def res_dtheta_growth(self, u_, p_, ivar, theta_old_, thres, dt, rquant):
theta_ = ivar["theta"]
grfnc = growthfunction(theta_, self.I)
try:
omega = self.gandrparams['thres_tol']
except:
omega = 0
try:
reduc = self.gandrparams['trigger_reduction']
except:
reduc = 1
assert (reduc <= 1 and reduc > 0)
# threshold should not be lower than specified (only relevant for multiscale analysis, where threshold is set element-wise)
threshold = conditional(gt(thres, self.gandrparams['growth_thres']),
(1. + omega) * thres,
self.gandrparams['growth_thres'])
# trace of elastic Mandel stress
if self.growth_trig == 'volstress':
trigger = reduc * tr(self.M_e(u_, p_, self.kin.C(u_), ivar))
# elastic fiber stretch
elif self.growth_trig == 'fibstretch':
trigger = reduc * self.fibstretch_e(self.kin.C(u_), theta_,
self.kin.fib_funcs[0])
else:
raise NameError("Unknown growth_trig!")
# growth function
ktheta = grfnc.grfnc1(trigger, threshold, self.gandrparams)
# growth residual
r_growth = theta_ - theta_old_ - ktheta * (trigger - threshold) * dt
# tangent
K_growth = diff(r_growth, theta_)
# increment
del_theta = -r_growth / K_growth
if rquant == 'res_del':
return r_growth, del_theta
elif rquant == 'ktheta':
return ktheta
elif rquant == 'tang':
return K_growth
else:
raise NameError("Unknown return quantity!")
def increase_conductivity(self, cond, e):
# Set up the three way choice function
intermediate = e * self.k + self.h
not_less_than = ufl.conditional(ufl.gt(e, self.threshold_irreversible), self.sigma_end, intermediate)
cond_expression = ufl.conditional(ufl.lt(e, self.threshold_reversible), self.sigma_start, not_less_than)
# Project this onto the function space
cond_function = d.project(ufl.Max(cond_expression, cond), cond.function_space())
cond.assign(cond_function)
def test_cpp_formatting_of_conditionals():
x, y = ufl.SpatialCoordinate(ufl.triangle)
# Test conditional expressions
assert expr2cpp(ufl.conditional(ufl.lt(x, 2), y, 3)) \
== "x[0] < 2 ? x[1]: 3"
assert expr2cpp(ufl.conditional(ufl.gt(x, 2), 4 + y, 3)) \
== "x[0] > 2 ? 4 + x[1]: 3"
assert expr2cpp(ufl.conditional(ufl.And(ufl.le(x, 2), ufl.ge(y, 4)), 7, 8)) \
== "x[0] <= 2 && x[1] >= 4 ? 7: 8"
assert expr2cpp(ufl.conditional(ufl.Or(ufl.eq(x, 2), ufl.ne(y, 4)), 7, 8)) \
== "x[0] == 2 || x[1] != 4 ? 7: 8"
def nn(eps, u, p, v, q):
#return inner(grad(p), grad(q)) * dx
def sigma_(vec, func=ufl.tanh):
v = [func(vec[i]) for i in range(vec.ufl_shape[0])]
return ufl.as_vector(v)
relu = lambda vec: conditional(ufl.gt(vec, 0), vec, (ufl.exp(vec) - 1))
sigma = lambda vec: sigma_(vec, func=relu)
nn = dot(W_3, sigma(ufl.transpose(as_vector([W_1, W_2])) * eps * grad(p) + b_1)) + b_2
return inner(nn, grad(q))*dx, inner(nn, nn)*dx
def increase_conductivity(self, cond, e):
# Set up the three way choice function
intermediate = e * self.k + self.h
not_less_than = ufl.conditional(ufl.gt(e, self.threshold_irreversible),
self.sigma_end, intermediate)
cond_expression = ufl.conditional(ufl.lt(e, self.threshold_reversible),
self.sigma_start, not_less_than)
# Project this onto the function space
cond_function = d.project(ufl.Max(cond_expression, cond),
cond.function_space())
cond.assign(cond_function)
def smooth(up_u_adv, up_u, o):
# Calculate averaged velocities
# Define the indicator function of the turbine support
chi = ufl.conditional(ufl.gt(tf, 0), 1, 0)
c_diff = Constant(1e6)
F1 = chi * (inner(up_u - up_u_adv, o) + c_diff * inner(grad(up_u), grad(o))) * dx
invchi = 1 - chi
F2 = inner(invchi * up_u, o) * dx
F = F1 + F2
return F
def smooth(up_u_adv, up_u, o):
# Calculate averaged velocities
# Define the indicator function of the turbine support
chi = ufl.conditional(ufl.gt(tf, 0), 1, 0)
c_diff = Constant(1e6)
F1 = chi * (inner(up_u - up_u_adv, o) +
c_diff * inner(grad(up_u), grad(o))) * dx
invchi = 1 - chi
F2 = inner(invchi * up_u, o) * dx
F = F1 + F2
return F
def nn(u, p, v, q):
# return inner(grad(p), grad(q)) * dx, None, None
def sigma_(vec, func=ufl.tanh):
v = [func(vec[i]) for i in range(vec.ufl_shape[0])]
return ufl.as_vector(v)
relu = lambda vec: conditional(ufl.gt(vec, 0), vec, (ufl.exp(vec) - 1))
sigma = lambda vec: sigma_(vec, func=relu)#lambda x:x)
nn_p = dot(W_3, sigma(ufl.transpose(as_vector([W_1, W_2])) * u + b_1)) + b_2
#nn_q = dot(W_3, sigma(ufl.transpose(as_vector([W_1, W_2])) * grad(q) + b_1)) + b_2
return inner(nn_p, v)*dx, inner(nn_p, nn_p)*dx, nn_p
def smooth(u, up_u, o):
# Calculate averaged velocities
# Define the indicator function of the turbine support
chi = ufl.conditional(ufl.gt(tf, 0), 1, 0)
# Solve the Helmholtz equation in each turbine area to obtain averaged velocity values
c_diff = Constant(1e6)
F1 = chi * (inner(up_u - norm_approx(u), o) + Constant(distance_to_upstream) / norm_approx(u) * (inner(dot(grad(norm_approx(u)), u), o) + c_diff * inner(grad(up_u), grad(o)))) * dx
