Nonlinear decoding outperforms linear decoding.

<p><b>A</b>: Luminance trace (red) with linear (blue) and nonlinear KRR (green) and neural network (grey) predictions. <b>B</b>: Average decoder performance (± SD across sites), achievable using increasing numbers of cells with highest L1 filter norm. For nonlinear decoding, “All” is the optimal subset that maximizes performance (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006057#pcbi.1006057.s007" target="_blank">S7 Fig</a>). Since the neural network (grey point with an error bar) simultaneously decodes the movie at all sites, it only makes sense to train it using “All” cells. <b>C</b>: Average ROC across all testing movie frames. <b>D</b>: Fractional improvement (average ± SEM across sites) of nonlinear KRR versus linear decoders for test stimuli with different numbers of discs. All decoders were trained only on the 10-disc stimulus. <b>E</b>: Decoding error (MSE; average ± SEM across sites) in fluctuating and constant epochs is significantly larger for linear decoders (p<0.001) relative to nonlinear KRR and the neural network.</p

Linear decoding of a complex movie.

<p><b>A</b>: An example stimulus frame. At each site (red dots = partially shown 20×20 grid) the stimulus was convolved with a spatial gaussian filter (red circle = 1<i>σ</i>). Typical RGC receptive field center size shown in gray. <b>B</b>: Responses of 91 RGCs with 750 <i>ms</i> decoding window overlaid in blue. <b>C</b>: Three example luminance traces (red) and the linear decoders’ predictions (blue). <b>D</b>: Decoded frame (same as in <b>A</b>) reconstructed from 20×20 separately decoded traces. Disc contours of the original frame shown for reference in green. <b>E</b>: RF centers of the 91 cells (black dots = centers of fitted ellipses). RF centers overlapping a chosen site (red dot) are highlighted in blue. <b>F</b>: Performance of the linear decoders across space, as Fraction of Variance Explained (FVE). Black dots as in <b>E</b>; black contour is the boundary <i>FVE</i> = 0.4. <b>G</b>: Performance of the linear decoders (FVE) across sites as a function of cell coverage (grayscale = conditional histograms, red dots = means, error bars = ± SD). <b>H</b>: Average decoding error across sites (MSE ± SD) of 10-disc-trained decoders, tested on withheld stimuli with different numbers of discs. <b>I</b>: Cells (black dots = RF center positions) contributing to the decoding at two example sites (red circles); decoding filters shown below. For each site, contributing cells (highlighted in red and joined to the site) account for at least half of the total L1 norm. <b>J</b>: Decoding field of a single cell (here, evaluated over a denser 50×50 grid and normalized to unit maximal variance); the cell’s RF center shown in black.</p

Spike-history dependencies affect decoding performance.

<p><b>A</b>: Shuffles of responses to repeated stimulus presentations remove different types of correlations, but preserve average locking to the stimulus (PSTH), and thus stimulus-induced correlations. <b>B</b>: A repeated stimulus fragment (red trace), nonlinear kernelized decoder predictions using real responses (green), and using responses without different types of correlations (gray); shown is the prediction mean ± SD over repeats. <b>C</b>: Increase in decoding error (MSE) when spike-history dependencies or noise correlations are removed (average ± SEM across sites); percentages report fractional differences relative to the original performance. <b>D</b>: Spike count distributions for a single example cell. Removing spike-history dependencies broadens the distributions, in particular in constant epochs. Dashed line = expectation for a fully randomized spike train with a matched firing rate. <b>E</b>: Variance-to-mean ratio <i>F</i> of spike count distributions for spike trains with and without spike-history dependencies. Each point is a cell that contributes most to decoding at a particular site (when the same cell contributes to multiple sites, average ± SD across sites is shown).</p

Spike-history dependencies of intermediate strength facilitate nonlinear decoding in simple models of neural processing.

<p><b>A</b>: Schematic of a single-cell Generalized Linear Model (see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006057#sec002" target="_blank">Methods</a>). The neuron’s sensitivity to the stimulus is determined by a radially symmetric difference-of-Gaussians spatial filter that has a monophasic timecourse (), and combines additively with the neuron’s sensitivity to its own past spiking, given by filter (with strong refractoriness followed by weak facilitation). Importantly, shapes spike-history dependencies in the resulting spike trains. A nonlinear function <i>f</i>(⋅) (here, threshold-linear) of the combined sensitivities gives the neuron’s instantaneous firing rate that can be used to generate individual spike train instances. Shapes, as well as the temporal and spatial scales of the filters, were realistic for our data. <b>B</b>: Example rasters (50 repeats) generated with the encoding model for a given intensity trace and different magnitudes (<i>α</i>) of spiking history filter . The rasters are matched in PSTH (bottom) but differ in temporal noise correlations. <b>C</b>: Average spike count variance-to-mean ratio, <i>F</i>, (± SD) of the model as a function of <i>α</i> in fluctuating and constant epochs. <b>D</b>: Decoding error as a function of <i>α</i>. Decoders are trained for each separate <i>α</i> and tested on withheld stimuli; shade = SD over 10 spike train realizations.</p

Distributions of mode T_h/d and study of the correlation between hyperpolarization and depolarization times.

