Fig. 4: Curvilinear manifolds predict systematic biases in decoding.

a A curvilinear geometry would produce biases in the accuracy of a linear decoder. This cartoon illustrates how the projection of an arc onto a line compresses the values at the tails of the range. b A linear decoder trained on the vmPFC population and used to predict value. Note that predicted low values are higher than the true values and predicted high values are lower. c (top) Comparing the accuracy of a decoder trained on the real vmPFC data (vertical line) with decoders trained on linearized control populations (purple bars), 3.49 vs 2.82, 95% CI = [2.64, 2.99], bootstrapped test, p < 0.001. RMSE: root mean square error, higher = less accurate. c (bottom) Points projected from an arc onto a line, as visualized in (a), would have residual errors that follow a sine function. Difference in the quality of a fit of a linear and sine function to the residuals from vmPFC (vertical line) and the linearized controls (purple bars), 0.001 vs −0.003, 95% CI = [−0.006, 0.002], bootstrapped test, p = 0.057. SSE - sum of squared estimate of errors. d Another bias predicted by a curved–but not by a linear–manifold affects out-of-range observations. A decoder trained on a portion of the curved function would not make accurate predictions about the values in the other portion of the curve. e Decoders trained on the population response to one-half of the values (filled circles) and used to predict values outside of this range (open circles). Red = trained on high values; blue = trained on low values. f (top) The angle between the high- and low-value-trained decoders in real vmPFC population (vertical line) and in linearized controls (purple bars), 22.33 vs 13.97, 95% CI = [12.27, 15.76], bootstrapped test, p < 0.001. f (bottom) The change in slope in (e) between within-range and out-of-range values for each decoder (red = trained on high values; blue = trained on low), high-trained: 0.37 vs 0.12, 95% CI = [0.06, 0.18], bootstrapped test, p < 0.001; low-trained: 0.36 vs 0.1, 95% CI = [0.04, 0.17], bootstrapped test, p < 0.001. Vertical lines correspond to the vmPFC data and distributions to the control populations. Error bars indicate ± standard error of the mean across neurons (SEM), n = 121. ****p < 0.001.