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A mean-field model of neural networks with PV and SOM interneurons reveals connectivity-based mechanisms of gamma oscillations.

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Jupyter notebook · 306 lines · 7.9 KB · CC-BY-4.0

  1. # %%
  2. import numpy as np
  3. import matplotlib.pyplot as plt
  4. from mpmath import *
  5. from scipy.stats import norm
  6. from scipy.optimize import minimize
  7. import scipy.special as sp_spec
  9. # %%
  10. PPV=[-0.05233495, 0.00547203, -0.01033892, -0.01446707, -0.00092874,
  11. 0.00032317, 0.0025617 , 0.0231148 , -0.0234656 , -0.0348654 ]
  13. PRS=[-0.0497335 , 0.00799996, 0.00923167, -0.00277308, 0.02764466,
  14. -0.03766599, 0.0298739 , -0.02301237, -0.00580462, 0.02240026]
  16. PSST=[-4.67628863e-02, 6.30755857e-05, -3.22476692e-03, -1.14962091e-02,
  17. 9.68216410e-04, 1.39817715e-03, 9.39652694e-03, 1.30900273e-02,
  18. 2.24057900e-02, 1.07883016e-02]
  20. # %%
  21. # Parameters
  22. Gl=10*1.e-9; Cm=200*1.e-12; Ee=0; Ei=-80*1.e-3; bRS=50*1.e-12; ars=4*1.e-9; bsst=80*1.e-12; asst=4*1.e-9; twRS=500*1.e-3;
  24. Qi1=5*1.e-9; Qi2=5.*1.e-9; Ti=8*1.e-3; Te=2*1.e-3;
  26. fext = 4; Qext = 2.5*1.e-9;
  28. Elsst=-55*1.e-3;
  29. Elrs=-65*1.e-3;
  30. Elpv=-65*1.e-3;
  32. pconnec = .07;
  34. NRS, NPV, NLTS = 8000 , 750 , 750
  36. # %%
  37. def meanv_rs(fexc,finh1,finh2,adaptation):
  39. El=-65*1.e-3;
  40. Qe=2*1.e-9
  42. fe = fexc*pconnec*NRS;
  43. fi1 = finh1*pconnec*NPV;
  44. fi2 = finh2*pconnec*NLTS;
  48. finp = fext*pconnec*NRS;
  51. muGi1 = Qi1*Ti*fi1;
  52. muGi2 = Qi2*Ti*fi2;
  53. muGe = Qe*Te*fe;
  55. muGext = Qext*Te*finp;
  57. muG = Gl+muGe+muGi1+muGi2+muGext;
  58. muV = (muGe*Ee+muGext*Ee+(muGi1+muGi2)*Ei+Gl*El-adaptation)/(muG);
  61. return muV
  63. # %%
  64. def meanv_sst(fexc,finh1,finh2,adaptation):
  67. El=-55*1.e-3;
  68. Qe=3*1.e-9
  69. fe = fexc*pconnec*NRS;
  70. fi1 = finh1*pconnec*NPV;
  71. fi2 = finh2*pconnec*NLTS;
  73. Qi1 = 0.*1.e-9
  74. Qi2 = 0*1.e-9
  77. finp = 0.;
  80. muGi1 = Qi1*Ti*fi1;
  81. muGi2 = Qi2*Ti*fi2;
  82. muGe = Qe*Te*fe;
  84. muGext = 0.;
  86. muG = Gl+muGe+muGi1+muGi2+muGext;
  87. muV = (muGe*Ee+muGext*Ee+(muGi1+muGi2)*Ei+Gl*El-adaptation)/(muG);
  90. return muV
  92. # %%
  93. def TF_RS(P,fexc,finh1,finh2,adapt):
