Model Mathematics

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Component: environment

Component: membrane

I_st = \{\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{- 1}{Cm} (i_Na + i_Ca_L + i_Ca_T + i_Kr + i_Ks + i_K_Na + i_K_ATP + i_to + i_K1 + i_Kp + i_NaCa + i_p_Ca + i_Na_b + i_Ca_b + i_NaK + i_ns_Ca + I_st) dVdt = \frac{- 1}{Cm} (i_Na + i_Ca_L + i_Ca_T + i_Kr + i_Ks + i_K_Na + i_K_ATP + i_to + i_K1 + i_Kp + i_NaCa + i_p_Ca + i_Na_b + i_Ca_b + i_NaK + i_ns_Ca + I_st)

Component: fast_sodium_current

i_Na = g_Na P_O_Na (V - E_Na) E_Na = \frac{R T}{F} \ln (\frac{Nao}{Nai})

Component: Na_channel_states

\frac{d}{d time} (P_C3) = (- (beta_3 + alpha_11)) P_C3 + alpha_3 P_IC3 + beta_11 P_C2 \frac{d}{d time} (P_C2) = (- (beta_11 + beta_3 + alpha_12)) P_C2 + alpha_11 P_C3 + beta_12 P_C1 + alpha_3 P_IC2 \frac{d}{d time} (P_C1) = (- (beta_12 + alpha_13 + beta_3)) P_C1 + alpha_12 P_C2 + beta_13 P_O_Na + alpha_3 P_IF \frac{d}{d time} (P_O_Na) = (- (alpha_2 + beta_13)) P_O_Na + beta_2 P_IF + alpha_13 P_C1 \frac{d}{d time} (P_IF) = (- (beta_2 + alpha_3 + alpha_4 + beta_12)) P_IF + beta_3 P_C1 + beta_4 P_IM1 + alpha_2 P_O_Na + alpha_12 P_IC2 \frac{d}{d time} (P_IC3) = (- (alpha_3 + alpha_11)) P_IC3 + beta_3 P_C3 + beta_11 P_IC2 \frac{d}{d time} (P_IC2) = (- (alpha_3 + alpha_12 + beta_11)) P_IC2 + beta_3 P_C2 + beta_12 P_IF + alpha_11 P_IC3 \frac{d}{d time} (P_IM1) = (- (alpha_5 + beta_4)) P_IM1 + beta_5 P_IM2 + alpha_4 P_IF \frac{d}{d time} (P_IM2) = alpha_5 P_IM1 - (beta_5 P_IM2) alpha_11 = \frac{3802}{0.1027 ⅇ^{\frac{- V}{17}} + 0.2 ⅇ^{\frac{- V}{150}}} alpha_12 = \frac{3802}{0.1027 ⅇ^{\frac{- V}{15}} + 0.23 ⅇ^{\frac{- V}{150}}} alpha_13 = \frac{3802}{0.1027 ⅇ^{\frac{- V}{12}} + 0.25 ⅇ^{\frac{- V}{150}}} beta_11 = 191.7 ⅇ^{\frac{- V}{20.3}} beta_12 = 200 ⅇ^{\frac{- (V - 5)}{20.3}} beta_13 = 220 ⅇ^{\frac{- (V - 10)}{20.3}} alpha_2 = 9178 ⅇ^{\frac{V}{29.68}} beta_2 = \frac{alpha_13 alpha_2 alpha_3}{beta_13 beta_3} alpha_3 = 3.7933 e -4 ⅇ^{\frac{- V}{7.7}} beta_3 = 8.4 + 0.02 V alpha_4 = \frac{alpha_2}{100} beta_4 = alpha_3 alpha_5 = \frac{alpha_2}{9.5 e 4} beta_5 = \frac{alpha_3}{50}

