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1Department of Physiology and Biophysics, University of Washington, Seattle, Washington; 2Department of Molecular Biology, Princeton University, Princeton, New Jersey; 3Department of Life Sciences, University of Sussex, United Kingdom
Submitted 21 September 2005; accepted in final form 10 August 2006
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIXGRANTSACKNOWLEDGMENTSREFERENCES |
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIXGRANTSACKNOWLEDGMENTSREFERENCES |
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; Segev et al. 2004). This provides the opportunity for unprecedented understanding of the coding properties of ganglion cells, individually and as a population. Retinal image processing is sophisticated with many complex visual patterns modulating the firing of individual ganglion cells, such as direction of motion (Barlow and Levick 1965), object motion against a background (Lettvin et al. 1959; Olveczky et al. 2003), looming objects (Ishikane et al. 1999), static edges (Caldwell and Daw 1978), spatial generalization over local subunits (Victor and Shapley 1979), and deviations from temporal periodicity (Bullock et al. 1990). Systematic studies of the ganglion cell population using multi-electrode arrays have shown great functional diversity (DeVries and Baylor 1997; Schneidman et al. 2002) and extensive receptive field overlap (Segev et al. 2004), allowing high-order visual features to be represented by multineuronal firing patterns (Puchalla et al. 2005; Schnitzer and Meister 2003).
Many studies have focused on measurement of the spatiotemporal receptive field, an approximation of ganglion cell function that implicitly assumes that the cell represents only one visual feature. However, retinal ganglion cell spike trains can represent multiple stimulus features, so more complete models are needed. The computation of Wiener kernels provides an estimate of the firing rate in an expansion in successive orders of the input (Wiener 1958). In practice, however, data limitations restrict the computation of the kernels to second order (Naka and Sakai 1991). Cascade models, where the stimulus is filtered through a linear stage then nonlinearly thresholded, provide a general framework for forward models, aimed at predicting a spike train given the stimulus. Elaborations of the cascade model include a further linear feedback stage that modulates the firing output depending on recent spike activity, thus including the effects of refractoriness (Gerstner and Kistler 2002; Keat et al. 2001; Paninski et al. 2004; Shapley and Victor 1979; Victor 1987, 1988). The linear filtering stage models the receptive field, whereas the threshold and the second linear filter are associated with the mechanism of spike generation.
Here, we apply second-order reverse correlation methods to discover new, detailed information about receptive field structure, in particular, the representation of multiple visual features by each spike (Aguera y Arcas et al. 2003; Brenner et al. 2000a; de Ruyter van Steveninck and Bialek 1988; Schwartz et al. 2002). While considering the units of representation in the spike train to consist of sequences of spikes has been shown to be fruitful (Berry and Meister 1998; Metzner et al. 1998; Reinagel and Reid 1998; Reinagel et al. 1999; Strong et al. 1998), here we will limit our discussion to the representation of the stimulus by single spikes only. Reverse models, describing what a spike reveals about the stimulus, offer a simple interpretation of the neural code of a ganglion cell: the visual features they identify directly illustrate the meaning of the neuron's spikes (Rieke et al. 1997). At the same time, the reverse model can be inverted using Bayes' rule to make a prediction of the neuron's output for arbitrary visual inputs (Aguera y Arcas et al. 2003; Brenner et al. 2000a). Furthermore, reverse models allow the use of information theory as an error measure that is well-matched to the aspects of a ganglion cell spike train that are important for conveying visual information and makes few other assumptions about the nature of the neural code (de Ruyter van Steveninck et al. 1997; Gawne and Richmond 1993; Panzeri and Schultz 2001; Panzeri et al. 1999; Reich et al. 2001; Reinagel and Reid 1998; Schneidman et al. 2003).
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METHODS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIXGRANTSACKNOWLEDGMENTSREFERENCES |
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Eyes were dissected from larval tiger salamanders (Ambystoma tigrinum) that had been light adapted for 1–3 h. The cornea was removed, and the remaining eyecup was cut into sections of quarter size or less. This preparation left the pigment epithelium and sclera intact for better stability during electrical recordings. Sections were placed with the ganglion cells facing a multi-electrode array (MultiChannel Systems MEA 60) and were perfused with oxygenated Ringer medium at room temperature (Balasubramanian and Berry 2002). Stable recordings of >12 h were achieved under these conditions. For a given stimulus condition, between 2 and 5 h of data were acquired.
Extracellular voltages were amplified (1,200-fold), filtered (200–5,000 Hz), and streamed to disk for off-line analysis at 10 KSamples ·s–1 · channel–1. Spike waveforms were sorted using the spike peak and width in a 2.5-ms window. Only well-isolated spike waveforms were used: we required that spike clusters were clearly visible above the noise and that the sorted spike trains had <0.3% and more typically between 0 and 0.1% of all interspike intervals <2 ms. This study is based on measurements of 59 cells recorded from seven retinas.
Visual stimulation
Retinas were stimulated with spatially uniform flicker presented using a white light-emitting diode (LED) diffused over the area of the array. Light intensities were chosen randomly at 120 or 180 Hz from a Gaussian distribution with a SD equal to 24–36% of the mean. Mean light levels ranged from 12 to 24 mW/m2, corresponding to photopic vision. The relative absorption ratios for salamander red rods:red cones:blue cones are 1:0.96:0.61.
