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 \land time - stim_start - (⌊ \frac{time - stim_start}{stim_period} ⌋ stim_period) ≦ stim_duration \\ 0 otherwise \end{matrix} \frac{d}{d time} (V) = \frac{- (i_Na + i_Ca_L + i_t + i_ss + i_f + i_K1 + i_B + i_NaK + i_NaCa + i_Ca_P + i_Stim + I_inter)}{Cm}

Component: sodium_current

E_Na = \frac{R T}{F} \ln (\frac{Na_o}{Na_i}) i_Na = g_Na m^{3} h j (V - E_Na)

Component: sodium_current_m_gate

m_infinity = \frac{1}{1 + ⅇ^{\frac{V + 45}{- 6.5}}} tau_m = \frac{0.00136}{\frac{0.32 (V + 47.13)}{1 - (ⅇ^{(- 0.1) (V + 47.13)})} + 0.08 ⅇ^{\frac{- V}{11}}} \frac{d}{d time} (m) = \frac{m_infinity - m}{tau_m}

Component: sodium_current_h_gate

h_infinity = \frac{1}{1 + ⅇ^{\frac{V + 76.1}{6.07}}} tau_h = \{\begin{matrix} 0.0004537 (1 + ⅇ^{\frac{- (V + 10.66)}{11.1}}) if V \geq - 40 \\ \frac{0.00349}{0.135 ⅇ^{\frac{- (V + 80)}{6.8}} + 3.56 ⅇ^{0.079 V} + 310000 ⅇ^{0.35 V}} otherwise \end{matrix} \frac{d}{d time} (h) = \frac{h_infinity - h}{tau_h}

Component: sodium_current_j_gate

j_infinity = \frac{1}{1 + ⅇ^{\frac{V + 76.1}{6.07}}} tau_j = \{\begin{matrix} \frac{0.01163 (1 + ⅇ^{(- 0.1) (V + 32)})}{ⅇ^{(- 0.0000002535) V}} if V \geq - 40 \\ \frac{0.00349}{\frac{V + 37.78}{1 + ⅇ^{0.311 (V + 79.23)}} ((- 127140) ⅇ^{0.2444 V} - (0.00003474 ⅇ^{(- 0.04391) V})) + \frac{0.1212 ⅇ^{(- 0.01052) V}}{1 + ⅇ^{(- 0.1378) (V + 40.14)}}} otherwise \end{matrix} \frac{d}{d time} (j) = \frac{j_infinity - j}{tau_j}

Component: L_type_Ca_channel

i_Ca_L = g_Ca_L d ((0.9 + \frac{Ca_inact}{10}) f_11 + (0.1 - (\frac{Ca_inact}{10})) f_12) (V - E_Ca_L)

Component: L_type_Ca_channel_d_gate

d_infinity = \frac{1}{1 + ⅇ^{\frac{V + 15.3}{- 5}}} tau_d = 0.00305 ⅇ^{(- 0.0045) {(V + 7)}^{2}} + 0.00105 ⅇ^{(- 0.002) {(V - 18)}^{2}} + 0.00025 \frac{d}{d time} (d) = \frac{d_infinity - d}{tau_d}

Component: L_type_Ca_channel_f_11_gate

f_11_infinity = \frac{1}{1 + ⅇ^{\frac{V + 26.7}{5.4}}} tau_f_11 = 0.105 ⅇ^{- ({(\frac{V + 45}{12})}^{2})} + \frac{0.04}{1 + ⅇ^{\frac{(- V) + 25}{25}}} + \frac{0.015}{1 + ⅇ^{\frac{V + 75}{25}}} + 0.0017 \frac{d}{d time} (f_11) = \frac{f_11_infinity - f_11}{tau_f_11}

Component: L_type_Ca_channel_f_12_gate

f_12_infinity = \frac{1}{1 + ⅇ^{\frac{V + 26.7}{5.4}}} tau_f_12 = 0.041 ⅇ^{- ({(\frac{V + 47}{12})}^{2})} + \frac{0.08}{1 + ⅇ^{\frac{V + 55}{- 5}}} + \frac{0.015}{1 + ⅇ^{\frac{V + 75}{25}}} + 0.0017 \frac{d}{d time} (f_12) = \frac{f_12_infinity - f_12}{tau_f_12}