  95. El=-65*1.e-3;
  96. Qe=2*1.e-9
  99. fe = fexc*pconnec*NRS;
  100. fi1 = finh1*pconnec*NPV;
  101. fi2 = finh2*pconnec*NLTS;
  104. finp = fext*pconnec*NRS;
  107. muGi1 = Qi1*Ti*fi1;
  108. muGi2 = Qi2*Ti*fi2;
  109. muGe = Qe*Te*fe;
  111. muGext = Qext*Te*finp;
  113. muG = Gl+muGe+muGi1+muGi2+muGext;
  114. muV = (muGe*Ee+muGext*Ee+(muGi1+muGi2)*Ei+Gl*El-adapt)/(muG);
  117. muGn = muG/Gl;
  118. Tm = Cm/muG;
  120. Ue = Qe/muG*(Ee-muV);
  121. Ui1 = Qi1/muG*(Ei-muV);
  122. Ui2 = Qi2/muG*(Ei-muV);
  123. Uext = Qext/muG*(Ee-muV);
  125. sV = sqrt(fe*(Ue*Te)*(Ue*Te)/2./(Te+Tm)+finp*(Uext*Te)*(Uext*Te)/2./(Te+Tm)+fi1*(Ui1*Ti)*(Ui1*Ti)/2./(Ti+Tm)+fi2*(Ui2*Ti)*(Ui2*Ti)/2./(Ti+Tm));
  126. fe= fe+1e-9;
  127. fi1=fi1+1e-9;
  128. fi2=fi2+1e-9;
  129. Tv = ( fe*(Ue*Te)*(Ue*Te) + finp*(Uext*Te)*(Uext*Te) + fi1*(Qi1*Ui1)*(Qi1*Ui1)+fi2*(Qi2*Ui2)*(Qi2*Ui2)) /( fe*(Ue*Te)*(Ue*Te)/(Te+Tm) + finp*(Uext*Te)*(Uext*Te)/(Te+Tm) +fi1*(Qi1*Ui1)*(Qi1*Ui1)/(Ti+Tm) + fi2*(Qi2*Ui2)*(Qi2*Ui2)/(Ti+Tm));
  130. TvN = Tv*Gl/Cm;
  131. muV0 = -60e-3;
  132. DmuV0 = 50e-3;
  133. sV0 = 4e-3;
  134. DsV0 = 6e-3;
  135. TvN0 = 0.5;
  136. DTvN0 = 1.;
  137. vthr = P[0] + P[1]*(muV-muV0)/DmuV0 + P[2]*(sV-sV0)/DsV0 + P[3]*(TvN-TvN0)/DTvN0 \
  138. + P[4]*((muV-muV0)/DmuV0)*((muV-muV0)/DmuV0) + P[5]*(muV-muV0)/DmuV0*(sV-sV0)/DsV0 + P[6]*(muV-muV0)/DmuV0*(TvN-TvN0)/DTvN0 \
  139. + P[7]*((sV-sV0)/DsV0)*((sV-sV0)/DsV0) + P[8]*(sV-sV0)/DsV0*(TvN-TvN0)/DTvN0 + P[9]*((TvN-TvN0)/DTvN0)*((TvN-TvN0)/DTvN0);
  140. vthr = float(vthr)
  141. muV = float(muV)
  142. sV = float(sV)
  143. #####
  144. frout=.5/TvN*Gl/Cm*sp_spec.erfc((vthr-muV)/np.sqrt(2)/sV)
  145. return frout;
  147. # %%
  148. def TF_PV(P,fexc,finh1,finh2,adapt):
  150. El=-65*1.e-3;
  151. Qe=3*1.e-9
  153. fe = fexc*pconnec*NRS;
  154. fi1 = finh1*pconnec*NPV;
  155. fi2 = finh2*pconnec*NLTS;
  159. finp = fext*pconnec*NRS;
  162. muGi1 = Qi1*Ti*fi1;
  163. muGi2 = Qi2*Ti*fi2;
  164. muGe = Qe*Te*fe;
  166. muGext = Qext*Te*finp;
  168. muG = Gl+muGe+muGi1+muGi2+muGext;
  169. muV = (muGe*Ee+muGext*Ee+(muGi1+muGi2)*Ei+Gl*El-adapt)/(muG);
  172. muGn = muG/Gl;
  173. Tm = Cm/muG;
  175. Ue = Qe/muG*(Ee-muV);
  176. Ui1 = Qi1/muG*(Ei-muV);
  177. Ui2 = Qi2/muG*(Ei-muV);
  178. Uext = Qext/muG*(Ee-muV);
