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Data-driven reduced modeling of neural dynamics.

Code ↔ Paper

4 matches between paragraphs of the paper and lines of its authors' code, computed by the harvester (lexical-v1). Click a colored paragraph or line to see its counterpart.

The 4 matches
  1. [1] § Results › A model for context-dependent decision-making ↔ src/rnn_paper/SSMfunctions.py, lines 93–102 · score 0.58 · Manifold Fitting Error, trajectories projected, spectral subspace, MFE, RNN, SSM
  2. [2] § Methods › The Vanilla RNN models › Multitasking RNN: memory-pro task ↔ mt demo.ipynb, lines 42–84 · score 0.57 · Memory Pro task, memory period, stimulus, vector, sensory
  3. [3] § Methods › The theory of spectral submanifolds › Data-driven computations of SSMs ↔ src/rnn_paper/SSMfunctions.py, lines 93–102 · score 0.52 · manifold fitting error, lifts, MFE, trajectories, SSM
  4. [4] § Methods › The Vanilla RNN models › Sine-wave generator RNN ↔ swgRNN/utils/UtilsSamplingLocs.py, lines 140–171 · score 0.50 · sine wave generator, network, RNN

Paper

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The authors' code

Python · 289 lines · 11 KB · MIT · 2 matches

  1. import numpy as np
  2. import scipy as sp
  3. import sympy as sym
  4. from IPython.display import display, Math, Latex
  5. from matplotlib import pyplot as plt
  6. from rnn_paper.utils import GenerateSymbols
  8. # Given coefficients and exponents of the polynomial expansion, computes the polynomial expansion
  9. def evaluate_polynomial(C, I, P, symbolic = False ):
  11. # C is the coefficient (dim(SSM) \times m_order)-matrix
  12. # I is the exponent (m_order \times dim(SSM))-matrix
  13. # P is the d-dimensional vector to expand in monomials
  15. if len(np.shape(P))!= 1:
  16. PHI = np.ones((np.shape(I)[0],*np.shape(P)[1:]), dtype = complex)
  17. else:
  18. PHI = np.ones((np.shape(I)[0]),dtype =complex)
  19. if symbolic:
  20. PHI = sym.MutableDenseNDimArray(PHI)
  21. C = np.round(C, 3)
  22. for i, exp in enumerate(I):
  24. for j in range(np.shape(P)[0]):
  25. PHI[i] = PHI[i]*np.power(P[j], exp[j])
  26. if len(np.shape(P))!= 1:
  27. return (np.tensordot(C, PHI, axes = 1))
  28. else:
  29. return np.dot(C, PHI)
  32. # Given coefficients and exponents of the (autonomous) SSM, computes the SSM expansion
  33. def construct_SSM(y, mfldCoeffs, mfldExps, symbolic = False):
  35. if symbolic == True:
  36. return evaluate_polynomial(mfldCoeffs, mfldExps, y, symbolic = symbolic)
  37. return evaluate_polynomial(mfldCoeffs, mfldExps,y)
  40. # Given coefficients and exponents of the (autonomous) ROM, computes the ROM expansion
  41. def ROM0(eta, rdCoeffs, rdExps, symbolic = False):
  43. if symbolic == True:
  44. return evaluate_polynomial(rdCoeffs,rdExps, eta, symbolic = symbolic)
  45. return np.real(evaluate_polynomial(rdCoeffs,rdExps, eta, symbolic = symbolic))
  46. # To integrate with scipy
  47. def ROM(t,eta, rdCoeffs, rdExps):
  49. return evaluate_polynomial(rdCoeffs,rdExps, eta)
  53. def ROM1D(t, eta, rdCoeffs, rdExps, symbolic = False):
  54. rhs = 0
  55. for i in range(rdCoeffs.shape[1]):
  56. rhs+= eta**rdExps[i,0]*rdCoeffs[0,i]
  58. return rhs
  60. def ROM1D0(eta, rdCoeffs, rdExps, symbolic = False):
  61. rhs = 0
  62. for i in range(rdCoeffs.shape[1]):
  63. rhs+= eta**rdExps[i]*rdCoeffs[0,i]
  64. return rhs
  69. # Print the ROM
  70. def printROM(d,rdCoeffs, rdExps):
  72. etas, detas = GenerateSymbols(d)
  73. # Construct ROM symbolic equations
  74. if d ==1:
  75. rhs = ROM1D0(etas,rdCoeffs, rdExps,symbolic=True)
  76. else:
  77. rhs = ROM0(etas,rdCoeffs, rdExps,symbolic=True)
  79. for i in range(d):
  80. display(Math(sym.latex(sym.Eq(detas[i], rhs[i]))))
  83. # Calculate the Normal Mean Trajectory Error
  84. def calculate_NMTE(nTest, DataTestTrunc, RomTraj):
  86. NMTE = 0
  87. for i in range(nTest):
  88. NMTE += np.mean(np.linalg.norm(DataTestTrunc[i,:,:]-RomTraj[i][:], axis = 0), axis = -1)/np.max(DataTestTrunc[i])
  90. return NMTE/nTest
  93. # Calculate the Manifold Fitting Error
  94. def calculate_MFE(nTest, DataTestTrunc, E, mfldCoeffs, mfldExps):
  96. MFE = 0
  97. for i in range(nTest):
