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SparseNetReciprocal_PositiveTF_Fig9.py
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SparseNetReciprocal_PositiveTF_Fig9.py
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# -*- coding: utf-8 -*-
"""
@author: Yuxiu Shao
Sparse EI Network with reciprocal connectivity, Full-rank Gaussian Network and Rank-one Mixture of Gaussian Approximation
Connectivity, Dynamics
"""
''' HELP FUNCTIONS '''
import numpy as np
import matplotlib.pylab as plt
import matplotlib
from numpy import linalg as la
# import seaborn as sb
# from sympy import *
extras_require = {'PLOT':['matplotlib>=1.1.1,<3.0']},
plt.style.use('seaborn-whitegrid')
# Set Matplotlib defaults
plt.rc('figure', autolayout=True)
plt.rc('axes', labelweight='bold', labelsize='large',titleweight='bold', titlesize=18, titlepad=10)
### import self-defined functions
from UtilFuncs import *
from Sparse_util import *
from Meanfield_Dyn_util import *
from Connect_util import *
def cal_outliers_general_complex(x, tau_set, coeffs, Jparams, Nparams, gmat, eigvAm):
"""
Parameters
----------
x :
complex eigenvalue solutions.
tau_set : array
motif statistics.
coeffs :
local statistics.
Jparams :
mean connectivity.
Nparams :
network scale.
gmat :
random variance.
eigvAm :
unperturbed eigenvaules.
Returns
-------
results :
solutions.
"""
NE, NI = int(Nparams[0]), int(Nparams[1])
N = NE+NI
alphaE,alphaI = NE/N, NI/N
JE, JI = Jparams[0],Jparams[1]
### mean connectivity
Mbar = np.ones((N,1))
Nbar = np.zeros((N,1))
Nbar[:NE,0],Nbar[NE:,0] = JE*N/NE, -JI*N/NI
gee, gei, gie, gii = gmat[0],gmat[1],gmat[2],gmat[3]
zchn = np.zeros((2,2))
zchn[0,0],zchn[0,1] = alphaE*tau_set['tchn'][0,0]*gee**2+alphaI*tau_set['tchn'][1,0]*gei*gie, alphaE*tau_set['tchn'][0,1]*gee*gei+alphaI*tau_set['tchn'][1,1]*gei*gii
zchn[1,0], zchn[1,1] = alphaE*tau_set['tchn'][0,0]*gie*gee+alphaI*tau_set['tchn'][1,0]*gii*gie, alphaE*tau_set['tchn'][0,1]*gie*gei+alphaI*tau_set['tchn'][1,1]*gii**2
Emean, Ediff = (zchn[0,0]+zchn[1,0])/2*N, (zchn[0,0]-zchn[1,0])/2*N
Imean, Idiff = (zchn[0,1]+zchn[1,1])/2*N, (zchn[0,1]-zchn[1,1])/2*N
# Imean *=(-1)
U,V = np.zeros((N,2)), np.zeros((N,2))
U[:,0], U[:NE,1], U[NE:,1] = 1, 1, -1
V[:NE,0],V[NE:,0] = Emean, Imean
V[:NE,1],V[NE:,1] = Ediff, Idiff
### negative
U *=(-1)
V /=(N)
### diagonal matrix A
#### upper
xall = complex(x[0],x[1])
# upper =xall**2-(alphaE*tau_set['trec'][0,0]*gee**2+alphaI*tau_set['trec'][1,0]*gei*gie)+(alphaE*tau_set['tchn'][0,0]*gee**2+alphaI*tau_set['tchn'][1,0]*gei*gie)
# bottom = xall**2-(alphaE*tau_set['trec'][0,1]*gie*gei+alphaI*tau_set['trec'][1,1]*gii**2)+(alphaE*tau_set['tchn'][0,1]*gie*gei+alphaI*tau_set['tchn'][1,1]*gii**2)
upper_r = (x[0]**2-x[1]**2)-(alphaE*tau_set['trec'][0,0]*gee**2+alphaI*tau_set['trec'][1,0]*gei*gie)+(alphaE*tau_set['tchn'][0,0]*gee**2+alphaI*tau_set['tchn'][1,0]*gei*gie)
upper_i = 2*x[0]*x[1]
bottom_r = (x[0]**2-x[1]**2)-(alphaE*tau_set['trec'][0,1]*gie*gei+alphaI*tau_set['trec'][1,1]*gii**2)+(alphaE*tau_set['tchn'][0,1]*gie*gei+alphaI*tau_set['tchn'][1,1]*gii**2)
bottom_i = 2*x[0]*x[1]
I2 = np.eye(2)
A_1 = np.zeros((N,N),dtype=complex)
for i in range(NE):
A_1[i,i] = complex(upper_r/(upper_r**2+upper_i**2),-upper_i/(upper_r**2+upper_i**2))
for i in range(NE,N):
A_1[i,i] = complex(bottom_r/(bottom_r**2+bottom_i**2),-bottom_i/(bottom_r**2+bottom_i**2))
# for i in range(NE):
# A_1[i,i] =1/upper
# for i in range(NE,N):
# A_1[i,i] = 1/bottom
### step-by-step
inverse = (I2+V.copy().T@A_1@U)
# print('........inverse:',inverse)
inverse = la.inv(inverse)
# psie,psii= inverse[0,0], inverse[-1,-1]
# diagpsi = np.eye(N)
# for i in range(NE):
# diagpsi[i,i] = 1/psie
# for j in range(NE,N):
# diagpsi[j,j] = 1/psii
# inverse = diagpsi.copy()
inverse = -A_1@U@[email protected]().T@A_1
inverse = A_1+ inverse
# inverse = A_1-A_1@U@(I2+V.copy().T@A_1@U)^{-1}@V.copy().T@A_1
recal = np.squeeze(Nbar.copy().T@[email protected]())#*(x**2)/N
recal_r,recal_i = recal.real,recal.imag
x2_r, x2_i = x[0]**2-x[1]**2, 2*x[0]*x[1]
result_r, result_i = recal_r*x2_r-recal_i*x2_i, recal_r*x2_i+recal_i*x2_r
results = [result_r/N - x[0],result_i/N-x[1]]
return results
## network params
nn,gnn=200,4
gEI,Cs,kE,kI = 3.35,60,gnn,1
Nt = np.array([gnn*nn,nn])
NE,NI = Nt[0],Nt[1]
N = NE+NI
Nparams = np.array([NE,NI])
Npercent = Nparams/N
nrank,ntrial,neta,nvec,nmaxFP=1,30,1,2,3
shiftx = 1.5
cuttrials=int(ntrial/2)
RERUN = 1
if RERUN == 0:
data_name = "/Users/yuxiushao/Public/DataML/EI_LOWRANK/dyns_spr_PositiveTF_reciprocal_kappa_1.npz"
# np.savez(data_name, **data)
data = np.load(data_name)
### sparse params
EPSP,a,b = 0.023,0,0#0.025,0,0# JE PER EPSP
ngEI = 21# number of eta(s)
# cell-type specific reciprocal connectivity
etaseries = np.linspace(0.0,1.0,ngEI)#
etassrecord = np.zeros((ngEI,3))
etauserecord = np.zeros((ngEI,3))
### recording variables
RAmvecseries,LAmvecseries = np.zeros((ngEI,N)),np.zeros((ngEI,N))
BAmvseries = np.zeros((ngEI,N),dtype=complex)
### original Sparse Matrix
Rsprvecseries,Lsprvecseries =np.zeros((ngEI,ntrial,N,nvec*2)),np.zeros((ngEI,ntrial,N,nvec*2))
RsprvecTseries,LsprvecTseries=np.zeros((ngEI,ntrial,N,nvec*2)),np.zeros((ngEI,ntrial,N,nvec*2))
Bsprvseries = np.zeros((ngEI,ntrial,N),dtype=complex)
Radiusseries = np.zeros(ngEI)
### eigenvector predictions obtained using perturbation theory
Restvecseries,Lestvecseries=np.zeros((ngEI,ntrial,N,nvec*2)),np.zeros((ngEI,ntrial,N,nvec*2))
### the equivalent gaussian connectivity
Riidvecseries,Liidvecseries =np.zeros((ngEI,ntrial,N,nvec*2)),np.zeros((ngEI,ntrial,N,nvec*2))
RiidvecTseries,LiidvecTseries=np.zeros((ngEI,ntrial,N,nvec*2)),np.zeros((ngEI,ntrial,N,nvec*2))
Biidvseries = np.zeros((ngEI,ntrial,N),dtype=complex)
### statistical properties
armu,sigrcov = np.zeros((ngEI,ntrial,2)),np.zeros((ngEI,ntrial,2)) # 2 for E and I
almu,siglcov = np.zeros((ngEI,ntrial,2)),np.zeros((ngEI,ntrial,2))
siglr = np.zeros((ngEI,ntrial,2))
if(RERUN==0):
Rsprvecseries[:,:,:,0] = data['Rsprvecseries']
Lsprvecseries[:,:,:,0] = data['Lsprvecseries']
Restvecseries[:,:,:,0] = data['Restvecseries']
Lestvecseries[:,:,:,0] = data['Lestvecseries']
Bsprvseries[:,:,0] = data['Bsprvseries']
sigrcov = data['sigrcov']
siglcov = data['siglcov']
for ig, eta in enumerate (etaseries):
Jbinary = np.zeros((N,N))