  180. sV = sqrt(fe*(Ue*Te)*(Ue*Te)/2./(Te+Tm)+finp*(Uext*Te)*(Uext*Te)/2./(Te+Tm)+fi1*(Ui1*Ti)*(Ui1*Ti)/2./(Ti+Tm)+fi2*(Ui2*Ti)*(Ui2*Ti)/2./(Ti+Tm));
  181. fe= fe+1e-9;
  182. fi1=fi1+1e-9;
  183. fi2=fi2+1e-9;
  184. Tv = ( fe*(Ue*Te)*(Ue*Te) + finp*(Uext*Te)*(Uext*Te) + fi1*(Qi1*Ui1)*(Qi1*Ui1)+fi2*(Qi2*Ui2)*(Qi2*Ui2)) /( fe*(Ue*Te)*(Ue*Te)/(Te+Tm) + finp*(Uext*Te)*(Uext*Te)/(Te+Tm) +fi1*(Qi1*Ui1)*(Qi1*Ui1)/(Ti+Tm) + fi2*(Qi2*Ui2)*(Qi2*Ui2)/(Ti+Tm));
  185. TvN = Tv*Gl/Cm;
  186. muV0 = -60e-3;
  187. DmuV0 = 10e-3;
  188. sV0 = 4e-3;
  189. DsV0 = 6e-3;
  190. TvN0 = 0.5;
  191. DTvN0 = 1.;
  192. vthr = P[0] + P[1]*(muV-muV0)/DmuV0 + P[2]*(sV-sV0)/DsV0 + P[3]*(TvN-TvN0)/DTvN0 \
  193. + P[4]*((muV-muV0)/DmuV0)*((muV-muV0)/DmuV0) + P[5]*(muV-muV0)/DmuV0*(sV-sV0)/DsV0 + P[6]*(muV-muV0)/DmuV0*(TvN-TvN0)/DTvN0 \
  194. + P[7]*((sV-sV0)/DsV0)*((sV-sV0)/DsV0) + P[8]*(sV-sV0)/DsV0*(TvN-TvN0)/DTvN0 + P[9]*((TvN-TvN0)/DTvN0)*((TvN-TvN0)/DTvN0);
  195. vthr = float(vthr)
  196. muV = float(muV)
  197. sV = float(sV)
  198. #####
  199. frout=.5/TvN*Gl/Cm*sp_spec.erfc((vthr-muV)/np.sqrt(2)/sV)
  200. return frout;
  202. # %%
  203. def TF_SST(P,fexc,finh1,finh2,adapt):
  205. El = -55*1.e-3;
  206. Qe = 3*1.e-9
  208. fe = fexc*pconnec*NRS;
  209. fi1 = finh1*pconnec*NPV;
  210. fi2 = finh2*pconnec*NLTS;
  212. Qi1 = 0*1.e-9
  213. Qi2 = 0*1.e-9
  216. finp = 0.;
  219. muGi1 = Qi1*Ti*fi1;
  220. muGi2 = Qi2*Ti*fi2;
  221. muGe = Qe*Te*fe;
  223. muGext = 0.;
  225. muG = Gl+muGe+muGi1+muGi2+muGext;
  226. muV = (muGe*Ee+muGext*Ee+(muGi1+muGi2)*Ei+Gl*El-adapt)/(muG);
  229. muGn = muG/Gl;
  230. Tm = Cm/muG;
  232. Ue = Qe/muG*(Ee-muV);
  233. Ui1 = Qi1/muG*(Ei-muV);
  234. Ui2 = Qi2/muG*(Ei-muV);
  235. Uext = 0.;
  237. sV = sqrt(fe*(Ue*Te)*(Ue*Te)/2./(Te+Tm)+finp*(Uext*Te)*(Uext*Te)/2./(Te+Tm)+fi1*(Ui1*Ti)*(Ui1*Ti)/2./(Ti+Tm)+fi2*(Ui2*Ti)*(Ui2*Ti)/2./(Ti+Tm));
  238. fe= fe+1e-9;
  239. fi1=fi1+1e-9;
  240. fi2=fi2+1e-9;
  241. Tv = ( fe*(Ue*Te)*(Ue*Te) + finp*(Uext*Te)*(Uext*Te) + fi1*(Qi1*Ui1)*(Qi1*Ui1)+fi2*(Qi2*Ui2)*(Qi2*Ui2)) /( fe*(Ue*Te)*(Ue*Te)/(Te+Tm) + finp*(Uext*Te)*(Uext*Te)/(Te+Tm) +fi1*(Qi1*Ui1)*(Qi1*Ui1)/(Ti+Tm) + fi2*(Qi2*Ui2)*(Qi2*Ui2)/(Ti+Tm));
  242. TvN = Tv*Gl/Cm;
  243. muV0=-50e-3;
  244. DmuV0 = 10e-3;
  245. sV0 = 4e-3;
  246. DsV0= 6e-3;
  247. TvN0=0.5;
  248. DTvN0 = 1.;
  249. vthr = P[0] + P[1]*(muV-muV0)/DmuV0 + P[2]*(sV-sV0)/DsV0 + P[3]*(TvN-TvN0)/DTvN0 \
