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Logo of nihpaAbout Author manuscriptsSubmit a manuscriptNIH Public Access; Author Manuscript; Accepted for publication in peer reviewed journal;
Neuron. Author manuscript; available in PMC Jan 10, 2009.
Published in final edited form as:
PMCID: PMC2344156
NIHMSID: NIHMS37911

A specialization for the statistics of binocular images in primate V1

Abstract

Sensory processing in the brain is thought to have evolved to encode naturally occurring stimuli efficiently. We report a novel adaptation in binocular cortical neurons that reflects the tight constraints imposed by the geometry of 3D vision. We show that the widely used binocular energy model predicts that neurons dedicate part of their dynamic range to impossible combinations of left and right images. Approximately 42% of the neurons we record from V1 of awake monkeys behave in this way (a powerful confirmation of the model), while about 58% deviate from the model in a manner that concentrates more of their dynamic range on stimuli that obey the constraints of binocular geometry. We propose a simple extension of the energy model, using multiple subunits, that explains the adaptation we observe, as well as other properties of binocular neurons that have been hard to account for, such as the response to anti-correlated stereograms.

Introduction

Evolutionary pressures have acted on a brain processing inputs from the natural world. Natural inputs are not random but are structured in particular ways. Visual inputs have significant spatial correlations and a characteristic Fourier power spectrum (1/f2). Auditory inputs, on the other hand, can be divided into separate classes, each with their own particular structure, e.g. vocalizations which tend to be harmonic and more steady-state or environmental sounds that are non-harmonic and more transient (Smith and Lewicki, 2006). One should therefore expect that instead of being equally capable of processing any inputs, the brain should have specialized to best deal with these natural stimuli. This is a key prediction of the efficient coding hypothesis (Attneave, 1954; Barlow, 1961), which has been assumed to be a general design principle for the brain. It states that one driving force behind the evolution of the brain is the goal to represent a maximum of information about the external world using a minimum of energy and neural resources.

Recent studies have provided evidence for this hypothesis in both the auditory (Smith and Lewicki, 2006) and visual domain (Field, 1987; Schwartz and Simoncelli, 2001; Simoncelli and Olshausen, 2001). In particular they show that visual processing is specialized for the Fourier statistics of monocular natural images and exploits the redundancy introduced by the spatial correlation in natural images. In this paper, we investigate an even more striking redundancy that is a consequence of stereovision. Our two eyes view the world from slightly different vantage points. The resulting differences between the two eyes' images (binocular disparities) are used by the brain to create a three-dimensional percept of the world. At the same time, the images in the left and the right eye are highly redundant: in most cases local regions of the right image will be closely matched by simply translating the appropriate part of the left image. What we call a ‘simple translation’ is a very specific subset of all the possible ways in which the image in the left eye could in principle differ from that in the right eye. In the laboratory it is possible to create binocular images that employ all possible left-right differences: those that occur in the real world – natural ones (or physical disparities) – and artificial ones (nonphysical disparities). Formalizing these differences allows us to embed both physical and nonphysical stimuli within the same continuous stimulus space and hence to explore the response of binocular neurons within that space.

At the earliest point where inputs from the two eyes converge onto single neurons (striate cortex), the firing rate of many neurons depends upon binocular disparity (Barlow et al., 1967; Nikara et al., 1968; Poggio and Fischer, 1977). The responses of disparity selective neurons are well explained by the very successful binocular energy model (Cumming and DeAngelis, 2001; Ohzawa et al., 1990) in which the output from receptive fields in both eyes is linearly combined by V1 simple cells and this sum is then passed through an output nonlinearity. This simple scheme naturally places limits on the extent to which binocular regularities in the inputs can be exploited. We explore these limits using a simple stimulus that allows the gradual transition from physical to nonphysical stimuli to be studied: the sum of two sinusoidal luminance gratings.

First, we show that the binocular energy model predicts maximum responsiveness to nonphysical left and right image pairs. Second, we recorded responses of disparity selective neurons in the striate cortex of awake monkeys. Approximately 42% of the neurons behave as the energy model predicts, with either their maximum or their minimum response elicited by nonphysical combinations of interocular phase difference. The other 58% of neurons deviate from the model's predictions, and these deviations are systematically in the direction that exploits the regularities in the natural inputs, as predicted by the efficient coding hypothesis.

Finally, we show how this specialization for physical disparities can be explained by extending the existing standard model, if real neurons represent the sum of multiple energy model-like subunits. This shows that combining multiple subunits can be a useful strategy for exploiting the statistical regularities of visual inputs. It might thereby provide an explanation for another recent observation (Rust et al., 2005) that indicates the presence of multiple excitatory and suppressive spatiotemporal subunits within V1 receptive fields.

