Model Mathematics

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Component: environment

Component: membrane

FCell = \{\begin{matrix} \frac{1.07 (3 dCell - 0.1)}{3 (1 + 0.7745 ⅇ^{\frac{- (3 dCell - 2.05)}{0.295}})} if Version = 0 \\ \frac{FCellConstant dCell}{1 + 0.7745 ⅇ^{\frac{- (3 dCell - 2.05)}{0.295}}} if Version = 1 \\ \frac{1.07 29 dCell}{30 (1 + 0.7745 ⅇ^{\frac{- (29 dCell - 24.5)}{1.95}})} otherwise \end{matrix} Cm = CmCentre + FCell (CmPeriphery - CmCentre) \frac{d}{d time} (V) = \frac{- 1}{Cm} (i_Na + i_Ca_L + i_Ca_T + i_to + i_sus + i_K_r + i_K_s + i_f_Na + i_f_K + i_b_Na + i_b_Ca + i_b_K + i_NaCa + i_p + i_Ca_p)

Component: sodium_current

g_Na = \{\begin{matrix} g_Na_Centre_Published + FCell (g_Na_Periphery_Published - g_Na_Centre_Published) if Version = 0 \\ g_Na_Centre_0DCapable + FCell (g_Na_Periphery_0DCapable - g_Na_Centre_0DCapable) if Version = 1 \\ g_Na_Centre_1DCapable + FCell (g_Na_Periphery_1DCapable - g_Na_Centre_1DCapable) otherwise \end{matrix} i_Na = \frac{\frac{g_Na m^{3} h Na_o F^{2}}{R T} (ⅇ^{\frac{(V - E_Na) F}{R T}} - 1)}{ⅇ^{\frac{V F}{R T}} - 1} V

Component: sodium_current_m_gate

\frac{d}{d time} (m) = \frac{m_infinity - m}{tau_m} m_infinity = \{\begin{matrix} {(\frac{1}{1 + ⅇ^{\frac{- V}{5.46}}})}^{\frac{1}{3}} if Version = 0 \\ {(\frac{1}{1 + ⅇ^{\frac{- (V + 30.32)}{5.46}}})}^{\frac{1}{3}} otherwise \end{matrix} tau_m = \{\begin{matrix} \frac{0.0006247}{0.832 ⅇ^{(- 0.335) (V + 56.7)} + 0.627 ⅇ^{0.082 (V + 65.01)}} + 4 e -5 if Version = 0 \\ \frac{0.0006247}{0.8322166 ⅇ^{(- 0.33566) (V + 56.7062)} + 0.6274 ⅇ^{0.0823 (V + 65.0131)}} + 4.569 e -5 otherwise \end{matrix}

Component: sodium_current_h_gate

F_Na = \{\begin{matrix} \frac{0.0952 ⅇ^{(- 0.063) (V + 34.4)}}{1 + 1.66 ⅇ^{(- 0.225) (V + 63.7)}} + 0.0869 if Version = 0 \\ \frac{0.09518 ⅇ^{(- 0.06306) (V + 34.4)}}{1 + 1.662 ⅇ^{(- 0.2251) (V + 63.7)}} + 0.08693 otherwise \end{matrix} h = (1 - F_Na) h1 + F_Na h2 \frac{d}{d time} (h1) = \frac{h1_infinity - h1}{tau_h1} \frac{d}{d time} (h2) = \frac{h2_infinity - h2}{tau_h2} h1_infinity = \frac{1}{1 + ⅇ^{\frac{V + 66.1}{6.4}}} h2_infinity = h1_infinity tau_h1 = \frac{3.717 e -6 ⅇ^{(- 0.2815) (V + 17.11)}}{1 + 0.003732 ⅇ^{(- 0.3426) (V + 37.76)}} + 0.0005977 tau_h2 = \frac{3.186 e -8 ⅇ^{(- 0.6219) (V + 18.8)}}{1 + 7.189 e -5 ⅇ^{(- 0.6683) (V + 34.07)}} + 0.003556

Component: L_type_Ca_channel

g_Ca_L = \{\begin{matrix} g_Ca_L_Centre_Published + FCell (g_Ca_L_Periphery_Published - g_Ca_L_Centre_Published) if Version = 0 \\ g_Ca_L_Centre_0DCapable + FCell (g_Ca_L_Periphery_0DCapable - g_Ca_L_Centre_0DCapable) if Version = 1 \\ g_Ca_L_Centre_1DCapable + FCell (g_Ca_L_Periphery_1DCapable - g_Ca_L_Centre_1DCapable) otherwise \end{matrix} i_Ca_L = g_Ca_L (f_L d_L + \frac{0.006}{1 + ⅇ^{\frac{- (V + 14.1)}{6}}}) (V - E_Ca_L)

