def extract_blocks(form,
test_functions: typing.List[ufl.Argument],
trial_functions: typing.List[ufl.Argument] = None):
"""Extract blocks from a monolithic UFL form.
Parameters
----------
form
test_functions
trial_functions: optional
Returns
-------
Splitted UFL form in the order determined by the passed test and trial functions.
If no `trial_functions` are provided returns a list, othwerwise returns list of lists.
"""
# Prepare empty block matrices list
if trial_functions is not None:
blocks = [[None for i in range(len(test_functions))]
for j in range(len(trial_functions))]
else:
blocks = [None for i in range(len(test_functions))]
for i, tef in enumerate(test_functions):
if trial_functions is not None:
for j, trf in enumerate(trial_functions):
to_null = dict()
# Dictionary mapping the other trial functions
# to zero
for item in trial_functions:
if item != trf:
to_null[item] = ufl.zero(item.ufl_shape)
# Dictionary mapping the other test functions
# to zero
for item in test_functions:
if item != tef:
to_null[item] = ufl.zero(item.ufl_shape)
blocks[i][j] = ufl.replace(form, to_null)
else:
to_null = dict()
# Dictionary mapping the other test functions
# to zero
for item in test_functions:
if item != tef:
to_null[item] = ufl.zero(item.ufl_shape)
blocks[i] = ufl.replace(form, to_null)
return blocks
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) == 3)
m, h, n = s
# Assign parameters
# Init return args
F_expressions = [ufl.zero()] * 3
# Expressions for the m gate component
alpha_m = (-5.0 - 0.1 * V) / (-1.0 + ufl.exp(-5.0 - V / 10.0))
beta_m = 4 * ufl.exp(-25.0 / 6.0 - V / 18.0)
F_expressions[0] = (1 - m) * alpha_m - beta_m * m
# Expressions for the h gate component
alpha_h = 0.07 * ufl.exp(-15.0 / 4.0 - V / 20.0)
beta_h = 1.0 / (1 + ufl.exp(-9.0 / 2.0 - V / 10.0))
F_expressions[1] = (1 - h) * alpha_h - beta_h * h
# Expressions for the n gate component
alpha_n = (-0.65 - 0.01 * V) / (-1.0 + ufl.exp(-13.0 / 2.0 - V / 10.0))
beta_n = 0.125 * ufl.exp(-15.0 / 16.0 - V / 80.0)
F_expressions[2] = (1 - n) * alpha_n - beta_n * n
# Return results
return dolfin.as_vector(F_expressions)
def _rush_larsen_step(rhs_exprs, diff_rhs_exprs, linear_terms,
system_size, y0, stage_solution, dt, time, a, c,
v, DX, time_dep_expressions):
# If we need to replace the original solution with stage solution
repl = None
if stage_solution is not None:
if system_size > 1:
repl = {y0: stage_solution}
else:
repl = {y0[0]: stage_solution}
# If we have time dependent expressions
if time_dep_expressions and abs(float(c)) > DOLFIN_EPS:
time_ = time
time = time + dt * float(c)
repl.update(_replace_dict_time_dependent_expression(time_dep_expressions,
time_, dt, float(c)))
repl[time_] = time
# If all terms are linear (using generalized=True) we add a safe
# guard to the linearized term. See below
safe_guard = sum(linear_terms) == system_size
# Add componentwise contribution to rl form
rl_ufl_form = ufl.zero()
num_rl_steps = 0
for ind in range(system_size):
# forward euler step
fe_du_i = rhs_exprs[ind] * dt * float(a)
# If exact integration
if linear_terms[ind]:
num_rl_steps += 1
# Rush Larsen step Safeguard the divisor: let's hope
# diff_rhs_exprs[ind] is never 1.0e-16! Let's get rid of
# this when the conditional fixes land properly in UFL.
eps = Constant(1.0e-16)
rl_du_i = rhs_exprs[ind] / (diff_rhs_exprs[ind] + eps) * (
ufl.exp(diff_rhs_exprs[ind] * dt) - 1.0)
# If safe guard
if safe_guard:
du_i = ufl.conditional(ufl.lt(abs(diff_rhs_exprs[ind]), 1e-8), fe_du_i, rl_du_i)
else:
du_i = rl_du_i
else:
du_i = fe_du_i
# If we should replace solution in form with stage solution
if repl:
du_i = ufl.replace(du_i, repl)
rl_ufl_form += (y0[ind] + du_i) * v[ind]
return rl_ufl_form * DX
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) == 2)
m, h = s
# Assign parameters
E_h = self._parameters["E_h"]
E_m = self._parameters["E_m"]
delta_h = self._parameters["delta_h"]
k_h = self._parameters["k_h"]
k_m = self._parameters["k_m"]
tau_h0 = self._parameters["tau_h0"]
tau_m = self._parameters["tau_m"]
# Init return args
F_expressions = [ufl.zero()]*2
# Expressions for the m gate component
m_inf = 1.0/(1.0 + ufl.exp((E_m - V)/k_m))
F_expressions[0] = (-m + m_inf)/tau_m
# Expressions for the h gate component
h_inf = 1.0/(1.0 + ufl.exp((-E_h + V)/k_h))
tau_h = 2*tau_h0*ufl.exp(delta_h*(-E_h + V)/k_h)/(1 + ufl.exp((-E_h +\
V)/k_h))
F_expressions[1] = (-h + h_inf)/tau_h
# Return results
return dolfin.as_vector(F_expressions)
def function_arg(self, g):
'''Set the value of this boundary condition.'''
if isinstance(g, firedrake.Function):
if g.function_space() != self.function_space():
raise RuntimeError("%r is defined on incompatible FunctionSpace!" % g)
self._function_arg = g
elif isinstance(g, ufl.classes.Zero):
if g.ufl_shape and g.ufl_shape != self.function_space().ufl_element().value_shape():
raise ValueError(f"Provided boundary value {g} does not match shape of space")
# Special case. Scalar zero for direct Function.assign.
self._function_arg = ufl.zero()
elif isinstance(g, ufl.classes.Expr):
if g.ufl_shape != self.function_space().ufl_element().value_shape():
raise RuntimeError(f"Provided boundary value {g} does not match shape of space")
try:
self._function_arg = firedrake.Function(self.function_space())
self._function_arg_update = firedrake.Interpolator(g, self._function_arg).interpolate
except (NotImplementedError, AttributeError):
# Element doesn't implement interpolation
self._function_arg = firedrake.Function(self.function_space()).project(g)
self._function_arg_update = firedrake.Projector(g, self._function_arg).project
else:
try:
g = as_ufl(g)
self._function_arg = g
except UFLException:
try:
# Recurse to handle this through interpolation.
self.function_arg = as_ufl(as_tensor(g))
except UFLException:
raise ValueError(f"{g} is not a valid DirichletBC expression")
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) == 7)
m, h, j, Cai, d, f, x1 = s
# Assign parameters
g_s = self._parameters["g_s"]
# Init return args
F_expressions = [ufl.zero()]*7
# Expressions for the Sodium current m gate component
alpha_m = (-47 - V)/(-1 + 0.0090952771017*ufl.exp(-0.1*V))
beta_m = 0.709552672749*ufl.exp(-0.056*V)
F_expressions[0] = -beta_m*m + (1 - m)*alpha_m
# Expressions for the Sodium current h gate component
alpha_h = 5.49796243871e-10*ufl.exp(-0.25*V)
beta_h = 1.7/(1 + 0.15802532089*ufl.exp(-0.082*V))
F_expressions[1] = (1 - h)*alpha_h - beta_h*h
# Expressions for the Sodium current j gate component
alpha_j = 1.86904730072e-10*ufl.exp(-0.25*V)/(1 +\
1.67882753e-07*ufl.exp(-0.2*V))
beta_j = 0.3/(1 + 0.0407622039784*ufl.exp(-0.1*V))
F_expressions[2] = (1 - j)*alpha_j - beta_j*j
# Expressions for the Slow inward current component
E_s = -82.3 - 13.0287*ufl.ln(0.001*Cai)
i_s = g_s*(-E_s + V)*d*f
F_expressions[3] = 7e-06 - 0.07*Cai - 0.01*i_s
# Expressions for the Slow inward current d gate component
alpha_d = 0.095*ufl.exp(1/20 - V/100)/(1 +\
1.43328813857*ufl.exp(-0.0719942404608*V))
beta_d = 0.07*ufl.exp(-44/59 - V/59)/(1 + ufl.exp(11/5 + V/20))
F_expressions[4] = -beta_d*d + (1 - d)*alpha_d
# Expressions for the Slow inward current f gate component
alpha_f = 0.012*ufl.exp(-28/125 - V/125)/(1 +\
66.5465065251*ufl.exp(0.149925037481*V))
beta_f = 0.0065*ufl.exp(-3/5 - V/50)/(1 + ufl.exp(-6 - V/5))
F_expressions[5] = (1 - f)*alpha_f - beta_f*f
# Expressions for the Time dependent outward current x1 gate component
alpha_x1 = 0.0311584109863*ufl.exp(0.0826446280992*V)/(1 +\
17.4117080633*ufl.exp(0.0571428571429*V))
beta_x1 = 0.000391646440562*ufl.exp(-0.0599880023995*V)/(1 +\
ufl.exp(-4/5 - V/25))
F_expressions[6] = (1 - x1)*alpha_x1 - beta_x1*x1
# Return results
return dolfin.as_vector(F_expressions)
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) == 7)
m, h, j, Cai, d, f, x1 = s
# Assign parameters
E_Na = self._parameters["E_Na"]
g_Na = self._parameters["g_Na"]
g_Nac = self._parameters["g_Nac"]
g_s = self._parameters["g_s"]
IstimAmplitude = self._parameters["IstimAmplitude"]
IstimPulseDuration = self._parameters["IstimPulseDuration"]
IstimStart = self._parameters["IstimStart"]
C = self._parameters["C"]
# Init return args
current = [ufl.zero()]*1
# Expressions for the Sodium current component
i_Na = (g_Nac + g_Na*(m*m*m)*h*j)*(-E_Na + V)
# Expressions for the Slow inward current component
E_s = -82.3 - 13.0287*ufl.ln(0.001*Cai)
i_s = g_s*(-E_s + V)*d*f
# Expressions for the Time dependent outward current component
i_x1 = 0.00197277571153*(-1 +\
21.7584023962*ufl.exp(0.04*V))*ufl.exp(-0.04*V)*x1
# Expressions for the Time independent outward current component
i_K1 = 0.0035*(-4 +\
119.85640019*ufl.exp(0.04*V))/(8.33113748769*ufl.exp(0.04*V) +\
69.4078518388*ufl.exp(0.08*V)) + 0.0035*(4.6 + 0.2*V)/(1 -\
0.398519041085*ufl.exp(-0.04*V))
# Expressions for the Stimulus protocol component
Istim = ufl.conditional(ufl.And(ufl.ge(time, IstimStart),\
ufl.le(time, IstimPulseDuration + IstimStart)), IstimAmplitude,\
0)
# Expressions for the Membrane component
current[0] = (-i_K1 + Istim - i_Na - i_x1 - i_s)/C
# Return results
return current[0]
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)
v, w = s
# Assign parameters
u_c = self._parameters["u_c"]
g_fi_max = self._parameters["g_fi_max"]
tau_0 = self._parameters["tau_0"]
tau_r = self._parameters["tau_r"]
k = self._parameters["k"]
tau_si = self._parameters["tau_si"]
u_csi = self._parameters["u_csi"]
Cm = self._parameters["Cm"]
V_0 = self._parameters["V_0"]
V_fi = self._parameters["V_fi"]
# Init return args
current = [ufl.zero()] * 1
# Expressions for the p component
p = ufl.conditional(ufl.lt((-V_0 + V) / (V_fi - V_0), u_c), 0, 1)
# Expressions for the Fast inward current component
tau_d = Cm / g_fi_max
J_fi = -(1 - (-V_0 + V)/(V_fi - V_0))*(-u_c + (-V_0 + V)/(V_fi -\
V_0))*p*v/tau_d
# Expressions for the Slow outward current component
J_so = p / tau_r + (1 - p) * (-V_0 + V) / (tau_0 * (V_fi - V_0))
# Expressions for the Slow inward current component
J_si = -(1 + ufl.tanh(k*(-u_csi + (-V_0 + V)/(V_fi -\
V_0))))*w/(2*tau_si)
# Expressions for the Stimulus protocol component
J_stim = 0
# Expressions for the Membrane component
current[0] = (V_0 - V_fi) * (J_stim + J_fi + J_si + J_so)
# Return results
return current[0]
def __init__(self,
mesh: df.Mesh,
time: df.Constant,
model: CellModel,
I_s: tp.Union[df.Expression, tp.Dict[int,
df.Expression]] = None,
parameters: df.Parameters = None) -> None:
"""Initialise parameters."""
super().__init__(mesh=mesh,
time=time,
cell_model=model,
parameters=parameters)
import ufl.classes # TODO Why?
self._I_s = I_s
# Initialize scheme
v, s = split_function(self.vs, self._num_states + 1)
w, q = split_function(df.TestFunction(self.VS), self._num_states + 1)
# Workaround to get algorithm in RL schemes working as it only works for scalar expressions
F_exprs = self._F(v, s, self._time)