Results

Energy model predictions

The energy model is part of a family of generalized Linear-Nonlinear (LN) models describing the responses of V1 neurons (Chichilnisky, 2001; Rust et al., 2005). During the linear stage of an LN neuron, the 2D image is multiplied with the neuron's 2D receptive field, yielding a single scalar value that represents the effect of the image on the RF. The scalar value is then passed through an output nonlinearity (typically parameterized with an exponent) the result of which determines the spike rate of the neuron. In the binocular energy model there are two separate receptive fields (RF), one for the left and one for the right eye, and their scalar responses are summed linearly, and this sum is then passed through an expansive output nonlinearity. The output nonlinearity means that the neuronal activity is enhanced when left and right RFs produce similar responses, and it is this that confers disparity selectivity on the cell. The top row of Fig. 1B shows two examples for such receptive fields – green for the left and red for the right eye (1D cross sections, the second dimension is typically well approximented by a Gaussian and can be safely ignored for the following discussion). The RFs are well-described by Gabor functions. There are two principle ways in which the cell can encode disparity: 1) the left and right receptive fields are identical up to a translation (“position disparity”, Fig. 1B top-left); or 2) receptive fields in the two eyes are related by a fixed phase shift applied to all frequency components (“phase disparity”, Fig. 1B top-right) (Ohzawa et al., 1990; DeAngelis et al., 1991; Fleet et al., 1996). Let's take the example of a position shift. At the bottom left of Fig. 1B we see the response (solid blue) to a broadband stimulus (stimuli in the real world are “broadband” in the sense that they consist of many Fourier components across a broad range of frequencies), with the displacement (“stimulus disparity”) between the left and the right plotted along the x-axis. The response peaks when the displacement between the images coincides with the displacement of the receptive fields. The model cell in the right column of Fig. 1B (solid blue) has a phase disparity, and this produces a characteristically asymmetrical shape in the disparity tuning curve. Fig. 1A illustrates the difference between position and phase shift in the stimulus. The solid lines in the top left panel in Fig. 1A show the profiles of a compound grating consisting of two sine-wave gratings with frequency ratio 1:2, displayed individually in the panels below. The profile in the left eye (green) is identical to that in the right eye (red) apart from a positional displacement of Δx. It is important to note that as a consequence of an identical positional displacement Δx in both component gratings, the phase shift is different for each component: for a fixed position shift Δx the phase shift Δϕi between the left and right Fourier component of frequency fi is proportional to their frequency: Δϕi = Δx · fi (Fleet et al., 1996). In the right column of Fig. 1A, on the other hand, the luminance profile in the left eye was phase shifted compared to the right eye (an identical interocular phase difference is added to each frequency component). However, as the two lower panels illustrate for F1 and F2, an identical phase shift implies different physical displacements. The implication for a broadband pattern can be seen in the top left and right panels of Fig. 1A (dashed): while a position-shifted broadband pattern (here an example bar) is identical in the left and right eye up to translation, the phase-shifted broadband pattern has a different shape in the left from the right eye, something that never occurs in the real world. Although real surfaces (e.g., tilted planes or curved surfaces) can give rise to disparities that are not simple translations, no solid surface produces equal interocular phase differences in all its Fourier components.

Figure 1
Geometric constraints

In order to explore the relationship of those two regimes – phase and position disparities – in both the stimulus and the RFs, we have used the simplest stimulus that allows both to be defined, a compound grating that consists of two frequency components, F1 and F2. Fig. 2 shows the energy model response for the two example neurons in Fig. 1B – one with RFs related by a pure position disparity (Fig. 2A) and one with RFs related by a pure phase disparity (Fig. 2B). The neuronal response is given in a 2D space spanned by the interocular phase differences for each component independently: the interocular phase difference for grating 1 is plotted on the x-axis and that for grating 2 on the y-axis. Each physical displacement of the image in the left with regard to the right eye implies a pair of interocular phase differences (Δϕ1ϕ2) where Δϕ1 / Δϕ2 = f1 / f2. We chose a value f2 / f1 = 2 for our empirical study. Disparities that result from the physical depth of planar patches in natural viewing therefore form a line of slope f2 / f1 = 2 in this space. We refer to this line as the “physical disparity line.” Stimuli away from this line do not occur in natural viewing (are nonphysical).

Figure 2
Energy model response

In the Methods (Energy model predictions) we show that the response of the energy model to a drifting compound grating is linear in its components: the response to a compound grating is simply the sum of the responses to the component gratings individually. Since the response of an energy model to a single grating is a sinusoidal function of disparity (Fleet et al., 1996), the peak and the trough of the combined response will be separated by 180 deg along each dimension (see Fig. 2A&B). Hence, the line connecting peak and trough (green in Fig. 2A&B) has a slope of 1. This means that peak and trough can never both lie on the physical disparity line at the same time – at least one of them has to occur for non-physical stimuli. The two examples in Fig. 2 illustrate this: for energy model neurons with no phase disparity, the maximum response is elicited by a naturally occurring disparity (Fig. 2A). However, in such neurons, eliciting the minimum response requires a nonphysical stimulus, one where each Fourier component is shifted by 180 deg, i.e. inverted (dashed lines in Fig. 2A&B). This means that only part of the dynamic range is dedicated to stimuli that actually occur in the real world. The same is true for an energy model constructed with a pure phase disparity (Fig. 2B). The maximum response is elicited when Δϕ1 = Δϕ2 and equals the RF phase disparity, a stimulus that is necessarily away from the physical disparity line. Note that the response in Fig. 2B is simply a translated version of Fig. 2A. This is true in general: any changes in either position or phase disparity correspond to a mere translation of the response surface, but do not alter the relative location of maximum and minimum responses. The relative location is also unaffected by changes in the static output nonlinearity of the energy model. Thus all variations on the energy model (including hybrid models that combine phase disparity and position disparity) exhibit a common feature: the separation between maximum response and minimum response is 180 degrees of phase on both axes. This shared feature of energy models implies that part of a neuron's dynamic range is devoted to parts of this space that do not correspond to realistic stimuli.

