DANDI:000404

Our tasks elicited up to 24 conditions of movement (with an average of 16–17 conditions per session)

Results, BMI to study neural population control of movement, paragraph beginning "To study control of diverse movements"; Methods, Neuroprosthetic tasks

Reading (a), conditions with at least one rewarded trial, gives a mean of 17.46 over the 13 sessions (Monkey G 18.00, Monkey J 16.25, at most 20), inside 14.4 to 18.7. Reading (b), conditions with a tuple of 15 or more occurrences, gives 16.23. Monkey J's 31 obstacle trials of the code's types 0 and 1 are left out.

How the test differed from the paper’s method

• Targets are read from the cursor position at reward, as the files' target series do not give the trial's target (see findings); obstacle trials are labelled clockwise or counterclockwise by the code's rule (util_fcns.CW_CCW_obs), with the code's fixed labels for Monkey J types 3, 4, 5, 8 and 9.

Each individual command was used with regularity during multiple conditions (on average ~7 conditions, Figure S1B)

Results, BMI to study neural population control of movement, paragraph beginning "Importantly, the BMI enabled us"; Figure S1B; Methods, Conditions that used a command regularly

Over commands with at least two conditions of 15 or more occurrences, the mean number of such conditions is 7.06 pooled over commands (reading a) and 6.98 over session means (reading b); the median is 7 (25th to 75th percentile 5 to 9). With one or more conditions the mean is 6.69.

How the test differed from the paper’s method

• The command is the decoder gain applied to the binned counts, with the gain recovered from the files by least squares (exact for Monkey G; for Monkey J the linear approximation the paper itself uses). The files hold no decoder matrices.

Condition-specific command subtrajectories are overall significantly different from the condition-pooled command subtrajectory: Monkey G [J]: p-value < 0.001 for 9/9 [4/4] sessions, p-value < 0.001 pooled over sessions (mean of command subtrajectory distances = 1.920 [2.342]

Figure S1D legend

What the paper reports

• each session: comparison with a bootstrapped distribution from the condition-pooled subtrajectories, p < 0.001 (n = 9 sessions of Monkey G and 4 of Monkey J, all significant)
• pooled over sessions, each monkey: comparison with a bootstrapped distribution from the condition-pooled subtrajectories, p < 0.001

Mean normalized subtrajectory distance 1.953 (G) and 2.330 (J), within 10% of 1.920 and 2.342; p < 0.001 in 9 of 9 and 4 of 4 sessions and pooled (no bootstrap average reached the data). With commands of one or more conditions, 1.951 and 2.322. Counting a tuple only when 15 or more occurrences remain after the edge occurrences are removed gives 2.026 and 2.481, with the same tests.

How the test differed from the paper’s method

• Commands from the recovered decoder gain (see figs1b-conditions-per-command).

Overall, neural activity issuing a given command significantly deviated across conditions relative to the shuffle distribution (Figure 3B–E). Distances averaged within-session ranged from 10% to 200% larger than shuffle distance

Results, The same command is issued by different neural activity patterns in different movements, paragraph 2; Figure 3D; Table S1

What the paper reports

• each session: comparison with behavior-preserving shuffle, p < 0.001 (n = 9 of 9 sessions of Monkey G and 4 of 4 of Monkey J)
• pooled over sessions, each monkey: comparison with behavior-preserving shuffle, p < 0.001

Measured under the first readings (behavior-preserving shuffle; commands with two or more qualifying conditions). All four combinations of readings give 1.2366 to 1.2380 for Monkey G and 1.7651 to 1.7713 for Monkey J, session means 1.10 to 1.98, and p < 0.001 in every session and both pooled tests (1000 shuffles, none reaching the data). No tuple was dropped by the matching.

How the test differed from the paper’s method

• Commands and the decoder-null space from the recovered decoder gain (see figs1b-conditions-per-command).

Distance was significantly greater than shuffle distance aggregating over all (command, condition, neuron) tuples: Monkey G [J]: p-value < 0.001 for 9/9 [4/4] sessions, p-value < 0.001 pooled over sessions.

Figure 3B legend; Table S1 (row 3B)

What the paper reports

• each session: comparison with behavior-preserving shuffle, p < 0.001 (n = 9 of 9 sessions of Monkey G and 4 of 4 of Monkey J)
• pooled over sessions, each monkey: comparison with behavior-preserving shuffle, p < 0.001

Mean single-neuron distance 1.166 to 1.167 Hz (G) and 1.221 to 1.222 Hz (J) under the readings, within 10% of 1.167 and 1.235; p < 0.001 in every session and pooled. The shuffle means, 1.002 to 1.004 Hz and 0.751 to 0.752 Hz, match the 1.004 and 0.745 the paper's code prints.

