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Sensitivity analysis enlightens effects of connectivity in a Neural Mass Model under Control-Target mode.

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  1. #!/usr/bin/env python3
  2. # -*- coding: utf-8 -*-
  3. """
  4. Created on Tue Jan 20 14:39:12 2026
  6. @author: anais
  7. """
  9. from numpy import *
  10. import math
  13. #%% parameters for the neural model
  15. def get_FixedParametersForCurrents() :
  18. Jnmda=0.15
  19. G=0.69
  20. Wplus = 1.4
  21. Wminus = 1
  22. Wi = 0.7
  23. We = 1
  24. I0 = 0.382
  27. global autocoupling_E
  28. global basal_forcing_E
  29. global coupling_EtoI
  30. global autocoupling_I_abs
  31. global basal_forcing_I
  33. autocoupling_E = Wplus*Jnmda
  35. basal_forcing_E = We * I0
  37. coupling_EtoI = Jnmda
  39. autocoupling_I_abs = Wminus #inhibitory autocoupling (in absolute value)
  41. basal_forcing_I = Wi*I0
  44. return(autocoupling_E,basal_forcing_E,coupling_EtoI,autocoupling_I_abs,basal_forcing_I,Jnmda,G)
  47. def get_FixedParametersForStates() :
  50. BetaE=6.6
  51. AlphaE=72
  53. BetaI=180
  54. AlphaI=530
  56. return(BetaE,AlphaE,BetaI,AlphaI)
  60. def get_FixedParametersForExcFiringRates() :
  62. a=310.
  63. b=125
  64. d=0.16
  65. return(a,b,d)
  71. def get_FixedParametersForInhFiringRates() :
  73. a=615.
  74. b=177
  75. d=0.087
  76. return(a,b,d)
  79. #%% parameters for the hemodynamic and bold models
  81. def get_FixedParametersForHemodynamics() :
  83. alpha=0.32
  84. gamma=0.41
  85. kappa=0.65
  86. tau=0.98
  87. rho=0.34
  89. return(alpha,gamma,kappa,tau,rho)
  92. def get_FixedParametersForBold() :
  95. Theta0=28.265*3
  96. rho=0.34
  97. TE=0.03 #30ms
  98. epsilon=0.47
  99. r0=110
  100. V0=0.02
  103. k1 = 4.3*Theta0*rho*TE
  104. k2 = epsilon*r0*rho*TE
  105. k3 = 1-epsilon
  107. return(V0,k1,k2,k3)
  110. #%% Fixing the type of connectivity
  112. def get_Matrix_kE_ForATypeOfConnectivity(TypeOfConnectivity):
  115. if TypeOfConnectivity=='EE':
  116. k = array([[0., 1.],
  117. [1., 0.]])
  120. elif TypeOfConnectivity=='II':
  121. k = array([[0., 0.],
  122. [0., 0.]])
  125. elif TypeOfConnectivity=='EI':
  126. k = array([[0., 1.],
  127. [0., 0.]])
  129. elif TypeOfConnectivity == 'IE':
  130. k = array([[0., 0.],
  131. [1., 0.]])
  134. return(k)
  137. def get_k_inh(k): # for the inhibitory pool. k_inh is a matrix that contains 0 on the diagonal and (1-k) everywhere else.
  139. mat1_0diago = ones(k.shape)
  140. for i in range(len(k)):
  141. mat1_0diago[i,i] = 0 #mat1_0diago contains 0 on the diagonal and 1 everywhere else
  143. k_inh = mat1_0diago-k
  145. return(k_inh)
  149. #%% Simulations with the neural model
  151. def FindFixedPoint(autocoupling_E,coupling_ItoE_abs,basal_forcing_E,coupling_EtoI,autocoupling_I_abs,basal_forcing_I,M,G,k,Jnmda,Tglu,Tgaba,N): #cf fonction EtudeStabilite papier tuning
  154. #----------Duration of the first simulation and timeline
  155. T0=0
  156. Tmax = 10
  157. (Time,dt) = get_Timeline(T0,Tmax)
