Extended Data Fig. 5: Decoder initialization slightly influences decoder-encoder pairs.
a. Model predictions. All panels show the gradient field of the user and decoder cost functions. Purple (user) and orange (decoder) curves show nullclines (where the agent’s gradient equals 0) that intersect at stationary points (black stars). Decoder initialization D1 (light orange) and initialization D2 (dark orange) are noted on the vertical axis of the diagram. Assumed user initialization (purple) is noted on the horizontal axis. b. Left: Average magnitude of the decoder matrix (norm) as a function of time in the trial for D1 (solid light orange) and D2 (dashed dark orange) initializations in slow learning rate conditions (N = 14, median; shading shows the 25th - 75th percentile). Right: Boxplots are average decoder effort across the trial for each subject (N = 14, center shows median; box shows 25th-75th percentiles; whiskers extend to 1.5 × this interquartile range; two-sided Wilcoxon signed-rank test, ns = 0.27). c. Left: Average magnitude of the user encoder matrix (norm) as a function of time in the trial for D1 (solid dark purple) and D2 (dashed light purple) decoder initializations (N = 14, median; shading shows the 25th - 75th percentile). Right: Boxplots are average effort for each subject across the trial (N = 14, center shows median; box shows 25th-75th percentiles; whiskers extend to 1.5 × this interquartile range; two-sided Wilcoxon signed-rank test, ns = 0.86). d. Percent change in error from the start of the trial (first 30 seconds) to the end of the trial (last 30 seconds), separated by D1 (dark gray) and D2 (light gray) initializations conditions (N = 14, center shows median; box shows 25th-75th percentiles; whiskers extend to 1.5 × this interquartile range; two-sided Wilcoxon signed-rank test, ns = 0.33). e. Last minute in trial of product of the average decoder matrix of each initialization with first-order feedforward (F1) contributions of the average encoder matrix of each initialization, separated by learning rate. Fast learning rate conditions are shaded gray. The matched conditions are the decoders and encoders of the same initialization and the mismatched conditions are the decoders and encoders of different initializations (N = 28, center shows median; box shows 25th-75th percentiles; whiskers extend to 1.5 × this interquartile range; two-sided Wilcoxon signed-rank test, ns > 0.05; *p < 0.05; **p < 0.001). f. Last minute in trial of product of the average decoder matrix of each initialization with first-order feedback (B1) contributions of the average encoder matrix of each initialization, separated by learning rate. Fast learning rate conditions are shaded gray. The matched conditions are the decoders and encoders of the same initialization and the mismatched conditions are the decoders and encoders of different initializations (N = 28, center shows median; box shows 25th-75th percentiles; whiskers extend to 1.5 × this interquartile range; two-sided Wilcoxon signed-rank test, ns > 0.05; *p < 0.05; **p < 0.001).
