def problem(): mesh = dolfin.UnitCubeMesh(MPI.comm_world, 10, 10, 10) cell = mesh.ufl_cell() vec_element = dolfin.VectorElement("Lagrange", cell, 1) # scl_element = dolfin.FiniteElement("Lagrange", cell, 1) Q = dolfin.FunctionSpace(mesh, vec_element) # Qs = dolfin.FunctionSpace(mesh, scl_element) # Coefficients v = dolfin.function.argument.TestFunction(Q) # Test function du = dolfin.function.argument.TrialFunction(Q) # Incremental displacement u = dolfin.Function(Q) # Displacement from previous iteration B = dolfin.Constant((0.0, -0.5, 0.0), cell) # Body force per unit volume T = dolfin.Constant((0.1, 0.0, 0.0), cell) # Traction force on the boundary # B, T = dolfin.Function(Q), dolfin.Function(Q) # Kinematics d = u.geometric_dimension() F = ufl.Identity(d) + grad(u) # Deformation gradient C = F.T * F # Right Cauchy-Green tensor # Invariants of deformation tensors Ic = tr(C) J = det(F) # Elasticity parameters E, nu = 10.0, 0.3 mu = dolfin.Constant(E / (2 * (1 + nu)), cell) lmbda = dolfin.Constant(E * nu / ((1 + nu) * (1 - 2 * nu)), cell) # mu, lmbda = dolfin.Function(Qs), dolfin.Function(Qs) # Stored strain energy density (compressible neo-Hookean model) psi = (mu / 2) * (Ic - 3) - mu * ln(J) + (lmbda / 2) * (ln(J))**2 # Total potential energy Pi = psi * dx - dot(B, u) * dx - dot(T, u) * ds # Compute first variation of Pi (directional derivative about u in the direction of v) F = ufl.derivative(Pi, u, v) # Compute Jacobian of F J = ufl.derivative(F, u, du) return J, F
def freeEnergy(C, Cv): J = sqrt(det(C)) I1 = tr(C) Ce = C * inv(Cv) Ie1 = tr(Ce) Je = J / sqrt(det(Cv)) psiEq = (3**(1 - alph1) / (2.0 * alph1) * mu1 * (I1**alph1 - 3**alph1) + 3**(1 - alph2) / (2.0 * alph2) * mu2 * (I1**alph2 - 3**alph2) - (mu1 + mu2) * ln(J) + mu_pr / 2 * (J - 1)**2) psiNeq = (3**(1 - a1) / (2.0 * a1) * m1 * (Ie1**a1 - 3**a1) + 3**(1 - a2) / (2.0 * a2) * m2 * (Ie1**a2 - 3**a2) - (m1 + m2) * ln(Je)) return psiEq + psiNeq
def bessk0(x): """ Modified Bessel function of the second kind. Code taken from [Flannery et al. 1992] B.P. Flannery, W.H. Press, S.A. Teukolsky, W. Vetterling, "Numerical recipes in C", Press Syndicate of the University of Cambridge, New York (1992). """ y1 = x * x / 4.0 expr1 = -ufl.ln(x / 2.0) * bessi0(x) + ( -0.57721566 + y1 * (0.42278420 + y1 * (0.23069756 + y1 * (0.3488590e-1 + y1 * (0.262698e-2 + y1 * (0.10750e-3 + y1 * 0.74e-5)))))) y2 = 2.0 / x expr2 = (ufl.exp(-x) / ufl.sqrt(x) * (1.25331414 + y2 * (-0.7832358e-1 + y2 * (0.2189568e-1 + y2 * (-0.1062446e-1 + y2 * (0.587872e-2 + y2 * (-0.251540e-2 + y2 * 0.53208e-3))))))) return ufl.conditional(x > 2, expr2, expr1)
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 hyperelasticity_action_forms(mesh, vec_el): cell = mesh.ufl_cell() Q = dolfin.FunctionSpace(mesh, vec_el) # Coefficients v = dolfin.function.argument.TestFunction(Q) # Test function du = dolfin.function.argument.TrialFunction(Q) # Incremental displacement u = dolfin.Function(Q) # Displacement from previous iteration u.vector().set(0.5) B = dolfin.Constant((0.0, -0.5, 0.0), cell) # Body force per unit volume T = dolfin.Constant((0.1, 0.0, 0.0), cell) # Traction force on the boundary # Kinematics d = u.geometric_dimension() F = ufl.Identity(d) + grad(u) # Deformation gradient C = F.T * F # Right Cauchy-Green tensor # Invariants of deformation tensors Ic = tr(C) J = det(F) # Elasticity parameters E, nu = 10.0, 0.3 mu = dolfin.Constant(E / (2 * (1 + nu)), cell) lmbda = dolfin.Constant(E * nu / ((1 + nu) * (1 - 2 * nu)), cell) # Stored strain energy density (compressible neo-Hookean model) psi = (mu / 2) * (Ic - 3) - mu * ln(J) + (lmbda / 2) * (ln(J)) ** 2 # Total potential energy Pi = psi * dx - dot(B, u) * dx - dot(T, u) * ds # Compute first variation of Pi (directional derivative about u in the direction of v) F = ufl.derivative(Pi, u, v) # Compute Jacobian of F J = ufl.derivative(F, u, du) w = dolfin.Function(Q) w.vector().set(1.2) L = ufl.action(J, w) return None, L, None
def test_dolfin_expression_compilation_of_math_functions(dolfin): # Define some PyDOLFIN coefficients mesh = dolfin.UnitSquareMesh(3, 3) # Using quadratic element deliberately for accuracy V = dolfin.FunctionSpace(mesh, "CG", 2) u = dolfin.Function(V) u.interpolate(dolfin.Expression("x[0]*x[1]")) w0 = u # Define ufl expression with math functions v = abs(ufl.cos(u))/2 + 0.02 uexpr = ufl.sin(u) + ufl.tan(v) + ufl.exp(u) + ufl.ln(v) + ufl.atan(v) + ufl.acos(v) + ufl.asin(v) #print dolfin.assemble(uexpr**2*dolfin.dx, mesh=mesh) # 11.7846508409 # Define expected output from compilation ucode = 'v_w0[0]' vcode = '0.02 + fabs(cos(v_w0[0])) / 2' funcs = 'asin(%(v)s) + (acos(%(v)s) + (atan(%(v)s) + (log(%(v)s) + (exp(%(u)s) + (sin(%(u)s) + tan(%(v)s))))))' oneliner = funcs % {'u':ucode, 'v':vcode} # Oneliner version (ignoring reuse): expected_lines = ['double s[1];', 'Array<double> v_w0(1);', 'w0->eval(v_w0, x);', 's[0] = %s;' % oneliner, 'values[0] = s[0];'] #cppcode = format_dolfin_expression(classname="DebugExpression", shape=(), eval_body=expected_lines) #print '-'*100 #print cppcode #print '-'*100 #dolfin.plot(dolfin.Expression(cppcode=cppcode, mesh=mesh)) #dolfin.interactive() # Split version (handles reuse of v, no other reuse): expected_lines = ['double s[2];', 'Array<double> v_w0(1);', 'w0->eval(v_w0, x);', 's[0] = %s;' % (vcode,), 's[1] = %s;' % (funcs % {'u':ucode,'v':'s[0]'},), 'values[0] = s[1];'] # Define expected evaluation values: [(x,value), (x,value), ...] import math x, y = 0.6, 0.7 u = x*y v = abs(math.cos(u))/2 + 0.02 v0 = .52 expected0 = math.tan(v0) + 1 + math.log(v0) + math.atan(v0) + math.acos(v0) + math.asin(v0) expected = math.sin(u) + math.tan(v) + math.exp(u) + math.log(v) + math.atan(v) + math.acos(v) + math.asin(v) expected_values = [((0.0, 0.0), (expected0,)), ((x, y), (expected,)), ] # Execute all tests check_dolfin_expression_compilation(uexpr, expected_lines, expected_values, members={'w0':w0})