  160. #---- Stimulation forcing on the excitatory pools
  161. StimulationFunction = zeros((N,len(Time)))
  163. #------- Initial conditions
  165. dsn_dt = 1
  166. dsg_dt = 1
  168. sn_fp = 0.5*ones((N,1))
  169. sg_fp = 0.5*ones((N,1))
  172. sn_fp = 0.1*ones((N,1))
  173. sg_fp = 0.05*ones((N,1))
  176. simu=0
  177. while all(abs(dsn_dt)>1e-10) and all(abs(dsg_dt)>1e-10): #We continue the simulation until we get close enough to the fixed point
  179. simu=simu+1
  181. Sn0 = sn_fp
  182. Sg0 = sg_fp
  185. (Xn,Xg,Rn,Rg,dSndt,dSgdt,Sn,Sg) = get_Dynamics(autocoupling_E,coupling_ItoE_abs,basal_forcing_E,coupling_EtoI,autocoupling_I_abs,basal_forcing_I,M,G,k,Jnmda,Tglu,Tgaba,Sn0,Sg0,dt,Time,StimulationFunction)
  188. sn_fp = reshape(Sn[:,-1],(N,1))
  189. sg_fp = reshape(Sg[:,-1],(N,1))
  191. dsn_dt = reshape(dSndt[:,-1],(N,1))
  192. dsg_dt = reshape(dSgdt[:,-1],(N,1))
  196. xn_fp = reshape(Xn[:,-1],(N,1))
  197. xg_fp = reshape(Xg[:,-1],(N,1))
  199. rn_fp = reshape(Rn[:,-1],(N,1))
  200. rg_fp = reshape(Rg[:,-1],(N,1))
  203. return(sn_fp,sg_fp,xn_fp,xg_fp,rn_fp,rg_fp)
  206. def get_Timeline(T0,Tmax):
  209. dt=0.001
  211. sampling=round((Tmax-T0)/dt)
  212. Tmax = T0 + dt*sampling
  214. Time=linspace(T0,Tmax,int(sampling)+1)
  216. return(Time,dt)
  221. def get_Dynamics(autocoupling_E,coupling_ItoE_abs,basal_forcing_E,coupling_EtoI,autocoupling_I_abs,basal_forcing_I,M,G,k,Jnmda,Tglu,Tgaba,Sn0,Sg0,dt,Time,StimulationFunction):
  223. N= Sn0.shape[0]
  226. # memory allocation
  228. Sn = zeros((N,len(Time)))
  229. Sg = zeros((N,len(Time)))
  231. dSndt = zeros((N,len(Time)))
  232. dSgdt = zeros((N,len(Time)))
  235. Xn = zeros((N,len(Time)))
  238. Xg = zeros((N,len(Time)))
  240. Rn = zeros((N,len(Time)))
  241. Rg = zeros((N,len(Time)))
  244. # we save the initial condition
  246. Sn[:,0] = reshape(Sn0,(Sn0.T).shape)
  247. Sg[:,0] = reshape(Sg0,(Sg0.T).shape)
  252. for j in range(len(Time)):
  256. sn_past = reshape(Sn[:,j],(N,1))
  257. sg_past = reshape(Sg[:,j],(N,1))
  260. t=Time[j]
  263. (xn,xg,rn,rg,dsndt,dsgdt,sn_new,sg_new) = Integration(autocoupling_E,coupling_ItoE_abs,basal_forcing_E,coupling_EtoI,autocoupling_I_abs,basal_forcing_I,M,G,k,Jnmda,Tglu,Tgaba,sn_past,sg_past,dt,j,StimulationFunction)
  267. dSndt[:,j] = reshape(dsndt,(dsndt.T).shape)
  268. dSgdt[:,j] = reshape(dsgdt,(dsgdt.T).shape)
  272. Rn[:,j] = reshape(rn,(rn.T).shape)
  273. Rg[:,j] = reshape(rg,(rg.T).shape)
  275. Xn[:,j] = reshape(xn,(xn.T).shape)
  276. Xg[:,j] = reshape(xg,(xg.T).shape)
  279. if j<len(Time)-1: # We do not keep the last point (i.e. Sn(t+dt) obtained with dSndt(t), for t=Time[-1]). Thus, Sn and dSndt have the same dimensions and the last point of each array corresponds to Sn(t) and dSndt(t) with t=Time[-1]
  280. Sn[:,j+1] = reshape(sn_new,(sn_new.T).shape)
  281. Sg[:,j+1] = reshape(sg_new,(sg_new.T).shape)
  284. return(Xn,Xg,Rn,Rg,dSndt,dSgdt,Sn,Sg)
  288. def Integration(autocoupling_E,coupling_ItoE_abs,basal_forcing_E,coupling_EtoI,autocoupling_I_abs,basal_forcing_I,M,G,k,Jnmda,Tglu,Tgaba,sn,sg,dt,j,StimulationFunction):
