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Department of Molecular Biology, Princeton University, New Jersey
Submitted 7 September 2005; accepted in final form 21 November 2005
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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). The somas of neighboring cells tended to be spaced one dendritic diameter apart from one another, the tips of their dendrites just barely touching. A similar territorial organization and dendritic coverage factors close to one have been found in other, prominent morphological types of ganglion cells both in the monkey and the rabbit (Dacey 1993; Vaney 1994). Tight correspondences between anatomy and function have also been found (Wässle and Boycott 1991), along with an increasingly intricate functional segregation of axons and dendrites in the inner plexiform layer (MacNeil et al. 1999; Pang et al. 2002; Roska and Werblin 2001; Sterling 1983; Wassle 2004; Zhang et al. 2004). For all these reasons, tiling has come to be regarded as a fundamental principle of retinal organization.
However, this simple picture of efficient tiling does not hold for all of the ganglion cells. Several other anatomical types have coverage factors greater than two (Rodieck 1998; Wässle and Boycott 1991), and even as large as six (Berson 2003; Pu et al. 1994). In a recent anatomical classification of rabbit ganglion cells, Rockhill et al. (2002) found that their 13 morphological types needed to have an average coverage factor of 3.2 to account for the ganglion cell density that they measured. Interestingly, it has been suggested that a coverage of roughly three is necessary for avoiding aliasing (Wässle and Boycott 1991). Furthermore, the area of the receptive field measured by physiology can be significantly larger than the receptive field measured anatomically (Peichl and Wassle 1979, 1983).
While anatomical techniques have made the systematic study of entire neuronal populations possible, the results of electrophysiology have been more limited. Several multielectrode array studies have recorded simultaneously from many ganglion cells, reporting that brisk transient ganglion cells in the rabbit (DeVries and Baylor 1997) and parasol ganglion cells in the monkey (Frechette et al. 2005) show convincing tiling. However, these methods may not record from a large fraction of the ganglion cells over the array. This is important, because such methods may then sample only a few cell types that tile and overlook other cell types with different properties. We have developed a new method of multielectrode recording and spike sorting that allow us to record from all or nearly all of the ganglion cells in a small patch of the salamander retina (Segev et al. 2004), a species that is technically advantageous. Here, we perform a systematic study of the functional organization of this ganglion cell population. We use several quantitative clustering algorithms to resolve the ganglion cells into functional types and explore the spatial organization of their receptive fields. We also use a more general, information-theoretic method to assess the functional similarity between ganglion cells. Our results indicate that only one functional type tiles visual space in the salamander and that there is extensive mixing of visual information among cells of different functional type.
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METHODS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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Experiments were performed on the larval tiger salamander (Ambystoma tigrinum). Retinas were isolated from the eye in darkness and peeled from the pigment epithelium. Retinas were placed with the ganglion cell layer facing a multielectrode array (Multichannel Systems) and superfused with oxygenated Ringer medium at room temperature. Two array geometries were used: a hexagonal layout with 19 electrodes and 28-µm spacing between electrodes and a rectangular array (6 lines and 5 rows) with 30-µm spacing. For both arrays, the electrode diameter was 10 µm. Extracellularly recorded signals were digitized at 10 kSamples/s on a personal computer and stored for off-line analysis. We used a total of six retinas from six different animals. Our rectangular array had two separate regions of 30 densely spaced electrodes, which allowed us to record from two small patches in each retina. These two patches each had an area 
0.02 mm2 and were 500 µm apart. In some cases, only one patch yielded good signals, because of the curvature of the tissue. In our analysis, we combined the data from several experiments to obtain larger populations of ganglion cells. Before combining experiments, we checked that the overall firing rates and response latencies were the same. We also verified a posteriori that cell types identified by our clustering algorithm were not dominated by cells from a single patch. For the clustering analysis that combined four different response characteristics, we could use only three retinas because we did not have peri-stimulus time histogram data for the other three retinas.
Multichannel spike sorting
The spike sorting procedure was previously described in detail (Segev et al. 2004). Briefly, using the peak voltage on 30 electrodes, we identified examples where a ganglion cell fired an action potential without overlapping spikes from other cells. Such instances are easy to find, although they represent only a fraction of all spikes produced by a given cell. These isolated spikes were averaged together to form a 3.2-ms template of the voltage activity on the array. Templates having a range of time shifts were matched against putative spike patterns in the raw data using mean-squared error as a measure of goodness of fit. Typically, the three or four smallest values of mean-squared error resulted from the same template with successive time shifts, indicating that matches have a temporal precision of 0.1 ms. We subtracted the best fitting template from the raw data and repeated this procedure on that residual until the mean-squared error for the cumulative fit no longer decreased. Most spike waveforms had a peak amplitude >100 µV on a single channel and some were as big as 500 µV, whereas the root-mean-squared electrical noise was about 10 µV. Typically, the amplitude of the spike waveform from a single ganglion cell was larger than the electrical noise on 5–10 channels. Because of the high signal-to-noise ratio of signals on many channels, this iterative procedure robustly found the optimal match.
