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A minimal model of working memory in neural systems and neuromorphic circuits.

Code ↔ Paper

3 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 3 matches
  1. [1] § Results › Networks of self-excitatory neurons and mean-field models ↔ Network and Mean-field/network_analysis.ipynb, lines 285–373 · score 0.71 · Population firing rate, interval coefficient, Membrane potential, Raster, CV, variation
  2. [2] § Results › Networks of self-excitatory neurons and mean-field models ↔ Network and Mean-field/network_analysis.ipynb, lines 169–220 · score 0.70 · synaptic conductance, eQIF, gsyn, post, pre, quenched
  3. [3] § Results › Networks of self-excitatory neurons and mean-field models ↔ Network and Mean-field/network_analysis.ipynb, lines 285–373 · score 0.56 · inter spike intervals, coefficient, CV, variation, activity, firing

Paper

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The authors' code

Jupyter notebook · 373 lines · 13 KB · no license · 3 matches

  1. # %%
  2. import numpy as np
  3. import matplotlib.pyplot as plt
  4. from scipy.integrate import solve_ivp
  5. from brian2 import *
  6. import collections
  8. # %%
  9. def compute_chi_sliding_window(state_monitor, time_window, f_sample, window_size=50*ms, step_size=10*ms):
  10. """
  11. In each window, compute the population mean voltage and variance, and the mean of individual variances.
  12. χ(t) = sqrt(Var(V_pop) / mean(Var(V_i)))
  14. Input:
  15. state_monitor : StateMonitor, Brian2 StateMonitor with voltage recordings
  16. window_size : time of the sliding window
  17. step_size : time step between successive windows
  18. Returns:
  19. times : array, center times of each window
  20. chi_t : array, time-resolved synchrony measure
  21. """
  22. time_mask = (state_monitor.t/ms >= time_window[0]) & (state_monitor.t/ms < time_window[1])
  23. V_traces = state_monitor.V[:, time_mask] / mV
  24. effective_times = state_monitor.t[time_mask]
  26. # Convert to indices
  27. window_samples = int(window_size / f_sample)
  28. step_samples = int(step_size / f_sample)
  29. n_timepoints = V_traces.shape[1]
  31. chi_t = []
  32. times = []
  34. var_V = []
  35. mean_var_Vi = []
  37. for start_idx in range(0, n_timepoints, step_samples):
  38. end_idx = start_idx + window_samples
  40. # Extract window
  41. V_window = V_traces[:, start_idx:end_idx]
  43. # Compute χ for this window
  44. V_pop_window = np.mean(V_window, axis=0)
  45. sigma_V_squared = np.var(V_pop_window)
  46. sigma_Vi_squared = np.var(V_window, axis=1)
  47. mean_sigma_Vi_squared = np.sum(sigma_Vi_squared)/N
  49. if mean_sigma_Vi_squared > 0:
  50. chi = np.sqrt(sigma_V_squared / mean_sigma_Vi_squared)
  51. else:
  52. chi = 0.0
  54. time = effective_times[start_idx:end_idx].mean()
  55. chi_t.append(chi)
  56. times.append(time/ms)
  57. var_V.append(sigma_V_squared)
  58. mean_var_Vi.append(mean_sigma_Vi_squared)
  61. return np.array(times), np.array(chi_t), np.array(var_V), np.array(mean_var_Vi)
  63. # %%
  64. # Define parameters of the model and the default values for the mean field parameters
  65. DynMemory = collections.namedtuple(
  66. typename='DynMemory',
  67. field_names='C ,g_L ,E_L ,v_th ,T_w ,a ,b ,E_e ,Q_e ,T_s, Eta ,Delta ,v_reset, v_peak')
  69. default_adj_params = DynMemory(
  70. C=200, g_L=10, E_L=-62, v_th=-55, T_w=20, a=4, b=20, E_e=0, Q_e=4, T_s=6, Eta=10, Delta=.5, v_reset=-70., v_peak=-10.)
  72. # Define the adjusted mean field equations for a network of eQIF neurons
  73. # The correction term of the current is discussed in SuppmentaL Material section of the paper
  74. def adjusted_dmMF(t, y, p:DynMemory, I_func):
  75. r, v, w, s = y
  76. C ,g_L ,E_L ,v_th ,T_w ,a ,b ,E_e ,Q_e ,T_s, Eta ,Delta, v_reset, v_peak = p
  77. I0 = I_func(t)
  79. alpha = g_L*(v_th + E_L) + s
