<b>V</b>∣ and maximum path weight <i>w</i>_<i>crit</i>. Running times are shown for different graph types (SW: small-world, SF: scale-free, RN: random networks with density <i>ρ</i>). Red markers indicate cases of intractability (execution was aborted after a pre-defined limit of reconstructed alternative paths was reached).</p

Overview of the proposed algorithm.

<p>The algorithm expects a weighted and directed graph <b>G</b> = {<b>V</b>,<b>E</b>} and a threshold <i>θ</i> as input. In a preprocessing step, the algorithm creates graph <b>G</b>′ from input <b>G</b>, as an input for the dynamic programming algorithm, by removing edge (<i>v</i>_<i>a</i>, <i>v</i>_<i>b</i>) and by relabeling and reordering nodes. Then, in the next step, alternative paths for (<i>v</i>_<i>a</i>, <i>v</i>_<i>b</i>), are searched through dynamic programming (see also main text). If at least one alternative path is found, paths are reconstructed using a depth first search (DFS, [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0140530#pone.0140530.ref027" target="_blank">27</a>]) to ensure that alternative paths do not contain loops. If an alternative path contains no loops, the currently investigated edge (<i>v</i>_<i>a</i>, <i>v</i>_<i>b</i>) is tagged as potentially spurious. If no alternative edge is found, (<i>v</i>_<i>a</i>, <i>v</i>_<i>b</i>) is considered non-spurious. The algorithm then enters the next iteration, in which the next edge (<i>v</i>_<i>a</i>, <i>v</i>_<i>b</i>) ∈ <b>E</b> is investigated for alternative paths.</p

Visualization of the proposed algorithm.

<p>Search for alternative paths to edge (1, 6) (dotted arrow), i.e., <i>v</i>_<i>s</i> = 1 and <i>v</i>_<i>t</i> = 6. Solutions <math><mrow><msubsup><mi>L</mi><msub><mi>v</mi><mi>s</mi></msub><mi>n</mi></msubsup><mrow><mo>(</mo><msub><mi>w</mi><mi>i</mi></msub><mo>;</mo><mo>⇝</mo><msub><mi>v</mi><mi>j</mi></msub><mo>)</mo></mrow></mrow></math> to subproblems are managed in a two dimensional solution array indexed by path weight <i>w</i>_<i>i</i> and vertex number <i>v</i>_<i>j</i>. Solutions are calculated iteratively over <i>w</i>_<i>i</i> (rows) and <i>v</i>_<i>j</i> (columns). (A) Solution matrix after first iterative step (subproblem <math><mrow><msubsup><mi>L</mi><mrow><msub><mi>v</mi><mi>s</mi></msub></mrow><mn>1</mn></msubsup><mrow><mo>(</mo><mn>1</mn><mo>;</mo><mo>⇝</mo><msub><mi>v</mi><mn>2</mn></msub><mo>)</mo></mrow></mrow></math>): There are two edges leading to vertex 2 of which only edge (1, 2) yields a valid solution by pointing to an earlier solved subproblem (green box), whereas edge (5, 2) has a weight of 7 leading to a negative difference in weights <i>i</i> − <i>w</i>_(5,2) (red box) for which no earlier solution exists; (B) Solution matrix after third iterative step <math><mrow><msubsup><mi>L</mi><mrow><msub><mi>v</mi><mi>s</mi></msub></mrow><mn>3</mn></msubsup><mrow><mo>(</mo><mn>1</mn><mo>;</mo><mo>⇝</mo><msub><mi>v</mi><mn>3</mn></msub><mo>)</mo></mrow></mrow></math>: Here, no valid solution exists (none of the arrows leading to vertex 2 are part of a path with summed weight 1); (C) Solution array after iteration over all vertices <i>v</i>_<i>j</i> for <i>w</i>_<i>i</i> = 1 (all vertices have been checked for a path of weight 1, originating from the start vertex 1); (D) Solution to subproblem <math><mrow><msubsup><mi>L</mi><mrow><msub><mi>v</mi><mi>s</mi></msub></mrow><mi>n</mi></msubsup><mrow><mo>(</mo><mn>6</mn><mo>;</mo><mo>⇝</mo><msub><mi>v</mi><mn>6</mn></msub><mo>)</mo></mrow></mrow></math>: edge (4, 6) together with the solution <math><mrow><msub><mi>L</mi><msub><mi>v</mi><mi>s</mi></msub></msub><mrow><mo>(</mo><mn>5</mn><mo>;</mo><mo>⇝</mo><msub><mi>v</mi><mn>4</mn></msub><mo>)</mo></mrow></mrow></math> form a valid path, whereas edge (5, 6) is not part of a valid solution as <math><mrow><msub><mi>L</mi><msub><mi>v</mi><mi>s</mi></msub></msub><mrow><mo>(</mo><mn>3</mn><mo>;</mo><mo>⇝</mo><msub><mi>v</mi><mn>5</mn></msub><mo>)</mo></mrow></mrow></math> is empty; (E) The algorithm terminates after iteration over all vertices <i>v</i>_<i>j</i> and path weights up to <i>w</i>_{(<i>v</i>_<i>s</i>, <i>v</i>_<i>t</i>)} + <i>θ</i>, where <i>θ</i> is a user defined threshold of 1. Backtracking is conducted for all entries in the reconstruction interval <i>w</i>_{(<i>v</i>_<i>s</i>, <i>v</i>_<i>t</i>)} ± <i>θ</i> (entries marked blue); (F) Reconstructed alternative path by backtracking of subproblems.</p

Graph representation of neural data.