  292. StimulationFunction_at_t = reshape(StimulationFunction[:,j],sn.shape)
  296. (xn,xg,rn,rg,dsndt,dsgdt) = get_Derivatives(sn,sg,autocoupling_E,coupling_ItoE_abs,basal_forcing_E,coupling_EtoI,autocoupling_I_abs,basal_forcing_I,M,G,k,Jnmda,Tglu,Tgaba,j,StimulationFunction_at_t)
  300. N = sn.shape[0]
  302. sn_next=zeros(sn.shape, dtype=float)
  303. sg_next=zeros(sg.shape, dtype=float)
  307. sn_next = sn + dt*dsndt
  308. sg_next = sg + dt*dsgdt
  312. return(xn,xg,rn,rg,dsndt,dsgdt,sn_next,sg_next)
  317. def get_Derivatives(sn,sg,autocoupling_E,coupling_ItoE_abs,basal_forcing_E,coupling_EtoI,autocoupling_I_abs,basal_forcing_I,M,G,k,Jnmda,Tglu,Tgaba,j,StimulationFunction_at_t):
  321. #--- Currents
  323. xn = ExcitatoryCurrent(autocoupling_E,coupling_ItoE_abs,basal_forcing_E,sn,sg,j,StimulationFunction_at_t,M,G,k,Jnmda)
  325. xg = InhibitoryCurrent (coupling_EtoI,autocoupling_I_abs,basal_forcing_I,sn,sg,j,M,G,k,Jnmda)
  328. #--- Firing rates
  330. rn=ExcitatoryFiringRate(xn)
  332. rg=InhibitoryFiringRate(xg)
  334. #--- Differential equations on state variables
  336. (BetaE,AlphaE,BetaI,AlphaI) = get_FixedParametersForStates()
  338. dsndt = -BetaE*sn+AlphaE*Tglu*(1-sn)*rn
  339. dsgdt = -BetaI*sg+AlphaI*Tgaba*(1-sg)*rg
  343. return(xn,xg,rn,rg,dsndt,dsgdt)
  347. #--- functions for firing rates
  350. def ExcitatoryFiringRate(x):
  352. (a,b,d) = get_FixedParametersForExcFiringRates()
  354. y=a*x-b
  357. result=zeros(y.shape, dtype=float)
  358. for i in range(len(y)):
  359. result[i] = y[i]/(1-math.exp(-d*y[i]))
  361. # if y[i]!=0:
  362. # result[i] = y[i]/(1-math.exp(-d*y[i]))
  363. # else:
  364. # result[i]=0
  366. return(result)
  370. def InhibitoryFiringRate(x):
  372. (a,b,d) = get_FixedParametersForInhFiringRates()
  374. y=a*x-b
  377. result=zeros(y.shape, dtype=float)
  378. for i in range(len(y)):
  379. result[i] = y[i]/(1-math.exp(-d*y[i]))
  380. # if y[i]!=0:
  381. # result[i] = y[i]/(1-math.exp(-d*y[i]))
  382. # else:
  383. # result[i]=0
  384. return(result)
  388. #--- functions for currents
  390. def ExcitatoryCurrent (autocoupling_E,coupling_ItoE_abs,basal_forcing_E,sn,sg,j,StimulationFunction_at_t,M,G,k,Jnmda):
  393. xn = autocoupling_E*sn - coupling_ItoE_abs*sg + basal_forcing_E + StimulationFunction_at_t + G*Jnmda*dot( k*M,sn )
  396. return(xn)
  400. def InhibitoryCurrent (coupling_EtoI,autocoupling_I_abs,basal_forcing_I,sn,sg,j,M,G,k,Jnmda):
  404. k_inh = get_k_inh(k)
  406. xg = coupling_EtoI*sn - autocoupling_I_abs*sg + basal_forcing_I + G*Jnmda*dot( k_inh*M,sn )
  410. return(xg)
  414. #%% Simulations with the hemodynamic and bold models
  417. def HemoBoldDynamics(SnDyna,z0,f0,v0,q0,dt,Time):
  419. bold0=BoldFunction(v0,q0)
  420. N= SnDyna.shape[0]
  423. #allocation mémoire
  424. Z = zeros((N,len(Time)))
  425. F = zeros((N,len(Time)))
  426. V = zeros((N,len(Time)))
  427. Q = zeros((N,len(Time)))
  428. BOLD = zeros((N,len(Time)))
