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Association-sensory spatiotemporal hierarchy and functional gradient-regularised recurrent neural network with implications for schizophrenia.

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] § Methods › Recurrent neural network, working memory tasks, and regularisations ↔ tasks_memory.py, lines 12–137 · score 0.75 · post response, cost mask, sin, modalities, fixation, strength
  2. [2] § Results › Neural timescale estimation and spatial-temporal convergence ↔ ou_mle_tau.py, lines 2–20 · score 0.56 · Ornstein Uhlenbeck, maximum likelihood, MLE, OU
  3. [3] § Methods › Neural timescale estimation ↔ tau_experiments.ipynb, lines 161–190 · score 0.52 · negative log likelihood, NLL
  4. [4] § Methods › Neural timescale estimation ↔ tau_experiments.ipynb, lines 31–70 · score 0.50 · exponential decay, empirical ACF, autocorrelation, fit

Paper

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The authors' code

Jupyter notebook · 775 lines · 24 KB · no license · 2 matches

  1. # %%
  2. import numpy as np
  3. import pymc as pm
  4. import arviz as az
  5. import matplotlib.pyplot as plt
  7. from scipy.optimize import minimize, curve_fit
  8. import scipy.stats as stats
  9. from scipy.stats import shapiro
  11. import seaborn as sns
  13. from tqdm.notebook import tqdm
  14. from mpl_toolkits.mplot3d import Axes3D
  16. import pandas as pd
  18. from statsmodels.tsa.stattools import acf
  20. # %%
  21. def simulate_ou_process(tau, D, delta_t, N, A0):
  22. A = np.zeros(N + 1)
  23. A[0] = A0
  24. exp_dt_tau = np.exp(-delta_t / tau)
  25. sqrt_variance = np.sqrt(D * tau * (1 - exp_dt_tau**2))
  26. for i in range(N):
  27. mean = A[i] * exp_dt_tau
  28. A[i + 1] = mean + sqrt_variance * np.random.randn()
  29. return A
  31. # %% [markdown]
  32. # ### Compute empirical ACF with covariance and estimate τ
  34. # %%
  35. def compute_empirical_acf(A, max_lag):
  36. N = len(A)
  37. A_mean = np.mean(A)
  38. A_centered = A - A_mean
  39. var_A = np.var(A_centered)
  40. autocorr = []
  42. for lag in range(1, max_lag + 1):
  43. # autocovariance function
  44. cov = np.mean(A_centered[:-lag] * A_centered[lag:])
  45. autocorr.append(cov / var_A)
  46. lags = np.arange(1, max_lag + 1)
  47. return lags, np.array(autocorr)
  49. def exponential_decay(lag, tau, delta_t):
  50. return np.exp(-lag * delta_t / tau)
  52. def estimate_tau_from_acf(lags, autocorr, delta_t):
  53. initial_guess = [1.0]
  55. try:
  56. params, covariance = curve_fit(
  57. lambda lag, tau: exponential_decay(lag, tau, delta_t), # fix delta_t
  58. lags,
  59. autocorr,
  60. p0=initial_guess,
  61. bounds=(1e-6, np.inf),
  62. method='trf',
  63. )
  64. tau_est = params[0]
  66. except:
  67. print('offset fit failed \n')
  68. tau_est = np.nan
  70. return tau_est
  72. # %% [markdown]
  73. # ###### Estimate τ with exp decay with offset
  75. # %%
  76. def exponential_decay_woffset(lag, A, tau, B, delta_t):
  77. return A * (np.exp(-lag * delta_t / tau) + B)
  79. def estimate_tau_from_acf_woffset(lags, autocorr, delta_t):
  80. initial_guess = [0,np.random.rand(1)[0]+0.01,0]
  82. model_func = lambda lag, A, tau, B: exponential_decay_woffset(lag, A, tau, B, delta_t)
  84. try:
  85. params, covariance = curve_fit(
  86. model_func,
  87. lags,
  88. autocorr,
  89. p0=initial_guess,
  90. bounds=([0, 0, -np.inf], [np.inf, np.inf, np.inf]),
  91. method='trf',
  92. )
  93. A_est, tau_est, B_est = params
  95. except:
  96. print('offset fit failed \n')
  97. tau_est = np.nan
  99. return tau_est
  101. # %%
  102. # True params
  103. tau_true = 15.33 # timescale
  104. D_true = .07 # diffusion coefficient
  105. delta_t = 1.72 # rime step
  106. N = int(10000/delta_t) # # of data points
  107. A0 = -0.013 # init value of the process
  109. num_simulations = 200 # # of simulations
  110. max_lag = 5000
  112. A = simulate_ou_process(tau_true, D_true, delta_t, N, A0)
  114. # direct Fit using ACF
  115. lags, autocorr = compute_empirical_acf(A, max_lag)
  116. tau_est_acf = estimate_tau_from_acf(lags, autocorr, delta_t)
  117. tau_est_acf_woffset = estimate_tau_from_acf_woffset(lags, autocorr, delta_t)
  118. print(f'exp fit: {tau_est_acf}')
  119. print(f'exp fit woffset: {tau_est_acf_woffset}')
  121. # %%
  122. D = 0.07
  123. delta_t = 1.72