  98. # Lift trajectories projected to the spectral subspace E to the SSM
  99. LiftTraj = construct_SSM(np.dot(E.T,DataTestTrunc[i]),mfldCoeffs, mfldExps)
  100. MFE += np.mean(np.linalg.norm(DataTestTrunc[i,:,:]-LiftTraj[:], axis = 0), axis = -1)/np.max(DataTestTrunc[i])
  102. return MFE/nTest
  105. ####### Time-Dependent SSMs (Weak Forcing) #######
  109. # Anchor traj at order 1
  110. def compute_anchorO1(N, d, t0, nTimesteps, evecs, evals, ns, stable = True, tau = 0.01, dt = 1e-3, tfin = 700):
  112. Sevals = np.copy(evals)
  114. if stable == False:
  115. Sevals[:d] = np.zeros((d), dtype= complex)
  116. Uevals = np.copy(evals)
  118. Uevals[d:] = np.zeros((N-d), dtype= complex)
  119. At = np.exp(np.outer(np.arange(0,nTimesteps*dt, dt), (Sevals)))
  120. Gt = np.zeros((nTimesteps, N,N), dtype = complex)
  121. for i in range(nTimesteps):
  122. Gt[i] = np.diag(At[i])
  123. Gt = np.matmul(evecs,np.matmul((Gt),np.linalg.inv(evecs)))
  124. if stable == False:
  125. Atu = np.exp(np.outer(-np.arange(0,nTimesteps*dt, dt), (Uevals)))
  126. Gtu = np.zeros((nTimesteps, N,N), dtype = complex)
  127. for i in range(nTimesteps):
  128. Gtu[i] = np.diag(Atu[i])
  129. Gtu = np.matmul(evecs,np.matmul((Gtu),np.linalg.inv(evecs)))
  130. T, N, M = Gt.shape
  131. _, M2 = ns.T.shape
  132. assert M == M2, "Dimensions of G(t) and f(t) must align."
  134. # Initialize output
  135. h = np.zeros((T, N), dtype = complex)
  137. # Perform the convolution-like operation
  138. for t in range(T):
  139. for s in range(t ):
  140. h[t] += Gt[t - s] @ ns[:,s]*dt/tau
  141. if stable == False:
  142. for s in range(t,tfin):
  143. h[t] -= Gtu[s-t]@ ns[:,s]*dt/tau
  145. return h
  147. def plotParamSSM(m,tangent_space,eta1_vec,all_coeffsP, fixed_pointsP,ps, p_fpt0, p0, expsP,rd_coeffsP, rd_expsP,parname,colors =['purple','orange','darkgreen', 'darkcyan', 'salmon'], fsize = 17, tsize = 13, dpi = 100 ):
  148. # fig = plt.figure(figsize=(15,8))
  149. fig = plt.figure(layout='constrained', figsize=(15, 6), dpi = dpi)
  150. subfigs = fig.subfigures(1, 2, )
  151. #ax1 = fig.add_subplot(2, 2, 1, )
  152. ax2 = subfigs[1].subplots(subplot_kw={"projection": "3d"})
  153. axsLeft = subfigs[0].subplots(len(ps), 2, sharex=True)
  155. for e, epsilon in enumerate(ps):
  158. for pt in fixed_pointsP[e]:
  159. ptC = (np.tensordot(tangent_space.T,np.array(pt).T-p_fpt0 , axes =1))
  160. ptSSM = construct_SSM([ptC, (epsilon-p0)], all_coeffsP, expsP)
  161. #ax2.plot(ptC,(epsilon-p0),(np.array(pt).T-p_fpt0 )[m],'x', color = colors[e])
  162. ax2.plot(ptC,(epsilon-p0),ptSSM[m],'X', color = colors[e], markersize = 8)
  163. axsLeft[e,0].plot(ptC,(np.array(pt).T-p_fpt0 )[m],'X', color = colors[e], markersize = 8)
  164. #axsLeft[e,0].plot(ptC,ptSSM[m],'', color = colors[e])
  165. axsLeft[e,1].plot(ptC,0,'X', color = colors[e], markersize = 8)
  166. # plot the manifold
  168. ssm = construct_SSM([eta1_vec, np.ones(len(eta1_vec))*(epsilon-p0)], all_coeffsP, expsP)
  169. axsLeft[e,0].plot(eta1_vec,ssm[m], '-',color =colors[e], label = r'$%s =%s$'%(parname,epsilon))
  170. ax2.plot(eta1_vec,(epsilon-p0)*np.ones(len(eta1_vec)),ssm[m], '-',color =colors[e], label = r'$%s =%s$'%(parname,epsilon))
  171. eta1_dot = ROM0([eta1_vec, (epsilon-p0)*np.ones(len(eta1_vec))], rd_coeffsP, rd_expsP)
  172. axsLeft[e,1].plot(eta1_vec, eta1_dot[0], label = r'$%s =%s$'%(parname,epsilon), color = colors[e])
  173. axsLeft[e,1].plot(eta1_vec,np.zeros(len(eta1_vec)),'--', color = 'red',)
  174. axsLeft[-1,0].set_xlabel(r'$\eta_1$', fontsize = fsize)
  175. axsLeft[-1,1].set_xlabel(r'$\eta_1$', fontsize = fsize)
  176. axsLeft[e,0].set_ylabel(r'$y_%s$'%(m+1), fontsize = fsize)
  177. axsLeft[e,1].set_ylabel(r'$\dot{\eta}_1$', fontsize = fsize)
  178. axsLeft[e,0].tick_params(axis='both', which='major', labelsize=tsize)
  179. axsLeft[e,1].tick_params(axis='both', which='major', labelsize=tsize)
  180. axsLeft[e,1].legend(fontsize = fsize, loc = 'upper center')
  181. axsLeft[e,0].legend(fontsize = fsize, loc = 'upper center')
  182. axsLeft[e,1].set_ylim(-1,1)
  184. p_vals = np.linspace(ps[0]-0.001,ps[-1]+0.001,100)
  185. ax2.set_xlabel(r'$\eta_1$',fontsize = fsize)