  250. + P[4]*((muV-muV0)/DmuV0)*((muV-muV0)/DmuV0) + P[5]*(muV-muV0)/DmuV0*(sV-sV0)/DsV0 + P[6]*(muV-muV0)/DmuV0*(TvN-TvN0)/DTvN0 \
  251. + P[7]*((sV-sV0)/DsV0)*((sV-sV0)/DsV0) + P[8]*(sV-sV0)/DsV0*(TvN-TvN0)/DTvN0 + P[9]*((TvN-TvN0)/DTvN0)*((TvN-TvN0)/DTvN0);
  252. vthr = float(vthr)
  253. muV = float(muV)
  254. sV = float(sV)
  255. #####
  256. frout=.5/TvN*Gl/Cm*sp_spec.erfc((vthr-muV)/np.sqrt(2)/sV)
  257. return frout;
  259. # %%
  260. tfinal = 4 ; dt=0.0001; tsteps=int(tfinal/dt); t=np.linspace(0,tfinal,tsteps);
  261. tau_l = 0.002
  262. #Initial conditions
  263. fecont = 1 ; ficont1 = 10 ; ficont2 = 3 ; LSfe = [fecont]*(1+int(tau_l/dt)) ; LSfi1 = [ficont1]*(1+int(tau_l/dt)); LSfi2 = [ficont2]*(1+int(tau_l/dt))
  264. w = 5.39860858e-11; LSw = [w]*(1+int(tau_l/dt)); w2 = 2.00949144e-10 ; LSw2 = [w2]*(1+int(tau_l/dt))
  266. #Solving differential equations
  267. for i in range(int(tau_l/dt) , len(t)):
  269. muvrs=meanv_rs(LSfe[i - int(tau_l/dt)], LSfi1[i - int(tau_l/dt)],LSfi2[i - int(tau_l/dt)],w)
  270. muvsst=meanv_sst(LSfe[i - int(tau_l/dt)], LSfi1[i - int(tau_l/dt)] ,LSfi2[i - int(tau_l/dt)], w2)
  272. fecont += dt/0.015*(TF_RS(PRS, LSfe[i - int(tau_l/dt)], LSfi1[i - int(tau_l/dt)], LSfi2[i - int(tau_l/dt)], w) - LSfe[i]) #RS
  273. ficont1 += dt/0.015*(TF_PV(PPV, LSfe[i - int(tau_l/dt)], LSfi1[i - int(tau_l/dt)], LSfi2[i - int(tau_l/dt)], 0) - LSfi1[i]) #PV
  274. ficont2 += dt/0.015*(TF_SST(PSST, LSfe[i - int(tau_l/dt)], LSfi1[i - int(tau_l/dt)] , LSfi2[i - int(tau_l/dt)] , w2) - LSfi2[i]) #SOM
  276. w += dt/twRS*(-w+bRS*LSfe[i]*twRS+ars*(muvrs-Elrs))
  277. w2 += dt/twRS*(-w2+bsst*LSfi2[i]*twRS+asst*(muvsst-Elsst))
  280. #Appending
  281. LSfe.append(float(fecont))
  282. LSfi1.append(float(ficont1))
  283. LSfi2.append(float(ficont2))
  284. LSw.append(float(w))
  285. LSw2.append(float(w2))
  289. LSfe.pop(0)
  290. LSfi1.pop(0)
  291. LSfi2.pop(0)
  293. ####Plotting
  294. plt.figure(figsize = (19,5))
  295. plt.title('Mean-Field', fontsize=15)
  297. plt.plot(t, LSfe, label='E', color='green')
  298. plt.plot(t, LSfi1, label='PV', color='red')
  299. plt.plot(t, LSfi2, label='SOM', color='blue')
  300. plt.legend(loc='best', fontsize=10)
  302. plt.xlabel('Time(Second)', fontsize=10)
  303. plt.ylabel('Rate(HZ)', fontsize=10)
  305. # %%

