The results were obtained from 23 adult female cats, ranging from 2.5–4.5 kg. We recorded from 121 cells in the PGN, 60 of which we held long enough for thorough study. Of these we were certain that 24 were binocular; 23 of the remaining cells were driven by the contralateral eye and 13 by the ipsilateral eye. In addition, we used 10 relay cells recorded from the same animals for control studies.
Estimating receptive fields
Earlier studies described reticular receptive fields, qualitatively, as having diffuse shapes and having varying degrees of sensitivity to stimuli of opposite contrast (e.g., Sanderson, 1971
; Dubin and Cleland, 1977
; Uhlrich et al., 1991
; Funke and Eysel, 1998
). Our first goal was to provide a quantitative description of receptive fields in the PGN in order to specify the visual features that drive these cells. Thus we employed methods that had been successfully used to explore other stations in the early visual pathway including retina and cortex (Touryan et al., 2002
; Rust et al., 2005
; Fairhall et al., 2006
; Schwartz et al., 2006
). Specifically, we used STA and STC analysis of responses to dense noise to recover the first and second order features of the stimulus that neurons encode. As we will describe later, dense noise is an ideal stimulus for further types of analysis.
The STA reveals the first order patterns to which a neuron responds; see Schwartz et al. (2006
) for review. The STA can provide a fine account of the receptive field if On and Off subregions are spatially separated, as for retina, LGN and simple cells in cortex. However, if there are regions in the receptive field where On and Off response overlap, then the influence of the weaker contrast is masked in the STA computed from responses to dense noise. At the extreme, equally strong On and Off responses are cancelled.
STC analysis recovers second order features of the stimulus that drive a cell. For example, it provides a means to explore sensitivity to On and Off stimuli presented to the same part of the receptive field. To perform the analysis, the STA is projected out from the spike-triggered stimulus ensemble, after which the axes of high and low variance (eigenvectors) in the covariance matrix are found using principal component analysis. Transforming the eigenvectors back to the image space reveals the second order spatiotemporal features that a given cell encodes (Schwartz et al., 2006
). This approach has been successful in exploring complex cells in cortex, for example Rust et al. (2005
). Taken together, the results of STA and STC analysis provide sets of linear subunits (also called spatiotemporal filters) that characterize the first and second order visual features that reticular cells encode.
Figure illustrates this process of estimating the subunits of the receptive field for an example cell. One spatiotemporal frame of the STA is shown in Figure . In this and all remaining figures we display only the small window of the larger stimulus grid that contained the receptive field; in this case the STA had 80 dimensions (spatial, 4 × 4; temporal, 5). Subsequent STC analysis yielded a series of 79 eigenvectors, ranked according to their eigenvalues, black circles. The last (80th) eigenvalue, zero, is not shown since it was removed prior to the analysis; Figure . We used a nested bootstrapping method to determine the significance of the eigenvectors (Schwartz et al., 2006
). For this cell, only the first eigenvector was significant; that is, it lay beyond the confidence interval determined by the bootstrap, red circles, Figure . The spatial filter corresponding to the significant eigenvector is displayed at right, indicated by the arrow. We refer to this filter as STC1
; a second significant filter would be called STC2
and so on. Note we use a new color code for these subunits since STC analysis does not provide information about the absolute sign of preferred features.
Figure 4 Establishing the significance of the filters obtained with STC. (A) STA recovered for a sample cell, shown as 4 × 4 window of the entire stimulus grid. (B) Principal component analysis of a (4 × 4 × 5) spike-triggered stimulus (more ...)
Of the 60 cells whose responses to Gaussian noise we recorded, we recovered filters from both STA and STC analysis for 25 cells. The STC subunits were obtained with an average of 20–60 spikes per stimulus dimension. Obtaining even more spikes rarely seemed to improve the chances of recovering additional subunits. Also, substituting independent component analysis for principal component analysis did not increase the number of filters recovered (data not shown). Further, the presence or absence of significant STC subunits did not correlate with the changes in depth of anesthesia, as judged from the power spectrum of the EEG. Last, virtually all significant filters we extracted were excitatory rather than suppressive.
Spatial structure of reticular receptive fields
By recovering the first and second order subunits of the receptive field, we were able to ask if similar patterns emerged across cells or if there were great heterogeneity in the population. We measured diversity in the receptive field by evaluating the relative magnitude of the On and Off subregions in the STA and STC subunits recovered from single cells. Our metric used the ratio between the absolute maxima of the On and Off subregions, setting the numerator to the highest peak and the denominator to the weaker one. A small value, near 1, indicated the presence of spatially separate On and Off peaks. We refer to subunits whose values for the “peak dominance” ratio was >2 as single peaked and those with values ≤2 as double peaked for ease of description. (Note that the relationship between overlapped subregions in the STA is “winner-take-all,” so a single peaked STA need not indicate an exclusive preference for stimuli of only one contrast.)