How the test differed from the paper’s method

• Commands from the recovered decoder gain (see figs1b-conditions-per-command).

Full dynamics, command left-out dynamics, and condition left-out dynamics all predicted neural activity significantly better than shuffle dynamics. For each model: Monkey G [J]: p-value < 0.001 for 9/9 [4/4] sessions, p-value < 0.001 for sessions pooled.

Figure 4C legend; Results, Invariant dynamics predict the different neural activity patterns used to issue the same command, paragraph beginning "On the level of single time points"; Table S1 (rows 4C)

What the paper reports

• each model, each session: R2 compared with shuffle dynamics models, p < 0.001 (n = 9 of 9 sessions of Monkey G and 4 of 4 of Monkey J)
• each model, pooled over sessions: R2 compared with shuffle dynamics models, p < 0.001

Scored as the code scores it, R2 is 0.1655 and 0.2492 (full), 0.1624 and 0.2418 (command left out), 0.1631 and 0.2391 (condition left out), all within 10%, each beating the shuffle scored on the same pairs with p < 0.001 in 9 of 9 and 4 of 4 sessions and pooled. The shuffle's pooled 95th percentiles, 0.1282 and 0.1963, match Table S1's 0.130 and 0.196. On pairs whose command at t is analysed the tests are the same and the values within tolerance (full 0.1705 and 0.2574). On all pairs the tests are the same, the command left-out model beating the shuffle in 9 of 9 and 4 of 4 sessions, but the full and condition left-out R2 (0.1942 and 0.2913, 0.1916 and 0.2808) are outside tolerance. Reading (a) of the shuffle ridge parameter (chosen on one shuffled dataset, 200 shuffles) leaves every full-model test significant.

How the test differed from the paper’s method

• The scored pairs are taken from the code: each model's test pairs whose commands at t and t-1 are both below the 95th percentile of magnitude (generalization_plots.check_ix_lo, lines 588-593, "Remove mag command 4"), with the full model and every shuffle model scored on exactly those pairs (generalization_plots.plot_loo_r2_overall, lines 2785-2867). The readings on pairs whose command at t (or t-1) is analysed and on all test pairs are shown, each with the shuffle on the same pairs.
• Commands, the decoder-null space and the conditioning on the command use the recovered decoder gain (see figs1b-conditions-per-command).
• The left-out command model leaves out pairs whose first or second bin issues the command (Figure S4A legend) and is tested on pairs whose second bin issues it.
• Folds are random over pairs of bins, stratified by task, with a fixed seed; the shuffle models use the real data's ridge parameter (reading b) with 1000 shuffles, reading (a) is shown with 200. As in the code (generalization_plots.predict_shuffles_v2_with_cond, one session-wide shuffle index per shuffle passed with the fold test indices), each shuffle permutes the whole session before the folds are split.

The full dynamics model predicted individual neuron activity better than shuffle dynamics, aggregating over all (command, condition, neuron) tuples (Monkey G [J]: p-value < 0.001 for 9/9 [4/4] sessions, p-value < 0.001 for pooled sessions).

Figure 4D and 4E legends; Table S1 (rows 4D, 4E)

What the paper reports

• single neurons, each session: error compared with shuffle dynamics models, p < 0.001 (n = 9 of 9 sessions of Monkey G and 4 of 4 of Monkey J)
• population, each session: error compared with shuffle dynamics models, p < 0.001 (n = 9 of 9 sessions of Monkey G and 4 of 4 of Monkey J)
• single neurons and population, pooled over sessions: error compared with shuffle dynamics models, p < 0.001

Over all tuples (reading a), single-neuron error 1.241 (G) and 1.198 Hz (J), population error over the shuffle mean 0.892 and 0.833, all within 10%; below the shuffle with p < 0.001 in every session and pooled. Reading (b), tuples significant in fig3d, gives 1.295, 1.200, 0.868 and 0.810.

How the test differed from the paper’s method

• Tuple averages use the tuple's bins that have a previous bin in the trial (all tuples keep bins).
• Recovered decoder gain (see figs1b-conditions-per-command).

The model predicted neural activity for the vast majority of commands, conditions, and neurons (Figure 4G), revealing that dynamics were indeed invariant.

Results, Invariant dynamics predict the different neural activity patterns used to issue the same command, paragraph beginning "The invariant dynamics model also predicted"; Figure 4G

Mean of session fractions for tuples, commands and neurons, Monkey G then Monkey J; all above 0.5.