invchi = 1 - chi
F2 = inner(invchi * up_u, o) * dx
F = F1 + F2
return F
def nn(u, v):
inp = as_vector(
[avg(u), jump(u), *grad(avg(u)), *grad(jump(u)), *n('+')])
def sigma_(vec, func=ufl.tanh):
v = [func(vec[i]) for i in range(vec.ufl_shape[0])]
return ufl.as_vector(v)
relu = lambda vec: conditional(ufl.gt(vec, 0), vec, (ufl.exp(vec) - 1))
sigma = lambda vec: sigma_(vec, func=relu)
nn = dot(W_2, sigma(ufl.transpose(as_vector(W_1)) * inp + b_1)) + b_2
return inner(nn, jump(v) + avg(v)) * dS, inner(nn, nn) * dS
def smooth(u, up_u, o):
# Calculate averaged velocities
# Define the indicator function of the turbine support
chi = ufl.conditional(ufl.gt(tf, 0), 1, 0)
# Solve the Helmholtz equation in each turbine area to obtain averaged velocity values
c_diff = Constant(1e6)
F1 = chi * (inner(up_u - norm_approx(u), o) +
Constant(distance_to_upstream) / norm_approx(u) *
(inner(dot(grad(norm_approx(u)), u), o) +
c_diff * inner(grad(up_u), grad(o)))) * dx
invchi = 1 - chi
F2 = inner(invchi * up_u, o) * dx
F = F1 + F2
return F
def nn(u, v):
#return inner(grad(p), grad(q)) * dx
def sigma_(vec, func=ufl.tanh):
v = [func(vec[i]) for i in range(vec.ufl_shape[0])]
return ufl.as_vector(v)
relu = lambda vec: conditional(ufl.gt(vec, 0), vec, (ufl.exp(vec) - 1))
sigma = lambda vec: sigma_(vec, func=relu) #lambda x:x)
#from IPython import embed
#embed()
n1 = dot(W_2, sigma(W_1 * u) + b_1)
n2 = dot(W_3_1, sigma(W_3_2 * u.dx(0)) + b_2)
return inner(n1, v) * dx + inner(n2, v.dx(0)) * dx, inner(
n1, n1) * dx + inner(n2, n2) * dx, n1
def _I(self, v, s, time):
"""
Original gotran transmembrane current dV/dt
"""
time = time if time else Constant(0.0)
# Assign states
V = v
assert(len(s) == 2)
m, h = s
# Assign parameters
amp = self._parameters["amp"]
duration = self._parameters["duration"]
stimStart = self._parameters["stimStart"]
E_Na = self._parameters["E_Na"]
g_Na = self._parameters["g_Na"]
E_K = self._parameters["E_K"]
g_K = self._parameters["g_K"]
k_r = self._parameters["k_r"]
# Init return args
current = [ufl.zero()]*1
# Expressions for the Fast sodium current component
i_Na = g_Na*ufl.elem_pow(m, 3.0)*(-E_Na + V)*h
# Expressions for the Potassium current component
i_K = g_K*(-E_K + V)*ufl.exp((E_K - V)/k_r)
# Expressions for the Membrane component
i_stim = ufl.conditional(ufl.And(ufl.gt(time, stimStart),\
ufl.le(time, duration + stimStart)), amp, 0)
current[0] = -i_K - i_Na - i_stim
# Return results
return current[0]
def test_comparison_checker(self):
cell = triangle
element = FiniteElement("Lagrange", cell, 1)
u = TrialFunction(element)
v = TestFunction(element)
a = conditional(ge(abs(u), imag(v)), u, v)
b = conditional(le(sqrt(abs(u)), imag(v)), as_ufl(1), as_ufl(1j))
c = conditional(gt(abs(u), pow(imag(v), 0.5)), sin(u), cos(v))
d = conditional(lt(as_ufl(-1), as_ufl(1)), u, v)
e = max_value(as_ufl(0), real(u))
f = min_value(sin(u), cos(v))
g = min_value(sin(pow(u, 3)), cos(abs(v)))
assert do_comparison_check(a) == conditional(ge(real(abs(u)), real(imag(v))), u, v)
with pytest.raises(ComplexComparisonError):
b = do_comparison_check(b)
with pytest.raises(ComplexComparisonError):
c = do_comparison_check(c)
assert do_comparison_check(d) == conditional(lt(real(as_ufl(-1)), real(as_ufl(1))), u, v)
assert do_comparison_check(e) == max_value(real(as_ufl(0)), real(real(u)))
assert do_comparison_check(f) == min_value(real(sin(u)), real(cos(v)))
assert do_comparison_check(g) == min_value(real(sin(pow(u, 3))), real(cos(abs(v))))
) # want the fluid to occupy 1/3 of the domain
mesh = fenics_adjoint.Mesh(
fenics.RectangleMesh(fenics.Point(0.0, 0.0), fenics.Point(delta, 1.0), N, N)
)
A = fenics.FunctionSpace(mesh, "CG", 1) # control function space
U_h = fenics.VectorElement("CG", mesh.ufl_cell(), 2)
P_h = fenics.FiniteElement("CG", mesh.ufl_cell(), 1)
W = fenics.FunctionSpace(mesh, U_h * P_h) # mixed Taylor-Hood function space
# Define the boundary condition on velocity
(x, y) = ufl.SpatialCoordinate(mesh)
l = 1.0 / 6.0 # noqa: E741
gbar = 1.0
cond1 = ufl.And(ufl.gt(y, (1.0 / 4 - l / 2)), ufl.lt(y, (1.0 / 4 + l / 2)))
val1 = gbar * (1 - (2 * (y - 0.25) / l) ** 2)
cond2 = ufl.And(ufl.gt(y, (3.0 / 4 - l / 2)), ufl.lt(y, (3.0 / 4 + l / 2)))
val2 = gbar * (1 - (2 * (y - 0.75) / l) ** 2)
inflow_outflow = ufl.conditional(cond1, val1, ufl.conditional(cond2, val2, 0))
inflow_outflow_bc = fenics_adjoint.project(inflow_outflow, W.sub(0).sub(0).collapse())
solve_templates = (fenics_adjoint.Function(A),)
assemble_templates = (fenics_adjoint.Function(W), fenics_adjoint.Function(A))
@build_jax_fem_eval(solve_templates)
def forward(rho):
"""Solve the forward problem for a given fluid distribution rho(x)."""