E_PV_SOM_Meanfield.ipynb, under CC-BY-4.0 · at the source

Overview

Authors: Farzin Tahvili1, Martin Vinck2, Matteo Di Volo1
ORCID iDs: Martin Vinck, Matteo Di Volo
  1. Université Lyon 1, INSERM, SBRI, U1208, Lyon, France
  2. Donders Centre for Neuroscience, Department of Neurophysics, Radboud University Nijmegen, Nijmegen, Netherlands
Institutions: Lyon 1 Université (France); Inserm (France); Institut Cellule Souche et Cerveau (France); Radboud University Nijmegen (Netherlands); Donders Centre for Neuroscience (Netherlands)
Journal: PLoS computational biology, volume 22, issue 6, article e1014378
Dates: received 3 November 2025; accepted 28 May 2026; published online 10 June 2026
Type: Research article · Language: English
License: CC BY
Identifiers: DOI 10.1371/journal.pcbi.1014378 · PMID 42268928 · PMCID PMC13286281 · OpenAlex W4415504493
Open access: gold, a free copy (OpenAlex)
Preprint: doi:10.1101/2025.10.23.684108
Status: code verified
Categories: none (in silico) (organism), computational (subfield)
Methods: Single-unit activity, calcium imaging
MeSH: Gamma Rhythm*, Interneurons*, Models, Neurological*, Nerve Net*, Parvalbumins*, Somatostatin*, Animals, Computational Biology, Computer Simulation (* major topic)
Journal subjects: Biology and Life Sciences, Cell Biology, Cellular Types, Animal Cells, Neurons, Interneurons, Neuroscience, Cellular Neuroscience, Physical Sciences, Mathematics, Algebra, Linear Algebra, Eigenvalues, Physiology, Electrophysiology, Membrane Potential, Action Potentials, Neurophysiology, Computer and Information Sciences, Neural Networks, Research and Analysis Methods, Simulation and Modeling, Population Biology, Population Dynamics, Mathematical and Statistical Techniques, Mathematical Functions, Transfer Functions
Topic: stochastic dynamics and bifurcation (Statistical and Nonlinear Physics, Physics and Astronomy), according to OpenAlex
Funding: Agence Nationale de la Recherche (French National Research Agency) (ANR-11-LABX-0042); European Research Council (ERC starting grant (850861)); DFG VI Grants (908/5-1 and 908/7-1; 505660261; 520285844; SPP LOOPS); NWO VIDI Grant; Dutch Brain Interface Initiative (DBI2)
Citations: not cited yet (Europe PMC); 75 references in the paper

Abstract

Classic theoretical models of cortical oscillations are based on the interactions between two populations of excitatory and inhibitory neurons. Nevertheless, experimental studies and network simulations suggest that interneuron subclasses such as parvalbumin (PV) and somatostatin (SOM) exert distinct control over oscillatory dynamics. Yet, we lack a theoretical understanding of the mechanisms underlying oscillations in E-PV-SOM circuits and of the differences with respect to the classical mechanisms for oscillations in simpler E–I networks. Here, we derive a biologically realistic mean-field model of a canonical three-population E-PV-SOM circuit. This model robustly generates oscillations whose features are consistent with experimental observations, including the relative timing of PV and SOM activity and the effects of optogenetic perturbations. By reducing the model to a linear analytical form, we demonstrate that gamma oscillations emerge directly from the cell-specific connectivity of the three-population circuit. This connectivity motif alone accounts for experimentally observed phase relationships, with PV activity consistently leading that of SOM neurons. Together, this mean field model identifies a distinct structural mechanism giving rise to oscillations in canonical E–PV–SOM circuits and provides theoretical primitives for constructing large-scale, cell-type-specific models of cortical dynamics.

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Zenodo 20391404

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Data Availability

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Recorded: type, language, journal, volume, issue, pages, dates, 3 authors, 9 MeSH terms, 5 funders, 68 references.

Cite

This paper

Tahvili, F., Vinck, M., & Di Volo, M. (2026). A mean-field model of neural networks with PV and SOM interneurons reveals connectivity-based mechanisms of gamma oscillations. PLoS computational biology, 22(6), e1014378. https://doi.org/10.1371/journal.pcbi.1014378

BibTeX

@article{tahvili2026mean,
author = {Tahvili, Farzin and Vinck, Martin and Di Volo, Matteo},
title = {{A mean-field model of neural networks with PV and SOM interneurons reveals connectivity-based mechanisms of gamma oscillations}},
journal = {PLoS computational biology},
year = {2026},
month = jun,
volume = {22},
number = {6},
pages = {e1014378},
publisher = {PLOS},
issn = {1553-734X},
doi = {10.1371/journal.pcbi.1014378},
url = {https://doi.org/10.1371/journal.pcbi.1014378},
pmid = {42268928},
pmcid = {PMC13286281}
}

RIS

TY - JOUR
AU - Tahvili, Farzin
AU - Vinck, Martin
AU - Di Volo, Matteo
TI - A mean-field model of neural networks with PV and SOM interneurons reveals connectivity-based mechanisms of gamma oscillations
T2 - PLoS computational biology
J2 - PLoS Comput Biol
PY - 2026
DA - 2026/06/10
VL - 22
IS - 6
SP - e1014378
SN - 1553-734X
PB - PLOS
DO - 10.1371/journal.pcbi.1014378
UR - https://doi.org/10.1371/journal.pcbi.1014378
LA - en
ER -

CSL-JSON

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"publisher": "PLOS",
"URL": "https://doi.org/10.1371/journal.pcbi.1014378",
"language": "en",
"issued": {
"date-parts": [
[
2026,
6,
10
]
]
}
}

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