def ogden_vol(self, params, C): kappa = params['kappa'] psi_vol = (kappa / 4.) * (self.IIIc - 2. * ln(sqrt(self.IIIc)) - 1.) S = 2. * diff(psi_vol, C) return S
def test_assign_to_mfs_sub(cg1, vcg1): W = cg1*vcg1 w = Function(W) u = Function(cg1) v = Function(vcg1) u.assign(4) v.assign(10) w.sub(0).assign(u) assert np.allclose(w.sub(0).dat.data_ro, 4) assert np.allclose(w.sub(1).dat.data_ro, 0) w.sub(1).assign(v) assert np.allclose(w.sub(0).dat.data_ro, 4) assert np.allclose(w.sub(1).dat.data_ro, 10) Q = vcg1*cg1 q = Function(Q) q.assign(11) w.sub(1).assign(q.sub(0)) assert np.allclose(w.sub(1).dat.data_ro, 11) assert np.allclose(w.sub(0).dat.data_ro, 4) with pytest.raises(ValueError): w.sub(1).assign(q.sub(1)) with pytest.raises(ValueError): w.sub(1).assign(w.sub(0)) with pytest.raises(ValueError): w.sub(1).assign(u) with pytest.raises(ValueError): w.sub(0).assign(v) w.sub(0).assign(ufl.ln(q.sub(1))) assert np.allclose(w.sub(0).dat.data_ro, ufl.ln(11)) with pytest.raises(ValueError): w.assign(q.sub(1))
def test_assign_to_mfs_sub(cg1, vcg1): W = cg1 * vcg1 w = Function(W) u = Function(cg1) v = Function(vcg1) u.assign(4) v.assign(10) w.sub(0).assign(u) assert np.allclose(w.sub(0).dat.data_ro, 4) assert np.allclose(w.sub(1).dat.data_ro, 0) w.sub(1).assign(v) assert np.allclose(w.sub(0).dat.data_ro, 4) assert np.allclose(w.sub(1).dat.data_ro, 10) Q = vcg1 * cg1 q = Function(Q) q.assign(11) w.sub(1).assign(q.sub(0)) assert np.allclose(w.sub(1).dat.data_ro, 11) assert np.allclose(w.sub(0).dat.data_ro, 4) with pytest.raises(ValueError): w.sub(1).assign(q.sub(1)) with pytest.raises(ValueError): w.sub(1).assign(w.sub(0)) with pytest.raises(ValueError): w.sub(1).assign(u) with pytest.raises(ValueError): w.sub(0).assign(v) w.sub(0).assign(ufl.ln(q.sub(1))) assert np.allclose(w.sub(0).dat.data_ro, ufl.ln(11)) with pytest.raises(ValueError): w.assign(q.sub(1))
def test_latex_formatting_of_cmath(): x = ufl.SpatialCoordinate(ufl.triangle)[0] assert expr2latex(ufl.exp(x)) == r"e^{x_0}" assert expr2latex(ufl.ln(x)) == r"\ln(x_0)" assert expr2latex(ufl.sqrt(x)) == r"\sqrt{x_0}" assert expr2latex(abs(x)) == r"\|x_0\|" assert expr2latex(ufl.sin(x)) == r"\sin(x_0)" assert expr2latex(ufl.cos(x)) == r"\cos(x_0)" assert expr2latex(ufl.tan(x)) == r"\tan(x_0)" assert expr2latex(ufl.asin(x)) == r"\arcsin(x_0)" assert expr2latex(ufl.acos(x)) == r"\arccos(x_0)" assert expr2latex(ufl.atan(x)) == r"\arctan(x_0)"
def test_complex_algebra(self): z1 = ComplexValue(1j) z2 = ComplexValue(1+1j) # Remember that ufl.algebra functions return ComplexValues, but ufl.mathfunctions return complex Python scalar # Any operations with a ComplexValue and a complex Python scalar promote to ComplexValue assert z1*z2 == ComplexValue(-1+1j) assert z2/z1 == ComplexValue(1-1j) assert pow(z2, z1) == ComplexValue((1+1j)**1j) assert sqrt(z2) * as_ufl(1) == ComplexValue(cmath.sqrt(1+1j)) assert ((sin(z2) + cosh(z2) - atan(z2)) * z1) == ComplexValue((cmath.sin(1+1j) + cmath.cosh(1+1j) - cmath.atan(1+1j))*1j) assert (abs(z2) - ln(z2))/exp(z1) == ComplexValue((abs(1+1j) - cmath.log(1+1j))/cmath.exp(1j))
def test_cpp_formatting_of_cmath(): x, y = ufl.SpatialCoordinate(ufl.triangle) # Test cmath functions assert expr2cpp(ufl.exp(x)) == "exp(x[0])" assert expr2cpp(ufl.ln(x)) == "log(x[0])" assert expr2cpp(ufl.sqrt(x)) == "sqrt(x[0])" assert expr2cpp(abs(x)) == "fabs(x[0])" assert expr2cpp(ufl.sin(x)) == "sin(x[0])" assert expr2cpp(ufl.cos(x)) == "cos(x[0])" assert expr2cpp(ufl.tan(x)) == "tan(x[0])" assert expr2cpp(ufl.asin(x)) == "asin(x[0])" assert expr2cpp(ufl.acos(x)) == "acos(x[0])" assert expr2cpp(ufl.atan(x)) == "atan(x[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) == 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 V(self, Us): """Calculate potential due to this group Required argument: Us: an iterable producing the concentrations of the ligands in the group. These may be numbers, but in the most important applcation they are UFL expressions. """ if len(Us) != self.nligands: raise KSDGException('wrong number of ligands %d, should be %d' % (len(Us), self.nligands)) if self.nligands == 0: return (0.0) sU = sum(l.weight * U for l, U in zip(self.ligands, Us)) return (-self.beta * ufl.ln(self.alpha + sU))
def strain_energy(i1, i2, i3): """Strain energy function i1, i2, i3: principal invariants of the Cauchy-Green tensor """ # Determinant of configuration gradient F J = ufl.sqrt(i3) # noqa: F841 # # Classical St. Venant-Kirchhoff # Ψ = la / 8 * (i1 - 3)**2 + mu / 4 * ((i1 - 3)**2 + 4 * (i1 - 3) - 2 * (i2 - 3)) # Modified St. Venant-Kirchhoff # Ψ = la / 2 * (ufl.ln(J))**2 + mu / 4 * ((i1 - 3)**2 + 4 * (i1 - 3) - 2 * (i2 - 3)) # Compressible neo-Hooke Ψ = mu / 2 * (i1 - 3 - 2 * ufl.ln(J)) + la / 2 * (J - 1)**2 # Compressible Mooney-Rivlin (beta = 0) # Ψ = mu / 4 * (i1 - 3) + mu / 4 * (i2 - 3) - mu * ufl.ln(J) + la / 2 * (J - 1)**2 # return Ψ
def Vfunc(U): return -params['beta'] * ufl.ln(U + params['alpha'])
