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Cortical receptive fields. Cat cortical layer 4 simple cell RFs were modeled as Gabor functions (see Fig. 2A). A Gabor function is a two-dimensional Gaussian, here with peak value 1, multiplied by a sinusoid. Positive regions of the Gabor correspond to ON subregions and yield connections from ON-center LGN cells, and negative regions correspond to OFF subregions and yield OFF-center inputs; the strength of the connection depends on the magnitude of the Gabor. The number of subregions is defined as the ratio of the width of the Gaussian envelope (at 5% of peak) to the width of a half-cycle of the sinusoid. The aspect ratio of a single subfield is defined as the ratio of the Gaussian envelope length to the sinusoid half-cycle width. Two sets of Gabor parameters were used. "Default" parameters were the mean values for simple cell physiological RFs reported in Jones and Palmer (1987)
The Journal of Neuroscience, August 1, 1998, 18(15):5908-5927
Contrast-Invariant Orientation Tuning in Cat Visual Cortex: Thalamocortical Input Tuning and Correlation-Based Intracortical Connectivity
Todd W. Troyer2, 6, Anton E. Krukowski5, 6, Nicholas J. Priebe4, 6, and Kenneth D. Miller1, 3, 4, 5, 6, 7 Departments of 1 Physiology, 2 Psychiatry, and 3 Otolaryngology, 4 Neuroscience and 5 Biophysics Graduate Programs, 6 W. M. Keck Center for Integrative Neuroscience, 7 Sloan Center for Theoretical Neurobiology at UCSF, University of California, San Francisco, California 94143-0444
ABSTRACT
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Abstract
Introduction
The origin of orientation selectivity in visual cortical responses is a central problem for understanding cerebral cortical circuitry. In cats, many experiments suggest that orientation selectivity arises from the arrangement of lateral geniculate nucleus (LGN) afferents to layer 4 simple cells. However, this explanation is not sufficient to account for the contrast invariance of orientation tuning.
To understand contrast invariance, we first characterize the input to cat simple cells generated by the oriented arrangement of LGN afferents. We demonstrate that it has two components: a spatial-phase-specific component (i.e., one that depends on receptive field spatial phase), which is tuned for orientation, and a phase-nonspecific component, which is untuned. Both components grow with contrast.
Second, we show that a correlation-based intracortical circuit, in which connectivity between cell pairs is determined by the correlation of their LGN inputs, is sufficient to achieve well tuned, contrast-invariant orientation tuning. This circuit generates both spatially opponent, "antiphase" inhibition ("push-pull"), and spatially matched, "same-phase" excitation. The inhibition, if sufficiently strong, suppresses the untuned input component and sharpens responses to the tuned component at all contrasts. The excitation amplifies tuned responses. This circuit agrees with experimental evidence showing spatial opponency between, and similar orientation tuning of, the excitatory and inhibitory inputs received by a simple cell. Orientation tuning is primarily input driven, accounting for the observed invariance of tuning width after removal of intracortical synaptic input, as well as for the dependence of orientation tuning on stimulus spatial frequency.
The model differs from previous push-pull models in requiring dominant rather than balanced inhibition and in predicting that a population of layer 4 inhibitory neurons should respond in a contrast-dependent manner to stimuli of all orientations, although their tuning width may be similar to that of excitatory neurons. The model demonstrates that fundamental response properties of cortical layer 4 can be explained by circuitry expected to develop under correlation-based rules of synaptic plasticity, and shows how such circuitry allows the cortex to distinguish stimulus intensity from stimulus form.
Key words: visual cortex; LGN; contrast invariance; cerebral cortical circuitry; orientation selectivity; model; simple cell; layer 4; V1; push-pull; opponent inhibition; spatial phase
INTRODUCTION
Thirty-five years ago, Hubel and Wiesel (1962)
discovered that cells in cat primary visual cortex (V1) are tuned for the orientation of light/dark borders. The inputs to V1 come from the lateral geniculate nucleus (LGN), whose cells are not significantly orientation selective (Hubel and Wiesel, 1961
In cats, orientation selective responses appear in cortical layer 4. Cat layer 4 is composed of simple cells (Hubel and Wiesel, 1962
Two major approaches to achieving contrast invariance have been proposed. Many authors have suggested that responses in simple cells are approximately linear, i.e., the response can be predicted by linear summation of stimulus luminance (relative to background), weighted by the cell's RF (Movshon et al., 1978
Other authors have proposed that orientation tuning emerges from orientation-specific short-range excitation and longer-range inhibition in cortex (Ben-Yishai et al., 1995
We propose a new model for cat layer 4 cortical circuitry that yields contrast-invariant orientation tuning. Our model examines two basic questions. First, what is the nature of the thalamocortical input to cortical simple cells? We assume that thalamocortical connectivity can be modeled by a Gabor function: a two-dimensional Gaussian multiplied by a sinusoid (Jones et al., 1987
Second, what patterns of intracortical connectivity are sufficient to yield contrast-invariant orientation tuning? We arrive at a surprisingly simple answer: "correlation-based" connectivity yields contrast invariance. By correlation-based connectivity we mean that intracortical connection strengths between two cells are fixed on the basis of the correlation in their thalamocortical RFs. Thus, inhibitory connections occur between cells with anticorrelated RFs, whereas excitatory connections occur between cells with correlated RFs. The "antiphase" inhibition eliminates the untuned input component and sharpens responses to the tuned component, whereas "same-phase" intracortical excitation amplifies the tuned response. As a result, our model achieves contrast-invariant tuning in the presence of positive thresholds and LGN rectification.
Our model uses a form of push-pull circuitry but differs from other such models in that inhibition dominates rather than balances excitation, and responses are not linear. Furthermore, we predict that a population of inhibitory neurons in cat layer 4 should respond in a contrast-dependent manner to stimuli of all orientations, although they may be tuned for orientation. The model has both developmental and functional implications for understanding the layer 4 cortical circuit, and suggests a general means of separating stimulus intensity (here represented by contrast) from stimulus form (represented by orientation).