Component: L_type_Ca_channel

I_CaCa = \frac{\frac{P_Ca 2^{2} V F^{2}}{R T} (gamma_Cai Cai ⅇ^{\frac{2 V F}{R T}} - (gamma_Cao Cao))}{ⅇ^{\frac{2 V F}{R T}} - 1} I_CaNa = \frac{\frac{P_Na 1^{2} V F^{2}}{R T} (gamma_Nai Nai ⅇ^{\frac{1 V F}{R T}} - (gamma_Nao Nao))}{ⅇ^{\frac{1 V F}{R T}} - 1} I_CaK = \frac{\frac{P_K 1^{2} V F^{2}}{R T} (gamma_Ki Ki ⅇ^{\frac{1 V F}{R T}} - (gamma_Ko Ko))}{ⅇ^{\frac{1 V F}{R T}} - 1} i_CaCa = d f f_Ca I_CaCa i_CaNa = d f f_Ca I_CaNa i_CaK = d f f_Ca I_CaK i_Ca_L = i_CaCa + i_CaK + i_CaNa

Component: L_type_Ca_channel_d_gate

E0_d = V + 10 d_infinity = \frac{1}{1 + ⅇ^{\frac{- E0_d}{6.24}}} tau_d = \{\begin{matrix} \frac{0.001}{0.035 6.24} if | E0_d | < 1 e -5 \\ \frac{0.001 d_infinity (1 - (ⅇ^{\frac{- E0_d}{6.24}}))}{0.035 E0_d} otherwise \end{matrix} alpha_d = \frac{d_infinity}{tau_d} beta_d = \frac{1 - d_infinity}{tau_d} \frac{d}{d time} (d) = alpha_d (1 - d) - (beta_d d)

Component: L_type_Ca_channel_f_gate

f_infinity = \frac{1}{1 + ⅇ^{\frac{V + 32}{8}}} + \frac{0.6}{1 + ⅇ^{\frac{50 - V}{20}}} tau_f = \frac{0.001}{0.0197 ⅇ^{- ({(0.0337 (V + 10))}^{2})} + 0.02} alpha_f = \frac{f_infinity}{tau_f} beta_f = \frac{1 - f_infinity}{tau_f} \frac{d}{d time} (f) = alpha_f (1 - f) - (beta_f f)

Component: L_type_Ca_channel_f_Ca_gate

f_Ca = \frac{1}{1 + \frac{Cai}{Km_Ca}}

Component: T_type_Ca_channel

i_Ca_T = g_CaT b b g (V - E_Ca)

Component: T_type_Ca_channel_b_gate

b_inf = \frac{1}{1 + ⅇ^{\frac{- (V + 14)}{10.8}}} tau_b = 0.0037 + \frac{0.0061}{1 + ⅇ^{\frac{V + 25}{4.5}}} \frac{d}{d time} (b) = \frac{b_inf - b}{tau_b}

Component: T_type_Ca_channel_g_gate

g_inf = \frac{1}{1 + ⅇ^{\frac{V + 60}{5.6}}} tau_g = \{\begin{matrix} (- 0.000875) V + 0.012 if V ≦ 0 \\ 0.012 otherwise \end{matrix} \frac{d}{d time} (g) = \frac{g_inf - g}{tau_g}

Component: rapid_delayed_rectifier_potassium_current

i_Kr = g_Kr P_O (V - E_Kr) g_Kr = 0.0135 {Ko}^{0.59} E_Kr = \frac{R T}{F} \ln (\frac{Ko}{Ki})

Component: Kr_channel_states

\frac{d}{d time} (P_C3) = beta P_C2 - (alpha P_C3) \frac{d}{d time} (P_C2) = (- (beta + alpha_in)) P_C2 + alpha P_C3 + beta_in P_C1 \frac{d}{d time} (P_C1) = (- (beta_in + alpha_alpha + alpha_alpha)) P_C1 + alpha_in P_C2 + beta_beta P_O + mu P_I \frac{d}{d time} (P_O) = (- (beta_beta + beta_i)) P_O + alpha_alpha P_C1 + alpha_i P_I \frac{d}{d time} (P_I) = (- (mu + alpha_i)) P_I + alpha_alpha P_C1 + beta_i P_O alpha = 55.5 ⅇ^{0.05547153 (V - 12)} beta = 2.357 ⅇ^{(- 0.036588) V} alpha_alpha = 65.5 ⅇ^{0.05547153 (V - 36)} beta_beta = 2.9357 ⅇ^{(- 0.02158) V} alpha_i = \frac{439 ⅇ^{(- 0.02352) (V + 25)} 4.5}{Ko} beta_i = \frac{656 ⅇ^{0.000942 V} {4.5}^{0.3}}{{Ko}^{0.3}} mu = \frac{alpha_i beta_beta alpha_alpha}{alpha_alpha beta_i}

