def __call__(self, shape, dtype=None):
shape = [size2len(d) for d in shape]
fan_in, fan_out = _compute_fans(shape,
in_axis=self.in_axis,
out_axis=self.out_axis)
if self.mode == "fan_in":
denominator = fan_in
elif self.mode == "fan_out":
denominator = fan_out
elif self.mode == "fan_avg":
denominator = (fan_in + fan_out) / 2
else:
raise ValueError(
"invalid mode for variance scaling initializer: {}".format(
self.mode))
variance = math.array(self.scale / denominator, dtype=dtype)
if self.distribution == "truncated_normal":
# constant is stddev of standard normal truncated to (-2, 2)
stddev = math.sqrt(variance) / math.array(.87962566103423978,
dtype)
res = self.rng.truncated_normal(-2, 2, shape) * stddev
return math.asarray(res, dtype=dtype)
elif self.distribution == "normal":
res = self.rng.normal(size=shape) * math.sqrt(variance)
return math.asarray(res, dtype=dtype)
elif self.distribution == "uniform":
res = self.rng.uniform(low=-1, high=1, size=shape) * math.sqrt(
3 * variance)
return math.asarray(res, dtype=dtype)
else:
raise ValueError(
"invalid distribution for variance scaling initializer")
def make_conn(self, x):
assert bm.ndim(x) == 1
x_left = bm.reshape(x, (-1, 1))
x_right = bm.repeat(x.reshape((1, -1)), len(x), axis=0)
d = self.dist(x_left - x_right)
Jxx = self.J0 * bm.exp(
-0.5 * bm.square(d / self.a)) / (bm.sqrt(2 * bm.pi) * self.a)
return Jxx
def make_conn(self):
x1, x2 = bm.meshgrid(self.x, self.x)
value = bm.stack([x1.flatten(), x2.flatten()]).T
d = self.dist(bm.abs(value[0] - value))
d = bm.linalg.norm(d, axis=1)
d = d.reshape((self.length, self.length))
Jxx = self.J0 * bm.exp(
-0.5 * bm.square(d / self.a)) / (bm.sqrt(2 * bm.pi) * self.a)
return Jxx
def __init__(self,
num_input,
num_hidden,
num_output,
tau=1.0,
dt=0.1,
g=1.8,
alpha=1.0,
**kwargs):
super(EchoStateNet, self).__init__(**kwargs)
# parameters
self.num_input = num_input
self.num_hidden = num_hidden
self.num_output = num_output
self.tau = tau
self.dt = dt
self.g = g
self.alpha = alpha
# weights
self.w_ir = bm.random.normal(size=(num_input,
num_hidden)) / bm.sqrt(num_input)
self.w_rr = g * bm.random.normal(
size=(num_hidden, num_hidden)) / bm.sqrt(num_hidden)
self.w_or = bm.random.normal(size=(num_output, num_hidden))
w_ro = bm.random.normal(size=(num_hidden,
num_output)) / bm.sqrt(num_hidden)
self.w_ro = bm.Variable(w_ro)
# variables
self.h = bm.Variable(bm.random.normal(size=num_hidden) * 0.5) # hidden
self.r = bm.Variable(bm.tanh(self.h)) # firing rate
self.o = bm.Variable(bm.dot(self.r, w_ro)) # output unit
self.P = bm.Variable(bm.eye(num_hidden) *
self.alpha) # inverse correlation matrix
def make_conn(self): x_left = bm.reshape(self.x, (-1, 1)) x_right = bm.repeat(self.x.reshape((1, -1)), len(self.x), axis=0) d = self.dist(x_left - x_right) conn = self.J0 * bm.exp(-0.5 * bm.square(d / self.a)) / (bm.sqrt(2 * bm.pi) * self.a) return conn
def fV(self, V, t, y, z, Isyn):
# m channel
t1 = 13. - V + self.NaK_th
t1_exp = bm.exp(t1 / 4.)
m_alpha_by_V = 0.32 * t1 / (t1_exp - 1.) # \alpha_m(V)
m_alpha_by_V_diff = (-0.32 * (t1_exp - 1.) + 0.08 * t1 * t1_exp) / (
t1_exp - 1.)**2 # \alpha_m'(V)
t2 = V - 40. - self.NaK_th
t2_exp = bm.exp(t2 / 5.)
m_beta_by_V = 0.28 * t2 / (t2_exp - 1.) # \beta_m(V)
m_beta_by_V_diff = (0.28 * (t2_exp - 1) - 0.056 * t2 * t2_exp) / (
t2_exp - 1)**2 # \beta_m'(V)
m_tau_by_V = 1. / self.phi_m / (m_alpha_by_V + m_beta_by_V
) # \tau_m(V)
m_inf_by_V = m_alpha_by_V / (m_alpha_by_V + m_beta_by_V
) # \m_{\infty}(V)
m_inf_by_V_diff = (m_alpha_by_V_diff * m_beta_by_V - m_alpha_by_V * m_beta_by_V_diff) / \
(m_alpha_by_V + m_beta_by_V) ** 2 # \m_{\infty}'(V)
# h channel
h_alpha_by_y = 0.128 * bm.exp(
(17. - y + self.NaK_th) / 18.) # \alpha_h(y)
t3 = bm.exp((40. - y + self.NaK_th) / 5.)
