from brian import *
import sys, subprocess, os
import numpy as np
import matplotlib
from matplotlib import lines
import pylab as pl
# The potassium reversal potential is recovered
# from a pathological value (-60mV) toward physiological values:
# Eke for excitatory neurons and Eki for inhibitory neurons.
Eke = -70
Eki = -80
defaultclock.dt = 0.05 *ms
# simulation time
simt = 2500 *ms
# Number of neurons
M = 3000
Ne = int(M)
Ni = int(0.25*M)
p = 0.01
# Synaptic reversal potential
ve = 25 *mV
vi = -70 *mV
# Excitatory ML parameters
Ce = 5 *nF
gke = 22 *usiemens
gnae= 20 *usiemens
gle = 2 *usiemens
v1e = -7 *mV
v2e = 15 *mV
v3e = -8 *mV
v4e = 15 *mV
v3te = 0 *mV
v4te = 15 *mV
#vke = part of the neuron model eqse (see below)
vnae= 60 *mV
vle = -60 *mV
phie = 0.55 ###
# Inhibitory ML parameters
Ci = 1.0 *nF
gki = 7.0 *usiemens
gnai= 16.0 *usiemens
gli = 7.0 *usiemens
v1i = -7.35 *mV
v2i = 23.0 *mV
v3i = -12.0 *mV
v4i = 12.0 *mV
v3ti = -15.0 *mV
v4ti = 10.0 *mV
#vki = part of the neuron model eqsi (see below)
vnai= 60 *mV
vli = -60 *mV
phii = 1.0
# synaptic decay time constants
tauext = 3.0 *ms
taue = 3.0 *ms
taui = 7.0 *ms
# synaptic coupling strength
Jee = 90.0 *nsiemens
Jie = 190.0 *nsiemens
Jei = 50.0 *nsiemens
Jii = 15.0 *nsiemens
Jin = 200 *nsiemens
# neuron threshold, refractory period
v_the = 15 *mV
v_thi = 10 *mV
rfe = 2 *ms
rfi = 1 *ms
# External input to excitatory neurons.
# Given a time vector tvec = [t0, t1, ..., tn] and a rate vector rvec = [r0, r1, ..., rn],
# the function stim(t) generates a piecewise linear function that connects (t0,r0), (t1,r1), ..., (tn,rn).
exc_base = 400 *Hz #500
stim = 900 *Hz
tvec = [ 0.*ms, 1*ms, 11*ms, 12*ms, simt]
rvec = [exc_base, stim, stim, exc_base, exc_base]
def heavi(x):
if x > 0:
return 1
elif x <=0:
return 0
def stim(t):
r = 0.*Hz
for j in range(len(tvec)-1):
r = r + (rvec[j] + (rvec[j+1]-rvec[j])/(tvec[j+1]-tvec[j])*(t-tvec[j]))*(heavi(t-tvec[j]) - heavi(t-tvec[j+1]))
return r
excinput = lambda t: stim(t)
inhinput = 200.*Hz
# neuron models
# excitatory ML neurons
eqse = Equations('''
dv/dt = ( - gnae*minf*(v - vnae) - gke*w*(v - vke) - gle*(v - vle) \
- gext*(v - ve) - ge*(v - ve) - gi*(v - vi) )/Ce : volt
dw/dt = phie*(winf - w)/tauw : 1
dgext/dt = -gext/tauext : siemens
dge/dt = -ge/taue : siemens
dgi/dt = -gi/taui : siemens
minf = .5*(1 + tanh((v - v1e)/v2e)) : 1
winf = .5*(1 + tanh((v - v3e)/v4e)) : 1
tauw = 1/cosh((v - v3te)/(2*v4te))*ms : second
vke : volt
''')
# inhibitory ML neurons
eqsi = Equations('''
dv/dt = ( - gnai*minf*(v - vnai) - gki*w*(v - vki) - gli*(v - vli) \
- gext*(v - ve) - ge*(v - ve) - gi*(v - vi))/Ci : volt
dw/dt = phii*(winf - w)/tauw : 1
dgext/dt = -gext/tauext : siemens
dge/dt = -ge/taue : siemens
dgi/dt = -gi/taui : siemens
minf = .5*(1 + tanh((v - v1i)/v2i)) : 1
winf = .5*(1 + tanh((v - v3i)/v4i)) : 1
tauw = 1/cosh((v - v3ti)/(2*v4ti))*ms : second
vki : volt
''')
# Create population of neurons
Pe = NeuronGroup(Ne, model=eqse,
threshold='v>v_the',
refractory = rfe)
Pi = NeuronGroup(Ni, model=eqsi,
threshold='v>v_thi',
refractory = rfi)
# Recurrent connections
Cee = Connection(Pe, Pe, 'ge', weight=Jee, sparseness=p)
Cie = Connection(Pe, Pi, 'ge', weight=Jie, sparseness=p)
Cei = Connection(Pi, Pe, 'gi', weight=Jei, sparseness=p)
Cii = Connection(Pi, Pi, 'gi', weight=Jii, sparseness=p)
# External Poisson spikes
inpute = PoissonGroup(Ne, rates = excinput)
inputi = PoissonGroup(Ni, rates = inhinput)