### GENERATE SYMMETRIC, SPARSE SUBMATRIX.
p = kE*Cs/NE
# for NE
etaEE = eta#1.0
# for NI
etaII = 0# eta#0#1.0#0#
# for EI
etaEI = eta#eta#0.0#1.0
etassrecord[ig,0],etassrecord[ig,1],etassrecord[ig,2] = (etaEE)/(1),(etaEI)/(1),(etaII)/(1)
### generate corresponding gaussian random network with reciprocal connectivity
etaset = np.zeros(3)
etaset[0],etaset[1],etaset[2] = (etaEE-p)/(1-p),-(etaEI-p)/(1-p),(etaII-p)/(1-p)
etauserecord[ig,:]=etaset[:]
### covariance of eigenvectors
Sigmn_num = np.zeros((ngEI,ntrial,3)) ## sparse
Sigmniid_num = np.zeros((ngEI,ntrial,3)) ## Gaussian
Sigmn_theo = np.zeros((ngEI,ntrial,3)) ## theoretical
### cell-type-dependent reciprocal statistics
etaMat = np.zeros((ngEI,ntrial,2,2))
gRandom = np.zeros((ngEI,ntrial,2))
### fixed points of dynamical variable kappa
kappaintersect_Sparse = np.zeros((ngEI,ntrial,nmaxFP*2))
kappaintersect_Full = np.zeros((ngEI,ntrial,nmaxFP*2))
kappaintersect_R1 = np.zeros((ngEI,ntrial,nmaxFP*2))
if(RERUN==0):
kappaintersect_Sparse = data['kappaintersect_Sparse']
kappaintersect_Full = data['kappaintersect_Full']
kappaintersect_R1 = data['kappaintersect_R1']
### params for temporal dynamics
tt = np.linspace(0,100,500)
dt = tt[2]-tt[1]
ntt = len(tt)
## compare three neuron population activities
xfpseries_Sparse = np.zeros((ngEI,ntrial,N,ntt))
xfpseries_Full = np.zeros((ngEI,ntrial,N,ntt))
xfpseries_R1 = np.zeros((ngEI,ntrial,N,ntt))
if(RERUN==0):
xfpseries_Sparse[:,:,:,-1] = data['xfpseries_Sparse']
xfpseries_Full[:,:,:,-1] = data['xfpseries_Full']
xfpseries_R1[:,:,:,-1] = data['xfpseries_R1']
kappaseries_R1 = np.zeros((ngEI,ntrial,ntt))
### Am is fixed, mean matrix, reference eigenvalue and eigenvectors
Am = np.zeros((N,N))
IPSP = gEI*EPSP
JE0 = EPSP*kE*Cs
JI0 = IPSP*kI*Cs
Am[:,:NE],Am[:,NE:] = JE0/NE,-JI0/NI
xAm,yAm = np.zeros((N,1)),np.zeros((N,1))
eigvAm = np.zeros(N)
eigvAm[0]=(JE0-JI0)
xAm[:NE,0],xAm[NE:,0] = 1,1
yAm[:NE,0],yAm[NE:,0] = N*JE0/NE,-N*JI0/NI
RAmvecseries[0,:],LAmvecseries[0,:] = xAm[:,0],yAm[:,0]
BAmvseries[0,:] = eigvAm[:]
print('eigvAm:',eigvAm[0])
### preparation for the equivalent gaussian connectivity
gE2,gI2 = EPSP**2*(1-kE*Cs/NE)*kE*Cs*N/NE,IPSP**2*(1-kI*Cs/NI)*kI*Cs*N/NI
# print("ratio gE/gI ",np.sqrt(gE2/gI2),', ',1/gEI)
alphaNEI = Npercent.copy()
gMmat = np.array([[gE2*alphaNEI[0],gI2*alphaNEI[1]],[gE2*alphaNEI[0],gI2*alphaNEI[1]]])
eigvgm,eigvecgm=la.eig(gMmat)
r_g2 = np.max(eigvgm)
r_g = np.sqrt(r_g2)
if(RERUN):
''' Iterative Processing '''
for iktrial in range(ntrial):
print('---- simulating neuronal activity #trial-',iktrial,'------')
#### the equivalent gaussian connectivity
Xsym = iidGaussian([0,1.0/np.sqrt(N)],[N,N]) #### within the same trial, share the same Y
XsymT = Xsym.copy().T
Xgiid = Xsym.copy()
Xgiid[:NE,:NE]*=np.sqrt(gE2)
Xgiid[:NE,NE:]*=np.sqrt(gI2)
Xgiid[NE:,:NE]*=np.sqrt(gE2)
Xgiid[NE:,NE:]*=np.sqrt(gI2)
Jgiid = Xgiid.copy()+Am.copy()
#### the original Sparse Network
eigvJiid,leigveciid,reigveciid,xnorm0iid,ynorm0iid=decompNormalization(Jgiid,xAm,yAm,xAm,yAm,nparams = Nparams,sort=1)
## same sparse Random Connectivity for each trial
## EE-trial
n1,n2,prob= NE,NE,kE*Cs/NE
subProb = randbin(1,n1*n2,1-prob)
JEEsub = np.reshape(subProb,(n1,n2))
## EI-trial
n1,n2,prob= NI,NE,kE*Cs/NE
subProb = randbin(1,n1*n2,1-prob)
JEIsub = np.reshape(subProb,(n1,n2))
## II-trial
n1,n2,prob= NI,NI,kI*Cs/NI
subProb = randbin(1,n1*n2,1-prob)
JIIsub = np.reshape(subProb,(n1,n2))
## dilute/sparse matrix /attic
for ig, eta in enumerate (etaseries):
Jbinary = np.zeros((N,N))
#### setting reciprocal statistics
p = kE*Cs/NE
# for NE
etaEE = eta#1.0
# for NI
etaII = 0# eta#0#1.0#0#
# for EI
etaEI = eta#eta#0.0#1.0
etassrecord[ig,0],etassrecord[ig,1],etassrecord[ig,2] = (etaEE)/(1),(etaEI)/(1),(etaII)/(1)
#### the equivalent gaussian connectivity with reciprocal motifs
etaset = np.zeros(3)
etaset[0],etaset[1],etaset[2] = (etaEE-p)/(1-p),-(etaEI-p)/(1-p),(etaII-p)/(1-p)
etauserecord[ig,:]=etaset[:]
#### check the sign of the statistics(negative)
signeta_use = np.ones(3)
for i in range(3):
if etaset[i]<0:
signeta_use[i] *=(-1)
### E-E
probEE = kE*Cs/NE
JbEE,JbEET_=generate_SymSpr_Mat_trial(JEEsub,etaEE,kE*Cs/NE,np.array([NE,NE]))
JbEE_=JbEET_.copy().T
