I explore here whether linear mechanisms can explain directional selectivity (DS) in simple cells of the cat's striate cortex, a question suggested by a recent upswing in popularity of linear DS models. I chose a simple cell with a space-time inseparable receptive field (RF), i.e. one that shows gradually shifting latency across space, as the RF type most likely to depend on linear mechanisms of DS. However, measured responses of the cell to a moving bar were less modulated, and extended over a larger spatial region than predicted by two different popular “linear” models. They also were more DS in exhibiting a higher ratio of total spikes for the preferred direction. Each of the two models used for comparison has a single “branch” with a single spatiotemporally inseparable linear filter followed by a threshold, hence, a “1-branch” model. Nonlinear interactions between pairs of bars in a 2-bar linear superposition test of the cell also disagreed in time-course with those of the 1-branch models. The only model whose 1-bar and 2-bar predictions matched the measured cell (including a complete “4-branch” motion energy model that matches complex cells) has two branches that differ in phase by about 90 deg, i.e. in quadrature. Each branch has its own threshold that helps define the preceding spatiotemporal unit as a subunit even after the outputs of the two branches are summed. As subunit phases differ by only 90 deg, flashing bar responses of the 2-subunit model are similar to those of the 1-subunit model. Therefore, the number of subunits is hidden from view when testing with a conventional stationary bar. In summary, movement responses and nonlinear interactions between pairs of bars in the measured cell matched those of the 2-subunit model, while they disagreed with the popular 1-subunit model. Thus, multiple nonlinear subunits appear to be necessary for DS, even in simple cortical cells.

We have compared the responses of simple cells to laterally moving sinusoidal gratings and to stationary temporally-modulated gratings. From the amplitudes and temporal phases of the responses to stationary gratings of different spatial phases, it should be possible to predict the preferred direction of movement, the amplitudes of the responses to gratings moving in the preferred and nonpreferred directions and, thence, the degree of directional preference (Reid et al., 1987). The preferred direction can be predicted reliably. However, the magnitude of the directional preference cannot be predicted, since the measured amplitude of the response in the nonpreferred direction of movement is very much less than that predicted by a linear theory. Nonlinearities in the relationship between response amplitude and contrast may contribute to the failure of the predictions, but this contribution is small. We conclude that the magnitude of the directional preference seems to be determined predominantly by nonlinear suppression of the response in the nonpreferred direction of movement.

It is believed that spatial summation in most simple cells is a linear process. If this were so, then the Fourier transform of a simple cell's line weighting function should predict the cell's spatial frequency tuning curve. We have compared such predictions with experimental measurements and have found a consistent discrepancy: the predicted tuning curve is much too broad. We show qualitatively that this kind of discrepancy is consistent with the well-known threshold nonlinearity shown by most cortical cells. We have tested quantitatively whether a response threshold could explain the observed disagreements between predictions and measurements: a least-squares minimization routine was used to fit the inverse Fourier Transform of the measured frequency tuning curve to the measured line weighting function. The fitting procedure permitted us to introduce a threshold to the reconstructed line weighting function. The results of the analysis show that, for all of the cells tested, the Fourier transforms produced better predictions when a response threshold was included in the model. For some cells, the actual magnitude of the response threshold was measured independently and found to be compatible with that suggested by the model. The effects of nonlinearities of spatial summation are considered.

Extracellular single-unit recordings were made from simple cells in area 17 of anesthetized cats. Cells were tested with drifting gratings under control and adapted conditions. Response amplitude and phase were measured as a function of either contrast or temporal frequency. Adapting not only reduces amplitude, but also retards phase. Adaptation alters the responses of simple cells in a particular way: the onset of the response to each cycle of a sinusoidally modulated stimulus is delayed. Once cells start to respond during each cycle, however, they generally recover to control levels, and the offset of the response is unaffected by adapting. The timing aftereffects are independent of the amplitude aftereffects. Timing aftereffects are tuned around the adapting temporal frequency, with a bias toward lower temporal frequencies. Adaptation thus modifies cortical responses even more specifically then previously thought. Firing rates are depressed primarily at response onset, even after several stimulus cycles have occurred following the end of adapting. Because all cells appear to adapt in this way, the data offer an opportunity to theorize about cortical connectivity. One implication is that inhibition onto a simple cell arises from other simple cells with similar response properties that fire a half-cycle out of phase with the target cell.

In almost every study of the linearity of spatiotemporal summation in simple cells of the cat's visual cortex, there have been systematic mismatches between the experimental observations and the predictions of the linear theory. These mismatches have generally been explained by supposing that the initial spatiotemporal summation stage is strictly linear, but that the following output stage of the simple cell is subject to some contrast-dependent nonlinearity. Two main models of the output nonlinearity have been proposed: the threshold model (e.g. Tolhurst & Dean, 1987) and the contrast-normalization model (e.g. Heeger, 1992a, b). In this paper, the two models are fitted rigorously to a variety of previously published neurophysiological data, in order to determine whether one model is a better explanation of the data. We reexamine data on the interaction between two bar stimuli presented in different parts of the receptive field; on the relationship between the receptive-field map and the inverse Fourier transform of the spatial-frequency tuning curve; on the dependence of response amplitude and phase on the spatial phase of stationary gratings; on the relationships between the responses to moving and modulated gratings; and on the suppressive action of gratings moving in a neuron's nonpreferred direction. In many situations, the predictions of the two models are similar, but the contrast-normalization model usually fits the data slightly better than the threshold model, and it is easier to apply the equations of the normalization model. More importantly, the normalization model is naturally able to account very well for the details and subtlety of the results in experiments where the total contrast energy of the stimuli changes; some of these phenomena are completely beyond the scope of the threshold model. Rigorous application of the models' equations has revealed some situations where neither model fits quite well enough, and we must suppose, therefore, that there are some subtle nonlinearities still to be characterized.

