DOI: 10.5061/dryad.hx3ffbgq9
title: Code and data from: Representational geometry explains puzzling error distributions in behavioral tasks

description: Measuring and interpreting errors in behavioral tasks is critical for
 understanding cognition. Conventional wisdom assumes that
 encoding/decoding errors for continuous variables in behavioral tasks
 should naturally have Gaussian distributions, so that deviations from
 normality in the empirical data indicate the presence of more complex
 sources of noise. This line of reasoning has been central for prior
 research on working memory. Here we re-assess this assumption, and find
 that even in ideal observer models with Gaussian encoding noise, the error
 distribution is generally non-Gaussian, contrary to the commonly held
 belief. Critically, we find that the shape of the error distribution is
 determined by the geometrical structure of the encoding manifold via a
 simple rule. In the case of a high-dimensional geometry, the error
 distributions naturally exhibit flat tails. Using this novel insight, we
 apply our theory to visual short-term memory tasks, and find that it can
 account for a large array of experimental data with only two free
 parameters. Our results challenge the dominant view in the
  mechanisms and capacity constraints of working memory systems,
 and instead suggest that the Bayesian framework, which explains various
 aspects of perceptual behavior, also provides an excellent account of
 working memory.Overall, our results establish a new and direct connection
 between neural manifold geometry and behavior, and call attention to the
 geometry of the representation as a critically important, yet
 underappreciated factor in determining the character of errors in human
 behavior.
description: No new data were collected in this study. The analyses in this
 study were based on several previously published datasets collected from
 other labs. For convenience, we have included these published datasets
 here. Please refer to the original publications for detailed experiment
 procedures for the data collection. 
description: # **Code and data from: Representational geometry explains puzzling error
 distributions in behavioral tasks**
 [https://doi.org/10.5061/dryad.hx3ffbgq9](https://doi.org/10.5061/dryad.hx3ffbgq9) **Description of the data and file structure** This file contains information for using the MATLAB code associated with the paper entitled “Representational geometry explains puzzling error distributions in behavioral tasks” by Wei & Woodford, PNAS, 2025. Section 1: Explanations of the Datasets used in this paper No new experimental data were collected in this study. The analyses in the present study were based on several previously published datasets (from Refs 1,2,3,4,5; see the reference list below) collected in other labs.  The data in Ref 1  can be downloaded from [https://osf.io/j2h65/?view_only=fdd51dd775a945508c7cbbf25b662692](https://osf.io/j2h65/?view_only=fdd51dd775a945508c7cbbf25b662692) The data in Refs 2,3,4,5  can also be downloaded from [https://github.com/WeiJiMaLab/delayed_estimation_benchmark/tree/master/data](https://github.com/WeiJiMaLab/delayed_estimation_benchmark/tree/master/data) Please refer to the original publications for detailed experiment procedures for the data collection. References: 1.  MW Schurgin, JT Wixted, TF Brady, Psychophysical scaling reveals a unified theory of visual memory strength. Nature Human Behaviour 4, 1156–1172 (2020). 2\. W Zhang, SJ Luck, Discrete fixed-resolution representations in visual working memory. Nature 453, 233–235 (2008). 3.  Van den Berg, H Shin, WC Chou, R George, WJ Ma, Variability in encoding precision accounts for visual short-term memory limitations. Proceedings National Academy Sciences 109, 8780–8785 (2012). 4\. PM Bays, RF Catalao, M Husain, The precision of visual working memory is set by allocation of a shared resource. Journal of Vision 9, 7–7 (2009). 5\. RL Rademaker, CH Tredway, F Tong, Introspective judgments predict the precision and likelihood of successful maintenance of visual working memory. Journal of Vision 12, 21–21 (2012) Section 2: Code/Software/simulated data The analysis code was written in software MATLAB. (A) The model generated predicted error distributions for different combinations of parameters. The predictions are stored in the following data file: error_tensor.mat.zip Unzip the file would result in the following data file: “error_tensor.mat”, which stored the simulated data in an 21 *41 *360 tensor. (B) The code is organized according the list of figures reported in the paper. Each sub-folder contains the code/scripts for reproducing the analysis of that particular figure.  The scripts are stored in the zip file "Dryad_code_for_Wei_and_Woodford_PNAS_2025_Jan_26.zip". These files are summarized below:  \~/binning_function.m:    a function that bins an error vector into an error histogram \~/noise_distribution2.m:    a function that simulate the error distributions for an arbitrary noise parameter (kappa) and an arbitray noise level. \~/Figure 1/Fig1_geometry.m: a file that simulates figure panels in Figure 1. \~/Figure 2/Fig2_panels_e_to_h.m: a file that simulates Figure 2 panels e to h. \~/Figure 2/Fig2_panles_a_to_d.m: a file that simulates Figure 2 panels a to d. \~/Figure 3/Fig3_panel_a.m: a file that generates panel a of Figure 3. \~/Figure 3/Fig3_panel_b.m: a file that generates panel b of Figure 3. \~/Figure 3/simulate_error_density.m: a function that simulates error distribution for different combinations of parameters. \~/Figure 3/Generation_error_data_for_fitting.m: a file that generate the simtulated data by using function “simulate_error_density.m”. \~/Figure 4/Fig4_color: a file that generates panels in Figure 4. \~/Figure 5/Fig5_orientation: a file that generates panels in Figure 5. \~/Figure 6/twoAFC_predicting_NAFC.m: a file that generates Figure 6. \~/SI_fig 1_simulation_geometry and tuning/FigS1_different_tuning_same_geometry.m:  a file that generates SI Figure 1. \~/SI_fig 2_convolution/FigS2_kappa_and_geometry.m:  a file that generates SI Figure 2. \~/SI_fig 3_mixture of tuning widths/FigS3_RD_mixed_kappa.m:  a file that generates SI Figure 3. \~/SI_fig 4_dimensionality/FigS4_dimensionality.m: a file that generates SI Figure 4. \~/SI_fig 5_noise and decaying constant/FigS5_sigma_and_slope.m:  a file that generates SI Figure 5. \~/SI_Fig 6_power law/FigS6_powerLaw.m: a file that generates SI Figure 6. \~/SI_Fig 7_Fit to older color data/FigS7.m: a file that generates SI Figure 7. \~/SI_Fig 7_Fit to older color data/MaximumLikelihoodFitToDataFixedKappa: a function that fits our model to the data by assuming a given geometry paramter. \~/SI_Fig 8_noise and dprime/FigS8.m: a file that generates SI Figure 8. (C) There are a few functions that are useful to multiple figures. These functions are contained in files binning_function.m, noise_distribution2.m. If the code file for a certain figure cannot run properly, please check if these functions are included in the path.  Some figures involve data analysis. Please make sure the code has access to the relevant data files described above when running the code.