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We aimed to quantify the relationship between the response elicited by the bi-speed stimuli and the corresponding component responses. We first assumed that the response R of a neuron elicited by two component speeds can be described as a weighted sum of the component responses R_s and R_f elicited by the slower ($v_s$) and faster ($v_f$) component speed, respectively Equation 1.

(1)

${\displaystyle {\displaystyle R\left(v_s,v_f\right)=w_s\left(v_s,v_f\right)\bullet R_s+w_f\left(v_s,v_f\right)\bullet R_f,}}$

in which, w_s and w_f are the response weights for the slower and faster speed component $v_{s}and{v}_f$, respectively.

Our goal was to estimate the weights for each speed pair and determine whether the weights change with the stimulus speeds. In our main data set, the two speed components moved in the same direction. To determine the weights of $w_s$ and w_f for each neuron at each speed pair, we have three data points R, R_s, and R_f, which are trial-averaged responses. Since it is not possible to solve for both variables, $w_s$ and w_f, from a single equation Equation 1 with three data points, we introduced an additional constraint: $w_s$ + w_f = 1. With this constraint, the weighted sum becomes a weighted average. While this constraint may not yield the exact weights that would be obtained with a fully determined system, it nevertheless allows us to characterize how the relative weights vary with stimulus speed. As long as R_f ≠ R_s, R can be expressed as:

(2)

${\displaystyle {\displaystyle R=\frac{R_f-R}{R_f-R_s}{R}_s+\frac{R-R_s}{R_f-R_s}{R}_f,}}$

The response weights are $w_s=\frac{R_f-R}{R_f-R_s}$ , $w_f=\frac{R-R_s}{R_f-R_s}$. Intuitively, if R were closer to one component response, that stimulus component would have a higher weight. Note that Equation 2 is not intended for fitting the response R using $R_s$ and $R_f$, but rather to use the relationship among R, $R_s$, and $R_f$ to determine the weights for the faster and slower components.

Using this approach to estimate response weights for individual neurons can be unreliable, particularly when R_f and Rs are similar. This situation often arises when the two speeds fall on opposite sides of the neuron’s preferred speed, resulting in a small denominator (R_f – Rs) and consequently an artificially inflated weight estimate. We, therefore, used the neuronal responses across the population to determine the response weights (Figure 5). For each pair of stimulus speeds, we plotted (R−R_s) in the ordinate versus (R_f − R_s) in the abscissa. Figure 5A1–E1 shows the results obtained at 4x speed separation. Across the neuronal population, the relationship between (R – R_s) and (R_f − R_s) can be described by a linear equation (Equation 3) (see R² in Table 1). This linearity suggests that the response weights for each speed pair are roughly consistent across the neuronal population.

(3)

${\displaystyle {\displaystyle R-R_s=k\left(R_f-R_s\right)+b}}$

Because all the regression lines in Figure 5 nearly go through the origin (i.e. intercept b ≈ 0, Table 1), the slope k obtained from the linear regression approximates $\frac{R-R_s}{R_f-R_s}$, which is the response weight $w_f$ for the faster component (Equation 2). Hence, for each pair of stimulus speeds, we can estimate the response weight for the faster component using the slope of the linear regression of the responses from the neuronal population.

Our results showed that the bi-speed response showed a strong bias toward the faster component when the speeds were slow and changed progressively from a scheme of ‘faster-component-take-all’ to ‘response-averaging’ as the speeds of the two stimulus components increased (Figure 5F1). We found similar results when the speed separation between the stimulus components was small (2x), although the bias toward the faster component at low stimulus speeds was not as strong as 4x speed separation (Figure 5A2–F2 and Table 1).

In the regression between $\left(R{-R}_s\right)$ and $\left(R_f-R_s\right)$, $R_s$ (i.e. the firing rate to the slow component averaged across all trials for each neuron) was a common term and, therefore, could artificially introduce correlations. We wanted to determine whether our estimates of the regression slope ($w_f$) were confounded by this factor. We performed two additional analyses.

First, at each speed pair and for each of the 100 neurons in the data sample shown in Figure 5, we simulated the response to the bi-speed stimuli ($R_e$) as a randomly weighted average of $R_f$ and $R_s$ of the same neuron.

(4)

${\displaystyle {\displaystyle R_e=a{R}_f+\left(1-a\right)R_s,}}$

in which $a$ was a randomly generated weight (between 0 and 1) for $R_f$, and the weights for $R_f$ and $R_s$ summed to one. We then calculated the regression slope and the correlation coefficient between the simulated $R_e-R_s$ and $R_f-R_s$ across the 100 neurons. We repeated the process 1000 times and obtained the mean and 95% confidence interval (CI) of the regression slope and the R². The mean slope based on the simulated responses was 0.5 across all speed pairs. The estimated slope ($w_f$) from the data was significantly greater than the simulated slope at slow speeds of 1.25/5, 2.5/10 (Figure 5F1), and 1.25/2.5, 2.5/5, and 5/10°/s (Figure 5F2) (bootstrap test, see p-values in Table 1). The estimated R² based on the data was also significantly higher than the simulated R² for most of the speed pairs (Table 1).

Second, we calculated $R_s$ in the ordinate and abscissa of Figure 5A–E using responses averaged across different subsets of trials, such that $R_s$ was no longer a common term in the ordinate and abscissa. For each neuron, we determined $R_{s1}$ by averaging the firing rates of $R_s$ across half of the recorded trials, selected randomly. We also determined $R_{s2}$ by averaging the firing rates of $R_s$ across the rest of the trials. We regressed $\left(R{-R}_{s1}\right)$ on $\left(R_f{-R}_{s2}\right)$, as well as $\left(R{-R}_{s2}\right)$ on $\left(R_f{-R}_{s1}\right)$, and repeated the procedure 50 times. The averaged slopes obtained with $R_s$ from the split trials showed the same pattern as those using $R_s$ from all trials (Table 1 and Appendix 1—figure 1), although the coefficient of determination was slightly reduced (Table 1). For 4x speed separation, the slopes were nearly identical to those shown in Figure 5F1. For 2x speed separation, the slopes were slightly smaller than those in Figure 5F2, but followed the same pattern (Appendix 1—figure 1). Together, these analysis results confirmed the faster-speed bias at the slow stimulus speeds and the change of the response weights as stimulus speeds increased.