  124. N = int(300 / delta_t)
  125. A0 = -0.013
  126. max_lag = 100
  128. tau_high = 15.33
  129. tau_low = 2.0
  131. A_high = simulate_ou_process(tau_high, D, delta_t, N, A0)
  132. A_low = simulate_ou_process(tau_low, D, delta_t, N, A0)
  134. # Compute empirical ACFs
  135. lags, autocorr_high = compute_empirical_acf(A_high, max_lag)
  136. _, autocorr_low = compute_empirical_acf(A_low, max_lag)
  138. plt.figure(figsize=(10, 6))
  139. plt.plot(lags * delta_t, autocorr_high, label=f'High Timescale')
  140. plt.plot(lags * delta_t, autocorr_low, label=f'Low Timescale')
  141. plt.legend()
  142. plt.xlim([-1,50])
  144. plt.xticks(fontsize=8)
  145. plt.yticks(fontsize=8)
  146. plt.legend()
  147. plt.tight_layout()
  148. plt.show()
  150. # est tau from ACFs
  151. tau_est_high = estimate_tau_from_acf(lags, autocorr_high, delta_t)
  152. tau_est_low = estimate_tau_from_acf(lags, autocorr_low, delta_t)
  154. # # MLE
  155. # initial_guess = [1.0, 0.5]
  156. # tau_est_mle_high, _ = estimate_parameters_mle(A_high, delta_t, initial_guess)
  158. print(f"Estimated τ for high timescale process: {tau_est_high:.2f}")
  159. print(f"Estimated τ for low timescale process: {tau_est_low:.2f}")
  161. # %% [markdown]
  162. # ### MLE negative log-likelihood and τ, D estimation
  164. # %%
  165. def negative_log_likelihood(params, A, delta_t):
  166. tau, D = params
  167. if tau <= 0 or D <= 0:
  168. return np.inf
  169. N = len(A) - 1
  170. exp_dt_tau = np.exp(-delta_t / tau)
  171. variance = D * tau * (1 - exp_dt_tau**2)
  172. if variance <= 0:
  173. return np.inf
  174. residuals = A[1:] - A[:-1] * exp_dt_tau
  175. nll = 0.5 * N * np.log(2 * np.pi * variance) + np.sum(residuals**2) / (2 * variance)
  176. return nll
  178. def estimate_parameters_mle(A, delta_t, initial_guess):
  179. result = minimize(
  180. negative_log_likelihood,
  181. initial_guess,
  182. args=(A, delta_t),
  183. method='L-BFGS-B',
  184. bounds=((1e-6, None), (1e-6, None))
  185. )
  186. if result.success:
  187. tau_est, D_est = result.x
  188. return tau_est, D_est
  189. else:
  190. return np.nan, np.nan
  192. # %% [markdown]
  193. # ### Bayesian (w PyMC)
  195. # %%
  196. def estimate_parameters_bayesian(A, delta_t, mle_tau_estimate):
  197. with pm.Model() as ou_model:
  198. # Priors for tau and D
  199. # empirical bayes - use direct fit estimate as prior mean for tau
  200. # tau_prior_mean = mle_tau_estimate if not np.isnan(mle_tau_estimate) else 13.0
  202. # gaussian priors
  203. tau_prior_mean = 12. # prior mean for tau
  204. tau_prior_sd = 10. # prior standard deviation for tau
  205. D_prior_mean = 1.5 # prior mean for D
  206. D_prior_sd = 1 # prior standard deviation for D
  208. tau = pm.Normal('tau', mu=tau_prior_mean, sigma=tau_prior_sd)
  209. D = pm.Normal('D', mu=D_prior_mean, sigma=D_prior_sd)
  212. # gamma priors
  213. # tau = pm.Gamma('tau', alpha=2.0, beta=2.0 / tau_prior_mean)
  214. # D = pm.Gamma('D', alpha=2.0, beta=1.0)
  216. # compute variance and mean
  217. exp_dt_tau = pm.math.exp(-delta_t / tau)
  218. variance = D * tau * (1 - exp_dt_tau**2)
  219. mean = A[:-1] * exp_dt_tau
  221. # likelihood
  222. A_obs = pm.Normal('A_obs', mu=mean, sigma=pm.math.sqrt(variance), observed=A[1:])
  224. # sample from posterior
  225. try:
  226. trace = pm.sample(
  227. 1000, # Number of samples
  228. tune=500, # Number of tuning steps
  229. progressbar=False,
  230. target_accept=0.9
  231. )
  232. summary = az.summary(trace, var_names=['tau', 'D'], round_to=4)
  233. tau_est = summary.loc['tau', 'mean']
  234. D_est = summary.loc['D', 'mean']
  235. except:
  236. tau_est = np.nan
  237. D_est = np.nan
  238. return tau_est, D_est
  240. # %% [markdown]
  241. # ### Simulations
  243. # %%
  244. # True parameters
  245. delta_t = 1.72
  246. # N = int(310/delta_t) # Number of data points
  247. N = 210
  249. tau_true = 4.748 # True intrinsic timescale
  250. D_true = .06967 # True diffusion coefficient
  251. A0 = -0.013 # Initial value of the process
  253. # Estimation settings
  254. num_simulations = 200 # Number of simulations
  255. max_lag = 50 # Maximum lag for ACF
  257. # %%
  258. exp_dt_tau = np.exp(-delta_t / tau_true)
  259. variance = D_true * tau_true * (1 - exp_dt_tau**2)
  261. print(exp_dt_tau)
  262. print(variance)
  264. # %%