<p><b>A.</b> Mode T_h/d Probability Density Functions of a subject and mode T_h/d for each of the events detected in a whole night of sleep (5685 events were detected in that particular night of which only the first 5000 are shown). mark the position of the typical fast/slow hyperpolarization/depolarization mode times of the events of the subject. <b>B. </b><i>Left:</i> T_d vs. T_h for a single subject. <i>Right:</i> mode T_d vs. mode T_h for a single subject. (A logarithmic scale has been used in both cases for the density color coding). <b>C.</b> Probability Density Function of (T_h+T_d) and (mode T_h+mode T_d). The continuous lines correspond to the estimated PDF using KDE (See <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0030757#s4" target="_blank">Materials and Methods</a>).</p

Propagation of the slow waves.

<p><b>A.</b> Filtered (0.1–4 Hz) signals of intracranial EEG contacts of one global slow wave event ordered as a function of the position of the minimum (red dots). The vertical dashed line marks the position of the minimum of the first detection. <b>B.</b> Three examples of local propagation in the temporal lobe. Each example corresponds to a different slow wave event. In all cases the signals are linked to the position of the recording contacts in the brain, represented by red cubes.</p

Figure 5

<p><a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0030757#s2" target="_blank"><b>Results</b></a><b> of the propagation study.. </b><b>A.. </b><i>Left:</i> First and last hubs of the H-wave propagation of all studied subjects superimposed on the same reference brain (Top and left views). <i>Right:</i> First and last hubs of the D-wave propagation of all studied subjects superimposed on the same reference brain (Top and left views). The little black dots correspond to the implanted contacts not presenting a first/last hub character. These are shown to illustrate the total coverage of the sum of the individual implantations. Please note the final coverage of the sum of all the implantations is not homogeneous with 295 (71%) contacts is the left hemisphere and 122 (29%) in the right hemisphere. <b>B.</b> Number of first/last hubs found in each of the following regions: Frontal (Fro), Parietal (Par), Insula (Ins), Cingulate cortex (Cin), Temporal (Tem), Occipital (Occ) and Striatum (Str).</p

Illustration of the slow wave events detection method and example of T_h/T_d distributions.

<p><b>A. </b><i>Left</i>: 3D reconstructed implantation of one subject. The red circle marks the approximate position of the scalp reference contact. <i>Right</i>: Example of scalp slow oscillation and one of its intracranial correlates (note the polarity inversion of the intracranial signal with respect to scalp). All signals are filtered in the band 0.1–4.0 Hz. The hyperpolarization and depolarization onsets have been marked for the scalp signal. The duration of the hyperpolarization (T_h) and depolarization (T_d) phases of each oscillation is shown as well (black and gray horizontal lines, respectively). <b>B.</b> Segment of scalp signal presenting six detected slow waves. For the second detection, the T_h and T_d of the detected intracranial correlates have been plotted. The numbers stand for contacts of the same electrode. This particular implantation consisted of 55 contacts distributed in 11 electrodes, 6 electrodes (n.1–6, 30 contacts) in the left frontal lobe and 5 (n.7–11, 25 contacts) in the left temporal lobe. The contacts in the frontal lobe cover from the F1, F2 and orbital regions to interior regions like the anterior part of the cingulated gyrus, the insula and the cortex subcallosum. The electrodes in the temporal lobe cover different cortical and white matter regions as well as deeper structures like the amygdala and the hippocampus. <b>C.</b> Examples of typical Probability Density Functions of T_h and T_d obtained for a single subject. For the construction of the pdfs only the intracranially measured T_h and T_d were taken into account. mark the position of the fast/slow hyperpolarization/depolarization times of the subject.</p

Examples of fast/slow hyperpolarization/depolarization slow waves.

<p><b>A.</b> Superposition of filtered signals (0.1–4 Hz) of scalp EEG recordings (FP1) corresponding to 300 slow waves showing fast hyperpolarization (top) and 300 slow waves showing slow hyperpolarization (bottom) of a single subject (S1). The thick red line is the average profile of the shown filtered signals. A vertical dashed line marks the position of the hyperpolarization onset of the average signal. The graphs in the right panels indicate the region in the <i>mode T_h</i> distribution from which the events in the figures have been drawn. <b>B.</b> Superposition of filtered signals (0.1–4 Hz) of scalp EEG recordings (FP1) corresponding to 300 slow waves showing fast depolarization (top) and 300 slow waves showing slow depolarization (bottom) of subject S1. The thick red line is the average profile of the shown filtered signals. The vertical dashed line marks the end of the depolarization (as has been defined in the text). The graphs in the right panels indicate the region in the <i>mode T_d</i> distribution from which the events in the figure have been drawn.</p

Responses to motion in the receptive field surround in salamander retina.

<p>Responses are shown for two example cells, cell 1 (first column) and cell 2 (second column). For each cell, PSTH of the response to the same trajectory is plotted when the trajectory is displayed over the receptive field center (blue, <b>A</b> and <b>B</b>); in the near surround (green, <b>C</b> and <b>D</b>); and in the far surround (red, <b>E</b> and <b>F</b>). <b>G, H:</b> average time course of the bar speed before a spike for different positions of the trajectory (color code matches that in A-F). <b>I, J:</b> Decoding filter of the same cells for different positions of the trajectory (same color code)</p