  186. ax2.set_ylabel(r'$%s-%s0$'%(parname, parname),fontsize =fsize)
  187. ax2.set_zlabel(r'$y_%s$'%(m+1), fontsize = fsize)
  188. #eta1_vec = np.linspace(lims[0]+xmargins[0],lims[1]+xmargins[1],100)
  189. ETA, P = np.meshgrid(eta1_vec, p_vals)
  191. ssm = construct_SSM([ETA, P-p0], all_coeffsP, expsP)
  193. ax2.plot_surface(ETA,P-p0,ssm[m],color = 'mintcream', alpha = 0.2, edgecolors='lightgray', lw = 0.01, zorder = 2)
  194. ax2.legend(fontsize = fsize , )
  196. ax2.tick_params(axis='both', which='major', labelsize=tsize )
  198. ax2.view_init(30, 40, )
  199. #ax2.set_yticks(np.linspace(p_vals[0], p_vals[-1],5))
  200. ax2.set_yticks([])
  201. return fig
  203. # Calculate first order coefficients
  204. '''def calculate_H11(N,d,t0, epsilon,evals, evecs, idx, anchor,d2fy2, ui, tsteps, dtInt, dt = 1e-3 ):
  206. P = evecs[:,np.flip(np.argsort(evals))]
  207. P_inv = np.linalg.inv(P)
  208. A1 = np.copy(evals[d:])
  209. A1 = A1-evals[idx]*np.identity(N-d)
  210. ki = 0
  211. summand = np.zeros(( N-d, int(tsteps+t0/dtInt)), dtype= complex)
  212. h11 = np.zeros((N-d,tsteps),dtype= complex)
  214. for i in np.arange(t0,tsteps+t0, 1):# t
  217. kj = 0
  218. for j in np.arange(t0,i, dtInt): # s
  219. G = np.exp(A1*(i-j)*dt)
  220. m11 = np.matmul(P_inv, np.dot(np.dot(d2fy2,anchor[:,int(j)]/epsilon), np.matmul(P,ui)))[d:]
  221. summand[:,kj] = np.matmul(G, m11)*dt*dtInt
  222. kj = kj+1
  224. h11[:,ki] = (np.sum(summand, axis = 1))
  225. ki = ki+1
  226. return h11'''
  230. def calculate_A1(d, evals, eig):
  231. Ak = np.copy(evals[d:])
  232. Ak = Ak-eig
  233. return Ak
  235. def calculate_H11_vectorized(N, d, evals_sorted, P, P_inv, d2fy2, anchor, ui,
  236. nTimesteps, div=1, dt=0.001, tau=0.01,
  237. order=2, h0='quasistatic'):
  238. """h11(t) solving dh/dt = (Lambda[d:] - lambda_1) h + m11(t),
  239. integrated by exponential time differencing. Returns (T, N-d), complex,
  240. i.e. the same convention as before -> caller still does `.T`.
  241. `tau` is unused (it sits inside evals_sorted); kept for signature compatibility."""
  242. if d != 1:
  243. raise NotImplementedError('written for a 1-D master subspace')
  244. step = dt / div # the *actual* grid step, not dt
  245. T = anchor.shape[1]
  247. # m11(t), contracting d2fy2 with the eigenvector first to avoid an (N,N,T) array
  248. m11 = np.einsum('ijk,j,kt->it', d2fy2, np.matmul(P, ui), anchor, optimize=True)
  249. m11 = np.matmul(P_inv, m11)[d:] # (N-d, T)
  251. A1 = evals_sorted[d:] - evals_sorted[0] # = np.diag(A10)
  252. if np.any(np.real(A1) >= 0):
  253. raise ValueError('no spectral gap: Re(lambda_n - lambda_1) >= 0')
  255. # ETD coefficients (A1 diagonal -> everything elementwise)
  256. z = A1 * step
  257. E = np.exp(z)
  258. small = np.abs(z) < 1e-6 # series guard
  259. zs = np.where(small, 1.0, z)
  260. phi1 = np.where(small, 1 + z/2 + z**2/6, (E - 1) / zs)
  261. phi2 = np.where(small, 0.5 + z/6 + z**2/24, (E - 1 - z) / zs**2)
  263. h = np.zeros((T, N - d), dtype=complex)
  264. h[0] = -m11[:, 0] / A1 if h0 == 'quasistatic' else 0.0 # slaved value; 'zero' also fine
  265. b1, b2 = step * phi1, step * phi2
  266. if order == 1: # m11 piecewise constant
  267. for k in range(T - 1):
  268. h[k+1] = E * h[k] + b1 * m11[:, k]
  269. else: # m11 piecewise linear
  270. for k in range(T - 1):
  271. h[k+1] = E * h[k] + b1 * m11[:, k] + b2 * (m11[:, k+1] - m11[:, k])
  272. return h
  275. def construct_SSM1Dt(uv, t, epsilon, *args, surf =True):
  276. # h0i = 0, h10 = 0, h20, h11,
  277. coeff11, coeff20, coeff30 = args
  279. if surf:
  280. u = np.tile(np.expand_dims(uv, 0), (len(t),1))
  281. return np.concatenate(([u],np.multiply(np.tile(np.expand_dims(coeff20, (1,2)),(1,np.shape(u)[0], np.shape(u)[1])),u**2)+ np.multiply(np.tile(np.expand_dims(coeff30, (1,2)),(1,np.shape(u)[0], np.shape(u)[1])),u**3)
  282. + epsilon*np.multiply(np.tile(np.expand_dims(coeff11[:,t],2),(1,1,np.shape(u)[1])),u)), axis =0)
  283. else:
  284. u = uv
  285. return np.concatenate(([u],np.multiply(np.tile(np.expand_dims(coeff20, (1)),(1,np.shape(u)[0])),u**2)+
  286. + np.multiply(coeff11[:,t],u)*epsilon), axis =0)