# MER: This looks much more complicated than it needs to be!
# If we have a as_vector expression
F_exprs_q = ufl.zero()
if isinstance(F_exprs, ufl.classes.ListTensor):
for i, expr_i in enumerate(F_exprs.ufl_operands):
F_exprs_q += expr_i * q[i]
else:
F_exprs_q = F_exprs * q
rhs = F_exprs_q - self._I_ion(v, s, self._time) * w
# Handle stimulus: only handle single function case for now
if self._I_s:
rhs += self._I_s * w
self._rhs = rhs * df.dP()
name = self._parameters["scheme"]
Scheme = self._name_to_scheme(name)
self._scheme = Scheme(self._rhs, self.vs, self._time)
# Initialize solver and update its parameters
self._pi_solver = df.PointIntegralSolver(self._scheme)
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)
s, m = s
# Assign parameters
Cm = self._parameters["Cm"]
E_L = self._parameters["E_L"]
g_L = self._parameters["g_L"]
# synapse components
alpha = self._parameters["alpha"]
g_S = self._parameters["g_S"]
t0 = self._parameters["t0"]
v_eq = self._parameters["v_eq"]
# Init return args
current = [ufl.zero()] * 1
# Expressions for the Membrane component
# FIXME: base on stim_type + add to Hodgkin
# if
# i_Stim = g_S*(-v_eq + V)*ufl.conditional(ufl.ge(time, t0), 1, 0)*ufl.exp((t0 - time)/alpha)
# elif sss:
# i_Stim = g_S*ufl.conditional(ufl.ge(time, t0), 1, 0)
# else:
# i_Stim = g_S*ufl.conditional(ufl.And(ufl.ge(time, t0),
# ufl.le(time, t1), 1, 0))
i_Stim = g_S * (-v_eq + V) * ufl.conditional(ufl.ge(
time, t0), 1, 0) * ufl.exp((t0 - time) / alpha)
i_L = g_L * (-E_L + V)
current[0] = (-i_L - i_Stim) / Cm
# Return results
return current[0]
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) == 3)
m, h, n = s
# Assign parameters
g_Na = self._parameters["g_Na"]
g_K = self._parameters["g_K"]
g_L = self._parameters["g_L"]
Cm = self._parameters["Cm"]
E_R = self._parameters["E_R"]
# Init return args
current = [ufl.zero()] * 1
# Expressions for the Sodium channel component
E_Na = 115.0 + E_R
i_Na = g_Na * (m * m * m) * (-E_Na + V) * h
# Expressions for the Potassium channel component
E_K = -12.0 + E_R
i_K = g_K * ufl.elem_pow(n, 4) * (-E_K + V)
# Expressions for the Leakage current component
E_L = 10.613 + E_R
i_L = g_L * (-E_L + V)
# Expressions for the Membrane component
# TODO can be defined outside
i_Stim = 0.0
current[0] = (-i_K - i_L - i_Na + i_Stim) / Cm
# Return results
return current[0]
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) == 2)
v, w = s
# Assign parameters
u_c = self._parameters["u_c"]
u_v = self._parameters["u_v"]
tau_v1_minus = self._parameters["tau_v1_minus"]
tau_v2_minus = self._parameters["tau_v2_minus"]
tau_v_plus = self._parameters["tau_v_plus"]
tau_w_minus = self._parameters["tau_w_minus"]
tau_w_plus = self._parameters["tau_w_plus"]
V_0 = self._parameters["V_0"]
V_fi = self._parameters["V_fi"]
# Init return args
F_expressions = [ufl.zero()] * 2
# Expressions for the p component
p = ufl.conditional(ufl.lt((-V_0 + V) / (V_fi - V_0), u_c), 0, 1)
# Expressions for the q component
q = ufl.conditional(ufl.lt((-V_0 + V) / (V_fi - V_0), u_v), 0, 1)
# Expressions for the v gate component
tau_v_minus = tau_v1_minus * q + tau_v2_minus * (1 - q)
F_expressions[0] = (1 - p) * (1 - v) / tau_v_minus - p * v / tau_v_plus
# Expressions for the w gate component
F_expressions[1] = (1 - p) * (1 - w) / tau_w_minus - p * w / tau_w_plus
# Return results
return dolfin.as_vector(F_expressions)
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) == 2)
s, m = s
# Assign parameters
# Init return args
F_expressions = [ufl.zero()] * 2
# Expressions for the Membrane component
F_expressions[0] = -m
F_expressions[1] = s
# Return results
return dolfin.as_vector(F_expressions)
def _I(self, v, s, time):
"""
Original gotran transmembrane current dV/dt
"""
time = time if time else Constant(0.0)
# Assign states
V = v
# Assign parameters
g_leak = self._parameters["g_L"]
Cm = self._parameters["Cm"]
E_leak = self._parameters["E_L"]
stim_type = self._parameters["stim_type"]
g_s = self._parameters["g_S"]
alpha = self._parameters["alpha"]
v_eq = self._parameters["v_eq"]
t0 = self._parameters["t_start"]
t1 = self._parameters["t_stop"]
# Init return args
current = [ufl.zero()] * 1
# Expressions for the Membrane component
if stim_type == 0:
i_stim = g_s * (-v_eq + V) * ufl.conditional(
ufl.ge(time, t0), 1, 0) * ufl.exp((t0 - time) / alpha)
elif stim_type == 1:
i_stim = -g_s * ufl.conditional(ufl.ge(time, t0), 1, 0)
elif stim_type == 2:
i_stim = -g_s * ufl.conditional(
ufl.And(ufl.ge(time, t0), ufl.le(time, t1)), 1, 0)
i_leak = g_leak * (-E_leak + V)
current[0] = (-i_leak - i_stim) / Cm
# Return results
return current[0]
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 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) == 18)
Xr1, Xr2, Xs, m, h, j, d, f, f2, fCass, s, r, Ca_SR, Ca_i, Ca_ss,\
R_prime, 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"]
Buf_ss = self._parameters["Buf_ss"]
Ca_o = self._parameters["Ca_o"]
EC = self._parameters["EC"]
K_buf_c = self._parameters["K_buf_c"]
K_buf_sr = self._parameters["K_buf_sr"]
K_buf_ss = self._parameters["K_buf_ss"]
K_up = self._parameters["K_up"]
V_leak = self._parameters["V_leak"]
V_rel = self._parameters["V_rel"]
V_sr = self._parameters["V_sr"]
V_ss = self._parameters["V_ss"]
V_xfer = self._parameters["V_xfer"]
Vmax_up = self._parameters["Vmax_up"]
k1_prime = self._parameters["k1_prime"]
k2_prime = self._parameters["k2_prime"]
k3 = self._parameters["k3"]
k4 = self._parameters["k4"]
max_sr = self._parameters["max_sr"]
min_sr = self._parameters["min_sr"]
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"]
K_o = self._parameters["K_o"]
# Init return args
F_expressions = [ufl.zero()] * 18
# 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 = 1400 / ufl.sqrt(1 + ufl.exp(5 / 6 - V / 6))
beta_xs = 1.0 / (1 + ufl.exp(-7 / 3 + V / 15))
tau_xs = 80 + 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 = ufl.conditional(ufl.lt(V, -40),\
4.43126792958e-07*ufl.exp(-0.147058823529*V), 0)
beta_h = ufl.conditional(ufl.lt(V, -40), 2.7*ufl.exp(0.079*V) +\
310000*ufl.exp(0.3485*V), 0.77/(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 = ufl.conditional(ufl.lt(V, -40), (37.78 +\
V)*(-25428*ufl.exp(0.2444*V) - 6.948e-06*ufl.exp(-0.04391*V))/(1 +\
50262745826.0*ufl.exp(0.311*V)), 0)
beta_j = ufl.conditional(ufl.lt(V, -40),\
0.02424*ufl.exp(-0.01052*V)/(1 +\
0.0039608683399*ufl.exp(-0.1378*V)), 0.6*ufl.exp(0.057*V)/(1 +\
0.0407622039784*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*g_CaL*(F*F)*(-15 + V)*(0.25*Ca_ss*ufl.exp(F*(-30 +\
2*V)/(R*T)) - Ca_o)*d*f*f2*fCass/(R*T*(-1 + ufl.exp(F*(-30 +\
2*V)/(R*T))))
# Expressions for the d gate component
d_inf = 1.0 / (1 + 0.344153786865 * 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 = 20 + 1102.5*ufl.exp(-((27 + V)*(27 + V))/225) + 180/(1 +\
ufl.exp(3 + V/10)) + 200/(1 + ufl.exp(13/10 - V/10))
F_expressions[7] = (f_inf - f) / tau_f
# Expressions for the F2 gate component
f2_inf = 0.33 + 0.67 / (1 + ufl.exp(5 + V / 7))
tau_f2 = 80/(1 + ufl.exp(3 + V/10)) + 562*ufl.exp(-((27 + V)*(27 +\
V))/240) + 31/(1 + ufl.exp(5/2 - V/10))
F_expressions[8] = (-f2 + f2_inf) / tau_f2
# Expressions for the FCass gate component
fCass_inf = 0.4 + 0.6 / (1 + 400.0 * (Ca_ss * Ca_ss))
tau_fCass = 2 + 80 / (1 + 400.0 * (Ca_ss * Ca_ss))
F_expressions[9] = (fCass_inf - fCass) / tau_fCass
# 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[10] = (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[11] = (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_up = Vmax_up / (1 + (K_up * K_up) / (Ca_i * Ca_i))
i_leak = V_leak * (Ca_SR - Ca_i)
i_xfer = V_xfer * (Ca_ss - Ca_i)
kcasr = max_sr - (-min_sr + max_sr) / (1 + (EC * EC) / (Ca_SR * Ca_SR))
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)))
Ca_ss_bufss = 1.0/(1 + Buf_ss*K_buf_ss/((Ca_ss + K_buf_ss)*(Ca_ss +\
K_buf_ss)))
F_expressions[13] = (i_xfer - Cm*(i_b_Ca + i_p_Ca -\
2*i_NaCa)/(2*F*V_c) + V_sr*(-i_up + i_leak)/V_c)*Ca_i_bufc
k1 = k1_prime / kcasr
k2 = k2_prime * kcasr
O = (Ca_ss * Ca_ss) * R_prime * k1 / ((Ca_ss * Ca_ss) * k1 + k3)
F_expressions[15] = -Ca_ss * R_prime * k2 + k4 * (1 - R_prime)
i_rel = V_rel * (Ca_SR - Ca_ss) * O
F_expressions[12] = (i_up - i_leak - i_rel) * Ca_sr_bufsr
F_expressions[14] = (-Cm*i_CaL/(2*F*V_ss) - V_c*i_xfer/V_ss +\
V_sr*i_rel/V_ss)*Ca_ss_bufss
# Expressions for the Sodium dynamics component
F_expressions[16] = Cm * (-i_b_Na - i_Na - 3 * i_NaCa -
3 * i_NaK) / (F * V_c)
# Expressions for the Membrane component
i_Stim = 0
# Expressions for the Potassium dynamics component
F_expressions[17] = Cm*(2*i_NaK - i_Ks - i_Stim - i_Kr - i_to - i_p_K\
- i_K1)/(F*V_c)
# Return results
return as_vector(F_expressions)
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) == 17)
m, h, j, d, f1, f2, fCa, Xr1, Xr2, Xs, Xf, q, r, Nai, g, Cai, Ca_SR = s
# Assign parameters
Chromanol_iKs30 = self._parameters["Chromanol_iKs30"]
Chromanol_iKs50 = self._parameters["Chromanol_iKs50"]
Chromanol_iKs70 = self._parameters["Chromanol_iKs70"]
Chromanol_iKs90 = self._parameters["Chromanol_iKs90"]
E4031_100nM = self._parameters["E4031_100nM"]
E4031_30nM = self._parameters["E4031_30nM"]
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"]
PkNa = self._parameters["PkNa"]
g_Na = self._parameters["g_Na"]
g_CaL = self._parameters["g_CaL"]
g_Kr = self._parameters["g_Kr"]
g_Ks = self._parameters["g_Ks"]
g_K1 = self._parameters["g_K1"]
E_f = self._parameters["E_f"]
g_f = self._parameters["g_f"]
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"]
g_to = self._parameters["g_to"]
Cao = self._parameters["Cao"]
F = self._parameters["F"]
Ki = self._parameters["Ki"]
Ko = self._parameters["Ko"]
Nao = self._parameters["Nao"]
R = self._parameters["R"]
T = self._parameters["T"]
# Init return args
current = [ufl.zero()] * 1
# Expressions for the Electric potentials component
E_Na = R * T * ufl.ln(Nao / Nai) / F
E_K = R * T * ufl.ln(Ko / Ki) / F
E_Ks = R * T * ufl.ln((Ko + Nao * PkNa) / (Ki + PkNa * 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 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 i_Kr component
E4031_coeff = ufl.conditional(ufl.eq(E4031_30nM, 1), 0.77,\
ufl.conditional(ufl.eq(E4031_100nM, 1), 0.5, 1))
i_Kr = 0.430331482912 * g_Kr * ufl.sqrt(Ko) * (
-E_K + V) * E4031_coeff * Xr1 * Xr2
# Expressions for the i_Ks component
Chromanol_coeff = ufl.conditional(ufl.eq(Chromanol_iKs30, 1), 0.7,\
ufl.conditional(ufl.eq(Chromanol_iKs50, 1), 0.5,\
ufl.conditional(ufl.eq(Chromanol_iKs70, 1), 0.3,\
ufl.conditional(ufl.eq(Chromanol_iKs90, 1), 0.1, 1))))
i_Ks = g_Ks*(Xs*Xs)*(1 + 0.6/(1 +\
6.48182102606e-07*ufl.elem_pow(1.0/Cai, 1.4)))*(-E_Ks +\
V)*Chromanol_coeff
# Expressions for the i_K1 component
alpha_K1 = 3.91 / (
1 + 2.44592857399e-52 * ufl.exp(594.2 * V - 594.2 * E_K))
beta_K1 = (0.00277806676906*ufl.exp(588.6*V - 588.6*E_K) -\
1.5394838221*ufl.exp(0.2*V - 0.2*E_K))/(1 + ufl.exp(454.7*V -\
454.7*E_K))
XK1_inf = alpha_K1 / (alpha_K1 + beta_K1)
i_K1 = 0.430331482912 * g_K1 * ufl.sqrt(Ko) * (-E_K + V) * XK1_inf
# Expressions for the i_f component
i_f = g_f * (-E_f + V) * 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 i_to component
i_to = g_to * (-E_K + V) * q * r
# Expressions for the Membrane component
current[0] = -i_CaL - i_K1 - i_Kr - i_Ks - i_Na - i_NaCa - i_NaK -\
i_PCa - i_b_Ca - i_b_Na - i_f - i_to
# Return results
return current[0]
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)
from dolfin import * from ufl import zero mesh = UnitSquareMesh(10, 10) V = FunctionSpace(mesh, 'CG', 1) u = TrialFunction(V) v = TestFunction(V) cs = map(Constant, range(4)) forms = [c*inner(u, v)*dx for c in cs] f0 = sum(forms) f1 = sum(forms, 0) f2 = sum(forms, zero()) f3 = sum(forms[1:], forms[0]) iforms = iter(forms) f4 = sum(iforms, next(iforms)) print len(set([f.signature() for f in (f0, f1, f2, f3, f4)]))
def __init__(self,
F_form: typing.List,
u: typing.List,
lmbda: dolfinx.fem.Function,
A_form=None,
B_form=None,
prefix=None):
"""SLEPc problem and solver wrapper.
Wrapper for a generalised eigenvalue problem obtained from UFL residual forms.
Parameters
----------
F_form
Residual forms
u
Current solution vectors
lmbda
Eigenvalue function. Residual forms must be linear in lmbda for
linear eigenvalue problems.
A_form, optional
Override automatically derived A
B_form, optional
Override automatically derived B
Note
----
In general, eigenvalue problems have form T(lmbda) * x = 0,
where T(lmbda) is a matrix-valued function.
Linear eigenvalue problems have T(lmbda) = A + lmbda * B, and if B is not identity matrix
then this problem is called generalized (linear) eigenvalue problem.