h_beta_by_y = 4. / (t3 + 1.) # \beta_h(y)
h_inf_by_y = h_alpha_by_y / (h_alpha_by_y + h_beta_by_y
) # h_{\infty}(y)
# n channel
t5 = (15. - y + self.NaK_th)
t5_exp = bm.exp(t5 / 5.)
n_alpha_by_y = 0.032 * t5 / (t5_exp - 1.) # \alpha_n(y)
t6 = bm.exp((10. - y + self.NaK_th) / 40.)
n_beta_y = self.b * t6 # \beta_n(y)
n_inf_by_y = n_alpha_by_y / (n_alpha_by_y + n_beta_y) # n_{\infty}(y)
# p channel
t7 = bm.exp((self.p_half - y + self.IT_th) / self.p_k)
p_inf_by_y = 1. / (1. + t7) # p_{\infty}(y)
t8 = bm.exp((self.q_half - z + self.IT_th) / self.q_k)
q_inf_by_z = 1. / (1. + t8) # q_{\infty}(z)
# x
gNa = self.g_Na * m_inf_by_V**3 * h_inf_by_y # gNa
gK = self.g_K * n_inf_by_y**4 # gK
gT = self.g_T * p_inf_by_y * p_inf_by_y * q_inf_by_z # gT
FV = gNa + gK + gT + self.g_L + self.g_KL # dF/dV
Fm = 3 * self.g_Na * h_inf_by_y * (
V -
self.E_Na) * m_inf_by_V * m_inf_by_V * m_inf_by_V_diff # dF/dvm
t9 = self.C / m_tau_by_V
t10 = FV + Fm
t11 = t9 + FV
rho_V = (t11 - bm.sqrt(bm.maximum(t11**2 - 4 * t9 * t10,
0.))) / 2 / t10 # rho_V
INa = gNa * (V - self.E_Na)
IK = gK * (V - self.E_KL)
IT = gT * (V - self.E_T)
IL = self.g_L * (V - self.E_L)
IKL = self.g_KL * (V - self.E_KL)
Iext = self.V_factor * Isyn
dVdt = rho_V * (-INa - IK - IT - IL - IKL + Iext) / self.C
return dVdt
def reduced_trn_derivative(V, y, z, t, Isyn, b, rho_p, g_T, g_L, g_KL,
E_L, E_KL, IT_th, NaK_th):
# m channel
t1 = 13. - V + NaK_th
t1_exp = bm.exp(t1 / 4.)
m_alpha_by_V = 0.32 * t1 / (t1_exp - 1.) # \alpha_m(V)
m_alpha_by_V_diff = (-0.32 *
(t1_exp - 1.) + 0.08 * t1 * t1_exp) / (
t1_exp - 1.)**2 # \alpha_m'(V)
t2 = V - 40. - NaK_th
t2_exp = bm.exp(t2 / 5.)
m_beta_by_V = 0.28 * t2 / (t2_exp - 1.) # \beta_m(V)
m_beta_by_V_diff = (0.28 * (t2_exp - 1) - 0.056 * t2 * t2_exp) / (
t2_exp - 1)**2 # \beta_m'(V)
m_tau_by_V = 1. / phi_m / (m_alpha_by_V + m_beta_by_V) # \tau_m(V)
m_inf_by_V = m_alpha_by_V / (m_alpha_by_V + m_beta_by_V
) # \m_{\infty}(V)
m_inf_by_V_diff = (m_alpha_by_V_diff * m_beta_by_V - m_alpha_by_V * m_beta_by_V_diff) / \
(m_alpha_by_V + m_beta_by_V) ** 2 # \m_{\infty}'(V)
# h channel
h_alpha_by_V = 0.128 * bm.exp(
(17. - V + NaK_th) / 18.) # \alpha_h(V)
h_beta_by_V = 4. / (bm.exp(
(40. - V + NaK_th) / 5.) + 1.) # \beta_h(V)
h_inf_by_V = h_alpha_by_V / (h_alpha_by_V + h_beta_by_V
) # h_{\infty}(V)
h_tau_by_V = 1. / phi_h / (h_alpha_by_V + h_beta_by_V) # \tau_h(V)
h_alpha_by_y = 0.128 * bm.exp(
(17. - y + NaK_th) / 18.) # \alpha_h(y)
t3 = bm.exp((40. - y + NaK_th) / 5.)