We stimulated the retina with white-noise segments of length 30 s repeated 10 times (5 min total), interspersed with random sequences of length 20 min. This basic pattern was repeated, using different random sequences but the same repeat sequence, for 
6 h. The analysis of a given cell was restricted to the portion of the experiment where the cell's mean firing rate was approximately constant.
Model simulation
We implemented the filter-and-fire model described in Keat et al. (2001). There are two noise processes, one of which is additive and is summed with the filtered stimulus. This noise was Gaussian with a mean of zero and a SD of 0.15 (in units of stimulus SD). There is also a multiplicative noise which independently multiplies the amplitude of the afterhyperpolarization following each spike; this was a Gaussian variable with mean of one and SD of 0.085. The afterhyperpolarization used in the simulations shown was given by A(t) = –0.6 exp(–t/0.44s), and the filters were constructed as a sum of two alpha functions with timescales of order several hundred milliseconds. The threshold 
took values of 1.5, 2, and 2.5. For each analysis, 30,000 spikes were used.
Covariance analysis
To estimate the temporal response properties of ganglion cells with a high degree of generality, we stimulated the retina with the wide variety of light patterns found in spatially uniform flicker and used covariance analysis to determine to which visual features each ganglion cell was sensitive. During this stimulus, denoted by s(t), we recorded the occurrence time of every spike elicited from a ganglion cell, {ti}, and selected the stimulus histories preceding every spike, si(
) = s(ti – ), where denotes the time index. We used 100 stimulus values preceding a spike. Thus for stimuli presented at a frame rate of 120 Hz, extended back to 833 ms before a spike, whereas for 180-Hz stimuli, extended back to 556 ms before a spike. The mean, 
i(), is the spike-triggered average. The covariance matrix is formed from the outer product of the stimulus histories averaged across all spikes
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...
is over all spike occurrences {ti}.
We were interested in learning what features in the stimulus are relevant to spiking. These are the features for which the variance is altered from that of the prior distribution of stimuli, which is Gaussian along all stimulus dimensions, independent of the response. Thus we subtracted the covariance matrix of the prior itself, Cprior, to obtain a matrix representing the covariance differences (Brenner et al. 2000a)
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to find its eigenmodes and corresponding eigenvalues. Each eigenmode can be thought of a visual "feature" that may drive ganglion cell spiking, and its eigenvalue is equal to the variance of stimuli along this feature axis when a spike occurs. Because the prior stimulus distribution was subtracted from the covariance matrix, an eigenvalue of zero corresponds to the prior variance and indicates that the corresponding visual feature was not relevant to ganglion cell spiking. The number of significant eigenvalues gives an estimate for the dimensionality of the relevant stimulus space, and the corresponding eigenmodes span that space. Therefore the response function of the neuron may be completely determined by sampling in this subspace.
The significance of eigenvalues depends on sampling. We assessed eigenvalue significance by computing the eigenvalue spectrum as a function of the number of spikes N for a range of data fractions (Aguera y Arcas et al. 2003). Eigenvalues which were stable with respect to N were judged to be significant.
Information conveyed by the spike train
The information in the spike train can be evaluated directly using methods introduced in Strong et al. (1998) and Brenner et al. (2000b). The method of Strong et al. (1998) uses the distribution of spike words created from discretizing the spike train in time bins and concatenating the spike counts in these bins into a spike word; by varying the length of the spike words and the size of the time bin, this method can evaluate the information encoded by a neuron in the limit of infinite word length or zero time bin.
As we are evaluating how much information the observation of each spike gives about the stimulus ensemble, we used instead the method of Brenner et al. (2000b) and measured the information that the time of occurrence of a single spike conveys about the visual stimulus. To do this, we randomly generated a stimulus segment s(t) of length T = 30 s from the ensemble of spatially uniform flicker, S. We repeated the same pseudo-random sequence many times to find r(t), the time-varying firing rate of a ganglion cell responding to s(t). These repetitions were interspersed between changing random sequences that were used to determine the overall mean firing rate 
in response to the entire stimulus ensemble S. The information per second is given by the entropy difference between the response, , where the stimulus is an unknown sequence drawn from the ensemble S, and the response, r(t), when the particular stimulus sequence s(t) of duration T is known (Brenner et al. 2000b). We then computed
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. The rate appears in the logarithm because this is the bestestimate of the firing rate averaged over the entire stimulus ensemble S. We divide by a factor of rmean, because the time integral in Eq. 1 is taken over the repeated stimulus segment, and the average firing rate over this segment is rmean. This puts the information in units of bits per spike rather than bits per second. See APPENDIX for a derivation of Eq. 1.
To compare this information benchmark against the information captured by models of a ganglion cell's response function, we needed to match rmean to . To this end, we randomly selected stimulus segments of duration slightly shorter than T (ranging from 0.75 T to 0.95 T), each having a different mean firing rate, and recalculated the information for all such segments. We plotted Ione spike versus rmean and linearly interpolated to find the information for rmean 
. We took the uncertainty in this interpolation to be our random error in measuring Ione spike; this random error was quite small, typically 
1–2%.