"""
self.F_form = F_form
self.u = u
self.lmbda = lmbda
self.comm = u[0].function_space.mesh.comm
self.ur = []
self.ui = []
for func in u:
self.ur.append(
dolfinx.fem.Function(func.function_space, name=func.name))
self.ui.append(
dolfinx.fem.Function(func.function_space, name=func.name))
# Prepare tangent form M0 which has terms involving lambda
self.M0 = [[None for i in range(len(self.u))]
for j in range(len(self.u))]
for i in range(len(self.u)):
for j in range(len(self.u)):
self.M0[i][j] = ufl.algorithms.expand_derivatives(
ufl.derivative(F_form[i], self.u[j],
ufl.TrialFunction(
self.u[j].function_space)))
if B_form is None:
B0 = [[None for i in range(len(self.u))]
for j in range(len(self.u))]
for i in range(len(self.u)):
for j in range(len(self.u)):
# Differentiate wrt. lambda and replace all remaining lambda with Zero
B0[i][j] = ufl.algorithms.expand_derivatives(
ufl.diff(self.M0[i][j], lmbda))
B0[i][j] = ufl.replace(B0[i][j], {lmbda: ufl.zero()})
if B0[i][j].empty():
B0[i][j] = None
self.B_form = B0
else:
self.B_form = B_form
if A_form is None:
A0 = [[None for i in range(len(self.u))]
for j in range(len(self.u))]
for i in range(len(self.u)):
for j in range(len(self.u)):
A0[i][j] = ufl.replace(self.M0[i][j], {lmbda: ufl.zero()})
if A0[i][j].empty():
A0[i][j] = None
continue
self.A_form = A0
else:
self.A_form = A_form
self.eps = SLEPc.EPS().create(self.comm)
self.eps.setOptionsPrefix(prefix)
self.eps.setFromOptions()
self.A_form = dolfinx.fem.form(self.A_form)
self.B_form = dolfinx.fem.form(self.B_form)
self.A = dolfinx.fem.petsc.create_matrix_block(self.A_form)
self.B = None
if not self.empty_B():
self.B = dolfinx.fem.petsc.create_matrix_block(self.B_form)
def _butcher_scheme_generator_tlm(a, b, c, time, solution, rhs_form,
perturbation):
"""
Generates a list of forms and solutions for a given Butcher tableau
*Arguments*
a (2 dimensional numpy array)
The a matrix of the Butcher tableau.
b (1-2 dimensional numpy array)
The b vector of the Butcher tableau. If b is 2 dimensional the
scheme includes an error estimator and can be used in adaptive
solvers.
c (1 dimensional numpy array)
The c vector the Butcher tableau.
time (_Constant_)
A Constant holding the time at the start of the time step
solution (_Function_)
The prognostic variable
rhs_form (ufl.Form)
A UFL form representing the rhs for a time differentiated equation
perturbation (_Function_)
The perturbation in the initial condition of the solution
"""
a = _check_abc(a, b, c)
size = a.shape[0]
DX = _check_form(rhs_form)
# Get test function
arguments, coefficients = ufl.algorithms.\
extract_arguments_and_coefficients(rhs_form)
v = arguments[0]
# Create time step
dt = Constant(0.1)
# rhs forms
dolfin_stage_forms = []
ufl_stage_forms = []
# Stage solutions
k = [
Function(solution.function_space(), name="k_%d" % i)
for i in range(size)
]
kdot = [Function(solution.function_space(), name="kdot_%d"%i) \
for i in range(size)]
# Create the stage forms
y_ = solution
ydot = solution.copy()
time_ = time
time_dep_expressions = _time_dependent_expressions(rhs_form, time)
zero_ = ufl.zero(*y_.shape())
forward_forms = []
stage_solutions = []
jacobian_indices = []
for i, ki in enumerate(k):
# Check whether the stage is explicit
explicit = a[i, i] == 0
# Evaluation arguments for the ith stage
evalargs = y_ + dt * sum([float(a[i,j]) * k[j] \
for j in range(i+1)], zero_)
time = time_ + dt * c[i]
replace_dict = _replace_dict_time_dependent_expression(time_dep_expressions, \
time_, dt, c[i])
replace_dict[y_] = evalargs
replace_dict[time_] = time
stage_form = ufl.replace(rhs_form, replace_dict)
forward_forms.append(stage_form)
# The recomputation of the forward run:
if explicit:
stage_forms = [stage_form]
jacobian_indices.append(-1)
else:
# Create a F=0 form and differentiate it
stage_form_implicit = stage_form - ufl.inner(ki, v) * DX
stage_forms = [
stage_form_implicit,
derivative(stage_form_implicit, ki)
]
jacobian_indices.append(0)
ufl_stage_forms.append(stage_forms)
dolfin_stage_forms.append([Form(form) for form in stage_forms])
stage_solutions.append(ki)
# And now the tangent linearisation:
stage_form_tlm = safe_action(derivative(stage_form, y_), perturbation) + \
sum([dt*float(a[i,j]) * safe_action(derivative(\
forward_forms[j], y_), kdot[j]) for j in range(i+1)])
if explicit:
stage_forms_tlm = [stage_form_tlm]
jacobian_indices.append(-1)
else:
# Create a F=0 form and differentiate it
stage_form_tlm -= ufl.inner(kdot[i], v) * DX
stage_forms_tlm = [
stage_form_tlm,
derivative(stage_form_tlm, kdot[i])
]
jacobian_indices.append(1)
ufl_stage_forms.append(stage_forms_tlm)
dolfin_stage_forms.append([Form(form) for form in stage_forms_tlm])
stage_solutions.append(kdot[i])
# Only one last stage
if len(b.shape) == 1:
last_stage = Form(ufl.inner(perturbation + sum(\
[dt*float(bi)*kdoti for bi, kdoti in zip(b, kdot)], zero_), v)*DX)
else:
raise Exception("Not sure what to do here")
human_form = []
for i in range(size):
kterm = " + ".join("%sh*k_%s" % ("" if a[i,j] == 1.0 else \
"%s*"% a[i,j], j) \
for j in range(size) if a[i,j] != 0)
if c[i] in [0.0, 1.0]:
cih = " + h" if c[i] == 1.0 else ""
else:
cih = " + %s*h" % c[i]
kdotterm = " + ".join("%(a)sh*action(derivative(f(t_n%(cih)s, y_n + "\
"%(kterm)s), kdot_%(i)s" % \
{"a": ("" if a[i,j] == 1.0 else "%s*"% a[i,j], j),
"i": i,
"cih": cih,
"kterm": kterm} \
for j in range(size) if a[i,j] != 0)
if len(kterm) == 0:
human_form.append("k_%(i)s = f(t_n%(cih)s, y_n)" % {
"i": i,
"cih": cih
})
human_form.append("kdot_%(i)s = action(derivative("\
"f(t_n%(cih)s, y_n), y_n), ydot_n)" % \
{"i": i, "cih": cih})
else:
human_form.append("k_%(i)s = f(t_n%(cih)s, y_n + %(kterm)s)" % \
{"i": i, "cih": cih, "kterm": kterm})
human_form.append("kdot_%(i)s = action(derivative(f(t_n%(cih)s, "\
"y_n + %(kterm)s), y_n) + %(kdotterm)s" % \
{"i": i, "cih": cih, "kterm": kterm, "kdotterm": kdotterm})
parentheses = "(%s)" if np.sum(b > 0) > 1 else "%s"
human_form.append("ydot_{n+1} = ydot_n + h*" + parentheses % (" + ".join(\
"%skdot_%s" % ("" if b[i] == 1.0 else "%s*" % b[i], i) \
for i in range(size) if b[i] > 0)))
human_form = "\n".join(human_form)
return ufl_stage_forms, dolfin_stage_forms, jacobian_indices, last_stage, \
stage_solutions, dt, human_form, perturbation
def __define_boundary_terms(self):
if self.verb > 2: print0("Defining boundary terms")
self.bnd_vector_form = numpy.empty(self.DD.G, dtype=object)
self.bnd_matrix_form = numpy.empty(self.DD.G, dtype=object)
n = FacetNormal(self.DD.mesh)
i,p,q = ufl.indices(3)
natural_boundaries = self.BC.vacuum_boundaries.union(self.BC.incoming_fluxes.keys())
nonzero = lambda x: numpy.all(x > 0)
nonzero_inc_flux = any(map(nonzero, self.BC.incoming_fluxes.values()))
if nonzero_inc_flux and not self.fixed_source_problem:
coupled_solver_error(__file__,
"define boundary terms",
"Incoming flux specified for an eigenvalue problem"+\
"(Q must be given whenever phi_inc is; it may possibly be zero everywhere)")
for g in range(self.DD.G):
self.bnd_vector_form[g] = ufl.zero()
self.bnd_matrix_form[g] = ufl.zero()
if natural_boundaries:
try:
ds = Measure("ds")[self.DD.boundaries]
except TypeError:
coupled_solver_error(__file__,
"define boundary terms",
"File assigning boundary indices to facets required if vacuum or incoming flux boundaries "
"are specified.")
for bnd_idx in natural_boundaries:
# NOTE: The following doesn't work because ufl.abs requires a function
#
#for g in range(self.DD.G):
# self.bnd_matrix_form[g] += \
# abs(self.tensors.G[p,q,i]*n[i])*self.u[g][q]*self.v[g][p]*ds(bnd_idx)
#
# NOTE: Instead, the following explicit loop has to be used; this makes tensors.G unneccessary
#
for pp in range(self.DD.M):
omega_p_dot_n = self.DD.ordinates_matrix[i,pp]*n[i]
for g in range(self.DD.G):
self.bnd_matrix_form[g] += \
abs(omega_p_dot_n)*self.tensors.Wp[pp]*self.u[g][pp]*self.v[g][pp]*ds(bnd_idx)
if nonzero_inc_flux:
for pp in range(self.DD.M):
omega_p_dot_n = self.DD.ordinates_matrix[i,pp]*n[i]
for bnd_idx, psi_inc in self.BC.incoming_fluxes.iteritems():
if psi_inc.shape != (self.DD.M, self.DD.G):
coupled_solver_error(__file__,
"define boundary terms",
"Incoming flux with incorrect number of groups and directions specified: "+
"{}, expected ({}, {})".format(psi_inc.shape, self.DD.M, self.DD.G))
for g in range(self.DD.G):
self.bnd_vector_form[g] += \
ufl.conditional(omega_p_dot_n < 0,
omega_p_dot_n*self.tensors.Wp[pp]*psi_inc[pp,g]*self.v[g][pp]*ds(bnd_idx),
ufl.zero()) # FIXME: This assumes zero adjoint outgoing flux
else: # Apply vacuum b.c. everywhere
ds = Measure("ds")
for pp in range(self.DD.M):
omega_p_dot_n = self.DD.ordinates_matrix[i,pp]*n[i]
for g in range(self.DD.G):
self.bnd_matrix_form[g] += abs(omega_p_dot_n)*self.tensors.Wp[pp]*self.u[g][pp]*self.v[g][pp]*ds()
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) == 18)
Xr1, Xr2, Xs, m, h, j, d, f, f2, fCass, s, r, Ca_SR, Ca_i, Ca_ss,\
R_prime, 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"]
Ca_o = self._parameters["Ca_o"]
Na_o = self._parameters["Na_o"]
F = self._parameters["F"]
R = self._parameters["R"]
T = self._parameters["T"]
K_o = self._parameters["K_o"]
# Init return args
current = [ufl.zero()] * 1
# 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 Slow time dependent potassium current component
i_Ks = g_Ks * (Xs * Xs) * (-E_Ks + V)
# Expressions for the Fast sodium current component
i_Na = g_Na * (m * m * m) * (-E_Na + V) * h * 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)*(-15 + V)*(0.25*Ca_ss*ufl.exp(F*(-30 +\
2*V)/(R*T)) - Ca_o)*d*f*f2*fCass/(R*T*(-1 + ufl.exp(F*(-30 +\
2*V)/(R*T))))
# 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 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 Membrane component
i_Stim = 0
current[0] = -i_CaL - i_Ks - i_NaCa - i_b_Na - i_Stim - i_Kr - i_p_Ca\
- i_to - i_b_Ca - i_Na - i_p_K - i_NaK - i_K1
# Return results
return current[0]
def _butcher_scheme_generator_tlm(a, b, c, time, solution, rhs_form, perturbation):
"""
Generates a list of forms and solutions for a given Butcher tableau
*Arguments*
a (2 dimensional numpy array)
The a matrix of the Butcher tableau.
b (1-2 dimensional numpy array)
The b vector of the Butcher tableau. If b is 2 dimensional the
scheme includes an error estimator and can be used in adaptive
solvers.
c (1 dimensional numpy array)
The c vector the Butcher tableau.
time (_Constant_)
A Constant holding the time at the start of the time step
solution (_Function_)
The prognostic variable
rhs_form (ufl.Form)
A UFL form representing the rhs for a time differentiated equation
perturbation (_Function_)
The perturbation in the initial condition of the solution
"""
a = _check_abc(a, b, c)
size = a.shape[0]
DX = _check_form(rhs_form)
# Get test function
arguments = rhs_form.arguments()
coefficients = rhs_form.coefficients()
v = arguments[0]
# Create time step
dt = Constant(0.1)
# rhs forms
dolfin_stage_forms = []
ufl_stage_forms = []
# Stage solutions
k = [Function(solution.function_space(), name="k_%d"%i) for i in range(size)]
kdot = [Function(solution.function_space(), name="kdot_%d"%i) \
for i in range(size)]
# Create the stage forms
y_ = solution
time_ = time
time_dep_expressions = _time_dependent_expressions(rhs_form, time)
zero_ = ufl.zero(*y_.ufl_shape)
forward_forms = []
stage_solutions = []
jacobian_indices = []
for i, ki in enumerate(k):
# Check whether the stage is explicit
explicit = a[i,i] == 0
# Evaluation arguments for the ith stage
evalargs = y_ + dt * sum([float(a[i,j]) * k[j] \
for j in range(i+1)], zero_)
time = time_ + dt*c[i]
replace_dict = _replace_dict_time_dependent_expression(time_dep_expressions,
time_, dt, c[i])
replace_dict[y_] = evalargs
replace_dict[time_] = time
stage_form = ufl.replace(rhs_form, replace_dict)
forward_forms.append(stage_form)
# The recomputation of the forward run:
if explicit:
stage_forms = [stage_form]
jacobian_indices.append(-1)
else:
# Create a F=0 form and differentiate it
stage_form_implicit = stage_form - ufl.inner(ki, v)*DX
stage_forms = [stage_form_implicit, derivative(stage_form_implicit, ki)]
jacobian_indices.append(0)
ufl_stage_forms.append(stage_forms)
dolfin_stage_forms.append([Form(form) for form in stage_forms])
stage_solutions.append(ki)
# And now the tangent linearisation:
stage_form_tlm = safe_action(derivative(stage_form, y_), perturbation) + \
sum([dt*float(a[i,j]) * safe_action(derivative(\
forward_forms[j], y_), kdot[j]) for j in range(i+1)])
if explicit:
stage_forms_tlm = [stage_form_tlm]
jacobian_indices.append(-1)
else:
# Create a F=0 form and differentiate it
stage_form_tlm -= ufl.inner(kdot[i], v)*DX
stage_forms_tlm = [stage_form_tlm, derivative(stage_form_tlm, kdot[i])]
jacobian_indices.append(1)
ufl_stage_forms.append(stage_forms_tlm)
dolfin_stage_forms.append([Form(form) for form in stage_forms_tlm])
stage_solutions.append(kdot[i])
# Only one last stage
if len(b.shape) == 1:
last_stage = Form(ufl.inner(perturbation + sum(\
[dt*float(bi)*kdoti for bi, kdoti in zip(b, kdot)], zero_), v)*DX)
else:
raise Exception("Not sure what to do here")
human_form = []
for i in range(size):
kterm = " + ".join("%sh*k_%s" % ("" if a[i,j] == 1.0 else \
"%s*"% a[i,j], j) \
for j in range(size) if a[i,j] != 0)
if c[i] in [0.0, 1.0]:
cih = " + h" if c[i] == 1.0 else ""
else:
cih = " + %s*h" % c[i]
kdotterm = " + ".join("%(a)sh*action(derivative(f(t_n%(cih)s, y_n + "\
"%(kterm)s), kdot_%(i)s" % \
{"a": ("" if a[i,j] == 1.0 else "%s*"% a[i,j], j),
"i": i,
"cih": cih,
"kterm": kterm} \
for j in range(size) if a[i,j] != 0)
if len(kterm) == 0:
human_form.append("k_%(i)s = f(t_n%(cih)s, y_n)" % {"i": i, "cih": cih})
human_form.append("kdot_%(i)s = action(derivative("\
"f(t_n%(cih)s, y_n), y_n), ydot_n)" % \
{"i": i, "cih": cih})
else:
human_form.append("k_%(i)s = f(t_n%(cih)s, y_n + %(kterm)s)" % \
{"i": i, "cih": cih, "kterm": kterm})
human_form.append("kdot_%(i)s = action(derivative(f(t_n%(cih)s, "\
"y_n + %(kterm)s), y_n) + %(kdotterm)s" % \
{"i": i, "cih": cih, "kterm": kterm, "kdotterm": kdotterm})
parentheses = "(%s)" if np.sum(b>0) > 1 else "%s"
human_form.append("ydot_{n+1} = ydot_n + h*" + parentheses % (" + ".join(\
"%skdot_%s" % ("" if b[i] == 1.0 else "%s*" % b[i], i) \
for i in range(size) if b[i] > 0)))
human_form = "\n".join(human_form)
return ufl_stage_forms, dolfin_stage_forms, jacobian_indices, last_stage, \
stage_solutions, dt, human_form, perturbation
def assemble_algebraic_system(self):
if self.verb > 1: print0(self.print_prefix + "Assembling algebraic system.")
