Model Mathematics

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Component: environment

Component: membrane

i_stim = \{\begin{matrix} stim_amplitude if time \geq stim_start \land time ≦ stim_end \\ 0 otherwise \end{matrix} \frac{d}{d time} (V) = - (i_tot + i_stim) i_tot = i_Na + i_Na_L + i_Ca_L + i_Ca_T + i_to_1 + i_to_2 + i_Kr + i_Ks + i_K1 + i_NaCa + i_NaK + i_Na_b + i_K_b + i_Ca_b + i_Cl_b + i_Ca_p + i_K_p

Component: equilibrium_potentials

E_Na = \frac{R T}{F} \ln (\frac{Na_o}{Na_i}) E_K = \frac{R T}{F} \ln (\frac{K_o}{K_i}) E_Ca = \frac{R T}{2 F} \ln (\frac{Ca_o}{Ca_i}) E_Cl = \frac{(- R) T}{F} \ln (\frac{Cl_o}{Cl_i}) E_Ks = \frac{R T}{F} \ln (\frac{K_o + r_NaK Na_o}{K_i + r_NaK Na_i})

Component: i_Na

i_Na = g_Na m^{3} (0.8 h + 0.2 j) (V - E_Na)

Component: i_Na_m_gate

alpha_m = \frac{0.32 (V + 47.13)}{1 - (ⅇ^{(- 0.1) (V + 47.13)})} beta_m = 0.08 ⅇ^{\frac{- V}{11}} m_infinity = \frac{alpha_m}{alpha_m + beta_m} tau_m = \frac{1}{alpha_m + beta_m} \frac{d}{d time} (m) = \frac{m_infinity - m}{tau_m}

Component: i_Na_h_gate

alpha_h = \{\begin{matrix} 0.135 ⅇ^{\frac{V + 80}{- 6.8}} if V < - 40 \\ 0 otherwise \end{matrix} beta_h = \{\begin{matrix} 3.56 ⅇ^{0.079 V} + 310000 ⅇ^{0.35 V} if V < - 40 \\ \frac{1}{0.13 (1 + ⅇ^{\frac{V + 10.66}{- 11.1}})} otherwise \end{matrix} h_infinity = \frac{alpha_h}{alpha_h + beta_h} tau_h = \frac{1}{alpha_h + beta_h} \frac{d}{d time} (h) = \frac{h_infinity - h}{tau_h}

Component: i_Na_j_gate

alpha_j = \{\begin{matrix} \frac{((- 127140) ⅇ^{0.2444 V} - (0.00003474 ⅇ^{(- 0.04391) V})) (V + 37.78)}{1 + ⅇ^{0.311 (V + 79.23)}} if V < - 40 \\ 0 otherwise \end{matrix} beta_j = \{\begin{matrix} \frac{0.1212 ⅇ^{(- 0.01052) V}}{1 + ⅇ^{(- 0.1378) (V + 40.14)}} if V < - 40 \\ \frac{0.3 ⅇ^{(- 0.0000002535) V}}{1 + ⅇ^{(- 0.1) (V + 32)}} otherwise \end{matrix} j_infinity = \frac{0.1 alpha_j}{alpha_j + beta_j} tau_j = \frac{0.1}{alpha_j + beta_j} \frac{d}{d time} (j) = \frac{j_infinity - j}{tau_j}

Component: i_Na_L

i_Na_L = g_Na_L {m_L}^{3} h_L (V - E_Na)

Component: i_Na_L_m_L_gate

alpha_m_L = \frac{0.32 (V + 47.13)}{1 - (ⅇ^{(- 0.1) (V + 47.13)})} beta_m_L = 0.08 ⅇ^{\frac{- V}{11}} m_L_infinity = \frac{alpha_m_L}{alpha_m_L + beta_m_L} tau_m_L = \frac{1}{alpha_m_L + beta_m_L} \frac{d}{d time} (m_L) = \frac{m_L_infinity - m_L}{tau_m_L}

Component: i_Na_L_h_L_gate

h_L_infinity = \frac{1}{1 + ⅇ^{\frac{V + 69}{6.1}}} tau_h_L = 175 + \frac{125}{1 + ⅇ^{\frac{- (V + 25)}{6}}} \frac{d}{d time} (h_L) = \frac{h_L_infinity - h_L}{tau_h_L}