Values were corrected for bias due to the finite number of stimulus repeats by dividing the data into fractions with respect to the number of repeats, repeating the calculation, and computing the information as a function of 1/(sample size). We obtained the best quadratic fit to this dependence and extrapolated to infinite sample size (Golomb et al. 1997; Strong et al. 1998). The quadratic dependence gave a good fit for all cells that we have included.
This information measure makes no assumptions about the relationship between stimulus and response, and is an upper bound for the information captured by any particular low-dimensional model of this response function (Adelman et al. 2003; Aguera y Arcas et al. 2003). Because ganglion cell spike trains were sparse and precise, each spike carried a great deal of information: 4.9 ± 1.0 (SD) bits.
Information captured by the model
To compare the model with the directly computed information content of the spike train, we follow the argument of (Aguera y Arcas et al. 2003): starting with Eq. 1, we note that
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). In forming a K-dimensional model defined by the filters {f1,..., fK}, we replace the stimulus 
with its reduced, lower dimensional description in terms of the K projections of the stimulus, denoted by si, onto the filters fi
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), the information in the reduced description must be bounded by that in the full description
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Fig. 6 is from the latter case. Treating the filters as vectors, they were normalized such that the squared sum over the vector components equaled unity, and the stimulus was sampled in time bins equal to the frame time so that there were no temporal correlations. Because the prior stimulus distribution was independent in all stimulus directions and had a variance of unity, we used the analytically computed Gaussian prior for both P(s1) and P(s1, s2).
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s. For 1D distributions, the value was very stable as a function of s, but the 2D models depended more strongly on the bin size. We corrected our information calculation for finite sampling bias by considering the information carried by the eigenmode with an eigenvalue closest to zero; this visual feature should be most completely irrelevant to the firing of the neuron. By itself, this eigenmode should carry no information. Instead, its information was measured to be between 0.002 and 0.014 bits, depending on the number of spikes recorded for that ganglion cell. Therefore this information, as a function of s, was subtracted from the values for all other information calculations for 1D models. Although this correction typically was small (<1%), it produced information values that were nearly constant down to a bin size of 0.05 times the prior SD (Fig. 1A).
To determine the bias for 2D models, we performed our calculation for the STA combined with the most insignificant eigenmode. Similarly, the information in this 2D model should be the same as the information in the STA alone but was larger due to finite sampling. We took the sampling bias to be equal to the difference between the information in this 2D model and the information about the STA alone, as a function of s. This bias was subtracted from all of our calculated information in 2D models, giving values that were nearly constant as a function of the bin size, s (Fig. 1A). For the final value, we averaged over bin sizes in the range of 0.15–0.4 (normalized units); the SD was taken as an estimate of the random error.
We define the synergy in the 2D model as the difference in information between the 2D model and the summed information from the corresponding 1D models
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In all of our information calculations, we used a time bin that was matched to the frame time of the stimulus, either 5.56 or 8.33 ms. The same time bin was used for both full and model information as well as for the covariance analysis. To explore the limit of finer time scales, we ran an experiment in which the stimulus frame rate was 1,000 Hz, but where the temporal sequence of light intensities was low-pass filtered with a cut-off frequency of 120 Hz. This allowed us to extend our covariance analysis down to a time bin of 1 ms. For time bins in the range of 1 ms up to 40 ms, the information fraction captured by our analysis was roughly constant (Fig. 1B).
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RESULTS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIXGRANTSACKNOWLEDGMENTSREFERENCES |
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Before describing the results of our covariance analysis for real ganglion cells, it is helpful to develop intuition using a simple model, where all elements of the spike generation are known and understood (Aguera y Arcas and Fairhall 2003). Keat et al. introduced an instantiation of the cascade model described in the preceding text—a spike response model (Gerstner and Kistler 2002)—which had considerable success in predicting the timing of spikes from a variety of ganglion cells (Keat et al. 2001). Therefore we first performed covariance analysis on this model of ganglion cell function. In the Keat model, the stimulus s(t) is convolved with a single linear filter, f+(t). We denote this filter with the superscript + to indicate that it has nonzero values only at positive times (due to causality). The output of this filtering stage, g(t), is passed through a static nonlinearity which is simply a threshold, . Each spike adds an afterhyperpolarization, A(t) = –Ao exp(–t/
), to the filtered stimulus g(t). Noise is added both to the filtered stimulus prior to thresholding and to the amplitude of the afterhyperpolarization to account for the timing jitter and spike count variation in experimentally measured spike output.
Covariance analysis, illustrated in Fig. 2, finds the directions in stimulus space along which the variance is altered with respect to the prior stimulus distribution (see METHODS). Each direction (or eigenmode) has a corresponding eigenvalue, which describes how the spike-triggered stimulus variance differs from the prior: negative eigenvalues indicate that the spike-triggered stimulus distribution is narrower than the prior, positive values indicate that it is wider, and zero indicates no difference. The directions with eigenvalue significantly different from zero constitute the visual "features" about which the occurrence of a spike conveys information. The set of all visual features revealed by covariance analysis constitute what we call a "reverse model" of the ganglion cell light response. These features describe what the brain can infer about the visual stimulus given the occurrence of a spike. This terminology is in contrast to a "forward model," such as that described by Keat et al. that predicts the probability of a spike given the visual stimulus. As we will determine the visual features using reverse correlation, they are indexed in the same sense as the stimulus, and have nonzero values only at negative times (again due to causality). We will therefore denote the visual features by {fi–(t)} to indicate this sense of the index t.