## combine two to get EE sub-matrix
JbEEt = np.zeros((NE,NE))
for i in range(NE):
JbEEt[i,i]=JbEE[i,i]
idxEEd, idyEEd = np.where(JbEE>0)
iddown = np.where(idxEEd>idyEEd)
idxEEd,idyEEd=idxEEd[iddown],idyEEd[iddown]
JbEEt[idxEEd,idyEEd]=JbEE[idxEEd,idyEEd]
idxEEu, idyEEu = np.where(JbEET_>0)
idup = np.where(idxEEu<idyEEu)
idxEEu,idyEEu=idxEEu[idup],idyEEu[idup]
JbEEt[idxEEu,idyEEu]=JbEET_[idxEEu,idyEEu]
# print('symmetryEE:',np.sum(np.sum(JbEEt*JbEEt.T))/kE/Cs/NE,etaEE)
# print('probEE:',np.sum(np.sum(JbEEt))/NE**2,kE*Cs/NE)
### I-I
probII = kE*Cs/NE
JbII,JbIIT_=generate_SymSpr_Mat_trial(JIIsub,etaII,kI*Cs/NI,np.array([NI,NI]))
JbII_=JbIIT_.copy().T
## combine two to get EE sub-matrix
JbIIt = np.zeros((NI,NI))
for i in range(NI):
JbIIt[i,i]=JbII[i,i]
idxIId, idyIId = np.where(JbII>0)
iddown = np.where(idxIId>idyIId)
idxIId,idyIId=idxIId[iddown],idyIId[iddown]
JbIIt[idxIId,idyIId]=JbII[idxIId,idyIId]
idxIIu, idyIIu = np.where(JbIIT_>0)
idup = np.where(idxIIu<idyIIu)##
idxIIu,idyIIu=idxIIu[idup],idyIIu[idup]
JbIIt[idxIIu,idyIIu]=JbIIT_[idxIIu,idyIIu]
#
JbIE,JbIET_=generate_SymSpr_MatEI_trial(JEIsub,etaEI,np.array([kE*Cs/NE,kI*Cs/NI]),np.array([NI,NE]))
## combine two to get EE sub-matrix
JbIEt,JbEIt = np.zeros((NI,NE)),np.zeros((NE,NI))
JbIEt,JbEIt = JbIE.copy(),JbIET_.copy()
### dalean network
Jbinary[:NE,:NE],Jbinary[NE:,NE:]=JbEEt,-JbIIt
Jbinary[NE:,:NE],Jbinary[:NE,NE:]=JbIEt,-JbEIt
### from attic to sparse connectivity
JE,JI = EPSP,IPSP
Jdilute = Jbinary.copy()
Jdilute[:,:NE] *= JE
Jdilute[:,NE:] *= JI
### overall sparse network
J = Jdilute.copy()
JT = Jdilute.copy().T
### subtracting thee mean connectivity
# X = Jdilute.copy() - Am.copy() # random matrix, with zero mean
X = Jdilute.copy() # random matrix, with zero mean
X[:NE,:NE] = X[:NE,:NE] - np.mean(X[:NE,:NE].flatten())
X[:NE,NE:] = X[:NE,NE:] - np.mean(X[:NE,NE:].flatten())
X[NE:,:NE] = X[NE:,:NE] - np.mean(X[NE:,:NE].flatten())
X[NE:,NE:] = X[NE:,NE:] - np.mean(X[NE:,NE:].flatten())
''' Original Sparse Network '''
eigvJ,leigvec,reigvec,xnorm0,ynorm0=decompNormalization(J,xnorm0iid,ynorm0iid,xAm,yAm,nparams = Nparams,sort=1)
# eigvJ,leigvec,reigvec,xnorm0,ynorm0=decompNormalization(J,xAm,yAm,xAm,yAm,nparams = Nparams,sort=1)
Rsprvecseries[ig,iktrial,:,0],Lsprvecseries[ig,iktrial,:,0]=xnorm0[:,0].copy(),ynorm0[:,0].copy()
Bsprvseries[ig,iktrial,:]=eigvJ.copy()
print('Sparse eigenvalue:',eigvJ[0],', ', np.sum(xnorm0*ynorm0)/N)
####
eigvnorm = np.real(Bsprvseries[ig,iktrial,0])#eigvAm[0]#
Zss = X.copy()# J.copy()-Am.copy()
Restvec,LestvecT = np.reshape(Zss@xAm/eigvnorm,(N,1)),np.reshape(yAm.copy().T@Zss/eigvnorm,(1,N))
Restvec,LestvecT = Restvec+xAm,LestvecT+yAm.T
Restvecseries[ig,iktrial,:,0],Lestvecseries[ig,iktrial,:,0] =np.squeeze(Restvec),np.squeeze(LestvecT)
# #### normalization(not necessary)
# Lestvec = np.reshape(LestvecT.T,(N,1))
# check_lambda = np.squeeze(np.reshape(Lestvec,(1,N))@np.reshape(Restvec,(N,1)))
# should_lambda = np.real(Bsprvseries[ig,iktrial,0]*N)
# Lestvec = Lestvec/check_lambda*should_lambda
# _,_,xnormt,ynormt=Normalization(Restvec.copy(),Lestvec.copy(),xAm.copy(),yAm.copy(),nparams=Nparams,sort=0,nrank=1)
# ### RECONSTRUCTED EIGENVECTORS
# Restvecseries[ig,iktrial,:,0],Lestvecseries[ig,iktrial,:,0] =xnormt[:,0],ynormt[:,0]
#### statistics of the eigenvectors
axrmu,aylmu,sigxr,sigyl,sigcov = numerical_stats(xnorm0,ynorm0,xAm,yAm,eigvJ,nrank,2,ppercent=Npercent)
armu[ig,iktrial,:],almu[ig,iktrial,:] = axrmu[:,0],aylmu[:,0]
sigrcov[ig,iktrial,:],siglcov[ig,iktrial,:]= sigxr[:,0],sigyl[:,0]
siglr[ig,iktrial,:] = sigcov[:,0,0]
Sigmn_num[ig,iktrial,:2] = siglr[ig,iktrial,:] #*np.real(eigvJ[0])#eigvAm[0] ## already considered
Sigmn_num[ig,iktrial,2] = Sigmn_num[ig,iktrial,0]*Npercent[0]+Sigmn_num[ig,iktrial,1]*Npercent[1]
## Rank One Approximation, Nullclines of \kappa
xnorm0, ynorm0 = Rsprvecseries[ig,iktrial,:,0].copy(),Lsprvecseries[ig,iktrial,:,0]
xnorm0, ynorm0 = np.reshape(xnorm0,(N,1)),np.reshape(ynorm0,(N,1))
xmu,ymu = armu[ig,iktrial,:].copy(),almu[ig,iktrial,:].copy()
xsig,ysig = sigrcov[ig,iktrial,:].copy(),siglcov[ig,iktrial,:].copy()
yxcov = siglr[ig,iktrial,:].copy()
### the equivalent gaussian connectivity
Xinit = Xsym.copy()