There was a wide range in the relative strength of On and Off subregions in both STA and STC subunits, Figure . We first illustrate two cells that had a double peaked STA and STC1 in which the On and Off subregions were of roughly similar strength for both subunits; in this case the peak dominance ratios were near one, Figure . Note, however, for each of these two cells, the shape of the STA and STC1 were different. Figure depicts two cases for which the STA and STC had only a single peak. Even though the STAs resembled those of relay cells in the LGN, these reticular cells had overlapping dark and bright responses, as captured by filters recovered with STC analysis. In other instances, the relative weights of On and Off peaks in STAs were different from those in the filters recovered with STC analysis. Examples of cells with a double peaked STA, but single peaked STCK's, are illustrated in Figure while cells with reverse arrangement of subunits are illustrated in Figure .
Figure 5 Receptive field diversity in the PGN. (A–D) Each panel illustrates spatial maps of subunits (STA, top and STC bottom) of the receptive field for two cells with similar receptive fields (yellow squares indicate 1°visual angle). The cells (more ...)
To assess the extent of spatial diversity of reticular receptive fields, we plotted a histogram of values for the peak dominance ratio for each STA and STCK recovered for the population, Figure . The plot of the STAs is color coded according to preference for stimulus contrast. It shows that strong responses to both bright and dark stimuli are common, Figure . The plot also suggests that the relative weight of On and Off subregions varies continuously across the population. The distribution of peak dominance ratios is similar for subunits obtained with STC analysis, Figure . Last, we compared the peak dominance of STA and STC subunits for the same cells, Figure . Most points lie at some distance above or below the line of unity slope, further supporting the idea that neural receptive fields in the PGN are selective for many different combinations of visual features.
Linear-non-linear models of the PGN
Components of the model
Quantitative maps of the spatiotemporal receptive fields in the retina and the LGN allow one to build simple computational models that predict neural responses reasonably well (Simoncelli et al., 2004
; Carandini et al., 2005
). But the complicated and diverse shapes of reticular receptive fields hinted that these simple models might not perform as well in the PGN. To determine how well subunits recovered from spike-triggered analyses help to predict neural responses in the PGN, we used simple LN models (Simoncelli et al., 2004
; Carandini et al., 2005
). The linear stage of the model was built using one or more filters (derived from STA or STC analysis) that were convolved with a time varying visual stimulus to generate a “filter output.” The non-linear stage comprised a static non-linearity, one for each subunit, which mapped the filter output to the strength of the corresponding neural response.
The components of the model are illustrated for a cell from which we recovered an STA and STC1 subunit, Figure . A snapshot in time for each linear spatiotemporal filter (STA and STC1) is shown as a contour plot at left, next to its associated non-linearity. The non-linearities are curves plotted as firing rate against the filter output (see “Methods” for greater detail). One can think of the non-linearity as a lookup table that maps a given value of the filter output to a (mean) firing rate.
Figure 6 1D and 2D LN models. (A) Subunits recovered using STA and STC analysis with the corresponding response non-linearities for an example cell. (B) Joint response non-linearity, displayed as grayscale map, with the corresponding 1D non-linearities shown at (more ...)
The non-linearities for the STA and STC subunits have different shapes. For the STA, stimuli with the same polarity as the filter produced positive outputs and led to elevated spike rates; further, firing rate grew with the magnitude of the filter output. By contrast, filter outputs for stimuli that had the opposite polarity produced negative values that were not associated with notable changes in firing rate. Thus the shape of the non-linearity for the STA subunit was one-sided, or (approximately) half-wave rectified. The shape of the non-linearity associated with the STC subunit was qualitatively different; it was U-shaped. This is because both positive and negative values of the filter output were associated with elevated firing rates. In other words, firing rate grew larger as the absolute value of the filter output increased. For similar examples in the literature, see Rust et al. (2005
), Touryan et al. (2005
) and see Schwartz et al. (2006
) for review.
These non-linearities were generated separately for each filter. However, interactions between subunits have the potential to influence neural responses. Such mutual influence can be estimated by constructing a joint non-linearity, Figure . Here, filter outputs for STA and STC1 are plotted alongside a square grid whose entries represent firing rates evoked by the coincident activation of both subunits as they are variously engaged by the stimulus.