We studied the retina of the salamander, a species that is especially well suited for recording from all of the ganglion cells for several reasons: its ganglion cells are large, they form a monolayer in the ganglion cell layer, and they occur at a moderate areal density that is uniform across the retina. When we applied our method to recordings from the salamander retina, we found that we could match spike patterns unambiguously to all of the signals significantly above the noise. Comparing the density of ganglion cells over the array using retrograde labeling with the number of isolated cells obtained from the spike sorting revealed that we have recorded from a large fraction of the ganglion cells over the array in the salamander retina (
80%), and the data are consistent with recording from 100% of the overlying cells (see Segev et al. 2004 for details). For comparison, previous methods that use a more widely spaced array of electrodes can only record from 10% of the cells over the array (Meister et al. 1994; Schnitzer and Meister 2003).
When observing ganglion cells with very similar function and nearly concentric receptive fields, we need to make sure that this is not the result of spike sorting errors. To this end, we checked that each sorted spike train did not have spikes within the short refractory period of the ganglion cell spike generator. Typically, the fraction of spikes within a 2-ms interval of the preceding spike was <0.1% (Segev et al. 2004). This indicates that our sorted spike trains were isolated to spikes primarily from a single ganglion cell. Another possible spike sorting error is to split the spikes of a single ganglion cell into two sorted spike trains. We screened for this possibility with two tests. First, we combined pairs of sorted spike trains into a single spike train. If the two spike trains came from the same ganglion cell, the combined spike train will still have a clean refractory period. Second, we checked whether the cross-correlation function between pairs of sorted spike trains had a clean refractory period, because this will only occur if the two spike trains are subsets of spikes from a single ganglion cell. This analysis was carried out for every pair of cells in each experiment (see Figs. 6 and 7 in Segev et al. 2004 for a more details), and whenever a pair failed this test, their spike trains were combined together.
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Random checkerboard stimuli were displayed on a cathode ray tube monitor at a frame rate of 120 Hz. The image from the monitor was focused onto the plane of the retina using standard optics (Meister et al. 1994). In this stimulus ensemble, visual space was divided into 50-µm squares on the retina, which allows many squares to fit inside the receptive field center of each ganglion cell. Within each square, the red, green, and blue levels of the monitor were randomly and independently turned on or off every four frames, resulting in a light intensity and color that flickered rapidly. In some experiments, we used a black and white checkerboard, in which the red, green, and blue guns were either all on or all off in each square. Directional selectivity was probed with drifting square-wave gratings of full contrast (wavelength 1.2 mm on retina; period, 0.33 s) presented in eight different directions.
In addition to artificial stimuli, we also used natural movie clips, which were acquired using a Canon Optura Pi video camera at 30 frames/s. Movies were taken of woodland and urban scenes and included several qualitatively different kinds of motion, such as optic flow and simulated saccades. Individual retinas were stimulated with a 16-min clip that was repeated several times. Further details about the statistics of these movie clips can be found in elsewhere (Puchalla et al. 2005). The mean intensity of the monitor was 12 mW/m2.
The stimulus sequence started with 5 min of spatially uniform light steps—1 s ON and 1 s OFF—to adapt the retina to the light intensities of the monitor. This part of the stimulus was not included in the analysis. Then, we used the flickering checkerboard for 60–120 min, followed by another 5 min of ON-OFF light steps. The second ON-OFF stimulus was used for our clustering analysis. In some of the experiments, we next stimulated the retina with natural movie clips to calculate the shared information. Our results did not change at all if we instead used the first segment of light ON-OFF rather than the second. Furthermore, our results did not change at all if we used only the first half or only the second half of the flickering checkerboard data. This indicates that retinal adaptation and other forms of nonstationarity were not significant in this study.
Receptive field analysis
The receptive field of each cell was mapped by calculating the average stimulus pattern preceding a spike under random checkerboard stimulation. This spike-triggered average (STA) is a function of spatial coordinates x and y (50-µm bin), time before the spike t (8.33-ms bin), and color index l (red, green, or blue). The central region of the receptive field was identified by finding all the squares with a time-course of the same shape as the square with the maximal response. The STA in the central region was closely approximated as the product of three functions: the temporal dynamics A(t), spatial profile B(x,y), and chromatic sensitivity, C(l) (Schnitzer and Meister 2003). The chromatic sensitivity was defined as the amplitude of the STA for each of the red, green, and blue gun (l=1, 2, and 3) of the computer monitor. For convenience, each of the three functions, A, B, and C, was separately normalized to have unit length (Schnitzer and Meister 2003). The spatial profile, B(x,y), was fit using a two-dimensional Gaussian, giving center coordinates, 
, one-sigma radius 
, and area 
. The eccentricity was 
, where 
max and min are the major and minor radii, respectively. The normalized receptive field distance between cells i and j was defined by 
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Classification of ganglion cells
We used a broad stimulus ensemble—spatial, temporal, and chromatic flicker—to classify cells into types having systematically different visual sensitivities (DeVries and Baylor 1997; Meister et al. 1994; Puchalla et al. 2005; Schnitzer and Meister 2003). In addition to the spatio-temporal receptive field, we supplemented our characterization of ganglion cell function with the firing rate in response to diffuse steps of light (Burkhardt et al. 1998; Carcieri et al. 2003; DeVries and Baylor 1997; Hartline 1938) as well as with responses to gratings drifting in different directions (Amthor et al. 1989; Barlow and Levick 1965). The functional similarity between two cells i and j was initially quantified by computing the mean-squared difference between the temporal dynamics of the centers of their receptive fields, 
, normalized by the median mean-squared difference between all pairs of single cells, 
. With this normalization, the typical mean-squared difference between two cells was roughly one, and differences much greater than one indicated that the cells were functionally very dissimilar.