  430. # we keep the initial condition
  431. Z[:,0] = reshape(z0,(z0.T).shape)
  432. F[:,0] = reshape(f0,(f0.T).shape)
  433. V[:,0] = reshape(v0,(v0.T).shape)
  434. Q[:,0] = reshape(q0,(q0.T).shape)
  435. BOLD[:,0] = reshape(bold0,(bold0.T).shape)
  438. for j in range(1,len(Time)):
  441. sn_past = SnDyna[0:N,j-1:j]
  444. z_past = reshape(Z[:,j-1],(N,1))
  445. f_past = reshape(F[:,j-1],(N,1))
  446. v_past = reshape(V[:,j-1],(N,1))
  447. q_past = reshape(Q[:,j-1],(N,1))
  450. bold_past=BoldFunction(v_past,q_past)
  452. (z_new,f_new,v_new,q_new) = HemodynamicsIntegration(sn_past,z_past,f_past,v_past,q_past,dt)
  454. bold_new=BoldFunction(v_new,q_new)
  457. Z[:,j] = reshape(z_new,(z_new.T).shape)
  458. F[:,j] = reshape(f_new,(f_new.T).shape)
  460. V[:,j] = reshape(v_new,(v_new.T).shape)
  461. Q[:,j] = reshape(q_new,(q_new.T).shape)
  463. BOLD[:,j] = reshape(bold_new,(bold_new.T).shape)
  466. return(Z,F,V,Q,BOLD)
  469. def HemodynamicsIntegration(sn,z,f,v,q,dt):
  472. (alpha,gamma,kappa,tau,rho) = get_FixedParametersForHemodynamics()
  475. dzdt = sn-kappa*z-gamma*(f-1)
  477. dfdt = z
  479. dvdt = (1/tau)*(f-pow(v,1/alpha))
  481. dqdt = (1/(tau*rho))*f*(1-pow((1-rho),1/f)) - (1/tau)*q*pow(v,(1/alpha)-1)
  484. z_next=zeros(z.shape, dtype=float)
  485. f_next=zeros(f.shape, dtype=float)
  486. v_next=zeros(v.shape, dtype=float)
  487. q_next=zeros(q.shape, dtype=float)
  490. z_next=z+dt*dzdt
  492. f_next=f+dt*dfdt
  494. v_next=v+dt*dvdt
  496. q_next=q+dt*dqdt
  498. return(z_next,f_next,v_next,q_next)
  503. def BoldFunction(v,q):
  505. (V0,k1,k2,k3) = get_FixedParametersForBold()
  508. y = V0*(k1*(1-q)+k2*(1-q*(1/v))+k3*(1-v))
  510. return(y)
  514. #%%% Sensitivities
  516. #%% Sensitivities of the isolated pools (with their autocoupling)
  518. def CloseLoopSensitivity_Pool_E(Sn,zE,autocoupling_E,Tglu):
  521. Sensi_Sn_zE = zeros(Sn.shape)
  523. (BetaE,AlphaE,BetaI,AlphaI) = get_FixedParametersForStates()
  525. (XN,RN,DRN_DXN) = Calculate_XN_RN_DRNDXN_PoolE(Sn,autocoupling_E,zE)
  527. # Sensitivity of sn w.r.t. the forcing zE
  528. Sensi_Sn_zE = 1/( ( (BetaE + AlphaE*Tglu*RN)/(AlphaE*Tglu*(1-Sn)*DRN_DXN) ) - autocoupling_E)
  530. # Feedback gain
  531. Gain = (AlphaE*Tglu*(1-Sn)*DRN_DXN*autocoupling_E)/(BetaE + AlphaE*Tglu*RN)
  533. return(Sensi_Sn_zE, Gain)
  538. def CloseLoopSensitivity_Pool_I(Sg,zI,autocoupling_I_abs,Tgaba):
  541. Sensi_Sg_zI = zeros(Sg.shape)
  543. (BetaE,AlphaE,BetaI,AlphaI) = get_FixedParametersForStates()
  545. (XG,RG,DRG_DXG) = Calculate_XG_RG_DRGDXG_PoolI(Sg,autocoupling_I_abs,zI)
  547. # Sensitivity of sg w.r.t. the forcing zI
  548. Sensi_Sg_zI = 1/( ( (BetaI + AlphaI*Tgaba*RG)/(AlphaI*Tgaba*(1-Sg)*DRG_DXG) ) + autocoupling_I_abs )
  550. # Feedback gain
  551. Gain = (AlphaI*Tgaba*(1-Sg)*DRG_DXG * (-autocoupling_I_abs))/(BetaI + AlphaI*Tgaba*RG)
  554. return(Sensi_Sg_zI, Gain)
  557. def Calculate_XN_RN_DRNDXN_PoolE(Sn,autocoupling_E,zE):
  559. XN = zeros(Sn.shape)
  560. RN = zeros(Sn.shape)
  561. DRN_DXN = zeros(Sn.shape)