This work has been published previously in abstract form (Krukowski et al., 1996
MATERIALS AND METHODS
We study both a very simple ("conceptual") model and a more realistic ("computational") model. We first present the elements common to both, and then present each model.
Elements common to both conceptual and computational models
LGN model. Our model was based on cat V1 at ~5° eccentricity. LGN spatial RFs were center-surround difference of Gaussians, with cells responding either to light onset (ON cells) or light offset (OFF cells) in their RF centers. LGN spatial filter parameters [(17/center2)e
x2/
2center
(16/
2surround;
Conceptual model
To explore the basic concepts underlying our results, we constructed a conceptual model designed to be as simple as possible. The model contains two "rate-coded" cortical neurons, one excitatory and one inhibitory; the inhibitory cell inhibits the excitatory cell. The activity of each cell is represented by a scalar value corresponding to average firing rate. The LGN was modeled as a uniform sheet of cells, approximated as a dense lattice (lattice spacing = 0.05°). The two cortical RFs were determined by Gabor RFs with identical Gaussian shape and location but having sinusoids of opposite spatial phase (thus, the inhibitory cell provides antiphase inhibition). For computational convenience in obtaining orientation tuning curves, rather than showing many gratings to one pair of cells, we showed one grating to many independent cell pairs. Thus, we constructed multiple pairs of cortical RFs with identical retinotopic positions and with orientation and spatial phases spaced at 10 and 20° intervals, respectively. For each time step, we first calculated the LGN input to each RF by summing LGN firing rates, weighted by the Gabor function, to give the excitatory input A(,
) to the cell of orientation
]+. For each set of Gabor parameters, the threshold
Computational model
Most simulations were carried out in a computational model incorporating details of cortical cells and maps. Computational LGN model. For the computational model, a realistically dense lattice of LGN cells was used. We restricted our attention to LGN X-cells, which dominate central cat V1 physiology (Ferster, 1990exff/npickff)
nffpicki=1 pi where npickff = 3,
, the RF becomes a perfect Gabor function. A typical sampled RF is shown in Figure 2B. With this sampling, cortical cells received input from 125 ± 8 (mean ± SD) LGN cells using the default Gabor. Using the more broadly tuned Gabor, cortical cells received input from 61 ± 5 LGN cells. Cortical model. Cortical cells were modeled as simple integrate-and-fire neurons as described in Troyer and Miller (1997a
fall
tdelay
exrise = 0.25 msec,
×
LGN g(i, a)g(j, b)c(i, j). Here, i, j are LGN cells, g(i, a) and g(j, b) are the thalamocortical weights from i to a and j to b, and c(i, j) is the cross-correlation of the spatial RFs of i and j, where OFF spatial RFs are negative of ON. Correlation is then c(a,b)=c'(a,b)/
. A connectivity function C(a, b)
roughly, the probability of a connection from a to b
RESULTS
Modeling approach
We pursued two parallel approaches to modeling contrast-invariant orientation tuning. To explore the basic ideas underlying such tuning, we constructed a conceptual model, designed to be as simple as possible. This model considered two cortical simple cells, one excitatory and one inhibitory, with a monosynaptic connection from inhibitory to excitatory. The RFs of the two cells had identical position and preferred orientation but opposite spatial phase (see Materials and Methods). The neurons were "rate-coded": the average firing rate of each cell was determined by a linear thresholding operation applied to the weighted sum of input cell firing rates. For simplicity, the inhibitory threshold was set to zero (i.e., the inhibitory cell's response was a linear function of its input). The excitatory cell's threshold was set automatically to the level that best produced contrast-invariant tuning for contrasts of 5% and above (see Materials and Methods). Therefore, after the structure of the cortical receptive fields was determined, the conceptual model had only a single free parameter: the strength of intracortical inhibition relative to the strength of thalamocortical excitation.
To study the robustness of our ideas to the complexity of real cortical circuits, we also constructed a computational model that incorporated known details of cortical cells and maps. The cortical component of this model consisted of 1600 excitatory and 400 inhibitory layer 4 simple cells, arranged in a
LGN input
We focused our research on the response to full-field sinusoidal gratings, because these are the only stimuli for which contrast dependence of orientation tuning has been studied (Sclar and Freeman, 1982
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Figure 1. LGN cell responses to 3 Hz, 0.8 cycles/degree moving gratings. A, Instantaneous firing rate. Straight line is background. B, Contrast response functions. Top shows amplitude of first harmonic (F1); bottom shows mean (DC) firing rate. The mean rate increases at contrasts >5%, attributable to rectification as seen in A. Data modified from Cheng et al. (1995)
LGN cells responded to sinusoidal grating stimuli with a sinusoidal modulation in firing rate (Fig. 1A). The temporal responses of ON-center and OFF-center cells with spatially overlapping RFs were 180° out of phase. Increasing the stimulus contrast resulted in a larger modulation of firing rate. At contrasts above ~5%, the spike rate modulation exceeds the background firing rate. For these contrasts, responses are no longer purely sinusoidal, because spike rate cannot be negative (Fig. 1A, solid lines); that is, LGN responses rectify. Once responses rectify, mean (DC) firing rates increase with increasing contrast (Fig. 1B, DC curves), because peak firing rates continue to increase and minimal firing rates cannot decrease below zero. This contrast-dependent increase in mean LGN firing rates has important consequences for contrast-invariant orientation tuning that will be discussed in detail below.
The oriented arrangement of LGN inputs to simple cell RF subregions was modeled using a Gabor function, a two-dimensional Gaussian multiplied by a sinusoid (Fig. 2A). In the conceptual model, the Gabor function directly determined the weights of geniculocortical connections: positive values corresponded to the weights of ON-center inputs, negative values to the weights of OFF-center inputs. In the computational model, geniculocortical synaptic strengths were determined by probabilistic sampling of the Gabor function from a realistically dense lattice of LGN cells (Fig. 2B).