### >>>>>>>>.Subcircuit Exc sym >>>>>>>>>.
## remaining EE
asqr=(1-np.sqrt(1-etaset[0]**2))/2.0 ## when eta = 0, asqr = 0, aamp = 0, XT-0, X-1
aamp=np.sqrt(asqr)
Xinit[:NE,:NE]=signeta_use[0]*aamp*XsymT[:NE,:NE].copy()+np.sqrt(1-aamp**2)*Xsym[:NE,:NE].copy()
### >>>>>>> Total >>>>>
# EI
asqr=(1-np.sqrt(1-(etaset[1])**2))/2.0 ## when eta = 0, asqr = 0, aamp = 0, XT-0, X-1
aamp=np.sqrt(asqr)
Xinit[NE:,:NE]=signeta_use[1]*aamp*XsymT[NE:,:NE].copy()+np.sqrt(1-aamp**2)*Xsym[NE:,:NE].copy()
Xinit[:NE,NE:]=signeta_use[1]*aamp*XsymT[:NE,NE:].copy()+np.sqrt(1-aamp**2)*Xsym[:NE,NE:].copy()
## II ##
asqr=(1-np.sqrt(1-etaset[2]**2))/2.0
aamp=np.sqrt(asqr)
Xinit[NE:,NE:]=signeta_use[2]*aamp*XsymT[NE:,NE:].copy()+np.sqrt(1-aamp**2)*Xsym[NE:,NE:].copy()
Xinit[:NE,:NE]*=np.sqrt(gE2)
Xinit[:NE,NE:]*=np.sqrt(gI2)
Xinit[NE:,:NE]*=np.sqrt(gE2)
Xinit[NE:,NE:]*=np.sqrt(gI2)
Jfull = Xinit.copy()+Am.copy()
eigvJiid,leigveciid,reigveciid,xnormgau,ynormgau=decompNormalization(Jfull,xnorm0iid,ynorm0iid,xAm,yAm,nparams = Nparams,sort=1)
# eigvJiid,leigveciid,reigveciid,xnormgau,ynormgau=decompNormalization(Jfull,xAm,yAm,xAm,yAm,nparams = Nparams,sort=1)
print('Gaussian eigenvalue:',eigvJiid[0],', ', np.sum(xnormgau*ynormgau)/N)
Riidvecseries[ig,iktrial,:,0],Liidvecseries[ig,iktrial,:,0]=xnormgau[:,0].copy(),ynormgau[:,0].copy()
Biidvseries[ig,iktrial,:]=eigvJiid.copy()
## lambda_change
KE,KI=kE*Cs,kI*Cs
p=KE/NE ## KI/NI
forJEE=EPSP**2*KE*etassrecord[ig,0]*(1-p)**2-2*EPSP**2*KE*(1-etassrecord[ig,0])*(1-p)*p+EPSP**2*p**2*(NE-KE*2+KE*etassrecord[ig,0])## sigma^E
forJEI=-EPSP*IPSP*KI*etassrecord[ig,1]*(1-p)**2+2*EPSP*IPSP*KI*(1-etassrecord[ig,1])*(1-p)*p-EPSP*IPSP*p**2*(NI-KI*2+KI*etassrecord[ig,1])## sigma^I
forJIE=-EPSP*IPSP*KE*etassrecord[ig,1]*(1-p)**2+2*EPSP*IPSP*KE*(1-etassrecord[ig,1])*(1-p)*p-EPSP*IPSP*p**2*(NE-KE*2+KE*etassrecord[ig,1])## signa^E
forJII=IPSP**2*KI*etassrecord[ig,2]*(1-p)**2-2*IPSP**2*KI*(1-etassrecord[ig,2])*(1-p)*p+IPSP**2*p**2*(NI-KI*2+KI*etassrecord[ig,2])
ssigE,ssigI = (forJEE*N*JE0/NE*NE+forJIE*(-N*JI0/NI*NI))/NE,(forJEI*N*JE0/NE*NE+forJII*(-N*JI0/NI*NI))/NI
# gsigE = JE0*EPSP**2*N*p*(1-p)*etassrecord[ig,0]-JI0*EPSP*IPSP*N*(1-p)*p*etassrecord[ig,1]
# gsigI = JE0*EPSP*IPSP*N*p*(1-p)*etassrecord[ig,1]-JI0*IPSP*IPSP*N*(1-p)*p*etassrecord[ig,2]
gsigE = JE0*EPSP**2*p*(etassrecord[ig,0]-p)*(1)*N+JE0*EPSP**2*p*(etassrecord[ig,1]-p)*(KI/KE*gEI**2)*N
gsigI = -(gEI*JE0*EPSP**2*p*(etassrecord[ig,1]-p)*(1)*N + gEI*JE0*EPSP**2*p*(etassrecord[ig,2]-p)*(KI/KE*gEI**2)*N)
# print('COMPARE-SPARSE:',ssigE,ssigI,ssigE*Npercent[0]+ssigI*Npercent[1])
# print('COMPARE-gau:',gsigE,gsigI)
# Sigmn_theo[ig,iktrial,0],Sigmn_theo[ig,iktrial,1]=ssigE/eigvJiid[0]**2, ssigI/eigvJiid[0]**2#ssigE/eigvAm[0]**2, ssigI/eigvAm[0]**2
### normalized by lambda_0
Sigmn_theo[ig,iktrial,0],Sigmn_theo[ig,iktrial,1]=ssigE/eigvAm[0]**2, ssigI/eigvAm[0]**2
Sigmn_theo[ig,iktrial,2] = Sigmn_theo[ig,iktrial,0]*Npercent[0]+Sigmn_theo[ig,iktrial,1]*Npercent[1]
axrmuiid,aylmuiid,sigxriid,sigyliid,sigcoviid = numerical_stats(xnormgau,ynormgau,xAm,yAm,eigvJiid,nrank,2,ppercent=Npercent)
Sigmniid_num[ig,iktrial,:2] = sigcoviid[:,0,0] # *np.real(eigvJiid[0])#eigvAm[0] ## already considered
Sigmniid_num[ig,iktrial,2] = Sigmniid_num[ig,iktrial,0]*Npercent[0]+Sigmniid_num[ig,iktrial,1]*Npercent[1]
### sparse connectivity - dynamics
Jpt = J.copy()
if(iktrial<cuttrials):
xinit = np.random.normal(5,1E-1,(1,N))
else:
xinit = np.random.normal(0,1E-2,(1,N))
xinit = np.squeeze(xinit)
xtemporal = odesimulationP(tt,xinit,Jpt,0)
xfpseries_Sparse[ig,iktrial,:,:] = xtemporal.T.copy()
### ------------ renew
kappanum = np.zeros(3)
if(1):
xact = np.squeeze(xfpseries_Sparse[ig,iktrial,:,-1])
kappanum[0] = np.squeeze(LestvecT.copy()@(1.0+np.tanh(xact.copy()-shiftx)))/N
else:
xact = np.squeeze(xfpseries_Sparse[ig,iktrial,:,-1])
mvec_norm = np.reshape(Restvec,(N,1))
kappanum[0] = np.squeeze(np.reshape(np.squeeze(xact),(1,N))@mvec_norm)/np.squeeze(mvec_norm.T@mvec_norm)
kappaintersect_Sparse[ig, iktrial, :3] = kappanum[:].copy()
''' Gaussian full Connectivity -- Dynamics '''
Jpt = Jfull.copy()
xtemporal = odesimulationP(tt,xinit,Jpt,0)
xfpseries_Full[ig,iktrial,:,:] = xtemporal.T.copy()
''' kappa dynamics '''
## 2 populations
kappanum = np.zeros(3)
xact = np.squeeze(xfpseries_Full[ig,iktrial,:,-1])