How the test differed from the paper’s method

• Recovered decoder gain (see figs1b-conditions-per-command).

All models predicted next neural activity significantly better than shuffle dynamics. For each model, Monkey G [J]: p-value < 0.001 for 9/9 [4/4] sessions, p-value < 0.001 for sessions pooled.

Figure 5C legend; Results, Invariant dynamics align with the decoder, propagating neural activity to issue the next command, paragraph 3; Table S1 (rows 5C)

What the paper reports

• each model, each session: R2 compared with shuffle dynamics models, p < 0.001 (n = 9 of 9 sessions of Monkey G and 4 of 4 of Monkey J)
• each model, pooled over sessions: R2 compared with shuffle dynamics models, p < 0.001

Scored as the code scores it, all eight R2 values are within 10%. Every model beats the shuffle scored on the same pairs pooled with p < 0.001 and in every session with p < 0.001, except decoder-null dynamics in Monkey G session 1 (p = 0.007), 8 of 9 as the criterion needs. On pairs whose command at t-1 is analysed, and on all pairs, decoder-null dynamics beat the shuffle in 7 of 9 G sessions, and on all pairs the full model's R2 (0.1194 and 0.1399) is outside tolerance. The outcome therefore rests on the code's scored pairs, whose shuffle 95th percentiles (0.0543 and 0.0527) match Table S1's 0.055 and 0.053.

How the test differed from the paper’s method

• Scored pairs from the code, as in fig4c-dynamics-predict-activity-given-command (the code removes pairs with a command above the 95th percentile at t or t-1 for next predictions too).
• The command left-out model is tested on pairs whose first bin issues the left-out command.
• Recovered decoder gain (see figs1b-conditions-per-command).

the full dynamics model was low-dimensional (Figure S3B) and predicted ~4 dimensions more accurately than the rest of neural activity (Figure 5D)

Results, Invariant dynamics align with the decoder, propagating neural activity to issue the next command, paragraph 3; Figure 5D

Under reading (a), dimensions with R2 above the session's pooled R2, the median over sessions is 4 (G) and 3 (J); under reading (b), above the shuffle's 95th percentile, 6 and 4.

How the test differed from the paper’s method

• The pooled R2 of reading (a) is the full model's on all bins.

The full dynamics model predicted condition-specific next command better than shuffle dynamics, aggregating over all (command, condition) tuples (Monkey G [J]: p-value < 0.001 for 9/9 [4/4] sessions, p-value < 0.001 for pooled sessions).

Figure 5F legend; Results, Invariant dynamics align with the decoder, propagating neural activity to issue the next command, paragraph 3; Table S1 (row labelled 5E)

What the paper reports

• each session: error compared with shuffle dynamics models, p < 0.001 (n = 9 of 9 sessions of Monkey G and 4 of 4 of Monkey J)
• pooled over sessions: error compared with shuffle dynamics models, p < 0.001

The prediction beats the shuffle with p < 0.001 in every session and pooled, and is significant for 0.805 (G) and 0.617 (J) of tuples on average. The errors, ten times the error in the command's own units as the amended criterion takes them, are 3.973 (G) and 7.052 (J), within tolerance of 3.956 and 7.324; in the command's own units they are 0.397 and 0.705. The pooled shuffle 5th percentiles are 5.011 (G) and 8.051 (J), against 5.38 and 9.240 in Table S1. The reader's answer in questions/000404.yaml settled the units (the code that prints Table S1 applies K to activity in Hz, counts per 100 ms bin times 10) and the claims file was amended to say so. Under the criterion as first written, in the command's own units, the outcome was partially reproduced; the tests and the fractions are unchanged, only the scale of the error and its shuffle.

How the test differed from the paper’s method

• Monkey J's commands are in cm/s summed over 20 frames of 5 ms (Methods; the code converts m to cm).
• Recovered decoder gain (see figs1b-conditions-per-command).

Incorporating invariant neural dynamics (cyan) significantly improves upon predictions from the model with all task-relevant variables

Figure S5B legend; Results, Invariant dynamics predict the different neural activity patterns used to issue the same command, paragraph beginning "On the level of single time points"

What the paper reports

• Monkey G: paired Student's t-test, T = -7.182, p = 9.4 × 10⁻⁵ (n = N = 18)
• Monkey J: paired Student's t-test, T = -5.141, p = 0.0143 (n = N = 8)

Adding previous activity raises the held-out R2 in 9 of 9 G sessions (paired T = -8.852, p = 2.09e-05) and 4 of 4 J sessions (T = -5.634, p = 0.0111); the paper gives T = -7.182 and -5.141.