The neuron fires whenever a decision variable, consisting of the stimulus convolved with the filter f+(t), plus an afterhyperpolarizing current convolved with the previous spike train, crosses threshold from below (see METHODS). Let us initially neglect the afterhyperpolarization and consider the case that the system simply fires a spike on an upward threshold crossing of g(t) = 
0
f+() s(t – )dt. Clearly, because spiking occurs where the filtered stimulus attains or exceeds the threshold value , f–(t) is itself a relevant feature for spiking: any spike-triggering stimulus projected onto f–(t) will be close to and will therefore have variance less than the prior and so should appear as a direction with negative eigenvalue.
A second feature is expected due to the threshold crossing: as the threshold is usually crossed from below, the time derivative g'(t) of the filtered stimulus at the time of the spike should be positive (DeWeese 1995). Equivalently, the projection of the stimulus s(t) onto the filter f+'(t) should be positive when there is a spike: if we take the time derivative of the filtered stimulus, we get g'(t) = 0 f+() s'(t– )d, where in practice the filter is taken to be zero at some finite time in positive t. Next, we can integrate by parts to get g'(t) = –f+()s(t – )|=0= + 0 f+'() s(t – )d. The boundary term drops out, because the filter f+(t) is zero both at time 0 and for large positive t. Thus the time derivative of the filtered stimulus g'(t) is seen to be a projection of the stimulus along a direction given by the time derivative of the filter, f+'(t). We note that f+'(t) = –f–'(–t)] because the time derivative operator is anti-symmetric under t –t. It is important to remember that the eigenmodes identified by covariance analysis are only determined up to an arbitrary sign. Therefore we choose a sign convention such that the second visual feature is the negative time derivative of the first feature. With this choice, the projection of the stimulus onto the second visual feature tends to have positive values at the time of threshold crossing. And as a result, the distribution of these projections given a spike will have less variance than the a priori stimulus distribution and hence appear with negative eigenvalue.
With the afterhyperpolarization, the condition that spikes are only fired on an upward threshold crossing of g(t) is weakened; the sum of the filtered stimulus and the combined AHPs from previous spikes, h(t) = g(t) + 0A()
(t – )d, must cross threshold, where (t) = 
i(t – ti) is the spike train. Thus subsequent spikes can occur either when the filtered stimulus is sufficiently large such that h(t) still exceeds or if g(t) is still above threshold when the AHP decays away. The AHP does not affect the form of the stimulus filtering so f(t) should still be a relevant mode. The dependence of spiking on the time derivative f'(t) will, however, be weakened, as h(t) may now cross threshold while g(t) itself is decreasing.
The filtered stimulus g(t) typically remains above threshold for a time on the order of the correlation time of g(t), 
. If the time scale of the AHP is very long, following an initial spike the AHP can keep the decision variable below threshold for a time longer than , preventing further spikes. For shorter , A(t) decays away and h(t) can recover to allow several spikes within time . Thus burst-like events appear as a consequence of the AHP recovery; the burst duration is established by the correlation time of the filtered stimulus, , and the typical burst interspike interval is governed by the AHP time scale,. For the first spike in a burst, the original criterion on f'(t) holds. For all subsequent spikes, h(t) will be at threshold while g(t) is larger than threshold, but g'(t) will be first positive and then negative within this time. Thus the relative importance of the derivative feature should depend on the time scale of h(t): for large , the AHP case is close to the pure upward-threshold crossing case, and the derivative will be highly significant; for smaller , the derivative condition will be weakened.
Indeed, covariance analysis does recover both of these visual features (Fig. 3). The eigenvalues corresponding to these features depend weakly on the threshold, but the features themselves are unaffected by the value of the threshold (Fig. 3, A and B). The spike-triggered average is precisely a linear combination of the filter and its time derivative. Note that these visual features are the time-reversed versions of the respective filters in a forward model, so that the second feature is actually the negative time derivative of the first feature. To understand the structure of the spike-triggered stimuli in terms of these two features, we plot each spike-triggering stimulus history projected onto features 1 and 2 (see METHODS). This shows clearly the thresholding: the projection onto the first feature, by construction, is always near or greater than threshold, and the projection onto the second feature is mostly positive (Fig. 3C).
The interactions between spikes have been shown, for single neurons, to have a profound influence on the output of a white noise analysis (Aguera y Arcas and Fairhall 2003; Aguera y Arcas et al. 2003; Pillow and Simoncelli 2003; Powers et al. 2005). This effect is particularly dramatic for an integrate-and-fire-like neuron, where the integrator is reset after a spike, losing memory of the filtered stimulus. The Keat model does not make this assumption. Here we can control the strength of the interspike interaction with two parameters, the amplitude A0 and time scale of the afterhyperpolarization. Varying the amplitude had no effect on the results of covariance analysis. As expected, when the time scale was reduced sufficiently, the second, derivative-like feature was no longer significant (data not shown). We have found no combinations of spike afterhyperpolarization parameters that give rise to more than two significant visual features.
