Fig 1.
Direction selective responses of MT cells were measured using different direction combinations of stimuli and different attentional conditions. The stimuli in the receptive field (RF) of the recorded cell were either two random dot patterns (RDP) moving in directions 120° apart and placed in spatially separated (panels A–B) or overlapping (panels C–D) apertures or just one single (unidirectional) RDP (panel E). A cue instructed the monkey to attend to either: a luminance change of the fixation point (FP), in the attend-fix condition (afix, panels A and C) and single stimulus (uni, panel E) conditions; or to changes of the direction or velocity of the cued RDP (orange) in the RF, in the attend-in conditions (ain, panels B and D). The transparent uni condition was taken to be the cue-period of the ain condition (panel D). F: Example of a “well-behaved” tuning curve from the spatially separated paradigm in the afix condition. Gray circles denote trial-averaged firing rates and error bars their standard deviation. A sum-of-two-gaussians fit is also shown (brown). The stimulus directions are aligned for each cell, so that the attended direction corresponds to the preferred direction in the uni condition at 240° (see Materials and Methods for details).
Fig 2.
Many tuning curves are not “well-behaved”.
A: typical example of tuning curve from the spatially-separated afix condition (compare with Fig 1F). The shape of the curve—including the position of the two peaks that should be elicited by the composite RDP stimulus—cannot reliably be inferred due to large error bars (std.). B: Histogram of estimated firing rate standard deviations (expressed in relative units, as ratios between std. and a matching mean), obtained by lumping together all stimulus directions and attentional conditions, for the spatially separated (left) and the transparent (right) paradigms. Both these histograms are strongly right-skewed, denoting the existence of cells with highly variable responses to certain stimuli.
Fig 3.
Many models are consistent with the data.
A) Eight model functions were fitted to an example tuning curve (gray error bars denote std.) from the transparent afix condition. Due to the tuning curve’s large error bars all models provided good fits even though they clearly differ. B) According to a goodness-of-fit score (see main text) all eight models provided good fits for almost all cells, independent of experimental condition and paradigm. C) The Akaike Information Criterion AIC was thus employed to select the best (ΔAIC = 0) or at least close to best (ΔAIC≤1) model for each cell. The fraction of cells for which each model constitutes the respective best or almost best model is illustrated with full and light bars. No model was chosen for all cells, still the most widely selected model was the fourth order Fourier series (F4). Both of these facts mirror the high heterogeneity in the data that is hard to capture in a single tuning curve shape. Color code red: 2nd order Fourier (F2); blue: 3rd order Fourier (F3); green: 4th order Fourier (F4); violet: symmetric Beta (sβ); orange—von Mises (vM); yellow: wrapped Cauchy (wC); brown: wrapped Gaussian (wG); pink: wrapped generalized bell-shaped membership function (wB).
Fig 4.
Alternative to fitting: algorithmic features.
A) We describe all tuning curve properties of interest by algorithmically extracted features, as opposed to model parameters. The panel gives pseudocode for three selected features, illustrated also in panels B,C in corresponding colors. B) Feature extraction algorithms take as input only the sampled fitted tuning curve. Features are thus defined independent of a model function, allowing for their comparison between models. Furthermore, all aspects of tuning curves can be described by suitably chosen features such as Maximumleft (dark red), InnerMinimum (orange), Maximumright (blue), (occuring at positions MaximumAngleleft, InnerMinimumAngle and MaximumAngleright, respectively), InnerWidthleft (pink), InnerWidthright (green). C) Due to their algorithmic nature feature extraction rules can equally well be applied directly to the coarse measured trial-averaged tuning curve. Thereby, tuning curve properties can be described and analyzed without referring to the fit.
Table 1.
Selected features for the description of tuning curve shape.
The complete list of features that we used as well as the algorithm for their computation can be found in S1–S4 Tables.
Fig 5.
Effects found in the data depend on model.