h_beta_by_y = 4. / (t3 + 1.) # \beta_h(y)
h_beta_by_y_diff = 0.8 * t3 / (1 + t3)**2 # \beta_h'(y)
h_inf_by_y = h_alpha_by_y / (h_alpha_by_y + h_beta_by_y
) # h_{\infty}(y)
h_alpha_by_y_diff = -h_alpha_by_y / 18. # \alpha_h'(y)
h_inf_by_y_diff = (h_alpha_by_y_diff * h_beta_by_y - h_alpha_by_y * h_beta_by_y_diff) / \
(h_beta_by_y + h_alpha_by_y) ** 2 # h_{\infty}'(y)
# n channel
t4 = (15. - V + NaK_th)
n_alpha_by_V = 0.032 * t4 / (bm.exp(t4 / 5.) - 1.) # \alpha_n(V)
n_beta_by_V = b * bm.exp((10. - V + NaK_th) / 40.) # \beta_n(V)
n_tau_by_V = 1. / (n_alpha_by_V + n_beta_by_V) / phi_n # \tau_n(V)
n_inf_by_V = n_alpha_by_V / (n_alpha_by_V + n_beta_by_V
) # n_{\infty}(V)
t5 = (15. - y + NaK_th)
t5_exp = bm.exp(t5 / 5.)
n_alpha_by_y = 0.032 * t5 / (t5_exp - 1.) # \alpha_n(y)
t6 = bm.exp((10. - y + NaK_th) / 40.)
n_beta_y = b * t6 # \beta_n(y)
n_inf_by_y = n_alpha_by_y / (n_alpha_by_y + n_beta_y
) # n_{\infty}(y)
n_alpha_by_y_diff = (0.0064 * t5 * t5_exp - 0.032 *
(t5_exp - 1.)) / (t5_exp -
1.)**2 # \alpha_n'(y)
n_beta_by_y_diff = -n_beta_y / 40 # \beta_n'(y)
n_inf_by_y_diff = (n_alpha_by_y_diff * n_beta_y - n_alpha_by_y * n_beta_by_y_diff) / \
(n_alpha_by_y + n_beta_y) ** 2 # n_{\infty}'(y)
# p channel
p_inf_by_V = 1. / (1. + bm.exp(
(p_half - V + IT_th) / p_k)) # p_{\infty}(V)
p_tau_by_V = (3 + 1. / (bm.exp(
(V + 27. - IT_th) / 10.) + bm.exp(-(V + 102. - IT_th) / 15.))
) / phi_p # \tau_p(V)
t7 = bm.exp((p_half - y + IT_th) / p_k)
p_inf_by_y = 1. / (1. + t7) # p_{\infty}(y)
p_inf_by_y_diff = t7 / p_k / (1. + t7)**2 # p_{\infty}'(y)
# p channel
q_inf_by_V = 1. / (1. + bm.exp(
(q_half - V + IT_th) / q_k)) # q_{\infty}(V)
t8 = bm.exp((q_half - z + IT_th) / q_k)
q_inf_by_z = 1. / (1. + t8) # q_{\infty}(z)
q_inf_diff_z = t8 / q_k / (1. + t8)**2 # q_{\infty}'(z)
q_tau_by_V = (85. + 1 / (bm.exp(
(V + 48. - IT_th) / 4.) + bm.exp(-(V + 407. - IT_th) / 50.))
) / phi_q # \tau_q(V)
# ----
# x
# ----
gNa = g_Na * m_inf_by_V**3 * h_inf_by_y # gNa
gK = g_K * n_inf_by_y**4 # gK
gT = g_T * p_inf_by_y * p_inf_by_y * q_inf_by_z # gT
FV = gNa + gK + gT + g_L + g_KL # dF/dV
Fm = 3 * g_Na * h_inf_by_y * (
V - E_Na) * m_inf_by_V * m_inf_by_V * m_inf_by_V_diff # dF/dvm
t9 = C / m_tau_by_V
t10 = FV + Fm
t11 = t9 + FV
rho_V = (t11 - bm.sqrt(bm.maximum(t11**2 - 4 * t9 * t10,
0.))) / 2 / t10 # rho_V
INa = gNa * (V - E_Na)
IK = gK * (V - E_KL)
IT = gT * (V - E_T)
IL = g_L * (V - E_L)
IKL = g_KL * (V - E_KL)
Iext = V_factor * Isyn
dVdt = rho_V * (-INa - IK - IT - IL - IKL + Iext) / C
# ----
# y
# ----
Fvh = g_Na * m_inf_by_V**3 * (V - E_Na) * h_inf_by_y_diff # dF/dvh
Fvn = 4 * g_K * (V -
E_KL) * n_inf_by_y**3 * n_inf_by_y_diff # dF/dvn
f4 = Fvh + Fvn
rho_h = (1 - rho_p) * Fvh / f4
rho_n = (1 - rho_p) * Fvn / f4
fh = (h_inf_by_V - h_inf_by_y) / h_tau_by_V / h_inf_by_y_diff
fn = (n_inf_by_V - n_inf_by_y) / n_tau_by_V / n_inf_by_y_diff
fp = (p_inf_by_V - p_inf_by_y) / p_tau_by_V / p_inf_by_y_diff
dydt = rho_h * fh + rho_n * fn + rho_p * fp
# ----
# z
# ----
dzdt = (q_inf_by_V - q_inf_by_z) / q_tau_by_V / q_inf_diff_z
return dVdt, dydt, dzdt