## use yAm -- unperturbed centre.
if(1):#(aylmuiid[0,0])*(yAm[0,0])>0): ## ((aylmuiid[0,0]*eigvJiid[0])*(yAm[0,0]*eigvAm[0])>0):
# do not change
kappanum[0] = yAm[0,0]*np.mean(1.0+np.tanh(xact[:NE]-shiftx))*Npercent[0]+yAm[NE,0]*np.mean(1.0+np.tanh(xact[NE:]-shiftx))*Npercent[1]
else:
kappanum[0] = -yAm[0,0]*np.mean(1.0+np.tanh(xact[:NE]-shiftx))*Npercent[0]-yAm[NE,0]*np.mean(1.0+np.tanh(xact[NE:]-shiftx))*Npercent[1]
kappanum[0]=kappanum[0]#*np.sqrt(N)*eigvAm[0] already considered
kappaintersect_Full[ig,iktrial,:3]=kappanum[:].copy()
### rank one approximation dynamics
r1Matt = np.real(Restvec@LestvecT)
r1Matt = r1Matt/N
## use the same initial values
xtemporal = odesimulationP(tt,xinit,r1Matt,0)
xfpseries_R1[ig,iktrial,:,:] = xtemporal.T.copy()
kappaseries_R1[ig, iktrial, :] = np.squeeze(
xtemporal.copy()@np.reshape(xnorm0, (-1, 1)))/np.sum(xnorm0**2)
### temporal evolution of kappa
kappanum = np.zeros(3)
xact = np.reshape(np.squeeze(xfpseries_R1[ig,iktrial,:,-1]),(N,1))
#### @18/08 - error
kappanum[0] = np.squeeze(LestvecT.copy()@(1.0+np.tanh(xact.copy()-shiftx)))/N#@np.tanh(xact.copy()))/N #
kappaintersect_R1[ig,iktrial,:3]=kappanum[:].copy()
def analysis():
'''
with symmetry, subcircuit
eigvJ0
'''
xticks = np.linspace(-1.0,1.0,3)
xlims = [-1.,1.8]
yticks = np.linspace(-1,1,2)
ylims = [-1.2,1.2]
theta = np.linspace(0, 2 * np.pi, 200)
idxc=0
cm='rbg'
Bsprvseries_theo = np.zeros((ngEI,3),dtype=complex)
Bsprvseries_theo_general = np.zeros((ngEI,3),dtype=complex) ### infinite -- first round revision
ig,iiitrial = 16,2#eta==0.8#8,2#3,2#
eta = etaseries[ig]
figtspt,axtspt=plt.subplots(2,1,figsize=(4,5),gridspec_kw = {'wspace':0, 'hspace':0.2})
for i, ax in enumerate(figtspt.axes):
ax.grid('on', linestyle='--')
# ax.set_xticklabels([])
ax.set_yticklabels([])
IPSP = gEI*EPSP
JE0=EPSP*kE*Cs
JI0=IPSP*kI*Cs
KE,KI=kE*Cs,kI*Cs
p=KE/NE
etaset = etassrecord[ig,:]
forJEE = EPSP**2*KE*etaset[0]*(1-p)**2-2*EPSP**2*KE*(1-etaset[0])*(1-p)*p+EPSP**2*p**2*(NE-KE*2+KE*etaset[0])## sigma^E
forJEI = -EPSP*IPSP*KI*etaset[1]*(1-p)**2+2*EPSP*IPSP*KI*(1-etaset[1])*(1-p)*p-EPSP*IPSP*p**2*(NI-KI*2+KI*etaset[1])## sigma^I
forJIE=-EPSP*IPSP*KE*etaset[1]*(1-p)**2+EPSP*IPSP*KE*(1-etaset[1])*(1-p)*p+EPSP*IPSP*KE*(1-etaset[1])*(1-p)*p-EPSP*IPSP*p**2*(NE-KE*2+KE*etaset[1])## signa^E
forJII=IPSP**2*KI*etaset[2]*(1-p)**2-2*IPSP**2*KI*(1-etaset[2])*(1-p)*p+IPSP**2*p**2*(NI-KI*2+KI*etaset[2])
totallambda=(forJEE+forJEI)*N*JE0/NE*NE/N-(forJIE+forJII)*JI0
Janalytical = JE0*EPSP**2*p*(etaset[0]-p)*(1)*NE+JE0*EPSP**2*p*(etaset[1]-p)*(KI/KE*gEI**2)*NE#JE0*EPSP**2*p*(eta-p)*(1+KI/KE*gEI**2)*NE
Janalytical -= (gEI*JE0*EPSP**2*p*(etaset[1]-p)*(1)*NI + gEI*JE0*EPSP**2*p*(etaset[2]-p)*(KI/KE*gEI**2)*NI)#gEI*JE0*EPSP**2*p*(eta-p)*(1+KI/KE*gEI**2)*NI
B2=totallambda
roots = Cubic([1,-BAmvseries[0,0],0,-B2])
Bsprvseries_theo[ig,:] = roots[:3]
#### @YX Oct 22 infinite series summing
tauset_series = {}
tau_set = np.zeros(14)
# # ~~~~~ reciprocal motisf ~~~~~~~~~~
tau_set[10:] = etauserecord[ig,0], etauserecord[ig,1],etauserecord[ig,1], etauserecord[ig,2]
tdiv, tcon, tchn, trec = tau_set[:2],np.reshape(tau_set[2:6],(2,2)), np.reshape(tau_set[6:10],(2,2)),np.reshape(tau_set[10:],(2,2))
tauset_series = {'tdiv':tdiv,
'tcon':tcon,
'tchn':tchn,
'trec':trec
}
Jparams = [JE0, JI0]
Nparams = [NE,NI]
gmat = np.array([np.sqrt(gE2),np.sqrt(gI2),np.sqrt(gE2),np.sqrt(gI2)])
INIT = [-0.5,1]
realg,imagg = fsolve(cal_outliers_general_complex,INIT,args=(tauset_series, [], Jparams, Nparams, gmat, eigvAm),xtol=10e-9,maxfev=1000)
Bsprvseries_theo_general[ig,1] = complex(realg, imagg)
INIT = [-0.5,-1] ### heterogeneous g
# INIT = [gaverage*np.sqrt(etaE),-0.] ### Fig3.1
# INIT = [-0.05,-0.3] ### Fig3.-1
realg,imagg = fsolve(cal_outliers_general_complex,INIT,args=(tauset_series, [], Jparams, Nparams, gmat, eigvAm),xtol=10e-9,maxfev=1000)
Bsprvseries_theo_general[ig,2] = complex(realg, imagg)
INIT = [1.5,0]
realg,imagg = fsolve(cal_outliers_general_complex,INIT,args=(tauset_series, [], Jparams, Nparams, gmat, eigvAm))#,xtol=10e-9,maxfev=1000)
Bsprvseries_theo_general[ig,0] = complex(realg, imagg)
### variance
alphaNEI = Npercent.copy()
gMmat = np.array([[gE2*alphaNEI[0],gI2*alphaNEI[1]],[gE2*alphaNEI[0],gI2*alphaNEI[1]]])
eigvgm,eigvecgm=la.eig(gMmat)
r_g2= np.max(eigvgm)
r_g = np.sqrt(r_g2)
### reciprocity
for idxtrial in range(ntrial):
imagsort = np.argsort(np.imag(Bsprvseries[ig,idxtrial,:]))
mostIm = Bsprvseries[ig,idxtrial,imagsort[0]]#(imagsort[:3])
axtspt[0].scatter(np.real(mostIm),np.imag(mostIm),s=2,c='blue',alpha=0.75) #
mostIm = Bsprvseries[ig,idxtrial,imagsort[-1]]#(imagsort[:3])
axtspt[0].scatter(np.real(mostIm),np.imag(mostIm),s=2,c='blue',alpha=0.75) #
imagsort = np.argsort(np.imag(Biidvseries[ig,idxtrial,:]))
mostIm = Biidvseries[ig,idxtrial,imagsort[0]]#np.mean(imagsort[:3])
axtspt[1].scatter(np.real(mostIm),np.imag(mostIm),s=2,c='green',alpha=0.75)
mostIm = Biidvseries[ig,idxtrial,imagsort[-1]]#np.mean(imagsort[:3])
axtspt[1].scatter(np.real(mostIm),np.imag(mostIm),s=2,c='green',alpha=0.75)