  80. beta = g_L*v_th*E_L + s*E_e + w + I0 + Eta
  82. mu = 4*beta/g_L - (alpha/g_L)**2
  84. if mu >= 0:
  85. gamma = np.arctan((2*v_peak - alpha/g_L)/np.sqrt(mu)) - np.arctan((2*v_reset - alpha/g_L)/np.sqrt(mu))
  86. I_adj = g_L*mu*np.pi**2 / (4*gamma**2) + alpha**2 / (4*g_L) -g_L*E_L*v_th - s*E_e - w - Eta
  87. else:
  88. I_adj = I0
  90. dr = r*(g_L*(2*v-v_th-E_L) - s) / C + Delta*g_L/(np.pi*C**2)
  91. dv = (g_L * (E_L - v) * (v_th - v) + w + I_adj + s*(E_e - v) + Eta - (np.pi*C*r)**2/g_L) / C
  92. dw = (a * (v - E_L) - w) / T_w + b*r
  93. ds = -s/T_s + Q_e*r
  95. return np.array([dr, dv, dw, ds])
  97. # %%
  98. # Define time parameters for integration
  99. dt = 0.01
  100. duration_mf = 1500
  101. dtime = dt * ms
  102. duration = duration_mf * ms
  103. defaultclock.dt = dtime
  105. # Set parameters for the mean-field model
  106. adj_params = default_adj_params
  108. # Define parameters with units for Brian2 simulation (same as in mean-field)
  109. N=5000
  111. C = adj_params.C * pF
  112. g_L = adj_params.g_L * nS/mV
  113. E_L = adj_params.E_L * mV
  114. V_T = adj_params.v_th * mV
  115. tau_w = adj_params.T_w * ms
  116. a = adj_params.a * nS
  117. b = adj_params.b * pA
  118. V_reset = adj_params.v_reset * mV
  119. V_peak = adj_params.v_peak * mV
  120. Ee = adj_params.E_e * mV
  121. Qe = adj_params.Q_e/N * nS
  122. Tsyn = adj_params.T_s * ms
  124. # %%
  125. # Define time-varying external current for mean-field simualtion
  126. AmpStep = 60
  127. BaseI = 90
  128. Pert = 10
  129. def input_current(t):
  130. # Definition of the external time-varying current
  131. if t>105 and t<250:
  132. return BaseI+AmpStep # 100-600 ms: 130 pA
  133. elif t>650 and t<720:
  134. return BaseI-Pert
  135. elif t>780 and t<850:
  136. return BaseI+Pert
  137. elif t>1000 and t<1150:
  138. return BaseI-AmpStep
  139. elif t>1250 and t<1320:
  140. return BaseI+Pert
  141. elif t>1350 and t<1420:
  142. return BaseI-Pert
  143. else:
  144. return BaseI
  146. time_array = np.arange(0, duration_mf, dt)
  147. current_array = []
  148. for step in time_array:
  149. current_array.append(input_current(step))
  151. # Define time-varying external current for Brian2 simulation using TimedArray
  152. time_steps = int(duration / dtime)
  153. current_array = np.full(time_steps, BaseI)
  154. current_array[int(100*ms/dtime):int(250*ms/dtime)] = BaseI+AmpStep
  155. current_array[int(650*ms/dtime):int(720*ms/dtime)] = BaseI-Pert
  156. current_array[int(780*ms/dtime):int(850*ms/dtime)] = BaseI+Pert
  157. current_array[int(1000*ms/dtime):int(1150*ms/dtime)] = BaseI-AmpStep
  158. current_array[int(1250*ms/dtime):int(1320*ms/dtime)] = BaseI+Pert
  159. current_array[int(1350*ms/dtime):int(1420*ms/dtime)] = BaseI-Pert
  160. I_t = TimedArray(current_array * pA, dt=dtime)
  162. # %%
  163. # Simulate the adjusted mean-field model with random initial conditions
  164. rnd = np.random.default_rng()
  165. y0 = [np.round(rnd.uniform(.001, .003), 3), np.round(rnd.uniform(-70., -65.), 3),
  166. np.round(rnd.uniform(1., 5.), 3), np.round(rnd.uniform(.001, .005), 3)]
  167. adj_sim = solve_ivp(adjusted_dmMF, (0, duration_mf), y0, args=(adj_params, lambda t: input_current(t)), max_step=dt)
  169. # %%
  170. # Define the spiking neural network in Brian2
  171. start_scope()
  173. # Set shared variable for synaptic conductance
  174. Gsyn = NeuronGroup(1, '''
  175. dGesyn/dt = -Gesyn/Tsyn : siemens
  176. ''', method='rk4')
  177. Gsyn.Gesyn = 0*siemens
  179. # eQIF model equations with synaptic input and external current
  180. adj_eqs = '''
  181. dV/dt = (g_L * (E_L - V) * (V_T - V) + w + I_ext + n - Gesyn*(V-Ee)) / C : volt
  182. dw/dt = (a * (V - E_L) - w) / tau_w : amp
  183. I_ext = I_t(t) : amp
  184. n : amp
  185. Gesyn : siemens (linked)
  186. '''