Neuronal responses

We examined the responses of disparity selective neurons (recorded from the striate cortex of two awake fixating monkeys) to such stimuli, constructed by summing two sinusoidal luminance gratings, with spatial frequencies in the ratio 1:2. The top row in Fig. 3 illustrates the responses of one example cell, which closely matches the predictions of the energy model. The response to all combinations of interocular phase difference is well predicted by the responses to individual components, as expected from the energy model. Its response peaks for a physical stimulus, as implied by a model with no phase disparity and, as a consequence, the lower end of its dynamical range is dedicated to non-physical stimuli. The bottom row in Fig. 3 shows responses of a different cell that did not follow the model predictions. This cell exploits the properties of naturally occurring disparities – both peak and trough lie close to the physical disparity line. Thus, all its dynamic range is devoted to physical stimuli.

Figure 3
Two example cells

The degree to which a single neuron exploits the properties of physical disparities can be conveniently quantified by the shortest distance from the physical disparity line to the location of the peak and of the trough (blue arrows in Fig. 2). The sum S of these distances is 0 if both peak and trough lie on the line. Since the responses of all energy models are identical up to translation in our space, S is the same for all of them. From the geometry of the space it follows that S = 90deg· sin arctan 2 or about 80.5 degrees for the energy model (note the shortest distance is not aligned with either axis, and the value of S depends upon the slope of the physical disparity line). Because both axes represent circular variables, the maximum possible value for this sum S is ~161 degrees, the shortest distance between adjacent sections of the physical disparity line. We can therefore use S as a measure for how well adapted any one neuron is to physical images. For the example neurons in Fig. 3 that resembles the energy model S = 81 degrees while for the other, adapted, neuron S = 16 degrees. Note that S is strictly positive and S = 0 is only achieved when both peak and trough lie exactly on the physical disparity line; any deviation due to sampling noise increases its value so that the observed value of S would be greater than 0 even for a perfectly adapted neuron.

We show the locations of peaks and troughs for our sample of 45 neurons in Fig. 4. It is striking that the locations of the peaks are clustered closely around the physical disparity line, indicating some specialization for the properties of natural binocular images. This observation alone can readily be explained by the energy model, as long as the majority of neurons have only small phase disparities. However, if that were the case, then the response troughs would necessarily cluster away from the physical disparity line as explained above. What we find, however, is a distribution of troughs with two maxima, one near the physical disparity line and one far away (Fig. 4C, blue). This implies that there is a subpopulation of neurons that has both peaks and troughs near the physical disparity line (like our example in Fig. 3B). The deeper implication is a specialization for natural binocular images in those neurons – dedicating the majority of the dynamic range to physical disparities – one that the energy model cannot explain. Despite this phenomenon, the clustering of peak responses around physical disparities is much stronger than the clustering of troughs. This suggests that the coding scheme used by the brain may attach a special significance to response maxima. The distribution of troughs shows a second peak around ±80.5 deg compatible with an energy-model-like population of cells with peaks near the physical disparity line, and troughs therefore far away from it.

Figure 4
Population plot of peak and trough locations

The adaptation to physical disparities that we observe automatically implies a deviation from the energy model, but the converse is not true: deviations from the energy model do not in general produce this adaptation – deviations can also increase the distance to the physical disparity line so that less, not more, of the dynamic range is devoted to naturally occurring stimuli. In order to quantify deviation from the energy model independent of adaptation, we use another measure of distance in the interocular phase space: the Euclidian distance of the observed peak from the energy model prediction, given the observed trough. In Fig. 5A we compare those two measures – the distance to the energy model prediction, and the total distance to the physical disparity line. First we note that the upper half of the space potentially occupied by neurons (borders are dashed) is almost completely empty and all cells lie either close to the centerline (energy model) or below it. This demonstrates the power of the pressure for disparity-selective neurons to be adapted in this particular way. We find that whenever neurons deviate from the energy model, they do so to decrease S and systematically devote more of their dynamic range to signaling naturally occurring disparities. The correlation between the two measures is highly significant (Spearman's r = −0.77; p = 4 · 10−9). In our sample of 45 neurons, 17 (38%) show a significant adaptation (p < 0.05, by resampling, magenta dots in Fig. 5). Of the 20 neurons that deviate significantly from the energy model, all but one do so in the direction of increased adaptation. The marginal distribution of neurons along the y-axis (Fig. 5B) confirms this finding on the population level. The blue solid curve shows the distribution of the 1000x resampled data (raw data in cyan). By assuming that values S > 80.5deg are due to sampling noise around energy-model-like neurons (true S = 80.5deg), we can estimate the overall distribution of energy-model-like cells (dashed blue curve, see Methods). The area under the dashed curve yields the total percentage of energy-model-like neurons in our population: 42%. The remaining 58% are at least partially adapted (S < 80.5deg).

Figure 5
Population summary of adaptation

A new model

The new specialization of disparity selective neurons we find can be achieved by a relatively simple extension of the energy model (illustrated in Fig.6). Conceptually, an adapted cell can be constructed by combining two cells (C+ and C-), each of which is a traditional energy model plus an expansive static output nonlinearity. Here, input from one of the cells (C-) is subtracted from the response of the other (C+). The result is that the location of the response maximum is largely determined by the position of the peak for C+, and the response minimum is largely determined by the location of the peak for C-. Fig. 6A demonstrates this for compound gratings: if the peaks for C+ and C- lie close to the physical disparity line (produced by position disparity), then the model cell reproduces the adaptation we observe. Fig. 6B shows the response of the same model to random dot stereograms (RDS, red lines). Both C+ and C- are Tuned-Excitatory (TE) and incorporate an expansive output nonlinearity (an exponent of 3 in this example). The third panel of Fig. 6B shows the total response of the new model: its preferred and anti-preferred disparities are inherited from its subunits and its overall tuning curve is odd-symmetrical, very much as expected from an energy model with phase disparity. We refer to these disparity selective input neurons (C+ and C-) as “subunits” below. In order to avoid confusion, we shall refer to the inputs to the energy model itself as binocular simple cells.