There are two additional model parameters, governing the amplitude of both additive and multiplicative sources of noise. The noise adds variability in the spike times. As long as the temporal jitter is smaller than the time scale of the filter itself, jitter in spike timing results only in a blurring of the threshold and weak dependence on the derivative mode. Thus there is some dependence of the eigenvalues on the amplitude of the noise, although noise levels appropriate for matching retinal timing jitter do not change the overall structure of the results (see DISCUSSION).
Feature selectivity of ganglion cells
We recorded spike trains from ganglion cells in the salamander retina while stimulating with spatially uniform flicker, a stimulus ensemble that contains a rich variety of temporal patterns of light (see METHODS). As shown in previous studies, ganglion cell spike trains were found to consist of sparse and reliable firing events under these stimulus conditions (Berry et al. 1997). Covariance analysis revealed a great diversity of temporal features encoded by retinal ganglion cells (Fig. 4). We found at least two and as many as six significant visual features for individual cells. (Because the resolution of eigenvalues was limited by sampling noise, we only included ganglion cells from which we recorded at least 2,000 spikes.) Within the ganglion cell population, a wide variety of feature selectivities was observed. Although the properties of individual cells did not fall into distinct classes, it is useful to divide the population into five qualitatively different types to describe this diversity. Figure 4 shows an example cell from each such response type. This shorthand description should not be taken as a claim that the ganglion cells could be clustered into five distinct functional classes. Instead, their response functions seemed to be distributed more continuously, with individual cells often having properties intermediate between these five types and the examples shown in Fig. 4 representing the range of possibilities that we observed.
The first type is the filter-and-fire cell (Fig. 4A; n = 11 cells), where the covariance analysis closely resembled that of the Keat model. These cells had two strongly negative eigenvalues (left). The corresponding visual features included one that resembled the STA and a second feature that was proportional to the time derivative of the first feature (middle). As discussed in the preceding text, we chose a sign convention in which this second feature was the negative time derivative of the first. Finally, projections of the stimulus onto these visual features at the occurrence time of a spike showed a threshold-like nonlinearity along both feature directions (right). Similar to the Keat model, the first feature had a slower time course than the STA. As for the Keat model, the STA is a linear combination of these two visual features, and it is sharpened in time by the addition of the derivative-like feature. All but one of these cells had a spike-triggered stimulus average that was either biphasic OFF or standard OFF-type (Puchalla et al. 2005). These cells have also been described as fast OFF in previous studies (Schnitzer and Meister 2003).
Bimodal cells (Fig. 4E; n = 7 cells) displayed two separate clouds in the plane formed by the projections onto two visual features. Such cells were strongly ON-OFF in their response to steps of light. If the STA was calculated separately for the two clouds seen in the projection space, one cloud had an ON-type STA and the other cloud was OFF-type (see Fig. 9). For these cells, the leading eigenvalue was positive. As this was the direction along which the variance change from the prior was most increased, this is the direction which maximized the bimodality of the spike-triggered distribution. One can think of this as the direction that best differentiates between the individual responses making up the two clouds. Although this feature is not the same as the STA, the STA was also always effective in separating the two clouds. For the example cell shown in Fig. 4E, there were many additional features that show repeated oscillations over a range of frequencies, like a Fourier basis. This structure was often seen for ganglion cells with higher average firing rates. These cells all had a fast OFF-type spike-triggered average, like the filter-and-fire cells.
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). Instead, this structure is more typical of a stimulus feature that causes a spike timing shift as seen in simulations of a model neuron with Hodgkin and Huxley currents (Aguera y Arcas et al. 2003). Consequently, we call these Hodgkin-Huxley-like (HH-like) cells. Most of these cells had a medium OFF-type spike-triggered average (Puchalla et al. 2005) with a roughly monophasic time course and a longer latency to peak than for filter-and-fire cells. The two other response types had more complex structure. Ring cells (Fig. 4D; n = 6 cells) had several positive eigenvalues, and in the space of the first and second features, they showed a projection cloud resembling a ring. All slow OFF-type cells in our data set (n = 3) were ring cells; the other ring cells were fast OFF-type. Another group of cells was similar to filter-and-fire cells but had additional features or nontrivial structure in their multi-dimensional projections (complex FF; n = 14 cells). In the example shown (Fig. 4B), the third visual feature had oscillations well prior to the main lobe of the STA. In many other cases, complex-FF cells were weakly bimodal, where the projection cloud closely resembled the cell shown in Fig. 4B but also had a small fraction of the spikes elicited by an ON-type stimulus feature. Such weakly bimodal cells typically had a mixture of positive and negative eigenvalues.