w = fenics_adjoint.Function(W)
(u, p) = fenics.split(w)
def F(self, v, s, time=None):
"""
Right hand side for ODE system
"""
time = time if time else Constant(0.0)
# Assign states
V = v
assert(len(s) == 16)
Xr1, Xr2, Xs, m, h, j, d, f, fCa, s, r, Ca_SR, Ca_i, g, Na_i, K_i = s
# Assign parameters
P_kna = self._parameters["P_kna"]
g_K1 = self._parameters["g_K1"]
g_Kr = self._parameters["g_Kr"]
g_Ks = self._parameters["g_Ks"]
g_Na = self._parameters["g_Na"]
g_bna = self._parameters["g_bna"]
g_CaL = self._parameters["g_CaL"]
g_bca = self._parameters["g_bca"]
g_to = self._parameters["g_to"]
K_mNa = self._parameters["K_mNa"]
K_mk = self._parameters["K_mk"]
P_NaK = self._parameters["P_NaK"]
K_NaCa = self._parameters["K_NaCa"]
K_sat = self._parameters["K_sat"]
Km_Ca = self._parameters["Km_Ca"]
Km_Nai = self._parameters["Km_Nai"]
alpha = self._parameters["alpha"]
gamma = self._parameters["gamma"]
K_pCa = self._parameters["K_pCa"]
g_pCa = self._parameters["g_pCa"]
g_pK = self._parameters["g_pK"]
Buf_c = self._parameters["Buf_c"]
Buf_sr = self._parameters["Buf_sr"]
Ca_o = self._parameters["Ca_o"]
K_buf_c = self._parameters["K_buf_c"]
K_buf_sr = self._parameters["K_buf_sr"]
K_up = self._parameters["K_up"]
V_leak = self._parameters["V_leak"]
V_sr = self._parameters["V_sr"]
Vmax_up = self._parameters["Vmax_up"]
a_rel = self._parameters["a_rel"]
b_rel = self._parameters["b_rel"]
c_rel = self._parameters["c_rel"]
tau_g = self._parameters["tau_g"]
Na_o = self._parameters["Na_o"]
Cm = self._parameters["Cm"]
F = self._parameters["F"]
R = self._parameters["R"]
T = self._parameters["T"]
V_c = self._parameters["V_c"]
stim_amplitude = self._parameters["stim_amplitude"]
stim_duration = self._parameters["stim_duration"]
stim_start = self._parameters["stim_start"]
K_o = self._parameters["K_o"]
# Init return args
F_expressions = [ufl.zero()]*16
# Expressions for the Reversal potentials component
E_Na = R*T*ufl.ln(Na_o/Na_i)/F
E_K = R*T*ufl.ln(K_o/K_i)/F
E_Ks = R*T*ufl.ln((Na_o*P_kna + K_o)/(K_i + P_kna*Na_i))/F
E_Ca = 0.5*R*T*ufl.ln(Ca_o/Ca_i)/F
# Expressions for the Inward rectifier potassium current component
alpha_K1 = 0.1/(1 + 6.14421235333e-06*ufl.exp(-0.06*E_K + 0.06*V))
beta_K1 = (3.06060402008*ufl.exp(0.0002*V - 0.0002*E_K) +\
0.367879441171*ufl.exp(0.1*V - 0.1*E_K))/(1 + ufl.exp(0.5*E_K -\
0.5*V))
xK1_inf = alpha_K1/(alpha_K1 + beta_K1)
i_K1 = 0.430331482912*g_K1*ufl.sqrt(K_o)*(-E_K + V)*xK1_inf
# Expressions for the Rapid time dependent potassium current component
i_Kr = 0.430331482912*g_Kr*ufl.sqrt(K_o)*(-E_K + V)*Xr1*Xr2
# Expressions for the Xr1 gate component
xr1_inf = 1.0/(1 + ufl.exp(-26/7 - V/7))
alpha_xr1 = 450/(1 + ufl.exp(-9/2 - V/10))
beta_xr1 = 6/(1 + 13.5813245226*ufl.exp(0.0869565217391*V))
tau_xr1 = alpha_xr1*beta_xr1
F_expressions[0] = (xr1_inf - Xr1)/tau_xr1
# Expressions for the Xr2 gate component
xr2_inf = 1.0/(1 + ufl.exp(11/3 + V/24))
alpha_xr2 = 3/(1 + ufl.exp(-3 - V/20))
beta_xr2 = 1.12/(1 + ufl.exp(-3 + V/20))
tau_xr2 = alpha_xr2*beta_xr2
F_expressions[1] = (xr2_inf - Xr2)/tau_xr2
# Expressions for the Slow time dependent potassium current component
i_Ks = g_Ks*(Xs*Xs)*(-E_Ks + V)
# Expressions for the Xs gate component
xs_inf = 1.0/(1 + ufl.exp(-5/14 - V/14))
alpha_xs = 1100/ufl.sqrt(1 + ufl.exp(-5/3 - V/6))
beta_xs = 1.0/(1 + ufl.exp(-3 + V/20))
tau_xs = alpha_xs*beta_xs
F_expressions[2] = (xs_inf - Xs)/tau_xs
# Expressions for the Fast sodium current component
i_Na = g_Na*(m*m*m)*(-E_Na + V)*h*j
# Expressions for the m gate component
m_inf = 1.0/((1 + 0.00184221158117*ufl.exp(-0.110741971207*V))*(1 +\
0.00184221158117*ufl.exp(-0.110741971207*V)))
alpha_m = 1.0/(1 + ufl.exp(-12 - V/5))
beta_m = 0.1/(1 + ufl.exp(-1/4 + V/200)) + 0.1/(1 + ufl.exp(7 + V/5))
tau_m = alpha_m*beta_m
F_expressions[3] = (-m + m_inf)/tau_m
# Expressions for the h gate component
h_inf = 1.0/((1 + 15212.5932857*ufl.exp(0.134589502019*V))*(1 +\
15212.5932857*ufl.exp(0.134589502019*V)))
alpha_h = 4.43126792958e-07*ufl.exp(-0.147058823529*V)/(1 +\
2.35385266837e+17*ufl.exp(1.0*V))
beta_h = (2.7*ufl.exp(0.079*V) + 310000*ufl.exp(0.3485*V))/(1 +\
2.35385266837e+17*ufl.exp(1.0*V)) + 0.77*(1 - 1/(1 +\
2.35385266837e+17*ufl.exp(1.0*V)))/(0.13 +\
0.0497581410839*ufl.exp(-0.0900900900901*V))
tau_h = 1.0/(alpha_h + beta_h)
F_expressions[4] = (-h + h_inf)/tau_h
# Expressions for the j gate component
j_inf = 1.0/((1 + 15212.5932857*ufl.exp(0.134589502019*V))*(1 +\
15212.5932857*ufl.exp(0.134589502019*V)))
alpha_j = (37.78 + V)*(-25428*ufl.exp(0.2444*V) -\
6.948e-06*ufl.exp(-0.04391*V))/((1 +\
2.35385266837e+17*ufl.exp(1.0*V))*(1 +\
50262745826.0*ufl.exp(0.311*V)))
beta_j = 0.6*(1 - 1/(1 +\
2.35385266837e+17*ufl.exp(1.0*V)))*ufl.exp(0.057*V)/(1 +\
0.0407622039784*ufl.exp(-0.1*V)) +\
0.02424*ufl.exp(-0.01052*V)/((1 +\