Figure 6
Proposed subunit model

Note that although the parameters for our example model shown in Fig. 6 were not fitted to real data, they were chosen to yield realistic responses to gratings and RDS. The particular values for the disparities and spatial frequency tuning were chosen such as to resemble the data of the example cell in Fig. 3B. The preferred disparity of subunit 1 is given by the location of the peak in the 2D response surface in Fig. 3B and the preferred disparity of subunit 2 by the location of the trough. The RFs of both subunits overlap extensively in each eye, but the exact location does not affect disparity responses. We also emphasize that the spatial frequency tuning curve is identical for both subunits: the adaptation is purely due to combining subunits followed by an additional output nonlinearity, not because of different frequency responses of the subunits.

It is not essential that one subunit provide a subtractive input. A similar result is obtained if a binocular cell of the Tuned-Inhibitory (TI) type (one whose disparity tuning function is dominated by a trough) (Poggio and Fischer, 1977) is summed with a cell C+ (Tuned-Excitatory), as shown in Supplementary Figure S1. In this case the Tuned-Inhibitory cell must incorporate a compressive output nonlinearity. What both implementations have in common is that one subunit contributes a pronounced peak in the response, and a separate subunit contributes the pronounced trough. We find that when linearly combining two energy model cells, it is important that we include a static output nonlinearity in addition to the squaring inherent to the energy model for the two cells – linearly combining two energy model units without an additional output nonlinearity cannot produce an adaptation to encoding physical disparities.

The simplest form of our subtractive model assumes that all the RFs in both subunits are identical up to a translation, and assumes that all subunits have exactly balanced inputs from both eyes. Deviations from these simplifying assumptions have little impact on the binocular responses we predict, but can profoundly affect other properties of model neurons, such as the RF properties expected during monocular stimulation. In our simplest subtractive model, cancelation of the inputs from two subunits can even lead to no monocular responses in either eye (resembling the striking property of many disparity selective neurons in which the binocular response is much greater than either monocular response). However, if the model simple cells do not have exactly balanced inputs from the two eyes (commonplace in real neurons (Hubel and Wiesel, 1962; Prince et al., 2002; LeVay and Voigt, 1988; Smith et al., 1997; Ohzawa and Freeman, 1986b)) it is to easy produce monocular receptive fields that resemble real complex cells. If the simple cells contributing to the subtractive subunit have stronger inputs from one eye, while the inputs to the excitatory subunit are dominated by the other eye, the result will be a neuron that is disparity selective yet appears monocular (no response to stimulation in one of the eyes). Such responses have been noted in many studies (Ohzawa and Freeman, 1986b; Read and Cumming, 2004a; Smith et al., 1997), and are naturally explained in our new model. Ocular imbalance in not necessary to produce monocular responses in the additive version of our model (TE plus TI).

So far the only assumptions we have made about the nature of C+ and C- concern their disparity selectivity. Even if C+ and C- were binocular simple cells, out model still produces an adaptation. Such a neuron would be classified as simple by the criteria used in physiological experiments (Kagan et al., 2002; Skottun et al., 1991). As a result, at least two different implementations can yield adapted complex cells. In the first, hierarchical one (illustrated in Fig.7A), the adapted neuron receives its input from two non-adapted complex cells, which each receive inputs from binocular simple cells. A second implementation does not require an additional hierarchy of complex cells as shown in Fig. 7A. If all simple cell subunits, used to construct the two complex cells, are combined in one step to form a complex cell, an adaptation is produced, provided each simple cell subunit is augmented by an output nonlinearity. The additional output nonlinearity of the simple cell enhances the peak (or trough) of the disparity response. Note that only two traditional binocular simple cells of different preferred disparity (S1a and S2a) are strictly necessary to produce an adapted simple cell in either implementation. Four simple cells (S1a&b and S2a&b) are necessary to create a complex cell.

Figure 7
Two implementations of subunit model

In both formulations, two key features of the original energy model are preserved: processing is linear up to the point of binocular combination, and a static output nonlinearity generates sensitivity to disparity. However, the original energy model also assumes (primarily for mathematical tractability) that all simple cell subunits share the same phase disparity and position disparity (Ohzawa et al., 1990). The empirical deviations shown in Fig. 5 force us to abandon this assumption.

The extensions we propose have a profound impact on the long-standing debate concerning the relative roles of position disparity and phase disparity in generating the responses of complex cells (Anzai et al., 1999c; Anzai et al., 1999a; Cumming and DeAngelis, 2001; Fleet et al., 1996; Nieder and Wagner, 2000; Ohzawa et al., 1997; Prince et al., 2002; Qian, 1997; Wagner and Frost, 1993). Our results demonstrate for the first time that the responses of many neurons cannot be summarized with a single pair of values for phase and position disparities. The second example in Fig. 3 illustrates this point: an analysis of responses to physical disparities (a cross section along the physical disparity line) reveals a pattern of odd symmetry that has been used to infer the existence of a phase disparity (Cumming and DeAngelis, 2001; Ohzawa et al., 1990). Our examination of responses to other combinations of interocular phase differences reveals that this inference is incorrect when the responses are better described as the result of more than one position disparity, or more than one phase disparity.