mat_timer = Timer("---- MTX: Complete construction")
ass_timer = Timer("---- MTX: Assembling")
add_values_A = False
add_values_B = False
add_values_Q = False
for gto in range(self.DD.G):
#=============================== LOOP OVER GROUPS AND ASSEMBLE ================================
spc = self.print_prefix + " "
if self.verb > 3 and self.DD.G > 1:
print spc + 'GROUP [', gto, ',', gto, '] :'
pres_fiss = self.PD.get_xs('chi', self.chi, gto)
self.PD.get_xs('D', self.D, gto)
self.PD.get_xs('St', self.R, gto)
i,j,p,q,k1,k2 = ufl.indices(6)
form = ( self.D*self.tensors.T[p,q,i,j]*self.u[gto][q].dx(j)*self.v[gto][p].dx(i) +
self.R*self.tensors.W[p,q]*self.u[gto][q]*self.v[gto][p] ) * dx + self.bnd_matrix_form[gto]
ass_timer.start()
assemble(form, tensor=self.A, finalize_tensor=False, add_values=add_values_A)
add_values_A = True
ass_timer.stop()
if self.fixed_source_problem:
if self.PD.isotropic_source_everywhere:
self.PD.get_Q(self.fixed_source, 0, gto)
# FIXME: This assumes that adjoint source == forward source
form = self.fixed_source * self.tensors.Wp[p] * self.v[gto][p] * dx + self.bnd_vector_form[gto]
else:
form = ufl.zero()
for n in range(self.DD.M):
self.PD.get_Q(self.fixed_source, n, gto)
# FIXME: This assumes that adjoint source == forward source
form += self.fixed_source[n,gto] * self.tensors.Wp[n] * self.v[gto][n] * dx + self.bnd_vector_form[gto]
ass_timer.start()
assemble(form, tensor=self.Q, finalize_tensor=False, add_values=add_values_Q)
add_values_Q = True
ass_timer.stop()
for gfrom in range(self.DD.G):
if self.verb > 3 and self.DD.G > 1: print spc + 'GROUP [', gto, ',', gfrom, '] :'
pres_Ss = False
# TODO: Enlarge self.S and self.C to (L+1)^2 (or 1./2.*(L+1)*(L+2) in 2D) to accomodate for anisotropic
# scattering (lines below using CC, SS are correct only for L = 0, when the following inner loop runs only
# once.
for l in range(self.L+1):
for m in range(-l, l+1):
if self.DD.angular_quad.get_D() == 2 and divmod(l+m,2)[1] == 0:
continue
pres_Ss |= self.PD.get_xs('Ss', self.S[l], gto, gfrom, l)
self.PD.get_xs('C', self.C[l], gto, gfrom, l)
if pres_Ss:
Sd = ufl.diag(self.S)
SS = self.tensors.QT[p,k1]*Sd[k1,k2]*self.tensors.Q[k2,q]
Cd = ufl.diag(self.C)
CC = self.tensors.QtT[p,i,k1]*Cd[k1,k2]*self.tensors.Qt[k2,q,j]
ass_timer.start()
form = ( CC[p,i,q,j]*self.u[gfrom][q].dx(j)*self.v[gto][p].dx(i) -
SS[q,p]*self.u[gfrom][q]*self.v[gto][p] ) * dx
assemble(form, tensor=self.A, finalize_tensor=False, add_values=add_values_A)
ass_timer.stop()
if pres_fiss:
pres_nSf = self.PD.get_xs('nSf', self.R, gfrom)
if pres_nSf:
ass_timer.start()
if self.fixed_source_problem:
form = -self.chi*self.R/(4*numpy.pi)*\
self.tensors.QT[p,0]*self.tensors.Q[0,q]*self.u[gfrom][q]*self.v[gto][p]*dx
assemble(form, tensor=self.A, finalize_tensor=False, add_values=add_values_A)
else:
form = self.chi*self.R/(4*numpy.pi)*\
self.tensors.QT[p,0]*self.tensors.Q[0,q]*self.u[gfrom][q]*self.v[gto][p]*dx
assemble(form, tensor=self.B, finalize_tensor=False, add_values=add_values_B)
add_values_B = True
ass_timer.stop()
#============================= END LOOP OVER GROUPS AND ASSEMBLE ===============================
self.A.apply("add")
if self.fixed_source_problem:
self.Q.apply("add")
elif self.eigenproblem:
self.B.apply("add")
def F(self, V, s, time=None):
m, h, n, NKo, NKi, NNao, NNai, NClo, NCli, voli, O = s
O = dolfin.conditional(dolfin.ge(O, 0), O, 0)
G_K = self._parameters["G_K"]
G_Na = self._parameters["G_Na"]
G_ClL = self._parameters["G_ClL"]
G_Kl = self._parameters["G_KL"]
G_NaL = self._parameters["G_NaL"]
C = self._parameters["C"]
# Ion Concentration related Parameters
eps_K = self._parameters["eps_K"]
G_glia = self._parameters["G_glia"]
eps_O = self._parameters["eps_O"]
Ukcc2 = self._parameters["Ukcc2"]
Unkcc1 = self._parameters["Unkcc1"]
rho_max = self._parameters["rho_max"]
# Volume
vol = 1.4368e-15 # unit:m^3, when r=7 um,v=1.4368e-15 m^3 # TODO: add to parameters
beta0 = self._parameters["beta0"]
# Time Constant
tau = self._parameters["tau"]
KBath = self._parameters["KBath"]
OBath = self._parameters["OBath"]
gamma = self._gamma(voli)
volo = (1 + 1 / beta0) * vol - voli # Extracellular volume
beta = voli / volo # Ratio of intracelluar to extracelluar volume
# Gating variables
alpha_m = 0.32 * (54 + V) / (1 - exp(-(V + 54) / 4))
beta_m = 0.28 * (V + 27) / (exp((V + 27) / 5) - 1)
alpha_h = 0.128 * exp(-(V + 50) / 18)
beta_h = 4 / (1 + exp(-(V + 27) / 5))
alpha_n = 0.032 * (V + 52) / (1 - exp(-(V + 52) / 5))
beta_n = 0.5 * exp(-(V + 57) / 40)
dotm = alpha_m * (1 - m) - beta_m * m
doth = alpha_h * (1 - h) - beta_h * h
dotn = alpha_n * (1 - n) - beta_n * n
Ko, Ki, Nao, Nai, Clo, Cli = self._get_concentrations(V, s)
fo = 1 / (1 + exp((2.5 - OBath) / 0.2))
fv = 1 / (1 + exp((beta - 20) / 2))
eps_K *= fo * fv
G_glia *= fo
rho = rho_max / (1 + exp((20 - O) / 3)) / gamma
I_glia = G_glia / (1 + exp((18 - Ko) / 2.5))
Igliapump = rho / 3 / (1 + exp((25 - 18) / 3)) / (1 + exp(3.5 - Ko))
I_diff = eps_K * (Ko - KBath) + I_glia + 2 * Igliapump * gamma
I_K = self._I_K(V, n, Ko, Ki)
I_Na = self._I_Na(V, m, h, Nao, Nai)
I_Cl = self._I_Cl(V, Clo, Cli)
I_pump = self._I_pump(Nai, Ko, O, gamma)
# Cloride transporter (mM/s)
fKo = 1 / (1 + exp(16 - Ko))
FKCC2 = Ukcc2 * ln((Ki * Cli) / (Ko * Clo))
FNKCC1 = Unkcc1 * fKo * (ln((Ki * Cli) / (Ko * Clo)) + ln(
(Nai * Cli) / (Nao * Clo)))
dotNKo = tau * volo * (gamma * beta * (I_K - 2.0 * I_pump) - I_diff +
FKCC2 * beta + FNKCC1 * beta)
dotNKi = -tau * voli * (gamma * (I_K - 2.0 * I_pump) + FKCC2 + FNKCC1)
dotNNao = tau * volo * beta * (gamma * (I_Na + 3.0 * I_pump) + FNKCC1)
dotNNai = -tau * voli * (gamma * (I_Na + 3.0 * I_pump) + FNKCC1)
dotNClo = beta * volo * tau * (FKCC2 - gamma * I_Cl + 2 * FNKCC1)
dotNCli = voli * tau * (gamma * I_Cl - FKCC2 - 2 * FNKCC1)
r1 = vol / voli
r2 = 1 / beta0 * vol / ((1 + 1 / beta0) * vol - voli)
pii = Nai + Cli + Ki + 132 * r1
pio = Nao + Ko + Clo + 18 * r2
vol_hat = vol * (1.1029 - 0.1029 * exp((pio - pii) / 20))
dotVoli = -(voli - vol_hat) / 0.25 * tau
dotO = tau * (-5.3 * (I_pump + Igliapump) * gamma + eps_O *
(OBath - O))
F_expressions = [ufl.zero()] * self.num_states()
F_expressions[0] = dotm
F_expressions[1] = doth
F_expressions[2] = dotn
F_expressions[3] = dotNKo
F_expressions[4] = dotNKi
F_expressions[5] = dotNNao
F_expressions[6] = dotNNai
F_expressions[7] = dotNClo
F_expressions[8] = dotNCli
F_expressions[9] = dotVoli
F_expressions[10] = dotO
return dolfin.as_vector(F_expressions)
from dolfin import * from ufl import zero mesh = UnitSquareMesh(10, 10) V = FunctionSpace(mesh, 'CG', 1) u = TrialFunction(V) v = TestFunction(V) cs = map(Constant, range(4)) forms = [c * inner(u, v) * dx for c in cs] f0 = sum(forms) f1 = sum(forms, 0) f2 = sum(forms, zero()) f3 = sum(forms[1:], forms[0]) iforms = iter(forms) f4 = sum(iforms, next(iforms)) print len(set([f.signature() for f in (f0, f1, f2, f3, f4)]))
def _butcher_scheme_generator(a, b, c, time, solution, rhs_form):
"""
Generates a list of forms and solutions for a given Butcher tableau
*Arguments*
a (2 dimensional numpy array)
The a matrix of the Butcher tableau.
b (1-2 dimensional numpy array)
The b vector of the Butcher tableau. If b is 2 dimensional the
scheme includes an error estimator and can be used in adaptive
solvers.
c (1 dimensional numpy array)
The c vector the Butcher tableau.
time (_Constant_)
A Constant holding the time at the start of the time step
solution (_Function_)
The prognostic variable
rhs_form (ufl.Form)
A UFL form representing the rhs for a time differentiated equation
"""
a = _check_abc(a, b, c)
size = a.shape[0]
DX = _check_form(rhs_form)
# Get test function
arguments, coefficients = ufl.algorithms.\
extract_arguments_and_coefficients(rhs_form)
v = arguments[0]
# Create time step
dt = Constant(0.1)
# rhs forms
dolfin_stage_forms = []
ufl_stage_forms = []
# Stage solutions
k = [
Function(solution.function_space(), name="k_%d" % i)
for i in range(size)
]
jacobian_indices = []
# Create the stage forms
y_ = solution
time_ = time
time_dep_expressions = _time_dependent_expressions(rhs_form, time)
zero_ = ufl.zero(*y_.shape())
for i, ki in enumerate(k):
# Check whether the stage is explicit
explicit = a[i, i] == 0
# Evaluation arguments for the ith stage
evalargs = y_ + dt * sum([float(a[i,j]) * k[j] \
for j in range(i+1)], zero_)
time = time_ + dt * c[i]
replace_dict = _replace_dict_time_dependent_expression(time_dep_expressions, \
time_, dt, c[i])
replace_dict[y_] = evalargs
replace_dict[time_] = time
stage_form = ufl.replace(rhs_form, replace_dict)
if explicit:
stage_forms = [stage_form]
jacobian_indices.append(-1)
else:
# Create a F=0 form and differentiate it
stage_form -= ufl.inner(ki, v) * DX
stage_forms = [stage_form, derivative(stage_form, ki)]
jacobian_indices.append(0)
ufl_stage_forms.append(stage_forms)
dolfin_stage_forms.append([Form(form) for form in stage_forms])
# Only one last stage
if len(b.shape) == 1:
last_stage = Form(ufl.inner(y_+sum([dt*float(bi)*ki for bi, ki in \
zip(b, k)], zero_), v)*DX)
else:
# FIXME: Add support for addaptivity in RKSolver and MultiStageScheme
last_stage = [Form(ufl.inner(y_+sum([dt*float(bi)*ki for bi, ki in \
zip(b[0,:], k)], zero_), v)*DX),
Form(ufl.inner(y_+sum([dt*float(bi)*ki for bi, ki in \
zip(b[1,:], k)], zero_), v)*DX)]
# Create the Function holding the solution at end of time step
#k.append(solution.copy())
# Generate human form of MultiStageScheme
human_form = []
for i in range(size):
kterm = " + ".join("%sh*k_%s" % ("" if a[i,j] == 1.0 else \
"%s*"% a[i,j], j) \
for j in range(size) if a[i,j] != 0)
if c[i] in [0.0, 1.0]:
cih = " + h" if c[i] == 1.0 else ""
else:
cih = " + %s*h" % c[i]
if len(kterm) == 0:
human_form.append("k_%(i)s = f(t_n%(cih)s, y_n)" % {
"i": i,
"cih": cih
})
else:
human_form.append("k_%(i)s = f(t_n%(cih)s, y_n + %(kterm)s)" % \
{"i": i, "cih": cih, "kterm": kterm})
parentheses = "(%s)" if np.sum(b > 0) > 1 else "%s"
human_form.append("y_{n+1} = y_n + h*" + parentheses % (" + ".join(\
"%sk_%s" % ("" if b[i] == 1.0 else "%s*" % b[i], i) \
for i in range(size) if b[i] > 0)))
human_form = "\n".join(human_form)
return ufl_stage_forms, dolfin_stage_forms, jacobian_indices, last_stage, \
k, dt, human_form, None
def F(self, v, s, time=None):
"""
Right hand side for ODE system
"""