Component: slow_delayed_rectifier_potassium_current

E_Ks = \frac{R T}{F} \ln (\frac{Ko + PNaK Nao}{Ki + PNaK Nai}) g_Ks = 0.968 0.433 (1 + \frac{0.6}{1 + {(\frac{3.8 e -5}{Cai})}^{1.4}}) i_Ks = g_Ks xs1 xs2 (V - E_Ks)

Component: slow_delayed_rectifier_potassium_current_xs1_gate

xs1_infinity = \frac{1}{1 + ⅇ^{\frac{- (V - 1.5)}{16.7}}} tau_xs1 = \frac{0.001}{\frac{7.19 e -5 (V + 30)}{1 - (ⅇ^{(- 0.148) (V + 30)})} + \frac{0.000131 (V + 30)}{ⅇ^{0.0687 (V + 30)} - 1}} \frac{d}{d time} (xs1) = \frac{xs1_infinity - xs1}{tau_xs1}

Component: slow_delayed_rectifier_potassium_current_xs2_gate

xs2_infinity = \frac{1}{1 + ⅇ^{\frac{- (V - 1.5)}{16.7}}} tau_xs2 = \frac{4 0.001}{\frac{7.19 e -5 (V + 30)}{1 - (ⅇ^{(- 0.148) (V + 30)})} + \frac{0.000131 (V + 30)}{ⅇ^{0.0687 (V + 30)} - 1}} \frac{d}{d time} (xs2) = \frac{xs2_infinity - xs2}{tau_xs2}

Component: time_independent_potassium_current

g_K1 = 0.75 \sqrt{\frac{Ko}{5.4}} E_K = \frac{R T}{F} \ln (\frac{Ko}{Ki}) i_K1 = g_K1 K1_infinity (V - E_K)

Component: time_independent_potassium_current_K1_gate

alpha_K1 = \frac{1020}{1 + ⅇ^{0.2385 (V - E_K - 59.215)}} beta_K1 = \frac{1000 (0.49124 ⅇ^{0.08032 ((V - E_K) + 5.476)} + ⅇ^{0.06175 (V - E_K - 594.31)})}{1 + ⅇ^{(- 0.5143) ((V - E_K) + 4.753)}} K1_infinity = \frac{alpha_K1}{alpha_K1 + beta_K1}

Component: plateau_potassium_current

Kp = \frac{1}{1 + ⅇ^{\frac{7.488 - V}{5.98}}} i_Kp = g_Kp Kp (V - E_K)

Component: sodium_activated_potassium_current

g_K_Na = 0 0.12848 pona = \frac{0.85}{1 + {(\frac{kdKNa}{Nai})}^{nKNa}} pov = 0.8 - (\frac{0.65}{1 + ⅇ^{\frac{V + 125}{15}}}) i_K_Na = g_K_Na pona pov (V - E_K)

Component: ATP_sensitive_potassium_current

g_K_ATP = \frac{i_K_ATP_on 0.000193}{nicholsarea} pATP = \frac{1}{1 + {(\frac{ATPi}{kATP})}^{hATP}} GKbaraATP = g_K_ATP pATP {(\frac{Ko}{4})}^{nATP} i_K_ATP = GKbaraATP (V - E_K)

Component: transient_outward_current

g_to = 0.05 rvdv = ⅇ^{\frac{V}{100}} i_to = g_to {zdv}^{3} ydv rvdv (V - E_K)

