MODELING SIMPLE-CELL DIRECTION SELECTIVITY WITH NORMALIZED, HALF-SQUARED, LINEAR-OPERATORS

1. A longstanding view of simple cells is that they sum their inputs l inearly. However, the linear model falls short of a complete account o f simple-cell direction selectivity. We have developed a nonlinear mod el of simple-cell responses (hereafter referred to as the normalizatio n model) to explain a larger body of physiological data. 2. The normal ization model consists of an underlying linear stage along with two ad ditional nonlinear stages. The first is a half-squaring nonlinearity; half-squaring is half-wave rectification followed by squaring. The sec ond is a divisive normalization nonlinearity in which each model cell is suppressed by the pooled activity of a large number of cells. 3. By comparing responses with counterphase (flickering) gratings and drift ing gratings, researchers have demonstrated that there is a nonlinear contribution to simple-cell responses. Specifically they found 1) that the linear prediction from counterphase grating responses underestima tes a direction index computed from drifting grating responses, 2) tha t the linear prediction correctly estimates responses to gratings drif ting in the preferred direction, and 3) that the linear prediction ove restimates responses to gratings drifting in the nonpreferred directio n. 4. We have simulated model cell responses and derived mathematical expressions to demonstrate that the normalization model accounts for t his empirical data. Specifically the model behaves as follows. 1) The linear prediction from counterphase data underestimates the direction index computed from drifting grating responses. 2) The linear predicti on from counterphase data overestimates the response to gratings drift ing in the nonpreferred direction. The discrepancy between the linear prediction and the actual response is greater when using higher contra st stimuli. 3) For an appropriate choice of contrast, the linear predi ction from counterphase data correctly estimates the response to grati ngs drifting in the preferred direction. For higher contrasts the line ar prediction overestimates the actual response, and for lower contras ts the linear prediction underestimates the actual response. 5. In add ition, the normalization model is qualitatively consistent with data o n the dynamics of simple-cell responses. Tolhurst et al. found that si mple cells respond with an initial transient burst of activity when a stimulus first appears. The normalization model behaves similarly; it takes some time after a stimulus first appears before the model cells are fully normalized. We derived the dynamics of the model and found t hat the transient burst of activity in model cells depends in 'a parti cular way on stimulus contrast. The burst is short for high-contrast s timuli and longer for low-contrast stimuli. 6. The importance of these results is that the normalization model preserves the essential featu res of linearity in the face of apparently contradictory behavior. Acc ording to the model, a cell's direction selectivity is attributed to t he underlying linear stage, and a cell's nonlinear behavior is attribu ted to half-squaring and normalization.