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Figure 2. Gabor-shaped cortical RFs. Lighter grays to white indicate positive values of Gabor function, corresponding to weights of ON-center LGN cells with centers at corresponding spatial positions; darker grays to black indicate negative values of Gabor function, corresponding to weights of OFF-center cells. A, A full Gabor function, used to determine LGN inputs to a cortical cell in the conceptual model. B, Typical LGN inputs to a cortical cell in the computational model, after probabilistic sampling from the full Gabor (see Materials and Methods). These receptive fields are typical; different cortical cells may have different preferred orientations, spatial phase (relative locations of ON or OFF subregions), spatial location, and, in the computational model, different outcomes of the probabilistic sampling. Spatial frequency of sinusoid in Gabor function is 0.8 cycles/degree.
We considered two different sets of Gabor parameters to describe geniculocortical connections. The first set was matched to RF parameters taken from physiological measurements of cat simple cells (Jones and Palmer, 1987
In the conceptual model, the excitatory and inhibitory cells had identical Gabor RFs, except that their sinusoids were 180° out of phase. In the computational model, a distribution of receptive fields was obtained from variations in three parameters: preferred orientation, determined by a measured cortical map (see Fig. 8A); retinotopic position, progressing uniformly across the sheet; and spatial phase, assigned randomly to each cell (DeAngelis et al., 1992
Tuning of the LGN input to a simple cell
At the preferred orientation, the bright and dark portions of a sinusoidal grating stimulus align with the cortical cell's ON and OFF subregions simultaneously. Thus, all of the cortical cell's LGN inputs fire relatively synchronously and the temporal modulation of this input is large (Fig. 3A). At the null orientation, the inputs are stimulated asynchronously, so the temporal modulation of the total input is small. Note that the mean rate of LGN input does not depend on stimulus orientation. This follows from the assumption that LGN cells are untuned for orientation: because the mean LGN input received by a simple cell is the (weighted) sum of the mean rates of the LGN cells projecting to it, this mean input must also be untuned for orientation (Ferster, 1987
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Figure 3. Tuning of total LGN input. A, Input to cortical cells in response to high (50%) and low (2.5%) contrast gratings at the preferred and null (orthogonal to preferred) orientations. High (50%) contrast and low (2.5%) contrast are shown. Curved traces show input in response to preferred orientation; black traces, average input (40 presentations) from computational model, using a sampled Gabor RF (as in B); gray curves, input for conceptual model, using connections from the full Gabor function (A). Gray straight lines show response in the conceptual model to a stimulus at the null orientation; in inset, these lines are repeated and compared to average input to null stimuli in computational model (black traces). Note that input to null stimulus at 50% contrast typically exceeds peak input to preferred stimulus at 2.5% contrast. Agreement of the two models for both preferred and null stimuli indicates that RF sampling and Poisson firing of LGN inputs have little effect. B, Tuning of mean (dashed lines) and mean plus first harmonic (solid lines) of thalamic input conductance. Lines show results from the conceptual model; solid circles show results from the computational model; error bars represent ±1 SD. Sum of mean plus first harmonic represents peak input during a cycle of the grating stimulus. Note that mean input is untuned for orientation, and mean input at high contrasts exceeds peak input to preferred orientation at low contrasts. Thus, although the first harmonic is well tuned, no single spike threshold can give tuned responses at both high and low contrasts. In this and subsequent figures showing orientation tunings, cells are grouped by preferred orientation in 10° bins, and orientation axis represents difference of stimulus orientation from preferred.
The untuned mean input presents the primary problem for a purely thalamocortical explanation of contrast-invariant orientation tuning. As a result of LGN rectification, mean LGN firing rates increase with increasing contrast (Fig. 1). This contrast-dependent increase in firing rate is sufficiently large that the mean LGN input at the null orientation at high contrasts exceeds the peak LGN input at the preferred orientation at low contrast (Fig. 3). No single-spiking threshold level can yield well tuned responses for stimuli of all contrasts.
Therefore, to achieve contrast-invariant orientation tuning in response to sinusoidal gratings, the cortex must cancel the untuned, mean input component in a contrast-dependent manner, while it extracts the tuned, modulated component. We will show that this decomposition of the input into a tuned and an untuned component generalizes to stimuli such as flashed and moving bars.
Antiphase inhibition can achieve contrast-invariant orientation tuning
The main purpose of this paper is to demonstrate that correlation-based intracortical inhibition can achieve contrast-invariant orientation tuning (the effects of correlation-based intracortical excitation will also be considered below). By correlation-based inhibition, we mean that the probability of a connection from an inhibitory cell to an excitatory cell is an increasing function of the degree of anticorrelation between their RFs, i.e., the strongest inhibitory connections are made between cells with the most anticorrelated RFs (see Materials and Methods). This implies that an excitatory cell receives the strongest inhibition from inhibitory cells with identical Hubel-Wiesel RFs but of opposite spatial phase. We will call such an inhibitory neuron the cell's "antiphase partner." (By "spatial phase" of an RF, we refer to absolute position in visual space of the ON or OFF subregions, rather than to their position relative to each cell's Gabor function; thus, two RFs have "opposite spatial phase" if the ON subregions of one tend to overlap the OFF subregions of the other in visual space.) The existence of such "spatially opponent" or antiphase inhibition in cat layer 4 is well supported experimentally: at ON locations, where a light stimulus evokes excitation (EPSPs), dark stimuli evoke inhibition (IPSPs), and vice versa for OFF locations (Palmer and Davis, 1981
Our model is not a developmental model: we first determined the pattern of LGN input to cortical cells and then fixed the pattern of intracortical connections according to the above correlation-based rule. However, this pattern of inhibition would be expected to arise from a Hebb-type synaptic modification rule, generalized to apply to inhibitory synapses. Such a rule states that synaptic strengths grow more negative (more strongly inhibitory) when presynaptic and postsynaptic firings are anticorrelated, or equivalently, that synapses strengthen when they are effective, i.e., when the inhibitory presynaptic cell is active and the postsynaptic cell is inactive. Such generalization of Hebb-type learning rules to inhibitory synapses is only a hypothesis; plasticity of inhibitory synapses is not well understood [but see Komatsu (1996)
The sufficiency of correlation-based inhibition for contrast-invariant tuning is demonstrated in Fig. 4, which shows tuning curves for both the computational and conceptual models, for gratings of 2.5, 5, 10, 25, and 50% contrast. Both models display contrast-invariant orientation tuning above 5% contrast. By choosing the appropriate level of inhibition, both models were able to match experimental estimates of mean orientation tuning width for simple cells. For example, Heggelund and Albus (1978)
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Figure 4. Contrast-invariant tuning. Response versus orientation for gratings of 2.5, 5, 10, 25, and 50% contrast. A, Computational model. B, Conceptual model. Both models yield contrast-invariant tuning at 5% contrast and above.