2.35385266837e+17*ufl.exp(1.0*V))*(1 +\
0.0039608683399*ufl.exp(-0.1378*V)))
tau_j = 1.0/(alpha_j + beta_j)
F_expressions[5] = (-j + j_inf)/tau_j
# Expressions for the Sodium background current component
i_b_Na = g_bna*(-E_Na + V)
# Expressions for the L_type Ca current component
i_CaL = 4*g_CaL*(F*F)*(Ca_i*ufl.exp(2*F*V/(R*T)) -\
0.341*Ca_o)*V*d*f*fCa/(R*T*(-1 + ufl.exp(2*F*V/(R*T))))
# Expressions for the d gate component
d_inf = 1.0/(1 + 0.513417119033*ufl.exp(-0.133333333333*V))
alpha_d = 0.25 + 1.4/(1 + ufl.exp(-35/13 - V/13))
beta_d = 1.4/(1 + ufl.exp(1 + V/5))
gamma_d = 1.0/(1 + ufl.exp(5/2 - V/20))
tau_d = alpha_d*beta_d + gamma_d
F_expressions[6] = (-d + d_inf)/tau_d
# Expressions for the f gate component
f_inf = 1.0/(1 + ufl.exp(20/7 + V/7))
tau_f = 80 + 1125*ufl.exp(-((27 + V)*(27 + V))/240) + 165/(1 +\
ufl.exp(5/2 - V/10))
F_expressions[7] = (f_inf - f)/tau_f
# Expressions for the FCa gate component
alpha_fCa = 1.0/(1 + 8.03402376702e+27*ufl.elem_pow(Ca_i, 8))
exp_arg_0 = -5.0 + 10000.0*Ca_i
exp_arg_00 = ufl.conditional(ufl.lt(exp_arg_0, 500.0), exp_arg_0,\
500.0)
beta_fCa = 0.1/(1 + ufl.exp(exp_arg_00))
exp_arg_1 = -0.9375 + 1250.0*Ca_i
exp_arg_11 = ufl.conditional(ufl.lt(exp_arg_1, 500.0), exp_arg_1,\
500.0)
gama_fCa = 0.2/(1 + ufl.exp(exp_arg_11))
fCa_inf = 0.157534246575 + 0.684931506849*gama_fCa +\
0.684931506849*alpha_fCa + 0.684931506849*beta_fCa
tau_fCa = 2
d_fCa = (-fCa + fCa_inf)/tau_fCa
F_expressions[8] = ufl.conditional(ufl.And(ufl.gt(V, -60),\
ufl.gt(fCa_inf, fCa)), 0, d_fCa)
# Expressions for the Calcium background current component
i_b_Ca = g_bca*(-E_Ca + V)
# Expressions for the Transient outward current component
i_to = g_to*(-E_K + V)*r*s
# Expressions for the s gate component
s_inf = 1.0/(1 + ufl.exp(4 + V/5))
tau_s = 3 + 5/(1 + ufl.exp(-4 + V/5)) + 85*ufl.exp(-((45 + V)*(45 +\
V))/320)
F_expressions[9] = (s_inf - s)/tau_s
# Expressions for the r gate component
r_inf = 1.0/(1 + ufl.exp(10/3 - V/6))
tau_r = 0.8 + 9.5*ufl.exp(-((40 + V)*(40 + V))/1800)
F_expressions[10] = (r_inf - r)/tau_r
# Expressions for the Sodium potassium pump current component
i_NaK = K_o*P_NaK*Na_i/((K_mNa + Na_i)*(K_mk + K_o)*(1 +\
0.0353*ufl.exp(-F*V/(R*T)) + 0.1245*ufl.exp(-0.1*F*V/(R*T))))
# Expressions for the Sodium calcium exchanger current component
i_NaCa = K_NaCa*(-alpha*(Na_o*Na_o*Na_o)*Ca_i*ufl.exp(F*(-1 +\
gamma)*V/(R*T)) +\
Ca_o*(Na_i*Na_i*Na_i)*ufl.exp(F*gamma*V/(R*T)))/((1 +\
K_sat*ufl.exp(F*(-1 + gamma)*V/(R*T)))*(Km_Ca +\
Ca_o)*((Na_o*Na_o*Na_o) + (Km_Nai*Km_Nai*Km_Nai)))
# Expressions for the Calcium pump current component
i_p_Ca = g_pCa*Ca_i/(Ca_i + K_pCa)
# Expressions for the Potassium pump current component
i_p_K = g_pK*(-E_K + V)/(1 + 65.4052157419*ufl.exp(-0.167224080268*V))
# Expressions for the Calcium dynamics component
i_rel = (c_rel + a_rel*(Ca_SR*Ca_SR)/((b_rel*b_rel) +\
(Ca_SR*Ca_SR)))*d*g
i_up = Vmax_up/(1 + (K_up*K_up)/(Ca_i*Ca_i))
i_leak = V_leak*(Ca_SR - Ca_i)
g_inf = 1/((1 + 0.0301973834223*ufl.exp(10000.0*Ca_i))*(1 +\
5.43991024148e+20*ufl.elem_pow(Ca_i, 6))) + (1 - 1/(1 +\
0.0301973834223*ufl.exp(10000.0*Ca_i)))/(1 +\
1.9720198874e+55*ufl.elem_pow(Ca_i, 16))
d_g = (-g + g_inf)/tau_g
F_expressions[13] = (1 - 1.0/((1 + ufl.exp(60 + V))*(1 +\
ufl.exp(10.0*g_inf - 10.0*g))))*d_g
Ca_i_bufc = 1.0/(1 + Buf_c*K_buf_c/((Ca_i + K_buf_c)*(Ca_i + K_buf_c)))
Ca_sr_bufsr = 1.0/(1 + Buf_sr*K_buf_sr/((Ca_SR + K_buf_sr)*(Ca_SR +\
K_buf_sr)))
F_expressions[12] = (i_rel - i_up - Cm*(i_b_Ca + i_p_Ca - 2*i_NaCa +\
i_CaL)/(2*F*V_c) + i_leak)*Ca_i_bufc
F_expressions[11] = V_c*(i_up - i_leak - i_rel)*Ca_sr_bufsr/V_sr
# Expressions for the Sodium dynamics component
F_expressions[14] = Cm*(-i_b_Na - i_Na - 3*i_NaCa - 3*i_NaK)/(F*V_c)
# Expressions for the Membrane component
i_Stim = -stim_amplitude*(1 - 1/(1 + ufl.exp(5.0*time -\
5.0*stim_start)))/(1 + ufl.exp(5.0*time - 5.0*stim_start -\
5.0*stim_duration))
# Expressions for the Potassium dynamics component
F_expressions[15] = Cm*(2*i_NaK - i_Ks - i_Kr - i_to - i_Stim - i_p_K\
- i_K1)/(F*V_c)
# Return results
return dolfin.as_vector(F_expressions)
def xtest_latex_formatting_of_conditionals():
# Test conditional expressions
assert expr2latex(ufl.conditional(ufl.lt(x, 2), y, 3)) == "x_0 < 2 ? x_1: 3"
assert expr2latex(ufl.conditional(ufl.gt(x, 2), 4 + y, 3)) == "x_0 > 2 ? 4 + x_1: 3"
assert expr2latex(ufl.conditional(ufl.And(ufl.le(x, 2), ufl.ge(y, 4)), 7, 8)) == "x_0 <= 2 && x_1 >= 4 ? 7: 8"
assert expr2latex(ufl.conditional(ufl.Or(ufl.eq(x, 2), ufl.ne(y, 4)), 7, 8)) == "x_0 == 2 || x_1 != 4 ? 7: 8"
def F(self, v, s, time=None):
"""
Right hand side for ODE system
"""
time = time if time else Constant(0.0)
# Assign states
V = v
assert(len(s) == 16)
Xr1, Xr2, Xs, m, h, j, d, f, fCa, s, r, g, Ca_i, Ca_SR, Na_i, K_i = s