We included three other response characteristics to our quantitative measure of functional similarity. The difference between the receptive field size of two cells was measured by computing 
, where i is the radius of the receptive field center for cell i. Again, we normalized by the median difference between all cells, 
. The difference between two cell’s auto-correlation functions F(t) (calculated 
20 ms with a 1-ms time bin) was measured by 
and normalized by the median difference between all cells to obtain fij. We use the checkerboard stimulus for calculation of F(t). The difference between two cell’s firing rate in response to steps of light R(t) was measured by 
and again normalized in the same fashion to obtain rij. We combined multiple measures of functional similarity by averaging the normalized values together; for instance, combining all four measures gives 
. Notice that combinations are formed using normalized measures of function difference, such as aij and not Aij. Normalization is necessary because individual measures have different units, such as spikes/s for Rij and microns for Bij. We used the median for our normalization to reduce the susceptibility to the outliers of the distribution. Because each measure was normalized, the resulting average effectively weighted each response characteristic equally. We experimented with changes in the relative weight of each response characteristic, but did not find any significant changes in the resulting functional classification.
We did not include the difference in cell’s chromatic profiles, because the chromatic sensitivity was found to be essentially identical for every cell, or the directional index, as salamander ganglion cells did not show significant directionality. In addition, the surround was left out of the classification process because anatomical studies use only the shape of the dendritic tree to classify cells. Furthermore, when we did measure the temporal dynamics of the receptive field surround, we found that its dynamics closely reflect the dynamics of the center. This implies that adding the temporal dynamics of the surround to our classification would not change the resulting clusters.
Broad cell types
Functional classification was carried out using the well-known method of agglomerative clustering, which is an iterative algorithm (Duda et al. 2000). At the outset, each cell formed its own cluster. First, we found the pair of clusters that had the smallest mean-squared difference, Dij. Then, these clusters were merged into a single cluster by averaging all of their properties, weighted by the number of cells in each cluster. This procedure was repeated until all cells were merged into a single cluster. The significance of each merger was evaluated using the merger score, which is the functional difference between the two clusters that were merged together. By looking at the merger score as fewer and fewer clusters remain (Figs. 1C, 2, and 3), we can find that the differences between clusters suddenly become large, which indicates that these clusters are significant.
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Our choices of significance threshold are in every case indicated as a dashed line on plots of the merger score versus number of clusters (Figs. 2 and 3). Further discussion of the issue of choosing the number of significant clusters in a data set can be found in several interesting books and articles (Duda et al. 2000; Jain et al. 1999; Kleinberg 2002). For broad functional types, the algorithm was applied to all of the cells recorded from multiple retinas.
Similarity matrix
To visualize the patterns of functional segregation present in the ganglion cell population, we generated a similarity matrix between cells. In this matrix, each element is the similarity Dij between cells i and j, as defined above for each response characteristic or for averages over several response characteristics. Initially, the cells were in a random order, and therefore no features can be observed in the matrix (Fig. 2A, middle). After applying the clustering algorithm, we reorganized the matrix such that neurons that belong to the same class were grouped together in adjacent rows/columns (Fig. 2A, right). Because the cells that belong to a single group are functionally similar, they should have small difference values between all pairs in the group. This will appear as a block of low difference values (shown by colors close to red) in the reordered similarity matrix. Conversely, the regions that correspond to ganglion cells from different functional types should all have high difference values (shown by colors close to blue). Therefore if the ganglion cell population exhibits clustering into distinct functional types, the reordered matrix will have a clear block structure. If instead the functional distinctions within the population are mostly subtle, the reordered matrix will not show clear block structure.
Fine cell types
To split the ganglion cell population into as many types as could be justified by the data, we used K-means clustering to define cluster boundaries, because this method is known to be biased toward forming extra clusters when used on data with relatively few examples (Duda et al. 2000). In K-means clustering, we first decide how many clusters the data will be divided into and randomly assign one ganglion cell to each cluster. All remaining ganglion cells are assigned to the nearest cluster, based on the (normalized) mean-squared difference between cell i and cluster k, aik. For this analysis, we used only the temporal dynamics of the receptive field center, Aij. Next, the cluster waveform is computed by averaging the temporal waveforms of all members. At this point, a goodness-of-fit measure is obtained by calculating the total mean-squared difference between all cells and their respective clusters, 
. The algorithm iterates by starting with the new cluster waveforms and reassigning all ganglion cells to the nearest cluster. This iteration is continued until the total difference between cells and clusters, 
, no longer decreases. Because the resulting cluster structure depends on the choice of initial clusters, we repeated this algorithm with 1,000 different random choices of initial cluster definitions for each value of K and selected the final cluster partition that had the smallest total difference, .