This model explains earlier results

Finally, these data and the extended energy model we propose throw new light on one of the few existing observations concerning nonphysical disparities. Using stimuli in which the left eye's image is the photographic negative of the right eye's image (anticorrelated stereograms), several groups have reported that neurons still show disparity selectivity, but that the strength of this selectivity is weaker than for correlated stereograms (Cumming and Parker, 1997; Nieder and Wagner, 2001; Ohzawa et al., 1997). Two modifications of the energy model have been proposed to explain this attenuation. One model (Lippert and Wagner, 2001) pointed out that passing the output of an energy model through an additional static nonlinearity can explain the result in some cells (although this cannot work for cells with odd-symmetric tuning (Read et al., 2002)). A second model proposed that a threshold is applied to monocular responses before binocular combination (Read et al., 2002).

Our data provide a powerful test of these models because responses to anticorrelated stereograms are part of the same continuous space as responses to correlated (physical) stimuli. They are a cross section through our data running parallel to the physical disparity line (dashed lines in Fig. 2). Models with thresholded monocular responses generate diagonally elongated regions in plots like Figs. 2 and and3,3, unlike any we observed in real neurons. Simply adding an output exponent to the energy model does not change the locations of the peaks and troughs and so cannot explain the results shown in Figs. 3, ,44 and and5.5. Thus our data demonstrate that current explanations of the responses to anticorrelated stereograms are inadequate. Our model (Fig. 6), by virtue of placing both response peak and trough on the physical disparity line, inevitably produces an attenuated response to anticorrelated compared to correlated patterns (blue vs red lines in Fig. 6B). Thus this same model also captures the main features of the same neuron's response to anticorrelated RDS (figure 8B).

Figure 8
A&B: Responses to anticorrelated stimuli

Therefore a further test of our model is provided by comparing the responses to anticorrelated compound gratings with the responses to anticorrelated RDS. In Fig. 8A&B we compare the responses to correlated and anticorrelated compound gratings by taking cross sections of the response along the correlated (physical disparities) and the anticorrelated line (solid and dashed, respectively, in Fig. 2). The first example cell shows a very good agreement between the compound gratings and the RDS response in accordance with the energy model. The second example neuron shows an attenuation for both stimulus types; however, the attenuation for the anticorrelated RDS response is significantly stronger than that for the anticorrelated compound grating response. Figure 8 shows that this second pattern was typically observed across the population of our cells: the attenuation is generally larger for RDS than it is for compound gratings and the two are correlated (Spearman's r=0.38; p<0.02). This stronger attenuation for RDS results because RDS contain many more frequency components than the two in our compound gratings, allowing the nonlinear interaction between them to further accentuate the effect. Our model replicates this behavior. The red line in Fig 8 represents one example family of our models, obtained by starting with the example model in Fig. 6 and varying the output exponent for each complex cell subunit (from 0.9 in the upper right to 3.9 in the lower left). This line illustrates how our model replicates two aspects of the data: 1) the attenuation for RDS is stronger than for compound gratings and 2) when keeping everything else constant and varying only the output nonlinearity both attenuations are correlated. Please note, however, that the line we show represents only one of many families: the location and slope depend upon other model parameters (including the shape of the nonlinearity, the spatial frequency tuning of the RFs, and the baseline firing rate). Consequently, even our model does not predict a strong correlation between the attenuation for RDS and for compound gratings for a randomly sampled population of neurons. Nonetheless, a number of neurons lie above the line produced by our model. Whether they can be explained by invoking more than two subunits or by including other known properties like contrast-gain control requires further investigation.

Discussion

Over small regions of the visual field, the difference between the two eyes' images is well described by a simple translation. This means that, when decomposing the two images into their different frequency components, the interocular difference in the spatial phase of each frequency component is proportional to its frequency. Existing models of disparity selectivity, in particular the energy model, do not take this restriction into account, devoting part of their dynamic range to stimuli that never occur in the real world. We find that some neurons share this property of the energy model, and are best tuned to nonphysical stimuli. This striking failure to adapt responses to the properties of physical binocular images is inevitable in the energy model, and provides a powerful confirmation of it. However, we also find that more than half of our V1 neurons deviate from the energy model, and do so in a way that exploits the properties of physical binocular images. We show that they concentrate their dynamic range in the regions occupied by realistic stimuli, demonstrating a novel adaptation of cortical neurons.

Furthermore, neurons that show no such adaptation (or incomplete adaptation) tend to have their maximal responses elicited by physical stimuli (Fig 4). This necessarily results in response minima elicited by non-physical stimuli. This difference between the locations of minima and maxima suggests that the brain attaches special significance to high firing rates.

We propose two alternative models to explain the adaptation we observe: (1) combining the outputs of two (or more) energy model-like complex cells, with different phase or position disparities, in a new stage of binocular processing, or (2) allowing complex cells to receive input from binocular simple cells with different phase disparities, position disparities, and output nonlinearities. Model (1) has the advantage that it can explain why almost half of our neurons are not adapted to natural stimuli – they are necessary as building blocks. Model (2) does not need an additional processing stage and might provide a more natural explanation for the fact that our distribution of neurons displays a continuum of degrees of adaptation. Although model (2) does not explain why so many cells fail to show adaptation, it may be that responses to unnatural combinations of interocular phase may be useful for some other functions. One possibility is that these cells help solve the correspondence problem (determining how to match image features between left and right images), precisely because they identify signals that cannot correspond to a real world object (Read and Cumming, 2007). Both extensions of the energy model also reconcile the model with the earlier observation that responses to disparity in anticorrelated stereograms are weaker than responses to correlated stereograms (Cumming and Parker, 1997; Ohzawa et al., 1997; Nieder and Wagner, 2001).