# ### original sparse
# axtspt[0].scatter(np.real(Bsprvseries[ig,:,0]),np.imag(Bsprvseries[ig,:,0])-0.2*idxc,s=10,c='blue',marker='o',alpha=0.75) #
# axtspt[1].scatter(np.real(Biidvseries[ig,:,0]),np.imag(Biidvseries[ig,:,0])-0.2*idxc,s=10,c='green',marker='o',alpha=0.75)
axtspt[1].scatter(np.real(roots[0]),np.imag(roots[0]),s=50,c='',marker='o',edgecolor='purple')
axtspt[1].scatter(np.real(roots[1]),np.imag(roots[1]),s=50,c='',marker='o',edgecolor='red') #
axtspt[1].scatter(np.real(roots[2]),np.imag(roots[2]),s=50,c='',marker='o',edgecolor='orange') # #
axtspt[0].scatter(np.real(roots[0]),np.imag(roots[0]),s=50,c='',marker='o',edgecolor='purple')
axtspt[0].scatter(np.real(roots[1]),np.imag(roots[1]),s=50,c='',marker='o',edgecolor='red')
axtspt[0].scatter(np.real(roots[2]),np.imag(roots[2]),s=50,c='',marker='o',edgecolor='orange') # #
idrands=np.random.choice(np.arange(nrank,N),size=600,replace=False)
axtspt[0].scatter(np.real(Bsprvseries[ig,iiitrial,idrands]),np.imag(Bsprvseries[ig,iiitrial,idrands]),s=2,c='blue',alpha=0.25) #
axtspt[1].scatter(np.real(Biidvseries[ig,iiitrial,idrands]),np.imag(Biidvseries[ig,iiitrial,idrands]),s=2,c='green',alpha=0.25)
### first round revision -- S6 Appendix
axtspt[0].scatter(np.real(Bsprvseries_theo_general[ig,0]),np.imag(Bsprvseries_theo_general[ig,0]),s=50,c='',marker='^',edgecolor='purple')
axtspt[0].scatter(np.real(Bsprvseries_theo_general[ig,1]),np.imag(Bsprvseries_theo_general[ig,1]),s=50,c='',marker='^',edgecolor='red')
axtspt[0].scatter(np.real(Bsprvseries_theo_general[ig,2]),np.imag(Bsprvseries_theo_general[ig,2]),s=50,c='',marker='^',edgecolor='orange')
axtspt[1].scatter(np.real(Bsprvseries_theo_general[ig,0]),np.imag(Bsprvseries_theo_general[ig,0]),s=50,c='',marker='^',edgecolor='purple')
axtspt[1].scatter(np.real(Bsprvseries_theo_general[ig,1]),np.imag(Bsprvseries_theo_general[ig,1]),s=50,c='',marker='^',edgecolor='red')
axtspt[1].scatter(np.real(Bsprvseries_theo_general[ig,2]),np.imag(Bsprvseries_theo_general[ig,2]),s=50,c='',marker='^',edgecolor='orange')
for i in range(2):
axtspt[i].set_aspect('equal')
axtspt[i].scatter(np.ones(1)*BAmvseries[0,0],np.zeros(1),s=50,c='',marker='o',edgecolor='red') #
axtspt[i].spines['right'].set_color('none')
axtspt[i].spines['top'].set_color('none')
axtspt[i].xaxis.set_ticks_position('bottom')
axtspt[i].spines['bottom'].set_position(('data', 0))
axtspt[i].set_xlim(xlims)
axtspt[i].set_ylim(ylims)
axtspt[i].set_xticks(xticks)
axtspt[i].set_yticks(yticks)
axtspt[i].set_aspect('equal')
### ~~~~~~~~~~~~~~ raw iid sparse ~~~~~~~~~~~
# eta = 0.0
# longaxis,shortaxis=(1+eta)*r_g ,(1-eta)*r_g
# xr = longaxis*np.cos(theta)
# yr = shortaxis*np.sin(theta)
# axtspt[0].plot(xr, yr, color="gray", linewidth=0.5,linestyle='--')
# axtspt[1].plot(xr, yr, color="gray", linewidth=0.5,linestyle='--')
### transition dynamics
IPSP = gEI*EPSP
JE0=EPSP*kE*Cs
JI0=IPSP*kI*Cs
KE,KI=kE*Cs,kI*Cs
p=KE/NE
if(etassrecord[-1,2]==0):
coeffs= [1, 1, 0]
else:
coeffs=[1,1,1]
# gee,gei,gie,gii
gmat = np.array([np.sqrt(gE2), np.sqrt(gI2), np.sqrt(gE2), np.sqrt(gI2)])
Jparams = np.array([JE0, JI0])
aprob=p
int_eta = 0.1
transitioneta = fsolve(TransitionEta_spr, int_eta, args=(
Jparams, yAm, xAm, eigvAm, gmat, coeffs, aprob, Nparams,))
targetvalue = (1.0-eigvAm[0])
transitioneta_ = fsolve(TransitionOver_spr, int_eta, args=(
targetvalue, Jparams, yAm, xAm, eigvAm, gmat, coeffs, aprob, Nparams,))
print("transition eta LAMBDA:", transitioneta)
print("transition eta OVERLAP:", transitioneta_)
#### B. theoretical value of $\lambda$ v.s. numerical of the original sparse network
xticks = np.linspace(0,1.0,2)
xlims = [-0.,1.]
theta = np.linspace(0, 2 * np.pi, 200)
idxc=0
cm=['b','g','c']#'bgc'
envelopeRe = np.zeros(ngEI)
envelopeIm = np.zeros(ngEI)
# Bsprvseries_theo = np.zeros((ngEI,3),dtype=complex)
IPSP = gEI*EPSP
JE0=EPSP*kE*Cs
JI0=IPSP*kI*Cs
KE,KI=kE*Cs,kI*Cs
p=KE/NE
for ig,eta in enumerate (etaseries):
etaset = etassrecord[ig,:]
forJEE=EPSP**2*KE*etaset[0]*(1-p)**2-2*EPSP**2*KE*(1-etaset[0])*(1-p)*p+EPSP**2*p**2*(NE-KE*2+KE*etaset[0])## sigma^E
forJEI=-EPSP*IPSP*KI*etaset[1]*(1-p)**2+2*EPSP*IPSP*KI*(1-etaset[1])*(1-p)*p-EPSP*IPSP*p**2*(NI-KI*2+KI*etaset[1])## sigma^I
forJIE=-EPSP*IPSP*KE*etaset[1]*(1-p)**2+EPSP*IPSP*KE*(1-etaset[1])*(1-p)*p+EPSP*IPSP*KE*(1-etaset[1])*(1-p)*p-EPSP*IPSP*p**2*(NE-KE*2+KE*etaset[1])## signa^E
forJII=IPSP**2*KI*etaset[2]*(1-p)**2-2*IPSP**2*KI*(1-etaset[2])*(1-p)*p+IPSP**2*p**2*(NI-KI*2+KI*etaset[2])
totallambda=(forJEE+forJEI)*N*JE0/NE*NE/N-(forJIE+forJII)*JI0
Janalytical = JE0*EPSP**2*p*(etaset[0]-p)*(1)*NE+JE0*EPSP**2*p*(etaset[1]-p)*(KI/KE*gEI**2)*NE#JE0*EPSP**2*p*(eta-p)*(1+KI/KE*gEI**2)*NE
Janalytical -= (gEI*JE0*EPSP**2*p*(etaset[1]-p)*(1)*NI + gEI*JE0*EPSP**2*p*(etaset[2]-p)*(KI/KE*gEI**2)*NI)#gEI*JE0*EPSP**2*p*(eta-p)*(1+KI/KE*gEI**2)*NI
B2=totallambda
roots = Cubic([1,-BAmvseries[0,0],0,-B2])
#### SORT THE ROOTS @YX 07DEC
realroots=np.real(roots)
sidx = np.argsort(realroots)
roots_uno = roots.copy()
for i in range(len(sidx)):
roots[i]=roots_uno[sidx[2-i]]
###--------------------------------