The number of clusters K is a parameter of this algorithm, and as more clusters are used to describe the population, the total difference (K) must decrease. To determine what value of K resolves significant clusters, we plotted the decrease in the total difference as new clusters were added, 
. When this decrease is large, clusters are significant, and when the decrease is small, the new clusters resolve only minor details in the ganglion cell population (Fig. 6A).
Coverage factor
The coverage factor of each cell type was defined as C=Atype
type, where Atype is the average area within the one-sigma receptive field radius of all cells of a given type, and type is the density of that cell type. Cell type density was determined by multiplying the total ganglion cell density by the fraction of all recorded cells from that cell type. For salamander, we assumed a ganglion cell density of 1,400 cells/mm2 in this calculation (Segev et al. 2004). The average eccentricity of ganglion cell receptive fields was 
=0.46; the minimal and maximal eccentricities were 0.09 and 0.81, respectively. Our definition of the coverage factor is an average that does not reflect the detailed orientation of each cell’s receptive field.
Shared information between ganglion cells
To measure the functional similarity between ganglion cells in a manner that does not rely on prior functional classification or on strong assumptions about how ganglion cells encode visual scenes, we computed the shared information I(R1;R2) between the responses R1 and R2 of two ganglion cells. This quantity is an upper bound on the redundancy between cell pairs (Schneidman et al. 2003) and has values that quite closely correspond with the redundancy. We estimated the shared information using a procedure very similar to that described in (Puchalla et al. 2005) for the redundancy. In brief, the response of each cell was mapped into spike words by binning the spike train in 10-ms time windows and concatenating K time bins to form word, W (Puchalla et al. 2005; Strong et al. 1998). The joint distribution of spike words (W1, W2) for each cell pair was compiled during the entire experiment and used to evaluate 
. The entropy of each cell’s response is given by 
, where pi(Wj) is the probability of spike word j for cell i, and the joint entropy of both cell’s responses H(R1;R2) has an analogous formula using the probability distribution of joint spike words. The firing rates of individual ganglion cells varied by a factor of >100 within the population, and the shared information depended strongly on this firing rate. Therefore, to put all cell pairs on a single, comparable scale, we normalized by min[H(R1), H(R2)]. This normalization factor is an upper bound on the mutual information, so that the normalized shared information, 
, ranges between 0 and 1. Cells that are completely independent in their function have zero shared information, and two cells that are identical have = 1. Note that this normalization is different from that used in Puchalla et al. (2005), so that the value of the shared information is not necessarily larger than the redundancy. A comparison of the two quantities indicated that they correspond quite closely: cells with large redundancy had large shared information, and cells with zero redundancy also had zero shared information.
Correction for finite data size was made by extrapolating the trend in spike train entropies from one-quarter of the data set to all of the data. To best capture the effect of correlations between successive time bins, the entropy trend was extrapolated from words of K = 3 digits up to K = 6 digits. A final bias correction was made by randomly shifting one cell’s spike train relative to the other cell. The shared information between shifted spike trains should be zero, but instead had values ranging up to 3% for strongly correlated cell pairs because of finite sampling. We subtracted the value for shifted spike from the value for the original spike trains. As a test of our bias correction, we note that ON-OFF cell pairs from different retinal patches (with spacing D' > 2) had an average shared information of = 0.0004 ± 0.002 (n = 62 pairs). Because such cell pairs are expected to have zero shared information, this result implies that our estimate has a remaining upward bias of 0.04% and a random error of 0.2%. The overall pattern of shared information within the population (Fig. 9) was not qualitatively changed either by this normalization or by our bias correction.
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RESULTS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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60 functional types assuming all cell types tile or that there are several types that have significant overlap. Classification of ganglion cells
Previous studies have classified ganglion cells into functional types using a variety of methods, relying on different measures of functional similarity and on different ensembles of visual stimuli (Caldwell and Daw 1978; Carcieri et al. 2003; Cleland and Levick 1974a,b; DeVries and Baylor 1997; Grusser-Cornehls and Himstedt 1973; Hochstein and Shapley 1976; Lettvin et al. 1959; Maturana et al. 1960; Roth 1987; Schnitzer and Meister 2003; Stone 1983). Our objective here was to use quantitative clustering techniques along with a very broad set of stimuli, so that our results better reflect the information encoded by ganglion cells under natural visual conditions. Because there exists no a priori method of functional classification, we have made several choices of stimulus set and clustering method to divide the ganglion cells into broader or finer groupings.