Simple cells in striate cortex have been depicted as rectified linear operators, and complex cells have been depicted as energy mechanisms (constructed from the squared sums of linear operator outputs). This paper discusses two essential hypotheses of the linear/energy model: (1) that a cell's selectivity is due to an underlying (spatiotemporal and binocular) linear stage; and (2) that a cell's firing rate depends on the squared output of the underlying linear stage. This paper reviews physiological measurements of cat striate cell responses, and concludes that both of these hypotheses are supported by the data.

A reverse correlation technique, which permits estimation of three-dimensional first-order properties of receptive fields (RFs), was applied to simple cells in areas 17 and 18 of cat. Two classes of simple cells were found. For one class, the spatial and temporal RF characteristics were Separable, i.e. they could be synthesized as the product of spatial and temporal weighting functions. RFs in the other class were Inseparable, i.e. bright and dark subregions comprising each field were obliquely oriented in space-time. Based on a linear superposition model, these observations led to testable hypotheses: (1) simple cells with separable space-time characteristics should be speed but not direction selective and (2) simple cells with inseparable space-time characteristics should be direction selective and the optimal velocity of moving stimuli should be predictable from the slope of the oriented subregions. These hypotheses were tested by comparing responses to moving bars with those predicted by application of the convolution integral. Linear predictions accounted for waveforms of responses to moving bars in detail. For cells with oriented space-time characteristics, the preferred direction was always predicted correctly and the optimal speed was predicted quite well. Most cells with separable space-time characteristics were not direction selective as predicted. The major discrepancies between measured and predicted behavior were twofold. First, 8/32 cells with separable space-time RFs were direction selective. Second, predicted directional indices were weakly correlated with actual measurements. These conclusions hold for simple cells in both areas 17 and 18. The major difference between simple RFs in these areas is the coarser spatial scale seen in area 18. These results demonstrate a significant linear contribution to the speed and direction selectivity of simple cells in areas 17 and 18. Where additional, nonlinear mechanisms are inferred, they appear to act synergistically with the linear mechanism.

Previous tests of the linearity of spatiotemporal summation in cat simple cells have compared the responses to moving sinusoidal gratings and to gratings whose contrast was modulated sinusoidally in time. In particular, since a moving grating can be expressed as a sum of modulated gratings, the response to a moving grating should be predictable (assuming linearity) from the responses to modulated gratings. However, these simple linear predictions have shown varying degrees of failure (e.g. Reid et al., 1987, 1991), depending on the directional selectivity of the neurons (Tolhurst & Dean, 1991). We demonstrate here that the failures of these linear predictions are, in fact, explained by the contrast-normalization model of Heeger (1993). We concentrate on the ratio of the measured to predicted moving grating responses. In the context of the contrast-normalization model, calculating this ratio turns out to be particularly appropriate, since the ratio is independent of the precise details of the linear front-end mechanisms ultimately responsible for directional selectivity. Hence, the contrast-normalization model can be compared quantitatively with this ratio measure, by varying only one free parameter. When account is taken both of the expansive output nonlinearity and of contrast normalization, the directional selectivity of simple cells seems to be dependent only on linear spatiotemporal filtering.

Simple cells in the striate cortex have been depicted as half-wave-rectified linear operators. Complex cells have been depicted as energy mechanisms, constructed from the squared sum of the outputs of quadrature pairs of linear operators. However, the linear/energy model falls short of a complete explanation of striate cell responses. In this paper, a modified version of the linear/energy model is presented in which striate cells mutually inhibit one another, effectively normalizing their responses with respect to stimulus contrast. This paper reviews experimental measurements of striate cell responses, and shows that the new model explains a significantly larger body of physiological data.

The goal of this study was to determine the effects of inactivating layer A or the C layers of the cat lateral geniculate nucleus on the supragranular layers of area 18, including cells antidromically activated from the lateral suprasylvian visual area (LS). Isolated cells were visually driven via the contralateral eye while the retinotopically corresponding regions of layer A or, in some cases, the C layers were reversibly inactivated with injections of cobaltous chloride. Simple cells were frequently encountered and were on average more dependent upon layer A than were complex cells, a result qualitatively similar to that found previously in area 17 (Malpeli, 1983; Malpeli et al., 1986). However, the influence of the C layers on area 18 was much more apparent than for area 17. In area 18, as in area 17, the dependence of simple cells on particular geniculate layers appears to follow the terminal patterns of the major direct geniculate inputs. Those simple cells most dependent on layer A were located in lower layer 3. Simple cells in upper layer 3, like complex cells, showed little dependence on layer A, but were strongly dependent upon the C layers. All cells antidromically activated from LS were simple cells with rapidly conducting axons. They had, on average, the same moderately strong dependence on layer A as the patches of LS receiving area 18 input (Lee et al., 1997), supporting the conclusion that the influence of layer A in these patches is largely transmitted via association inputs from area 18. These results demonstrate that simple cells play a major role in association pathways.