Bsprvseries_theo[ig,:] = roots[:3]
### @YX 07DEC --- note here we use abs??? np.sqrt(re^2+im^2)
# envelopeRe[idxeta]=np.mean(np.abs(np.squeeze(Beigvseries[idxeta,:,1])))
# envelopeRe[ig]=np.mean(np.abs(np.real(Bsprvseries[ig,:,1])))
##### @YX SORT AND THEN FIND ENVELOPEre and imag
realsort = np.sort(np.real(Bsprvseries[ig,:,:]),axis=1)
envelopeRe[ig]=np.mean(realsort[:,2])
# amax_value = np.amax(arr, axis=0)
imagsort = np.sort(np.imag(Bsprvseries[ig,:,:]),axis=1)
envelopeIm[ig]=np.mean(imagsort[:,2])
figtspt,axtspt=plt.subplots(figsize=(5,3))
axtspt.plot(etaseries,np.real(Bsprvseries_theo[:,0]),color='black',linestyle='--',linewidth=1.5)
#### @YX variance of the eigenvalue outlier
mean_eigv_Jss,std_eigv_Jss = np.mean(np.real(Bsprvseries[:,:,0]),axis=1),np.std(np.real(Bsprvseries[:,:,0]),axis=1)
# #### sparse symmetry
axtspt.fill_between(etaseries,mean_eigv_Jss+std_eigv_Jss,mean_eigv_Jss-std_eigv_Jss, alpha=0.3,facecolor='tab:green')
axtspt.plot(etaseries,np.ones(ngEI)*BAmvseries[0,0],linestyle='--',linewidth=1.5,color='grey')
axtspt.set_xlim(xlims)
axtspt.set_ylim(ylims)
axtspt.set_xticks(xticks)
axtspt.set_yticks(yticks)
axtspt.legend()
cm = ['purple','red','orange']
# figR,axR=plt.subplots(figsize=(5,3))
figR,axR=plt.subplots(2,1,figsize=(4,4),sharex=True, tight_layout=True)
#### @YX mean and variance of the eigenvalue outlier
for i in range(3):
axR[0].plot(etaseries,Bsprvseries_theo[:,i].real,color=cm[i],linewidth=1.5,label='theo')
mean_eigv_Jss,std_eigv_Jss = np.mean(np.real(Bsprvseries[:,:,0]),axis=1),np.std(np.real(Bsprvseries[:,:,0]),axis=1)
axR[0].fill_between(etaseries,mean_eigv_Jss+std_eigv_Jss,mean_eigv_Jss-std_eigv_Jss, alpha=0.25,facecolor='purple')
axR[0].fill_between(etaseries,-(envelopeRe),envelopeRe, color='gray',alpha=0.2)
axR[0].plot(etaseries,np.ones(ngEI)*BAmvseries[0,0],linestyle='--',linewidth=1.5,color='black')
# yticks = np.linspace(-1.0,1.5,3)
# ylims = [-1.0,1.5]
yticks = [-0.6,0,2]#np.linspace(-0.6,2.0,3)
ylims = [-0.6,2.0]
axR[0].set_xlim(xlims)
axR[0].set_ylim(ylims)
axR[0].set_xticks(xticks)
axR[0].set_yticks(yticks)
# yticks = np.linspace(-1.5,1.5,2)
# ylims = [-1.5,1.5]
yticks = np.linspace(-1.5,1.5,2)
ylims = [-1.5,1.5]
#### @YX 0409 modify -- imaginary part
# figI,axI=plt.subplots(figsize=(5,3))
for i in range(3):
axR[1].plot(etaseries,np.imag(Bsprvseries_theo[:,i]),color=cm[i],linewidth=1.5,label='theo')
axR[1].fill_between(etaseries,-(envelopeIm),envelopeIm, color='gray',alpha=0.2)
mean_lambda_num = np.mean(Bsprvseries[:,:,:2].imag,axis=1)
std_lambda_num = np.std(Bsprvseries[:,:,:2].imag,axis=1)
# axI.plot(etaseries,mean_lambda_num[:,0],c=cm[0],linewidth=1.5,)
axR[1].fill_between(etaseries,mean_lambda_num[:,0]+std_lambda_num[:,0],mean_lambda_num[:,0]-std_lambda_num[:,0],alpha=0.25,facecolor='tab:blue')
axR[1].set_xlim(xlims)
axR[1].set_ylim(ylims)
axR[1].set_xticks(xticks)
axR[1].set_yticks(yticks)
# ### scatter with colormap
# xticks = np.linspace(0.5,1.5,2)
# xlims = [0.5,1.5]
# yticks = np.linspace(0.5,1.5,2)
# ylims = [0.5,1.5]
### ~~~~~~~~~~ NEW HIGH ~~~~~~~~
### scatter with colormap
xticks = [0.5,2.0]
xlims = [0.5,2.0]
yticks = [0.5,2.0]
ylims = [0.5,2.0]
fig,ax=plt.subplots(figsize=(4,4))
# scatter(X, Y, c = angle, cmap = cm.hsv)
for i in range(ntrial):
ax.scatter(Bsprvseries[:,i,0].real,Biidvseries[:,i,0].real,c=etaseries,cmap=matplotlib.cm.Blues,s=22,marker='o')
ax.set_xlim(ylims)
ax.set_ylim(ylims)
ax.set_xticks(yticks)
ax.set_yticks(yticks)
fig,ax=plt.subplots(figsize=(4,4))
# xticks = np.linspace(0.0,1.5,2)
# xlims = [-0.2,1.7]
# yticks = np.linspace(0.0,1.5,2)
# ylims = [-0.2,1.7]
###~~~~~~~~~~~ new one
xticks = [-0.5,1.0]
xlims = [-0.5,1.0]
yticks = [-0.5,1.0]
ylims = [-0.5,0.8]
mean_DeltaLR_num,std_DeltaLR_num = np.mean(Sigmn_num[:,:,2],axis=1),np.std(Sigmn_num[:,:,2],axis=1)
x = np.real(Bsprvseries_theo[:,0])-eigvAm[0]
y1 = mean_DeltaLR_num+std_DeltaLR_num
y2 = mean_DeltaLR_num-std_DeltaLR_num
polygon = ax.fill_between(x, y1, y2, lw=0, color='none')
xlim = plt.xlim()
ylim = plt.ylim()
verts = plt.vstack([p.vertices for p in polygon.get_paths()])
gradient = ax.imshow(np.linspace(0, 1, 256).reshape(1, -1), cmap='Blues', aspect='auto',
extent=[verts[:, 0].min(), verts[:, 0].max(), verts[:, 1].min(), verts[:, 1].max()])
gradient.set_clip_path(polygon.get_paths()[0], transform=plt.gca().transData)
ax.set_xlim(ylims)
ax.set_ylim(ylims)
ax.set_xticks(yticks)
ax.set_yticks(yticks)
### D. overlap between m and n
### eigenvalue
xticks = np.linspace(0.0,1.0,2)
xlims = [0.0,1.0]
yticks = np.linspace(0.0,1.0,3)
ylims = [-0.2,1.20]
theta = np.linspace(0, 2 * np.pi, 200)
idxc=0
cm=['r','b','g']#rbg'
Sigmn_theo_ = np.zeros((ngEI,3))
IPSP = gEI*EPSP
JE0=EPSP*kE*Cs
JI0=IPSP*kI*Cs
KE,KI=kE*Cs,kI*Cs
p=KE/NE
for ig,eta in enumerate (etaseries):
etaset = etassrecord[ig,:]#eta*np.ones(3)#
# etaset[2:]=0
forJEE=EPSP**2*KE*etaset[0]*(1-p)**2-2*EPSP**2*KE*(1-etaset[0])*(1-p)*p+EPSP**2*p**2*(NE-KE*2+KE*etaset[0])## sigma^E
forJEI=-EPSP*IPSP*KI*etaset[1]*(1-p)**2+2*EPSP*IPSP*KI*(1-etaset[1])*(1-p)*p-EPSP*IPSP*p**2*(NI-KI*2+KI*etaset[1])## sigma^I