# Assign parameters
P_kna = self._parameters["P_kna"]
g_K1 = self._parameters["g_K1"]
g_Kr = self._parameters["g_Kr"]
g_Ks = self._parameters["g_Ks"]
g_Na = self._parameters["g_Na"]
g_bna = self._parameters["g_bna"]
g_CaL = self._parameters["g_CaL"]
g_bca = self._parameters["g_bca"]
g_to = self._parameters["g_to"]
K_mNa = self._parameters["K_mNa"]
K_mk = self._parameters["K_mk"]
P_NaK = self._parameters["P_NaK"]
K_NaCa = self._parameters["K_NaCa"]
K_sat = self._parameters["K_sat"]
Km_Ca = self._parameters["Km_Ca"]
Km_Nai = self._parameters["Km_Nai"]
alpha = self._parameters["alpha"]
gamma = self._parameters["gamma"]
K_pCa = self._parameters["K_pCa"]
g_pCa = self._parameters["g_pCa"]
g_pK = self._parameters["g_pK"]
Buf_c = self._parameters["Buf_c"]
Buf_sr = self._parameters["Buf_sr"]
Ca_o = self._parameters["Ca_o"]
K_buf_c = self._parameters["K_buf_c"]
K_buf_sr = self._parameters["K_buf_sr"]
K_up = self._parameters["K_up"]
V_leak = self._parameters["V_leak"]
V_sr = self._parameters["V_sr"]
Vmax_up = self._parameters["Vmax_up"]
a_rel = self._parameters["a_rel"]
b_rel = self._parameters["b_rel"]
c_rel = self._parameters["c_rel"]
tau_g = self._parameters["tau_g"]
Na_o = self._parameters["Na_o"]
Cm = self._parameters["Cm"]
F = self._parameters["F"]
R = self._parameters["R"]
T = self._parameters["T"]
V_c = self._parameters["V_c"]
stim_amplitude = self._parameters["stim_amplitude"]
stim_duration = self._parameters["stim_duration"]
stim_period = self._parameters["stim_period"]
stim_start = self._parameters["stim_start"]
K_o = self._parameters["K_o"]
# Init return args
F_expressions = [ufl.zero()]*16
# Expressions for the Reversal potentials component
E_Na = R*T*ufl.ln(Na_o/Na_i)/F
E_K = R*T*ufl.ln(K_o/K_i)/F
E_Ks = R*T*ufl.ln((K_o + Na_o*P_kna)/(P_kna*Na_i + K_i))/F
E_Ca = 0.5*R*T*ufl.ln(Ca_o/Ca_i)/F
# Expressions for the Inward rectifier potassium current component
alpha_K1 = 0.1/(1.0 + 6.14421235332821e-06*ufl.exp(0.06*V - 0.06*E_K))
beta_K1 = (0.36787944117144233*ufl.exp(0.1*V - 0.1*E_K) +\
3.0606040200802673*ufl.exp(0.0002*V - 0.0002*E_K))/(1.0 +\
ufl.exp(0.5*E_K - 0.5*V))
xK1_inf = alpha_K1/(alpha_K1 + beta_K1)
i_K1 = 0.4303314829119352*g_K1*ufl.sqrt(K_o)*(-E_K + V)*xK1_inf
# Expressions for the Rapid time dependent potassium current component
i_Kr = 0.4303314829119352*g_Kr*ufl.sqrt(K_o)*(-E_K + V)*Xr1*Xr2
# Expressions for the Xr1 gate component
xr1_inf = 1.0/(1.0 +\
0.02437284407327961*ufl.exp(-0.14285714285714285*V))
alpha_xr1 = 450.0/(1.0 + 0.011108996538242306*ufl.exp(-0.1*V))
beta_xr1 = 6.0/(1.0 +\
13.581324522578193*ufl.exp(0.08695652173913043*V))
tau_xr1 = 1.0*alpha_xr1*beta_xr1
F_expressions[0] = (-Xr1 + xr1_inf)/tau_xr1
# Expressions for the Xr2 gate component
xr2_inf = 1.0/(1.0 + 39.12128399815321*ufl.exp(0.041666666666666664*V))
alpha_xr2 = 3.0/(1.0 + 0.049787068367863944*ufl.exp(-0.05*V))
beta_xr2 = 1.12/(1.0 + 0.049787068367863944*ufl.exp(0.05*V))
tau_xr2 = 1.0*alpha_xr2*beta_xr2
F_expressions[1] = (-Xr2 + xr2_inf)/tau_xr2
# Expressions for the Slow time dependent potassium current component
i_Ks = g_Ks*ufl.elem_pow(Xs, 2.0)*(-E_Ks + V)
# Expressions for the Xs gate component
xs_inf = 1.0/(1.0 + 0.6996725373751304*ufl.exp(-0.07142857142857142*V))
alpha_xs = 1100.0/ufl.sqrt(1.0 +\
0.18887560283756186*ufl.exp(-0.16666666666666666*V))
beta_xs = 1.0/(1.0 + 0.049787068367863944*ufl.exp(0.05*V))
tau_xs = 1.0*alpha_xs*beta_xs
F_expressions[2] = (-Xs + xs_inf)/tau_xs
# Expressions for the Fast sodium current component
i_Na = g_Na*ufl.elem_pow(m, 3.0)*(-E_Na + V)*h*j
# Expressions for the m gate component
m_inf = 1.0*ufl.elem_pow(1.0 +\
0.0018422115811651339*ufl.exp(-0.1107419712070875*V), -2.0)
alpha_m = 1.0/(1.0 + 6.14421235332821e-06*ufl.exp(-0.2*V))
beta_m = 0.1/(1.0 + 1096.6331584284585*ufl.exp(0.2*V)) + 0.1/(1.0 +\
0.7788007830714049*ufl.exp(0.005*V))
tau_m = 1.0*alpha_m*beta_m
F_expressions[3] = (-m + m_inf)/tau_m
# Expressions for the h gate component
h_inf = 1.0*ufl.elem_pow(1.0 +\
15212.593285654404*ufl.exp(0.13458950201884254*V), -2.0)
alpha_h = ufl.conditional(ufl.lt(V, -40.0),\
4.4312679295805147e-07*ufl.exp(-0.14705882352941177*V), 0)
beta_h = ufl.conditional(ufl.lt(V, -40.0), 310000.0*ufl.exp(0.3485*V)\
+ 2.7*ufl.exp(0.079*V), 0.77/(0.13 +\
0.049758141083938695*ufl.exp(-0.0900900900900901*V)))
tau_h = 1.0/(alpha_h + beta_h)
F_expressions[4] = (-h + h_inf)/tau_h
# Expressions for the j gate component
j_inf = 1.0*ufl.elem_pow(1.0 +\
15212.593285654404*ufl.exp(0.13458950201884254*V), -2.0)
alpha_j = ufl.conditional(ufl.lt(V, -40.0), 1.0*(37.78 +\
V)*(-25428.0*ufl.exp(0.2444*V) -\
6.948e-06*ufl.exp(-0.04391*V))/(1.0 +\
50262745825.95399*ufl.exp(0.311*V)), 0)
beta_j = ufl.conditional(ufl.lt(V, -40.0),\
0.02424*ufl.exp(-0.01052*V)/(1.0 +\
0.003960868339904256*ufl.exp(-0.1378*V)),\