Component: i_Ca_L

i_Ca_L = g_Ca_L d f f2 f_Ca f_Ca2 i_Ca_L_max i_Ca_L_max = \frac{\frac{\frac{1 p_Ca}{Cm} {z_Ca}^{2} (V - 15) F^{2}}{R T} (gamma_Cai Ca_r ⅇ^{\frac{z_Ca F (V - 15)}{R T}} - (gamma_Cao Ca_o))}{ⅇ^{\frac{z_Ca F (V - 15)}{R T}} - 1}

Component: i_Ca_L_d_gate

\frac{d}{d time} (d) = \frac{d_infinity - d}{tau_d} d_infinity = \frac{1}{1 + ⅇ^{\frac{- (V - 4)}{6.74}}} tau_d = 0.59 + \frac{0.8 ⅇ^{0.052 (V + 13)}}{1 + ⅇ^{0.132 (V + 13)}}

Component: i_Ca_L_f_gate

\frac{d}{d time} (f) = \frac{f_infinity - f}{tau_f} f_infinity = \frac{1}{1 + ⅇ^{\frac{V + 18}{10}}} tau_f = 4 + 0.005 {(V - 2.5)}^{2}

Component: i_Ca_L_f2_gate

\frac{d}{d time} (f2) = \frac{f2_infinity - f2}{tau_f2} f2_infinity = \frac{1}{1 + ⅇ^{\frac{V + 18}{10}}} tau_f2 = 38 + 0.07 {(V - 18.6)}^{2}

Component: i_Ca_L_f_Ca_gate

\frac{d}{d time} (f_Ca) = \frac{f_Ca_infinity - f_Ca}{tau_f_Ca} f_Ca_infinity = \frac{0.3}{1 - (\frac{i_Ca_L}{0.05})} + \frac{0.55}{1 + \frac{Ca_r}{0.003}} + 0.15 tau_f_Ca = 0.5 + \frac{10 1 Ca_MK_act}{1 Ca_MK_act + km_Ca_MK} + \frac{1}{1 + \frac{Ca_r}{0.003}}

Component: i_Ca_L_f_Ca2_gate

\frac{d}{d time} (f_Ca2) = \frac{f_Ca2_infinity - f_Ca2}{tau_f_Ca2} f_Ca2_infinity = \frac{1}{1 - (\frac{i_Ca_L}{0.01})} tau_f_Ca2 = 125 + \frac{300}{1 + ⅇ^{\frac{- i_Ca_L - 0.175}{0.04}}}

Component: i_Ca_T

i_Ca_T = g_Ca_T b g (V - 50)

Component: i_Ca_T_b_gate

alpha_b = 1.068 ⅇ^{\frac{V + 16.3}{30}} beta_b = 1.068 ⅇ^{\frac{- (V + 16.3)}{30}} b_infinity = \frac{1}{1 + ⅇ^{\frac{- (V + 33)}{6.1}}} tau_b = \frac{1}{alpha_b + beta_b} \frac{d}{d time} (b) = \frac{b_infinity - b}{tau_b}

Component: i_Ca_T_g_gate

alpha_g = 0.015 ⅇ^{\frac{- (V + 71.7)}{83.3}} beta_g = 0.015 ⅇ^{\frac{V + 71.7}{15.4}} g_infinity = \frac{1}{1 + ⅇ^{\frac{V + 60}{6.6}}} tau_g = \frac{1}{alpha_g + beta_g} \frac{d}{d time} (g) = \frac{g_infinity - g}{tau_g}

Component: i_to_1

i_to_1 = g_to_1 a (0.8 i + 0.2 i2) (V - E_K)

Component: i_to_1_a_gate

\frac{d}{d time} (a) = \frac{a_infinity - a}{tau_a} alpha_a = \frac{25 ⅇ^{\frac{V - 76}{20}}}{1 + ⅇ^{\frac{V - 76}{20}}} beta_a = \frac{25 ⅇ^{\frac{- (V + 54)}{20}}}{1 + ⅇ^{\frac{- (V + 54)}{20}}} tau_a = \frac{1}{alpha_a + beta_a} a_infinity = \frac{alpha_a}{alpha_a + beta_a}