Component: L_type_Ca_channel_Ca_inact_gate

Ca_inact_infinity = \frac{1}{1 + \frac{Ca_ss}{0.01}} \frac{d}{d time} (Ca_inact) = \frac{Ca_inact_infinity - Ca_inact}{tau_Ca_inact}

Component: Ca_independent_transient_outward_K_current

E_K = \frac{R T}{F} \ln (\frac{K_o}{K_i}) i_t = g_t r (a s + b s_slow) (V - E_K)

Component: Ca_independent_transient_outward_K_current_r_gate

r_infinity = \frac{1}{1 + ⅇ^{\frac{V + 10.6}{- 11.42}}} tau_r = \frac{1}{45.16 ⅇ^{0.03577 (V + 50)} + 98.9 ⅇ^{(- 0.1) (V + 38)}} \frac{d}{d time} (r) = \frac{r_infinity - r}{tau_r}

Component: Ca_independent_transient_outward_K_current_s_gate

s_infinity = \frac{1}{1 + ⅇ^{\frac{V + 45.3}{6.8841}}} tau_s = 0.35 ⅇ^{- ({(\frac{V + 70}{15})}^{2})} + 0.035 \frac{d}{d time} (s) = \frac{s_infinity - s}{tau_s}

Component: Ca_independent_transient_outward_K_current_s_slow_gate

s_slow_infinity = \frac{1}{1 + ⅇ^{\frac{V + 45.3}{6.8841}}} tau_s_slow = 3.7 ⅇ^{- ({(\frac{V + 70}{30})}^{2})} + 0.035 \frac{d}{d time} (s_slow) = \frac{s_slow_infinity - s_slow}{tau_s_slow}

Component: steady_state_outward_K_current

i_ss = g_ss r_ss s_ss (V - E_K)

Component: steady_state_outward_K_current_r_ss_gate

r_ss_infinity = \frac{1}{1 + ⅇ^{\frac{V + 11.5}{- 11.82}}} tau_r_ss = \frac{10}{45.16 ⅇ^{0.03577 (V + 50)} + 98.9 ⅇ^{(- 0.1) (V + 38)}} \frac{d}{d time} (r_ss) = \frac{r_ss_infinity - r_ss}{tau_r_ss}

Component: steady_state_outward_K_current_s_ss_gate

s_ss_infinity = \frac{1}{1 + ⅇ^{\frac{V + 87.5}{10.3}}} tau_s_ss = 2.1 \frac{d}{d time} (s_ss) = \frac{s_ss_infinity - s_ss}{tau_s_ss}

Component: inward_rectifier

i_K1 = \frac{(\frac{48}{ⅇ^{\frac{V + 37}{25}} + ⅇ^{\frac{V + 37}{- 25}}} + 10) 0.001}{1 + ⅇ^{\frac{V - (E_K + 76.77)}{- 17}}} + \frac{g_K1 (V - (E_K + 1.73))}{(1 + ⅇ^{\frac{1.613 F (V - (E_K + 1.73))}{R T}}) (1 + ⅇ^{\frac{K_o - 0.9988}{- 0.124}})}

Component: hyperpolarisation_activated_current

f_K = 1 - f_Na i_f_Na = g_f y f_Na (V - E_Na) i_f_K = g_f y f_K (V - E_K) i_f = i_f_Na + i_f_K

Component: hyperpolarisation_activated_current_y_gate

y_infinity = \frac{1}{1 + ⅇ^{\frac{V + 138.6}{10.48}}} tau_y = \frac{1}{0.11885 ⅇ^{\frac{V + 80}{28.37}} + 0.5623 ⅇ^{\frac{V + 80}{- 14.19}}} \frac{d}{d time} (y) = \frac{y_infinity - y}{tau_y}