def test_diff_then_integrate(): # Define 1D geometry n = 21 mesh = UnitIntervalMesh(MPI.comm_world, n) # Shift and scale mesh x0, x1 = 1.5, 3.14 mesh.coordinates()[:] *= (x1 - x0) mesh.coordinates()[:] += x0 x = SpatialCoordinate(mesh)[0] xs = 0.1 + 0.8 * x / x1 # scaled to be within [0.1,0.9] # Define list of expressions to test, and configure # accuracies these expressions are known to pass with. # The reason some functions are less accurately integrated is # likely that the default choice of quadrature rule is not perfect F_list = [] def reg(exprs, acc=10): for expr in exprs: F_list.append((expr, acc)) # FIXME: 0*dx and 1*dx fails in the ufl-ffc-jit framework somewhere # reg([Constant(0.0, cell=cell)]) # reg([Constant(1.0, cell=cell)]) monomial_list = [x**q for q in range(2, 6)] reg(monomial_list) reg([2.3 * p + 4.5 * q for p in monomial_list for q in monomial_list]) reg([x**x]) reg([x**(x**2)], 8) reg([x**(x**3)], 6) reg([x**(x**4)], 2) # Special functions: reg([atan(xs)], 8) reg([sin(x), cos(x), exp(x)], 5) reg([ln(xs), pow(x, 2.7), pow(2.7, x)], 3) reg([asin(xs), acos(xs)], 1) reg([tan(xs)], 7) try: import scipy except ImportError: scipy = None if hasattr(math, 'erf') or scipy is not None: reg([erf(xs)]) else: print( "Warning: skipping test of erf, old python version and no scipy.") # if 0: # print("Warning: skipping tests of bessel functions, doesn't build on all platforms.") # elif scipy is None: # print("Warning: skipping tests of bessel functions, missing scipy.") # else: # for nu in (0, 1, 2): # # Many of these are possibly more accurately integrated, # # but 4 covers all and is sufficient for this test # reg([bessel_J(nu, xs), bessel_Y(nu, xs), bessel_I(nu, xs), bessel_K(nu, xs)], 4) # To handle tensor algebra, make an x dependent input tensor # xx and square all expressions def reg2(exprs, acc=10): for expr in exprs: F_list.append((inner(expr, expr), acc)) xx = as_matrix([[2 * x**2, 3 * x**3], [11 * x**5, 7 * x**4]]) x3v = as_vector([3 * x**2, 5 * x**3, 7 * x**4]) cc = as_matrix([[2, 3], [4, 5]]) reg2([xx]) reg2([x3v]) reg2([cross(3 * x3v, as_vector([-x3v[1], x3v[0], x3v[2]]))]) reg2([xx.T]) reg2([tr(xx)]) reg2([det(xx)]) reg2([dot(xx, 0.1 * xx)]) reg2([outer(xx, xx.T)]) reg2([dev(xx)]) reg2([sym(xx)]) reg2([skew(xx)]) reg2([elem_mult(7 * xx, cc)]) reg2([elem_div(7 * xx, xx + cc)]) reg2([elem_pow(1e-3 * xx, 1e-3 * cc)]) reg2([elem_pow(1e-3 * cc, 1e-3 * xx)]) reg2([elem_op(lambda z: sin(z) + 2, 0.03 * xx)], 2) # pretty inaccurate... # FIXME: Add tests for all UFL operators: # These cause discontinuities and may be harder to test in the # above fashion: # 'inv', 'cofac', # 'eq', 'ne', 'le', 'ge', 'lt', 'gt', 'And', 'Or', 'Not', # 'conditional', 'sign', # 'jump', 'avg', # 'LiftingFunction', 'LiftingOperator', # FIXME: Test other derivatives: (but algorithms for operator # derivatives are the same!): # 'variable', 'diff', # 'Dx', 'grad', 'div', 'curl', 'rot', 'Dn', 'exterior_derivative', # Run through all operators defined above and compare integrals debug = 0 for F, acc in F_list: # Apply UFL differentiation f = diff(F, SpatialCoordinate(mesh))[..., 0] if debug: print(F) print(x) print(f) # Apply integration with DOLFIN # (also passes through form compilation and jit) M = f * dx f_integral = assemble_scalar(M) # noqa f_integral = MPI.sum(mesh.mpi_comm(), f_integral) # Compute integral of f manually from anti-derivative F # (passes through PyDOLFIN interface and uses UFL evaluation) F_diff = F((x1, )) - F((x0, )) # Compare results. Using custom relative delta instead # of decimal digits here because some numbers are >> 1. delta = min(abs(f_integral), abs(F_diff)) * 10**-acc assert f_integral - F_diff <= delta
def F(self, v, s, time=None): """ Right hand side for ODE system """ time = time if time else Constant(0.0) # Assign states V = v assert(len(s) == 16) Xr1, Xr2, Xs, m, h, j, d, f, fCa, s, r, g, Ca_i, Ca_SR, Na_i, K_i = s # Assign parameters P_kna = self._parameters["P_kna"] g_K1 = self._parameters["g_K1"] g_Kr = self._parameters["g_Kr"] g_Ks = self._parameters["g_Ks"] g_Na = self._parameters["g_Na"] g_bna = self._parameters["g_bna"] g_CaL = self._parameters["g_CaL"] g_bca = self._parameters["g_bca"] g_to = self._parameters["g_to"] K_mNa = self._parameters["K_mNa"] K_mk = self._parameters["K_mk"] P_NaK = self._parameters["P_NaK"] K_NaCa = self._parameters["K_NaCa"] K_sat = self._parameters["K_sat"] Km_Ca = self._parameters["Km_Ca"] Km_Nai = self._parameters["Km_Nai"] alpha = self._parameters["alpha"] gamma = self._parameters["gamma"] K_pCa = self._parameters["K_pCa"] g_pCa = self._parameters["g_pCa"] g_pK = self._parameters["g_pK"] Buf_c = self._parameters["Buf_c"] Buf_sr = self._parameters["Buf_sr"] Ca_o = self._parameters["Ca_o"] K_buf_c = self._parameters["K_buf_c"] K_buf_sr = self._parameters["K_buf_sr"] K_up = self._parameters["K_up"] V_leak = self._parameters["V_leak"] V_sr = self._parameters["V_sr"] Vmax_up = self._parameters["Vmax_up"] a_rel = self._parameters["a_rel"] b_rel = self._parameters["b_rel"] c_rel = self._parameters["c_rel"] tau_g = self._parameters["tau_g"] Na_o = self._parameters["Na_o"] Cm = self._parameters["Cm"] F = self._parameters["F"] R = self._parameters["R"] T = self._parameters["T"] V_c = self._parameters["V_c"] stim_amplitude = self._parameters["stim_amplitude"] stim_duration = self._parameters["stim_duration"] stim_period = self._parameters["stim_period"] stim_start = self._parameters["stim_start"] K_o = self._parameters["K_o"] # Init return args F_expressions = [ufl.zero()]*16 # Expressions for the Reversal potentials component E_Na = R*T*ufl.ln(Na_o/Na_i)/F E_K = R*T*ufl.ln(K_o/K_i)/F E_Ks = R*T*ufl.ln((K_o + Na_o*P_kna)/(P_kna*Na_i + K_i))/F E_Ca = 0.5*R*T*ufl.ln(Ca_o/Ca_i)/F # Expressions for the Inward rectifier potassium current component alpha_K1 = 0.1/(1.0 + 6.14421235332821e-06*ufl.exp(0.06*V - 0.06*E_K)) beta_K1 = (0.36787944117144233*ufl.exp(0.1*V - 0.1*E_K) +\ 3.0606040200802673*ufl.exp(0.0002*V - 0.0002*E_K))/(1.0 +\ ufl.exp(0.5*E_K - 0.5*V)) xK1_inf = alpha_K1/(alpha_K1 + beta_K1) i_K1 = 0.4303314829119352*g_K1*ufl.sqrt(K_o)*(-E_K + V)*xK1_inf # Expressions for the Rapid time dependent potassium current component i_Kr = 0.4303314829119352*g_Kr*ufl.sqrt(K_o)*(-E_K + V)*Xr1*Xr2 # Expressions for the Xr1 gate component xr1_inf = 1.0/(1.0 +\ 0.02437284407327961*ufl.exp(-0.14285714285714285*V)) alpha_xr1 = 450.0/(1.0 + 0.011108996538242306*ufl.exp(-0.1*V)) beta_xr1 = 6.0/(1.0 +\ 13.581324522578193*ufl.exp(0.08695652173913043*V)) tau_xr1 = 1.0*alpha_xr1*beta_xr1 F_expressions[0] = (-Xr1 + xr1_inf)/tau_xr1 # Expressions for the Xr2 gate component xr2_inf = 1.0/(1.0 + 39.12128399815321*ufl.exp(0.041666666666666664*V)) alpha_xr2 = 3.0/(1.0 + 0.049787068367863944*ufl.exp(-0.05*V)) beta_xr2 = 1.12/(1.0 + 0.049787068367863944*ufl.exp(0.05*V)) tau_xr2 = 1.0*alpha_xr2*beta_xr2 F_expressions[1] = (-Xr2 + xr2_inf)/tau_xr2 # Expressions for the Slow time dependent potassium current component i_Ks = g_Ks*ufl.elem_pow(Xs, 2.0)*(-E_Ks + V) # Expressions for the Xs gate component xs_inf = 1.0/(1.0 + 0.6996725373751304*ufl.exp(-0.07142857142857142*V)) alpha_xs = 1100.0/ufl.sqrt(1.0 +\ 0.18887560283756186*ufl.exp(-0.16666666666666666*V)) beta_xs = 1.0/(1.0 + 0.049787068367863944*ufl.exp(0.05*V)) tau_xs = 1.0*alpha_xs*beta_xs F_expressions[2] = (-Xs + xs_inf)/tau_xs # Expressions for the Fast sodium current component i_Na = g_Na*ufl.elem_pow(m, 3.0)*(-E_Na + V)*h*j # Expressions for the m gate component m_inf = 1.0*ufl.elem_pow(1.0 +\ 0.0018422115811651339*ufl.exp(-0.1107419712070875*V), -2.0) alpha_m = 1.0/(1.0 + 6.14421235332821e-06*ufl.exp(-0.2*V)) beta_m = 0.1/(1.0 + 1096.6331584284585*ufl.exp(0.2*V)) + 0.1/(1.0 +\ 0.7788007830714049*ufl.exp(0.005*V)) tau_m = 1.0*alpha_m*beta_m F_expressions[3] = (-m + m_inf)/tau_m # Expressions for the h gate component h_inf = 1.0*ufl.elem_pow(1.0 +\ 15212.593285654404*ufl.exp(0.13458950201884254*V), -2.0) alpha_h = ufl.conditional(ufl.lt(V, -40.0),\ 4.4312679295805147e-07*ufl.exp(-0.14705882352941177*V), 0) beta_h = ufl.conditional(ufl.lt(V, -40.0), 310000.0*ufl.exp(0.3485*V)\ + 2.7*ufl.exp(0.079*V), 0.77/(0.13 +\ 0.049758141083938695*ufl.exp(-0.0900900900900901*V))) tau_h = 1.0/(alpha_h + beta_h) F_expressions[4] = (-h + h_inf)/tau_h # Expressions for the j gate component j_inf = 1.0*ufl.elem_pow(1.0 +\ 15212.593285654404*ufl.exp(0.13458950201884254*V), -2.0) alpha_j = ufl.conditional(ufl.lt(V, -40.0), 1.0*(37.78 +\ V)*(-25428.0*ufl.exp(0.2444*V) -\ 6.948e-06*ufl.exp(-0.04391*V))/(1.0 +\ 50262745825.95399*ufl.exp(0.311*V)), 0) beta_j = ufl.conditional(ufl.lt(V, -40.0),\ 0.02424*ufl.exp(-0.01052*V)/(1.0 +\ 0.003960868339904256*ufl.exp(-0.1378*V)),\ 0.6*ufl.exp(0.057*V)/(1.0 +\ 0.040762203978366204*ufl.exp(-0.1*V))) tau_j = 1.0/(alpha_j + beta_j) F_expressions[5] = (-j + j_inf)/tau_j # Expressions for the Sodium background current component i_b_Na = g_bna*(-E_Na + V) # Expressions for the L_type Ca current component i_CaL = 4.0*g_CaL*ufl.elem_pow(F, 2.0)*(-0.341*Ca_o +\ Ca_i*ufl.exp(2.0*F*V/(R*T)))*V*d*f*fCa/(R*T*(-1.0 +\ ufl.exp(2.0*F*V/(R*T)))) # Expressions for the d gate component d_inf = 1.0/(1.0 + 0.513417119032592*ufl.exp(-0.13333333333333333*V)) alpha_d = 0.25 + 1.4/(1.0 +\ 0.0677244716592409*ufl.exp(-0.07692307692307693*V)) beta_d = 1.4/(1.0 + 2.718281828459045*ufl.exp(0.2*V)) gamma_d = 1.0/(1.0 + 12.182493960703473*ufl.exp(-0.05*V)) tau_d = 1.0*alpha_d*beta_d + gamma_d F_expressions[6] = (-d + d_inf)/tau_d # Expressions for the f gate component f_inf = 1.0/(1.0 + 17.411708063327644*ufl.exp(0.14285714285714285*V)) tau_f = 80.0 + 165.0/(1.0 + 12.182493960703473*ufl.exp(-0.1*V)) +\ 1125.0*ufl.exp(-0.004166666666666667*ufl.elem_pow(27.0 + V, 2.0)) F_expressions[7] = (-f + f_inf)/tau_f # Expressions for the FCa gate component alpha_fCa = 1.0/(1.0 + 8.03402376701711e+27*ufl.elem_pow(Ca_i, 8.0)) beta_fCa = 0.1/(1.0 + 0.006737946999085467*ufl.exp(10000.0*Ca_i)) gama_fCa = 0.2/(1.0 + 0.391605626676799*ufl.exp(1250.0*Ca_i)) fCa_inf = 0.15753424657534246 + 0.684931506849315*alpha_fCa +\ 0.684931506849315*beta_fCa + 0.684931506849315*gama_fCa tau_fCa = 2.0 d_fCa = (-fCa + fCa_inf)/tau_fCa F_expressions[8] = ufl.conditional(ufl.And(ufl.gt(V, -60.0),\ ufl.gt(fCa_inf, fCa)), 0, d_fCa) # Expressions for the Calcium background current component i_b_Ca = g_bca*(-E_Ca + V) # Expressions for the Transient outward current component i_to = g_to*(-E_K + V)*r*s # Expressions for the s gate component s_inf = 1.0/(1.0 + 54.598150033144236*ufl.exp(0.2*V)) tau_s = 3.0 + 5.0/(1.0 + 0.01831563888873418*ufl.exp(0.2*V)) +\ 85.0*ufl.exp(-0.003125*ufl.elem_pow(45.0 + V, 2.0)) F_expressions[9] = (-s + s_inf)/tau_s # Expressions for the r gate component r_inf = 1.0/(1.0 + 28.031624894526125*ufl.exp(-0.16666666666666666*V)) tau_r = 0.8 + 9.5*ufl.exp(-0.0005555555555555556*ufl.elem_pow(40.0 +\ V, 2.0)) F_expressions[10] = (-r + r_inf)/tau_r # Expressions for the Sodium potassium pump current component i_NaK = K_o*P_NaK*Na_i/((K_mNa + Na_i)*(K_mk + K_o)*(1.0 +\ 0.0353*ufl.exp(-F*V/(R*T)) + 0.1245*ufl.exp(-0.1*F*V/(R*T)))) # Expressions for the Sodium calcium exchanger current component i_NaCa = K_NaCa*(Ca_o*ufl.elem_pow(Na_i,\ 3.0)*ufl.exp(F*gamma*V/(R*T)) - alpha*ufl.elem_pow(Na_o,\ 3.0)*Ca_i*ufl.exp(F*(-1.0 + gamma)*V/(R*T)))/((1.0 +\ K_sat*ufl.exp(F*(-1.0 + gamma)*V/(R*T)))*(Ca_o +\ Km_Ca)*(ufl.elem_pow(Km_Nai, 3.0) + ufl.elem_pow(Na_o, 3.0))) # Expressions for the Calcium pump current component i_p_Ca = g_pCa*Ca_i/(K_pCa + Ca_i) # Expressions for the Potassium pump current component i_p_K = g_pK*(-E_K + V)/(1.0 +\ 65.40521574193832*ufl.exp(-0.16722408026755853*V)) # Expressions for the Calcium dynamics component i_rel = (c_rel + a_rel*ufl.elem_pow(Ca_SR, 2.0)/(ufl.elem_pow(b_rel,\ 2.0) + ufl.elem_pow(Ca_SR, 2.0)))*d*g i_up = Vmax_up/(1.0 + ufl.elem_pow(K_up, 2.0)*ufl.elem_pow(Ca_i, -2.0)) i_leak = V_leak*(-Ca_i + Ca_SR) g_inf = ufl.conditional(ufl.lt(Ca_i, 0.00035), 1.0/(1.0 +\ 5.439910241481018e+20*ufl.elem_pow(Ca_i, 6.0)), 1.0/(1.0 +\ 1.9720198874049195e+55*ufl.elem_pow(Ca_i, 16.0))) d_g = (-g + g_inf)/tau_g F_expressions[11] = ufl.conditional(ufl.And(ufl.gt(V, -60.0),\ ufl.gt(g_inf, g)), 0, d_g) Ca_i_bufc = 1.0/(1.0 + Buf_c*K_buf_c*ufl.elem_pow(K_buf_c + Ca_i,\ -2.0)) Ca_sr_bufsr = 1.0/(1.0 + Buf_sr*K_buf_sr*ufl.elem_pow(K_buf_sr +\ Ca_SR, -2.0)) F_expressions[12] = (-i_up - 0.5*Cm*(1.0*i_CaL + 1.0*i_b_Ca +\ 1.0*i_p_Ca - 2.0*i_NaCa)/(F*V_c) + i_leak + i_rel)*Ca_i_bufc F_expressions[13] = V_c*(-i_leak - i_rel + i_up)*Ca_sr_bufsr/V_sr # Expressions for the Sodium dynamics component F_expressions[14] = 1.0*Cm*(-1.0*i_Na - 1.0*i_b_Na - 3.0*i_NaCa -\ 3.0*i_NaK)/(F*V_c) # Expressions for the Membrane component i_Stim = ufl.conditional(ufl.And(ufl.ge(time -\ stim_period*ufl.floor(time/stim_period), stim_start), ufl.le(time\ - stim_period*ufl.floor(time/stim_period), stim_duration +\ stim_start)), -stim_amplitude, 0) # Expressions for the Potassium dynamics component F_expressions[15] = 1.0*Cm*(2.0*i_NaK - 1.0*i_K1 - 1.0*i_Kr -\ 1.0*i_Ks - 1.0*i_Stim - 1.0*i_p_K - 1.0*i_to)/(F*V_c) # Return results return dolfin.as_vector(F_expressions)
chi = 0.2 po = Gdry/lamo muo = Rg*T*(math.log(Omega*cod/(1+Omega*cod))+1/(1+Omega*cod)+ chi/pow((1+Omega*cod),2)) \ +Omega*po mus = muo mutop = -10000 # Kinematics I = Identity(3) F = I + grad(u) J = det(F) # Constitutive equations S = (Gdry/lamo)*F-p*cofac(F) mu = Rg*T*(ufl.ln(Omega*Jo*c/(1+Omega*Jo*c)) + 1./(1.+Omega*Jo*c)+ \ chi/pow((1.+Omega*Jo*c),2))+Omega*p h = g(c)*grad(c)+(-c*D/(Rg*T))*Omega*grad(p) dmudp = Omega dmu = mu-mus # Boundary conditions def left_boundary(x, on_boundary): # x = 0 return on_boundary and abs(x[0]) < DOLFIN_EPS def back_boundary(x, on_boundary): # y = 0 return on_boundary and abs(x[1]) < DOLFIN_EPS
def test_div_grad_then_integrate_over_cells_and_boundary(): # Define 2D geometry n = 10 mesh = RectangleMesh(Point(0.0, 0.0), Point(2.0, 3.0), 2 * n, 3 * n) x, y = SpatialCoordinate(mesh) xs = 0.1 + 0.8 * x / 2 # scaled to be within [0.1,0.9] # ys = 0.1 + 0.8 * y / 3 # scaled to be within [0.1,0.9] n = FacetNormal(mesh) # Define list of expressions to test, and configure accuracies # these expressions are known to pass with. The reason some # functions are less accurately integrated is likely that the # default choice of quadrature rule is not perfect F_list = [] def reg(exprs, acc=10): for expr in exprs: F_list.append((expr, acc)) # FIXME: 0*dx and 1*dx fails in the ufl-ffc-jit framework somewhere # reg([Constant(0.0, cell=cell)]) # reg([Constant(1.0, cell=cell)]) monomial_list = [x**q for q in range(2, 6)] reg(monomial_list) reg([2.3 * p + 4.5 * q for p in monomial_list for q in monomial_list]) reg([xs**xs]) reg( [xs**(xs**2)], 8 ) # Note: Accuracies here are from 1D case, not checked against 2D results. reg([xs**(xs**3)], 6) reg([xs**(xs**4)], 2) # Special functions: reg([atan(xs)], 8) reg([sin(x), cos(x), exp(x)], 5) reg([ln(xs), pow(x, 2.7), pow(2.7, x)], 3) reg([asin(xs), acos(xs)], 1) reg([tan(xs)], 7) # To handle tensor algebra, make an x dependent input tensor # xx and square all expressions def reg2(exprs, acc=10): for expr in exprs: F_list.append((inner(expr, expr), acc)) xx = as_matrix([[2 * x**2, 3 * x**3], [11 * x**5, 7 * x**4]]) xxs = as_matrix([[2 * xs**2, 3 * xs**3], [11 * xs**5, 7 * xs**4]]) x3v = as_vector([3 * x**2, 5 * x**3, 7 * x**4]) cc = as_matrix([[2, 3], [4, 5]]) reg2( [xx] ) # TODO: Make unit test for UFL from this, results in listtensor with free indices reg2([x3v]) reg2([cross(3 * x3v, as_vector([-x3v[1], x3v[0], x3v[2]]))]) reg2([xx.T]) reg2([tr(xx)]) reg2([det(xx)]) reg2([dot(xx, 0.1 * xx)]) reg2([outer(xx, xx.T)]) reg2([dev(xx)]) reg2([sym(xx)]) reg2([skew(xx)]) reg2([elem_mult(7 * xx, cc)]) reg2([elem_div(7 * xx, xx + cc)]) reg2([elem_pow(1e-3 * xxs, 1e-3 * cc)]) reg2([elem_pow(1e-3 * cc, 1e-3 * xx)]) reg2([elem_op(lambda z: sin(z) + 2, 0.03 * xx)], 2) # pretty inaccurate... # FIXME: Add tests for all UFL operators: # These cause discontinuities and may be harder to test in the # above fashion: # 'inv', 'cofac', # 'eq', 'ne', 'le', 'ge', 'lt', 'gt', 'And', 'Or', 'Not', # 'conditional', 'sign', # 'jump', 'avg', # 'LiftingFunction', 'LiftingOperator', # FIXME: Test other derivatives: (but algorithms for operator # derivatives are the same!): # 'variable', 'diff', # 'Dx', 'grad', 'div', 'curl', 'rot', 'Dn', 'exterior_derivative', # Run through all operators defined above and compare integrals debug = 0 if debug: F_list = F_list[1:] for F, acc in F_list: if debug: print('\n', "F:", str(F)) # Integrate over domain and its boundary int_dx = assemble(div(grad(F)) * dx(mesh)) # noqa int_ds = assemble(dot(grad(F), n) * ds(mesh)) # noqa if debug: print(int_dx, int_ds) # Compare results. Using custom relative delta instead of # decimal digits here because some numbers are >> 1. delta = min(abs(int_dx), abs(int_ds)) * 10**-acc assert int_dx - int_ds <= delta
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)
# Before defining the energy density and thus the total potential # energy, it only remains to specify constants for the elasticity # parameters:: # Elasticity parameters E = 10.0 nu = 0.3 mu = E/(2*(1 + nu)) lmbda = E*nu/((1 + nu)*(1 - 2*nu)) # Both the first variation of the potential energy, and the Jacobian of # the variation, can be automatically computed by a call to # ``derivative``:: # Stored strain energy density (compressible neo-Hookean model) psi = (mu/2)*(Ic - 3) - mu*ln(J) + (lmbda/2)*(ln(J))**2 # Total potential energy Pi = psi*dx # - inner(B, u)*dx - inner(T, u)*ds # First variation of Pi (directional derivative about u in the direction of v) F_form = derivative(Pi, u, v) # Compute Jacobian of F J_form = derivative(F_form, u, du) # Compute Cauchy stress sigma = (1/J)*diff(psi, F)*F.T forms = [F_form, J_form] elements = [(element)]
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 _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) == 