Broad types
Following previous studies (DeVries and Baylor 1997; Schnitzer and Meister 2003), we initially used the temporal dynamics of the receptive field center to classify ganglion cells into functional types. We formalized classification using an iterative algorithm that at each step merged the most functionally similar cells into the same cluster and averaged their receptive fields together; similarity was judged by the mean-squared difference between the receptive field center dynamics of two cells, normalized by the median difference between all cells (see METHODS). With this normalization, the typical difference between cells was one. By examining the similarity of the clusters that are merged at each step of this algorithm, we can assess the significance of the merger: when two clusters are very similar, their merger score will be close to zero, and when two clusters are very different, their score will be greater than one.
Figure 1C shows the results of this clustering algorithm when applied to 103 ganglion cells recorded from three retinas in the salamander, whose temporal profiles are shown in Fig. 1D. There was a clear break in the similarity of cell clusters, shown by a dashed line. At this point, there were six distinct clusters with multiple members—fast ON, slow ON, biphasic OFF, monophasic OFF, medium OFF, and slow OFF (shown in colors)—as well as six cells that belonged to their own cluster (shown in gray). The distinct clusters had members recorded from all three retinas used in this study and were routinely observed in other experiments, so we treated them as broad types. Unique cells were observed in a single retinal patch and were not commonly seen in other experiments, so we treated them as unclassified. The classification of cells into six broad types is a robust property of the temporal dynamics of the receptive field center. When we used a different measure of the similarity of ganglion cells—the overlap between two cell’s temporal profile rather than the mean-squared difference—the same six classes emerged from our clustering algorithm (Puchalla et al. 2005).
Other response measures
There are many other aspects of a ganglion cell’s light response on which to base functional classification. To explore whether other response measures could help resolve additional functional classes, we added to our analysis the radius of the spatial profile of the receptive field center, the auto-correlation function measured during random flicker stimulation, and the firing rate during diffuse ON and OFF steps of light (see METHODS). These same three measures were used to resolve functional types in a systematic, multielectrode array study of ganglion cells in the rabbit retina by DeVries and Baylor (1997). For each measure, we calculated the mean-squared difference between ganglion cells and normalized by the typical distance, as we had for the temporal dynamics of the receptive field center. Individual measures were averaged together to give a total difference score between ganglion cells (see labels on each panel of Figs. 2 and 3), and the same agglomerative clustering algorithm described above was used to assign cells to functional classes (see METHODS).
Figure 2 shows the results of this clustering analysis on a different set of 99 ganglion cells from three retinas, where we measured multiple response characteristics. On the left is the merger score as a function of the number of clusters, and on the right is the matrix of similarities between all pairs of cells both before and after the clustering process (see METHODS). When using only the temporal dynamics of the receptive field center (Fig. 2A), the postclustering matrix was clearly organized into five or six large blocks, where all the members of the block have high similarity with each other and lower similarity with cells in other blocks. Each block corresponded with a distinct, significant cluster found by the clustering algorithm. If we take the number of broad types to be six, those types are the same as those shown in Fig. 1: fast ON, slow ON, biphasic OFF, monophasic OFF, medium OFF, and slow OFF.
When clustering was performed with the size of the receptive field center, five clear blocks were evident. However, membership in these clusters did not always correspond with membership in the clusters formed by the receptive field dynamics. (We return to this point in more detail in Fig. 4.) When using the auto correlation function for clustering (Fig. 2B), we found three main blocks corresponding to three classes, along with many outliers. However, one class contained almost all of the cells. Again, the correspondence between membership in classes based on the auto-correlation function and the previous classes was not precise (see Fig. 5). Finally, when we clustered using the response to diffuse steps of light, we again detected three classes. One class was dominant and would be classically defined as an OFF class according to its response to diffuse flash, and two would be defined as ON-OFF classes. Again, these classes did not always correspond with the previous classes, and we also saw response patterns more complex than typically described (see Fig. 6).
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Why do these additional response characteristics blur the distinctions between broad functional types formed using the receptive field dynamics alone? First, each of these response characteristics had great diversity within the ganglion cell population, such that relatively few broad classes could be defined using each characteristic individually. This diversity is perhaps better described as a functional continuum than as several distinct classes. Second, the classes defined by the clustering algorithm did not fall precisely within the classes defined by the temporal dynamics of the receptive field. For instance, the distribution of receptive field radii had only three extensively overlapping peaks (Fig. 4). While both slow ON and slow OFF cells tended to have large receptive fields (Fig. 4A), the monophasic OFF cells had both small- and medium-sized receptive fields (Fig. 4B), and the medium OFF cells could have any size of receptive field (Fig. 4C).