The adaptation we describe cannot be explained by other, already known effects like dynamic changes in spatial frequency tuning (Bredfeldt and Ringach, 2002). As we show, the energy model response to a drifting compound grating is the sum of its responses to the component gratings individually, each of which is sinusoidal. As the spatial frequency tuning changes over time, only the amplitude of those sinusoidal responses is affected, not their shape. This means that the relative location of peak and trough in the 2D response to compound gratings is also unaffected – separated from each other by 180 deg along each dimension. Indeed, this is true for any frequency combination, any spatial frequency tuning and any output nonlinearity.

Another feature of real neurons that our model explains naturally concerns the binocular responses and ocularity of neurons. It has long been known that many neurons only respond to stimuli in one eye when probed monocularly, but are actually disparity-selective when tested binocularly (LeVay and Voigt, 1988; Ohzawa and Freeman, 1986a; Ohzawa and Freeman, 1986b; Read and Cumming, 2004a; Smith et al., 1997). This is a direct consequence of the subtractive implementation of our model. Consider the following case: two TE cells with identical RFs at identical locations in both eyes are subtracted from each other. The result is that both subunits will cancel each other perfectly (C+ minus C-) in both eyes. If we then allow C+ to have stronger input from the left eye, while C- has stronger input from the right eye, the neuron will respond to monocular stimulation in the left eye, but not in the right eye.

A quantitative and more detailed examination of the relationship between the monocular and binocular responses can be achieved by an analysis of Spike-Triggered Covariance (de Ruyter van Steveninck, 1988; Rust et al., 2005) which reveals functional subunits (eigenvectors). This is possible even for model neurons that would be classified as simple cells. Such an analysis can also distinguish between the additive and subtractive version of our model. We have confirmed in simulations that the additive version produces only excitatory subunits, while the subtractive version also produces inhibitory subunits. This is in agreement with recent physiological data (Rust et al., 2005) that found excitatory and inhibitory eigenvectors for most simple and most complex cells in monkey V1. Since those experiments were conducted with monocular stimuli, it is not possible to say whether the inhibitory eigenvectors had the binocular properties predicted by our model. A spike-triggered analysis of binocular interactions in complex cells of the cat (Anzai et al., 1999c; Anzai et al., 1999b) did not include an analysis of the monocular responses, so it is not clear that those data are at odds with our model.

Hence, allowing for multiple subunits that differ from each other in their disparity tuning changes our understanding of the relative roles played by position and phase disparity in complex cells. In the context of the original energy model, responses to a range of disparities can be used to deduce phase and position disparity (even in hybrid models that contain a mixture of both) (Anzai et al., 1999c; Cumming and DeAngelis, 2001; Fleet et al., 1996; Nieder and Wagner, 2000; Ohzawa et al., 1997; Prince et al., 2002; Qian, 1997; Wagner and Frost, 1993). Our new data show that for many disparity-selective cells it is not possible to summarize their responses with a single pair of phase and position disparities, and that attempts to do so can yield misleading results. For example, odd-symmetric tuning curves (“Near/Far” types) have generally been thought to indicate the presence of phase disparity (Fleet et al., 1996; Freeman and Ohzawa, 1990; Ohzawa et al., 1990). However, we show that odd symmetry can be achieved by combining subunits with different position disparities and data like that shown in Figs. 3, ,44 and and55 suggest that this combination is important, even in the striate cortex. If our model uses position disparities that differ only in their horizontal components, it resembles one that was proposed to explain the observation that V1 neurons encode a wider range of horizontal than vertical disparities (Cumming, 2002; Read and Cumming, 2004b) – another specialization for the statistics of the binocular input.

We further note the similarities between our subunit model for disparity-selective V1 neurons and the original motion energy model for direction selectivity (Adelson and Bergen, 1985). In both cases, the first stage is provided by the output of V1 simple cells selective to a particular stimulus feature (there a difference of position in time, here a difference of position between the eyes). And in both cases, the next step pools over a population of those simple cells. However, in the last step of the motion energy model, the responses of two pools of neurons signaling opposite directions are subtracted from each other (“opponency”). This opponent stage has not been used in previous versions of the binocular energy model. Our model incorporating subtraction is closely similar to the motion energy model with opponency, although differs somewhat because our two subunits are not in general symmetrical about zero disparity. Interestingly this agrees with psychophysical measures of disparity opponency, in which opponent channels were not symmetrical about the fixation plane (Cormack et al., 1993). Although this similarity is interesting it is important to note that the psychophysical result could still occur even if V1 contained only energy model-like neurons: one simply has to propose that opponency occurs later in the pathway. Similarly, the observation that stereoscopic adaptation produces a depth aftereffect (Blakemore and Julesz, 1971) is a necessary feature of our model, but is compatible with phase disparity encoding in V1.

The model we propose may also be relevant to understanding some of the transformations applied to disparity signals in extrastriate cortex. It has long been recognized that odd-symmetric tuning curves are more frequently encountered in extrastriate cortex than in striate cortex (Cumming and DeAngelis, 2001; Poggio, 1995; Poggio et al., 1988). Our model provides an explanation for how this could be achieved in the projection from striate cortex to extrastriate cortex, and our data demonstrate a way in which this suggestion can be tested.