  265. sns.set_theme(style="darkgrid")
  267. A = simulate_ou_process(tau_true, D_true, delta_t, N, A0)
  269. plt.figure(figsize=(14, 6))
  270. plt.plot(A)
  272. # %%
  273. f, (ax1, ax2) = plt.subplots(1, 2, figsize=(10, 4), sharex=True)
  275. ax1.hist(A, bins=15, label='A');
  276. ax1.legend()
  277. ax2.hist(np.diff(A), bins=15, label='dA', color='r');
  278. ax2.legend()
  280. # %% [markdown]
  281. # #### compare direct exp fit, MLE, Bayes
  283. # %%
  284. tau_true = 15.33
  285. D_true = .07
  286. delta_t = 1.72
  287. N = int(310/delta_t)
  288. A0 = -0.013
  290. # est settings
  291. num_simulations = 200
  292. max_lag = 50
  294. A = simulate_ou_process(tau_true, D_true, delta_t, N, A0)
  296. # %%
  298. tau_estimates_acf = []
  299. tau_estimates_acf_woffset = []
  301. tau_estimates_mle = []
  302. tau_estimates_bayes = []
  304. D_estimates_mle = []
  305. D_estimates_bayes = []
  307. # %%
  308. for sim in tqdm(range(num_simulations)):
  309. # sim the OU process
  310. A = simulate_ou_process(tau_true, D_true, delta_t, N, A0)
  312. # dir fit using ACF
  313. lags, autocorr = compute_empirical_acf(A, max_lag)
  314. tau_est_acf = estimate_tau_from_acf(lags, autocorr, delta_t)
  315. tau_estimates_acf.append(tau_est_acf)
  317. # # direct fit woffset
  318. # lags, autocorr = compute_empirical_acf(A, max_lag)
  319. # tau_est_acf_woffset = estimate_tau_from_acf_woffset(lags, autocorr, delta_t)
  320. # tau_estimates_acf_woffset.append(tau_est_acf_woffset)
  322. # MLE
  323. initial_guess = [1.0, 0.5]
  324. tau_est_mle, D_est_mle = estimate_parameters_mle(A, delta_t, initial_guess)
  325. tau_estimates_mle.append(tau_est_mle)
  326. D_estimates_mle.append(D_est_mle)
  328. # Bayes
  329. tau_est_bayes, D_est_bayes = estimate_parameters_bayesian(A, delta_t, tau_est_mle)
  330. tau_estimates_bayes.append(tau_est_bayes)
  331. D_estimates_bayes.append(D_est_bayes)
  333. print(f"Simulation {sim + 1}/{num_simulations} completed.")
  335. # %%
  336. tau_estimates_acf = np.array(tau_estimates_acf)
  337. # tau_estimates_acf_woffset = np.array(tau_estimates_acf_woffset)
  338. tau_estimates_mle = np.array(tau_estimates_mle)
  339. tau_estimates_bayes = np.array(tau_estimates_bayes)
  341. D_estimates_mle = np.array(D_estimates_mle)
  342. D_estimates_bayes = np.array(D_estimates_bayes)
  344. # %%
  345. bias_acf = tau_estimates_acf - tau_true
  346. bias_mle = tau_estimates_mle - tau_true
  347. bias_bayes = tau_estimates_bayes - tau_true
  349. # %%
  350. all_data = np.concatenate((tau_estimates_acf,
  351. tau_estimates_mle, tau_estimates_bayes))
  352. all_data = all_data[~np.isnan(all_data)]
  353. bin_edges = np.linspace(np.min(all_data), np.max(all_data), num=50) # 10 bins creates 11 edges
  355. sns.set_theme(style="white", palette=None)
  356. plt.figure(figsize=(8, 5))
  358. plt.hist(tau_estimates_acf, bins=bin_edges, color='#4888A3', alpha=.5, edgecolor='none', label='Direct exponential fit')
  359. # plt.hist(tau_estimates_acf_woffset, bins=bin_edges, alpha=.4, edgecolor='none', label='Direct Exp Fit with Offset')
  361. plt.hist(tau_estimates_mle, bins=bin_edges, color='#D32F26', alpha=.6, edgecolor='none', label='MLE')
  362. plt.hist(tau_estimates_bayes, bins=bin_edges, color='#DDC332', alpha=.4, edgecolor='none', label='Bayes with gaussian priors')
  363. plt.axvline(tau_true, color='k', linestyle='dashed', linewidth=1, label='True τ')
  364. plt.xlim([0, 50])
  365. plt.xlabel('τ Estimates')
  366. plt.ylabel('Counts')
  367. plt.title('')
  368. plt.legend()
  369. plt.show()
  371. # %%
  372. print(np.var(tau_estimates_acf), np.nanvar(tau_estimates_mle), np.var(tau_estimates_bayes))
  374. # %%
  375. print(np.mean(tau_estimates_acf), np.nanmean(tau_estimates_mle), np.mean(tau_estimates_bayes))
  377. # %%
  378. all_data = np.concatenate((D_estimates_mle, D_estimates_bayes))
  379. all_data = all_data[~np.isnan(all_data)]
  380. bin_edges = np.linspace(np.min(all_data), np.max(all_data), num=30)
  382. plt.figure(figsize=(8, 5))
  384. plt.hist(D_estimates_mle, bins=bin_edges, color='#D32F26', edgecolor='none', alpha=0.6, label='MLE')
  385. plt.hist(D_estimates_bayes, bins=bin_edges, color='#DDC332', edgecolor='none', alpha=0.4, label='Bayes with gaussian priors')
  386. plt.axvline(D_true, color='k', linestyle='dashed', linewidth=1, label='True D')