By itself, the autocorrelation function split the ganglion cells into only three broad types: very bursty cells (Fig. 5A) with a short refractory period (1–2 ms) under our stimulus conditions and an abundance of interspike intervals around 2–3 ms; bursty cells (Fig. 5B) with a longer refractory period and many intervals around 5–10 ms; and sustained cells (Fig. 5C), with a much longer refractory period and relatively few short interspike intervals. Again, it should also be kept in mind that both the bursty and sustained classes possessed considerable diversity. Very bursty cells were often fast ON, but could also be OFF-type (Fig. 5A). Most cells fell into the bursty category, making this group quite diverse. Bursty cells tended to come from three of the broad types—biphasic OFF, monophasic OFF, or medium OFF, but the correspondence was not exact. For example, Fig. 5B shows four cells with identical autocorrelation functions; two of them are monophasic OFF and two are slow OFF. Sustained cells were often slow OFF or slow ON, but Fig. 5C again shows cells with nearly identical autocorrelation functions, where one of them is medium OFF. As a result of this inconsistent correspondence between types of autocorrelation functions and types of receptive field dynamics, the autocorrelation function tended to blur the functional categories. The reader might be tempted to think that the data of Fig. 5, B and C, provide evidence that our six broad types have clear subtypes. This is not the case, because there are many intermediate cases that are not shown in the figure.
The light step response showed a broad continuum of ratios between the number of spikes elicited by ON steps versus OFF steps. While most biphasic OFF, monophasic OFF, and medium OFF cells responded to both ON and OFF steps to some degree, the pattern was not precise. Figure 6A shows an example of three biphasic OFF cells; two of them responded transiently to both ON and OFF steps (red), whereas the other responded only to OFF steps (black). The population also had a range of responses that were more transient (Fig. 6A) versus those that were more sustained (Fig. 6B). Transient cells tended to come from biphasic and monophasic OFF-types, whereas sustained cells tended to be slow or medium OFF-type. Again, many exceptions existed. Figure 6B shows an example of four ganglion cells with similar, sustained responses to steps of light; two cells are slow OFF-type, but the other two are monophasic OFF-type. In addition, some ganglion cells had more complex responses, showing multiple bursts of firing (Fig. 6C). As a result of the great diversity of responses to steps of light, this response measure significantly blurred the functional classes identified using other response characteristics.
We conclude that, for the salamander, the response characteristic that divides the population into the most clear functional classes is the temporal dynamics of the receptive field center, Ai(t). This measure resolves six broad functional types: fast ON, slow ON, biphasic OFF, monophasic OFF, medium OFF, and slow OFF. Adding more response characteristics tended to blur these classes rather than resolve clear subclasses, so we treat the clusters resolved by the receptive field center dynamics as the six broad functional types of ganglion cells in the salamander.
Fine types
Because agglomerative clustering algorithms lump the ganglion cell population into no more than six broad functional types, we wanted to explore other clustering schemes that might resolve more types. Our approach was motivated by the observation that for data recorded from a single patch of the retina, where ganglion cells presumably shared inputs from some of the same presynaptic neurons, we often found several ganglion cells with exceptionally similar functional properties. Building on this observation, we used a fine classification scheme, where we considered only cells from a single retinal patch, and used only the receptive field dynamics, as other response characteristics did not help to resolve more classes. To divide the population into the maximal number of cell types allowed by the data, we used K-means clustering to define cluster boundaries (see METHODS). This algorithm is known to be biased toward forming extra clusters when used on data with relatively few examples (Duda et al. 2000). As a result, our definition of fine functional types probably split the ganglion cell population into too many types. Oversplitting will lead to a decrease in the coverage factor and a greater chance of observing territoriality, but it will also give rise to cells of different type that encode very similar visual features.
Figure 7 shows examples of fine types formed from ganglion cells recorded in two different retinal patches (left column, 21 cells; right column, 29 cells). As more clusters were formed, the total difference between the temporal profile of individual ganglion cells and cluster averages decreased (Fig. 7, A and B). For the first retinal patch (left column), this decrease 
(K) was large when the number of clusters was 10 or less, and dropped significantly when >10 clusters were formed. As a result, we divided this group of 21 ganglion cells into 10 fine types (Fig. 7C, color). For the second retinal patch, a transition in the clustering score (K) was found after seven clusters. Consistent results were found for other retinal patches, with a total number of fine types ranging 12, depending in part on how many cells were recorded in a single patch. Our fine classification scheme was consistent with the broad scheme: fine functional types were either the same as a broad type or they were subtypes within a single broad type; the fine types never combined cells from different broad types. Ganglion cells of only three of the broad types—monophasic OFF, medium OFF, and slow OFF—were resolved into multiple categories by our fine clustering. Biphasic OFF cells emerged as a single, clear fine type. Both fast ON and slow ON cells tended to fall into a single fine type, although these classes might split into two or more fine types if we could record more examples in a single retinal patch.