# Connect the input and neurons.
input_co1 = IdentityConnection(inpute, Pe, 'gext', weight=Jin)
input_co2 = IdentityConnection(inputi, Pi, 'gext', weight=Jin)
# Gradual decrease (1000 - 2000ms) of potassium reversal potentials.
# Excitatory neurons: from Ek=-60mV to Ek=-70mV
# Inhibitory neurons: from Ek=-60mV to Ek=-80mV
tm_e = [ 0.*ms, 1000.*ms, 2000.*ms, simt]
EKe_arr = [-60*mV, -60*mV, Eke*mV, Eke*mV]
tm_i = [ 0.*ms, 1000.*ms, 2000.*ms, simt]
EKi_arr = [-60*mV, -60*mV, Eki*mV, Eki*mV]
def EKe_updated(t):
v = 0.*mV
for j in range(len(tm_e)-1):
v = v + (EKe_arr[j] + (EKe_arr[j+1]-EKe_arr[j])/(tm_e[j+1]-tm_e[j])*(t-tm_e[j]))*(heavi(t-tm_e[j]) - heavi(t-tm_e[j+1]))
return v
def EKi_updated(t):
v = 0.*mV
for j in range(len(tm_i)-1):
v = v + (EKi_arr[j] + (EKi_arr[j+1]-EKi_arr[j])/(tm_i[j+1]-tm_i[j])*(t-tm_i[j]))*(heavi(t-tm_i[j]) - heavi(t-tm_i[j+1]))
return v
@network_operation
def VKe(v):
Pe.vke = EKe_updated(v.t)
@network_operation
def VKi(v):
Pi.vki = EKi_updated(v.t)
# Initialization
Pe.v = -40*mV
Pe.w = 0.1
Pi.v = -40*mV
Pi.w = 0.1
# Record population activity
Me = MultiStateMonitor(Pe, record=True)
Mi = MultiStateMonitor(Pi, record=True)
Re = PopulationRateMonitor(Pe,bin=1*ms)
Ri = PopulationRateMonitor(Pi,bin=1*ms)
net = Network(VKi,VKe,Pe,Pi,inpute,inputi,input_co1,input_co2,Cee,Cei,Cie,Cii,Me,Mi,Re,Ri)
net.run(simt)
#==== excitatory and inhibitory firing rates ====#
pl.figure(1,figsize=(10,4))
axfig2 = pl.subplot(111)
pl.plot(Re.times[500:]/ms,Re.rate[500:],'b',linewidth=2,label='exc')
pl.plot(Re.times[500:]/ms,Ri.rate[500:],'r',linewidth=2,label='inh')
pl.ylabel('Hz',fontsize=30)
pl.xlabel('ms',fontsize=30)
pl.xlim([500,2500])
pl.ylim([0,400])
pl.yticks([0,200,400])
# remove boundary
axfig2.spines['top'].set_visible(False)
axfig2.spines['right'].set_visible(False)
axfig2.get_xaxis().tick_bottom()
axfig2.get_yaxis().tick_left()
# display the period Ek changes.
line = lines.Line2D([1000,2000],[400,400],lw=4,color='k',alpha=1)
line.set_clip_on(False)
axfig2.add_line(line)
# legend
pl.legend(loc=2,fontsize=24,frameon=False)
leg = pl.gca().get_legend()
llines=leg.get_lines()
pl.setp(llines,linewidth=3)
#==== voltage trace of sample neurons ====#
pl.figure(2,facecolor='white',figsize=(10,4))
# excitatory neuron
ax1 = subplot(211,frameon=False)
ax1.axes.get_xaxis().set_visible(False)
ax1.axes.get_yaxis().set_visible(False)
pl.plot(Me['v'].times/ms,Me['v'][0],'b',linewidth=2)
pl.xlim([500.,2500.])
# inhibitory neuron
ax2 = subplot(212,frameon=False)
ax2.axes.get_xaxis().set_visible(False)
ax2.axes.get_yaxis().set_visible(False)
pl.plot(Mi['v'].times/ms,Mi['v'][0],'r',linewidth=2)
pl.xlim([500.,2500.])
matplotlib.rcParams.update({'font.size':24})
pl.show()