<p>(A) Recorded signals from various sources in the brain; (B) Pairwise estimation of transfer entropy (TE) and reconstruction of interaction delays <i>u</i> between any two sources; (C) Adjacency matrix: representation of estimated delay times between all source combinations, every entry represents an information transfer from the <i>i</i>th row to the <i>j</i>th column; (D) Adjacency matrix after test for statistical significance; (E) Visualization of the graph represented by the connectivity matrix: every source is represented by a vertex, every significant information transfer is represented by an edge. (The blue circle indicates the respective representation of an exemplary interaction between source 1 and source 3 throughout all steps of graph reconstruction.)</p

Directed interactions in the turtle brain during visual stimulation with random light pulses (modified from [19], creative common attribution license CC BY).

<p>(A) Raw traces recorded in the tectum (blue) and from the retina (green) overlaid on the light pulses (yellow). (B) Turtle brain explant with eyes attached. Transfer entropy was found from the retina of the right eye to the left tectum, as well as from the light source (yellow) to the retina and to the tectum (***** denotes <i>p</i> < 10⁽⁻⁵⁾). P-values for the opposite directions were not significant (<i>n</i>.<i>s</i>.). Note, that the interaction between light source and optic tectum shows a interaction delay roughly equal to the summed interaction delay between light source and retina and retina and optic tectum (deviation ≤ 5%).</p

Notation.

<p>Notation.</p

Definition of neuronal avalanches.

<p>Black traces show LFP from 44 parallel intracranial depth recording sites in one patient. For each recording site the area under the deflection lobe between two zero crossings was calculated (see green box – blue indicates the area under the deflection lobe). A binary event (red dot) was counted by selecting the biggest area values such, that each recording site during each phase of constant sleep stage had an event rate of exactly ¼ Hz (in this example). The binary events across recording sites occurred in clusters (yellow background). These clusters are called neuronal avalanches. The avalanches were separated by pauses of no activity (white background). The avalanche size <i>s</i> is defined as the total number of binary events in one cluster. As examples, the sizes <i>s</i> of three avalanches were indicated above the raw traces.</p

The neuronal avalanche size distribution <i>f(s)</i> for humans approximated a power law.

<p><b>A.</b> The colored line shows <i>f(s)</i> for all avalanches across all 10 nights, evaluated at different event rates and at <i>bs = 1</i>·. The gray lines show <i>f(s)</i> at <i>r</i> = ¼Hz separately for each of the nights to indicate the variability between recording nights and patients. For better visibility, the gray distributions have some offset, while the colored distributions all are in absolute counts. <i>f(s)</i> approximated a power law (τ = <i>1.5</i> was indicated by the dotted line). The cut off around <i>s</i> = <i>50</i> is known to coincide with the number of recording electrodes, 51 on average. <b>B.</b> The slope of <i>f(s)</i> changed with the temporal scale or bin sizes (<i>bs</i>). The <i>bs</i> was between <i>1/32</i> and <i>4</i>, while here <i>r</i> was fixed at <i>r</i> = ¼ Hz. With larger <i>bs</i>, the slope of <i>f(s)</i> became flatter, but the distributions always resembled a power law. <b>C.</b> The slope τ of <i>f(s)</i> depended on the bin size (<i>bs</i>), but little on the rate (colored lines). The full lines show τ from fitting a power law, while the dashed lines show τ and α for a power law with cutoff (see inset for α). τ and α for small <i>bs</i> at high rates are not defined, because the <i>bs</i> there became smaller than the time resolution from sampling (<i>2.5 ms</i>). Estimation errors for τ and α scale with <i>n</i>^−½ where <i>n</i> is the number of samples <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002985#pcbi.1002985-Clauset1" target="_blank">[24]</a>. Here, <i>n</i>≈<i>10⁶</i>, and thus the error is of the order <i>10⁻³</i>, and thus error bars are close to line thickness. For details on the fitting parameters and quality, see also Supplementary <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002985#pcbi.1002985.s007" target="_blank">Table S1</a> and <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002985#pcbi.1002985.s001" target="_blank">Figure S1</a>. <b>D.</b> The branching parameter <i>σ</i> was plotted over the <i>bs</i>. <i>σ</i> changed with the <i>bs</i>, but was similar across event rates (colored lines). The (+) depicts [<i>σ = 1, bs = 1</i>] for visual guidance.</p