0.6*ufl.exp(0.057*V)/(1.0 +\
0.040762203978366204*ufl.exp(-0.1*V)))
tau_j = 1.0/(alpha_j + beta_j)
F_expressions[5] = (-j + j_inf)/tau_j
# Expressions for the Sodium background current component
i_b_Na = g_bna*(-E_Na + V)
# Expressions for the L_type Ca current component
i_CaL = 4.0*g_CaL*ufl.elem_pow(F, 2.0)*(-0.341*Ca_o +\
Ca_i*ufl.exp(2.0*F*V/(R*T)))*V*d*f*fCa/(R*T*(-1.0 +\
ufl.exp(2.0*F*V/(R*T))))
# Expressions for the d gate component
d_inf = 1.0/(1.0 + 0.513417119032592*ufl.exp(-0.13333333333333333*V))
alpha_d = 0.25 + 1.4/(1.0 +\
0.0677244716592409*ufl.exp(-0.07692307692307693*V))
beta_d = 1.4/(1.0 + 2.718281828459045*ufl.exp(0.2*V))
gamma_d = 1.0/(1.0 + 12.182493960703473*ufl.exp(-0.05*V))
tau_d = 1.0*alpha_d*beta_d + gamma_d
F_expressions[6] = (-d + d_inf)/tau_d
# Expressions for the f gate component
f_inf = 1.0/(1.0 + 17.411708063327644*ufl.exp(0.14285714285714285*V))
tau_f = 80.0 + 165.0/(1.0 + 12.182493960703473*ufl.exp(-0.1*V)) +\
1125.0*ufl.exp(-0.004166666666666667*ufl.elem_pow(27.0 + V, 2.0))
F_expressions[7] = (-f + f_inf)/tau_f
# Expressions for the FCa gate component
alpha_fCa = 1.0/(1.0 + 8.03402376701711e+27*ufl.elem_pow(Ca_i, 8.0))
beta_fCa = 0.1/(1.0 + 0.006737946999085467*ufl.exp(10000.0*Ca_i))
gama_fCa = 0.2/(1.0 + 0.391605626676799*ufl.exp(1250.0*Ca_i))
fCa_inf = 0.15753424657534246 + 0.684931506849315*alpha_fCa +\
0.684931506849315*beta_fCa + 0.684931506849315*gama_fCa
tau_fCa = 2.0
d_fCa = (-fCa + fCa_inf)/tau_fCa
F_expressions[8] = ufl.conditional(ufl.And(ufl.gt(V, -60.0),\
ufl.gt(fCa_inf, fCa)), 0, d_fCa)
# Expressions for the Calcium background current component
i_b_Ca = g_bca*(-E_Ca + V)
# Expressions for the Transient outward current component
i_to = g_to*(-E_K + V)*r*s
# Expressions for the s gate component
s_inf = 1.0/(1.0 + 54.598150033144236*ufl.exp(0.2*V))
tau_s = 3.0 + 5.0/(1.0 + 0.01831563888873418*ufl.exp(0.2*V)) +\
85.0*ufl.exp(-0.003125*ufl.elem_pow(45.0 + V, 2.0))
F_expressions[9] = (-s + s_inf)/tau_s
# Expressions for the r gate component
r_inf = 1.0/(1.0 + 28.031624894526125*ufl.exp(-0.16666666666666666*V))
tau_r = 0.8 + 9.5*ufl.exp(-0.0005555555555555556*ufl.elem_pow(40.0 +\
V, 2.0))
F_expressions[10] = (-r + r_inf)/tau_r
# Expressions for the Sodium potassium pump current component
i_NaK = K_o*P_NaK*Na_i/((K_mNa + Na_i)*(K_mk + K_o)*(1.0 +\
0.0353*ufl.exp(-F*V/(R*T)) + 0.1245*ufl.exp(-0.1*F*V/(R*T))))
# Expressions for the Sodium calcium exchanger current component
i_NaCa = K_NaCa*(Ca_o*ufl.elem_pow(Na_i,\
3.0)*ufl.exp(F*gamma*V/(R*T)) - alpha*ufl.elem_pow(Na_o,\
3.0)*Ca_i*ufl.exp(F*(-1.0 + gamma)*V/(R*T)))/((1.0 +\
K_sat*ufl.exp(F*(-1.0 + gamma)*V/(R*T)))*(Ca_o +\
Km_Ca)*(ufl.elem_pow(Km_Nai, 3.0) + ufl.elem_pow(Na_o, 3.0)))
# Expressions for the Calcium pump current component
i_p_Ca = g_pCa*Ca_i/(K_pCa + Ca_i)
# Expressions for the Potassium pump current component
i_p_K = g_pK*(-E_K + V)/(1.0 +\
65.40521574193832*ufl.exp(-0.16722408026755853*V))
# Expressions for the Calcium dynamics component
i_rel = (c_rel + a_rel*ufl.elem_pow(Ca_SR, 2.0)/(ufl.elem_pow(b_rel,\
2.0) + ufl.elem_pow(Ca_SR, 2.0)))*d*g
i_up = Vmax_up/(1.0 + ufl.elem_pow(K_up, 2.0)*ufl.elem_pow(Ca_i, -2.0))
i_leak = V_leak*(-Ca_i + Ca_SR)
g_inf = ufl.conditional(ufl.lt(Ca_i, 0.00035), 1.0/(1.0 +\
5.439910241481018e+20*ufl.elem_pow(Ca_i, 6.0)), 1.0/(1.0 +\
1.9720198874049195e+55*ufl.elem_pow(Ca_i, 16.0)))
d_g = (-g + g_inf)/tau_g
F_expressions[11] = ufl.conditional(ufl.And(ufl.gt(V, -60.0),\
ufl.gt(g_inf, g)), 0, d_g)
Ca_i_bufc = 1.0/(1.0 + Buf_c*K_buf_c*ufl.elem_pow(K_buf_c + Ca_i,\
-2.0))
Ca_sr_bufsr = 1.0/(1.0 + Buf_sr*K_buf_sr*ufl.elem_pow(K_buf_sr +\
Ca_SR, -2.0))
F_expressions[12] = (-i_up - 0.5*Cm*(1.0*i_CaL + 1.0*i_b_Ca +\
1.0*i_p_Ca - 2.0*i_NaCa)/(F*V_c) + i_leak + i_rel)*Ca_i_bufc
F_expressions[13] = V_c*(-i_leak - i_rel + i_up)*Ca_sr_bufsr/V_sr
# Expressions for the Sodium dynamics component
F_expressions[14] = 1.0*Cm*(-1.0*i_Na - 1.0*i_b_Na - 3.0*i_NaCa -\
3.0*i_NaK)/(F*V_c)
# Expressions for the Membrane component
i_Stim = ufl.conditional(ufl.And(ufl.ge(time -\
stim_period*ufl.floor(time/stim_period), stim_start), ufl.le(time\
- stim_period*ufl.floor(time/stim_period), stim_duration +\
stim_start)), -stim_amplitude, 0)
# Expressions for the Potassium dynamics component
F_expressions[15] = 1.0*Cm*(2.0*i_NaK - 1.0*i_K1 - 1.0*i_Kr -\
1.0*i_Ks - 1.0*i_Stim - 1.0*i_p_K - 1.0*i_to)/(F*V_c)
# Return results
return dolfin.as_vector(F_expressions)
def eval(self, values, x):
values[0] = (zmax * ufl.conditional(
ufl.gt(x[0], 0),
np.exp(-(((x[0] - x0)**2 / (2 * sigma_x1**2)))),
np.exp(-(((x[0] - x0)**2 / (2 * sigma_x2**2)))),
) + zmin)
def __call__(self, x, *args, **kwargs):
return conditional(ufl.gt(x, 0), x, self.alpha * (ufl.exp(x) - 1))
def F(self, v, s, time=None):
"""