Component: transient_outward_current_zdv_gate

alpha_zdv = \frac{10000 ⅇ^{\frac{V - 40}{25}}}{1 + ⅇ^{\frac{V - 40}{25}}} beta_zdv = \frac{10000 ⅇ^{\frac{- (V + 90)}{25}}}{1 + ⅇ^{\frac{- (V + 90)}{25}}} tau_zdv = \frac{1}{alpha_zdv + beta_zdv} zdv_ss = \frac{alpha_zdv}{alpha_zdv + beta_zdv} \frac{d}{d time} (zdv) = \frac{zdv_ss - zdv}{tau_zdv}

Component: transient_outward_current_ydv_gate

alpha_ydv = \frac{15}{1 + ⅇ^{\frac{V + 60}{5}}} beta_ydv = \frac{100 ⅇ^{\frac{V + 25}{5}}}{1 + ⅇ^{\frac{V + 25}{5}}} tau_ydv = \frac{1}{alpha_ydv + beta_ydv} ydv_ss = \frac{alpha_ydv}{alpha_ydv + beta_ydv} \frac{d}{d time} (ydv) = \frac{ydv_ss - ydv}{tau_ydv}

Component: sarcolemmal_calcium_pump

i_p_Ca = \frac{I_pCa Cai}{K_mpCa + Cai}

Component: sodium_background_current

i_Na_b = g_Nab (V - E_Na)

Component: calcium_background_current

E_Ca = \frac{R T}{2 F} \ln (\frac{Cao}{Cai}) i_Ca_b = g_Cab (V - E_Ca)

Component: sodium_potassium_pump

sigma = \frac{1}{7} (ⅇ^{\frac{Nao}{67.3}} - 1) f_NaK = \frac{1}{1 + 0.1245 ⅇ^{\frac{(- 0.1) V F}{R T}} + 0.0365 sigma ⅇ^{\frac{(- V) F}{R T}}} i_NaK = \frac{\frac{I_NaK f_NaK 1}{1 + {(\frac{K_mNai}{Nai})}^{2}} Ko}{Ko + K_mKo}

Component: non_specific_calcium_activated_current

P_ns_Ca = 0 1.75 e -7 I_ns_Na = \frac{\frac{P_ns_Ca 1^{2} V F^{2}}{R T} (gamma_Nai Nai ⅇ^{\frac{1 V F}{R T}} - (gamma_Nao Nao))}{ⅇ^{\frac{1 V F}{R T}} - 1} I_ns_K = \frac{\frac{P_ns_Ca 1^{2} V F^{2}}{R T} (gamma_Ki Ki ⅇ^{\frac{1 V F}{R T}} - (gamma_Ko Ko))}{ⅇ^{\frac{1 V F}{R T}} - 1} i_ns_Na = \frac{I_ns_Na 1}{1 + {(\frac{K_m_ns_Ca}{Cai})}^{3}} i_ns_K = \frac{I_ns_K 1}{1 + {(\frac{K_m_ns_Ca}{Cai})}^{3}} i_ns_Ca = i_ns_Na + i_ns_K

Component: Na_Ca_exchanger

i_NaCa = \frac{c1 ⅇ^{\frac{(gamma - 1) V F}{R T}} (ⅇ^{\frac{V F}{R T}} {Nai}^{3} Cao - ({Nao}^{3} Cai))}{1 + c2 ⅇ^{\frac{(gamma - 1) V F}{R T}} (ⅇ^{\frac{V F}{R T}} {Nai}^{3} Cao + {Nao}^{3} Cai)}

