(A) Input-output relationship showing a vertex (i.e., a sudden change in slope) (black curve). If we assume that the input is normally distributed with low intensity (i.e., standard deviation) such that all the input values are to the right of the vertex (light green distribution on *x*-axis), then the corresponding output distribution will also be normally distributed (light purple distribution on *y*-axis). The mean output (light purple circle on *y*-axis) corresponds to the image of the mean input (dashed purple circle on *y*-axis; note that the light purple and dashed purple circles were offset for clarity) as both input and output are linearly related. In contrast, for a higher intensity input that extends significantly past the vertex (dark green distribution on *x*-axis), the corresponding output distribution (dark purple on *y*-axis) is skewed with respect to the linear prediction (dashed purple on *y*-axis). The mean output (dark purple circle on *y*-axis) is thus greater than the linear prediction (dashed purple circle on *y*-axis). (Note that here and below, we represented the distributions to have the same maximum value in order to emphasize the fact that we are changing the standard deviation.) (B) Increasing the input distribution intensity for a given mean (compare red, yellow, and blue distributions) causes a greater skew in the corresponding output distribution (unpublished data) and thus an increased bias in their means (red, yellow, and blue dots on the *y*-axis and inset) as compared to the linear prediction (dashed yellow and blue dots on the *y*-axis). (C) Shifting the mean of the high intensity input distribution to the left (compare points 1, 2, and 3 on the *x*-axis and the inset) makes it extend to the left of the vertex more and more (compare the green curves on the *x*-axis), causing greater skewness in the corresponding output distributions (purple curves on the *y*-axis), which creates a greater bias in the mean (dark purple points on *y*-axis) with respect to the linear prediction (light purple points on *y*-axis). As a result, the mean output in response to a given value of the low intensity input (points 1, 2, and 3 on the *x*-axis) when the high intensity signal is present (dark purple line) has a lower slope (i.e., gain) than when the high intensity signal is absent (light purple line). (D) Shifting the mean of the high intensity input distribution to the left (compare points 1, 2, and 3 on the *x*-axis and the inset) makes the corresponding distributions of the low intensity input extend to the left of the vertex more and more (green curves on the *x*-axis), causing greater skewness in the output distribution (purple curves on the *y*-axis), which creates a greater bias in the mean (dark purple points on *y*-axis) with respect to the linear prediction (light purple points on *y*-axis). Note, however, that the bias in the mean will be lower than in (C) since the input distributions now have a lower intensity as explained in (B). Thus, the input-output relationship when the low intensity signal is present (dark purple line) will have a lower slope (i.e., gain) than when the low intensity signal is absent (light purple line) but the effect will be weaker than in (C).

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