  188. # Set neuron group
  189. G = NeuronGroup(N, adj_eqs, threshold='V > V_peak', reset='V = V_reset; w += b', method='rk4')
  191. # Initialize variables for quenched heterogeneity
  192. e = adj_params.Eta
  193. d = adj_params.Delta
  194. x = np.linspace(0+1/N,1-1/N,N)
  195. rng = np.random.default_rng()
  196. adj_etas = e + d*np.tan(np.pi*(x-0.5))
  197. rng.shuffle(adj_etas)
  199. # Initialize variables with some randomness
  200. Vinit = np.round(rnd.uniform(-70., -60., size=N), 3)
  201. Winit = np.round(rnd.uniform(1., 5., size=N), 3)
  202. G.n = adj_etas * pA
  203. G.V = Vinit * mV
  204. G.w = Winit * pA
  205. G.Gesyn = linked_var(Gsyn, 'Gesyn')
  207. # Connect neurons
  208. S = Synapses(G, Gsyn, on_pre='Gesyn_post += Qe')
  209. S.connect()
  211. # Monitor variables
  212. adj_M_spike = SpikeMonitor(G)
  213. adj_M_FR = PopulationRateMonitor(G)
  214. # Optional: record membrane potential with a specific dt
  215. # instead of default one (for reduce memory usage)
  216. sample_rate = 0.1*ms
  217. adj_M_voltage = StateMonitor(G, 'V', record=True, dt=sample_rate)
  219. # Continue without recording
  220. run(duration)
  222. # %%
  223. # Safety check: plot raster to verify activity and membrane potential of a few neurons to verify dynamics
  224. raster_activity = np.array([adj_M_spike.t/ms, adj_M_spike.i])
  225. index_mask = (raster_activity[1] <= 1000)
  227. plt.figure(figsize=(15, 5))
  228. plt.plot(raster_activity[0][index_mask], raster_activity[1][index_mask], ',k')
  229. plt.show()
  231. plt.figure(figsize=(12, 6))
  232. for i in range(5):
  233. plt.plot(adj_M_voltage.t/ms, adj_M_voltage.V[i]/mV + i*20, label=f'Neuron {i}') # Offset for visibility
  234. plt.xlim(50, 1100)
  235. plt.show()
  237. # %%
  238. # Compute ISI Coefficient of Variation (CV) for each neuron within stimulation
  239. # peak period (50-1100 ms). Sliding windows used to capture time-varying behaviors
  240. time_interval = 75
  241. time_windows = np.arange(50, 1100 + time_interval, time_interval)
  243. N = int(np.max(raster_activity[1])) + 1
  244. cv_measure = [[] for _ in range(len(time_windows) - 1)]
  246. for i in range(len(time_windows) - 1):
  247. mask = (raster_activity[0] >= time_windows[i]) & (raster_activity[0] < time_windows[i+1])
  249. N = int(np.max(raster_activity[1])) + 1
  250. results = [[] for _ in range(N)]
  252. for time, idx in zip(raster_activity[0][mask], raster_activity[1][mask].astype(int)):
  253. results[idx].append(np.round(time,2))
  255. results = [times for times in results if len(times) > 0]
  256. cvs = [np.std(np.diff(times)) / np.mean(np.diff(times)) if len(times) > 2 else np.nan for times in results]
  258. cv_measure[i] = cvs
  260. # Extract mean, min, max, std for each time window and create arrays for plotting
  261. time_bin_len = int(time_interval / (dtime/ms))
  262. cv_means = np.concatenate([np.ones(time_bin_len) * mean
  263. for mean in [np.nanmean(cvs) for cvs in cv_measure]])
  264. cv_mins = np.concatenate([np.ones(time_bin_len) * mn
  265. for mn in [np.nanmin(cvs) for cvs in cv_measure]])
  266. cv_maxs = np.concatenate([np.ones(time_bin_len) * mx
  267. for mx in [np.nanmax(cvs) for cvs in cv_measure]])
  268. cv_stds = np.concatenate([np.ones(time_bin_len) * std
  269. for std in [np.nanstd(cvs) for cvs in cv_measure]])
  271. # %%
  272. # Compute time-resolved synchrony measure based on membrane potential and plot it.
  273. # NOTE: the more fsample to store the membrane values is small the more the figure is refined
  274. times, chi_t, var_V, mean_var_Vi = compute_chi_sliding_window(adj_M_voltage, [50, 1100], f_sample=sample_rate, window_size=10*ms, step_size=2*ms)
  276. plt.figure(figsize=(12, 4))
  277. ax = plt.gca()
  278. ax.plot(adj_M_FR.t[int((50/dtime)*ms):]/ms, current_array[int((50/dtime)*ms):])
  279. ax2 = ax.twinx()