  563. (Aexc,Bexc,Dexc) = get_FixedParametersForExcFiringRates()
  566. XN = autocoupling_E*Sn + zE
  568. RN = (Aexc * XN-Bexc) /(1-exp(-Dexc*(Aexc * XN - Bexc)))
  570. DRN_DXN = (Aexc/2) * ( (exp(Dexc*(Aexc * XN - Bexc)) -1 - Dexc*(Aexc * XN - Bexc) ) / (cosh(Dexc*(Aexc * XN - Bexc)) -1) )
  572. return(XN,RN,DRN_DXN)
  576. def Calculate_XG_RG_DRGDXG_PoolI(Sg,autocoupling_I_abs,zI):
  578. XG = zeros(Sg.shape)
  579. RG = zeros(Sg.shape)
  580. DRG_DXG = zeros(Sg.shape)
  582. (Ainh,Binh,Dinh) = get_FixedParametersForInhFiringRates()
  585. XG = - autocoupling_I_abs*Sg + zI
  587. RG = (Ainh * XG - Binh) /(1-exp(-Dinh*(Ainh * XG - Binh)))
  589. DRG_DXG =(Ainh/2) * ( (exp(Dinh*(Ainh * XG - Binh)) -1 - Dinh*(Ainh * XG - Binh) ) / (cosh(Dinh*(Ainh * XG - Binh)) -1) )
  591. return(XG,RG,DRG_DXG)
  597. #%% Sensitivities of the isolated regions (as functions of open loop sensitivities of the pools)
  599. #---- Open loop sensitivities of the pools
  601. def OpenLoopSensitivities_Pools(Sn,Sg,zE,zI,autocoupling_E,autocoupling_I_abs,Tglu,Tgaba):
  603. (Sensi_Sn_zE, Gain_E) = CloseLoopSensitivity_Pool_E(Sn,zE,autocoupling_E,Tglu)
  605. (Sensi_Sg_zI, Gain_I) = CloseLoopSensitivity_Pool_I(Sg,zI,autocoupling_I_abs,Tgaba)
  608. return(Sensi_Sn_zE, Sensi_Sg_zI)
  611. #---- Close loop sensitivities of the isolated area
  613. def CloseLoopSensitivities_OneRegion(Sn,Sg,xE,xI,autocoupling_E,coupling_ItoE_abs,coupling_EtoI,autocoupling_I_abs,Tglu,Tgaba):
  616. (BetaE,AlphaE,BetaI,AlphaI) = get_FixedParametersForStates()
  618. # ---initialisation
  620. # open loop sensitivities
  621. Sensi_Sn_xE_O = zeros(Sn.shape)
  622. Sensi_Sg_xI_O = zeros(Sn.shape)
  624. # close loop sensitivities
  625. Sensi_Sn_xE = zeros(Sn.shape) # sensitivity of sn w.r.t. the forcing xE
  626. Sensi_Sg_xE = zeros(Sn.shape) # sensitivity of sg w.r.t. the forcing xE
  628. Sensi_Sn_xI = zeros(Sn.shape) # sensitivity of sn w.r.t. the forcing xI
  629. Sensi_Sg_xI = zeros(Sn.shape) # sensitivity of sg w.r.t. the forcing xI
  632. # We evaluate the open loop sensitivities using the fixed point in the close loop system
  633. ZE = - coupling_ItoE_abs*Sg + xE
  634. ZI = coupling_EtoI*Sn + xI
  635. (Sensi_Sn_xE_O, Sensi_Sg_xI_O) = OpenLoopSensitivities_Pools(Sn,Sg,ZE,ZI,autocoupling_E,autocoupling_I_abs,Tglu,Tgaba)
  638. # Feedback gain
  640. Gain = -coupling_EtoI * coupling_ItoE_abs * Sensi_Sn_xE_O * Sensi_Sg_xI_O
  642. # We calculate the close loop sensitivities, using the open loop sensitivities and the feedback gain
  644. Sensi_Sn_xE = Sensi_Sn_xE_O/(1-Gain)
  645. Sensi_Sg_xE = (coupling_EtoI * Sensi_Sn_xE_O * Sensi_Sg_xI_O )/(1-Gain)
  647. Sensi_Sn_xI = (-coupling_ItoE_abs * Sensi_Sn_xE_O * Sensi_Sg_xI_O)/(1-Gain)
  648. Sensi_Sg_xI = Sensi_Sg_xI_O/(1-Gain)
  651. return(Sensi_Sn_xE,Sensi_Sg_xE,Sensi_Sn_xI,Sensi_Sg_xI,Sensi_Sn_xE_O,Sensi_Sg_xI_O,Gain)
  654. #%% Sensitivities of two regions (as functions of open loop sensitivitie of each region)
  656. #---- Open loop sensitivities of one region