How the test differed from the paper’s method

• Target position is the cursor at reward (inside the target); the categorical variable is the condition (target, task and direction around the obstacle) one-hot coded.

The residual is larger in the early trial period than the late trial period.

Figure S3F legend

What the paper reports

• Monkey G: linear mixed effect model, session as random effect, early vs late as fixed effect, slope = -0.242, t(4746) = -24.926, p = 3.9 × 10⁻¹³⁷
• Monkey J: linear mixed effect model, session as random effect, early vs late as fixed effect, slope = -0.0625, t(1161) = -5.354, p = 8.6 × 10⁻⁸

The held-out residual is smaller late than early, LME slope -0.0752 (z = -24.55, p = 3.97e-133, N = 2774) for Monkey G and -0.0261 (z = -4.63, p = 3.65e-06, N = 948) for Monkey J; the paper gives -0.242 and -0.0625. Every session's late mean is below its early mean except Monkey J session 1 (0.7656 against 0.7623).

How the test differed from the paper’s method

• The residual is of the next-activity prediction on held-out folds, its norm over the square root of the number of neurons, in counts per bin.

All commands in a particular angular bin are left out and predicted significantly better than shuffle dynamics

Figure S4B legend

What the paper reports

• angle bin left out, Monkey G and J sessions: R2 compared with shuffle dynamics models, p < 0.001 (n = 9 of 9 sessions of Monkey G and 3 of 4 of Monkey J (1 not significant))
• magnitude bin left out, each session: R2 compared with shuffle dynamics models, p < 0.001 (n = 9 of 9 sessions of Monkey G and 4 of 4 of Monkey J)
• each, pooled over sessions: R2 compared with shuffle dynamics models, p < 0.001

Scored as the code scores it, the angle left-out models give 0.1571 (G) and 0.2241 (J) and the magnitude left-out models 0.1589 and 0.2367, all within 10%. Against the shuffle on the same pairs the angle models beat it with p < 0.001 in 9 of 9 G and 3 of 4 J sessions (J session 3, p = 0.321), the magnitude models in 9 of 9 and 4 of 4, and both pooled. The other two readings give the same session counts and values within tolerance.

How the test differed from the paper’s method

• Scored pairs from the code, as in fig4c-dynamics-predict-activity-given-command.
• Recovered decoder gain (see figs1b-conditions-per-command).

All neural activity in a particular set of left-out conditions is left out and predicted significantly better than shuffle dynamics.

Figure S4G legend

What the paper reports

• each class, each session: R2 compared with shuffle dynamics models, p < 0.001 (n = 9 of 9 sessions of Monkey G and 4 of 4 of Monkey J (diagonal 2, Monkey G only))
• each class, pooled over sessions: R2 compared with shuffle dynamics models, p < 0.001

Scored as the code scores it, all seven R2 values are within 10% (vertical 0.1621 and 0.2458, horizontal 0.1556 and 0.2341, diagonal 1 0.1675 and 0.2053, diagonal 2 0.1593), and every class beats the shuffle scored on the same pairs with p < 0.001 in 9 of 9 G and 4 of 4 J sessions and pooled. The other two readings give the same tests; on all pairs five of the seven values are outside tolerance. The first run's failure of Monkey J's diagonal 1 class (2 of 4 sessions) came from comparing the class's R2 with the shuffle's R2 on all analysed pairs.

How the test differed from the paper’s method

• Scored pairs from the code, as in fig4c-dynamics-predict-activity-given-command.
• Classes as the code defines them; Monkey G's by target direction from the cursor at reward, Monkey J's by the code's center-out indices and obstacle trial types (Monkey J's center-out targets 3 and 7 and obstacle type 7 are in no class, as in the code).