The width of the tuning is largely determined by the strength of the inhibition and the tuning of the LGN input. Fig. 5 shows cortical tuning half-widths, using either narrowly or broadly tuned LGN inputs, for various levels of inhibition. Tuning narrows with stronger inhibition but remains contrast-invariant above 5% contrast. Tuning to a long moving bar (width 0.62°, velocity 3.75°/sec) is slightly broader but shows identical sharpening with increasing levels of inhibition: filled circles show bar tuning at 50% contrast for the narrowly tuned LGN inputs.
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Figure 5. Increasing inhibition leads to sharper tuning. Tuning half-width at half-height (HWHH) versus level of inhibition for gratings of 2.5, 5, 10, 25, and 50% contrast. Thick solid (bottom) curve shows mean tuning HWHH above 5% for RFs with large subfield aspect ratios and narrow LGN tuning (matched to data from Jones and Palmer, 1987
For higher levels of inhibition and broadly tuned input, tuning at 5% contrast narrows slightly in the conceptual model. This is attributable to the fact that spike threshold was optimized for the default level of inhibition (see Materials and Methods) and could be corrected if spike thresholds were separately optimized for each set of parameters. At 2.5% contrast and high levels of inhibition (Fig. 5B, thin lines), the conceptual model predicts much narrower tuning, for reasons that are more general (see below).
The conceptual and computational models yield qualitatively similar results. Simple additions to the conceptual model led to progressively closer quantitative matches to computational model behavior. A significantly improved match was obtained by adding inhibitory thresholds and using correlation-based inhibitory connectivity from cells with a range of RF properties (rather than from only the single cell with precisely opposite spatial phase). Using simulated synaptic noise (and hence changing the threshold linear function to a smoother function near spike threshold) led to an even closer match between the models and nearly eliminated the difference in responses to 2.5% contrast gratings (see below). However, incorporation of these features required additional unconstrained parameters, and we began to lose the simplicity that was the strength of the conceptual model. Therefore, the results of these investigations are not further reported.
The behavior of our correlation-based model is presented below, in three steps. First, we analyze the reasons why antiphase inhibition achieves contrast-invariant tuning, using the simple conceptual version of the model. Second, we incorporate correlation-based intracortical excitation into the computational model and present results from this completed model. Finally, we explore the robustness of this computational model to variations in the key parameters controlling model behavior. A schematic representing the behavior of both models is shown in Fig. 6, and will be referred to throughout the text.
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Figure 6. Behavior of model using correlation-based connectivity. Schematic representing behavior of the model in response to preferred (A) and null (B) stimuli. The excitatory cell described in Results is in the top left; its inhibitory antiphase partner is in the bottom right. E, Excitatory cells; I, inhibitory cells. Solid lines represent excitation and depolarization; open lines represent inhibition and hyperpolarization. Line thickness and size of RF icon represent magnitude of activity. Dashed lines represent correlation-based excitation, which is included in the complete computational model only (see Figs. 8-11). Some simulations were performed without cortical excitatory projections onto inhibitory neurons (gray dashed lines), but this did not substantially affect network behavior (see Fig. 13B).
Conceptual model: antiphase inhibition cancels the untuned component of the input
Recall that the main obstacle to achieving contrast-invariant tuning is the untuned component of the LGN input, which increases with contrast as a result of the rectification of LGN responses at higher contrasts. The ability of antiphase inhibition to overcome this problem is most easily demonstrated in the context of the two-cell conceptual model. Here we introduce the term feedforward, by which we mean input from LGN to cortical cells not mediated by cortical excitatory cells. Thus, the geniculocortical input represents feedforward excitation, whereas the pathway from LGN to cortical inhibitory cell to cortical excitatory cell represents feedforward inhibition.
Suppose an excitatory simple cell receives total input Ae from the LGN, and its inhibitory antiphase partner receives LGN input Ai. Assuming for simplicity that inhibitory cell response is linear, the total feedforward input to the excitatory cell is Ae
Ai, and the inhibition wAi > Ae is sufficient to prevent excitatory cell spiking (Figs. 6B, 7A, bottom). Because Ai and Ae both rise with contrast at the same rate, the dominance of inhibition over excitation is maintained for null stimuli of all contrasts.
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Figure 7. Inputs to a cortical cell given antiphase inhibition (inputs shown relative to background). A, Averaged computational model responses (40 presentations) to 50% contrast gratings. Excitatory LGN input is marked Ex.; intracortical inhibitory input is marked Inh. To compare excitatory and inhibitory inputs, synaptic conductances were converted to currents obtained if the cell was voltage-clamped at threshold. B, Peak synaptic current versus orientation for computational model. Responses are to single presentations of 50, 10, and 5% contrast gratings at 128°. Peak current is the first harmonic (F1) plus the mean (DC) of the stimulus-induced current (including excitation and inhibition). Error bars for 50% contrast are ±1 SD. Dotted line shows approximate threshold level that would lead to contrast-invariant tuning; actual threshold in computational model is determined independently from in vitro data (see Materials and Methods). C, Peak synaptic current versus orientation for conceptual model. Because there is no noise, true peak current is shown. Dotted line shows automatically selected threshold (see Materials and Methods). For both models, mean input decreases and modulation increases with contrast. Thresholds near the crossover point of net input tuning curves result in sharp, contrast-invariant tuning.