Information conveyed about visual features
To determine how completely our reverse models can describe ganglion cell function, we compared the information captured by a given model to the total information available in the cell's spike train (Adelman et al. 2003; Aguera y Arcas et al. 2003). The available information was estimated directly by alternating between many nonrepeated segments of random flicker to sample the variety of ganglion cell responses and many repeats of a single, 30-s stimulus segment to sample the noise in that response (see METHODS, Eq. 1). The information captured by a given model of ganglion cell feature selectivity was calculated by comparing the distribution of stimuli given a spike to the prior stimulus distribution (see METHODS, Eq. 2). Because the models we considered ignored most of the possible features in the stimulus, the information they capture must be less than the information available in the spike train. However, if the ganglion cell is only sensitive to a small number of visual features, the model may be nearly complete.
First, we consider how much information we obtain from models that include only a single visual feature. Figure 5 plots the fraction of the available information captured by a single visual feature versus the corresponding eigenvalue from the covariance analysis. All of the significant visual features for the 59 cells in our study are displayed. It is worth noting a general feature of this plot: the distribution of eigenvalues has to be asymmetric because eigenvalues are bounded from below by –1, which can occur if the variance along the spike-triggered distribution is reduced to 0. In contrast, there is no a priori limit to the increase in variance shown by the spike-triggered distribution. We see that features with negative eigenvalues conveyed a large amount of visual information, in some cases up to almost 90% of the information encoded by the ganglion cell. Features with positive eigenvalues also made significant contributions, ranging up to >60% of a cell's available information. In general, the information conveyed by a visual feature increased with the absolute value of the eigenvalue, and visual features with negative eigenvalues tended to convey more information than those with positive eigenvalues of the same magnitude. Most of the points lying in the center of the plot between the positive and negative branches came from bimodal cells.
We have highlighted visual features from four example cells (Fig. 5B). The filter-and-fire cell (red) had a primary visual feature with an eigenvalue of –0.86 that captured 60% of the information in its spike train. The projection of stimuli along this direction at the time of a spike formed a narrow Gaussian distribution displaced far from the mean of the prior stimulus distribution (Fig. 5B, i). The Hodgkin-Huxley-like cell (yellow) also had a negative eigenvalue, but it captured less information; its distribution of projections was not as narrow or as nearly Gaussian as the FF cell (Fig. 5B, iii). Both the ring (green) and the bimodal (blue) cells had a visual feature with a positive eigenvalue, indicating that the variance along this direction of stimulus space was greater than for the prior. Their projections had bimodal distributions, indicating that both ON- and OFF-type visual features could elicit a spike from these cells (Fig. 5B, ii and iv). The bimodal cell can be seen to have cleaner separation between the two peaks in its projections; it also had a more positive eigenvalue, which captured more visual information.
Next we evaluated the completeness of reverse models that include two visual features. We formed such two-dimensional models either from all pairs of relevant visual features defined by covariance analysis or from the STA combined with the component of a relevant visual feature orthogonal to the STA. We found that the best model in almost all cases was the STA combined with the component of the visual feature that individually captured the most information, orthogonalized with respect to the STA. The two-feature models so constructed were highly successful descriptions of ganglion cell light responses: over the population of cells, they captured 89 ± 11% of the information; for some cells, the model captured 100% of the available information (Fig. 6B). We compared these two-feature models to one-feature models formed using the STA as this was often the most informative single stimulus feature. The STA alone was also very successful, capturing 78 ± 14% of the information (Fig. 6C). But in almost all cases, the two-feature model was significantly better than the STA alone (Fig. 6A). For some cells, the two-feature model captured more than twice as much information as the STA alone. For instance, for the ring cell shown in Figs. 4 and 5, the ratio of information captured by the two-feature model to that captured by the STA alone is a factor of 2.5.
How do the two most important features contribute to produce the neuron's output? In the most complete model, the firing probability depends jointly on the projection of the stimulus onto two visual features, P(spike at t|s1, s2) (see METHODS). However, sampling such a two-dimensional distribution is necessarily more difficult than sampling in one dimension. Can one instead assume that the two features contribute independently? We answered this question by calculating the synergy in the two-feature description constructed as described in the preceding text from the STA and the leading eigenmodes orthogonalized with respect to the STA. The synergy is the difference between the information in the full 2D description and the sum of information in the two 1D descriptions P(spike at t|s1) and P(spike at t|s2) (see METHODS, Eq. 4). If the neuron's firing rate depended independently on the two visual features, the information would sum and the synergy would be zero. In Fig. 7, we plotted the synergy versus the information contained in the second visual feature alone. The comparison indicates that most cells had significant synergy, 0.5 bits or more, even when the second visual feature contributed very little information by itself. This indicates that the ganglion cell spikes provide significantly more information about the visual scene if the receptive field is measured over the joint space of two relevant features. Synergy typically was found for cells that had a curved or tilted shape to the geometry of their spike-triggered stimulus space.
Predicting the spiking rate
We can invert our reverse model using Bayes' rule to form a forward model that can predict the output of the ganglion cell for any arbitrary visual stimulus contained in the ensemble. Because the reverse model describes the stimulus features that lead to the occurrence of a single spike, the corresponding forward model predicts the probability that a spike will occur given any stimulus (see METHODS, Eq. 2). Our reverse model was formed using the nonrepeated portion of the visual stimulus, but we tested the forward model on a different 30-s stimulus segment that was repeated many times. Therefore we are assessing the ability of the reverse model to generalize to arbitrary stimuli.