Dynamics-predicted distances across condition-pairs are significantly correlated with true distances

Figure S6C and S6D legends; Results, Invariant dynamics predict the different neural activity patterns used to issue the same command, paragraph beginning "Finally, the dynamics model preserved structure"

What the paper reports

• single neurons, Monkey G, pooled: linear regression (Wald test), slope = 0.221, r = 0.584, N = 377489, p < 0.001
• single neurons, Monkey J, pooled: linear regression (Wald test), slope = 0.317, r = 0.693, N = 34643, p < 0.001
• population, Monkey G, pooled: linear regression (Wald test), slope = 0.350, r = 0.638, N = 5112, p < 0.001
• population, Monkey J, pooled: linear regression (Wald test), slope = 0.449, r = 0.800, N = 1714, p < 0.001

With the code's pairwise command matching (7310 of 7925 pairs of conditions keep 15 or more occurrences of each), pooled r of predicted on true distance is 0.586 and 0.655 (neurons) and 0.650 and 0.766 (population), with slopes 0.225, 0.301, 0.357 and 0.434 (paper 0.221, 0.317, 0.350, 0.449). Every session and both pooled regressions give a positive slope with p < 0.001. N is 399524 and 37627 neuron pairs, 5448 and 1862 population pairs (paper 377489, 34643, 5112, 1714). Without the matching r is 0.594, 0.680, 0.646 and 0.789.

How the test differed from the paper’s method

• Recovered decoder gain (see figs1b-conditions-per-command). Each pair's means are over the matched bins that have a previous bin in the trial.

Over all commands, command subtrajectory distance across a condition-pair is significantly correlated with population activity distance

Results, The same command is issued by different neural activity patterns in different movements, paragraph 2; Figure S6G legend

What the paper reports

• true distances, Monkey G, pooled: linear mixed effect model, command identity and session as random effects, N = 4674, z = 31.04, p = 1.4 × 10⁻²¹¹
• true distances, Monkey J, pooled: linear mixed effect model, command identity and session as random effects, N = 1629, z = 14.18, p = 1.2 × 10⁻⁴⁵
• dynamics-predicted distances, Monkey G, pooled: linear mixed effect model, command identity and session as random effects, N = 4674, z = 33.1, p = 3.9 × 10⁻²⁴⁰
• dynamics-predicted distances, Monkey J, pooled: linear mixed effect model, command identity and session as random effects, N = 1629, z = 22.5, p = 5.9 × 10⁻¹¹²

Pooled LME slopes are positive with p < 0.001 for true distances (G slope 0.251, z = 38.69, N = 5509; J 0.129, z = 16.66, N = 1844) and predicted distances (z = 42.27 and 24.37); the paper gives z = 31.04 and 14.18 (true), 33.1 and 22.5 (predicted).

How the test differed from the paper’s method

• Session as the grouping factor with command as a variance component nested in it (statsmodels mixedlm); no pairwise command matching.

Dynamics-predicted distance between next commands across a condition-pair is significantly correlated with the true distance

Figure S6K legend; Results, Invariant dynamics align with the decoder, propagating neural activity to issue the next command, paragraph 3

What the paper reports

• Monkey G, pooled: linear regression, slope = 0.263, r = 0.76, p < 0.001 (n = significant in 8 of 9 sessions)
• Monkey J, pooled: linear regression, slope = 0.178, r = 0.63, p < 0.001 (n = significant in 4 of 4 sessions)

Pooled r is 0.774 (G) and 0.654 (J), slopes positive, p < 0.001; all 9 G sessions (session 7 at p = 0.046) and 4 of 4 J sessions are significant at p < 0.05.

Condition-specific next commands are overall significantly different from the condition-pooled next command: Monkey G [J]: p-value < 0.01 for 9/9 [4/4] sessions, p-value < 0.001 for 8/9 [4/4] sessions, p-value < 0.001 pooled over sessions (mean of next command distances = 1.676 [2.178]

Figure S1E legend

What the paper reports

• each session: comparison with a bootstrapped distribution from the condition-pooled next commands, p < 0.01 (n = 9 of 9 sessions of Monkey G and 4 of 4 of Monkey J (p < 0.001 in 8 of 9 and 4 of 4))
• pooled over sessions, each monkey: comparison with a bootstrapped distribution from the condition-pooled next commands, p < 0.001

Every session and both pooled tests give p < 0.001. Monkey J's mean normalized next-command distance, 2.218, is within 10% of 2.178; Monkey G's, 1.873, is above the upper bound of 1.844. Counting a tuple only when 15 or more occurrences remain after a trial's last bin is removed gives 1.870 (G, still outside) and 2.236 (J), with the same tests.

How the test differed from the paper’s method

• Commands from the recovered decoder gain (see figs1b-conditions-per-command).