  280. ax2.plot(times, chi_t)
  281. ax2.set_ylabel('χ(t)')
  282. ax2.set_xlim(50, 1500)
  283. plt.show()
  285. # %%
  286. # Final figure: top three panels full range, bottom two zoomed and aligned
  289. # Grid and style
  290. #grid_scheme = GridSpec(nrows=5, ncols=1, height_ratios=[.5, 1, 1, .5, .5], hspace=0.5)
  291. plt.rcParams.update({
  292. 'font.size': 20, # Controls default text size
  293. 'axes.titlesize': 20, # Title font size
  294. 'axes.labelsize': 20, # X/Y label font size
  295. 'xtick.labelsize': 18, # X tick labels
  296. 'ytick.labelsize': 18, # Y tick labels
  297. 'legend.fontsize': 14, # Legend font size
  298. })
  299. fig, axs = plt.subplots(5, 1 , height_ratios=[.5, 1, 1, .5, .5], figsize=(12, 12), sharex=True)
  300. ax0, ax1, ax2, ax3, ax4 = axs[0], axs[1], axs[2], axs[3], axs[4]
  303. # Plot time ranges (ms)
  304. full_start, full_end = 0, 1500
  305. zoom_start, zoom_end = 50, 1100
  307. # (a) Current array ---
  308. ax0.plot(adj_M_FR.t/ms, current_array, color='purple')
  309. ax0.set_xlim(full_start, full_end)
  310. ax0.set_ylabel('$I_{ext}$ (pA)')
  311. ax0.xaxis.set_visible(False)
  312. ax0.spines['top'].set_visible(False)
  313. ax0.spines['right'].set_visible(False)
  315. # (b) Population firing rate
  316. adj_M_FRsmt = adj_M_FR.smooth_rate(window='flat', width=1.01*ms)
  317. ax1.plot(adj_M_FR.t/ms, adj_M_FRsmt/Hz, 'k', label='SNN')
  318. ax1.plot(adj_sim.t, adj_sim.y[0]*1000, 'r', lw=2, label='MF')
  319. ax1.set_ylabel('$FR$ (Hz)')
  320. ax1.set_xlim(full_start, full_end)
  321. ax1.set_ylim(0,150)
  322. ax1.legend(loc='upper right')
  323. ax1.xaxis.set_visible(False)
  324. ax1.spines['top'].set_visible(False)
  325. ax1.spines['right'].set_visible(False)
  327. # (c) Raster plot
  328. # (green shaded area highlight the region were the measures were computed)
  329. index_mask = (raster_activity[1] <= 1000)
  330. ax2.plot(raster_activity[0][index_mask], raster_activity[1][index_mask], ',k')
  331. ax2.fill([zoom_start, zoom_start, zoom_end, zoom_end], [0, 1000, 1000, 0], color='lightgreen', alpha=0.5)
  332. ax2.set_xlim(full_start, full_end)
  333. ax2.set_ylabel('Neuron #')
  334. ax2.spines['top'].set_visible(False)
  335. ax2.spines['right'].set_visible(False)
  337. # (d) χ(t) synchrony measure (based on membrane potential)
  338. ax3.plot(times, chi_t, 'k')
  339. ax3.set_ylabel('χ(t)')
  340. ax3.xaxis.set_visible(False)
  341. ax3.spines['top'].set_visible(False)
  342. ax3.spines['right'].set_visible(False)
  344. # (e) Mean inter-spike intervals Coefficient of variation (CV) across the population
  345. cv_time = np.linspace(zoom_start, zoom_end, len(cv_means))
  346. yerr_lower = cv_means - cv_mins
  347. yerr_upper = cv_maxs - cv_means
  348. ax4.errorbar(cv_time, cv_means, yerr=[yerr_lower, yerr_upper], fmt='none', ecolor='orange', alpha=0.01)
  349. ax4.plot(cv_time, cv_means, 'k')
  350. ax4.set_ylabel('CV')
  351. ax4.set_xlabel('t (ms)')
  352. ax4.spines['top'].set_visible(False)
  353. ax4.spines['right'].set_visible(False)
  356. # Reposition bottom axes to match the highlighted horizontal region under the full-width axes
  357. # Align y-labels across subplots and ensure left margin is sufficient
  358. fig.align_ylabels(axs)
  359. fig.subplots_adjust(left=0.12)
  360. fig.canvas.draw()
  361. full_pos = ax1.get_position()
  363. # Place aligned subplot letters using figure coordinates (left of y-axis labels)
  364. # compute a common x in figure coords slightly left of the full axis left edge
  365. x_fig = full_pos.x0 - .12
  366. axes_list = [ax0, ax1, ax2, ax3, ax4]
  367. letters = ['a', 'b', 'c', 'd', 'e']
  368. for ax, letter in zip(axes_list, letters):
  369. pos = ax.get_position()
  370. y_fig = pos.y0 + pos.height * 0.98
  371. fig.text(x_fig, y_fig, letter, va='top', ha='left')
  373. plt.show()