Component: i_to_1_i_gate

\frac{d}{d time} (i) = \frac{i_infinity - i}{tau_i} alpha_i = \frac{0.03}{1 + ⅇ^{\frac{V + 25}{15}}} beta_i = \frac{0.1 ⅇ^{\frac{V - 40}{15}}}{1 + ⅇ^{\frac{V - 40}{15}}} tau_i = 6 + \frac{5}{1 + ⅇ^{\frac{V - 16.5}{10}}} i_infinity = \frac{alpha_i}{alpha_i + beta_i}

Component: i_to_1_i2_gate

\frac{d}{d time} (i2) = \frac{i2_infinity - i2}{tau_i2} alpha_i2 = \frac{0.00442}{1 + ⅇ^{\frac{V + 26}{15}}} beta_i2 = \frac{0.05 ⅇ^{\frac{V - 10}{15}}}{1 + ⅇ^{\frac{V - 10}{15}}} tau_i2 = 21.5 + \frac{30}{1 + ⅇ^{\frac{V - 25}{10}}} i2_infinity = \frac{alpha_i2}{alpha_i2 + beta_i2}

Component: i_Kr

i_Kr = g_Kr xr rr_infinity (V - E_K) g_Kr = 0.040008488 \sqrt{\frac{K_o}{5.4}} rr_infinity = \frac{1}{1 + ⅇ^{\frac{V - 5.4}{20.4}}}

Component: i_Kr_xr_gate

\frac{d}{d time} (xr) = \frac{xr_infinity - xr}{tau_xr} tau_xr = \frac{900}{1 + ⅇ^{\frac{V}{5}}} + 100 xr_infinity = \frac{1}{1 + ⅇ^{\frac{- (V + 0.085)}{12.25}}}

Component: i_Ks

i_Ks = g_Ks xs1 xs2 (V - E_Ks) g_Ks = 0.052581329 (1 + \frac{0.6}{1 + {(\frac{0.000038}{Ca_i})}^{1.4}})

Component: i_Ks_xs1_gate

\frac{d}{d time} (xs1) = \frac{xs1_infinity - xs1}{tau_xs1} tau_xs1 = \frac{1}{\frac{0.0000761 (V + 44.6)}{1 - (ⅇ^{(- 9.97) (V + 44.6)})} + \frac{0.00036 (V - 0.55)}{ⅇ^{0.128 (V - 0.55)} - 1}} xs1_infinity = \frac{1}{1 + ⅇ^{\frac{- (V - 9)}{13.7}}}

Component: i_Ks_xs2_gate

\frac{d}{d time} (xs2) = \frac{xs2_infinity - xs2}{tau_xs2} tau_xs2 = \frac{2 1}{\frac{0.0000761 (V + 44.6)}{1 - (ⅇ^{(- 9.97) (V + 44.6)})} + \frac{0.00036 (V - 0.55)}{ⅇ^{0.128 (V - 0.55)} - 1}} xs2_infinity = \frac{1}{1 + ⅇ^{\frac{- (V - 9)}{13.7}}}

Component: i_K1

i_K1 = (g_K1 xK1 + 0.004) (V - E_K) g_K1 = 0.25 \sqrt{\frac{K_o}{5.4}}

Component: i_K1_xK1_gate

xK1 = \frac{alpha_xK1}{alpha_xK1 + beta_xK1} beta_xK1 = \frac{0.49124 ⅇ^{0.08032 (V + 5.476 - E_K)} + ⅇ^{0.06175 (V - (594.31 + E_K))}}{1 + ⅇ^{(- 0.5143) (V + 4.753 - E_K)}} alpha_xK1 = \frac{1.02}{1 + ⅇ^{0.2385 (V - (E_K + 59.215))}}

Component: i_K_p

i_K_p = g_K_p kp (V - E_K) kp = \frac{1}{1 + ⅇ^{\frac{7.488 - V}{5.98}}}

Component: i_to_2

i_to_2 = 20 i_to_2_max a i_to_2_max = \frac{\frac{\frac{1 p_Cl}{Cm} {z_Cl}^{2} V F^{2}}{R T} (Cl_i - (Cl_o ⅇ^{\frac{(- z_Cl) V F}{R T}}))}{1 - (ⅇ^{\frac{(- z_Cl) V F}{R T}})}