Component: background_currents

E_Ca = \frac{0.5 R T}{F} \ln (\frac{Ca_o}{Ca_i}) i_B_Na = g_B_Na (V - E_Na) i_B_Ca = g_B_Ca (V - E_Ca) i_B_K = g_B_K (V - E_K) i_B = i_B_Na + i_B_Ca + i_B_K

Component: sodium_potassium_pump

sigma = \frac{ⅇ^{\frac{Na_o}{67.3}} - 1}{7} i_NaK = \frac{\frac{\frac{i_NaK_max 1}{1 + 0.1245 ⅇ^{\frac{(- 0.1) V F}{R T}} + 0.0365 sigma ⅇ^{\frac{(- V) F}{R T}}} K_o}{K_o + K_m_K} 1}{1 + {(\frac{K_m_Na}{Na_i})}^{1.5}}

Component: sarcolemmal_calcium_pump_current

i_Ca_P = \frac{i_Ca_P_max Ca_i}{Ca_i + 0.0004}

Component: Na_Ca_ion_exchanger_current

i_NaCa = \frac{K_NaCa ({Na_i}^{3} Ca_o ⅇ^{0.03743 V gamma_NaCa} - ({Na_o}^{3} Ca_i ⅇ^{0.03743 V (gamma_NaCa - 1)}))}{1 + d_NaCa (Ca_i {Na_o}^{3} + Ca_o {Na_i}^{3})}

Component: SR_Ca_release_channel

\frac{d}{d time} (P_C1) = (- k_a_plus) {(\frac{Ca_ss}{1})}^{n} P_C1 + k_a_minus P_O1 \frac{d}{d time} (P_O1) = (k_a_plus {(\frac{Ca_ss}{1})}^{n} P_C1 - (k_a_minus P_O1 + k_b_plus {(\frac{Ca_ss}{1})}^{m} P_O1 + k_c_plus P_O1)) + k_b_minus P_O2 + k_c_minus P_C2 \frac{d}{d time} (P_O2) = k_b_plus {(\frac{Ca_ss}{1})}^{m} P_O1 - (k_b_minus P_O2) \frac{d}{d time} (P_C2) = k_c_plus P_O1 - (k_c_minus P_C2) J_rel = v1 (P_O1 + P_O2) (Ca_JSR - Ca_ss)

Component: SERCA2a_pump

fb = {(\frac{Ca_i}{K_fb})}^{N_fb} rb = {(\frac{Ca_NSR}{K_rb})}^{N_rb} J_up = \frac{K_SR (Vmaxf fb - (Vmaxr rb))}{1 + fb + rb}

Component: intracellular_and_SR_Ca_fluxes

J_tr = \frac{Ca_NSR - Ca_JSR}{tau_tr} J_xfer = \frac{Ca_ss - Ca_i}{tau_xfer} J_HTRPNCa = k_htrpn_plus Ca_i (HTRPN_tot - HTRPNCa) - (k_htrpn_minus HTRPNCa) J_LTRPNCa = k_ltrpn_plus Ca_i (LTRPN_tot - LTRPNCa) - (k_ltrpn_minus LTRPNCa) J_trpn = J_HTRPNCa + J_LTRPNCa

Component: intracellular_ion_concentrations

beta_i = \frac{1}{1 + \frac{CMDN_tot K_mCMDN}{{(K_mCMDN + Ca_i)}^{2}} + \frac{EGTA_tot K_mEGTA}{{(K_mEGTA + Ca_i)}^{2}}} beta_SS = \frac{1}{1 + \frac{CMDN_tot K_mCMDN}{{(K_mCMDN + Ca_ss)}^{2}}} beta_JSR = \frac{1}{1 + \frac{CSQN_tot K_mCSQN}{{(K_mCSQN + Ca_JSR)}^{2}}} \frac{d}{d time} (Ca_i) = beta_i (J_xfer - (J_up + J_trpn + \frac{((i_B_Ca - (2 i_NaCa)) + i_Ca_P) 1}{2 V_myo F})) \frac{d}{d time} (Na_i) = \frac{(- (i_Na + i_B_Na + i_NaCa 3 + i_NaK 3 + i_f_Na)) 1}{V_myo F} \frac{d}{d time} (K_i) = \frac{(- (i_ss + i_B_K + i_t + i_K1 + i_f_K + i_NaK (- 2))) 1}{V_myo F} \frac{d}{d time} (Ca_ss) = beta_SS (\frac{J_rel V_JSR}{V_SS} - (\frac{J_xfer V_myo}{V_SS}) - (\frac{i_Ca_L 1}{2 V_SS F})) \frac{d}{d time} (Ca_JSR) = beta_JSR (J_tr - J_rel) \frac{d}{d time} (Ca_NSR) = \frac{J_up V_myo}{V_NSR} - (\frac{J_tr V_JSR}{V_NSR})