  387. plt.xlabel('Estimated D')
  388. plt.ylabel('Counts')
  389. plt.legend()
  390. plt.show()
  392. # %% [markdown]
  393. # ##### effect of timeseries length
  395. # %%
  396. tau_true = 2.1030
  397. D_true = 0.4623
  398. delta_t = 1
  399. A0 = -0.013
  401. max_lag = 50
  403. N_values = np.arange(60, 1001, 5)
  404. num_simulations = 100
  406. results = {
  407. 'N': [],
  408. 'mean_tau_acf': [],
  409. 'var_tau_acf': [],
  410. 'mean_tau_mle': [],
  411. 'var_tau_mle': []
  412. }
  414. # %%
  415. for N in tqdm(N_values, desc="Processing N values"):
  416. tau_estimates_acf = []
  417. tau_estimates_mle = []
  419. for sim in range(num_simulations):
  420. # sim OU
  421. A = simulate_ou_process(tau_true, D_true, delta_t, N, A0)
  423. # dir fit exp to ACF
  424. max_lag = min(100, N//2)
  425. lags, autocorr = compute_empirical_acf(A, max_lag)
  426. tau_est_acf = estimate_tau_from_acf(lags, autocorr, delta_t)
  427. tau_estimates_acf.append(tau_est_acf)
  429. # MLE
  430. initial_guess = [np.random.rand(1)[0]+0.01, np.random.rand(1)[0]+0.01]
  431. tau_est_mle, D_est_mle = estimate_parameters_mle(A, delta_t, initial_guess)
  432. tau_estimates_mle.append(tau_est_mle)
  434. mean_tau_acf = np.nanmean(tau_estimates_acf)
  435. var_tau_acf = np.nanvar(tau_estimates_acf)
  436. mean_tau_mle = np.nanmean(tau_estimates_mle)
  437. var_tau_mle = np.nanvar(tau_estimates_mle)
  439. results['N'].append(N)
  440. results['mean_tau_acf'].append(mean_tau_acf)
  441. results['var_tau_acf'].append(var_tau_acf)
  442. results['mean_tau_mle'].append(mean_tau_mle)
  443. results['var_tau_mle'].append(var_tau_mle)
  445. # %%
  446. df_results = pd.DataFrame(results)
  448. plt.figure(figsize=(12, 6))
  450. # ACF
  451. plt.plot(df_results['N'], df_results['mean_tau_acf'], label='Exp fit', color='#4888A3', lw=1.5)
  452. plt.fill_between(df_results['N'],
  453. df_results['mean_tau_acf'] - np.sqrt(df_results['var_tau_acf']),
  454. df_results['mean_tau_acf'] + np.sqrt(df_results['var_tau_acf']),
  455. color='#4888A3', alpha=0.25)
  457. # MLE
  458. plt.plot(df_results['N'], df_results['mean_tau_mle'], label='MLE', color='#D32F26', lw=1.5)
  459. plt.fill_between(df_results['N'],
  460. df_results['mean_tau_mle'] - np.sqrt(df_results['var_tau_mle']),
  461. df_results['mean_tau_mle'] + np.sqrt(df_results['var_tau_mle']),
  462. color='#D32F26', alpha=0.25)
  464. plt.axhline(y=tau_true, color='green', lw=2,linestyle='--', label='True Tau')
  465. plt.xlabel('Time Length')
  466. plt.ylabel('τ Estimates')
  467. plt.legend()
  468. plt.grid(True)
  469. plt.show()
  471. # %% [markdown]
  472. # ##### effect of Δt
  474. # %%
  475. tau_true = 15.33
  476. D_true = .07
  477. A0 = -0.013
  478. N = 200
  480. max_lag = 50
  482. delta_t_values = np.arange(1, 5, .01)
  483. num_simulations = 100
  485. results = {
  486. 'delta_t': [],
  487. 'mean_tau_acf': [],
  488. 'var_tau_acf': [],
  489. 'mean_tau_mle': [],
  490. 'var_tau_mle': []
  491. }
  493. # %%
  494. for delta_t in tqdm(delta_t_values, desc="Processing delta_t values"):
  495. tau_estimates_acf = []
  496. tau_estimates_mle = []
  498. for sim in range(num_simulations):
  499. A = simulate_ou_process(tau_true, D_true, delta_t, N, A0)
  501. max_lag = 100
  502. lags, autocorr = compute_empirical_acf(A, max_lag)
  503. tau_est_acf = estimate_tau_from_acf(lags, autocorr, delta_t)
  504. tau_estimates_acf.append(tau_est_acf)
  506. # MLE
  507. initial_guess = [np.random.rand(1)[0]+0.01, np.random.rand(1)[0]+0.01]
  508. tau_est_mle, D_est_mle = estimate_parameters_mle(A, delta_t, initial_guess)
  509. tau_estimates_mle.append(tau_est_mle)
  511. mean_tau_acf = np.nanmean(tau_estimates_acf)
  512. var_tau_acf = np.nanvar(tau_estimates_acf)
  513. mean_tau_mle = np.nanmean(tau_estimates_mle)
  514. var_tau_mle = np.nanvar(tau_estimates_mle)
  516. results['delta_t'].append(delta_t)
  517. results['mean_tau_acf'].append(mean_tau_acf)
  518. results['var_tau_acf'].append(var_tau_acf)
  519. results['mean_tau_mle'].append(mean_tau_mle)
  520. results['var_tau_mle'].append(var_tau_mle)
  522. # %%
  523. df_results = pd.DataFrame(results)
  525. plt.figure(figsize=(12, 6))
  527. plt.plot(df_results['delta_t'], df_results['mean_tau_acf'], label='Exp fit', color='#4888A3', lw=1.5)
  528. plt.fill_between(df_results['delta_t'],