SSMfunctions.py at commit f670273, under MIT · at the source

Overview

Authors: A. Marraffa1,2,3, R. Krause4,5, V. Mante4,5, G. Haller1
ORCID iDs: A. Marraffa, V. Mante, G. Haller
  1. Institute for Mechanical Systems, ETH Zurich,Zurich, Switzerland
  2. VIB Center for AI and Computational Biology, VIB,Leuven, Belgium
  3. Present Address: Department of Electrical Engineering, KU Leuven,Leuven, Belgium
  4. University of Zurich & ETH Zurich,Zurich, Switzerland
  5. Neuroscience Center Zurich, University of Zurich & ETH Zurich,Zurich, Switzerland
Institutions: ETH Zurich (Switzerland); VIB.AI (Belgium); KU Leuven (Belgium); University of Zurich (Switzerland)
Journal: Nature communications, volume 17, issue 1, article 9123
Dates: received 1 October 2025; accepted 15 July 2026; published online 28 July 2026
Type: Research article · Language: English
License: CC BY
Identifiers: DOI 10.1038/s41467-026-75924-7 · PMID 42649198 · PMCID PMC13519027 · OpenAlex W7171348837
Open access: gold, a free copy (OpenAlex)
Status: code verified
Categories: computational modeling (no new data) (modality), none (in silico) (organism)
Methods: Smoothing, state filtering, decompositions, Spectral & time-frequency
Keywords: Dynamical systems, Network models
Topic: Model Reduction and Neural Networks (Statistical and Nonlinear Physics, Physics and Astronomy), according to OpenAlex
Funding: Eidgenössische Technische Hochschule Zürich
Citations: not cited yet (Europe PMC); 61 references in the paper

