Fig. 3: Representational geometry of value is curvilinear at the population level.

a The neuronal population response to an offer value is a pattern of firing rates across neurons (top left). These patterns can also be understood as vectors in a neuron-dimensional space, or, equivalently, as points in the neuronal state space (top right). We probe the representational geometry of these neuronal population patterns through measuring the distance between neural responses (bottom left) or via examining the major axes of co-variability between neurons with principal components analysis (PCA; bottom right). b–d We considered three hypotheses about how value could be represented at the population level. First, value representations could be “tangled” (b): if population value representations are very high dimensional at the population level, nearby values might not even be represented by nearby patterns of activity. Second, value representations could be “linear” (c): population value representations could follow a single straight line through the neural state space or, equivalently, occupy a single dimension in the neural state space. Third, value representations could be “curved” (d): value representations could be structured–with nearby values represented by nearby patterns of activity–but the population manifold could still occupy more than one dimension. e The mean distance between neuronal states corresponding to different values. f The projection of the neural population onto the first 2 principal components (PCs). Shades of gray = value bins from low (light gray) to high (dark gray). Dotted line = best linear fit. Solid line = best quadratic fit. g Percent variance explained by each PC. Capturing the variance in a curved function would require more than one PC. h Same as (f, g) for one example linearized dataset (see “Methods” section). i A comparison of the variance explained by the first 2 PCs in the real population (vertical line) against bootstrapped distributions of linearized datasets, PC1: 27.08% vs 37.66%, 95% CI = [36.36%, 41.24%], bootstrapped estimate, p < 0.001; PC2: 8.60% vs 5.43%, 95% CI = [4.73%, 5.92%], bootstrapped estimate, p < 0.001. Note that third and higher order PCs also continue to explain more variance in the vmPFC data compared to linearized controls (see “Results” section). j, k Same as (f, i), but for accepted offers only, PC1: 27.97% vs 41.29%, 95% CI = [37.0%, 44.78%], bootstrapped estimate, p < 0.001; PC2: 14.04% vs 10.63%, 95% CI = [9.61%, 12.82%], bootstrapped estimate, p = 0.001. l, m Same as (f, i), but for rejected offers only, PC1: 22.69% vs 36.86%, 95% CI = [36.23%, 43.73%], bootstrapped estimate, p < 0.001; PC2: 15.27% vs 12.43%, 95% CI = [9.65%, 12.67%], bootstrapped estimate, p < 0.001. n = 121. ****p ≤ 0.001.