Distances were significantly larger than shuffle distances for a large fraction of individual (command, condition) tuples (~30% for Monkey G, ~70% for Monkey J), individual commands (~65% for G, ~90% for J) when aggregating over conditions, and individual neurons (~40% for G, ~80% for J)

Results, The same command is issued by different neural activity patterns in different movements, paragraph 2; Figure 3E

Measured under the behavior-preserving shuffle, commands with two or more conditions, neurons over all tuples. Every combination of readings puts three or four of the six fractions within tolerance, none all six. Three with the shuffle null and neurons over significant tuples (Monkey G's neurons then 0.455 and 0.461), four under every other combination. Monkey G's fraction of significant tuples is 0.373 to 0.375 under every reading (claim 0.27 to 0.33). Monkey J's neurons are 0.890 or 0.903 over all tuples (claim 0.72 to 0.88), 0.878 over population-significant tuples with the subsampling null, where Monkey G's neurons are then 0.471 (claim 0.36 to 0.44). Commands are 0.672 to 0.700 (G) and 0.921 to 0.950 (J), tuples for J 0.707 to 0.710, neurons for G over all tuples 0.401 to 0.409, all within tolerance. All fractions are far above 0.05. The second reading, fractions pooled over all items of a monkey's sessions, also gives four of six within tolerance under every combination; Monkey G's tuples are 0.389 to 0.392 under it.

How the test differed from the paper’s method

• Commands from the recovered decoder gain (see figs1b-conditions-per-command).

It predicted 20–40% of the condition-specific component of neural activity

Results, Invariant dynamics predict the different neural activity patterns used to issue the same command, paragraph beginning "The invariant dynamics model also predicted"; Figure 4F; Table S1 (row 4F)

What the paper reports

• each session: variance explained compared with shuffle dynamics models, p < 0.001 (n = 9 of 9 sessions of Monkey G and 4 of 4 of Monkey J)
• pooled over sessions: variance explained compared with shuffle dynamics models, p < 0.001

Reading (a) gives 0.215 (G) and 0.303 (J), reading (b) 0.243 and 0.333, all within 10% of 0.226 and 0.330, against shuffle means of -0.010 to 0.005, with p < 0.001 in every session and pooled. The criterion's per-session range fails. Under reading (a) Monkey G sessions 5, 6, 7 and 8 (0.121 to 0.160) and Monkey J session 3 (0.155) are below 0.18; under reading (b) Monkey G sessions 5 to 8 (0.130 to 0.168) are. The criterion has no partial clause; partially reproduced is a judgment, as the means and tests hold and only the per-session range fails.

How the test differed from the paper’s method

• Recovered decoder gain (see figs1b-conditions-per-command).

All models except decoder-null dynamics predicted next command significantly better than shuffle dynamics.

Figure 5E legend; Results, Invariant dynamics align with the decoder, propagating neural activity to issue the next command, paragraph 3; Table S1 (rows labelled 5D)

What the paper reports

• full and command left-out dynamics, each session: R2 compared with shuffle dynamics models, p < 0.001 (n = 9 of 9 sessions of Monkey G and 4 of 4 of Monkey J)
• condition left-out dynamics, each session: R2 compared with shuffle dynamics models, p < 0.05 (n = 3 of 4 sessions of Monkey J (p < 0.001 in 2), 9 of 9 of Monkey G at p < 0.001)
• full, command and condition left-out dynamics, pooled over sessions: R2 compared with shuffle dynamics models, p < 0.001

Scored as the code scores it, every R2 is within tolerance. The full and command left-out models beat the shuffle with p < 0.001 in 9 of 9 and 4 of 4 sessions and pooled; decoder-null dynamics in no session. The condition left-out model beats it in 9 of 9 G sessions and pooled for G, but for Monkey J in 2 of 4 sessions (p = 0.560 and 1.000 in sessions 1 and 3) and pooled with p = 0.003, where the paper gives p < 0.001. On pairs whose command at t-1 is analysed the tests hold (pooled p < 0.001) but Monkey J's 0.2535, 0.2372 and 0.2326 are above the upper bounds 0.233, 0.232 and 0.212; on all pairs they are higher still. Every reading falls short in one part, so the outcome does not rest on the choice of pairs.

How the test differed from the paper’s method

• Scored pairs from the code, as in fig5c-next-activity.
• Recovered decoder gain (see figs1b-conditions-per-command).

Specifically, the dynamics model predicted whether the turn would be clockwise or counter clockwise (Figure 5H left) and the angle of turn (Fig 5H right) better than shuffle dynamics.