time = time if time else Constant(0.0)
# Assign states
V_m = v
assert (len(s) == 38)
h, j, m, x_kr, x_ks, x_to_f, x_to_s, y_to_f, y_to_s, d, f, f_Ca_Bj,\
f_Ca_Bsl, Ry_Ri, Ry_Ro, Ry_Rr, Na_Bj, Na_Bsl, CaM, Myo_c, Myo_m,\
SRB, Tn_CHc, Tn_CHm, Tn_CL, SLH_j, SLH_sl, SLL_j, SLL_sl, Ca_sr,\
Csqn_b, Na_i, Na_j, Na_sl, K_i, Ca_i, Ca_j, Ca_sl = s
# Assign parameters
Fjunc = self._parameters["Fjunc"]
Fjunc_CaL = self._parameters["Fjunc_CaL"]
cellLength = self._parameters["cellLength"]
cellRadius = self._parameters["cellRadius"]
GNa = self._parameters["GNa"]
GNaB = self._parameters["GNaB"]
IbarNaK = self._parameters["IbarNaK"]
KmKo = self._parameters["KmKo"]
KmNaip = self._parameters["KmNaip"]
Q10CaL = self._parameters["Q10CaL"]
pCa = self._parameters["pCa"]
pNa = self._parameters["pNa"]
IbarNCX = self._parameters["IbarNCX"]
Kdact = self._parameters["Kdact"]
KmCai = self._parameters["KmCai"]
KmCao = self._parameters["KmCao"]
KmNai = self._parameters["KmNai"]
KmNao = self._parameters["KmNao"]
Q10NCX = self._parameters["Q10NCX"]
ksat = self._parameters["ksat"]
nu = self._parameters["nu"]
IbarSLCaP = self._parameters["IbarSLCaP"]
KmPCa = self._parameters["KmPCa"]
Q10SLCaP = self._parameters["Q10SLCaP"]
GCaB = self._parameters["GCaB"]
Kmf = self._parameters["Kmf"]
Kmr = self._parameters["Kmr"]
MaxSR = self._parameters["MaxSR"]
MinSR = self._parameters["MinSR"]
Q10SRCaP = self._parameters["Q10SRCaP"]
Vmax_SRCaP = self._parameters["Vmax_SRCaP"]
ec50SR = self._parameters["ec50SR"]
hillSRCaP = self._parameters["hillSRCaP"]
kiCa = self._parameters["kiCa"]
kim = self._parameters["kim"]
koCa = self._parameters["koCa"]
kom = self._parameters["kom"]
ks = self._parameters["ks"]
Bmax_Naj = self._parameters["Bmax_Naj"]
Bmax_Nasl = self._parameters["Bmax_Nasl"]
koff_na = self._parameters["koff_na"]
kon_na = self._parameters["kon_na"]
Bmax_CaM = self._parameters["Bmax_CaM"]
Bmax_SR = self._parameters["Bmax_SR"]
Bmax_TnChigh = self._parameters["Bmax_TnChigh"]
Bmax_TnClow = self._parameters["Bmax_TnClow"]
Bmax_myosin = self._parameters["Bmax_myosin"]
koff_cam = self._parameters["koff_cam"]
koff_myoca = self._parameters["koff_myoca"]
koff_myomg = self._parameters["koff_myomg"]
koff_sr = self._parameters["koff_sr"]
koff_tnchca = self._parameters["koff_tnchca"]
koff_tnchmg = self._parameters["koff_tnchmg"]
koff_tncl = self._parameters["koff_tncl"]
kon_cam = self._parameters["kon_cam"]
kon_myoca = self._parameters["kon_myoca"]
kon_myomg = self._parameters["kon_myomg"]
kon_sr = self._parameters["kon_sr"]
kon_tnchca = self._parameters["kon_tnchca"]
kon_tnchmg = self._parameters["kon_tnchmg"]
kon_tncl = self._parameters["kon_tncl"]
Bmax_SLhighj0 = self._parameters["Bmax_SLhighj0"]
Bmax_SLhighsl0 = self._parameters["Bmax_SLhighsl0"]
Bmax_SLlowj0 = self._parameters["Bmax_SLlowj0"]
Bmax_SLlowsl0 = self._parameters["Bmax_SLlowsl0"]
koff_slh = self._parameters["koff_slh"]
koff_sll = self._parameters["koff_sll"]
kon_slh = self._parameters["kon_slh"]
kon_sll = self._parameters["kon_sll"]
Bmax_Csqn0 = self._parameters["Bmax_Csqn0"]
J_ca_juncsl = self._parameters["J_ca_juncsl"]
J_ca_slmyo = self._parameters["J_ca_slmyo"]
koff_csqn = self._parameters["koff_csqn"]
kon_csqn = self._parameters["kon_csqn"]
J_na_juncsl = self._parameters["J_na_juncsl"]
J_na_slmyo = self._parameters["J_na_slmyo"]
Nao = self._parameters["Nao"]
Ko = self._parameters["Ko"]
Cao = self._parameters["Cao"]
Mgi = self._parameters["Mgi"]
Cmem = self._parameters["Cmem"]
Frdy = self._parameters["Frdy"]
R = self._parameters["R"]
Temp = self._parameters["Temp"]
g_K1_factor = self._parameters["g_K1_factor"]
g_CaL_factor = self._parameters["g_CaL_factor"]
g_Kr_factor = self._parameters["g_Kr_factor"]
g_Ks_factor = self._parameters["g_Ks_factor"]
g_to_factor = self._parameters["g_to_factor"]
SR_Ca_release_ks_factor = self._parameters["SR_Ca_release_ks_factor"]
# Init return args
F_expressions = [ufl.zero()] * 38
# Expressions for the Geometry component
Vcell = 1e-15 * ufl.pi * cellLength * (cellRadius * cellRadius)
Vmyo = 0.65 * Vcell
Vsr = 0.035 * Vcell
Vsl = 0.02 * Vcell
Vjunc = 0.000539 * Vcell
Fsl = 1 - Fjunc
Fsl_CaL = 1 - Fjunc_CaL
# Expressions for the Reversal potentials component
FoRT = Frdy / (R * Temp)
ena_junc = ufl.ln(Nao / Na_j) / FoRT
ena_sl = ufl.ln(Nao / Na_sl) / FoRT
eca_junc = ufl.ln(Cao / Ca_j) / (2 * FoRT)
eca_sl = ufl.ln(Cao / Ca_sl) / (2 * FoRT)
Qpow = -31 + Temp / 10
# Expressions for the I_Na component
mss = 1.0/((1 + 0.00184221158117*ufl.exp(-0.110741971207*V_m))*(1 +\
0.00184221158117*ufl.exp(-0.110741971207*V_m)))
taum = 0.1292*ufl.exp(-((2.94658944659 +\
0.0643500643501*V_m)*(2.94658944659 + 0.0643500643501*V_m))) +\
0.06487*ufl.exp(-((-0.0943466353678 +\
0.0195618153365*V_m)*(-0.0943466353678 + 0.0195618153365*V_m)))
ah = ufl.conditional(ufl.ge(V_m, -40), 0,\
4.43126792958e-07*ufl.exp(-0.147058823529*V_m))
bh = ufl.conditional(ufl.ge(V_m, -40), 0.77/(0.13 +\
0.0497581410839*ufl.exp(-0.0900900900901*V_m)),\
310000.0*ufl.exp(0.3485*V_m) + 2.7*ufl.exp(0.079*V_m))
tauh = 1.0 / (bh + ah)
hss = 1.0/((1 + 15212.5932857*ufl.exp(0.134589502019*V_m))*(1 +\
15212.5932857*ufl.exp(0.134589502019*V_m)))
aj = ufl.conditional(ufl.ge(V_m, -40), 0, (37.78 +\
V_m)*(-25428.0*ufl.exp(0.2444*V_m) -\
6.948e-06*ufl.exp(-0.04391*V_m))/(1 +\
50262745826.0*ufl.exp(0.311*V_m)))
bj = ufl.conditional(ufl.ge(V_m, -40), 0.6*ufl.exp(0.057*V_m)/(1 +\
0.0407622039784*ufl.exp(-0.1*V_m)),\
0.02424*ufl.exp(-0.01052*V_m)/(1 +\
0.0039608683399*ufl.exp(-0.1378*V_m)))
tauj = 1.0 / (bj + aj)
jss = 1.0/((1 + 15212.5932857*ufl.exp(0.134589502019*V_m))*(1 +\
15212.5932857*ufl.exp(0.134589502019*V_m)))
F_expressions[2] = (-m + mss) / taum
F_expressions[0] = (hss - h) / tauh
F_expressions[1] = (-j + jss) / tauj
I_Na_junc = Fjunc * GNa * (m * m * m) * (-ena_junc + V_m) * h * j
I_Na_sl = GNa * (m * m * m) * (-ena_sl + V_m) * Fsl * h * j
# Expressions for the I_NaBK component
I_nabk_junc = Fjunc * GNaB * (-ena_junc + V_m)
I_nabk_sl = GNaB * (-ena_sl + V_m) * Fsl
# Expressions for the I_NaK component
sigma = -1 / 7 + ufl.exp(0.0148588410104 * Nao) / 7
fnak = 1.0/(1 + 0.1245*ufl.exp(-0.1*FoRT*V_m) +\
0.0365*ufl.exp(-FoRT*V_m)*sigma)
I_nak_junc = Fjunc*IbarNaK*Ko*fnak/((1 + ufl.elem_pow(KmNaip,\
4)/ufl.elem_pow(Na_j, 4))*(KmKo + Ko))
I_nak_sl = IbarNaK*Ko*Fsl*fnak/((1 + ufl.elem_pow(KmNaip,\
4)/ufl.elem_pow(Na_sl, 4))*(KmKo + Ko))
# Expressions for the I_Kr component
xrss = 1.0 / (1 + ufl.exp(-2 - V_m / 5))
tauxr = 230/(1 + ufl.exp(2 + V_m/20)) + 3300/((1 + ufl.exp(-22/9 -\
V_m/9))*(1 + ufl.exp(11/9 + V_m/9)))
F_expressions[3] = (-x_kr + xrss) / tauxr
# Expressions for the I_Ks component
xsss = 1.0 / (1 + 0.765928338365 * ufl.exp(-0.0701754385965 * V_m))
tauxs = 990.1 / (1 + 0.841540408868 * ufl.exp(-0.070821529745 * V_m))
F_expressions[4] = (-x_ks + xsss) / tauxs
# Expressions for the I_to component
xtoss = 1.0 / (1 + ufl.exp(19 / 13 - V_m / 13))
ytoss = 1.0 / (1 + 49.4024491055 * ufl.exp(V_m / 5))
tauxtos = 0.5 + 9 / (1 + ufl.exp(1 / 5 + V_m / 15))
tauytos = 30 + 800 / (1 + ufl.exp(6 + V_m / 10))
F_expressions[6] = (-x_to_s + xtoss) / tauxtos
F_expressions[8] = (ytoss - y_to_s) / tauytos
tauxtof = 0.5 + 8.5 * ufl.exp(-((9 / 10 + V_m / 50) *
(9 / 10 + V_m / 50)))
tauytof = 7 + 85 * ufl.exp(-((40 + V_m) * (40 + V_m)) / 220)
F_expressions[5] = (xtoss - x_to_f) / tauxtof
F_expressions[7] = (ytoss - y_to_f) / tauytof
# Expressions for the I_Ca component
fss = 0.6 / (1 + ufl.exp(5 / 2 - V_m / 20)) + 1.0 / (
1 + ufl.exp(35 / 9 + V_m / 9))
dss = 1.0 / (1 + ufl.exp(-5 / 6 - V_m / 6))
taud = (1 - ufl.exp(-5 / 6 - V_m / 6)) * dss / (0.175 + 0.035 * V_m)
tauf = 1.0/(0.02 + 0.0197*ufl.exp(-((0.48865 + 0.0337*V_m)*(0.48865 +\
0.0337*V_m))))
F_expressions[9] = (-d + dss) / taud
F_expressions[10] = (fss - f) / tauf
F_expressions[11] = 1.7 * (1 - f_Ca_Bj) * Ca_j - 0.0119 * f_Ca_Bj
F_expressions[12] = -0.0119 * f_Ca_Bsl + 1.7 * (1 - f_Ca_Bsl) * Ca_sl
fcaCaMSL = 0
fcaCaj = 0
ibarca_j = 4*Frdy*pCa*(-0.341*Cao +\
0.341*Ca_j*ufl.exp(2*FoRT*V_m))*FoRT*V_m/(-1 +\
ufl.exp(2*FoRT*V_m))
ibarca_sl = 4*Frdy*pCa*(-0.341*Cao +\
0.341*Ca_sl*ufl.exp(2*FoRT*V_m))*FoRT*V_m/(-1 +\
ufl.exp(2*FoRT*V_m))
ibarna_j = Frdy*pNa*(-0.75*Nao +\
0.75*Na_j*ufl.exp(FoRT*V_m))*FoRT*V_m/(-1 + ufl.exp(FoRT*V_m))
ibarna_sl = Frdy*pNa*(0.75*Na_sl*ufl.exp(FoRT*V_m) -\
0.75*Nao)*FoRT*V_m/(-1 + ufl.exp(FoRT*V_m))
I_Ca_junc = g_CaL_factor*0.45*Fjunc_CaL*ufl.elem_pow(Q10CaL, Qpow)*(1 - f_Ca_Bj +\
fcaCaj)*d*f*ibarca_j
I_Ca_sl = g_CaL_factor*0.45*ufl.elem_pow(Q10CaL, Qpow)*(1 - f_Ca_Bsl +\
fcaCaMSL)*Fsl_CaL*d*f*ibarca_sl
I_CaNa_junc = g_CaL_factor*0.45*Fjunc_CaL*ufl.elem_pow(Q10CaL, Qpow)*(1 - f_Ca_Bj\
+ fcaCaj)*d*f*ibarna_j
I_CaNa_sl = g_CaL_factor*0.45*ufl.elem_pow(Q10CaL, Qpow)*(1 - f_Ca_Bsl +\
fcaCaMSL)*Fsl_CaL*d*f*ibarna_sl
# Expressions for the I_NCX component
Ka_junc = 1.0 / (1 + (Kdact * Kdact) / (Ca_j * Ca_j))
Ka_sl = 1.0 / (1 + (Kdact * Kdact) / (Ca_sl * Ca_sl))
s1_junc = Cao * (Na_j * Na_j * Na_j) * ufl.exp(nu * FoRT * V_m)
s1_sl = Cao * (Na_sl * Na_sl * Na_sl) * ufl.exp(nu * FoRT * V_m)
s2_junc = (Nao * Nao * Nao) * Ca_j * ufl.exp((-1 + nu) * FoRT * V_m)
s3_junc = KmCao*(Na_j*Na_j*Na_j) + (Nao*Nao*Nao)*Ca_j +\
Cao*(Na_j*Na_j*Na_j) + KmCai*(Nao*Nao*Nao)*(1 +\
(Na_j*Na_j*Na_j)/(KmNai*KmNai*KmNai)) + (KmNao*KmNao*KmNao)*(1 +\
Ca_j/KmCai)*Ca_j
s2_sl = (Nao * Nao * Nao) * Ca_sl * ufl.exp((-1 + nu) * FoRT * V_m)
s3_sl = KmCai*(Nao*Nao*Nao)*(1 +\
(Na_sl*Na_sl*Na_sl)/(KmNai*KmNai*KmNai)) + (Nao*Nao*Nao)*Ca_sl +\
(KmNao*KmNao*KmNao)*(1 + Ca_sl/KmCai)*Ca_sl +\
Cao*(Na_sl*Na_sl*Na_sl) + KmCao*(Na_sl*Na_sl*Na_sl)
I_ncx_junc = Fjunc*IbarNCX*ufl.elem_pow(Q10NCX, Qpow)*(-s2_junc +\
s1_junc)*Ka_junc/((1 + ksat*ufl.exp((-1 + nu)*FoRT*V_m))*s3_junc)
I_ncx_sl = IbarNCX*ufl.elem_pow(Q10NCX, Qpow)*(-s2_sl +\
s1_sl)*Fsl*Ka_sl/((1 + ksat*ufl.exp((-1 + nu)*FoRT*V_m))*s3_sl)
# Expressions for the I_PCa component
I_pca_junc = Fjunc*IbarSLCaP*ufl.elem_pow(Q10SLCaP,\
Qpow)*ufl.elem_pow(Ca_j, 1.6)/(ufl.elem_pow(Ca_j, 1.6) +\
ufl.elem_pow(KmPCa, 1.6))
I_pca_sl = IbarSLCaP*ufl.elem_pow(Q10SLCaP, Qpow)*ufl.elem_pow(Ca_sl,\
1.6)*Fsl/(ufl.elem_pow(Ca_sl, 1.6) + ufl.elem_pow(KmPCa, 1.6))
# Expressions for the I_CaBK component
I_cabk_junc = Fjunc * GCaB * (-eca_junc + V_m)
I_cabk_sl = GCaB * (-eca_sl + V_m) * Fsl