  529. df_results['mean_tau_acf'] - np.sqrt(df_results['var_tau_acf']),
  530. df_results['mean_tau_acf'] + np.sqrt(df_results['var_tau_acf']),
  531. color='#4888A3', alpha=0.25)
  533. plt.plot(df_results['delta_t'], df_results['mean_tau_mle'], label='MLE', color='#D32F26', lw=1.5)
  534. plt.fill_between(df_results['delta_t'],
  535. df_results['mean_tau_mle'] - np.sqrt(df_results['var_tau_mle']),
  536. df_results['mean_tau_mle'] + np.sqrt(df_results['var_tau_mle']),
  537. color='#D32F26', alpha=0.25)
  539. plt.axhline(y=tau_true, color='green', lw=2,linestyle='--', label='True τ')
  540. plt.xlabel('Δt')
  541. plt.ylabel('τ Estimates')
  542. plt.legend()
  543. plt.grid(True)
  544. plt.show()
  546. # %% [markdown]
  547. # ##### joint effects of N and delta t
  549. # %%
  550. np.random.seed(0)
  552. tau_true = 10.0
  553. D_true = .08
  554. A0 = -0.013
  556. max_lag = 50
  557. N_values = np.arange(100, 501, 50)
  558. delta_t_values = np.arange(1, 5, .5)
  559. num_simulations = 100
  561. mean_tau_acf = np.zeros((len(N_values), len(delta_t_values)))
  562. mean_tau_mle = np.zeros((len(N_values), len(delta_t_values)))
  564. var_tau_acf = np.zeros((len(N_values), len(delta_t_values)))
  565. var_tau_mle = np.zeros((len(N_values), len(delta_t_values)))
  567. mse_tau_acf = np.zeros((len(N_values), len(delta_t_values)))
  568. mse_tau_mle = np.zeros((len(N_values), len(delta_t_values)))
  570. # sim loop
  571. for i, N in enumerate(tqdm(N_values, desc="Processing N values")):
  572. for j, delta_t in enumerate(tqdm(delta_t_values, desc=f"N={N}", leave=False)):
  573. tau_estimates_acf = []
  574. tau_estimates_mle = []
  576. for sim in range(num_simulations):
  577. A = simulate_ou_process(tau_true, D_true, delta_t, N, A0)
  579. # dir ACF Fit
  580. max_lag_current = N // 2
  581. lags, autocorr = compute_empirical_acf(A, max_lag_current)
  582. tau_est_acf = estimate_tau_from_acf(lags, autocorr, delta_t)
  583. tau_estimates_acf.append(tau_est_acf)
  585. # MLE
  586. initial_guess = [1.0, 0.5] # [tau_guess, D_guess]
  587. tau_est_mle, D_est_mle = estimate_parameters_mle(A, delta_t, initial_guess)
  588. tau_estimates_mle.append(tau_est_mle)
  590. tau_estimates_acf = np.array(tau_estimates_acf)
  591. tau_estimates_mle = np.array(tau_estimates_mle)
  593. mean_tau_acf[i, j] = np.nanmean(tau_estimates_acf)
  594. mean_tau_mle[i, j] = np.nanmean(tau_estimates_mle)
  596. # compute var of Estimated Tau
  597. var_tau_acf[i, j] = np.nanvar(tau_estimates_acf)
  598. var_tau_mle[i, j] = np.nanvar(tau_estimates_mle)
  600. # compute MSE
  601. # mse_tau_acf[i, j] = np.nanmean((tau_estimates_acf - tau_true) ** 2)
  602. # mse_tau_mle[i, j] = np.nanmean((tau_estimates_mle - tau_true) ** 2)
  603. mse_tau_acf[i, j] = np.nanmean(abs(tau_estimates_acf - tau_true))
  604. mse_tau_mle[i, j] = np.nanmean(abs(tau_estimates_mle - tau_true))
  606. overall_mean_mse_acf = np.mean(mse_tau_acf)
  607. overall_mean_mse_mle = np.mean(mse_tau_mle)
  609. overall_mean_var_acf = np.mean(var_tau_acf)
  610. overall_mean_var_mle = np.mean(var_tau_mle)
  612. print("\nAggregated Results for Tau Estimation Methods:")
  613. print("------------------------------------------------")
  614. print(f"ACF Method:")
  615. print(f" Mean MSE: {overall_mean_mse_acf:.4f}")
  616. print(f" Mean Variance: {overall_mean_var_acf:.4f}\n")
  618. print(f"MLE Method:")
  619. print(f" Mean MSE: {overall_mean_mse_mle:.4f}")
  620. print(f" Mean Variance: {overall_mean_var_mle:.4f}\n")
  622. # %%
  623. print(mean_tau_acf.mean())
  624. print(mean_tau_mle.mean())
  626. # %%
  627. import matplotlib.colors as mcolors
  629. def hex_to_rgba(hex_color, alpha=1):
  630. return (*mcolors.hex2color(hex_color), alpha)
  633. zmin = min(np.nanmin(mean_tau_acf), np.nanmin(mean_tau_mle), tau_true)
  634. zmax = max(np.nanmax(mean_tau_acf), np.nanmax(mean_tau_mle), tau_true)
  636. zmin = 5
  637. zmax = 10
  639. # Meshgrid
  640. N_grid, delta_t_grid = np.meshgrid(N_values, delta_t_values, indexing='ij')
  642. # Dir fit
  643. fig = plt.figure(figsize=(14, 6))
  644. true_tau_surface = tau_true * np.ones_like(N_grid)
  646. #
  647. ax1 = fig.add_subplot(121, projection='3d')
  648. surf1 = ax1.plot_surface(N_grid, delta_t_grid, mean_tau_acf, cmap='cividis',
  649. edgecolor='none', linewidth=0,
