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Manifold-constrained plasticity enables stable learning in recurrent neural circuits.

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The 5 matches · 3 of them tie a paragraph to a whole file, not to given lines: weak matches, whose lines are not tinted
  1. [1] § Results › Decoder performance depends on alignment with the intrinsic activity manifold ↔ SPLIT2.m, lines 1–140 · score 0.55 · random rotation, outside manifold, QR, orthogonal, matrix, orthonormal
  2. [2] § Results › Manifold-constrained learning stabilizes decoder adaptation in low-dimensional networks ↔ SPLIT3.m, the whole file · a weak match · score 0.53 · neural space, low dimensional manifold, high dimensional, decoder updates, reconstruct, adaptation
  3. [3] § Results › Manifold-constrained learning stabilizes decoder adaptation in low-dimensional networks ↔ SPLIT1.m, the whole file · a weak match · score 0.53 · neural space, low dimensional manifold, high dimensional, decoder updates, reconstruct, adaptation
  4. [4] § Results › SPLiT outperforms existing drift-adaptation rules under constrained dynamics ↔ ideal_observer1.m, the whole file · a weak match · score 0.53 · postsynaptic neuron, inverse correlation, readout, dynamics
  5. [5] § Results › Decoder performance depends on alignment with the intrinsic activity manifold ↔ SPLIT2.m, lines 1–140 · score 0.52 · orthogonal rotation, latent coordinates, remapped, subspace, perturbations, activity