A) Nine features, measuring aspects of firing rate (first column), width (second column) and global shape (third column) in all three conditions were calculated for each cell on the basis of eight fitted models (model abbreviations are as in Fig 3) and the “best model” (according to the ΔAIC = 0 criterion, see main text; abbreviated “bM”). Red (blue) indicates a statistically significant (not sig.) difference between two models’ values of that feature. Results from the spatially separated and transparent paradigm are plotted below and above the diagonal, respectively. The panels indicate that, in general, models disagree on the value of a feature, and, in particular, might contradict the optimal (bM) model. B) Histograms count for all feature pairs (depending on their category “sp.sep vs trans”, “sp.sep vs sp.sep” or “trans vs trans”) the number of model functions that find a significant difference (“effect”) between the pair. While mostly all models agree (counts 0 and 9) there are also numerous cases in which the presence of an effect depends on the chosen model.
Fig 6.
Direct method yields very similar results as fits.
A) Layout is similar to Fig 5B. Each bar from there was split into two, depending on if a considered feature pair was judged significantly different (blue) or not (green) when evaluated with the direct method. The panel illustrates a strong tendency to find a significant effect with either both the direct method and all nine models, or with neither the direct method and none of the models. B) z-scored quantitative differences between direct and fitted method’s feature values is less than one standard deviation for almost all features independent of the model indicating a considerable quantitative agreement between the methods. Solid lines in violines mark 2.5%, 50% and 97.5% quantiles. Color code and model abbreviations are as in Figs 3 and 5.
Table 2.
Median values and significant differences for selected features.
Numbers (numbers in parentheses) denote the median value over the population of cells from the corresponding paradigm and condition, calculated with the direct method (best model “bM”). Significant differences between conditions or paradigms (abbreviated here as “sp” and “tr”) are summarized (if present) in the line below the medians. <,≪,⋘ denotes a p-value < 0.05,0.01,0.001, respectively, of a Kruskal-Wallis test applied to the feature values of the direct method. The condition listed before the relation sign had a smaller median than the one after it (if the medians were identical, means were compared). “NA” denotes features that were not defined for the corresponding condition.
Fig 7.
Effects of attention on tuning curves.
Scatter plots of various features in the afix versus the ain condition. Orange error ellipses are centered on the mean feature values with half-axes corresponding to eigenvalues and -vectors of the feature-pair’s covariance matrix. Some outliers were omitted for better visualization. The indicated p-value in each panel corresponds to a Kruskal-Wallis test. A) Attention decreased (increased) the left peak—as measured by the feature normalizedPeakToPeakleft—in the spatially separated (transparent) paradigm. B) Attention increased the right peak in both paradigms according to the feature normalizedPeakToPeakright. C) Attention significantly increased the difference between left and right peak’s inner width—ΔInnerWidth—only for the spatially separated paradigm. Size of circles in panel C illustrates density of points at each particular coordinate (note that values of ΔInnerWidth from the direct method are quantized in steps of 30°due to the design of experimentally used stimuli). Altogether panels indicate that attention asymetrically expanded the right at the expense of the left peak for the spatially separated paradigm, but increased both peaks similarly for the transparent paradigm.
Fig 8.
Effects of adding a second stimulus or attention to the receptive field.
A) Each dot in this cartoon (not based on measured data) represents the observed spike count in one trial. For a given stimulus, spike count distributions can differ between experimental conditions either significantly (e. g. at 60°) or not (e. g. at 240°). B) Distribution of the proportion of cells with a significant difference between conditions for a given number of stimuli (maximum 12). The green histograms represent the two conditions where a second stimulus was added and pink histograms the conditions where attention was switched. C-F) Histograms show the stimulus-dependent fraction of cells with a non-significant response modulation (blue), a significant response enhancement (green) or response suppression (pink). The dotted and orange arrows along the x-axes in E and F indicate the RDP direction not present in the uni condition and the attended RDP in ain condition, respectively. Across the population a second stimulus tended to increase firing rates around 120°(C,D) and to decrease them around 240°. Attention asymmetrically affected the left and right peak in the spatially separated paradigm (E) whereas it symmetrically increased both peaks for the transparent paradigm (F). These stimulus-specific changes were compatible with the results of the direct method discussed in the text.