network_analysis.ipynb at commit 5021ac2, no license · at the source

Overview

Authors: Damien Depannemaecker1, Adrien d’Hollande2, Gabriele Casagrande1, Jiaming Wu2, Marcelo J. Rozenberg2,3,4
ORCID iDs: Damien Depannemaecker, Adrien d’Hollande, Gabriele Casagrande, Marcelo J. Rozenberg
  1. Aix Marseille Université, INSERM, Institut de Neurosciences des Systèmes (INS),Marseille, France
  2. Université Paris-Saclay, CNRS, Laboratoire de Physique des Solides,Orsay, France
  3. Université Paris-Cité, CNRS, Integrative Neuroscience and Cognition Center,Paris, France
  4. Department of Physics, University of California,San Diego, CA USA
Institutions: Institut de Neurosciences des Systèmes (France); Laboratoire de physique des Solides (France); Centre Neurosciences intégratives et Cognition (France); University of California San Diego (United States)
Journal: Nature communications, volume 17, issue 1, article 9867
Dates: received 10 November 2025; accepted 9 July 2026; published online 17 August 2026
Type: Research article · Language: English
License: CC BY
Identifiers: DOI 10.1038/s41467-026-75704-3 · PMID 42744781 · PMCID PMC13578268 · OpenAlex W7203623913
Open access: gold, a free copy (OpenAlex)
Status: code verified
Categories: computational modeling (no new data) (modality), human (organism)
Keywords: Computational neuroscience, Electronics, photonics and device physics
MeSH: Memory, Short-Term*, Models, Neurological*, Nerve Net*, Neural Networks, Computer*, Neurons*, Action Potentials, Animals, Computer Simulation, Humans (* major topic)
Topic: Advanced Memory and Neural Computing (Electrical and Electronic Engineering, Engineering), according to OpenAlex
Funding: EC | Horizon 2020 Framework Programme (EU Framework Programme for Research and Innovation H2020) (SGA 101147319); Agence Nationale de la Recherche (French National Research Agency) (ANR-23-CE30-0040)
Citations: not cited yet (Europe PMC); 59 references in the paper