Paper

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The authors' code

MATLAB · 259 lines · 9.3 KB · no license · 2 matches

  1. clear all;
  2. warning off all
  4. rng(1);
  6. % == Parameters ===
  7. N = 25; %(25)
  8. dt = 0.01; %0.01
  9. tau = 0.3; %(0.3)
  10. eta_r = 0.001; % step size for random drift in activity (0.001)
  11. proj_dim = 1; % dimension of low-D manifold (1)
  12. g = 1.5; % gain for initial weights (1.5)
  13. alpha = 1.5; % feedback strength (1.5)
  14. eta_U = 0.01; %Oja manifold adaptation rate (0.01)
  15. feedback_ratio = 1; %increase when increasing proj_dim (1)
  16. lambda = 0.1; %decoder learning rate (0.1)
  18. use_within_manifold = false;
  19. perturb_mode = 'outside'; % 'within' or 'outside' or 'none'
  20. theta = pi/2; % "outside-ness": 0 = fully within; pi/2 = fully orthogonal
  22. target_noise_var = 0;
  24. % Target output: center out task
  25. centerout;
  26. z2 = repmat(z_target,1,3);
  27. z_target = z2;
  29. steps = size(z_target,2);
  30. T = steps*dt;
  32. % === Fixed manifold basis ===
  33. U_fixed = orth(randn(N, proj_dim)); % low-D manifold
  35. % === Initialization ===
  36. x = 0.1 * randn(N,1);
  37. r = tanh(x);
  38. s = U_fixed' * r; % low-D coordinates if using manifold
  40. % Fixed recurrent weights
  41. W = g * randn(N, N) / sqrt(N);
  42. W_red = U_fixed' * W * U_fixed; % projected weights for low-D dynamics
  43. desired_rho = 1;
  44. rho_red = max(abs(eig(W_red)));
  45. W_red = W_red / max(1, rho_red / desired_rho);
  47. % Low-D basis (Oja's rule)
  48. U = orth(randn(N, proj_dim));
  49. D_proj = zeros(2, proj_dim); % decoder for 2D output
  51. % Logging
  52. r_log = zeros(N, steps);
  53. z_out = zeros(2, steps);
  55. % ============================
  56. % Perturbation configuration
  57. % ============================
  58. out_dim = size(z_target,1); % 1 for sine, 2 for cursor, etc.
  59. D_proj = zeros(out_dim, proj_dim); % decoder
  61. z_out = zeros(out_dim, steps);
  63. k_pert = round(steps/2); % apply perturbation halfway
  65. % --- Within-manifold: latent coordinate remapping P ---
  66. % Use either a permutation or a random rotation.
  67. P = eye(proj_dim); % no perturbation initially
  68. use_rotation = true; % true = random orthogonal rotation; false = permutation
  70. % --- Outside-manifold: build a subspace U_perp orthogonal to U_fixed ---
  71. % (same dimension as manifold)
  72. U_perp = randn(N, proj_dim);
  73. U_perp = U_perp - U_fixed*(U_fixed' * U_perp); % remove components in span(U_fixed)
  74. U_perp = orth(U_perp); % orthonormalize
  76. % --- I/O basis used for decoding + feedback injection (starts unperturbed) ---
  77. U_io = U; % by default, same as your learned Oja basis
  79. % === Freeze Oja for a short window after perturbation ===
  80. oja_freeze_start = 1601;
  81. oja_freeze_end = 1600*2;
  83. freeze_dec_len = round(0.1*steps); % x% window
  84. dec_freeze_end = oja_freeze_end;
  85. lambda0 = lambda;
  87. % === Weight-change logging ===
  88. dD_norm = zeros(1, steps); % magnitude of decoder update
  89. for k = 1:steps
  91. % ================================================================
  92. % Apply perturbation at k_pert (BCI mapping change)
  93. % ================================================================
  94. if k == k_pert
  95. switch lower(perturb_mode)
  96. case 'within'
  97. % Keep U_io the same, change only latent coordinates via P
  98. if use_rotation
  99. % Random orthogonal rotation: P'P = I
  100. [Q,~] = qr(randn(proj_dim));
  101. P = Q;
  102. D_proj = D_proj * P'; %optional
  103. else
  104. % Permutation matrix
  105. p = randperm(proj_dim);
  106. P = eye(proj_dim);
  107. P = P(:,p);
  108. end
  110. case 'outside'
  111. % Keep P = I; change the basis used for readout/feedback to include orthogonal component
  112. P = eye(proj_dim);
  114. % Mix current U with an orthogonal-to-U_fixed subspace
  115. % (and re-orthonormalize)
  116. U_io = orth(cos(theta)*U + sin(theta)*U_perp);
  118. otherwise
  119. % 'none'
  120. P = eye(proj_dim);
  121. U_io = U;
  122. end
  123. use_within_manifold = false; %MODIFIED HERE: SETTING OUTSIDE MANIFOLD AFTER PERTURBATION
  124. end
  127. if use_within_manifold
  128. % ================================================================
  129. % 1) Compute feedback from current activity (pre-update)
  130. % ================================================================
  131. % Current low-D readout coordinates (for decoding only)
  132. y_fb = U_io' * r; % proj_dim x 1
  133. y_fb_eff = P * y_fb; % perturbed latent coords (within-manifold)
  134. z_fb = D_proj * y_fb_eff; % out_dim x 1
  136. % Mix target and current output (same as before)
  137. feedback = (1 - feedback_ratio) * z_fb + feedback_ratio * z_target(:,k);
  138. feedback = min(max(feedback, -2), 2);
  140. %MODIFIED HERE: SETTING FEEDBACK TO ZERO
  141. %%%%%%%%%%%%%%%%%%%%%
  142. % feedback = zeros(1,2)';
  143. %%%%%%%%%%%%%%%%%%%%%
  145. % Map feedback back into neural space, then project into manifold
  146. % Effective decoder under within-manifold perturbation:
  147. % z = D_proj * (P * (U_io' r)) => D_eff = D_proj * P
  148. D_eff = D_proj * P;
  150. I_fb_full = alpha * U_io * (D_eff') * feedback; % N x 1
  151. I_fb_low = U_fixed' * I_fb_full; % proj_dim x 1
  154. % ================================================================
  155. % 2) Low-dimensional manifold dynamics for s
  156. % ================================================================
  157. d_s_rec = (-s + W_red * s) / tau; % recurrent term
  158. d_s_noise = (eta_r * sqrt(N/proj_dim)) * randn(proj_dim,1); % noise
  160. % Add feedback as an extra driving term in low-D
  161. s = s + dt * (d_s_rec + I_fb_low) + d_s_noise;
  163. % ================================================================
  164. % 3) Reconstruct full activity strictly within manifold
  165. % ================================================================
  166. x = U_fixed * s; % latent variable along manifold
  167. r = tanh(x); % N x 1 activity on manifold
  169. % Optional numerical safety on r/x (not used for dynamics here)
  170. r_clipped = max(min(r, 0.999), -0.999);
  171. x = atanh(r_clipped);
  173. else
  174. % ================================================================
  175. % Unconstrained high-dimensional dynamics (original branch)
  176. % ================================================================
  177. d_r = eta_r * randn(N,1);
  178. r = tanh(r + dt * (-r + W * r) / tau + d_r);
  180. r_clipped = max(min(r, 0.999), -0.999);
  181. x = atanh(r_clipped);
  182. end
  184. % ================================================================
  185. % 4) Oja's Rule for low-D decoder basis U
  186. % Freeze temporarily after perturbation
  187. % ================================================================
  188. if ~(k >= oja_freeze_start && k < oja_freeze_end)
  190. r_centered = r - mean(r);
  191. for i = 1:proj_dim
  192. u = U(:,i);
  193. % (r_proj/var_proj not needed for your delta_u form; can remove if you want)
  194. delta_u = eta_U * (r_centered * (r_centered' * u) - (u' * u) * u);
  195. U(:,i) = u + delta_u;
  196. end
  197. U = orth(U); % re-orthonormalize decoder basis
  198. end
  200. % Keep default I/O basis tied to current U unless we're in "outside" mode
  201. % where U_io is explicitly set at perturbation and should remain fixed.
  202. if ~strcmpi(perturb_mode,'outside')
  203. U_io = U;
  204. end
  207. % ================================================================
  208. % 5) Update decoder D_proj using OP-LMS (unchanged)
  209. % Uses updated r (post-dynamics) but same time index k
  210. % ================================================================
  211. y = U_io' * r; % proj_dim x 1
  212. y_eff = P * y; % perturbed latent coords
  213. z = D_proj * y_eff; % out_dim x 1
  214. z_out(:,k) = z;
  216. if k >= k_pert && k < dec_freeze_end
  217. lambda_eff = 0; % no learning immediately after perturbation
  218. else
  219. lambda_eff = lambda0; % relearn afterwards
  220. end
  222. err = z - z_target(:,k) + randn(out_dim,1).*target_noise_var;
  224. y_eff_norm = y_eff / (norm(y_eff) + 1e-8);
  225. dD = -lambda_eff * err * (y_eff_norm)'; % out_dim x proj_dim
  227. D_proj = D_proj + dD;
  228. dD_norm(k) = norm(dD, 'fro');
  231. % ================================================================
  232. % 6) Logging
  233. % ================================================================
  234. r_log(:,k) = r;
  235. end
  237. z = z_target(:,1600*2+1:end);
  238. o = z_out(:,1600*2+1:end);
  239. final_err = mean((z(:)-o(:)).^2);
  241. %plot center-out task
  243. %initial learning
  244. figure;
  245. plot(z_target(1,1:1600), z_target(2,1:1600), 'Color', [181 59 59]./255,'linewidth',12);
  246. hold on;
  247. plot(z_out(1,1:1600), z_out(2,1:1600),'Color', [158 159 43]./255,'linewidth',6);
  248. axis off
  250. %recovery after perturbation
  251. figure;
  252. plot(z_target(1,1600*2+1:end), z_target(2,1600*2+1:end), 'Color', [181 59 59]./255,'linewidth',12);
  253. hold on;
  254. plot(z_out(1,1600*2+1:end), z_out(2,1600*2+1:end),'Color', [158 159 43]./255,'linewidth',6);
  255. axis off