  650. vmin=zmin, vmax=zmax
  651. )
  652. ax1.plot_surface(N_grid, delta_t_grid, true_tau_surface, color='gray', alpha=0.24, linewidth=0)
  654. ax1.set_zlim(zmin, zmax)
  655. # fig.colorbar(surf1, ax=ax1, shrink=0.4, aspect=20)
  656. ax1.view_init(elev=25, azim=140)
  658. hexcol = "#f3f3f3"
  659. ax1.xaxis.set_pane_color(hex_to_rgba(hexcol))
  660. ax1.yaxis.set_pane_color(hex_to_rgba(hexcol))
  661. ax1.zaxis.set_pane_color(hex_to_rgba(hexcol))
  663. # ax1.set_xticks([200, 400])
  664. # ax1.set_yticks([2, 4])
  665. # ax1.set_zticks([3.5, 4.5])
  668. # MLE Method Plot
  669. ax2 = fig.add_subplot(122, projection='3d')
  670. surf2 = ax2.plot_surface(N_grid, delta_t_grid, mean_tau_mle, cmap='cividis',
  671. edgecolor='none',linewidth=0,
  672. vmin=zmin,
  673. vmax=zmax
  674. )
  675. ax2.plot_surface(N_grid, delta_t_grid, true_tau_surface, color='gray', alpha=0.24)
  677. ax2.set_zlim(zmin, zmax)
  678. fig.colorbar(surf2, ax=ax2, shrink=0.4, aspect=20)
  679. ax2.view_init(elev=25, azim=140)
  680. ax2.xaxis.set_pane_color(hex_to_rgba(hexcol))
  681. ax2.yaxis.set_pane_color(hex_to_rgba(hexcol))
  682. ax2.zaxis.set_pane_color(hex_to_rgba(hexcol))
  684. # ax2.set_xticks([200, 400])
  685. # ax2.set_yticks([2, 4])
  686. # ax2.set_zticks([3.5, 4.5])
  688. plt.tight_layout()
  689. # plt.savefig("/Users/subat/Desktop/datan/bsnip/st1_28_oct/strnn/tf/network/space_smry/tau_estimation.png", dpi=400)
  691. plt.show()
  693. # %% [markdown]
  694. # ##### small tau
  696. # %%
  697. tau_true = 4.748
  698. D_true = .06967
  699. A0 = -0.013
  701. max_lag = 50
  702. N_values = np.arange(100, 1101, 100)
  703. delta_t_values = np.arange(1, 5, .05)
  704. num_simulations = 100
  706. mean_tau_acf = np.zeros((len(N_values), len(delta_t_values)))
  707. mean_tau_mle = np.zeros((len(N_values), len(delta_t_values)))
  709. #
  710. var_tau_acf = np.zeros((len(N_values), len(delta_t_values)))
  711. var_tau_mle = np.zeros((len(N_values), len(delta_t_values)))
  713. # %%
  714. for i, N in enumerate(tqdm(N_values, desc="Processing N values")):
  715. for j, delta_t in enumerate(tqdm(delta_t_values, desc=f"N={N}", leave=False)):
  716. tau_estimates_acf = []
  717. tau_estimates_mle = []
  719. for sim in range(num_simulations):
  720. A = simulate_ou_process(tau_true, D_true, delta_t, N, A0)
  722. max_lag = min(100, N//2)
  723. lags, autocorr = compute_empirical_acf(A, max_lag)
  724. tau_est_acf = estimate_tau_from_acf(lags, autocorr, delta_t)
  725. tau_estimates_acf.append(tau_est_acf)
  727. initial_guess = [1.0, 0.5]
  728. tau_est_mle, D_est_mle = estimate_parameters_mle(A, delta_t, initial_guess)
  729. tau_estimates_mle.append(tau_est_mle)
  731. mean_tau_acf[i, j] = np.nanmean(tau_estimates_acf)
  732. var_tau_acf[i, j] = np.nanvar(tau_estimates_acf)
  733. mean_tau_mle[i, j] = np.nanmean(tau_estimates_mle)
  734. var_tau_mle[i, j] = np.nanvar(tau_estimates_mle)
  736. # %%
  737. zmin = min(np.nanmin(mean_tau_acf), np.nanmin(mean_tau_mle), tau_true)
  738. zmax = max(np.nanmax(mean_tau_acf), np.nanmax(mean_tau_mle), tau_true)
  740. N_grid, delta_t_grid = np.meshgrid(N_values, delta_t_values, indexing='ij')
  742. fig = plt.figure(figsize=(14, 6))
  743. true_tau_surface = tau_true * np.ones_like(N_grid)
  746. ax1 = fig.add_subplot(121, projection='3d')
  747. surf1 = ax1.plot_surface(N_grid, delta_t_grid, mean_tau_acf, cmap='viridis',
  748. edgecolor='none',
  749. # vmin=zmin, vmax=zmax
  750. )
  751. ax1.plot_surface(N_grid, delta_t_grid, true_tau_surface, color='gray', alpha=0.24)
  753. ax1.set_title('Exponential fit')
  754. ax1.set_xlabel('Time Length')
  755. ax1.set_ylabel('Δt')
  756. ax1.set_zlabel('τ Estimates')
  757. ax1.set_zlim(zmin, zmax)
  758. fig.colorbar(surf1, ax=ax1, shrink=0.4, aspect=20)
  760. ax2 = fig.add_subplot(122, projection='3d')
  761. surf2 = ax2.plot_surface(N_grid, delta_t_grid, mean_tau_mle, cmap='plasma',
  762. edgecolor='none',
  763. # vmin=zmin,
  764. # vmax=zmax
  765. )
  766. ax2.plot_surface(N_grid, delta_t_grid, true_tau_surface, color='gray', alpha=0.4)
  767. ax2.set_title('OU MLE')
  768. ax2.set_xlabel('Time Length')
  769. ax2.set_ylabel('Δt')
  770. ax2.set_zlabel('τ Estimates')
  771. ax2.set_zlim(zmin, zmax)
  772. fig.colorbar(surf2, ax=ax2, shrink=0.4, aspect=20)
  774. plt.tight_layout()
  775. plt.show()