Component: i_to_2_a_gate

\frac{d}{d time} (a) = \frac{a_infinity - a}{tau_a} a_infinity = \frac{1}{1 + \frac{km_to_2}{Ca_r}} tau_a = 1

Component: i_NaCa

i_NaCa = \frac{X_NaCa i_NaCa_max {Na_i}^{3} Ca_o ⅇ^{\frac{0.35 F V}{R T}} - (1.5 {Na_o}^{3} Ca_i ⅇ^{\frac{(- 0.65) F V}{R T}})}{(1 + {(\frac{km_Ca_act}{1.5 Ca_i})}^{2}) (1 + k_sat ⅇ^{\frac{(- 0.65) V F}{R T}}) (dNaCa_1 + dNaCa_2)} dNaCa_1 = km_Ca_o {Na_i}^{3} + 1.5 {km_Na_o}^{3} Ca_i + {km_Na_i_1}^{3} Ca_o (1 + \frac{1.5 Ca_i}{km_Ca_i}) dNaCa_2 = km_Ca_i {Na_o}^{3} (1 + \frac{Na_i}{km_Na_i_1}) + {Na_i}^{3} Ca_o + 1.5 {Na_o}^{3} Ca_i

Component: i_NaK

i_NaK = \frac{\frac{g_NaK f_NaK 1}{1 + {(\frac{km_Na_i_2}{Na_i})}^{2}} K_o}{K_o + km_K_o} f_NaK = \frac{1}{1 + 0.1245 ⅇ^{\frac{(- 0.1) F V}{R T}} + 0.0365 sigma ⅇ^{\frac{(- F) V}{R T}}} sigma = \frac{1}{7} (ⅇ^{\frac{Na_o}{67.3}} - 1)

Component: i_Ca_p

i_Ca_p = \frac{i_Ca_p_max Ca_i}{km_Ca_p + Ca_i}

Component: CT_K_Cl

CT_K_Cl = \frac{CT_K_Cl_max (E_K - E_Cl)}{E_K + 87.8251 - E_Cl}

Component: CT_Na_Cl

CT_Na_Cl = \frac{CT_Na_Cl_max {(E_Na - E_Cl)}^{4}}{{(E_Na - E_Cl)}^{4} + {87.8251}^{4}}

Component: background_currents

i_Na_b = g_Na_b (V - E_Na) i_K_b = g_K_b (V - E_K) i_Cl_b = g_Cl_b (V - E_Cl) i_Ca_b = \frac{\frac{\frac{1 p_Ca_b}{Cm} {z_Ca}^{2} V F^{2}}{R T} (gamma_Cai Ca_i ⅇ^{\frac{z_Ca V F}{R T}} - (gamma_Cao Ca_o))}{ⅇ^{\frac{z_Ca V F}{R T}} - 1}

Component: intracellular_ion_concentrations

\frac{d}{d time} (Na_i) = \frac{(- Cm) (i_Na + i_Na_L + i_Na_b + 3 i_NaK + 3 i_NaCa) a_cap}{Vol_myo F} + CT_Na_Cl \frac{d}{d time} (K_i) = \frac{(- Cm) (i_to_1 + i_K1 + i_Kr + i_Ks + i_K_p + i_K_b - (2 i_NaK)) a_cap}{Vol_myo F} + CT_K_Cl \frac{d}{d time} (Cl_i) = \frac{(- Cm) (i_to_2 + i_Cl_b) a_cap}{Vol_myo F} + CT_Na_Cl + CT_K_Cl

Component: Ca_i

\frac{d}{d time} (Ca_i) = (- b_myo) (\frac{Cm (i_Ca_b + i_Ca_p - (2 i_NaCa)) a_cap}{2 Vol_myo F} + \frac{(q_up - q_leak) Vol_nsr}{Vol_myo} - (\frac{q_diff Vol_ss}{Vol_myo})) CMDN = \frac{2 CMDN_max Ca_i}{{(Ca_i + km_CMDN)}^{2}} TRPN = \frac{2 TRPN_max Ca_i}{{(Ca_i + km_TRPN)}^{2}} b_myo = \frac{1}{1 + TRPN + CMDN}