  658. def OpenLoopSensitivities_OneRegion(Sni,Sgi,xEi,xIi,autocoupling_E_i,coupling_ItoE_abs_i,coupling_EtoI_i,autocoupling_I_abs_i,Tglui,Tgabai): #give the input arguments only for the region of interest
  661. (Sensi_Sni_xEi,Sensi_Sgi_xEi,Sensi_Sni_xIi,Sensi_Sgi_xIi,Sensi_Sni_xEi_O,Sensi_Sgi_xIi_O,Gain) = CloseLoopSensitivities_OneRegion(Sni,Sgi,xEi,xIi,autocoupling_E_i,coupling_ItoE_abs_i,coupling_EtoI_i,autocoupling_I_abs_i,Tglui,Tgabai)
  664. return(Sensi_Sni_xEi,Sensi_Sgi_xEi,Sensi_Sni_xIi,Sensi_Sgi_xIi)
  667. #---- Close loop sensitivities of two regions
  670. def CloseLoopSensitivities_TwoRegions(Sn_fp,Sg_fp,autocoupling_E,coupling_ItoE_abs,basal_forcing_E,coupling_EtoI,autocoupling_I_abs,basal_forcing_I,Tglu,Tgaba,G,Jnmda,k,M,control,target):
  673. Kappa = G*Jnmda*M
  675. Currents_OtherRegions_ToE = dot(k*Kappa,Sn_fp)
  678. k_inh = get_k_inh(k)
  679. Currents_OtherRegions_ToI = dot(k_inh*Kappa,Sn_fp)
  682. #--- Region 1 (open loop)
  684. i=control
  686. (SnC,SgC,coupling_ItoE_abs_C,TgabaC,TgluC) = get_ParametersForRegion_i(i,Sn_fp,Sg_fp,coupling_ItoE_abs,Tgaba,Tglu)
  688. xEC = basal_forcing_E[i] + Currents_OtherRegions_ToE[i]
  689. xIC = basal_forcing_I + Currents_OtherRegions_ToI[i]
  691. (Sensi_SnC_xEC,Sensi_SgC_xEC,Sensi_SnC_xIC,Sensi_SgC_xIC) = OpenLoopSensitivities_OneRegion(SnC,SgC,xEC,xIC,autocoupling_E,coupling_ItoE_abs_C,coupling_EtoI,autocoupling_I_abs,TgluC,TgabaC)
  694. #--- Region 2 (open loop)
  696. i=target
  698. (SnT,SgT,coupling_ItoE_abs_T,TgabaT,TgluT) = get_ParametersForRegion_i(i,Sn_fp,Sg_fp,coupling_ItoE_abs,Tgaba,Tglu)
  700. xET = basal_forcing_E[i] + Currents_OtherRegions_ToE[i]
  701. xIT = basal_forcing_I + Currents_OtherRegions_ToI[i]
  703. (Sensi_SnT_xET,Sensi_SgT_xET,Sensi_SnT_xIT,Sensi_SgT_xIT) = OpenLoopSensitivities_OneRegion(SnT,SgT,xET,xIT,autocoupling_E,coupling_ItoE_abs_T,coupling_EtoI,autocoupling_I_abs,TgluT,TgabaT)
  706. #--- Feedback gain
  708. (G12,G21,Gain) = FeedbackGain_TwoRegions(k,Kappa,Sensi_SnC_xEC,Sensi_SnC_xIC,Sensi_SnT_xET,Sensi_SnT_xIT)
  712. #--- Close loop system
  714. (dSnC_dBEC,dSnT_dBEC) = SensitivitiesTwoRegions_Sn_wrt_BEC(Sensi_SnC_xEC,Gain,G21)
  718. return(dSnC_dBEC, dSnT_dBEC, Gain, Sensi_SnC_xEC, Sensi_SgC_xEC, Sensi_SnC_xIC, Sensi_SgC_xIC, Sensi_SnT_xET, Sensi_SgT_xET, Sensi_SnT_xIT, Sensi_SgT_xIT)
  722. def get_ParametersForRegion_i(i,Sn,Sg,b,Tgaba,Tglu):
  724. Sni = Sn[i]
  725. Sgi = Sg[i]
  728. bi = b[i]
  729. Tgabai = Tgaba[i]
  730. Tglui = Tglu[i]
  732. return(Sni,Sgi,bi,Tgabai,Tglui)
  736. def FeedbackGain_TwoRegions(k,Kappa,Sensi_Sn1_xE1,Sensi_Sn1_xI1,Sensi_Sn2_xE2,Sensi_Sn2_xI2):
  738. G12 = Sensi_Sn1_xE1 * k[0,1]* Kappa[0,1] + Sensi_Sn1_xI1 * (1-k[0,1])* Kappa[0,1]
  741. G21 = Sensi_Sn2_xE2 * k[1,0] * Kappa[1,0] + Sensi_Sn2_xI2 * (1-k[1,0]) * Kappa[1,0]
  743. Gain = G12 * G21
  745. return(G12,G21,Gain)
  749. def SensitivitiesTwoRegions_Sn_wrt_BEC(Sensi_Sn1_xE1,Gain,G21):
  752. dSn1_dBE1 = (Sensi_Sn1_xE1)/(1-Gain)
  754. dSn2_dBE1 = G21 * dSn1_dBE1
  756. return(dSn1_dBE1,dSn2_dBE1)