It was necessary to estimate the joint non-linearity separately for each model because interactions between the STA and STC subunits were not merely additive or multiplicative. In other words, if the joint non-linearity simply resulted from the point-wise products of the two individual non-linearities, then the stronger entries in the grid would have formed rectangular contours. However, these stronger entries assumed curved patterns, see Rust et al. (2005
1D and 2D models
Using these components, we were able to build two types of LN models to explore the relative contributions of each different subunit of the receptive field. These were 1D models that included only the STA, Figure , and 2D models that also incorporated subunits derived from STC analysis, Figure . For cells with two or more significant STC subunits, we made pair-wise assessments between each one and the STA. This was because it is difficult to collect enough data to estimate the joint non-linearity for more than two subunits at a time because of the very high number of filter combinations.
Assessing the performance of the models and exploring interactions between subunits using explained variance
How well did these models predict neural responses? Because the models were fitted using only half of the data recorded for each cell, we were able to test the performance using the remaining data, cross-validation (Hastie et al., 2001
). We first compared the performance of the 2D to the 1D models, Figure . The assessments were based on explained variance. This quantity is the amount of variance in the stimulus-driven neural response (i.e., spike rate), that the model predicts. The performance of the more elaborate models was always best. This is seen in Figure , where all points fell above the line of unity slope in a plot of explained variance for the 1D vs. 2D models. Even the best 2D models, however, predicted only about 30% of the neural response; that is, they improved average values (21%) for the 1D models by approximately 50%. It is important to mention that these data exclude cells for which additional STC subunits were not significant, based on the bootstrap; for these cases, the 2D models were not better than those made with the STA alone.
Figure 7 Assessing model performance using explained variance. (A) Explained variance for the 2D vs. 1D model, all the points lie above the line of unity slope. (n = 20). (B) Comparison of explained variance for 1D models of cells in the PGN (n = 20), shaded, (more ...)
Our initial calculations of explained variance used a method developed by others (Haefner and Cumming, 2009
). One might fear that the low values we obtained resulted from an error in our implementation of these methods. Hence, we also used a different method that quantified the amount of stimulus-related signal power, see “Methods” (Sahani and Linden, 2003
). This metric gave similar values; 22% mean explained variance for the 1D model and 30.6% for the 2D model.
Controls using data from the LGN
One might also wonder if our values of explained variance were low because of a problem unique to our preparation. To address this concern we compared predictions of 1D models for reticular cells to those made for relay cells in the LGN that we recorded in the same animals. We used only 1D models since we recovered only STAs from relay cells. The explained variance for relay cells ranged between 60 and 70%, Figure , as has been reported elsewhere (Mante et al., 2008
). Thus, the low values of explained variance we obtained for reticular cells seemed to reflect intrinsic properties of the PGN.
Exploring interactions between subunits using measures of mutual information
Does each subunit make an independent contribution to the neural response or might their interactions reveal redundancy or synergy? To address this question, we explored interactions between subunits using information theory. We assessed the amount of encoded information about single subunits vs. jointly about multiple subunits by estimating the mutual information between the component filters and the neural response. This was done by calculating the Kullback–Leibler divergence between the distribution of filter outputs for the entire length of the stimulus period (the prior distribution) and the distribution of filter outputs just before each spike (the spike conditional distribution), using methods developed previously (Aguera Y Arcas et al., 2003
; Fairhall et al., 2006
). A difference between the two distributions indicates that the filtered stimulus provides information about the neural spike train.
The initial step in the analysis was to address the problem of finite sampling of a continuous signal, which we did by resampling the filter output at different bin sizes, as described by Fairhall et al. (2006
) (see “Methods”). Then, we calculated the information from the STA alone, I(STA)
, as well as the joint information available from the pairs of two subunits, the STA and a (significant) STC subunit, I(STA, STCK)
. The mutual information (between stimulus and response) obtained from the paired subunits I(STA, STCK)
, was significantly greater than that available from the STA alone, I(STA)
, Figure . So far, these results parallel those for explained variance in Figure and seem straightforward.
Figure 8 Assessing model performance by using information theory. (A) Comparison of mutual information estimated from one vs. jointly by two subunits; as in Figure . The error bars depict the standard deviation of the information value (circles) (more ...)
Next, we assessed the possibility of synergistic or redundant influences of one subunit on another. Thus, we determined if the amount of joint information, I(STA, STCK)
, exceeded that from a simple combination of I(STA)
. Synergy was defined by values >0 in an index that subtracts (I(STA)
) from I(STA, STCK)
, as outlined in previous work (Fairhall et al., 2006
). Independent contributions would equal zero whereas redundant interactions would score <0. The plot for the population, Figure , shows that there are synergistic interactions between subunits recovered from most cells and only rare cases of redundancy.