Coverage and territoriality of receptive fields
Tiling consists of two different properties. First, ganglion cells of each class should be territorial, meaning that the location of each cell’s soma avoids other cells of the same type (Wässle and Boycott 1991). Second, the coverage factor of each type of ganglion cell should be roughly equal to one. When we compared the spatial profiles of ganglion cells recorded from the same patch and corresponding to the same fine type, we often found extensive overlap. Several examples are shown in Fig. 7, E and F, with cells of the same fine type displayed in the same color. To quantify the territorial organization of each cell type, we calculated the distance between ganglion cells and normalized that distance by the sum of their receptive field radii; this measure gives a value of one when adjacent receptive fields just barely touch (see METHODS). The distribution of normalized receptive field distances D’ was statistically the same between cells of the same fine type as between cells of different fine type (Fig. 8A). This indicates that most ganglion cell types in the salamander retina do not have a territorial spatial organization. However, we did find one exception. The biphasic OFF cell had a low coverage factor, and its receptive fields just barely touched in most cases (Fig. 8, B and C). In addition, these cells had the largest action potentials recorded and appeared in our measurements with a fraction (7%) close to the fraction of myelinated axons in the optic nerve. These data suggest that the biphasic OFF cell may be analogous to the alpha ganglion cell found in the mammalian retina, a cell type that has been shown to have a territorial organization (Wässle and Boycott 1991).
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60, and many fewer functional types could be resolved. For the broad types, the coverage factors were as follows: fast ON, 5.1; slow ON, 5.6; biphasic OFF, 2.5; monophasic OFF, 13; medium OFF, 21; slow OFF, 10 (see METHODS). Dividing cells in the last three types finely rather than broadly still resulted in coverage factors of 2.5–5 for each fine type, although such fine types did not necessarily correspond from one retinal patch to the next. Therefore regardless of what method of classification we used to decompose the ganglion cell population into cell types, extensive overlap of receptive fields was found for cells of the same functional type. This indicates that within each "channel" of visual information, defined as all of the ganglion cells of the same functional type, multiple cells sample the same features of a visual scene. Shared information between ganglion cells
The definition of coverage that we have discussed so far is completely dependent on the results of our functional classification. Because there are many possible algorithms one might use to perform functional classification as well as many possible response characteristics, we would like to have an assessment of functional overlap that is less dependent on such arbitrary choices. To this end, we used information theory to quantify the functional similarity between ganglion cells. We calculated the shared information, which is closely related to the redundancy between two cells (Schneidman et al. 2003). Because of the wide variety of average firing rates found in the ganglion cell population, we normalized the shared information, , to have a value between zero for independent cells and one for identical cells (see METHODS).
The shared information can be evaluated between cells of either the same or different functional type, and it does not rely on making assumptions about the neural code, such as that a ganglion cell’s function is completely described by its classical receptive field or that each spike conveys the same visual message. In addition, the shared information can be calculated during stimulation with long sequences of natural movies (see METHODS), allowing us to evaluate functional similarity under the most realistic operating conditions possible. Natural stimuli may evoke different patterns of functional overlap between ganglion cells both because they contain visual features, like wide field motion or optic flow, not contained in simpler, artificial stimulus ensembles as well as because they have complex statistics that can evoke different mechanisms of retinal adaptation, which alter a ganglion cell’s spatio-temporal receptive field and could lead to differences in the outcome of a cluster analysis based on the receptive field.
We can use the shared information to evaluate how sharply the ganglion cell population clusters into distinct functional types and to test whether more functional types can be resolved. If two cells with overlapping receptive fields encode completely independent visual features, those cells will have zero shared information. In this case, the neurons should clearly belong to different functional types. However, we might also want to assign cells to different functional types if a less extreme condition is met. Cells of the same functional type should at least share more visual information, at equivalent spatial overlap, than pairs formed from cells of different functional types. This is fundamentally what it means to assign cells to distinct functional types: namely, that cells of the same type have greater functional similarity than cells of different type.
The shared information is a much more abstract quantity than the similarity between receptive fields. To gain intuition about it, we show four specific examples of the shared information between ganglion cells and compare it with the cross-correlation function. For cells of the same functional type and having large spatial receptive field overlap, there was often a prominent peak in the cross-correlation function near zero time lag, and the shared information was relatively high (Fig. 9A, 2 monophasic OFF cells; = 19%). For pairs formed by one ON cell and one OFF cell, there was no peak in the cross-correlation function, and the shared information was close to zero (Fig. 9B; = 0.5%). However, the pattern of information sharing in the ganglion cell population was quite complex. In some cases, cell pairs formed from very different function types had high shared information (Fig. 9C, biphasic OFF/slow OFF; = 14%). In other cases, cell pairs of the same functional type and large spatial overlap had much lower values of the shared information (Fig. 9D, 2 monophasic OFF cells; = 7.5%).
In all cases, there was a correspondence between the value of the shared information and the cross-correlation function: cell pairs with high shared information had a large peak in the cross-correlation function near zero time lag (although this peak varied in width, shape, and exact time lag), and cell pairs that shared no visual information had no such peak. This comparison indicates that the dominant form of correlation between ganglion cells is a tendency to fire spikes synchronously, as described in many previous studies (Brivanlou et al. 1998; DeVries 1999; Mastronarde 1989; Meister et al. 1995), and that this correlation is closely related to the functional similarity between ganglion cells (Puchalla et al. 2005). Anticorrelation was found only very rarely, presumably because salamander ganglion cells have very sparse spike trains (Berry et al. 1997).