# Expressions for the SR Fluxes component
kCaSR = MaxSR - (-MinSR + MaxSR) / (1 +
ufl.elem_pow(ec50SR / Ca_sr, 2.5))
koSRCa = koCa / kCaSR
kiSRCa = kiCa * kCaSR
RI = 1 - Ry_Ro - Ry_Ri - Ry_Rr
F_expressions[15] = -(Ca_j*Ca_j)*Ry_Rr*koSRCa + kom*Ry_Ro + kim*RI -\
Ca_j*Ry_Rr*kiSRCa
F_expressions[14] = -kom*Ry_Ro - Ca_j*Ry_Ro*kiSRCa + kim*Ry_Ri +\
(Ca_j*Ca_j)*Ry_Rr*koSRCa
F_expressions[13] = -kim*Ry_Ri + Ca_j*Ry_Ro*kiSRCa - kom*Ry_Ri +\
(Ca_j*Ca_j)*RI*koSRCa
J_SRCarel = SR_Ca_release_ks_factor * ks * (Ca_sr - Ca_j) * Ry_Ro
J_serca = Vmax_SRCaP*ufl.elem_pow(Q10SRCaP,\
Qpow)*(-ufl.elem_pow(Ca_sr/Kmr, hillSRCaP) +\
ufl.elem_pow(Ca_i/Kmf, hillSRCaP))/(1 + ufl.elem_pow(Ca_sr/Kmr,\
hillSRCaP) + ufl.elem_pow(Ca_i/Kmf, hillSRCaP))
J_SRleak = 5.348e-06 * Ca_sr - 5.348e-06 * Ca_j
# Expressions for the Na Buffers component
F_expressions[16] = -koff_na * Na_Bj + kon_na * (-Na_Bj +
Bmax_Naj) * Na_j
F_expressions[17] = kon_na * (-Na_Bsl +
Bmax_Nasl) * Na_sl - koff_na * Na_Bsl
# Expressions for the Cytosolic Ca Buffers component
F_expressions[24] = kon_tncl*(Bmax_TnClow - Tn_CL)*Ca_i -\
koff_tncl*Tn_CL
F_expressions[22] = -koff_tnchca*Tn_CHc + kon_tnchca*(-Tn_CHc +\
Bmax_TnChigh - Tn_CHm)*Ca_i
F_expressions[23] = Mgi*kon_tnchmg*(-Tn_CHc + Bmax_TnChigh - Tn_CHm)\
- koff_tnchmg*Tn_CHm
F_expressions[18] = kon_cam * (-CaM + Bmax_CaM) * Ca_i - koff_cam * CaM
F_expressions[19] = -koff_myoca*Myo_c + kon_myoca*(-Myo_c +\
Bmax_myosin - Myo_m)*Ca_i
F_expressions[20] = Mgi*kon_myomg*(-Myo_c + Bmax_myosin - Myo_m) -\
koff_myomg*Myo_m
F_expressions[21] = kon_sr * (Bmax_SR - SRB) * Ca_i - koff_sr * SRB
J_CaB_cytosol = -koff_tnchca*Tn_CHc - koff_myoca*Myo_c +\
Mgi*kon_myomg*(-Myo_c + Bmax_myosin - Myo_m) +\
Mgi*kon_tnchmg*(-Tn_CHc + Bmax_TnChigh - Tn_CHm) -\
koff_tnchmg*Tn_CHm + kon_tncl*(Bmax_TnClow - Tn_CL)*Ca_i +\
kon_sr*(Bmax_SR - SRB)*Ca_i - koff_myomg*Myo_m + kon_cam*(-CaM +\
Bmax_CaM)*Ca_i - koff_cam*CaM - koff_tncl*Tn_CL +\
kon_myoca*(-Myo_c + Bmax_myosin - Myo_m)*Ca_i +\
kon_tnchca*(-Tn_CHc + Bmax_TnChigh - Tn_CHm)*Ca_i - koff_sr*SRB
# Expressions for the Junctional and SL Ca Buffers component
Bmax_SLlowsl = Bmax_SLlowsl0 * Vmyo / Vsl
Bmax_SLlowj = Bmax_SLlowj0 * Vmyo / Vjunc
Bmax_SLhighsl = Bmax_SLhighsl0 * Vmyo / Vsl
Bmax_SLhighj = Bmax_SLhighj0 * Vmyo / Vjunc
F_expressions[27] = kon_sll * (Bmax_SLlowj -
SLL_j) * Ca_j - koff_sll * SLL_j
F_expressions[28] = kon_sll*(-SLL_sl + Bmax_SLlowsl)*Ca_sl -\
koff_sll*SLL_sl
F_expressions[25] = kon_slh*(Bmax_SLhighj - SLH_j)*Ca_j -\
koff_slh*SLH_j
F_expressions[26] = kon_slh*(-SLH_sl + Bmax_SLhighsl)*Ca_sl -\
koff_slh*SLH_sl
J_CaB_junction = kon_slh*(Bmax_SLhighj - SLH_j)*Ca_j +\
kon_sll*(Bmax_SLlowj - SLL_j)*Ca_j - koff_slh*SLH_j -\
koff_sll*SLL_j
J_CaB_sl = kon_sll*(-SLL_sl + Bmax_SLlowsl)*Ca_sl + kon_slh*(-SLH_sl\
+ Bmax_SLhighsl)*Ca_sl - koff_sll*SLL_sl - koff_slh*SLH_sl
# Expressions for the SR Ca Concentrations component
Bmax_Csqn = Bmax_Csqn0 * Vmyo / Vsr
F_expressions[30] = -koff_csqn*Csqn_b + kon_csqn*(Bmax_Csqn -\
Csqn_b)*Ca_sr
F_expressions[29] = -kon_csqn*(Bmax_Csqn - Csqn_b)*Ca_sr -\
J_SRleak*Vmyo/Vsr + koff_csqn*Csqn_b - J_SRCarel + J_serca
# Expressions for the Na Concentrations component
I_Na_tot_junc = 3*I_nak_junc + 3*I_ncx_junc + I_CaNa_junc + I_Na_junc\
+ I_nabk_junc
I_Na_tot_sl = I_Na_sl + I_nabk_sl + 3 * I_nak_sl + I_CaNa_sl + 3 * I_ncx_sl
F_expressions[32] = -Cmem*I_Na_tot_junc/(Frdy*Vjunc) +\
J_na_juncsl*(Na_sl - Na_j)/Vjunc - F_expressions[16]
F_expressions[33] = -F_expressions[17] + J_na_slmyo*(-Na_sl +\
Na_i)/Vsl + J_na_juncsl*(-Na_sl + Na_j)/Vsl -\
Cmem*I_Na_tot_sl/(Frdy*Vsl)
F_expressions[31] = J_na_slmyo * (Na_sl - Na_i) / Vmyo
# Expressions for the K Concentration component
F_expressions[34] = Constant(0.0)
# Expressions for the Ca Concentrations component
I_Ca_tot_junc = I_pca_junc + I_cabk_junc + I_Ca_junc - 2 * I_ncx_junc
I_Ca_tot_sl = -2 * I_ncx_sl + I_pca_sl + I_cabk_sl + I_Ca_sl
F_expressions[36] = J_ca_juncsl*(Ca_sl - Ca_j)/Vjunc +\
J_SRCarel*Vsr/Vjunc - J_CaB_junction -\
Cmem*I_Ca_tot_junc/(2*Frdy*Vjunc) + J_SRleak*Vmyo/Vjunc
F_expressions[37] = -J_CaB_sl + J_ca_juncsl*(Ca_j - Ca_sl)/Vsl -\
Cmem*I_Ca_tot_sl/(2*Frdy*Vsl) + J_ca_slmyo*(Ca_i - Ca_sl)/Vsl
F_expressions[35] = -J_CaB_cytosol + J_ca_slmyo*(Ca_sl - Ca_i)/Vmyo -\
J_serca*Vsr/Vmyo
# Return results
return dolfin.as_vector(F_expressions)
def _butcher_scheme_generator_adm(a, b, c, time, solution, rhs_form, adj):
"""
Generates a list of forms and solutions for a given Butcher tableau
*Arguments*
a (2 dimensional numpy array)
The a matrix of the Butcher tableau.
b (1-2 dimensional numpy array)
The b vector of the Butcher tableau. If b is 2 dimensional the
scheme includes an error estimator and can be used in adaptive
solvers.
c (1 dimensional numpy array)
The c vector the Butcher tableau.
time (_Constant_)
A Constant holding the time at the start of the time step
solution (_Function_)
The prognostic variable
rhs_form (ufl.Form)
A UFL form representing the rhs for a time differentiated equation
adj (_Function_)
The derivative of the functional with respect to y_n+1
"""
a = _check_abc(a, b, c)
size = a.shape[0]
DX = _check_form(rhs_form)
# Get test function
arguments, coefficients = ufl.algorithms.\
extract_arguments_and_coefficients(rhs_form)
v = arguments[0]
# Create time step
dt = Constant(0.1)
# rhs forms
dolfin_stage_forms = []
ufl_stage_forms = []
# Stage solutions
k = [
Function(solution.function_space(), name="k_%d" % i)
for i in range(size)
]
kbar = [Function(solution.function_space(), name="kbar_%d"%i) \
for i in range(size)]
# Create the stage forms
y_ = solution
ydot = solution.copy()
time_ = time
time_dep_expressions = _time_dependent_expressions(rhs_form, time)
zero_ = ufl.zero(*y_.shape())
forward_forms = []
stage_solutions = []
jacobian_indices = []
# The recomputation of the forward run:
for i, ki in enumerate(k):
# Check whether the stage is explicit
explicit = a[i, i] == 0
# Evaluation arguments for the ith stage
evalargs = y_ + dt * sum([float(a[i,j]) * k[j] \
for j in range(i+1)], zero_)
time = time_ + dt * c[i]
replace_dict = _replace_dict_time_dependent_expression(\
time_dep_expressions, time_, dt, c[i])
replace_dict[y_] = evalargs
replace_dict[time_] = time
stage_form = ufl.replace(rhs_form, replace_dict)
forward_forms.append(stage_form)
if explicit:
stage_forms = [stage_form]
jacobian_indices.append(-1)
else:
# Create a F=0 form and differentiate it
stage_form_implicit = stage_form - ufl.inner(ki, v) * DX
stage_forms = [stage_form_implicit, derivative(\
stage_form_implicit, ki)]
jacobian_indices.append(0)
ufl_stage_forms.append(stage_forms)
dolfin_stage_forms.append([Form(form) for form in stage_forms])
stage_solutions.append(ki)
for i, kbari in reversed(list(enumerate(kbar))):
# Check whether the stage is explicit
explicit = a[i, i] == 0
# And now the adjoint linearisation:
stage_form_adm = ufl.inner(dt * b[i] * adj, v)*DX + sum(\
[dt * float(a[j,i]) * safe_action(safe_adjoint(derivative(\
forward_forms[j], y_)), kbar[j]) for j in range(i, size)])
if explicit:
stage_forms_adm = [stage_form_adm]
jacobian_indices.append(-1)
else:
# Create a F=0 form and differentiate it
stage_form_adm -= ufl.inner(kbar[i], v) * DX
stage_forms_adm = [
stage_form_adm,
derivative(stage_form_adm, kbari)
]
jacobian_indices.append(1)
ufl_stage_forms.append(stage_forms_adm)
dolfin_stage_forms.append([Form(form) for form in stage_forms_adm])
stage_solutions.append(kbari)
# Only one last stage
if len(b.shape) == 1:
last_stage = Form(ufl.inner(adj, v)*DX + sum(\
[safe_action(safe_adjoint(derivative(forward_forms[i], y_)), kbar[i]) \
for i in range(size)]))
else:
raise Exception("Not sure what to do here")
human_form = "unimplemented"
return ufl_stage_forms, dolfin_stage_forms, jacobian_indices, last_stage,\
stage_solutions, dt, human_form, adj
def F(self, V, s, time=None):
"""ds/dt = F(v, s)."""
GCa = self._parameters["GCa"]
gamma1 = self._parameters["gamma1"]
Gglia = self._parameters["Gglia"]
tau = self._parameters["tau"]
control = self._parameters["control"]
GNa = self._parameters["GNa"]
GNaL = self._parameters["GNaL"]
GK = self._parameters["GK"]
GKL = self._parameters["GKL"]
GAHP = self._parameters["GAHP"]
Koinf = self._parameters["Koinf"]
# Define some parameters
ECa = 120
phi = 3
rho = 1.25
eps0 = 1.2
beta0 = 7
Cli = 6
Clo = 130
Nao = 144 - beta0*(s[5] - 18)
ENa = 26.64*df.ln(Nao/s[5])
Ki = 140 + (18 - s[5])
EK = 26.64*df.ln(control*s[4]/Ki)
INa = GNa*s[0]**3*s[2]*(V - ENa) + GNaL*(V - ENa)
IK = (GK*s[1]**4 + GAHP*s[3]/(1 + s[3]) + GKL)*(V - EK)
a_m = (3.0 + (0.1)*V)*(1 - df.exp(-3 -1/10*V))**(-1)
b_m = 4*df.exp(-55/18 - 1/18*V)
ah = (0.07)*df.exp(-11/5 - 1/20*V)
bh = (1 + df.exp(-7/5 - 1/10*V))**(-1)
an = (0.34 + (0.01)*V)*(1 - df.exp(-17/5 - 1/10*V))**(-1)
bn = (0.125)*df.exp(-11/20 - 1/80*V)
taum = (a_m + b_m)**(-1)
minf = a_m*(a_m + b_m)**(-1)
h_inf = (bh + ah)**(-1)*ah
tauh = (bh + ah)**(-1)
ninf = (an + bn)**(-1)*an
taun = (an + bn)**(-1)
dot_m = phi*(minf - s[0])*taum**(-1)
dot_h = phi*(ninf - s[1])/taun
dot_n = phi*(h_inf - s[2])/tauh
Ipump = rho*(1/(1 + df.exp((25 - s[5])/3)))*(1/(1 + df.exp(5.5 - s[4])))
IGlia = Gglia/(1 + df.exp((18 - s[4])/2.5))
Idiff = eps0*(s[4] - Koinf)
dot_Ca = -s[3]/80 - 0.002*GCa*(V - ECa)/(1 + df.exp(-(V + 25)/2.5))
dot_K = (gamma1*beta0*IK - 2*beta0*Ipump - IGlia - Idiff)/tau
# dot_Na = (gamma1*INa + 3*Ipump)/tau
dot_Na = -(gamma1*INa + 3*Ipump)/tau # Pretty sure it should me minus
F_expressions = [ufl.zero()]*self.num_states()
F_expressions[0] = dot_m
F_expressions[1] = dot_h
F_expressions[2] = dot_n
F_expressions[3] = dot_Ca
F_expressions[4] = dot_K
F_expressions[5] = dot_Na
return df.as_vector(F_expressions)
def solve_group_GS(self, it=0, init_slns_ary=None):
if self.verb > 1: print0(self.print_prefix + "Solving..." )
if self.eigenproblem:
coupled_solver_error(__file__,
"solve using group GS",
"Group Gauss-Seidel for eigenproblem not yet supported")
sol_timer = Timer("-- Complete solution")
mat_timer = Timer("---- MTX: Complete construction")
ass_timer = Timer("---- MTX: Assembling")
sln_timer = Timer("---- SOL: Solving")
# To simplify the weak forms
u = self.u[0]
v = self.v[0]
if init_slns_ary is None:
init_slns_ary = numpy.zeros((self.DD.G, self.local_sln_size))
for g in range(self.DD.G):
self.slns_mg[g].vector()[:] = init_slns_ary[g]
err = 0.