Using the projection onto two visual features as the reverse model captured 100% of the encoded visual information for some cells. In this case, the corresponding forward model gave a very accurate prediction of the firing rate over the entire 30-s stimulus segment (Fig. 8A; filter-and-fire cell). Essentially every firing event produced by the ganglion cell was also predicted by the model: predicted events matched the occurrence of real events to within the event's temporal width for > 99% of all of the cell's spikes. All of the periods of reliable ganglion cell silence were also correctly predicted: falsely predicted events accounted for <1% of the predicted spikes. On an expanded scale, we can see that the reverse model with two visual features gave an excellent prediction of the detailed firing rate (Fig. 8B). Remaining errors were small deviations that were of the same magnitude as the trial-to-trial variability of the cell's response, which is precisely why the model captures all of the cell's visual information.
We can compare the performance of a model formed using only the STA as a visual feature. For this neuron, the STA model was very successful, capturing 86% of the encoded information. The corresponding prediction also was quite good, with predicted firing events for essentially all real events and silence elsewhere. However, a detailed comparison revealed systematic errors: the peak firing rate was generally too low, firing events were too wide, and in some cases, there was a systematic temporal offset (Fig. 8B).
For bimodal cells, the two-feature model gave accurate predictions, including both ON- and OFF-type stimulus events (Fig. 8C). However, the STA model could not predict ON-type events because the STA was always OFF type for these cells. For strongly bimodal cells, the prediction of the STA model often had a low peak firing rate and included some predicted firing events where there was no real event (Fig. 8C). This occurred in part because the STA could be quite unlike either of the visual features in the full model (Fig. 4E).
Ring cells exhibited slower modulations in their firing rate with considerably lower peak rate than other cell types (Fig. 8D). Although the model with two visual features typically captured much less of these cell's visual information, the prediction matched most events in the real firing rate with some errors in the peak rate. In contrast, the STA alone predicted maintained firing from the cell as well as many events where the real cell was silent. For this cell, the STA only captured 25% of the total visual information. These errors result from the ring-like structure of its stimulus projections at the time of a spike: stimuli with zero projection onto both features do not lead to spikes (Fig. 4D), but stimuli with zero projection onto a single feature do lead to spikes (Fig. 5B, ii). Thus the STA alone can give a quite adequate description of the light responses of some ganglion cells, particularly filter-and-fire cells, but for other cells with more exotic response properties, the prediction is qualitatively wrong.
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DISCUSSION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIXGRANTSACKNOWLEDGMENTSREFERENCES |
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A substantial fraction of the cells displayed the computational structure of a filter-and-fire neuron. An important aspect of such a model is the explicit separation between the filtering properties of the retinal circuit and spike generation in the ganglion cell (often implemented in models by a simple threshold-crossing). For filter-and-fire cells, the spike-triggered average alone generally gave a very good description of the cell's spiking. However, the two visual features revealed by covariance analysis, along with the nonlinear decision function formed by the projection of the stimulus onto these features, show us this structure directly with no fitting required. We can interpret the features immediately as a filter and its derivative, and the plot of projections provides us with an estimate of the threshold, or the probability of spiking as a function of the projection. In contrast, Keat et al. had to perform nonlinear function optimization with 26 parameters, a daunting task (Keat et al. 2001). However, with a slight modification of the form of the cascade model, one can find globally optimal parameters, perhaps with an equally good fit to data and also recover the form of the spike afterhyperpolarization (Paninski et al. 2004).
Selectivity for more complex features
Many cells, however, show additional features, demonstrating that the filter-and-fire model does not capture all the relevant dynamics for these cells. Additional features may have two sources: the subset of the stimulus space that lead to a spike may be more complex than a single filter can describe, or the effects of the interaction between spikes—the effect of silence as well as spiking—may add complexity to the meaning of individual spikes, as has been seen for single neuron models (Aguera y Arcas and Fairhall 2003; Aguera y Arcas et al. 2001, 2003; Paninski et al. 2004). The complex FF cells show several deviations from the filter-and-fire picture, such as sensitivity to an additional visual feature having a negative eigenvalue. In the example shown in Fig. 4B, this extra visual feature involves a slow change in the light level at times prior to when the STA is significantly different from the mean light level. This additional feature can be thought of as modulating the firing rate depending on the recent visual context and may describe a form of local contrast adaptation. In other cases, these cells were weakly bimodal or had only a single significant feature with a positive eigenvalue.
Another subset of ganglion cells illustrate that the derivative-like visual feature need not be associated with a reduction in variance, as is the case for filter-and-fire models with threshold-crossing. For these cells, the additional visual feature had a positive eigenvalue, corresponding to an increase in the variance relative to the prior (Fig. 4C). This leads to an alternative interpretation of how a derivative-like visual feature arises: namely, from spike timing jitter. Imagine that a ganglion cell is performing template matching to one stimulus feature but that there is either noise or an additional stimulus feature that causes its spike to be shifted in time. This temporal jitter induces a time translation of the spike relative to the template, and the operation of time translation is equivalent to a time derivative, for small translations. This was analyzed in the case of a noisy temporal jitter in (Aldworth et al. 2005). In such a case, the distribution of projections onto the second, derivative-like feature is determined by the distribution of jitters, which may have a large variance. This behavior was observed in the Hodgkin-Huxley neuron (Aguera y Arcas et al. 2003), which leads us to speculate that for these cells the additional features that we observe may arise from the ion channels that generate spikes in the ganglion cell rather than from the filtering accomplished by the cell's presynaptic circuitry.