Last, the information theoretic analysis served a second function, as a control for our methods of estimating the performance of the 1D and 2D models. Specifically, we compared values of explained variance obtained earlier with those for mutual information. A ratio of the explained variance for the 2D vs. 1D models is plotted against the ratio of the I(STA, STCK)vs. I(STA) for each reticular cell, Figure . Most points fell along the unity line (R2 = 0.91), indicating that both measures are equally good at assessing the performance of the LN models we made.
Can STAs mapped with sparse noise (individual bright or dark stimuli) predict STA and STC subunits obtained using dense (gaussian) noise?
So far, we have discussed how we recovered subunits of reticular receptive fields, how we assessed their predictive power and estimated the amounts of information they encode. However, we have not addressed the question of how these subunits might be formed. In other words, we wondered how the subunits recovered from responses to Gaussian noise might reflect the On and Off inputs that relay cells supply. Recall that our previous analyses revealed a wide range in the relative strengths of On and Off contributions to filters recovered using STA and STC analysis, Figure . For example, some cells seemed to respond well to On or to Off stimuli while others strongly preferred stimuli of just one polarity.
The simplest explanation for the shapes and signs of the various filters is that these were built from a linear combination of On and Off subregions whose peaks were spatially displaced to varying degrees. For many of the cells we mapped with Gaussian noise, we also collected companion datasets with sparse noise, individually flashed bright and dark pixels. The sparse noise allowed us to recover separate STAs for bright and dark responses; we refer to these specifically as (STAON) and (STAOFF), respectively (we continue to use the term STA, without subscript, to refer to maps made with Gaussian noise). With the results of the sparse noise mapping, it was a simple matter to determine if the subunits (STA and STCK) recovered using Gaussian noise could be simulated by the weighted sum or product of the maps acquired with sparse noise, STAON and STAOFF.
The results of this analysis are depicted for two different cells in Figures . The top row shows the STAON and STAOFF maps; overlaid red and blue ellipses that were fit to the peaks illustrate the spatial offset between On and Off subregions. The middle rows compare the actual STAs obtained with Gaussian noise to those modeled by taking the weighted sum of those acquired with sparse noise (STAON and STAOFF). Specifically, the model was STA = a × STAON + b × STAOFF, where the coefficients, a and b, were optimized to reduce the mean square error between the model STA and the actual STA made from responses to Gaussian noise. The main features of the modeled STAs, prominent On and Off peaks and their relative positions, were similar to the actual STAs acquired using Gaussian noise, suggesting that the maps recovered from Gaussian noise approximate the weighted sum of On and Off inputs. This rough match between the modeled and real STAs was seen for all cells we were able to test, regardless of the relative strength or spatial displacement between On and Off subregions. The sparse noise analysis also allowed us to assess the degree of spatial and temporal overlap between STAON and STAOFF, which we did by calculating the normalized dot product of STAON and STAOFF. For our sample of eight cells, the spatial overlap was 64.26 ± 23.42%, mean ± standard deviation. The peaks of both STAON and STAOFF occurred during the same time interval.
Figure 9 Simulations to explore the construction of receptive fields. STAs for sparse noise are used to predict the STA and STC subunits recovered from responses to dense (Gaussian) noise. (A) Top, STAs made for bright (STAON, left) and dark (STAOFF, right) spots, (more ...)
The subunits recovered using STC analysis of responses to Gaussian noise reflect sensitivity to On and Off inputs. To model these subunits, we used a multiplicative model. Specifically, we modeled each STC as the weighted product of the STAON and STAOFF acquired using sparse noise(c × STAON × STAOFF). Non-zero values of the resulting product indicate the regions where On and Off subregions overlap. These models failed to reproduce the spatial structure of the STCs. At best, the filters produced by this simple operation captured one peak of the actual STC; even then, the modeled peak was often displaced from the actual. This result suggests that reticular cells do not pool their inputs in a simple fashion.
The next step in this analysis was to quantify the degree to which the real and modeled filters matched, which we did by calculating the signal-to-noise ratio. This metric is the logarithm of the ratio of power in the real subunit over the power of the error, with the latter defined as pixel-wise difference between the real and modeled subunit. As expected from visual inspection of the maps displayed in Figures , the performance of the modeled STA was better than that of the modeled STC; Figure . Thus, the weighted summation of On and Off inputs is able to explain overall spatial structure of the linear filters (STAs) recovered from response to a rich stimulus. On the contrary, a simple product of On and Off maps cannot reliably estimate filters generated by non-linear interactions (STCs) between bright and dark stimuli.