functions.py, no license · at the source

Overview

Authors: Anaïs Vallet1, Stéphane Blanco2, Coline Chevallier2,3, Francis Eustache1, Jacques Gautrais2,3, Jean-Yves Grandpeix4, Jean-Louis Joly2, Shailendra Segobin1, Pierre Gagnepain1
ORCID iDs: Anaïs Vallet, Coline Chevallier, Francis Eustache, Jacques Gautrais, Shailendra Segobin, Pierre Gagnepain
  1. Normandie Univ, UNICAEN, PSL Research University, EPHE, INSERM, U1077, CHU de Caen, GIP Cyceron, Neuropsychologie et Imagerie de la Mémoire Humaine, Université de Normandie, Caen, France
  2. CNRS, INPT, UPS, LAPLACE, Université de Toulouse, Toulouse, France
  3. Centre de Recherches sur la Cognition Animale (CRCA), Centre de Biologie Intégrative (CBI), CNRS, UPS, Université de Toulouse, Toulouse, France
  4. LMD/IPSL, CNRS, École Polytechnique, ENS, Sorbonne Université, Paris, France
Institutions: Inserm (France); École Pratique des Hautes Études (France); Université Paris Sciences et Lettres (France); Centre Hospitalier Universitaire de Caen Normandie (France); Cyceron (France); Neuropsychologie et Imagerie de la Mémoire Humaine (France); Université de Caen Normandie (France); Centre National de la Recherche Scientifique (France); Université Toulouse III - Paul Sabatier (France); Institut National Polytechnique de Toulouse (France); Laboratoire Plasma et Conversion d'Energie (France); Université de Toulouse (France); Centre de Recherches sur la Cognition Animale (France); Centre de Biologie Intégrative de Toulouse (France); École Polytechnique (France); École Normale Supérieure - PSL (France); Sorbonne Université (France)
Journal: PLoS computational biology, volume 22, issue 3, article e1014035
Dates: received 30 April 2025; accepted 17 February 2026; published online 23 March 2026
Type: Research article · Language: English
License: CC BY
Identifiers: DOI 10.1371/journal.pcbi.1014035 · PMID 41871132 · PMCID PMC13008111 · OpenAlex W7140113130
Open access: gold, a free copy (OpenAlex)
Status: code verified (how the code is found and verified)
Categories: human (organism), computational (subfield)
MeSH: Brain*, Models, Neurological*, Nerve Net*, Computational Biology, Computer Simulation, Humans, Neural Inhibition, Neurons (* major topic)
Journal subjects: Biology and Life Sciences, Neuroscience, Brain Mapping, Functional Magnetic Resonance Imaging, Medicine and Health Sciences, Diagnostic Medicine, Diagnostic Radiology, Magnetic Resonance Imaging, Research and Analysis Methods, Imaging Techniques, Radiology and Imaging, Neuroimaging, Biophysics, Physical Sciences, Physics, Cell Biology, Cellular Types, Animal Cells, Neurons, Cellular Neuroscience, Computer and Information Sciences, Neural Networks, Psychology, Behavior, Social Sciences, Signal Transduction, Cell Signaling, Signal Inhibition, Engineering and Technology, Signal Processing, Peak Values, Signal Amplification
Topic: Functional Brain Connectivity Studies (Cognitive Neuroscience, Neuroscience), according to OpenAlex
Funding: Agence Nationale de la Recherche (ANR-10-EQPX-0021); Institut National de la Santé et de la Recherche Médicale; Région Normandie
Citations: not cited yet (Europe PMC); 36 references in the paper