Two additional classes of cells emerged, ring and bimodal cells. For both of these types, the covariance analysis revealed many visual features. For the ring cells, the projections of the stimulus onto their two most informative features takes the form of a ring around the origin. The resulting bimodality along any one visual feature leads to positive eigenvalues or increased variance compared with the original stimulus. This structure is reminiscent of results for neurons in rat barrel cortex (Petersen and Diamond 2003), which are frequency sensitive but phase invariant and encode the power in a particular frequency band, as well as complex cells in V1 (Rust et al. 2004, 2005; Touryan et al. 2002), which exhibit invariance to the precise location of an oriented bar. It appears that a similar kind of phase invariance or sensitivity to stimulus power is being computed by ring cells in the retina.
Most strikingly, we observed a number of cells that showed a strong bimodality, with projections falling into two distinct clouds. Such cells are ON-OFF-type, with transient responses to both increases and decreases in illumination. The OFF-type responses were always seen to dominate, so that the STA was also OFF type. However, the multidimensional representation allows a clean separation of the two responses. If we then calculate the spike-triggered stimulus average separately for the two clouds, we see that one cloud of points represents an OFF-type stimulus feature, whereas the other cloud is ON type (Fig. 9). This analysis shows that there is a lack of exact symmetry between the ON and the OFF features. The generation of these two features may be the outcome of filtering through separate ON and OFF bipolar pathways, which have been shown to be asymmetric (Chichilnisky and Kalmar 2002; Rieke 2001). In other cascade models, these ON-OFF cells would require two separate channels of filtering with many more parameters and a corresponding increase in the complexity of finding optimal values for those parameters. In particular, the class of cascade models shown to have parameters that can be globally optimized cannot model bimodal cells (Paninski et al. 2004). On the contrary, this structure naturally emerges from covariance analysis with no fit parameters.
Diversity in the ganglion cell population
Although we have divided the ganglion cell population into several functional types for the purpose of discussion, we emphasize again that we saw no evidence that the population actually can be rigorously clustered into these five types. Based on the spike-triggered average alone, salamander ganglion cells can be divided into six broad functional types (Puchalla et al. 2005; Segev et al. 2006). We found a correspondence between these broad types and the full feature selectivity: for instance, ring-like cells tended to be slow OFF-type and Hodgkin-Huxley-like cells tended to be medium OFF-type. However, this correspondence was not exact, indicating that ganglion cell selectivity to multiple stimulus features does not resolve functional subtypes. In a similar fashion, combining responses to other stimuli, such as spatially uniform steps of light, with the spike-triggered average also served to blur broad functional types rather than delineate subtypes (Segev et al. 2006).
One can also assign ganglion cells to functional classes using an information theoretic method that forms clusters by maximizing the information that spike trains convey about cell identity (Schneidman et al. 2002). Although this method is highly sensitive to the detailed spiking pattern produced by each ganglion cell, it is important to note that this technique assigns cells to functional classes without providing any direct insight into each cell's response function. After functional classes were defined, we examined the spike-triggered averages of all the cells. We found that this method distinguished all of the broad functional types that could be resolved by clustering the spatiotemporal receptive field (Puchalla et al. 2005; Segev et al. 2006) as well as some additional subtypes. However, these subtypes did not correspond well from one preparation to the next. Overall, the impression formed by previous studies of functional classification in the salamander retina is that ganglion cells of the same response polarity (e.g., ON or OFF) form a functional continuum. The present study only reinforces this conclusion.
Different models of complex feature selectivity
When ganglion cell spikes represent two or more visual features, one should bear in mind that those features define a stimulus subspace that is relevant for ganglion cell spiking. Any change of basis that remains in that same subspace provides an equivalent description of the neuron's response function, but the new basis will be spanned by different visual features. Illustrating this point, Schwartz et al. performed a slightly different kind of feature analysis on salamander ganglion cells (Schwartz et al. 2002). Where here we have simply compared the variance of all spike-triggered stimuli to the prior to identify features that influence the probability of firing, Schwartz et al. first removed the spike-triggered average from the stimulus by orthogonalizing the stimulus with respect to the STA. Then covariance analysis was performed between spikes and the altered stimulus. In this case, the STA feature was, by definition, excitatory. All other visual features appeared with negative eigenvalues in this analysis, and these were interpreted as suppressive stimulus axes.
As we have discussed with reference to the Keat model analysis, the STA is in general a linear combination of relevant visual features and time derivatives. Projecting the STA out of the spike-triggered stimulus distribution removes a component of the true excitatory feature as well as a component of the time derivative; it also preserves a component of both these features. The remaining components along the directions f(t) and f'(t) will in general have reduced variance, and so will appear with negative eigenvalues and thus be classified as "inhibitory." It is thus possible that the STA-centered analysis may confound some of the geometrical s