SPLIT2.m, no license · at the source

Overview

Authors: Camille Godin1, Jean-Philippe Thivierge1,2
  1. School of Psychology, University of Ottawa, Ottawa, Canada
  2. Brain and Mind Research Institute, University of Ottawa, Ottawa, Canada
Institutions: University of Ottawa (Canada); uOttawa Brain and Mind Research Institute
Journal: PLoS computational biology, volume 22, issue 8, article e1014719
Dates: received 2 March 2026; accepted 17 August 2026; published online 25 August 2026
Type: Research article · Language: English
License: CC BY
Identifiers: DOI 10.1371/journal.pcbi.1014719 · PMID 42640956 · PMCID PMC13529248 · OpenAlex W7204199017
Open access: gold, a free copy (OpenAlex)
Status: code verified
Categories: human (organism)
Methods: Machine learning, Connectivity
MeSH: Learning*, Models, Neurological*, Nerve Net*, Neural Networks, Computer*, Neuronal Plasticity*, Algorithms, Animals, Computational Biology, Humans, Neurons, Recurrent Neural Networks (* major topic)
Topic: Advanced Memory and Neural Computing (Electrical and Electronic Engineering, Engineering), according to OpenAlex
Funding: Natural Sciences and Engineering Research Council of Canada (NSERC); natural sciences and engineering council of canada (210977)
Citations: not cited yet (Europe PMC); 85 references in the paper