Right hand side for ODE system
"""
time = time if time else Constant(0.0)
# Assign states
V = v
assert (len(s) == 17)
m, h, j, d, f1, f2, fCa, Xr1, Xr2, Xs, Xf, q, r, Nai, g, Cai, Ca_SR = s
# Assign parameters
TTX_10uM = self._parameters["TTX_10uM"]
TTX_30uM = self._parameters["TTX_30uM"]
TTX_3uM = self._parameters["TTX_3uM"]
nifed_100nM = self._parameters["nifed_100nM"]
nifed_10nM = self._parameters["nifed_10nM"]
nifed_30nM = self._parameters["nifed_30nM"]
nifed_3nM = self._parameters["nifed_3nM"]
g_Na = self._parameters["g_Na"]
g_CaL = self._parameters["g_CaL"]
tau_fCa = self._parameters["tau_fCa"]
L0 = self._parameters["L0"]
Q = self._parameters["Q"]
g_b_Na = self._parameters["g_b_Na"]
g_b_Ca = self._parameters["g_b_Ca"]
Km_K = self._parameters["Km_K"]
Km_Na = self._parameters["Km_Na"]
PNaK = self._parameters["PNaK"]
KmCa = self._parameters["KmCa"]
KmNai = self._parameters["KmNai"]
Ksat = self._parameters["Ksat"]
alpha = self._parameters["alpha"]
gamma = self._parameters["gamma"]
kNaCa = self._parameters["kNaCa"]
KPCa = self._parameters["KPCa"]
g_PCa = self._parameters["g_PCa"]
Cao = self._parameters["Cao"]
Cm = self._parameters["Cm"]
F = self._parameters["F"]
Ko = self._parameters["Ko"]
Nao = self._parameters["Nao"]
R = self._parameters["R"]
T = self._parameters["T"]
V_SR = self._parameters["V_SR"]
Vc = self._parameters["Vc"]
Buf_C = self._parameters["Buf_C"]
Buf_SR = self._parameters["Buf_SR"]
Kbuf_C = self._parameters["Kbuf_C"]
Kbuf_SR = self._parameters["Kbuf_SR"]
Kup = self._parameters["Kup"]
V_leak = self._parameters["V_leak"]
VmaxUp = self._parameters["VmaxUp"]
a_rel = self._parameters["a_rel"]
b_rel = self._parameters["b_rel"]
c_rel = self._parameters["c_rel"]
tau_g = self._parameters["tau_g"]
# Init return args
F_expressions = [ufl.zero()] * 17
# Expressions for the Electric potentials component
E_Na = R * T * ufl.ln(Nao / Nai) / F
E_Ca = 0.5 * R * T * ufl.ln(Cao / Cai) / F
# Expressions for the i_Na component
TTX_coeff = ufl.conditional(ufl.eq(TTX_3uM, 1), 0.18,\
ufl.conditional(ufl.eq(TTX_10uM, 1), 0.06,\
ufl.conditional(ufl.eq(TTX_30uM, 1), 0.02, 1)))
i_Na = g_Na * (m * m * m) * (-E_Na + V) * TTX_coeff * h * j
# Expressions for the m gate component
m_inf = 1.0*ufl.elem_pow(1 +\
0.00308976260789*ufl.exp(-169.491525424*V), -0.333333333333)
alpha_m = 1.0 / (1 + 6.14421235333e-06 * ufl.exp(-200.0 * V))
beta_m = 0.1/(1 + 1096.63315843*ufl.exp(200.0*V)) + 0.1/(1 +\
0.778800783071*ufl.exp(5.0*V))
tau_m = 0.001 * alpha_m * beta_m
F_expressions[0] = (-m + m_inf) / tau_m
# Expressions for the h gate component
h_inf = 1.0 / ufl.sqrt(1 + 311490.091283 * ufl.exp(175.438596491 * V))
alpha_h = ufl.conditional(ufl.lt(V, -0.04),\
4.43126792958e-07*ufl.exp(-147.058823529*V), 0)
beta_h = ufl.conditional(ufl.lt(V, -0.04), 310000.0*ufl.exp(348.5*V)\
+ 2.7*ufl.exp(79.0*V), 0.77/(0.13 +\
0.0497581410839*ufl.exp(-90.0900900901*V)))
tau_h = ufl.conditional(ufl.lt(V, -0.04), 1.5/(1000*alpha_h +\
1000*beta_h), 0.002542)
F_expressions[1] = (-h + h_inf) / tau_h
# Expressions for the j gate component
j_inf = 1.0 / ufl.sqrt(1 + 311490.091283 * ufl.exp(175.438596491 * V))
alpha_j = ufl.conditional(ufl.lt(V, -0.04), (37.78 +\
1000*V)*(-25428*ufl.exp(244.4*V) -\
6.948e-06*ufl.exp(-43.91*V))/(1 +\
50262745826.0*ufl.exp(311.0*V)), 0)
beta_j = ufl.conditional(ufl.lt(V, -0.04),\
0.02424*ufl.exp(-10.52*V)/(1 +\
0.0039608683399*ufl.exp(-137.8*V)), 0.6*ufl.exp(57.0*V)/(1 +\
0.0407622039784*ufl.exp(-100.0*V)))
tau_j = 7.0 / (1000 * alpha_j + 1000 * beta_j)
F_expressions[2] = (-j + j_inf) / tau_j
# Expressions for the i_CaL component
nifed_coeff = ufl.conditional(ufl.eq(nifed_3nM, 1), 0.93,\
ufl.conditional(ufl.eq(nifed_10nM, 1), 0.79,\
ufl.conditional(ufl.eq(nifed_30nM, 1), 0.56,\
ufl.conditional(ufl.eq(nifed_100nM, 1), 0.28, 1))))
i_CaL = 4*g_CaL*(F*F)*(-0.341*Cao +\
Cai*ufl.exp(2*F*V/(R*T)))*V*d*f1*f2*fCa*nifed_coeff/(R*T*(-1 +\
ufl.exp(2*F*V/(R*T))))
# Expressions for the d gate component
d_infinity = 1.0 / (1 + 0.272531793034 * ufl.exp(-1000 * V / 7))
alpha_d = 0.25 + 1.4 / (1 + ufl.exp(-35 / 13 - 1000 * V / 13))
beta_d = 1.4 / (1 + ufl.exp(1 + 200 * V))
gamma_d = 1.0 / (1 + ufl.exp(5 / 2 - 50 * V))
tau_d = 0.001 * gamma_d + 0.001 * alpha_d * beta_d
F_expressions[3] = (-d + d_infinity) / tau_d
# Expressions for the F1 gate component
f1_inf = 1.0 / (1 + ufl.exp(26 / 3 + 1000 * V / 3))
constf1 = ufl.conditional(ufl.gt(-f1 + f1_inf, 0), 0.92835 +\
1433*Cai, 1)
tau_f1 = 0.001*(20 + 200.0/(1 + ufl.exp(13/10 - 100*V)) + 180.0/(1 +\
ufl.exp(3 + 100*V)) +\
1102.5*ufl.exp(-0.00444444444444*ufl.elem_pow(27 + 1000*V,\
4)))*constf1
F_expressions[4] = (-f1 + f1_inf) / tau_f1
# Expressions for the F2 gate component
f2_inf = 0.33 + 0.67 / (1 + ufl.exp(35 / 4 + 250 * V))
constf2 = 1
tau_f2 = 0.001*constf2*(600*ufl.exp(-((25 + 1000*V)*(25 +\