Abstract

Neural ordinary differential equations (ODEs) are widely used in neuroscience to model the collective activity of neurons during behavioral tasks. The high dimensionality of their parameter and activity spaces, however, often make it challenging to infer and interpret the fundamental features of their dynamics.

In this study, we employ recent nonlinear dynamical system techniques to uncover the core dynamics of several Neural ODEs used in contemporary neuroscience. Specifically, using a data-driven approach, we identify Spectral Submanifolds (SSMs), i.e., low-dimensional attracting invariant manifolds tangent to the eigenspaces of fixed points. The internal dynamics of SSMs serve as nonlinear models that reduce the dimensionality of the full RNNs by orders of magnitude. Through low-dimensional, SSM-reduced models, we give mathematically precise definitions of line and ring attractors, which are intuitive concepts commonly used to explain decision-making and working memory. This unprecedented level of understanding of Neural ODEs obtained from SSM reduction enables the interpretation of mathematically well-defined and robust structures in neuronal dynamics, leading to predictions about the neural computations underlying behavior.

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

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haller-group/SSMTool-2.4

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Languages: MATLAB (2875), C++ (1)
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haller-group/SSMLearn

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alicemarr/RNN-paper

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Zenodo 20623670

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Data availability

The data is available at this URL https://github.com/alicemarr/RNN-paper under the MIT license. 10.5281/zenodo.20623670.

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Version 2, 28 September 2026

  • Funding: added Eidgenössische Technische Hochschule Zürich

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Recorded: type, language, journal, volume, issue, pages, dates, 4 authors, 2 keywords, 53 references.

Cite

This paper

Marraffa, A., Krause, R., Mante, V., & Haller, G. (2026). Data-driven reduced modeling of neural dynamics. Nature communications, 17(1), 9123. https://doi.org/10.1038/s41467-026-75924-7

BibTeX

@article{marraffa2026data,
author = {Marraffa, A. and Krause, R. and Mante, V. and Haller, G.},
title = {{Data-driven reduced modeling of neural dynamics}},
journal = {Nature communications},
year = {2026},
month = jul,
volume = {17},
number = {1},
pages = {9123},
publisher = {Nature Publishing Group},
issn = {2041-1723},
doi = {10.1038/s41467-026-75924-7},
url = {https://doi.org/10.1038/s41467-026-75924-7},
pmid = {42649198},
pmcid = {PMC13519027}
}

RIS

TY - JOUR
AU - Marraffa, A.
AU - Krause, R.
AU - Mante, V.
AU - Haller, G.
TI - Data-driven reduced modeling of neural dynamics
T2 - Nature communications
J2 - Nat Commun
PY - 2026
DA - 2026/07/28
VL - 17
IS - 1
SP - 9123
SN - 2041-1723
PB - Nature Publishing Group
DO - 10.1038/s41467-026-75924-7
UR - https://doi.org/10.1038/s41467-026-75924-7
LA - en
ER -

CSL-JSON

{
"id": "10.1038/s41467-026-75924-7",
"type": "article-journal",
"title": "Data-driven reduced modeling of neural dynamics",
"container-title": "Nature communications",
"author": [
{
"family": "Marraffa",
"given": "A."
},
{
"family": "Krause",
"given": "R."
},
{
"family": "Mante",
"given": "V."
},
{
"family": "Haller",
"given": "G."
}
],
"container-title-short": "Nat Commun",
"volume": "17",
"issue": "1",
"page": "9123",
"DOI": "10.1038/s41467-026-75924-7",
"PMID": "42649198",
"PMCID": "PMC13519027",
"ISSN": "2041-1723",
"publisher": "Nature Publishing Group",
"URL": "https://doi.org/10.1038/s41467-026-75924-7",
"language": "en",
"issued": {
"date-parts": [
[
2026,
7,
28
]
]
}
}

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