The ability of antiphase inhibition to achieve contrast-invariant tuning for a wide variety of stimuli can be best understood by dividing the LGN input into two components: the phase-nonspecific component Anon = (Ae + Ai)/2
This can be summarized by noting that the schematic circuit (Fig. 6) acts as a "differential phase filter": with inhibition sufficiently large, any stimulus that gives similar excitation to each of two opposite phases will cause more inhibition than excitation in excitatory cells and hence will be "filtered out." Only stimuli that predominantly excite one phase and not its opposite can "pass" through this "filter" and cause the excitatory cells to fire. The only stimuli that can accomplish this are stimuli near the preferred orientation; stimuli far from the preferred will give similar input to both phases. This argument applies to any type of oriented stimulus.
Conceptual model: dominant antiphase inhibition provides a contrast-dependent effective threshold
Although the most important effect of antiphase inhibition is to eliminate the phase-nonspecific component of the LGN input, this is not sufficient to achieve contrast-invariant tuning. This can be seen by setting w = 1, thereby causing (1
This problem is remedied by including relatively strong inhibition, (w > 1). Then the phase-nonspecific component (1
In the conceptual model, spike threshold for excitatory cells was automatically set at this crossover point in the input current (see Materials and Methods). Somewhat surprisingly, we have found that in the computational model, simply using a physiologically based spiking neuron model (Troyer and Miller, 1997a
At very low contrasts, the conceptual model predicts that orientation tuning will narrow. Below ~5% contrast, LGN responses do not rectify and therefore the plateau, (1
Computational model: adding correlation-based excitation
Up to this point, we have not considered the effect of intracortical excitation. We have seen that correlation-based inhibition is sufficient to achieve sharp, contrast-invariant tuning. Here we show that the addition of correlation-based excitation "amplifies" these contrast-invariant responses, without altering their tuning. The conceptual model, which contains only two cortical neurons, is too simple to explore the effects of intracortical excitation in any meaningful way. Hence, the remainder of this paper will present results from the computational model only.
Intracortical excitation was incorporated using a correlation-based rule analogous to that used for intracortical inhibition: excitatory connections were determined probabilistically, such that the strongest connections are found between cells whose RFs are most strongly correlated, i.e., those with similar preferred orientation and similar spatial phase. This is illustrated schematically by the dashed lines in Fig. 6. That intracortical excitation comes primarily from cells of similar orientation preference and similar spatial phase is supported by the fact that EPSPs are evoked only by stimuli of appropriate position and phase, with opposite phase to the stimuli that evoke IPSPs (Ferster, 1988
Because the dependence on correlation of intracortical inhibition and excitation differs only in sign, excitatory and inhibitory connections in our model have precisely the same average distribution in terms of orientation preference; they differ only in spatial phase. An example is shown in Fig. 8A, which illustrates the experimental V1 orientation map used to assign preferred orientations to cortical cells in the computational model. In this figure, white squares show the locations of cells making excitatory connections to the excitatory cell at the X, whereas black squares show the locations of cells making inhibitory connections. Excitatory and inhibitory connections to this cell have similar distributions as a function of orientation. Fig. 8B shows the theoretical average distribution of connections for retinotopically identical RFs, as a function of orientation difference (top) and spatial phase difference (bottom). The tightness of tuning as a function of correlation is determined by the parameter npow (see Materials and Methods). Large values of npow lead to tighter connectivity as a function of correlation, whereas smaller values of npow lead to broader connectivity. Increasing and decreasing npow had only minor effects on the behavior of the model.
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Figure 8. Behavior of the full computational model. A, Orientation map used, and typical pattern of intracortical connections. There is one excitatory cell in every location of the cortical map (40 × 40 lattice) and one inhibitory cell in every fourth location (20 × 20 lattice). Cells were assigned preferred orientation according to illustrated 40 × 40 color map (red-green-blue-red representing 0-60-120-180°) (map is
For the full computational model, we reduced the synaptic strengths of the LGN input by a factor of 2, relative to the previous simulations without intracortical excitation. We thus relied on the positive feedback from same-phase intracortical excitation to amplify the response to suprathreshold stimuli. We also increased the default level of inhibition (i
Intracortical amplification of an effective stimulus by feedback excitation has been used in many other models (Douglas et al., 1989
Computational model: tuning
For the full circuit, typical currents and voltages from excitatory cells at various orientations are shown in Fig. 9. The behavior of a cell tuned to the stimulus is shown at the top left. At the antipreferred temporal phase, large inhibitory currents hyperpolarize the cell to near the chloride reversal potential. As the preferred phase is approached, the inhibitory currents drop and the excitatory currents rise, depolarizing the membrane to spike threshold. Spikes in turn evoke adaptation currents, which help to shut off the cell's response. Adaptation currents in other cells across the network lead to a lowering of the intracortically evoked excitatory current. For a cell tuned to stimuli perpendicular to that presented (Fig. 9, top right), inhibition dominates at all phases and the cell is prevented from firing.
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Figure 9. Example traces. Voltage (Vm) and conductance in response to a 3 Hz sinusoidal grating at 50% contrast. AMPA, GABA-A, synaptic conductances with background subtracted (converted to currents at threshold as in Fig. 7A). AHP, Spike-triggered potassium conductance. Bars show stimulus orientation relative to preferred (vertical). Top row shows orientation differences spaced at 30° intervals; bottom row shows model behavior at orientations between 0 and 30°. Excitation and inhibition arrive out of phase and have similar orientation tuning. Inhibition dominates at the null.
The full model, including same-phase excitation, achieves contrast invariant tuning to sinusoidal gratings (Fig. 8C, colored lines). Orientation tuning half-widths at half-height of the mean firing rates of excitatory cells were between 19 and 21° for contrasts ranging from 2.5 to 50%. The model was also well tuned to a moving oriented bar; tuning width for the default parameters was 27.5°, slightly broader than for gratings.