Abstract

Biophysical models of human brain represent the latter as a graph of inter-connected neural regions. Building from the model by Naskar et al. (Network Neuroscience, 2021), our motivation was to understand how these brain regions can be connected at neural level to implement some inhibitory control, which calls for inhibitory connectivity rarely considered in such models. In this model, regions are made of inter-connected excitatory and inhibitory pools of neurons, but are long-range connected only via excitatory pools (mutual excitation). We thus extend this model by generalizing connectivity, and we analyse how connectivity affects the behaviour of this model. Focusing on the simplest paradigm made of a Control area and a Target area, we explore four typical kinds of connectivity: mutual excitation, Target inhibition by Control, Control inhibition by Target, and mutual inhibition. For this, we build an analytical sensitivity framework, nesting up sensitivities of isolated pools, of isolated regions, and of the full system. We show that inhibitory control can emerge only in Target inhibition by Control and mutual inhibition connectivities. We next offer an analysis of how the model sensitivities depends on connectivity structure, depending on a parameter controling the strength of the self-inhibition within Target region. Finally, we illustrate the effect of connectivity structure upon control effectivity in response to an external forcing in the Control area. Beyond the case explored here, our methodology to build analytical sensitivities by nesting up levels (pool, region, system) lays the groundwork for expressing nested sensitivities for more complex network configurations, either for this model or any other one.

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Repository

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supp:PMC13008111/pcbi.1014035.s003.zip

License: none: the authors keep all their rights
State: the link answers, verified on 30 September 2026
Evidence: files inventoried
Languages: Python (11), Shell (1)
Size: 12 files, 12 scripts
Software Heritage: not checked
Found in: the supplementary material
Not found: README, license file, CITATION.cff, environment file, tests, continuous integration, documentation
Tools: NumPy (11 files), Matplotlib (10 files)
Availability: 1 check, the latest on 30 September 2026: the link answers (HTTP 200)
  • 30 September 2026: the link answers (HTTP 200)
12 files
At the source: pmc-oa-opendata.s3.amazonaws.com/PMC13008111.1/pcbi.1014035.s003.zip

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Version 1, 30 September 2026: the first record

Recorded: type, language, journal, volume, issue, pages, dates, 9 authors, 8 MeSH terms, 3 funders, 36 references.

Cite

This paper

Vallet, A., Blanco, S., Chevallier, C., Eustache, F., Gautrais, J., Grandpeix, J.-Y., Joly, J.-L., Segobin, S., & Gagnepain, P. (2026). Sensitivity analysis enlightens effects of connectivity in a Neural Mass Model under Control-Target mode. PLoS computational biology, 22(3), e1014035. https://doi.org/10.1371/journal.pcbi.1014035

BibTeX

@article{vallet2026sensitivity,
author = {Vallet, Anaïs and Blanco, Stéphane and Chevallier, Coline and Eustache, Francis and Gautrais, Jacques and Grandpeix, Jean-Yves and Joly, Jean-Louis and Segobin, Shailendra and Gagnepain, Pierre},
title = {{Sensitivity analysis enlightens effects of connectivity in a Neural Mass Model under Control-Target mode}},
journal = {PLoS computational biology},
year = {2026},
month = mar,
volume = {22},
number = {3},
pages = {e1014035},
publisher = {PLOS},
issn = {1553-734X},
doi = {10.1371/journal.pcbi.1014035},
url = {https://doi.org/10.1371/journal.pcbi.1014035},
pmid = {41871132},
pmcid = {PMC13008111}
}

RIS

TY - JOUR
AU - Vallet, Anaïs
AU - Blanco, Stéphane
AU - Chevallier, Coline
AU - Eustache, Francis
AU - Gautrais, Jacques
AU - Grandpeix, Jean-Yves
AU - Joly, Jean-Louis
AU - Segobin, Shailendra
AU - Gagnepain, Pierre
TI - Sensitivity analysis enlightens effects of connectivity in a Neural Mass Model under Control-Target mode
T2 - PLoS computational biology
J2 - PLoS Comput Biol
PY - 2026
DA - 2026/03/23
VL - 22
IS - 3
SP - e1014035
SN - 1553-734X
PB - PLOS
DO - 10.1371/journal.pcbi.1014035
UR - https://doi.org/10.1371/journal.pcbi.1014035
LA - en
ER -

CSL-JSON

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"title": "Sensitivity analysis enlightens effects of connectivity in a Neural Mass Model under Control-Target mode",
"container-title": "PLoS computational biology",
"author": [
{
"family": "Vallet",
"given": "Anaïs"
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"container-title-short": "PLoS Comput Biol",
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"issued": {
"date-parts": [
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23
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