# forJIE=-EPSP*IPSP*KE*eta*(1-p)**2+2*EPSP*IPSP*KE*(1-eta)*(1-p)*p-EPSP*IPSP*p**2*(NE-KE*2+KE*eta)## ERROR!!!
forJIE=-EPSP*IPSP*KE*etaset[1]*(1-p)**2+EPSP*IPSP*KE*(1-etaset[1])*(1-p)*p+EPSP*IPSP*KE*(1-etaset[1])*(1-p)*p-EPSP*IPSP*p**2*(NE-KE*2+KE*etaset[1])## signa^E
forJII=IPSP**2*KI*etaset[2]*(1-p)**2-2*IPSP**2*KI*(1-etaset[2])*(1-p)*p+IPSP**2*p**2*(NI-KI*2+KI*etaset[2])
totallambda=(forJEE+forJEI)*N*JE0/NE*NE/N-(forJIE+forJII)*JI0
B2=totallambda
roots = Cubic([1,-BAmvseries[0,0],0,-B2])
### @YX -- SORT THE ROOTS
realroots= [np.real(x) for x in roots]
# print(realroots)
sidx = np.argsort(realroots)
roots_uno = roots.copy()
for i in range(len(sidx)):
roots[i]=roots_uno[sidx[2-i]]
###--------------------------------
JanalyticalE = (forJEE*N*JE0/NE*NE+forJIE*(-N*JI0/NI*NI))/NE
JanalyticalI = (forJEI*N*JE0/NE*NE+forJII*(-N*JI0/NI*NI))/NI
Sigmn_theo_[ig,0],Sigmn_theo_[ig,1] = JanalyticalE/roots[0]**2,JanalyticalI/roots[0]**2#JanalyticalE/eigvAm[0]**2,JanalyticalI/eigvAm[0]**2#JanalyticalE/Biidvseries[ig,0,0]**2,JanalyticalI/Biidvseries[ig,0,0]**2#
Sigmn_theo_[ig,2] = Sigmn_theo_[ig,0]*Npercent[0]+Sigmn_theo_[ig,1]*Npercent[1]
#### ~~~~~ previous scatter plot
# ax.plot(np.real(Bsprvseries_theo[:,0])-eigvAm[0],Sigmn_theo[:,0,2],c='black',linewidth=2.0)
ax.plot(np.real(Bsprvseries_theo[:,0])-eigvAm[0],Sigmn_theo_[:,2],c='black',linewidth=2.0)
titlestr='unperturbed eigenvalue'
fig,ax2 = plt.subplots(figsize=(5,3))
ax2.fill_between(etaseries,np.mean(Sigmn_num[:,:,0],axis=1)+np.std(Sigmn_num[:,:,0],axis=1),\
np.mean(Sigmn_num[:,:,0],axis=1)-np.std(Sigmn_num[:,:,0],axis=1),color='tab:red',alpha=0.3,label=r'numerical')
ax2.plot(etaseries,Sigmn_theo_[:,0],'tab:red',linewidth =1.5,label=r'theoretical')
ax2.fill_between(etaseries,np.mean(Sigmn_num[:,:,1],axis=1)+np.std(Sigmn_num[:,:,1],axis=1),\
np.mean(Sigmn_num[:,:,1],axis=1)-np.std(Sigmn_num[:,:,1],axis=1),color='tab:blue',alpha=0.3,label=r'numerical')
ax2.plot(etaseries,Sigmn_theo_[:,1],'tab:blue',linewidth =1.5,label=r'theoretical')
ax2.fill_between(etaseries,np.mean(Sigmn_num[:,:,2],axis=1)+np.std(Sigmn_num[:,:,2],axis=1),\
np.mean(Sigmn_num[:,:,2],axis=1)-np.std(Sigmn_num[:,:,2],axis=1),color='black',alpha=0.3,label=r'numerical')
ax2.plot(etaseries,Sigmn_theo_[:,2],'black',linewidth =1.5,label=r'theoretical')
yticks = [-2,0,3]#np.linspace(-2,5,7)
ylims = [-2,3]
ax2.set_xlim(xlims)
ax2.set_ylim(ylims)
ax2.set_xticks(xticks)
ax2.set_yticks(yticks)
##### ----------------------- 3*1 subplots
yticks = np.linspace(-2,1,3)
ylims = [-2,1]
fig,ax2 = plt.subplots(3,1,figsize=(5,4),sharex=True,tight_layout=True)
ax2[0].fill_between(etaseries,np.mean(Sigmn_num[:,:,0],axis=1)+np.std(Sigmn_num[:,:,0],axis=1),\
np.mean(Sigmn_num[:,:,0],axis=1)-np.std(Sigmn_num[:,:,0],axis=1),color='tab:red',alpha=0.25,label=r'numerical')
ax2[0].plot(etaseries,Sigmn_theo_[:,0],'tab:red',linewidth =0.75,linestyle='--',label=r'theoretical')
# yticks = np.linspace(-5,10,4)
# ylims = [-5,10]
ax2[0].set_xlim(xlims)
ax2[0].set_ylim(ylims)
ax2[0].set_xticks(xticks)
ax2[0].set_yticks(yticks)
# fig,ax2 = plt.subplots(figsize=(5,2))
yticks = np.linspace(-0.5,1,2)
ylims = [-0.5,1]
ax2[2].fill_between(etaseries,np.mean(Sigmn_num[:,:,2],axis=1)+np.std(Sigmn_num[:,:,2],axis=1),\
np.mean(Sigmn_num[:,:,2],axis=1)-np.std(Sigmn_num[:,:,2],axis=1),color='black',alpha=0.25,label=r'numerical')
# ax2[2].plot(etaseries,Sigmn_theo_[:,2],'k',linewidth =0.75,label=r'theoretical')
# # ax2.plot(etaseries,np.mean(Bsprvseries[:,:,0],axis=1)-eigvAm[0],'orange',linewidth =1.5,label=r'theoretical')
ax2[2].plot(etaseries,Sigmn_theo_[:,1]*1.0/5+Sigmn_theo_[:,0]*4.0/5,'k',linestyle='--',linewidth =1.5,label=r'theoretical')
ax2[2].set_xlim(xlims)
ax2[2].set_ylim(ylims)
ax2[2].set_xticks(xticks)
ax2[2].set_yticks(yticks)
yticks = np.linspace(-0.5,4,2)
ylims = [-0.5,4]
# fig,ax2 = plt.subplots(figsize=(5,2))
ax2[1].fill_between(etaseries,np.mean(Sigmn_num[:,:,1],axis=1)+np.std(Sigmn_num[:,:,1],axis=1),\
np.mean(Sigmn_num[:,:,1],axis=1)-np.std(Sigmn_num[:,:,1],axis=1),color='tab:blue',alpha=0.25,label=r'numerical')
ax2[1].plot(etaseries,Sigmn_theo_[:,1],'tab:blue',linewidth =0.75,linestyle='--',label=r'theoretical')
ax2[1].set_xlim(xlims)
ax2[1].set_ylim(ylims)
ax2[1].set_xticks(xticks)
ax2[1].set_yticks(yticks)
#### >>>>>>>>> individual perturbations >>>>>>>>>
### C. COMPARE EACH ENTRY ON THE EIGENVECTORS (RECONSTRUCT V.S. EIGENDECOMPOSITION)
xtickms = np.linspace(-2,4,2)
xlimms = [-2,4]
ytickms = np.linspace(-2,4,2)
ylimms = [-2,4]
xticks = np.linspace(-50,20,2)
xlims = [-50,20]
yticks = np.linspace(-50,20,2)
ylims = [-50,20]
'''# CHOOSE ONE TRIAL'''
idxtrial,idxtrial_ = 8,3 # ### @YX original
fig,ax=plt.subplots(2,2,figsize=(4,4))
idxtrial=0
### c SAMPLE
gEIsamples=np.array([16,20])
for i in range(len(gEIsamples)):
ax[0,i].plot(xticks,yticks,color='darkred',linestyle='--')
ax[1,i].plot(xticks,yticks,color='darkred',linestyle='--')
#### @YX modify 2508 -- redundancy
#### @YX modify 2508 -- from Reigvecseries[i,...] to Reigvecseries[etaseriesples[i]...]
idrandsE=np.random.choice(np.arange(0,NE),size=100,replace=False)
idrandsI=np.random.choice(np.arange(NE,N),size=100,replace=False)
ax[0,i].scatter(np.real(Rsprvecseries[gEIsamples[i],idxtrial,idrandsE]),np.real(Restvecseries[gEIsamples[i],idxtrial,idrandsE]),s=2,c='red',alpha=0.5)
ax[1,i].scatter(np.real(Lsprvecseries[gEIsamples[i],idxtrial,idrandsE]),np.real(Lestvecseries[gEIsamples[i],idxtrial,idrandsE]),s=2,c='red',alpha=0.5)
ax[0,i].scatter(np.real(Rsprvecseries[gEIsamples[i],idxtrial,idrandsI]),np.real(Restvecseries[gEIsamples[i],idxtrial,idrandsI]),s=2,c='tab:blue',alpha=0.5)
ax[1,i].scatter(np.real(Lsprvecseries[gEIsamples[i],idxtrial,idrandsI]),np.real(Lestvecseries[gEIsamples[i],idxtrial,idrandsI]),s=2,c='tab:blue',alpha=0.5)
# ax[0,i].scatter(np.real(Reigvecseries[idxgsamples[i],idxtrial,:int(NE/2)]),np.real(ReigvecTseries[idxgsamples[i],idxtrial,:int(NE/2)]),s=2,c='red',alpha=0.5)
# ax[1,i].scatter(np.real(Leigvecseries[idxgsamples[i],idxtrial,:int(NE/2)]),np.real(LeigvecTseries[idxgsamples[i],idxtrial,:int(NE/2)]),s=2,c='red',alpha=0.5)
# ax[0,i].scatter(np.real(Reigvecseries[idxgsamples[i],idxtrial,int(NE*3/2):]),np.real(ReigvecTseries[idxgsamples[i],idxtrial,int(NE*3/2):]),s=2,c='blue',alpha=0.5)
# ax[1,i].scatter(np.real(Leigvecseries[idxgsamples[i],idxtrial,int(NE*3/2):]),np.real(LeigvecTseries[idxgsamples[i],idxtrial,int(NE*3/2):]),s=2,c='blue',alpha=0.5)
ax[0,i].set_xticks([])
ax[0,i].set_yticks([])
ax[1,i].set_xticks([])
ax[1,i].set_yticks([])
for i in range(2):
ax[1,i].set_xlim(xlims)
ax[1,i].set_ylim(ylims)
ax[1,0].set_yticks(yticks)
for i in range(2):
ax[0,i].set_xlim(xlimms)
ax[0,i].set_ylim(ylimms)