16) Xr1, Xr2, Xs, m, h, j, d, f, fCa, s, r, g, Ca_i, Ca_SR, Na_i, K_i = s # Assign parameters P_kna = self._parameters["P_kna"] g_K1 = self._parameters["g_K1"] g_Kr = self._parameters["g_Kr"] g_Ks = self._parameters["g_Ks"] g_Na = self._parameters["g_Na"] g_bna = self._parameters["g_bna"] g_CaL = self._parameters["g_CaL"] g_bca = self._parameters["g_bca"] g_to = self._parameters["g_to"] K_mNa = self._parameters["K_mNa"] K_mk = self._parameters["K_mk"] P_NaK = self._parameters["P_NaK"] K_NaCa = self._parameters["K_NaCa"] K_sat = self._parameters["K_sat"] Km_Ca = self._parameters["Km_Ca"] Km_Nai = self._parameters["Km_Nai"] alpha = self._parameters["alpha"] gamma = self._parameters["gamma"] K_pCa = self._parameters["K_pCa"] g_pCa = self._parameters["g_pCa"] g_pK = self._parameters["g_pK"] Ca_o = self._parameters["Ca_o"] Na_o = self._parameters["Na_o"] F = self._parameters["F"] R = self._parameters["R"] T = self._parameters["T"] stim_amplitude = self._parameters["stim_amplitude"] stim_duration = self._parameters["stim_duration"] stim_period = self._parameters["stim_period"] stim_start = self._parameters["stim_start"] K_o = self._parameters["K_o"] # Init return args 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((K_o + Na_o*P_kna)/(P_kna*Na_i + K_i))/F E_Ca = 0.5*R*T*ufl.ln(Ca_o/Ca_i)/F # Expressions for the Inward rectifier potassium current component alpha_K1 = 0.1/(1.0 + 6.14421235332821e-06*ufl.exp(0.06*V - 0.06*E_K)) beta_K1 = (0.36787944117144233*ufl.exp(0.1*V - 0.1*E_K) +\ 3.0606040200802673*ufl.exp(0.0002*V - 0.0002*E_K))/(1.0 +\ ufl.exp(0.5*E_K - 0.5*V)) xK1_inf = alpha_K1/(alpha_K1 + beta_K1) i_K1 = 0.4303314829119352*g_K1*ufl.sqrt(K_o)*(-E_K + V)*xK1_inf # Expressions for the Rapid time dependent potassium current component i_Kr = 0.4303314829119352*g_Kr*ufl.sqrt(K_o)*(-E_K + V)*Xr1*Xr2 # Expressions for the Slow time dependent potassium current component i_Ks = g_Ks*ufl.elem_pow(Xs, 2.0)*(-E_Ks + V) # Expressions for the Fast sodium current component i_Na = g_Na*ufl.elem_pow(m, 3.0)*(-E_Na + V)*h*j # Expressions for the Sodium background current component i_b_Na = g_bna*(-E_Na + V) # Expressions for the L_type Ca current component i_CaL = 4.0*g_CaL*ufl.elem_pow(F, 2.0)*(-0.341*Ca_o +\ Ca_i*ufl.exp(2.0*F*V/(R*T)))*V*d*f*fCa/(R*T*(-1.0 +\ ufl.exp(2.0*F*V/(R*T)))) # Expressions for the 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 +\ 0.0353*ufl.exp(-F*V/(R*T)) + 0.1245*ufl.exp(-0.1*F*V/(R*T)))) # Expressions for the Sodium calcium exchanger current component i_NaCa = K_NaCa*(Ca_o*ufl.elem_pow(Na_i,\ 3.0)*ufl.exp(F*gamma*V/(R*T)) - alpha*ufl.elem_pow(Na_o,\ 3.0)*Ca_i*ufl.exp(F*(-1.0 + gamma)*V/(R*T)))/((1.0 +\ K_sat*ufl.exp(F*(-1.0 + gamma)*V/(R*T)))*(Ca_o +\ Km_Ca)*(ufl.elem_pow(Km_Nai, 3.0) + ufl.elem_pow(Na_o, 3.0))) # Expressions for the Calcium pump current component i_p_Ca = g_pCa*Ca_i/(K_pCa + Ca_i) # Expressions for the Potassium pump current component i_p_K = g_pK*(-E_K + V)/(1.0 +\ 65.40521574193832*ufl.exp(-0.16722408026755853*V)) # Expressions for the Membrane component i_Stim = ufl.conditional(ufl.And(ufl.ge(time -\ stim_period*ufl.floor(time/stim_period), stim_start), ufl.le(time\ - stim_period*ufl.floor(time/stim_period), stim_duration +\ stim_start)), -stim_amplitude, 0) current[0] = -1.0*i_CaL - 1.0*i_K1 - 1.0*i_Kr - 1.0*i_Ks - 1.0*i_Na -\ 1.0*i_NaCa - 1.0*i_NaK - 1.0*i_Stim - 1.0*i_b_Ca - 1.0*i_b_Na -\ 1.0*i_p_Ca - 1.0*i_p_K - 1.0*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_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)
phi = TestFunction(V) rho_0_lotnisko_inne = Expression("exp(-a*pow(x[0]-7, 2) - a*pow(x[1]-4, 2))", degree=2, a=0.01) rho_0_lotnisko = Expression("x[1] < 35 ? 1.0 : 0.0", degree=2, a=0.01) rho_0 = Expression("exp(-a*pow(x[0]-7, 2) - a*pow(x[1]-4, 2))", degree=2, a=1) F = u * v * dx + sigma * sigma * dot(grad(u), grad(v)) * dx a, L = lhs(F), rhs(F) u = Function(V) velocity = Function(V) solve(a == L, u, bc_lotnisko) phi = uf.ln(u) dx0 = project(phi.dx(0)) dx1 = project(phi.dx(1)) unnormed_grad_phi = project(grad(phi)) module = sqrt(dx0 * dx0 + dx1 * dx1) normed_phi = unnormed_grad_phi / module plot(normed_phi) plt.show() rhoold = interpolate(rho_0_lotnisko, V) plot(rhoold) plt.show()
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]
def test_neohooke(): mesh = dolfinx.mesh.create_unit_cube(MPI.COMM_WORLD, 7, 7, 7) V = dolfinx.fem.VectorFunctionSpace(mesh, ("P", 1)) P = dolfinx.fem.FunctionSpace(mesh, ("P", 1)) L = dolfinx.fem.FunctionSpace(mesh, ("DG", 0)) u = dolfinx.fem.Function(V, name="u") v = ufl.TestFunction(V) p = dolfinx.fem.Function(P, name="p") q = ufl.TestFunction(P) lmbda0 = dolfinx.fem.Function(L) d = mesh.topology.dim Id = ufl.Identity(d) F = Id + ufl.grad(u) C = F.T * F J = ufl.det(F) E_, nu_ = 10.0, 0.3 mu, lmbda = E_ / (2 * (1 + nu_)), E_ * nu_ / ((1 + nu_) * (1 - 2 * nu_)) psi = (mu / 2) * (ufl.tr(C) - 3) - mu * ufl.ln(J) + lmbda / 2 * ufl.ln(J)**2 + (p - 1.0)**2 pi = psi * ufl.dx F0 = ufl.derivative(pi, u, v) F1 = ufl.derivative(pi, p, q) # Number of eigenvalues to find nev = 8 opts = PETSc.Options("neohooke") opts["eps_smallest_magnitude"] = True opts["eps_nev"] = nev opts["eps_ncv"] = 50 * nev opts["eps_conv_abs"] = True # opts["eps_non_hermitian"] = True opts["eps_tol"] = 1.0e-14 opts["eps_max_it"] = 1000 opts["eps_error_relative"] = "ascii::ascii_info_detail" opts["eps_monitor"] = "ascii" slepcp = dolfiny.slepcblockproblem.SLEPcBlockProblem([F0, F1], [u, p], lmbda0, prefix="neohooke") slepcp.solve() # mat = dolfiny.la.petsc_to_scipy(slepcp.eps.getOperators()[0]) # eigvals, eigvecs = linalg.eigsh(mat, which="SM", k=nev) with dolfinx.io.XDMFFile(MPI.COMM_WORLD, "eigvec.xdmf", "w") as ofile: ofile.write_mesh(mesh) for i in range(nev): eigval, ur, ui = slepcp.getEigenpair(i) # Expect first 6 eignevalues 0, i.e. rigid body modes if i < 6: assert np.isclose(eigval, 0.0) for func in ur: name = func.name func.name = f"{name}_eigvec_{i}_real" ofile.write_function(func) func.name = name