Results, Invariant dynamics align with the decoder, propagating neural activity to issue the next command, paragraph 4; Figure 5H; Table S1 (rows labelled 5G)

What the paper reports

• sign of turn and angle error, each session: compared with shuffle dynamics models, p < 0.001 (n = 9 of 9 sessions of Monkey G and 4 of 4 of Monkey J)
• sign of turn and angle error, pooled over sessions: compared with shuffle dynamics models, p < 0.001

All four values are within 10%. The angle error beats the shuffle in every session and pooled (p < 0.001). The sign beats it pooled and in 4 of 4 J sessions with p < 0.001, and in 7 of 9 G sessions (p = 0.005 and 0.001 in sessions 6 and 7), where the criterion needs 8. The partial clause covers values outside tolerance with the tests holding, not the reverse; partially reproduced here is a judgment, as one test falls one session short.

How the test differed from the paper’s method

• The angle error is taken modulo 360 degrees (the smaller of the two differences).
• Recovered decoder gain (see figs1b-conditions-per-command).

The dimensionality of the full dynamics matrix (decoder-null dynamics matrix) ranged from 2-4 (2-5) when considering the eigenvalues that had timescales greater than the BMI update’s timescale of 0.1 seconds.

Results, Invariant dynamics predict the different neural activity patterns used to issue the same command, paragraph beginning "We found that the dynamics were low-dimensional"; Figure S3B and S3C legends

Long-timescale eigenvalues number 2 to 5 for the full model in Monkey G (5 in session 4, outside 2 to 4) and 2 to 3 in Monkey J; for decoder-null dynamics 2 to 5 in G and 1 to 4 in J (1 in session 3, outside 2 to 5). Monkey G's mean frequencies are 0.083 Hz (full) and 0.068 Hz (decoder-null), outside 0.09 to 0.11 and 0.054 to 0.066. Monkey J's long-timescale eigenvalues are real except one pair in session 0 at 0.036 Hz. Decay times are at most 0.38 s. The 99% step changes no count. Fitting the full model on rates in Hz with the same ridge grid leaves every full-model count unchanged and Monkey G's mean frequency at 0.083 Hz; the ridge fit is invariant to the scale where the grid holds the scaled parameter. That variant refits only the full model, so the decoder-null counts are not tested under it.

How the test differed from the paper’s method

• In-sample ridge fits on all pairs with the session's cross-validated parameter; decoder-null dynamics fit on activity projected with the recovered decoder's null space.

for all four analyzed sessions, neural activity mapped through could account for 99.6, 99.6, 99.5, and 99.8 percent of the variance of the command respectively

Methods, Command definition, Monkey J

The criterion needs the point process filter's velocity update or its parameters and says the claim is blocked when the dataset has neither; the files hold neither (see findings). The notebook shows estimates in which the cursor dynamics are fitted jointly with the map, which favours the fit, so none is the criterion's test. Over all frames with the offset column in the fitted dynamics the map accounts for 98.92, 98.89, 99.57 and 99.25%; over the frames of rewarded trials 99.63, 99.63, 99.57 and 99.80% with the offset column and 99.10, 99.09, 99.09 and 99.63% without it; with the update taken as the change in velocity, 87.05 to 93.89%. The paper gives 99.6, 99.6, 99.5 and 99.8%.

How the test differed from the paper’s method

• Not tested. The estimated readings fit the cursor dynamics jointly with the map on the 5 ms decoder state, as the files hold no PPF parameters.

These simulations revealed that the population required significantly less input in the Full Dynamics Model than in the No Dynamics Model (Figure 6C).

Figure 6C legend; Results, An OFC model reveals that invariant dynamics reduce the input, paragraph 4; Table S2

What the paper reports

• Monkey G, pooled over sessions: linear mixed effect model, session as random effect, z = 10.49, N = 432, p = 9.7 × 10⁻²⁶
• Monkey J, pooled over sessions: linear mixed effect model, session as random effect, z = 5.20, N = 192, p = 1.9 × 10⁻⁷
• each session: Wilcoxon signed-rank test with conditions paired, p < 0.05 (n = 9 of 9 sessions of Monkey G and 4 of 4 of Monkey J)

The Full Dynamics Model needs less input than the No Dynamics Model in every session (ratio 0.44 to 0.91; Wilcoxon p < 0.05 in 9 of 9 and 4 of 4 sessions); LME z = -11.27 (p = 1.87e-29, N = 432) for G and -14.94 (p = 1.87e-50, N = 192) for J. Every simulated trial ends within 0.365 cm of its target. With Monkey J's velocity gain fixed to the summed 5 ms gain (sensitivity reading), the effect holds (LME z = -16.61, Wilcoxon p < 0.05 in 4 of 4 sessions).