To study how visual information is distributed within the entire population, we plotted the shared information versus the receptive field distance D' between 604 pairs of ganglion cells recorded from two retinas under naturalistic visual stimulation (Fig. 10A). Pairs formed from ganglion cells of the same broad functional type based on their receptive field center dynamics are shown in color corresponding to their functional type, and cell pairs of different type are shown in gray. Shared information values were larger for nearby ganglion cells and decayed to zero for separations greater than D' 
2 receptive field spacings, indicating that ganglion cells were systematically independent at this distance apart. Under natural visual conditions, the stimulus has long-range spatial correlations and lots of wide-field motion. However, ganglion cells manage to reduce these long-range correlations and act independently when they are far apart. One should keep in mind that independence of neurons’ activity leads to independence of the information being conveyed by these cells. This is an indication that ganglion cells represent information only about a restricted spatial region. Biphasic and monophasic OFF pairs tended to share more information than medium or slow OFF cells, but exceptions to this pattern were found. Most notably, many cell pairs of the same functional type shared less information than pairs of different type at similar spacings (color vs. gray). Figure 10B shows a specific example of this mixing of information between different broad functional types: pairs of monophasic OFF cells (yellow dots) had values of shared information comparable with that of pairs formed from one monophasic OFF cell and one medium OFF cell (green diamonds). Similar results were found for other combinations of functional types; similar results also were found under stimulation with flickering checkerboards (data not shown).
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). In addition, the largest values of the shared information were only 25%, implying that cells of the same functional type still had subtle but significant functional differences. Together, the analysis of shared information confirms and strengthens the conclusions of our functional classification, namely that multiple ganglion cells with subtly different properties jointly participate in encoding the same features in a visual scene. |
DISCUSSION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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). We relied on quantitative methods of functional classification, involving several choices of clustering algorithm as well as several choices of response characteristic. Because any method of functional classification requires some arbitrary choices, we supplemented this approach by analyzing the shared information between ganglion cells, a measure of functional similarity that can be calculated during stimulation with natural scenes and makes minimal assumptions about how ganglion cell spike trains encode the visual world. These two approaches gave consistent and mutually reinforcing results: namely, that the ganglion cell population in the salamander can only be resolved into a small number of functional types, and all but one have highly overlapping receptive fields. Our data even suggest that these six broad functional types are not completely distinct from each other (Schneidman et al. 2002). Functional classification
Our classification of salamander ganglion cells into six broad functional types (4 OFF-types, 2 ON-types) is in good agreement with the number of cell types defined in previous classifications performed in the salamander. Physiological classification studies using either simple stimuli (Grusser-Cornehls and Himstedt 1973) or white noise (Warland et al. 1997) have described four functional types (3 OFF-types, 1 ON-type). The slow ON cell type that we observed has not been previously reported, so the only difference with these studies is whether there are three or four OFF-types, which is somewhat ambiguous in our data as well. Quantitative anatomical studies have classified salamander ganglion cells into five morphological types (Costa Lda and Velte 1999; Toris et al. 1995), which agrees with our number of broad types. In this work, the primary feature that distinguished anatomical types was the size of the dendritic field, which fell into three groups. Similar to their results, we found three broad classes of receptive field size (Fig. 4). However, it is not clear how these morphological types map onto our functional types, in particular because receptive field size did not resolve more types than were found using the temporal dynamics of the receptive field center.
Burkhardt et al. (1998) classified salamander ganglion cells into three main types based on the response to steps of light: ON, OFF, and ON-OFF. When we looked at the responses to diffuse steps of light, these three categories were evident, but each contained a great variety of different responses. This variety included a continuous range of response latencies as well as examples of cells with two or three peaks of firing in response to a single step of light (Fig. 6C). Almost all of the cells of biphasic OFF-, monophasic OFF-, and medium OFF-type, in fact, had responses to both the onset and offset of light. This large group comprised 76% of all retinal ganglion cells in the salamander, which is consistent with the finding of Burkhardt et al. that 68% of all cells were ON-OFF and the finding of Toris et al. (1995) that 80% of all cells had dendrites that were bistratified in the inner plexiform layer. Interestingly, the response of salamander bipolar cells to steps of light exhibits considerable variety (Burkhardt and Fahey 1998; Pang et al. 2004). Presumably, this great functional diversity is the basis for much of the diversity we observe at the level of the ganglion cells.
Many studies have used anatomy or a combination of anatomy and physiology to classify retinal ganglion cells. However, it is important to keep in mind that anatomical and functional classification are distinct projects that need not give the same results. For instance, ganglion cells with nearly identical dendritic morphologies may in fact receive synapses from different bipolar and amacrine cells or may express different balances of ionic conductances. Furthermore, ganglion cells with subtle but distinguishable dendritic morphology (e.g., curved vs. straight dendrites) may actually instantiate virtually the same function. Despite the impressive evidence in favor of a close relationship between structure and function (Masland 2001; Sterling 1998