tau_experiments.ipynb at commit 6d83652, no license · at the source

Overview

Authors: Subati Abulikemu1,2, Puria Radmard3, Michail Mamalakis1,2,4, John Suckling1,2
  1. Department of Psychiatry, University of Cambridge,Cambridge, UK
  2. Centre for Human-Inspired Artificial Intelligence, University of Cambridge,Cambridge, UK
  3. Department of Engineering, University of Cambridge,Cambridge, UK
  4. Department of Computer Science and Technology, University of Cambridge,Cambridge, UK
Institutions: University of Cambridge (United Kingdom)
Journal: NPJ systems biology and applications, volume 12, issue 1, article 100
Dates: received 9 October 2025; accepted 15 April 2026; published online 30 April 2026
Type: Research article · Language: English
License: CC BY
Identifiers: DOI 10.1038/s41540-026-00727-x · PMID 42056136 · PMCID PMC13338132 · OpenAlex W7158966799
Open access: gold, a free copy (OpenAlex)
Status: code verified
Categories: human (organism), schizophrenia / psychosis (population), computational (subfield)
Methods: Spectral & time-frequency, Preprocessing, Connectivity, Machine learning, fMRI & imaging
Keywords: Computational biology and bioinformatics, Neuroscience
MeSH: Schizophrenia*, Adult, Brain, Brain Mapping, Female, Humans, Magnetic Resonance Imaging, Male, Memory, Short-Term, Neocortex, Nerve Net, Recurrent Neural Networks (* major topic)
Topic: Functional Brain Connectivity Studies (Cognitive Neuroscience, Neuroscience), according to OpenAlex
Funding: Cambridge Trust; Caius Engineering Trust Studentship; Horizon Europe research and innovation program
Citations: cited by 1 paper (Europe PMC); 96 references in the paper