Abstract

The activity of large neuronal populations is often confined to low-dimensional manifolds that can drift over time, posing a challenge for learning rules that assume stable, full-rank representations. Here, we introduce SPLiT (Synaptic Projection Learning with intrinsic Tracking), a synaptic plasticity rule for recurrent neural networks that combines unsupervised manifold tracking with supervised learning. SPLiT uses an online Oja rule to continuously estimate the intrinsic low-dimensional activity subspace and performs normalized least-mean-squares learning in manifold coordinates. In this way, SPLiT keeps synaptic weights aligned with evolving population dynamics. Using a rate-based recurrent network, we show that SPLiT reliably learns time-varying target signals under both constrained dynamics, where activity is restricted to a fixed low-dimensional manifold, and unconstrained dynamics exhibiting changes in the dominant activity subspace. We show that manifold-constrained learning with SPLiT yields faster convergence, less sensitivity to recurrent gain, smaller weight updates, and is more robust to noisy teaching signals. Analytical results show that SPLiT learns the optimal decoder projected onto the instantaneous principal subspace and maintains bounded error under drift. Together, these findings provide a mechanistic account of how synaptic plasticity can leverage the structure of neural manifolds to enable stable and efficient supervised learning despite representational drift.

Reproduced under the paper's license (CC BY), from the paper cited above.

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supp:PMC13529248/pcbi.1014719.s002.zip

License: none: the authors keep all their rights
State: the link answers, verified on 27 September 2026
Evidence: files inventoried
Languages: MATLAB (9)
Size: 10 files, 9 scripts
Software Heritage: not checked
Found in: the supplementary material
Holds: README
Not found: license file, CITATION.cff, environment file, tests, continuous integration, documentation
Availability: 1 check, the latest on 27 September 2026: the link answers (HTTP 200)
  • 27 September 2026: the link answers (HTTP 200)
10 files
At the source: pmc-oa-opendata.s3.amazonaws.com/PMC13529248.1/pcbi.1014719.s002.zip

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  • 5 matches between paragraphs of the paper and lines of the code (method lexical-v1);
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Data Availability

All relevant data are within the manuscript and its Supporting Information files.

Reproduced under the paper's license (CC BY), from the paper cited above.

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Version 1, 27 September 2026: the first record

Recorded: type, language, journal, volume, issue, pages, dates, 2 authors, 11 MeSH terms, 2 funders, 80 references.

Cite

This paper

Godin, C., & Thivierge, J.-P. (2026). Manifold-constrained plasticity enables stable learning in recurrent neural circuits. PLoS computational biology, 22(8), e1014719. https://doi.org/10.1371/journal.pcbi.1014719

BibTeX

@article{godin2026manifold,
author = {Godin, Camille and Thivierge, Jean-Philippe},
title = {{Manifold-constrained plasticity enables stable learning in recurrent neural circuits}},
journal = {PLoS computational biology},
year = {2026},
month = aug,
volume = {22},
number = {8},
pages = {e1014719},
publisher = {PLOS},
issn = {1553-734X},
doi = {10.1371/journal.pcbi.1014719},
url = {https://doi.org/10.1371/journal.pcbi.1014719},
pmid = {42640956},
pmcid = {PMC13529248}
}

RIS

TY - JOUR
AU - Godin, Camille
AU - Thivierge, Jean-Philippe
TI - Manifold-constrained plasticity enables stable learning in recurrent neural circuits
T2 - PLoS computational biology
J2 - PLoS Comput Biol
PY - 2026
DA - 2026/08/25
VL - 22
IS - 8
SP - e1014719
SN - 1553-734X
PB - PLOS
DO - 10.1371/journal.pcbi.1014719
UR - https://doi.org/10.1371/journal.pcbi.1014719
LA - en
ER -

CSL-JSON

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