The idea that complex and simple cells in striate cortex receive input from multiple subunits with different spatiotemporal properties recently received direct support from a spike-triggered covariance analysis (Rust et al., 2005). Our results suggest that one reason that multiple subunits are combined is to ensure that the dynamic range of neuronal signals is matched to the range of inputs encountered naturally. This requires a combination of subunits appropriate to match the properties of the binocular inputs. The same principle might explain the recent finding that neurons give stronger responses to natural than to artificial monocular images (Felsen et al., 2005). This could result if multiple subunits were similarly matched to the statistics of monocular natural images. Thus, the unexpected specialization we show here might provide a general mechanism by which neural signals can be matched to the properties of natural occurring inputs.

The efficient coding hypothesis is one of the few unifying, cross-modal theories for the sensory system. Its main prediction, that neurons in the sensory system should be adapted to natural inputs, has been tested for a number of specific features inherent to auditory and monocular visual signals (Smith and Lewicki, 2006; Field, 1987; Schwartz and Simoncelli, 2001; Simoncelli and Olshausen, 2001). The tests in the visual domain have been on comparatively subtle statistical aspects of the stimuli, like short-range spatial correlations and the shape of the power spectrum. In this study we tested a specialization to a particularly strong redundancy caused by the binocular nature of visual inputs. The results a) confirm the efficient coding hypothesis by way of a novel adaptation for some neurons in our sample and b) for the other cells provide a strong confirmation of the energy model mechanism in generating disparity signals in the brain. We propose an extension of the energy model as a unified account of the behavior of both types of neurons.

Methods

Stimulation and Recording

We recorded extracellularly from isolated V1 neurons in two awake, fixating rhesus macaque monkeys. Details of the experimental protocol are described elsewhere (Read and Cumming, 2003). All protocols were approved by the Institute Animal Care and Use Committee and complied with Public Health Service policy on the humane care and use of laboratory animals.

In brief, monkeys perched in a primate chair with their heads immobilized. A digitally controlled microdrive was used to lower electrodes (glass coated PtIr, FHC Inc.) through the dura into V1. Spikes were isolated based on the height and width of the electrical waveforms. All electrical activity crossing a manually set voltage threshold was saved for offline analysis with a precision of 0.1 ms. The positions of both eyes were recorded with scleral search coils (implanted under general anesthesia). Stimuli were presented on two Eizo Flexscan F980 monitors (96Hz framerate, mean luminance 41.1 cd/m2, contrast 99%, viewing distance 89cm, 1 pixel subtended 1.1 min arc at the eye) viewed through a Wheatstone stereoscope. Gamma correction was adjusted to achieve a linear luminance response.

Trials lasted approximately two seconds, during which the monkey was required to maintain fixation within a 0.8° window. If the monkey successfully maintained fixation throughout the trial, he received a water reward. Each trial consisted of four 420 ms stimulus presentations, separated by 100 ms blank periods. If the monkey broke fixation before the end of the trial, the successfully completed stimulus presentations were saved, but the monkey did not receive a reward. Responses were measured as the mean spike rate over the entire stimulus presentation, beginning 50 ms after stimulus onset. Minimum response fields (MRF) were mapped manually with bars and gratings, and all subsequent stimuli were centered over the MRF, and were made substantially larger than the MRF to ensure that the receptive field was covered. Preferred orientation and spatial frequency were measured with patches of grating presented to the dominant eye, and these values were then used when measuring disparity selectivity to a single binocular grating. Disparity selectivity was also examined with dynamic random dot stimuli. Neurons exhibiting disparity selectivity to either gratings or RDS were then examined with compound gratings. (We also examined a few cells that were not disparity selective to single gratings or RDS. Such cells never exhibited selectivity for interocular phase difference in compound gratings, so were excluded.)

Compound gratings were generally constructed from summing two 50% contrast gratings (frequencies in the ratio 1:2) that spanned the preferred spatial frequency. In a few cases it was necessary to alter the frequencies because the preferred spatial frequency defined with monocular gratings did not produce the strongest disparity selectivity. The stimuli were typically square patches 4°×4° across, and the spatial frequencies used ranged from 0.1 to 6 cpd (depending on cell preference). The distribution of preferred orientations was approximately uniform. Eccentricities were in the range 2.5° to 8.5°. Eight values of interocular phase difference were applied to each component (= 64 combinations), but the absolute phase (mean of the two eyes) for each component was set randomly at the start of each stimulus. Thus the absolute phase and the relative phase between components in either eye were unaffected by changes in the interocular phase difference. The temporal frequency for each component was inversely proportional to its spatial frequency so that each eye saw a fixed pattern drifting through an aperture. Responses to the same interocular phase differences were also measured for the individual component gratings. All 80 stimulus conditions were interleaved in a pseudo-random sequence. A minimum of five stimulus repetitions was required, and the mean value was 13.6 repetitions per stimulus.

Analysis

We examined responses in 64 neurons that showed significant selectivity to either RDS or single binocular gratings (p<0.05 on 1-way ANOVA). Nineteen of these were excluded from further study because they did not show significant disparity selectivity to one of the two component gratings when tested alone (p<0.05 by 1-way ANOVA). All 45 of the remaining neurons showed significant selectivity for interocular phase in the compound grating.

In order to determine the location of maxima and minima, the responses were smoothed with a two-dimensional Gaussian filter (σ = 36 degrees), and the extrema of the smoothed response were used. The locations of these extrema were then used to compute the two indices (described in the Results) quantifying deviations from the energy model.