def assemble_test(cell_batch_size: int): mesh = dolfin.UnitCubeMesh(MPI.comm_world, 40, 40, 40) def isochoric(F): C = F.T*F I_1 = tr(C) I4_f = dot(e_f, C*e_f) I4_s = dot(e_s, C*e_s) I8_fs = dot(e_f, C*e_s) def cutoff(x): return 1.0/(1.0 + ufl.exp(-(x - 1.0)*30.0)) def scaled_exp(a0, a1, argument): return a0/(2.0*a1)*(ufl.exp(b*argument) - 1) E_1 = scaled_exp(a, b, I_1 - 3.) E_f = cutoff(I4_f)*scaled_exp(a_f, b_f, (I4_f - 1.)**2) E_s = cutoff(I4_s)*scaled_exp(a_s, b_s, (I4_s - 1.)**2) E_3 = scaled_exp(a_fs, b_fs, I8_fs**2) E = E_1 + E_f + E_s + E_3 return E cell = mesh.ufl_cell() lamda = dolfin.Constant(0.48, cell) a = dolfin.Constant(1.0, cell) b = dolfin.Constant(1.0, cell) a_s = dolfin.Constant(1.0, cell) b_s = dolfin.Constant(1.0, cell) a_f = dolfin.Constant(1.0, cell) b_f = dolfin.Constant(1.0, cell) a_fs = dolfin.Constant(1.0, cell) b_fs = dolfin.Constant(1.0, cell) # For more fun, make these general vector fields rather than # constants: e_s = dolfin.Constant([0.0, 1.0, 0.0], cell) e_f = dolfin.Constant([1.0, 0.0, 0.0], cell) V = dolfin.FunctionSpace(mesh, ufl.VectorElement("CG", cell, 1)) u = dolfin.Function(V) du = dolfin.function.argument.TrialFunction(V) v = dolfin.function.argument.TestFunction(V) # Misc elasticity related tensors and other quantities F = grad(u) + ufl.Identity(3) F = ufl.variable(F) J = det(F) Fbar = J**(-1.0/3.0)*F # Define energy E_volumetric = lamda*0.5*ln(J)**2 psi = isochoric(Fbar) + E_volumetric # Find first Piola-Kircchoff tensor P = ufl.diff(psi, F) # Define the variational formulation F = inner(P, grad(v))*dx # Take the derivative J = ufl.derivative(F, u, du) a, L = J, F if cell_batch_size > 1: cxx_flags = "-O2 -ftree-vectorize -funroll-loops -march=native -mtune=native" else: cxx_flags = "-O2" assembler = dolfin.fem.assembling.Assembler([[a]], [L], [], form_compiler_parameters={"cell_batch_size": cell_batch_size, "enable_cross_cell_gcc_ext": True, "cpp_optimize_flags": cxx_flags}) t = -time.time() A, b = assembler.assemble( mat_type=dolfin.cpp.fem.Assembler.BlockType.monolithic) t += time.time() return A, b, t
chi = 0.2 po = Gdry / lamo muo = Rg*T*(math.log(Omega*cod/(1+Omega*cod))+1/(1+Omega*cod)+ chi/pow((1+Omega*cod),2)) \ +Omega*po mus = muo mutop = -10000 # Kinematics I = Identity(3) F = I + grad(u) J = det(F) # Constitutive equations S = (Gdry / lamo) * F - p * cofac(F) mu = Rg*T*(ufl.ln(Omega*Jo*c/(1+Omega*Jo*c)) + 1./(1.+Omega*Jo*c)+ \ chi/pow((1.+Omega*Jo*c),2))+Omega*p h = g(c) * grad(c) + (-c * D / (Rg * T)) * Omega * grad(p) dmudp = Omega dmu = mu - mus # Boundary conditions def left_boundary(x, on_boundary): # x = 0 return on_boundary and abs(x[0]) < DOLFIN_EPS def back_boundary(x, on_boundary): # y = 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)
def rhs(states, time, parameters, dy=None): """ Compute right hand side """ # Imports import ufl import dolfin # Assign states assert(isinstance(states, dolfin.Function)) assert(states.function_space().depth() == 1) assert(states.function_space().num_sub_spaces() == 17) Xr1, Xr2, Xs, m, h, j, d, f, fCa, s, r, Ca_SR, Ca_i, g, Na_i, V, K_i =\ dolfin.split(states) # Assign parameters assert(isinstance(parameters, (dolfin.Function, dolfin.Constant))) if isinstance(parameters, dolfin.Function): assert(parameters.function_space().depth() == 1) assert(parameters.function_space().num_sub_spaces() == 45) else: assert(parameters.value_size() == 45) P_kna, g_K1, g_Kr, g_Ks, g_Na, g_bna, g_CaL, g_bca, g_to, K_mNa, K_mk,\ P_NaK, K_NaCa, K_sat, Km_Ca, Km_Nai, alpha, gamma, K_pCa, g_pCa,\ g_pK, Buf_c, Buf_sr, Ca_o, K_buf_c, K_buf_sr, K_up, V_leak, V_sr,\ Vmax_up, a_rel, b_rel, c_rel, tau_g, Na_o, Cm, F, R, T, V_c,\ stim_amplitude, stim_duration, stim_period, stim_start, K_o =\ dolfin.split(parameters) # Reversal potentials 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)/(Na_i*P_kna + K_i))/F E_Ca = 0.5*R*T*ufl.ln(Ca_o/Ca_i)/F # Inward rectifier potassium current alpha_K1 = 0.1/(1.0 + 6.14421235332821e-6*ufl.exp(0.06*V - 0.06*E_K)) beta_K1 = (3.06060402008027*ufl.exp(0.0002*V - 0.0002*E_K) +\ 0.367879441171442*ufl.exp(0.1*V - 0.1*E_K))/(1.0 + ufl.exp(0.5*E_K -\ 0.5*V)) xK1_inf = alpha_K1/(alpha_K1 + beta_K1) i_K1 = 0.430331482911935*ufl.sqrt(K_o)*(-E_K + V)*g_K1*xK1_inf # Rapid time dependent potassium current i_Kr = 0.430331482911935*ufl.sqrt(K_o)*(-E_K + V)*Xr1*Xr2*g_Kr # Rapid time dependent potassium current xr1 gate xr1_inf = 1.0/(1.0 + 0.0243728440732796*ufl.exp(-0.142857142857143*V)) alpha_xr1 = 450.0/(1.0 + ufl.exp(-9/2 - V/10.0)) beta_xr1 = 6.0/(1.0 + 13.5813245225782*ufl.exp(0.0869565217391304*V)) tau_xr1 = alpha_xr1*beta_xr1 # Rapid time dependent potassium current xr2 gate xr2_inf = 1.0/(1.0 + 39.1212839981532*ufl.exp(0.0416666666666667*V)) alpha_xr2 = 3.0/(1.0 + 0.0497870683678639*ufl.exp(-0.05*V)) beta_xr2 = 1.12/(1.0 + 0.0497870683678639*ufl.exp(0.05*V)) tau_xr2 = alpha_xr2*beta_xr2 # Slow time dependent potassium current i_Ks = (Xs*Xs)*(V - E_Ks)*g_Ks # Slow time dependent potassium current xs gate xs_inf = 1.0/(1.0 + 0.69967253737513*ufl.exp(-0.0714285714285714*V)) alpha_xs = 1100.0/ufl.sqrt(1.0 +\ 0.188875602837562*ufl.exp(-0.166666666666667*V)) beta_xs = 1.0/(1.0 + 0.0497870683678639*ufl.exp(0.05*V)) tau_xs = alpha_xs*beta_xs # Fast sodium current i_Na = (m*m*m)*(-E_Na + V)*g_Na*h*j # Fast sodium current m gate m_inf = 1.0/((1.0 +\ 0.00184221158116513*ufl.exp(-0.110741971207087*V))*(1.0 +\ 0.00184221158116513*ufl.exp(-0.110741971207087*V))) alpha_m = 1.0/(1.0 + ufl.exp(-12.0 - V/5.0)) beta_m = 0.1/(1.0 + 0.778800783071405*ufl.exp(0.005*V)) + 0.1/(1.0 +\ ufl.exp(7.0 + V/5.0)) tau_m = alpha_m*beta_m # Fast sodium current h gate h_inf = 1.0/((1.0 + 15212.5932856544*ufl.exp(0.134589502018843*V))*(1.0 +\ 15212.5932856544*ufl.exp(0.134589502018843*V))) alpha_h = 4.43126792958051e-7*ufl.exp(-0.147058823529412*V)/(1.0 +\ 2.3538526683702e+17*ufl.exp(1.0*V)) beta_h = (310000.0*ufl.exp(0.3485*V) + 2.7*ufl.exp(0.079*V))/(1.0 +\ 2.3538526683702e+17*ufl.exp(1.0*V)) + 0.77*(1.0 - 1.0/(1.0 +\ 2.3538526683702e+17*ufl.exp(1.0*V)))/(0.13 +\ 0.0497581410839387*ufl.exp(-0.0900900900900901*V)) tau_h = 1.0/(alpha_h + beta_h) # Fast sodium current j gate j_inf = 1.0/((1.0 + 15212.5932856544*ufl.exp(0.134589502018843*V))*(1.0 +\ 15212.5932856544*ufl.exp(0.134589502018843*V))) alpha_j = (37.78 + V)*(-6.948e-6*ufl.exp(-0.04391*V) -\ 25428.0*ufl.exp(0.2444*V))/((1.0 +\ 2.3538526683702e+17*ufl.exp(1.0*V))*(1.0 +\ 50262745825.954*ufl.exp(0.311*V))) beta_j = 0.6*(1.0 - 1.0/(1.0 +\ 2.3538526683702e+17*ufl.exp(1.0*V)))*ufl.exp(0.057*V)/(1.0 +\ 0.0407622039783662*ufl.exp(-0.1*V)) +\ 0.02424*ufl.exp(-0.01052*V)/((1.0 +\ 2.3538526683702e+17*ufl.exp(1.0*V))*(1.0 +\ 0.00396086833990426*ufl.exp(-0.1378*V))) tau_j = 1.0/(alpha_j + beta_j) # Sodium background current i_b_Na = (-E_Na + V)*g_bna # L type ca current i_CaL = 4.0*(F*F)*(-0.341*Ca_o +\ Ca_i*ufl.exp(2.0*F*V/(R*T)))*V*d*f*fCa*g_CaL/((-1.0 +\ ufl.exp(2.0*F*V/(R*T)))*R*T) # L type ca current d gate d_inf = 1.0/(1.0 + 0.513417119032592*ufl.exp(-0.133333333333333*V)) alpha_d = 0.25 + 1.4/(1.0 +\ 0.0677244716592409*ufl.exp(-0.0769230769230769*V)) beta_d = 1.4/(1.0 + ufl.exp(1.0 + V/5.0)) gamma_d = 1.0/(1.0 + 12.1824939607035*ufl.exp(-0.05*V)) tau_d = gamma_d + alpha_d*beta_d # L type ca current f gate f_inf = 1.0/(1.0 + 17.4117080633276*ufl.exp(0.142857142857143*V)) tau_f = 80.0 + 165.0/(1.0 + ufl.exp(5/2 - V/10.0)) +\ 1125.0*ufl.exp(-0.00416666666666667*((27.0 + V)*(27.0 + V))) # L type ca current fca gate alpha_fCa = 1.0/(1.0 + 8.03402376701711e+27*ufl.elem_pow(Ca_i, 8.0)) beta_fCa = 0.1/(1.0 + 0.00673794699908547*ufl.exp(10000.0*Ca_i)) gama_fCa = 0.2/(1.0 + 0.391605626676799*ufl.exp(1250.0*Ca_i)) fCa_inf = 0.157534246575342 + 0.684931506849315*gama_fCa +\ 0.684931506849315*beta_fCa + 0.684931506849315*alpha_fCa tau_fCa = 2.0 d_fCa = (-fCa + fCa_inf)/tau_fCa # Calcium background current i_b_Ca = (V - E_Ca)*g_bca # Transient outward current i_to = (-E_K + V)*g_to*r*s # Transient outward current s gate s_inf = 1.0/(1.0 + ufl.exp(4.0 + V/5.0)) tau_s = 3.0 + 85.0*ufl.exp(-0.003125*((45.0 + V)*(45.0 + V))) + 5.0/(1.0 +\ ufl.exp(-4.0 + V/5.0)) # Transient outward current r gate r_inf = 1.0/(1.0 + 28.0316248945261*ufl.exp(-0.166666666666667*V)) tau_r = 0.8 + 9.5*ufl.exp(-0.000555555555555556*((40.0 + V)*(40.0 + V))) # Sodium potassium pump current i_NaK = K_o*Na_i*P_NaK/((K_mk + K_o)*(Na_i + K_mNa)*(1.0 +\ 0.0353*ufl.exp(-F*V/(R*T)) + 0.1245*ufl.exp(-0.1*F*V/(R*T)))) # Sodium calcium exchanger current i_NaCa = (-(Na_o*Na_o*Na_o)*Ca_i*alpha*ufl.exp((-1.0 + gamma)*F*V/(R*T))\ + (Na_i*Na_i*Na_i)*Ca_o*ufl.exp(F*V*gamma/(R*T)))*K_NaCa/((1.0 +\ K_sat*ufl.exp((-1.0 + gamma)*F*V/(R*T)))*((Na_o*Na_o*Na_o) +\ (Km_Nai*Km_Nai*Km_Nai))*(Km_Ca + Ca_o)) # Calcium pump current i_p_Ca = Ca_i*g_pCa/(K_pCa + Ca_i) # Potassium pump current i_p_K = (-E_K + V)*g_pK/(1.0 +\ 65.4052157419383*ufl.exp(-0.167224080267559*V)) # Calcium dynamics i_rel = ((Ca_SR*Ca_SR)*a_rel/((Ca_SR*Ca_SR) + (b_rel*b_rel)) + c_rel)*d*g i_up = Vmax_up/(1.0 + (K_up*K_up)/(Ca_i*Ca_i)) i_leak = (-Ca_i + Ca_SR)*V_leak g_inf = (1.0 - 1.0/(1.0 + 0.0301973834223185*ufl.exp(10000.0*Ca_i)))/(1.0 +\ 1.97201988740492e+55*ufl.elem_pow(Ca_i, 16.0)) + 1.0/((1.0 +\ 0.0301973834223185*ufl.exp(10000.0*Ca_i))*(1.0 +\ 5.43991024148102e+20*ufl.elem_pow(Ca_i, 6.0))) d_g = (-g + g_inf)/tau_g Ca_i_bufc = 1.0/(1.0 + Buf_c*K_buf_c/((K_buf_c + Ca_i)*(K_buf_c + Ca_i))) Ca_sr_bufsr = 1.0/(1.0 + Buf_sr*K_buf_sr/((K_buf_sr + Ca_SR)*(K_buf_sr +\ Ca_SR))) # Sodium dynamics # Membrane i_Stim = -(1.0 - 1.0/(1.0 + ufl.exp(-5.0*stim_start +\ 5.0*time)))*stim_amplitude/(1.0 + ufl.exp(-5.0*stim_start + 5.0*time\ - 5.0*stim_duration)) # Potassium dynamics # The ODE system: 17 states # Init test function _v = dolfin.TestFunction(states.function_space()) # Derivative for state Xr1 dy = ((-Xr1 + xr1_inf)/tau_xr1)*_v[0] # Derivative for state Xr2 dy += ((-Xr2 + xr2_inf)/tau_xr2)*_v[1] # Derivative for state Xs dy += ((-Xs + xs_inf)/tau_xs)*_v[2] # Derivative for state m dy += ((-m + m_inf)/tau_m)*_v[3] # Derivative for state h dy += ((-h + h_inf)/tau_h)*_v[4] # Derivative for state j dy += ((j_inf - j)/tau_j)*_v[5] # Derivative for state d dy += ((d_inf - d)/tau_d)*_v[6] # Derivative for state f dy += ((-f + f_inf)/tau_f)*_v[7] # Derivative for state fCa dy += ((1.0 - 1.0/((1.0 + ufl.exp(60.0 + V))*(1.0 + ufl.exp(-10.0*fCa +\ 10.0*fCa_inf))))*d_fCa)*_v[8] # Derivative for state s dy += ((-s + s_inf)/tau_s)*_v[9] # Derivative for state r dy += ((-r + r_inf)/tau_r)*_v[10] # Derivative for state Ca_SR dy += ((-i_leak + i_up - i_rel)*Ca_sr_bufsr*V_c/V_sr)*_v[11] # Derivative for state Ca_i dy += ((-i_up - (i_CaL + i_p_Ca + i_b_Ca - 2.0*i_NaCa)*Cm/(2.0*F*V_c) +\ i_leak + i_rel)*Ca_i_bufc)*_v[12] # Derivative for state g dy += ((1.0 - 1.0/((1.0 + ufl.exp(60.0 + V))*(1.0 + ufl.exp(-10.0*g +\ 10.0*g_inf))))*d_g)*_v[13] # Derivative for state Na_i dy += ((-3.0*i_NaK - 3.0*i_NaCa - i_Na - i_b_Na)*Cm/(F*V_c))*_v[14] # Derivative for state V dy += (-i_Ks - i_to - i_Kr - i_p_K - i_NaK - i_NaCa - i_Na - i_p_Ca -\ i_b_Na - i_CaL - i_Stim - i_K1 - i_b_Ca)*_v[15] # Derivative for state K_i dy += ((-i_Ks - i_to - i_Kr - i_p_K - i_Stim - i_K1 +\ 2.0*i_NaK)*Cm/(F*V_c))*_v[16] # Return dy return dy
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 _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]