Abstract

The human neocortex is functionally organised at its highest level along a continuous sensory-to-association (AS) hierarchy. This study investigates two questions—how this hierarchy is structurally altered in schizophrenia, and what these alterations imply for neural dynamics and cognitive computation. Using a large fMRI dataset (N = 355), we extracted individual AS gradients via spectral analysis of brain connectivity and quantified hierarchical organisation by the gradient range. Schizophrenia showed a compressed AS hierarchy, indicating reduced functional differentiation. Estimating neural timescale (autocorrelation decay constant) with the Ornstein-Uhlenbeck process, we observed that the most specialised, locally cohesive regions at the gradient extremes exhibit longer timescales, an empirical spatiotemporal mapping that is attenuated in schizophrenia. To probe the computational consequences of this compression, we used the gradients to regularise subject-specific recurrent neural networks (RNNs) trained on working memory tasks. Networks endowed with greater gradient range learned more efficiently, plateaued at lower task loss, and maintained stronger alignment to the prescribed AS hierarchical geometry. Fixed-point linearisation showed that high-range networks settled into more stable neural states during memory delay, evidenced by lower energy and smaller maximal Jacobian eigenvalues. This gradient-regularised RNN framework thereby links large-scale cortical architecture with fixed point stability, providing a computational hypothesis that AS gradient de-differentiation can destabilise neural computations in schizophrenia, convergently supported by empirical timescale flattening along AS gradient and model-based evidence of less stable fixed points.

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SubatA20/as_spec_rnn

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State: the link answers, verified on 30 September 2026
Evidence: files inventoried
Commit: 6d83652afab9121679262fb478dfe8f801e67dcc, 28 July 2025
Languages: Python (11), Jupyter (1)
Size: 13 files, 12 scripts
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Found in: “Code availability”
Holds: README, 1 notebook
Not found: license file, CITATION.cff, environment file, tests, continuous integration, documentation
Tools: NumPy (12 files), PyTorch (7 files), SciPy (6 files), h5py (3 files), BrainSpace (2 files), Matplotlib (2 files), seaborn (2 files), ArviZ (1 file), pandas (1 file), PyMC (1 file), statsmodels (1 file)
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At the source: github.com/SubatA20/as_spec_rnn

Code availability

All codes used in empirical and theoretical studies are available on the following GitHub repository: https://github.com/SubatA20/as_spec_rnn.

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

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Data used in the preparation of this manuscript were obtained from the National Institute of Mental Health (NIMH) Data Archive (NDA; https://nda.nih.gov/). NDA is a collaborative informatics system created by the National Institutes of Health to provide a national resource to support and accelerate research in mental health. Dataset identifiers: NDAR ID: 2274. This manuscript reflects the views of the authors and may not reflect the opinions or views of the NIH or of the Submitters submitting original data to NDA.

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Version 1, 30 September 2026: the first record

Recorded: type, language, journal, volume, issue, pages, dates, 4 authors, 2 keywords, 12 MeSH terms, 3 funders, 92 references.

Cite

This paper

Abulikemu, S., Radmard, P., Mamalakis, M., & Suckling, J. (2026). Association-sensory spatiotemporal hierarchy and functional gradient-regularised recurrent neural network with implications for schizophrenia. NPJ systems biology and applications, 12(1), 100. https://doi.org/10.1038/s41540-026-00727-x

BibTeX

@article{abulikemu2026association,
author = {Abulikemu, Subati and Radmard, Puria and Mamalakis, Michail and Suckling, John},
title = {{Association-sensory spatiotemporal hierarchy and functional gradient-regularised recurrent neural network with implications for schizophrenia}},
journal = {NPJ systems biology and applications},
year = {2026},
month = apr,
volume = {12},
number = {1},
pages = {100},
publisher = {Nature Publishing Group},
issn = {2056-7189},
doi = {10.1038/s41540-026-00727-x},
url = {https://doi.org/10.1038/s41540-026-00727-x},
pmid = {42056136},
pmcid = {PMC13338132}
}

RIS

TY - JOUR
AU - Abulikemu, Subati
AU - Radmard, Puria
AU - Mamalakis, Michail
AU - Suckling, John
TI - Association-sensory spatiotemporal hierarchy and functional gradient-regularised recurrent neural network with implications for schizophrenia
T2 - NPJ systems biology and applications
J2 - NPJ Syst Biol Appl
PY - 2026
DA - 2026/04/30
VL - 12
IS - 1
SP - 100
SN - 2056-7189
PB - Nature Publishing Group
DO - 10.1038/s41540-026-00727-x
UR - https://doi.org/10.1038/s41540-026-00727-x
LA - en
ER -

CSL-JSON

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"container-title": "NPJ systems biology and applications",
"author": [
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"given": "Subati"
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"PMID": "42056136",
"PMCID": "PMC13338132",
"ISSN": "2056-7189",
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"URL": "https://doi.org/10.1038/s41540-026-00727-x",
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"date-parts": [
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2026,
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