Component: standard_ionic_concentrations

Component: membrane_fibro

V_fibro = \{\begin{matrix} 1e-7 if | VmReal | < 1e-7 \\ VmReal otherwise \end{matrix} \frac{d}{d time} (VmReal) = \frac{- (I_Kir + I_Shkr + I_b + I_inter)}{Cm}

Component: I_Kir

I_Kir = GKir OKir \sqrt{Ko (1e-3)} (V_fibro - EK) OKir = \frac{1}{aKir + ⅇ^{\frac{bKir (V_fibro - EK) F}{R T}}} EK = \frac{R T}{F} \ln (\frac{Ko}{Ki})

Component: I_Shkr

I_Shkr = \frac{\frac{PShkr OShkr V_fibro F^{2}}{R T} (Ki - (Ko ⅇ^{\frac{(- V_fibro) F}{R T}}))}{1 - (ⅇ^{\frac{(- V_fibro) F}{R T}})} C0Shkr = \{\begin{matrix} 0 if C0ShkrReal < 0 \\ 1 if C0ShkrReal > 1 \\ C0ShkrReal otherwise \end{matrix} C1Shkr = \{\begin{matrix} 0 if C1ShkrReal < 0 \\ 1 if C1ShkrReal > 1 \\ C1ShkrReal otherwise \end{matrix} C2Shkr = \{\begin{matrix} 0 if C2ShkrReal < 0 \\ 1 if C2ShkrReal > 1 \\ C2ShkrReal otherwise \end{matrix} C3Shkr = \{\begin{matrix} 0 if C3ShkrReal < 0 \\ 1 if C3ShkrReal > 1 \\ C3ShkrReal otherwise \end{matrix} C4Shkr = \{\begin{matrix} 0 if C4ShkrReal < 0 \\ 1 if C4ShkrReal > 1 \\ C4ShkrReal otherwise \end{matrix} OShkr = \{\begin{matrix} 0 if OShkrReal < 0 \\ 1 if OShkrReal > 1 \\ OShkrReal otherwise \end{matrix} kv = kv0 ⅇ^{\frac{V_fibro zv F}{R T}} k_v = k_v0 ⅇ^{\frac{V_fibro z_v F}{R T}} \frac{d}{d time} (C0ShkrReal) = k_v C1Shkr - (4 kv C0Shkr) \frac{d}{d time} (C1ShkrReal) = 2 k_v C2Shkr + 4 kv C0Shkr - ((3 kv + k_v) C1Shkr) \frac{d}{d time} (C2ShkrReal) = 3 k_v C3Shkr + 3 kv C1Shkr - ((2 kv + 2 k_v) C2Shkr) \frac{d}{d time} (C3ShkrReal) = 4 k_v C4Shkr + 2 kv C2Shkr - ((kv + 3 k_v) C3Shkr) \frac{d}{d time} (C4ShkrReal) = k_o OShkr + kv C3Shkr - ((ko + 4 k_v) C4Shkr) \frac{d}{d time} (OShkrReal) = ko C4Shkr - (k_o OShkr)

Component: I_b

I_b = Gb (V_fibro - Eb)

Component: I_inter

I_inter_fibro = \frac{1e+06 (V_fibro - V)}{R_mf} I_inter_myo = (-1) I_inter_fibro number_of_fibroblasts