Abstract

Phenomenological spiking neuron models such as Izhikevich, adaptive quadratic integrate-and-fire (aQIF), and Adaptive Exponential (AdEx) are widely used because of their simplicity and numerical efficiency. These models reproduce diverse neuronal dynamics through a slow self-inhibitory adaptation variable. Here we introduce their symmetric counterpart by replacing adaptation with slow self-excitation, motivated by intrinsic calcium-mediated membrane currents. This minimal modification enables robust persistent spiking and working-memory dynamics without compromising computational efficiency. These properties remain in excitatory spiking neural networks. We then derive and validate a mean-field neural mass model that remains stable while retaining working-memory functionality. Additionally, we implement the single-neuron model in a minimal memristor-based neuromorphic circuit and experimentally confirm its dynamics. These results provide scalable tools for large-scale brain simulations and neuromorphic applications in robotics, brain-machine interfaces, and edge AI devices.

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GabrieleCasagrande/MinimalNeuro_WorkingMemo

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Cite

This paper

Depannemaecker, D., d’Hollande, A., Casagrande, G., Wu, J., & Rozenberg, M. J. (2026). A minimal model of working memory in neural systems and neuromorphic circuits. Nature communications, 17(1), 9867. https://doi.org/10.1038/s41467-026-75704-3

BibTeX

@article{depannemaecker2026minimal,
author = {Depannemaecker, Damien and d’Hollande, Adrien and Casagrande, Gabriele and Wu, Jiaming and Rozenberg, Marcelo J.},
title = {{A minimal model of working memory in neural systems and neuromorphic circuits}},
journal = {Nature communications},
year = {2026},
month = aug,
volume = {17},
number = {1},
pages = {9867},
publisher = {Nature Publishing Group},
issn = {2041-1723},
doi = {10.1038/s41467-026-75704-3},
url = {https://doi.org/10.1038/s41467-026-75704-3},
pmid = {42744781},
pmcid = {PMC13578268}
}

RIS

TY - JOUR
AU - Depannemaecker, Damien
AU - d’Hollande, Adrien
AU - Casagrande, Gabriele
AU - Wu, Jiaming
AU - Rozenberg, Marcelo J.
TI - A minimal model of working memory in neural systems and neuromorphic circuits
T2 - Nature communications
J2 - Nat Commun
PY - 2026
DA - 2026/08/17
VL - 17
IS - 1
SP - 9867
SN - 2041-1723
PB - Nature Publishing Group
DO - 10.1038/s41467-026-75704-3
UR - https://doi.org/10.1038/s41467-026-75704-3
LA - en
ER -

CSL-JSON

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"id": "10.1038/s41467-026-75704-3",
"type": "article-journal",
"title": "A minimal model of working memory in neural systems and neuromorphic circuits",
"container-title": "Nature communications",
"author": [
{
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"given": "Damien"
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{
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{
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},
{
"family": "Wu",
"given": "Jiaming"
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{
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"given": "Marcelo J."
}
],
"container-title-short": "Nat Commun",
"volume": "17",
"issue": "1",
"page": "9867",
"DOI": "10.1038/s41467-026-75704-3",
"PMID": "42744781",
"PMCID": "PMC13578268",
"ISSN": "2041-1723",
"publisher": "Nature Publishing Group",
"URL": "https://doi.org/10.1038/s41467-026-75704-3",
"language": "en",
"issued": {
"date-parts": [
[
2026,
8,
17
]
]
}
}

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