Component: Ca_MK_act

Ca_MK_act = Ca_MK_bound + Ca_MK_trap \frac{d}{d time} (Ca_MK_trap) = alpha_Ca_MK Ca_MK_bound (Ca_MK_bound + Ca_MK_trap) - (beta_Ca_MK Ca_MK_trap) Ca_MK_bound = \frac{Ca_MK_0 (1 - Ca_MK_trap)}{1 + \frac{km_Ca_MK}{Ca_r}}

Component: Ca_NSR

\frac{d}{d time} (Ca_NSR) = q_up - (q_leak + \frac{q_tr Vol_jsr}{Vol_nsr})

Component: Ca_JSR

\frac{d}{d time} (Ca_JSR) = \frac{q_tr - q_rel}{1 + \frac{CSQN_max km_CSQN}{{(km_CSQN + Ca_JSR)}^{2}}}

Component: Ca_r

\frac{d}{d time} (Ca_r) = Ca_r_tot (\frac{(- Cm) i_Ca_L a_cap}{Vol_ss z_Ca F} + \frac{q_rel Vol_jsr}{Vol_ss} - q_diff) q_diff = \frac{Ca_r - Ca_i}{tau_ss} b_SL = \frac{2 b_SL_max Ca_r}{{(Ca_r + km_b_SL)}^{2}} b_SR = \frac{2 b_SR_max Ca_r}{{(Ca_r + km_b_SR)}^{2}} Ca_r_tot = \frac{1}{1 + b_SR + b_SL}

Component: q_rel

q_rel = g_rel ro ri (Ca_JSR - Ca_r) g_rel = 3000 vg vg = \frac{1}{1 + ⅇ^{\frac{g_Ca_L i_Ca_L_max + 13}{5}}}

Component: q_rel_ri_gate

\frac{d}{d time} (ri) = \frac{ri_infinity - ri}{tau_ri} tau_ri = \frac{350 - tau_Ca_MK}{1 + ⅇ^{\frac{(Ca_r - 0.003) + 0.003 Ca_fac}{0.0002}}} + 3 + tau_Ca_MK ri_infinity = \frac{1}{1 + ⅇ^{\frac{(Ca_r - 0.0004) + 0.002 Ca_fac}{0.000025}}} Ca_fac = \frac{1}{1 + ⅇ^{\frac{i_Ca_L + 0.05}{0.015}}} tau_Ca_MK = \frac{tau_Ca_MK_max 1 Ca_MK_act}{km_Ca_MK + 1 Ca_MK_act}

Component: q_rel_ro_gate

\frac{d}{d time} (ro) = \frac{ro_infinity - ro}{tau_ro} ro_infinity = \frac{ro_infinity_JSR {i_Ca_L}^{2}}{{i_Ca_L}^{2} + 1} ro_infinity_JSR = \frac{{Ca_JSR}^{1.9}}{{Ca_JSR}^{1.9} + {(\frac{49.28 Ca_r}{Ca_r + 0.0028})}^{1.9}}

Component: q_leak

q_leak = \frac{q_leak_max Ca_NSR}{NSR_max}

Component: q_up

q_up = \frac{X_q_up (dq_up_Ca_MK + 1) q_up_max Ca_i}{Ca_i + km_up - dkm_plb} dq_up_Ca_MK = \frac{dq_up_Ca_MK_max Ca_MK_act 1}{km_Ca_MK + Ca_MK_act 1} dkm_plb = \frac{dkm_plb_max Ca_MK_act 1}{km_Ca_MK + Ca_MK_act 1}

Component: q_tr

q_tr = \frac{Ca_NSR - Ca_JSR}{tau_tr}

Component: model_parameters

a_geo = 2 3.14 {radius}^{2} + 2 3.14 radius length a_cap = rcg a_geo Vol_myo = Vol_cell 0.68 Vol_nsr = Vol_cell 0.0552 Vol_jsr = Vol_cell 0.0048 Vol_ss = Vol_cell 0.02