Component: calcium_dynamics

V_JSR = \frac{0.0048}{0.68} V_myo V_NSR = \frac{0.0552}{0.68} V_myo \frac{d}{d time} (APtrack) = \{\begin{matrix} 100000 (1 - APtrack) - (500 APtrack) if dVdt > 150000 \\ (- 500) APtrack otherwise \end{matrix} \frac{d}{d time} (APtrack2) = \{\begin{matrix} 100000 (1 - APtrack2) - (500 APtrack2) if APtrack < 0.2 \land APtrack > 0.18 \\ (- 500) APtrack2 otherwise \end{matrix} \frac{d}{d time} (APtrack3) = \{\begin{matrix} 100000 (1 - APtrack3) - (500 APtrack3) if APtrack < 0.2 \land APtrack > 0.18 \\ (- 10) APtrack3 otherwise \end{matrix} \frac{d}{d time} (Cainfluxtrack) = \{\begin{matrix} \frac{(- A_cap) ((i_CaCa + i_Ca_T - i_NaCa) + i_p_Ca + i_Ca_b)}{2 V_myo F} if APtrack > 0.2 \\ 0 if APtrack2 > 0.01 \land APtrack ≦ 0.2 \\ (- 500) Cainfluxtrack otherwise \end{matrix} \frac{d}{d time} (OVRLDtrack) = \{\begin{matrix} 50000 (1 - OVRLDtrack) if \frac{1}{1 + \frac{K_mCSQN}{Ca_NSR}} > CSQNthresh \land OVRLDtrack3 < 0.37 \land APtrack3 < 0.37 \\ (- 500) OVRLDtrack otherwise \end{matrix} \frac{d}{d time} (OVRLDtrack2) = \{\begin{matrix} 50000 (1 - OVRLDtrack2) if OVRLDtrack > Logicthresh \land OVRLDtrack2 < Logicthresh \\ (- 500) OVRLDtrack2 otherwise \end{matrix} \frac{d}{d time} (OVRLDtrack3) = \{\begin{matrix} 50000 (1 - OVRLDtrack3) if OVRLDtrack > Logicthresh \land OVRLDtrack3 < Logicthresh \\ (- 10) OVRLDtrack3 otherwise \end{matrix} G_rel = \{\begin{matrix} \frac{G_rel_max (Cainfluxtrack - delta_Ca_ith)}{K_mrel + Cainfluxtrack - delta_Ca_ith} (1 - APtrack2) APtrack2 if Cainfluxtrack > delta_Ca_ith \\ G_rel_overload (1 - OVRLDtrack2) OVRLDtrack2 if Cainfluxtrack ≦ delta_Ca_ith \land OVRLDtrack2 > 0 \\ 0 otherwise \end{matrix} i_rel = G_rel (Ca_JSR - Cai) i_up = \frac{I_up Cai}{Cai + K_mup} K_leak = \frac{I_up}{Ca_NSR_max} i_leak = K_leak Ca_NSR i_tr = \frac{Ca_NSR - Ca_JSR}{tau_tr} \frac{d}{d time} (Ca_JSR) = \frac{1}{1 + \frac{CSQN_max K_mCSQN}{{(K_mCSQN + Ca_JSR)}^{2}}} (i_tr - i_rel) \frac{d}{d time} (Ca_NSR) = (\frac{(- i_tr) V_JSR}{V_NSR} - i_leak) + i_up \frac{d}{d time} (Cai) = \frac{1}{1 + \frac{CMDN_max K_mCMDN}{{(K_mCMDN + Cai)}^{2}} + \frac{Tn_max K_mTn}{{(K_mTn + Cai)}^{2}}} (\frac{(- A_cap) ((i_CaCa + i_Ca_T - (2 i_NaCa)) + i_p_Ca + i_Ca_b)}{2 V_myo F} + \frac{i_rel V_JSR}{V_myo} + \frac{(i_leak - i_up) V_NSR}{V_myo})

Component: ionic_concentrations

volume = π preplength {radius}^{2} V_myo = 0.68 volume \frac{d}{d time} (Nai) = \frac{(- (i_Na + i_CaNa + i_Na_b + i_ns_Na + i_NaCa 3 + i_NaK 3)) A_cap}{V_myo F} \frac{d}{d time} (Ki) = \frac{(- (i_CaK + i_Kr + i_Ks + i_K1 + i_Kp + i_K_Na + i_K_ATP + i_to + i_ns_K + (- i_NaK) 2)) A_cap}{V_myo F}