Data on responses to RDS were available for 43 of the 45 neurons. Five of these did not show significant responses to RDS, so we were able to compare the effects of anticorrelation (in RDS and compound gratings) in 38 neurons. To quantify the response attenuation caused by anticorrelation in compound gratings, we simply compared peak-to-peak response magnitudes for the appropriate slices through smoothed response surfaces like those in Fig. 3. To quantify this attenuation in anticorrelated RDS we simultaneously fit two Gabor-functions, allowing only the amplitude and phase to differ between the two stimuli (Cumming and Parker, 1997):

fcorr,anti(x)=b+acorr,antiexp((xx0)22σ2)cos(2πfx(xxd)φcorr.anti)

where b is the response to uncorrelated stimuli, acorr and aanti are the amplitudes, x0 is the location of the centroid (equal to preferred disparity in the case of tuned even-symmetric cells), fx is the preferred spatial frequency and [var phi]corr and [var phi]anti are the phase disparities for the correlated and anticorrelated responses. The ratio aanti / acorr then defines the attenuation.

Energy model predictions

In the classical energy model for disparity-selective complex cells (Ohzawa et al., 1990; Ohzawa et al., 1997) the output of two pairs of binocular simple cells with different RFs in both eyes are combined. The RFs in each pair are the inverse of each other and the RFs of the two pairs are phase-shifted by 90 degrees. The preferred position and phase disparity of the cells is then given by the position and phase difference between the RFs in left and right eye which are assumed to be identical in all four binocular simple cells. Following the notation of (Read et al., 2002) we can write the output of a monocular receptive field as

ν=dxdyI(x,y)ρ(x,y)

where I (x,y) is the luminance of the image in that eye and ρ(x, y) is the receptive field. For a grating of frequency f and phase ϕ, this becomes

ν(f,ϕ)=ρO(f)sinϕ+ρE(f)cosϕ
(A1)

where ρE,O is the even/odd part of the Fourier transform of the receptive field. For clarity in the following derivation and without restricting generality, we assume position and phase disparity to be zero, one binocular pair to have RFs which are purely even (E), and one pair whose RFs are purely odd (O). Using the labels L and R to distinguish between left and right eye, the complex cell response to a compound grating becomes

C=(νLE+νRE)2+(νLO+νRO)2
(A2)

with

νLO=Asinϕ1+Bsinϕ2νRO=Asin(ϕ1+Δϕ1)+Bsin(ϕ2+Δϕ2)νLE=Acosϕ1+Bcosϕ2νRE=Acos(ϕ1+Δϕ1)+Bcos(ϕ2+Δϕ2)
(A3)

where indices 1 and 2 refer to the component gratings of frequencies f1 and f2 and Δϕ1,2 to the interocular phase difference for component grating 1 and 2 respectively. A=ρ(f1) and B=ρ(f2) are given by the Fourier transform of the two RFs evaluated at frequencies f1 and f2 (Eq. A1). Substituting (A3) into (A2) yields

C=2A2(1+cosΔϕ1)+2B2(1+cosΔϕ2)+2AB[cos(ϕ1ϕ2)+cos(ϕ1ϕ2+Δϕ1)+cos(ϕ1ϕ2+Δϕ2)+cos(ϕ1ϕ2+Δϕ1Δϕ2)]

The first two terms in this sum are the responses of the complex cell to each of the two component gratings alone. The third term depends both on the interocular phase differences Δϕ1,2 and the difference between the two absolute phases ϕ1ϕ2.

For a compound grating ϕ1ϕ2 varies as a function of physical displacement. Therefore the energy model response to compound gratings varies as a function of position, even though the response is position-invariant when either grating is presented alone. However, since ϕ1ϕ2 takes on all possible values between 0 and 360 degrees for each pair (Δϕ1ϕ2), the third term averages to zero for drifting compound gratings. Hence, the average response of the energy model is linear in drifting gratings of differing spatial frequencies:

C(Δϕ1,Δϕ2)t=2A2(1+cosΔϕ1)+2B2(1+cosΔϕ2)=C(Δϕ1)+C(Δϕ2)

where C(Δφ)=2ρ(f)2·(1+cosΔφ) is the energy model response to an individual grating with identical receptive fields in left and right eye. Because this result depends on averaging over absolute phases, it can be seen that the same holds for simple cells. Relaxing the assumption of identical RFs in left and right eyes and introducing a position and a phase disparity ΔxRF and Δ[var phi]RF between the RFs in the left and right eye has the effect of simply translating the response pattern (for an example see Fig. 2):

C(Δϕ1,Δϕ2)t=2A2(1+cos(Δϕ1ϕRF(f1)))+2B2(1+cos(Δϕ2ϕRF(f2)))

where the preferred interocular phase difference is [var phi]RF(f) = ΔxRF · f + Δ[var phi]RF.

Computational details for Figure 5

We obtain an estimate of the frequency distribution of our population by performing a 1000 times resampling with replacement on the original spike rates. The resulting distribution functions fi(S) obtained for every neuron i are summed to form the distribution function F(S) for our entire population (solid blue curve in Fig. 5). In order to estimate the proportion of adapted neurons, we assume that F(S>80.5deg) is entirely due to energy-model-like neurons plus sampling noise. This can only overestimate the true number of energy-model-like neurons. Since random noise gives rise to a symmetric distribution around 80.5deg, the distribution of energy-model-like neurons will be symmetric with respect to 80.5deg (dashed blue line in Fig.5). The difference between this and the total distribution represents the adapted subpopulation.

Supplementary Material

01

Acknowledgments

This research was supported by the Intramural Research Program of the NIH, National Eye Institute. We thank Denise Parker for excellent animal care.

Footnotes

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