def linear_response_1loop(w, W, phi_r):
'''
calculate one-loop correction to the propagator around mean-field theory
:param w: frequency
:param W: weight matrix, weighted by postsynaptic gain
:param phi_r: firing rates
:return: propagator matrix
'''
par = params.params()
b = par.b
gain = par.gain
N = par.N
Tmax = 100
dt_ccg = 1
wmax = 1. / dt_ccg
dw = 1. / Tmax
w_calc = np.arange(-wmax, wmax, dw) * math.pi
dw *= math.pi
Nw = w_calc.size
g0 = np.dot(W, phi_r) + b
phi_1 = phi_prime(g0, gain)
phi_1_diag = np.diag(phi_1)
phi_2 = phi_prime2(g0, gain)
phi_2_diag = np.diag(phi_2)
F1 = linear_response_fun(w, np.dot(phi_1_diag, W), phi_r)
Fbar = np.dot(g_fun(w) * W, F1)
Fbar_int = np.zeros((N, N), dtype='complex128')
for o in range(Nw):
Fbar1 = np.dot(
g_fun(w_calc[o]) * W,
linear_response_fun(w_calc[o], np.dot(phi_1_diag, W), phi_r))
Fbar2 = np.dot(
g_fun(w - w_calc[0]) * W,
linear_response_fun(w - w_calc[o], np.dot(phi_1_diag, W), phi_r))
Fbar_int += np.dot(Fbar1 * Fbar2, np.dot(phi_1_diag, Fbar)) * dw
linear_response_1loop = np.dot(np.dot(F1, phi_2_diag / 2.),
Fbar_int) / (2 * math.pi**1)
return linear_response_1loop
def rates_1loop(W):
"""
inputs: weight matrix
calculate one-loop correction for steady-state firing rates in fluctuation expansion
"""
import params
reload(params)
from phi import phi_prime
from phi import phi_prime2
par = params.params()
N = par.N
gain = par.gain
b = par.b
phi_r = rates_ss(W)
Tmax = 100
dt_ccg = 1.
wmax = 1. / dt_ccg
dw = 1. / Tmax
w = np.arange(-wmax, wmax, dw) * math.pi
dw = dw * math.pi
Nw = w.size
g0 = np.dot(W, phi_r) + b
phi_1 = phi_prime(g0, gain)
phi_1 = np.diag(phi_1)
phi_2 = phi_prime2(g0, gain)
Fbarsum = np.zeros((N, Nw), dtype=complex)
for o in range(Nw): # first compute Fbar over dummy frequency
Fbar1 = np.dot(
g_fun(w[o]) * W, linear_response_fun(w[o], np.dot(phi_1, W),
phi_r))
Fbarsum[:, o] = np.dot(Fbar1 * Fbar1.conj(),
phi_r) # sum over first inner vertex
Fbarsum_int = np.sum(Fbarsum,
axis=1) * dw # integrate over dummy frequency
F1 = linear_response_fun(0., np.dot(phi_1, W), phi_r)
r_1loop = np.dot(F1, .5 * phi_2 * Fbarsum_int) / (
(2 * math.pi)**1) # sum over second inner vertex
return r_1loop