for gsi in range(self.max_group_GS_it):
#========================================== GAUSS-SEIDEL LOOP ============================================
if self.verb > 2: print self.print_prefix + 'Gauss-Seidel iteration {}'.format(gsi)
for gto in range(self.DD.G):
#========================================= LOOP OVER GROUPS ===============================================
self.sln_vec = self.slns_mg[gto].vector()
prev_slng_vec = self.aux_slng.vector()
prev_slng_vec.zero()
prev_slng_vec.axpy(1.0, self.sln_vec)
prev_slng_vec.apply("insert")
spc = self.print_prefix + " "
if self.verb > 3 and self.DD.G > 1: print spc + 'GROUP [', gto, ',', gto, '] :'
#==================================== ASSEMBLE WITHIN-GROUP PROBLEM ========================================
mat_timer.start()
pres_fiss = self.PD.get_xs('chi', self.chi, gto)
self.PD.get_xs('D', self.D, gto)
self.PD.get_xs('St', self.R, gto)
i,j,p,q,k1,k2 = ufl.indices(6)
form = ( self.D*self.tensors.T[p,q,i,j]*u[q].dx(j)*v[p].dx(i) +
self.R*self.tensors.W[p,q]*u[q]*v[p] ) * dx + self.bnd_matrix_form[gto]
ass_timer.start()
add_values_A = False
add_values_Q = False
assemble(form, tensor=self.A, finalize_tensor=False, add_values=add_values_A)
add_values_A = True
ass_timer.stop()
if self.fixed_source_problem:
if self.PD.isotropic_source_everywhere:
self.PD.get_Q(self.fixed_source, 0, gto)
# FIXME: This assumes that adjoint source == forward source
form = self.fixed_source*self.tensors.Wp[p]*v[p]*dx + self.bnd_vector_form[gto]
else:
form = ufl.zero()
for n in range(self.DD.M):
self.PD.get_Q(self.fixed_source, n, gto)
# FIXME: This assumes that adjoint source == forward source
form += self.fixed_source[n,gto] * self.tensors.Wp[n] * v[n] * dx + self.bnd_vector_form[gto]
ass_timer.start()
assemble(form, tensor=self.Q, finalize_tensor=False, add_values=add_values_Q)
add_values_Q = True
ass_timer.stop()
for gfrom in range(self.DD.G):
if self.verb > 3 and self.DD.G > 1:
print spc + 'GROUP [', gto, ',', gfrom, '] :'
pres_Ss = False
# TODO: Enlarge self.S and self.C to (L+1)^2 (or 1./2.*(L+1)*(L+2) in 2D) to accomodate for anisotropic
# scattering (lines below using CC, SS are correct only for L = 0, when the following inner loop runs only
# once.
for l in range(self.L+1):
for m in range(-l, l+1):
if self.DD.angular_quad.get_D() == 2 and divmod(l+m,2)[1] == 0:
continue
pres_Ss |= self.PD.get_xs('Ss', self.S[l], gto, gfrom, l)
self.PD.get_xs('C', self.C[l], gto, gfrom, l)
if pres_Ss:
Sd = ufl.diag(self.S)
SS = self.tensors.QT[p,k1]*Sd[k1,k2]*self.tensors.Q[k2,q]
Cd = ufl.diag(self.C)
CC = self.tensors.QtT[p,i,k1]*Cd[k1,k2]*self.tensors.Qt[k2,q,j]
ass_timer.start()
if gfrom != gto:
form = ( SS[p,q]*self.slns_mg[gfrom][q]*v[p] - CC[p,i,q,j]*self.slns_mg[gfrom][q].dx(j)*v[p].dx(i) ) * dx
assemble(form, tensor=self.Q, finalize_tensor=False, add_values=add_values_Q)
else:
form = ( CC[p,i,q,j]*u[q].dx(j)*v[p].dx(i) - SS[q,p]*u[q]*v[p] ) * dx
assemble(form, tensor=self.A, finalize_tensor=False, add_values=add_values_A)
ass_timer.stop()
if pres_fiss:
pres_nSf = self.PD.get_xs('nSf', self.R, gfrom)
if pres_nSf:
ass_timer.start()
if gfrom != gto:
form = self.chi*self.R/(4*numpy.pi)*\
self.tensors.QT[p,0]*self.tensors.Q[0,q]*self.slns_mg[gfrom][q]*v[p]*dx
assemble(form, tensor=self.Q, finalize_tensor=False, add_values=add_values_Q)
else:
# NOTE: Fixed-source case (eigenproblems can currently be solved only by the coupled-group scheme)
if self.fixed_source_problem:
form = -self.chi*self.R/(4*numpy.pi)*self.tensors.QT[p,0]*self.tensors.Q[0,q]*u[q]*v[p]*dx
assemble(form, tensor=self.A, finalize_tensor=False, add_values=add_values_A)
ass_timer.stop()
#================================== END ASSEMBLE WITHIN-GROUP PROBLEM =======================================
self.A.apply("add")
self.Q.apply("add")
mat_timer.stop()
self.save_algebraic_system({'A':'A_{}'.format(gto), 'Q':'Q_{}'.format(gto)}, it)
#==================================== SOLVE WITHIN-GROUP PROBLEM ==========================================
sln_timer.start()
dolfin_solve(self.A, self.sln_vec, self.Q, "cg", "petsc_amg")
sln_timer.stop()
self.up_to_date["flux"] = False
err = max(err, delta(self.sln_vec.array(), prev_slng_vec.array()))
#==================================== END LOOP OVER GROUPS ====================================
if err < self.parameters["group_GS"]["tol"]:
break
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 _butcher_scheme_generator(a, b, c, time, solution, rhs_form):
"""
Generates a list of forms and solutions for a given Butcher tableau
*Arguments*
a (2 dimensional numpy array)
The a matrix of the Butcher tableau.
b (1-2 dimensional numpy array)
The b vector of the Butcher tableau. If b is 2 dimensional the
scheme includes an error estimator and can be used in adaptive
solvers.
c (1 dimensional numpy array)
The c vector the Butcher tableau.
time (_Constant_)
A Constant holding the time at the start of the time step
solution (_Function_)
The prognostic variable
rhs_form (ufl.Form)
A UFL form representing the rhs for a time differentiated equation
"""
a = _check_abc(a, b, c)
size = a.shape[0]
DX = _check_form(rhs_form)
# Get test function
arguments = rhs_form.arguments()
coefficients = rhs_form.coefficients()
v = arguments[0]
# Create time step
dt = Constant(0.1)
# rhs forms
dolfin_stage_forms = []
ufl_stage_forms = []
# Stage solutions
k = [Function(solution.function_space(), name="k_%d"%i) for i in range(size)]
jacobian_indices = []
# Create the stage forms
y_ = solution
time_ = time
time_dep_expressions = _time_dependent_expressions(rhs_form, time)
zero_ = ufl.zero(*y_.ufl_shape)
for i, ki in enumerate(k):
# Check whether the stage is explicit
explicit = a[i,i] == 0
# Evaluation arguments for the ith stage
evalargs = y_ + dt * sum([float(a[i,j]) * k[j] \
for j in range(i+1)], zero_)
time = time_ + dt*c[i]
replace_dict = _replace_dict_time_dependent_expression(time_dep_expressions,
time_, dt, c[i])
replace_dict[y_] = evalargs
replace_dict[time_] = time
stage_form = ufl.replace(rhs_form, replace_dict)
if explicit:
stage_forms = [stage_form]
jacobian_indices.append(-1)
else:
# Create a F=0 form and differentiate it
stage_form -= ufl.inner(ki, v)*DX
stage_forms = [stage_form, derivative(stage_form, ki)]
jacobian_indices.append(0)
ufl_stage_forms.append(stage_forms)
dolfin_stage_forms.append([Form(form) for form in stage_forms])
# Only one last stage
if len(b.shape) == 1:
last_stage = Form(ufl.inner(y_+sum([dt*float(bi)*ki for bi, ki in \
zip(b, k)], zero_), v)*DX)
else:
# FIXME: Add support for adaptivity in RKSolver and MultiStageScheme
last_stage = [Form(ufl.inner(y_+sum([dt*float(bi)*ki for bi, ki in \
zip(b[0,:], k)], zero_), v)*DX),
Form(ufl.inner(y_+sum([dt*float(bi)*ki for bi, ki in \
zip(b[1,:], k)], zero_), v)*DX)]
# Create the Function holding the solution at end of time step
#k.append(solution.copy())
# Generate human form of MultiStageScheme
human_form = []
for i in range(size):
kterm = " + ".join("%sh*k_%s" % ("" if a[i,j] == 1.0 else \
"%s*"% a[i,j], j) \
for j in range(size) if a[i,j] != 0)
if c[i] in [0.0, 1.0]:
cih = " + h" if c[i] == 1.0 else ""
else:
cih = " + %s*h" % c[i]
if len(kterm) == 0:
human_form.append("k_%(i)s = f(t_n%(cih)s, y_n)" % {"i": i, "cih": cih})
else:
human_form.append("k_%(i)s = f(t_n%(cih)s, y_n + %(kterm)s)" % \
{"i": i, "cih": cih, "kterm": kterm})
parentheses = "(%s)" if np.sum(b>0) > 1 else "%s"
human_form.append("y_{n+1} = y_n + h*" + parentheses % (" + ".join(\
"%sk_%s" % ("" if b[i] == 1.0 else "%s*" % b[i], i) \
for i in range(size) if b[i] > 0)))
human_form = "\n".join(human_form)
return ufl_stage_forms, dolfin_stage_forms, jacobian_indices, last_stage, \
k, dt, human_form, None
def _butcher_scheme_generator(a, b, c, solution, rhs_form):
"""
Generates a list of forms and solutions for a given Butcher tableau
*Arguments*
a (2 dimensional numpy array)
The a matrix of the Butcher tableau.
b (1-2 dimensional numpy array)
The b vector of the Butcher tableau. If b is 2 dimensional the
scheme includes an error estimator and can be used in adaptive
solvers.
c (1 dimensional numpy array)
The c vector the Butcher tableau.
solution (_Function_)
The prognastic variable
rhs_form (ufl.Form)
A UFL form representing the rhs for a time differentiated equation
"""
if not (isinstance(a, np.ndarray) and (len(a) == 1 or \
(len(a.shape)==2 and a.shape[0] == a.shape[1]))):
raise TypeError("Expected an m x m numpy array as the first argument")
if not (isinstance(b, np.ndarray) and len(b.shape) in [1,2]):
raise TypeError("Expected a 1 or 2 dimensional numpy array as the second argument")
if not (isinstance(c, np.ndarray) and len(c.shape) == 1):
raise TypeError("Expected a 1 dimensional numpy array as the third argument")
# Make sure a is a "matrix"
if len(a) == 1:
a.shape = (1, 1)
# Get size of system
size = a.shape[0]
# If b is a matrix we expect it to have two rows
if len(b.shape) == 2:
if not (b.shape[0] == 2 and b.shape[1] == size):
raise ValueError("Expected a 2 row matrix with the same number "\
"of collumns as the first dimension of the a matrix.")
elif len(b) != size:
raise ValueError("Expected the length of the b vector to have the "\
"same size as the first dimension of the a matrix.")
if len(c) != size:
raise ValueError("Expected the length of the c vector to have the "\
"same size as the first dimension of the a matrix.")
# Check if tableau is fully implicit
for i in range(size):
for j in range(i):
if a[j, i] != 0:
raise ValueError("Does not support fully implicit Butcher tableau.")
if not isinstance(rhs_form, ufl.Form):
raise TypeError("Expected a ufl.Form as the 5th argument.")
# Check if form contains a cell or point integral
if "cell" in rhs_form.integral_groups():
DX = ufl.dx
elif "point" in rhs_form.integral_groups():
DX = ufl.dP
else:
raise ValueError("Expected either a cell or point integral in the form.")
# Get test function
arguments, coefficients = ufl.algorithms.extract_arguments_and_coefficients(rhs_form)
if len(arguments) != 1:
raise ValueError("Expected the form to have rank 1")
v = arguments[0]
# Create time step
dt = Constant(0.1)
# rhs forms
dolfin_stage_forms = []
ufl_stage_forms = []
# Stage solutions
k = [solution.copy(deepcopy=True) for i in range(size)]
# Create the stage forms
y_ = solution
for i, ki in enumerate(k):
# Check wether the stage is explicit
explicit = a[i,i] == 0
# Evaluation arguments for the ith stage
evalargs = y_ + dt * sum([float(a[i,j]) * k[j] \
for j in range(i+1)], ufl.zero(*y_.shape()))
stage_form = ufl.replace(rhs_form, {y_:evalargs})
if explicit:
stage_forms = [stage_form]
else:
# Create a F=0 form and differentiate it
stage_form -= ufl.inner(ki, v)*DX
stage_forms = [stage_form, derivative(stage_form, ki)]
ufl_stage_forms.append(stage_forms)
dolfin_stage_forms.append([Form(form) for form in stage_forms])
# Only one last stage
if len(b.shape) == 1:
last_stage = cpp.FunctionAXPY([(float(bi), ki) for bi, ki in zip(b, k)])
else:
# FIXME: Add support for addaptivity in RKSolver and MultiStageScheme
last_stage = [cpp.FunctionAXPY([(float(bi), ki) for bi, ki in zip(b[0,:], k)]),
cpp.FunctionAXPY([(float(bi), ki) for bi, ki in zip(b[1,:], k)])]
# Create the Function holding the solution at end of time step
#k.append(solution.copy())
# Generate human form of MultiStageScheme
human_form = []
for i in range(size):
kterm = " + ".join("%sh*k_%s" % ("" if a[i,j] == 1.0 else \
"%s*"% a[i,j], j) \
for j in range(size) if a[i,j] != 0)
if c[i] in [0.0, 1.0]:
cih = " + h" if c[i] == 1.0 else ""
else:
cih = " + %s*h" % c[i]
if len(kterm) == 0:
human_form.append("k_%(i)s = f(t_n%(cih)s, y_n)" % {"i": i, "cih": cih})
else:
human_form.append("k_%(i)s = f(t_n%(cih)s, y_n + %(kterm)s)" % \
{"i": i, "cih": cih, "kterm": kterm})
parentheses = "(%s)" if np.sum(b>0) > 1 else "%s"
human_form.append("y_{n+1} = y_n + h*" + parentheses % (" + ".join(\
"%sk_%s" % ("" if b[i] == 1.0 else "%s*" % b[i], i) \
for i in range(size) if b[i] > 0)))
human_form = "\n".join(human_form)
return ufl_stage_forms, dolfin_stage_forms, last_stage, k, dt, human_form
def _butcher_scheme_generator_adm(a, b, c, time, solution, rhs_form, adj):
"""
Generates a list of forms and solutions for a given Butcher tableau
*Arguments*
a (2 dimensional numpy array)
The a matrix of the Butcher tableau.
b (1-2 dimensional numpy array)
The b vector of the Butcher tableau. If b is 2 dimensional the
scheme includes an error estimator and can be used in adaptive
solvers.
c (1 dimensional numpy array)
The c vector the Butcher tableau.