Our model also achieves nearly identical orientation tuning of the excitation and inhibition received by a simple cell, as observed in Ferster (1986)
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Figure 10. Orientation tuning of peak excitatory current (dot-dashed line) and inhibitory current (solid line) at 50% contrast (peak current equals DC+F1; see Fig. 7B). Dotted line shows excitatory current scaled and translated to match maxima and minima of inhibitory currents. Peak inhibition is larger than excitation at all orientations, but the tuned components of excitation and inhibition have nearly identical shape.
Orientation tuning is driven by LGN input
The tuning in our model is driven by the LGN inputs, despite the strong role of intracortical inhibition. In agreement with the cortical cooling experiments of Ferster et al. (1996)
A more rigorous test of the hypothesis that cortical tuning is driven by the LGN can be obtained by systematically varying the tuning of the LGN input. This can be accomplished by varying the spatial frequency of a sinusoidal grating (Fig. 11). As spatial frequency is increased, the orientation tuning of the F1 of the LGN inputs becomes narrower (Fig. 11A, middle). The F1 of the full circuit closely follows the LGN F1 across spatial frequencies (Fig. 11A, top), yielding a narrowing of orientation tuning of the spiking response with increasing spatial frequency (Fig. 11B). Note that there is an optimal spatial frequency, at which spike response to stimuli of the preferred orientation is maximal (Fig. 11B, inset); thus, as spatial frequency increases, the orientation tuning width monotonically narrows, while spike response to the preferred orientation rises to a maximum and falls again. This is precisely the behavior observed in cat cortical cells (Vidyasagar and Sigüenza, 1985
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Figure 11. Orientation tuning narrows with increasing spatial frequency of a sinusoidal grating (3 Hz). A, Tuning of the F1 voltage modulation for the full circuit (top) and with LGN excitation only (middle). Tuning curves, from widest to narrowest, represent response to spatial frequencies 0.4 (dot-dashed), 0.56 (thin), 0.8 (thick), and 1.13 (dashed) cycles/degree, respectively; each curve is normalized to its peak response. Thicker line is the spatial frequency used for simulations in other figures. The bottom shows the difference between the normalized tuning curves: LGN input F1 and full circuit F1 closely match. B, Half-width at half height versus spatial frequency for 5, 10, 25, and 50% contrast gratings. Orientation tuning remains contrast invariant over a broad range of spatial frequencies. Inset, Spatial frequency tuning curve at the preferred orientation and 50% contrast.
Inhibitory sharpening of orientation tuning is attributable exclusively to disynaptic inhibition driven by the LGN input, i.e., sharpening depends on feedforward inhibition. Feedback inhibition [resulting from cortical cell excitation of inhibitory cells (Fig. 6, gray dashed lines)] has little effect on model behavior. In simulations run after setting these connections to zero, tuning curve half-widths at the default parameters remain unchanged to within 1° for 10% contrast and above (4° change at 5% contrast), and peak spiking responses increased by <6.4% (see Fig. 13B). These small changes in output occur even though the mean inhibitory conductance in response to a 50% contrast grating at the preferred orientation is reduced by 38%.
This result can be understood by examining Figure 6. With same-phase excitation and antiphase inhibition, firing of excitatory cells of one phase increases the inhibition onto excitatory cells of the opposite phase, but excitatory cells with RFs of opposite spatial phase spike during opposite temporal phase of the stimulus. Therefore, feedback inhibition is directed onto cells that are already not spiking and has little effect on model behavior.
Inhibitory dominance and inhibitory blockade
The model operates in an inhibition-dominated regime, as revealed by various measures. For the default parameter settings used to measure tuning (Fig. 8), the total synaptic strength received by a cell (see Materials and Methods for definition) were 5, 4.25, and 7.5 nA msec for the LGN inputs, cortical excitation, and cortical inhibition, respectively. A null-oriented stimulus induces an inhibitory current that is ~3.9 times as large as the excitatory current for a cell clamped at threshold voltage, corresponding to an inhibitory conductance change that is 11.7 times greater than the change in excitatory conductance. Strong inhibition is consistent with the experimental result that nonspecific stimulation of LGN afferents (or cortical white matter) yields a short EPSP followed by a large IPSP (Ferster and Jagadeesh, 1992
Our model agrees with the experiments of Nelson et al. (1994)
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Figure 12. Orientation tuning after blocking inhibition in a single cell [as in Nelson et al. (1994)
The effects of global blockade of inhibition are discussed below (Fig. 13A).
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Figure 13. Robustness of model behavior to changes in various model parameters. Each subplot represents the effect of varying the strength of two variables out of the following four: the three types of synaptic connections (LGN input, intracortical excitation, and inhibition) and spike-rate adaptation in excitatory cells. Mean spike tuning half-width at half-height (HWHH) for 5, 10, 25, and 50% contrast gratings is represented by oval width. Oval height represents 30°. Mean spike rates (Hz) for preferred stimuli at 50% contrast are printed inside each oval. Darker ovals indicate a loss of contrast invariance, monitored as SD divided by the mean of the HWHH over the four contrasts sampled. Points with extremely broad tuning are contrast-invariant only because all contrasts give maximal HWHH. Lines show experimentally reported values for mean spike rate (bold line equals 20 Hz) (Albrecht, 1995
Robustness of model results
The effect of parameter variations on computational model behavior is shown in Figure 13. Each of the subplots represents the effect of varying the strength of two variables out of the following four: the three types of synaptic connections in the model [thalamocortical excitation (LGN), intracortical excitation (e
Simulations were run at four contrasts (5, 10, 25, and 50%) for each point in the 5 × 5 parameter grid. We monitored contrast-invariance of orientation tuning, average tuning width (HWHH), and firing rates at the preferred orientation and 50% contrast. Contrast-invariance is represented by color within the ovals; darker ovals represent loss of contrast invariance. Spike tuning HWHH is denoted by oval width, relative to oval height, which represents 30°. The light gray shaded regions indicate HWHH tuning of <22°. Spike rate to a 50% contrast preferred stimulus is printed inside each oval. The lines show the relation of model behavior to various experimental estimates and are discussed below.
