Conditioned Reflex

To quantify behavioral performance, we plotted the proportion of 'rightward' decisions as a function of heading (Fig. 1c), and we fit these psychometric functions with a cumulative Gaussian30 (link). The psychophysical threshold for each stimulus condition was taken as the standard deviation parameter of the Gaussian fit.
Predicted thresholds for the combined condition, assuming optimal (maximum likelihood) cue integration, were computed as5 (link): $${\sigma }_{\mathit{\text{prediction}}}=\sqrt{\frac{{\sigma }_{\mathit{\text{vestibular}}}^2\times {\sigma }_{\mathit{\text{visual}}}^2}{{\sigma }_{\mathit{\text{vestibular}}}^2+{\sigma }_{\mathit{\text{visual}}}^2}}$$ where σ_vestibular and σ_visual represent psychophysical thresholds in the vestibular and visual conditions, respectively.
Neural responses were quantified as mean firing rates over the middle 1 s interval of each stimulus presentation (see Fig. 8 for other time windows). To characterize neuronal sensitivity, we used receiver operating characteristic (ROC) analysis to compute the ability of an ideal observer to discriminate between two oppositely-directed headings (e.g. +1° vs. −1°) based solely on the firing rate of the recorded neuron and a presumed 'anti-neuron' with opposite tuning29 (link),31 (link). Neurometric functions were constructed from these ROC values and were fit with cumulative Gaussian functions to determine neuronal thresholds.
To quantify the relationship between MSTd responses and perceptual decisions, we computed “choice probabilities” using ROC analysis29 (link),34 (link). For each heading direction, neuronal responses were sorted into two groups based on the animal’s choice at the end of each trial (i.e., ‘preferred’ versus ‘null’ choices). ROC values were calculated from these two distributions whenever there were at least 3 choices in each group, and this yielded a choice probability (CP) for each heading direction. We combined data across headings (following z-score normalization) to compute a grand CP for each cue condition29 (link). The statistical significance of CPs (relative to the chance level of 0.5) was determined using permutation tests (1000 permutations).
Note that, for opposite cells, the definition of “preferred” and “null” choices is different for the vestibular and visual conditions. In computing CPs, we defined preferred and null choices according to the tuning of the neuron in each particular stimulus condition. Thus, if the opposite neuron of Fig. 2d consistently responds more strongly when the monkey reports rightward movement, it will have a CP > 0.5 for the vestibular condition and a CP < 0.5 for the visual condition.
To quantify the congruency between visual and vestibular tuning functions measured during discrimination, we calculated a congruency index (CI). A Pearson correlation coefficient was first computed for each single-cue condition. This quantified the strength of the linear trend between firing rate and heading for vestibular (R_vestibular) and visual (R_visual) stimuli. CI was defined as the product of these two correlation coefficients: $$CI=R_{\mathit{\text{vestibular}}}\times R_{\mathit{\text{visual}}}$$ CI ranges from −1 to 1 with values near 1 indicating that visual and vestibular tuning functions have a consistent slope (Fig. 2b), whereas values near −1 indicate opposite slopes (Fig. 2d). Note that CI reflects both the congruency of tuning and the steepness of the slopes of the tuning curves around straight ahead. CI was considered to be significantly different from zero when both of the constituent R values were significant (p < 0.05). We denote neurons having values of CI significantly different from zero as CI-congruent (CI > 0) or CI-opposite (CI < 0). We also examined a global measure of visual-vestibular congruency (see Supplementary Methods) and obtained similar results using this measure (Suppl. Figure 11 and Suppl. Figure 12).
We used a linear weighted summation model to predict responses during cue combination from responses to each single cue condition: $$R_{\mathit{\text{prediction}}}=w_{\mathit{\text{vestibular}}}\times R_{\mathit{\text{vestibular}}}+w_{\mathit{\text{visual}}}\times R_{\mathit{\text{visual}}}$$ where R_vestibular and R_visual are responses from the single-cue conditions, and w_vestibular and w_visual represent weights applied to the vestibular and visual responses, respectively. The weights were determined by minimizing the sum squared error between predicted responses and measured responses in the combined condition. Weights were constrained to lie between −20 and +20. The correlation coefficient (R) from a linear regression fit, which ranges from −1 to 1, was used to assess goodness of fit. We also evaluated three variants of the linear model, as described in Supplementary Methods and Suppl. Fig. 6.