time (_Constant_)
A Constant holding the time at the start of the time step
solution (_Function_)
The prognostic variable
rhs_form (ufl.Form)
A UFL form representing the rhs for a time differentiated equation
adj (_Function_)
The derivative of the functional with respect to y_n+1
"""
a = _check_abc(a, b, c)
size = a.shape[0]
DX = _check_form(rhs_form)
# Get test function
arguments = rhs_form.arguments()
coefficients = rhs_form.coefficients()
v = arguments[0]
# Create time step
dt = Constant(0.1)
# rhs forms
dolfin_stage_forms = []
ufl_stage_forms = []
# Stage solutions
k = [Function(solution.function_space(), name="k_%d"%i) for i in range(size)]
kbar = [Function(solution.function_space(), name="kbar_%d"%i) \
for i in range(size)]
# Create the stage forms
y_ = solution
time_ = time
time_dep_expressions = _time_dependent_expressions(rhs_form, time)
zero_ = ufl.zero(*y_.ufl_shape)
forward_forms = []
stage_solutions = []
jacobian_indices = []
# The recomputation of the forward run:
for i, ki in enumerate(k):
# Check whether the stage is explicit
explicit = a[i,i] == 0
# Evaluation arguments for the ith stage
evalargs = y_ + dt * sum([float(a[i,j]) * k[j] \
for j in range(i+1)], zero_)
time = time_ + dt*c[i]
replace_dict = _replace_dict_time_dependent_expression(\
time_dep_expressions, time_, dt, c[i])
replace_dict[y_] = evalargs
replace_dict[time_] = time
stage_form = ufl.replace(rhs_form, replace_dict)
forward_forms.append(stage_form)
if explicit:
stage_forms = [stage_form]
jacobian_indices.append(-1)
else:
# Create a F=0 form and differentiate it
stage_form_implicit = stage_form - ufl.inner(ki, v)*DX
stage_forms = [stage_form_implicit, derivative(\
stage_form_implicit, ki)]
jacobian_indices.append(0)
ufl_stage_forms.append(stage_forms)
dolfin_stage_forms.append([Form(form) for form in stage_forms])
stage_solutions.append(ki)
for i, kbari in reversed(list(enumerate(kbar))):
# Check whether the stage is explicit
explicit = a[i,i] == 0
# And now the adjoint linearisation:
stage_form_adm = ufl.inner(dt * b[i] * adj, v)*DX + sum(\
[dt * float(a[j,i]) * safe_action(safe_adjoint(derivative(\
forward_forms[j], y_)), kbar[j]) for j in range(i, size)])
if explicit:
stage_forms_adm = [stage_form_adm]
jacobian_indices.append(-1)
else:
# Create a F=0 form and differentiate it
stage_form_adm -= ufl.inner(kbar[i], v)*DX
stage_forms_adm = [stage_form_adm, derivative(stage_form_adm, kbari)]
jacobian_indices.append(1)
ufl_stage_forms.append(stage_forms_adm)
dolfin_stage_forms.append([Form(form) for form in stage_forms_adm])
stage_solutions.append(kbari)
# Only one last stage
if len(b.shape) == 1:
last_stage = Form(ufl.inner(adj, v)*DX + sum(\
[safe_action(safe_adjoint(derivative(forward_forms[i], y_)), kbar[i]) \
for i in range(size)]))
else:
raise Exception("Not sure what to do here")
human_form = "unimplemented"
return ufl_stage_forms, dolfin_stage_forms, jacobian_indices, last_stage,\
stage_solutions, dt, human_form, adj
def _I(self, v, s, time):
"""
Original gotran transmembrane current dV/dt
"""
time = time if time else Constant(0.0)
# Assign states
V_m = v
assert (len(s) == 38)
h, j, m, x_kr, x_ks, x_to_f, x_to_s, y_to_f, y_to_s, d, f, f_Ca_Bj,\
f_Ca_Bsl, Ry_Ri, Ry_Ro, Ry_Rr, Na_Bj, Na_Bsl, CaM, Myo_c, Myo_m,\
SRB, Tn_CHc, Tn_CHm, Tn_CL, SLH_j, SLH_sl, SLL_j, SLL_sl, Ca_sr,\
Csqn_b, Na_i, Na_j, Na_sl, K_i, Ca_i, Ca_j, Ca_sl = s
# Assign parameters
Fjunc = self._parameters["Fjunc"]
Fjunc_CaL = self._parameters["Fjunc_CaL"]
GNa = self._parameters["GNa"]
GNaB = self._parameters["GNaB"]
IbarNaK = self._parameters["IbarNaK"]
KmKo = self._parameters["KmKo"]
KmNaip = self._parameters["KmNaip"]
gkp = self._parameters["gkp"]
pNaK = self._parameters["pNaK"]
epi = self._parameters["epi"]
GClB = self._parameters["GClB"]
GClCa = self._parameters["GClCa"]
KdClCa = self._parameters["KdClCa"]
Q10CaL = self._parameters["Q10CaL"]
pCa = self._parameters["pCa"]
pK = self._parameters["pK"]
pNa = self._parameters["pNa"]
IbarNCX = self._parameters["IbarNCX"]
Kdact = self._parameters["Kdact"]
KmCai = self._parameters["KmCai"]
KmCao = self._parameters["KmCao"]
KmNai = self._parameters["KmNai"]
KmNao = self._parameters["KmNao"]
Q10NCX = self._parameters["Q10NCX"]
ksat = self._parameters["ksat"]
nu = self._parameters["nu"]
IbarSLCaP = self._parameters["IbarSLCaP"]
KmPCa = self._parameters["KmPCa"]
Q10SLCaP = self._parameters["Q10SLCaP"]
GCaB = self._parameters["GCaB"]
Nao = self._parameters["Nao"]
Ko = self._parameters["Ko"]
Cao = self._parameters["Cao"]
Cli = self._parameters["Cli"]
Clo = self._parameters["Clo"]
Frdy = self._parameters["Frdy"]
R = self._parameters["R"]
Temp = self._parameters["Temp"]
g_K1_factor = self._parameters["g_K1_factor"]
g_CaL_factor = self._parameters["g_CaL_factor"]
g_Kr_factor = self._parameters["g_Kr_factor"]
g_Ks_factor = self._parameters["g_Ks_factor"]
g_to_factor = self._parameters["g_to_factor"]
SR_Ca_release_ks_factor = self._parameters["SR_Ca_release_ks_factor"]
# Init return args
current = [ufl.zero()] * 1
# Expressions for the Geometry component
Fsl = 1 - Fjunc
Fsl_CaL = 1 - Fjunc_CaL
# Expressions for the Reversal potentials component
FoRT = Frdy / (R * Temp)
ena_junc = ufl.ln(Nao / Na_j) / FoRT
ena_sl = ufl.ln(Nao / Na_sl) / FoRT
ek = ufl.ln(Ko / K_i) / FoRT
eca_junc = ufl.ln(Cao / Ca_j) / (2 * FoRT)
eca_sl = ufl.ln(Cao / Ca_sl) / (2 * FoRT)
ecl = ufl.ln(Cli / Clo) / FoRT
Qpow = -31 + Temp / 10
# Expressions for the I_Na component
I_Na_junc = Fjunc * GNa * (m * m * m) * (-ena_junc + V_m) * h * j
I_Na_sl = GNa * (m * m * m) * (-ena_sl + V_m) * Fsl * h * j
# Expressions for the I_NaBK component
I_nabk_junc = Fjunc * GNaB * (-ena_junc + V_m)
I_nabk_sl = GNaB * (-ena_sl + V_m) * Fsl
# Expressions for the I_NaK component
sigma = -1 / 7 + ufl.exp(0.0148588410104 * Nao) / 7
fnak = 1.0/(1 + 0.1245*ufl.exp(-0.1*FoRT*V_m) +\
0.0365*ufl.exp(-FoRT*V_m)*sigma)
I_nak_junc = Fjunc*IbarNaK*Ko*fnak/((1 + ufl.elem_pow(KmNaip,\
4)/ufl.elem_pow(Na_j, 4))*(KmKo + Ko))
I_nak_sl = IbarNaK*Ko*Fsl*fnak/((1 + ufl.elem_pow(KmNaip,\
4)/ufl.elem_pow(Na_sl, 4))*(KmKo + Ko))
I_nak = I_nak_junc + I_nak_sl
# Expressions for the I_Kr component
gkr = g_Kr_factor * 0.0150616019019 * ufl.sqrt(Ko)
rkr = 1.0 / (1 + ufl.exp(37 / 12 + V_m / 24))
I_kr = (-ek + V_m) * gkr * rkr * x_kr
# Expressions for the I_Kp component
kp_kp = 1.0 / (1 + 1786.47556538 * ufl.exp(-0.167224080268 * V_m))
I_kp_junc = Fjunc * gkp * (-ek + V_m) * kp_kp
I_kp_sl = gkp * (-ek + V_m) * Fsl * kp_kp
I_kp = I_kp_sl + I_kp_junc
# Expressions for the I_Ks component
eks = ufl.ln((Nao * pNaK + Ko) / (pNaK * Na_i + K_i)) / FoRT
gks_junc = g_Ks_factor * 0.0035
gks_sl = g_Ks_factor * 0.0035
I_ks_junc = Fjunc * gks_junc * (x_ks * x_ks) * (-eks + V_m)
I_ks_sl = gks_sl * (x_ks * x_ks) * (-eks + V_m) * Fsl
I_ks = I_ks_sl + I_ks_junc
# Expressions for the I_to component
GtoSlow = ufl.conditional(ufl.eq(epi, 1), 0.0156, 0.037596)
GtoFast = ufl.conditional(ufl.eq(epi, 1), 0.1144, 0.001404)
I_tos = (-ek + V_m) * GtoSlow * x_to_s * y_to_s
I_tof = (-ek + V_m) * g_to_factor * GtoFast * x_to_f * y_to_f
I_to = I_tos + I_tof
# Expressions for the I_Ki component
aki = 1.02 / (1 +
7.35454251046e-07 * ufl.exp(0.2385 * V_m - 0.2385 * ek))
bki = (0.762624006506*ufl.exp(0.08032*V_m - 0.08032*ek) +\
1.15340563519e-16*ufl.exp(0.06175*V_m - 0.06175*ek))/(1 +\
0.0867722941577*ufl.exp(-0.5143*V_m + 0.5143*ek))
kiss = aki / (aki + bki)
I_ki = g_K1_factor * 0.150616019019 * ufl.sqrt(Ko) * (-ek + V_m) * kiss
# Expressions for the I_ClCa component
I_ClCa_junc = Fjunc * GClCa * (-ecl + V_m) / (1 + KdClCa / Ca_j)
I_ClCa_sl = GClCa * (-ecl + V_m) * Fsl / (1 + KdClCa / Ca_sl)
I_ClCa = I_ClCa_sl + I_ClCa_junc
I_Clbk = GClB * (-ecl + V_m)
# Expressions for the I_Ca component
fcaCaMSL = 0
fcaCaj = 0
ibarca_j = 4*Frdy*pCa*(-0.341*Cao +\
0.341*Ca_j*ufl.exp(2*FoRT*V_m))*FoRT*V_m/(-1 +\
ufl.exp(2*FoRT*V_m))
ibarca_sl = 4*Frdy*pCa*(-0.341*Cao +\
0.341*Ca_sl*ufl.exp(2*FoRT*V_m))*FoRT*V_m/(-1 +\
ufl.exp(2*FoRT*V_m))
ibark = Frdy*pK*(-0.75*Ko + 0.75*K_i*ufl.exp(FoRT*V_m))*FoRT*V_m/(-1 +\
ufl.exp(FoRT*V_m))
ibarna_j = Frdy*pNa*(-0.75*Nao +\
0.75*Na_j*ufl.exp(FoRT*V_m))*FoRT*V_m/(-1 + ufl.exp(FoRT*V_m))
ibarna_sl = Frdy*pNa*(0.75*Na_sl*ufl.exp(FoRT*V_m) -\
0.75*Nao)*FoRT*V_m/(-1 + ufl.exp(FoRT*V_m))
I_Ca_junc = g_CaL_factor*0.45*Fjunc_CaL*ufl.elem_pow(Q10CaL, Qpow)*(1 - f_Ca_Bj +\
fcaCaj)*d*f*ibarca_j
I_Ca_sl = g_CaL_factor*0.45*ufl.elem_pow(Q10CaL, Qpow)*(1 - f_Ca_Bsl +\
fcaCaMSL)*Fsl_CaL*d*f*ibarca_sl
I_CaK = g_CaL_factor*0.45*ufl.elem_pow(Q10CaL, Qpow)*(Fjunc_CaL*(1 - f_Ca_Bj +\
fcaCaj) + (1 - f_Ca_Bsl + fcaCaMSL)*Fsl_CaL)*d*f*ibark
I_CaNa_junc = g_CaL_factor*0.45*Fjunc_CaL*ufl.elem_pow(Q10CaL, Qpow)*(1 - f_Ca_Bj\
+ fcaCaj)*d*f*ibarna_j
I_CaNa_sl = g_CaL_factor*0.45*ufl.elem_pow(Q10CaL, Qpow)*(1 - f_Ca_Bsl +\
fcaCaMSL)*Fsl_CaL*d*f*ibarna_sl
# Expressions for the I_NCX component
Ka_junc = 1.0 / (1 + (Kdact * Kdact) / (Ca_j * Ca_j))
Ka_sl = 1.0 / (1 + (Kdact * Kdact) / (Ca_sl * Ca_sl))
s1_junc = Cao * (Na_j * Na_j * Na_j) * ufl.exp(nu * FoRT * V_m)
s1_sl = Cao * (Na_sl * Na_sl * Na_sl) * ufl.exp(nu * FoRT * V_m)
s2_junc = (Nao * Nao * Nao) * Ca_j * ufl.exp((-1 + nu) * FoRT * V_m)
s3_junc = KmCao*(Na_j*Na_j*Na_j) + (Nao*Nao*Nao)*Ca_j +\
Cao*(Na_j*Na_j*Na_j) + KmCai*(Nao*Nao*Nao)*(1 +\
(Na_j*Na_j*Na_j)/(KmNai*KmNai*KmNai)) + (KmNao*KmNao*KmNao)*(1 +\
Ca_j/KmCai)*Ca_j
s2_sl = (Nao * Nao * Nao) * Ca_sl * ufl.exp((-1 + nu) * FoRT * V_m)
s3_sl = KmCai*(Nao*Nao*Nao)*(1 +\
(Na_sl*Na_sl*Na_sl)/(KmNai*KmNai*KmNai)) + (Nao*Nao*Nao)*Ca_sl +\
(KmNao*KmNao*KmNao)*(1 + Ca_sl/KmCai)*Ca_sl +\
Cao*(Na_sl*Na_sl*Na_sl) + KmCao*(Na_sl*Na_sl*Na_sl)
I_ncx_junc = Fjunc*IbarNCX*ufl.elem_pow(Q10NCX, Qpow)*(-s2_junc +\
s1_junc)*Ka_junc/((1 + ksat*ufl.exp((-1 + nu)*FoRT*V_m))*s3_junc)
I_ncx_sl = IbarNCX*ufl.elem_pow(Q10NCX, Qpow)*(-s2_sl +\
s1_sl)*Fsl*Ka_sl/((1 + ksat*ufl.exp((-1 + nu)*FoRT*V_m))*s3_sl)
# Expressions for the I_PCa component
I_pca_junc = Fjunc*IbarSLCaP*ufl.elem_pow(Q10SLCaP,\
Qpow)*ufl.elem_pow(Ca_j, 1.6)/(ufl.elem_pow(Ca_j, 1.6) +\
ufl.elem_pow(KmPCa, 1.6))
I_pca_sl = IbarSLCaP*ufl.elem_pow(Q10SLCaP, Qpow)*ufl.elem_pow(Ca_sl,\
1.6)*Fsl/(ufl.elem_pow(Ca_sl, 1.6) + ufl.elem_pow(KmPCa, 1.6))
# Expressions for the I_CaBK component
I_cabk_junc = Fjunc * GCaB * (-eca_junc + V_m)
I_cabk_sl = GCaB * (-eca_sl + V_m) * Fsl
# Expressions for the Na Concentrations component
I_Na_tot_junc = 3*I_nak_junc + 3*I_ncx_junc + I_CaNa_junc + I_Na_junc\
+ I_nabk_junc
I_Na_tot_sl = I_Na_sl + I_nabk_sl + 3 * I_nak_sl + I_CaNa_sl + 3 * I_ncx_sl
# Expressions for the K Concentration component
I_K_tot = -2 * I_nak + I_ks + I_CaK + I_kr + I_kp + I_ki + I_to
# Expressions for the Ca Concentrations component
I_Ca_tot_junc = I_pca_junc + I_cabk_junc + I_Ca_junc - 2 * I_ncx_junc
I_Ca_tot_sl = -2 * I_ncx_sl + I_pca_sl + I_cabk_sl + I_Ca_sl
# Expressions for the Membrane potential component
i_Stim = 0
I_Na_tot = I_Na_tot_junc + I_Na_tot_sl
I_Cl_tot = I_Clbk + I_ClCa
I_Ca_tot = I_Ca_tot_junc + I_Ca_tot_sl
I_tot = I_Na_tot + I_K_tot + I_Cl_tot + I_Ca_tot
current[0] = -I_tot - i_Stim
# Return results
return current[0]