How the test differed from the paper’s method

• Assumption for Monkey J. The paper builds Monkey J's 100 ms decoder from the PPF's parameters, not in the files; here its 100 ms cursor update is the least-squares update of the decoder state over each 100 ms bin (variance accounted for above 99%). Monkey G's is recovered exactly. The notebook also shows Monkey J with the velocity gain fixed to the summed 5 ms gain and the velocity rows of the cursor dynamics refitted.
• The terminal weight 100 is the code's (behavior_co_obs.py); the dynamics are in-sample ridge fits on counts per bin.

There was no significant difference in input magnitude between Decoder-null Dynamics (pink ‘D’) and No Dynamics (black ‘ND’) when pooling across sessions (Monkey G [J] p-value > 0.05) and on individual sessions

Figure 6D legend; Table S2

What the paper reports

• Monkey G, pooled over sessions: linear mixed effect model, session as random effect, z = 0.002, N = 432, p = 0.998
• Monkey J, pooled over sessions: linear mixed effect model, session as random effect, z = -0.003, N = 192, p = 0.99

Claim of absence, judged on the paper's tests. The LME finds no difference (G z = -0.00, p = 0.998; J z = 0.02, p = 0.988) and no session's Wilcoxon test does (smallest p 0.069); the input ratios are 0.998 to 1.003. With Monkey J's velocity gain fixed to the summed 5 ms gain (sensitivity reading), the LME still finds no difference (p = 0.995) but Monkey J session 1's Wilcoxon test does (p = 0.0016), so under that reading the criterion, which fails on any session's Wilcoxon p < 0.05, would give not reproduced. The outcome therefore rests on the decoder reading for Monkey J; the fitted 100 ms update, which accounts for more of the 100 ms velocity (0.995 to 0.998 against 0.984 to 0.994), stays the adopted reading.

How the test differed from the paper’s method

• As fig6c-dynamics-reduce-input.

The population activity distances for (command, condition) tuples were significantly larger than shuffle in the Full Dynamics Model but not in the No Dynamics Model (Figure 6FG).

Figure 6G and 6H legends; Results, An OFC model reveals that invariant dynamics reduce the input, paragraph 5; Table S2

What the paper reports

• Full Dynamics Model against shuffle, Monkey G, pooled: linear mixed effect model, session as random effect, z = -23.09, N = 4906, p = 6.4 × 10⁻¹¹⁸
• Full Dynamics Model against shuffle, Monkey J, pooled: linear mixed effect model, session as random effect, z = -16.68, N = 2408, p = 1.8 × 10⁻⁶²
• No Dynamics Model against shuffle, Monkey G, pooled: linear mixed effect model, session as random effect, z = 0.168, N = 4334, p = 0.866
• No Dynamics Model against shuffle, Monkey J, pooled: linear mixed effect model, session as random effect, z = 0.462, N = 2188, p = 0.644
• Decoder-null Dynamics Model against shuffle, Monkey G, pooled: linear mixed effect model, session as random effect, z = 0.932, N = 4488, p = 0.351
• Decoder-null Dynamics Model against shuffle, Monkey J, pooled: linear mixed effect model, session as random effect, z = -1.490, N = 2252, p = 0.136
• No Dynamics Model of the decoder-null comparison against shuffle, Monkey G, pooled: linear mixed effect model, session as random effect, z = 0.611, N = 4482, p = 0.541
• No Dynamics Model of the decoder-null comparison against shuffle, Monkey J, pooled: linear mixed effect model, session as random effect, z = 0.449, N = 2250, p = 0.654
• Full Dynamics Model, each session: Mann-Whitney U test, p < 0.05 (n = 9 of 9 sessions of Monkey G and 4 of 4 of Monkey J (none for the other models))

Under both nulls (1000 draws), the Full Dynamics Model's distances exceed the null (LME z = 25.4 to 26.0 for G, 15.5 to 15.7 for J; Mann-Whitney p < 0.05 in 9 of 9 and 4 of 4 sessions). Both No Dynamics models show no difference (LME p 0.83 to 0.999; no session significant). Decoder-null dynamics show none for Monkey G (p = 0.18) or in any session, but Monkey J's LME finds one (effect 0.016, z = 2.30 to 2.33, p = 0.020 to 0.021, mean normalized distance 1.09), so that absence is not reproduced. The LME here codes the simulation as 1 and the null as 0, so its z has the opposite sign to the paper's. With Monkey J's velocity gain fixed to the summed 5 ms gain, the result is the same: the decoder-null LME finds a difference (p = 0.0059 and 0.0051) and the other tests hold.

How the test differed from the paper’s method

• As fig6c-dynamics-reduce-input. Commands of the simulations are discretized with the simulations' own magnitude percentiles.