The model robustly achieves sharp, contrast-invariant tuning. Thin white ovals within the gray shaded regions indicate that in the large regions of parameter space in which tuning is sharp, it also remains contrast invariant. Increasing inhibition leads to sharper tuning (narrower ovals) and has only moderate effects on spike rate, except with very strong levels of intracortical excitation.
This qualitative behavior is quite robust to changes in other model parameters. Increasing and decreasing the tightness of the correlation-based connectivity by changing the parameter npow (Fig. 8B) by a factor of 2 had only moderate effects on model behavior. When the more broadly tuned geniculocortical connectivity matched to the data of Ferster et al. (1996)
The roles of feedforward and feedback inhibition
High levels of intracortical excitation (Fig. 13A,B,D, top portions) lead to unstable feedback excitation, indicated by sharply increased spike rates, broadening of orientation tuning, and a loss of contrast invariance. Strong excitation is difficult to control, even for high levels of inhibition, because the model lacks the center-surround ("Mexican hat") intracortical connectivity commonly used to stabilize feedback excitation (Ben-Yishai et al., 1995Comparisons with experimental estimates
In addition to the width and invariance of orientation tuning, we monitored the conductance change to a null stimulus, the magnitude of the full circuit voltage F1, and the mean spike rate for a 50% contrast preferred stimulus. The lines in Figure 13 show the contours through parameter space corresponding to experimental estimates of these variables. First, some results (Douglas et al., 1988Varying neuronal gain
One obvious way out of this dilemma is that cortical cells may respond more strongly to weak thalamocortical input and moderate amplification than do our model neurons. We used extremely simple model neurons, including a simple, voltage-independent mechanism for spike rate adaptation. These were matched to experimental data on firing rate versus constant somatic current level in vitro (McCormick et al., 1985
DISCUSSION
Principal findings and predictions
We have constructed a simple model that accounts for contrast-invariant orientation tuning in layer 4 of cat visual cortex. First, we analyzed the LGN input to simple cells in response to drifting sinusoidal gratings. This input has two components: a temporally modulated, phase-specific component that is tuned for orientation, and an unmodulated, phase-nonspecific component that, assuming LGN cells are not orientation-tuned, must be completely untuned. Because of LGN rectification, this untuned input increases significantly with increasing contrast, implying that no simple threshold mechanism could correct for it across contrasts. This problem is general: the phase-nonspecific LGN input component for any stimulus is poorly tuned and increases with contrast. To counteract this increase in untuned feedforward excitation, stimulation of an excitatory cell at its null orientation must evoke increasing intracortical inhibition and/or increasing withdrawal of intracortical excitation with increasing contrast. Thus, the central questions for understanding contrast-invariant orientation tuning in cat layer 4 are how the level of this contrast-dependent "pull" is computed and which cells are its source.
Second, we have demonstrated that cortical circuitry relying on correlation-based connectivity is sufficient to robustly yield contrast-invariant orientation tuning. In our model, the contrast-dependent inhibition to layer 4 simple cells comes from inhibitory cells with similar orientation tuning and roughly opposite spatial phase (Fig. 8B). The tuned input component is amplified by excitatory connections between cells with similar orientation tuning and similar spatial phase. Although some details might vary, our key predictions are that the layer 4 circuitry is strongly phase-specific, inhibition-dominated, and localized in orientation, and has orientation tuning that depends on the LGN input tuning.
Beyond the circuit itself, our most striking prediction is the existence of inhibitory simple cells that are tuned for orientation but have a contrast-dependent response to null-oriented stimuli. We also predict that (1) the DC LGN input should increase with increasing contrast at all orientations, whereas (2) the dominance of inhibition should cause the net DC feedforward input (LGN plus feedforward inhibition) to decrease with increasing contrast at all orientations. These predictions are most easily tested with null-oriented stimuli, for which feedback excitation should be negligible. Prediction (1) could be tested by blocking inhibition intracellularly in a single layer 4 cell. Note that for (2) the predicted decrease is in net current at threshold voltage; this need not imply voltage decrease from rest, because of differences between excitatory and inhibitory reversal potentials. In addition, we predict that intracellular, or sufficiently weak extracellular, GABAergic blockade in layer 4 should reveal a tuned response component sitting on an untuned plateau response (Fig. 12).
Possible sources of contrast-dependent inhibition
Besides antiphase inhibition, what are other possible sources of the contrast-dependent pull required to achieve contrast-invariant tuning? Because the LGN input is exclusively excitatory (Ferster and Lindstrom, 1983
Another source of pull could be withdrawal of excitation from other cortical cells. Because layer 4 simple cells have very low spontaneous rates, significant withdrawal would most likely come from complex cells, which do show contrast-dependent inhibition to a null stimulus (Sclar and Freeman, 1982
One study has suggested that inhibition onto layer 4 simple cells with multiple ON/OFF subregions may come primarily from layer 4 simple cells with single-subregion RFs (Toyama et al., 1981
Carandini and Ferster (1997)
Our analysis shows that the untuned component of the LGN input will cause simple cells to spike in response to a high-contrast, null-oriented stimulus, unless they are inhibited. Our model relies on the simplest explanation for the source of this contrast-dependent inhibition: inhibitory neurons (or a subset of them) are not strongly inhibited and do spike, providing the inhibition necessary to prevent responses in the remaining layer 4 neurons.