1000*V))/170) + 16.0/(1 + ufl.exp(3 + 100*V)) + 31.0/(1 +\
ufl.exp(5/2 - 100*V)))
F_expressions[5] = (-f2 + f2_inf) / tau_f2
# Expressions for the FCa gate component
alpha_fCa = 1.0 / (1 + 5.95374180765e+25 * ufl.elem_pow(Cai, 8))
beta_fCa = 0.1 / (1 + 0.000123409804087 * ufl.exp(10000.0 * Cai))
gamma_fCa = 0.3 / (1 + 0.391605626677 * ufl.exp(1250.0 * Cai))
fCa_inf = 0.760109455762*alpha_fCa + 0.760109455762*beta_fCa +\
0.760109455762*gamma_fCa
constfCa = ufl.conditional(ufl.And(ufl.gt(V, -0.06), ufl.gt(fCa_inf,\
fCa)), 0, 1)
F_expressions[6] = (-fCa + fCa_inf) * constfCa / tau_fCa
# Expressions for the Xr1 gate component
V_half = -19.0 - 1000*R*T*ufl.ln(ufl.elem_pow(1 + 0.384615384615*Cao,\
4)/(L0*ufl.elem_pow(1 + 1.72413793103*Cao, 4)))/(F*Q)
Xr1_inf = 1.0 / (1 +
ufl.exp(0.204081632653 * V_half - 204.081632653 * V))
alpha_Xr1 = 450.0 / (1 + ufl.exp(-9 / 2 - 100 * V))
beta_Xr1 = 6.0 / (1 + 13.5813245226 * ufl.exp(86.9565217391 * V))
tau_Xr1 = 0.001 * alpha_Xr1 * beta_Xr1
F_expressions[7] = (-Xr1 + Xr1_inf) / tau_Xr1
# Expressions for the Xr2 gate component
Xr2_infinity = 1.0 / (1 + ufl.exp(44 / 25 + 20 * V))
alpha_Xr2 = 3.0 / (1 + ufl.exp(-3 - 50 * V))
beta_Xr2 = 1.12 / (1 + ufl.exp(-3 + 50 * V))
tau_Xr2 = 0.001 * alpha_Xr2 * beta_Xr2
F_expressions[8] = (-Xr2 + Xr2_infinity) / tau_Xr2
# Expressions for the Xs gate component
Xs_infinity = 1.0 / (1 + ufl.exp(-5 / 4 - 125 * V / 2))
alpha_Xs = 1100.0 / ufl.sqrt(1 + ufl.exp(-5 / 3 - 500 * V / 3))
beta_Xs = 1.0 / (1 + ufl.exp(-3 + 50 * V))
tau_Xs = alpha_Xs * beta_Xs / 1000
F_expressions[9] = (-Xs + Xs_infinity) / tau_Xs
# Expressions for the Xf gate component
Xf_infinity = 1.0 / (1 + 5780495.71031 * ufl.exp(200 * V))
tau_Xf = 1.9 / (1 + ufl.exp(3 / 2 + 100 * V))
F_expressions[10] = (-Xf + Xf_infinity) / tau_Xf
# Expressions for the i_b Na component
i_b_Na = g_b_Na * (-E_Na + V)
# Expressions for the i_b Ca component
i_b_Ca = g_b_Ca * (-E_Ca + V)
# Expressions for the i_NaK component
i_NaK = Ko*PNaK*Nai/((Km_K + Ko)*(Km_Na + Nai)*(1 +\
0.0353*ufl.exp(-F*V/(R*T)) + 0.1245*ufl.exp(-0.1*F*V/(R*T))))
# Expressions for the i_NaCa component
i_NaCa = kNaCa*(Cao*(Nai*Nai*Nai)*ufl.exp(F*gamma*V/(R*T)) -\
alpha*(Nao*Nao*Nao)*Cai*ufl.exp(F*(-1 + gamma)*V/(R*T)))/((1 +\
Ksat*ufl.exp(F*(-1 + gamma)*V/(R*T)))*(Cao +\
KmCa)*((KmNai*KmNai*KmNai) + (Nao*Nao*Nao)))
# Expressions for the i_PCa component
i_PCa = g_PCa * Cai / (KPCa + Cai)
# Expressions for the q gate component
q_inf = 1.0 / (1 + ufl.exp(53 / 13 + 1000 * V / 13))
tau_q = 0.00606 + 0.039102/(0.0168716780457*ufl.exp(-80.0*V) +\
6.42137321286*ufl.exp(100.0*V))
F_expressions[11] = (-q + q_inf) / tau_q
# Expressions for the r gate component
r_inf = 1.0 / (1 + 3.28489055021 * ufl.exp(-53.3333333333 * V))
tau_r = 0.00275352 + 0.01440516/(16.3010892258*ufl.exp(90.0*V) +\
0.0211152735604*ufl.exp(-120.0*V))
F_expressions[12] = (-r + r_inf) / tau_r
# Expressions for the Sodium dynamics component
F_expressions[13] = -1e+18*Cm*(3*i_NaCa + 3*i_NaK + i_Na +\
i_b_Na)/(F*Vc)
# Expressions for the Calcium dynamics component
i_rel = 0.0411*(c_rel + a_rel*(Ca_SR*Ca_SR)/((b_rel*b_rel) +\
(Ca_SR*Ca_SR)))*d*g
i_up = VmaxUp / (1 + (Kup * Kup) / (Cai * Cai))
i_leak = V_leak * (-Cai + Ca_SR)
g_inf = ufl.conditional(ufl.le(Cai, 0.00035), 1.0/(1 +\
5.43991024148e+20*ufl.elem_pow(Cai, 6)), 1.0/(1 +\
1.9720198874e+55*ufl.elem_pow(Cai, 16)))
const2 = ufl.conditional(ufl.And(ufl.gt(V, -0.06), ufl.gt(g_inf, g)),\
0, 1)
F_expressions[14] = (-g + g_inf) * const2 / tau_g
Cai_bufc = 1.0 / (1 + Buf_C * Kbuf_C / ((Kbuf_C + Cai) *
(Kbuf_C + Cai)))
Ca_SR_bufSR = 1.0/(1 + Buf_SR*Kbuf_SR/((Kbuf_SR + Ca_SR)*(Kbuf_SR +\
Ca_SR)))
F_expressions[15] = (-i_up - 5e+17*Cm*(-2*i_NaCa + i_CaL + i_PCa +\
i_b_Ca)/(F*Vc) + i_leak + i_rel)*Cai_bufc
F_expressions[16] = Vc * (-i_leak - i_rel + i_up) * Ca_SR_bufSR / V_SR
# Return results
return dolfin.as_vector(F_expressions)
def amp(self, t, lam, amp_old):
uabs_minus = Max(-Min(self.ua(t), 0), 0)
return conditional(gt(uabs_minus, 0.), self.g(lam), amp_old)
b_2.vector()[:] = 2 * rand(R2.dim())
c_vars = [W_1, b_1, W_2, b_2, W_3, W_4]
c_values = []
#for i, c in enumerate(c_vars):
# c.vector()[:] = numpy.load("test_poisson_nn_fail_c_{}.npy".format(i))
def sigma_(vec, func=ufl.tanh):
v = [func(vec[i]) for i in range(vec.ufl_shape[0])]
return ufl.as_vector(v)
a = 1.0
relu = lambda vec: conditional(ufl.gt(vec, 0), vec, a * (ufl.exp(vec) - 1))
sigma = lambda vec: sigma_(vec, func=relu)
U_ = Function(V)
from pyadjoint.placeholder import Placeholder
p = Placeholder(U_)
a1 = inner(
inner(
W_4,
sigma(
ufl.transpose(as_vector([W_1, W_2, W_3])) *
as_vector([U_, *X]) + b_1)) + b_2, v
) * dx + inner(grad(U), grad(v)) * dx - Constant(
1
) * v * dx #+ inner(inner(W_4, sigma(as_vector(W_3, W_4)*X + b_3)) + b_4, v)*dx