Experimental results related to inhibitory cell tuning
Several studies have reported that cat layer 4 inhibitory interneurons have simple cell RF structure much like that of excitatory neurons and are orientation-tuned (Gilbert and Wiesel, 1979
Our prediction for inhibitory cell tuning is also consistent with observations in other systems. In rabbit visual cortex, putative inhibitory neurons respond to all orientations, whereas other neurons respond only to a limited range of orientations (Swadlow, 1988
Input-driven tuning
Orientation tuning in our model is driven by the LGN inputs. The model yields voltage responses that have identical F1 tuning with and without intracortical synaptic input (Figs. 8, 10) (Ferster et al., 1996
Nonetheless, cortical circuitry plays a key role in our model. Inhibitory cells have an untuned component to their response that eliminates the untuned component of the LGN input and sharpens spike tuning. Studies in the rat whisker barrel system also indicate that the layer 4 computation is local, input-driven, and dependent on broad inhibitory tuning (Brumberg et al., 1996
Other models (Ben-Yishai et al., 1995
Model robustness and experimental constraints
Our model uses correlation-based inhibition to robustly achieve sharp, contrast-invariant, input-driven orientation tuning over a wide parameter range. The ability of a simple, two-cell conceptual model to capture the essential behavior of the more realistic computational model argues strongly for the robustness of the underlying mechanisms.
We examined the conditions under which this tuning could be achieved while satisfying several other constraints suggested by various experiments: (1) small (<20%) conductance changes to a null stimulus; (2) high (~20 Hz) steady-state firing rates to a high-contrast preferred stimulus; and (3) only moderate (~2.7 times) cortical amplification of the LGN input. We have found that these conditions can be satisfied simultaneously if the neuronal gain of our model excitatory neurons is increased, for example by reducing the magnitude of the spike-trigged adaptation (AHP) current. Alternately, by increasing feedforward synaptic strengths, all but the conductance condition can be robustly met.
The experimental support for these constraints in layer 4 is not entirely clear. With the exception of the amplification level, our numerical targets are based on average values across cortical simple cells from all layers, but properties of layer 4 cells may differ [Tolhurst and Thompson (1981)
Withdrawal of tonic input could also have an important effect on stimulus-induced conductance changes. Withdrawal of excitation from complex cells, discussed above, would contribute to low conductance change in response to a null stimulus. Withdrawal of tonic antiphase inhibition might allow a large spike response to a preferred stimulus without a large change in conductance. There are some indications from experiments that this change may be small during spike responses (Carandini and Ferster, 1997
Many factors not studied here might increase the stability of feedback excitation, enabling high firing rates with small increases in null conductance (at the cost of large amplification). These factors include frequency-dependent short-term synaptic depression (Thomson and Deuchars, 1994
Nonlinear response properties with local circuitry
Many groups have suggested that (1) important features of simple cell responses can be approximated using a linear function of stimulus contrast (Movshon et al., 1978
Our model uses opponent inhibition but assumes that inhibition dominates, rather than balances, excitation. The inhibition resulting from the phase-nonspecific LGN input depends strongly on contrast but not orientation and can produce nonlinear effects similar to the normalizing inhibition of Carandini and colleagues, although the biological substrate is quite different. For example, preliminary simulations indicate that one such nonlinearity, cross-orientation inhibition (suppression of response to a stimulus of the preferred orientation by simultaneous presentation of a null-oriented stimulus), arises naturally in our model from the antiphase inhibition (A. Hoffman and our unpublished observations).
Additional nonlinearities include effects of contrast on temporal responses. Preliminary results (Priebe et al., 1997
Developmental and functional implications
The LGN input to simple cells assumed here
The antiphase inhibition that results from such development allows cortical circuitry local to a small number of iso-orientation columns to distinguish the form (orientation) from the intensity (contrast) of an oriented stimulus. Suppose that spatial phase is ignored, so that all cortical cells preferring the same orientation receive the same single component of LGN input, an instantaneous synaptic input rate. Then the two input variables of contrast and orientation are confounded. An intermediate input rate might correspond to the preferred orientation at low contrast or to a nonpreferred orientation at higher contrast. To disambiguate these, a comparison across cells of all preferred orientations is needed, to determine for each cell whether it is receiving more or less input than cells of other preferred orientations. Recent models achieve this comparison through circuitries in which intracortical inhibition is broader in orientation than excitation (Ben-Yishai et al., 1995
Our model instead takes into account spatial phase and predicts that the computation in layer 4 is local. If the cell and its antiphase partners are receiving similar input (Fig. 6B), the preferred orientation is not being seen, and the cell does not respond, regardless of stimulus contrast (intensity). If the cell is receiving input and its antiphase partners are not (Fig. 6A), the preferred orientation (form) is present, and the cell responds; the strength of its response reflects the stimulus contrast.
This provides a specific example of a possible, more general principle of cortical organization that should arise from correlation-based development: a cortical cell should respond to the difference between its preferred stimulus (call it P) and its "antipreferred" stimulus (call it
), rather than to the preferred stimulus alone. By antipreferred stimulus, we mean the stimulus that evokes an input pattern most anticorrelated with that of the preferred. A stimulus that is uncorrelated with the preferred stimulus pattern (i.e., evokes an uncorrelated input pattern), while partially stimulating the cell, should also stimulate the cell's "antipreferred partners" (cells with preferred stimulus
Correlation-based connectivity also suggests a possible developmental explanation for columnar organization (the vertical invariance of RF properties). Why don't inhibitory and excitatory neurons in a given column take on opposite preferred orientations [or ocular dominance (OD)]? We hypothesize that interconnected excitatory and inhibitory neurons share RF properties that are shared by both P and
FOOTNOTES Received Jan. 30, 1998; revised May 4, 1998; accepted May 14, 1998.
This work was supported by a Howard Hughes Medical Institute predoctoral fellowship (A.E.K.), the McDonnell-Pew Program in Cognitive Neuroscience (T.W.T.), an NEI predoctoral training grant (N.J.P.), and a Whitaker Foundation Biomedical Engineering Research Grant, the Searles Scholars' Program, and the Lucille P. Markey Charitable Trust (K.D.M.). We thank P. Bush, A. J. Doupe, M. Crair, S. G. Lisberger, T. J. Sejnowski, and D. Buonomono for helpful comments on this manuscript, and P. Bush for many helpful discussions.
T.W.T. and A.E.K. contributed equally to this work.
Correspondence should be addressed to: Kenneth Miller, Department of Physiology, University of California, San Francisco, 513 Parnassus, San Francisco, CA 94143-0444.
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