Component: L_type_Ca_channel_d_gate

\frac{d}{d time} (d_L) = \frac{d_L_infinity - d_L}{tau_d_L} alpha_d_L = \{\begin{matrix} \frac{(- 28.38) (V + 35)}{ⅇ^{\frac{- (V + 35)}{2.5}} - 1} - (\frac{84.9 V}{ⅇ^{(- 0.208) V} - 1}) if Version = 0 \\ \frac{(- 28.39) (V + 35)}{ⅇ^{\frac{- (V + 35)}{2.5}} - 1} - (\frac{84.9 V}{ⅇ^{(- 0.208) V} - 1}) if Version = 1 \\ \frac{(- 28.4) (V + 35)}{ⅇ^{\frac{- (V + 35)}{2.5}} - 1} - (\frac{84.9 V}{ⅇ^{(- 0.208) V} - 1}) otherwise \end{matrix} beta_d_L = \{\begin{matrix} \frac{11.43 (V - 5)}{ⅇ^{0.4 (V - 5)} - 1} if Version = 1 \\ \frac{11.42 (V - 5)}{ⅇ^{0.4 (V - 5)} - 1} otherwise \end{matrix} tau_d_L = \frac{2}{alpha_d_L + beta_d_L} d_L_infinity = \{\begin{matrix} \frac{1}{1 + ⅇ^{\frac{- (V + 23.1)}{6}}} if Version = 0 \\ \frac{1}{1 + ⅇ^{\frac{- (V + 22.3 + 0.8 FCell)}{6}}} if Version = 1 \\ \frac{1}{1 + ⅇ^{\frac{- (V + 22.2)}{6}}} otherwise \end{matrix}

Component: L_type_Ca_channel_f_gate

\frac{d}{d time} (f_L) = \frac{f_L_infinity - f_L}{tau_f_L} alpha_f_L = \{\begin{matrix} \frac{3.75 (V + 28)}{ⅇ^{\frac{V + 28}{4}} - 1} if Version = 1 \\ \frac{3.12 (V + 28)}{ⅇ^{\frac{V + 28}{4}} - 1} otherwise \end{matrix} beta_f_L = \{\begin{matrix} \frac{30}{1 + ⅇ^{\frac{- (V + 28)}{4}}} if Version = 1 \\ \frac{25}{1 + ⅇ^{\frac{- (V + 28)}{4}}} otherwise \end{matrix} tau_f_L = \{\begin{matrix} \frac{1.2 - (0.2 FCell)}{alpha_f_L + beta_f_L} if Version = 1 \\ \frac{1}{alpha_f_L + beta_f_L} otherwise \end{matrix} f_L_infinity = \frac{1}{1 + ⅇ^{\frac{V + 45}{5}}}

Component: T_type_Ca_channel

g_Ca_T = \{\begin{matrix} g_Ca_T_Centre_Published + FCell (g_Ca_T_Periphery_Published - g_Ca_T_Centre_Published) if Version = 0 \\ g_Ca_T_Centre_0DCapable + FCell (g_Ca_T_Periphery_0DCapable - g_Ca_T_Centre_0DCapable) if Version = 1 \\ g_Ca_T_Centre_1DCapable + FCell (g_Ca_T_Periphery_1DCapable - g_Ca_T_Centre_1DCapable) otherwise \end{matrix} i_Ca_T = g_Ca_T d_T f_T (V - E_Ca_T)

Component: T_type_Ca_channel_d_gate

\frac{d}{d time} (d_T) = \frac{d_T_infinity - d_T}{tau_d_T} alpha_d_T = 1068 ⅇ^{\frac{V + 26.3}{30}} beta_d_T = 1068 ⅇ^{\frac{- (V + 26.3)}{30}} tau_d_T = \frac{1}{alpha_d_T + beta_d_T} d_T_infinity = \frac{1}{1 + ⅇ^{\frac{- (V + 37)}{6.8}}}

Component: T_type_Ca_channel_f_gate

\frac{d}{d time} (f_T) = \frac{f_T_infinity - f_T}{tau_f_T} alpha_f_T = \{\begin{matrix} 15.3 ⅇ^{\frac{- (V + 71 + 0.7 FCell)}{83.3}} if Version = 1 \\ 15.3 ⅇ^{\frac{- (V + 71.7)}{83.3}} otherwise \end{matrix} beta_f_T = \{\begin{matrix} 15 ⅇ^{\frac{V + 71}{15.38}} if Version = 1 \\ 15 ⅇ^{\frac{V + 71.7}{15.38}} otherwise \end{matrix} tau_f_T = \frac{1}{alpha_f_T + beta_f_T} f_T_infinity = \frac{1}{1 + ⅇ^{\frac{V + 71}{9}}}

Component: four_AP_sensitive_currents

g_to = \{\begin{matrix} g_to_Centre_Published + FCell (g_to_Periphery_Published - g_to_Centre_Published) if Version = 0 \\ g_to_Centre_0DCapable + FCell (g_to_Periphery_0DCapable - g_to_Centre_0DCapable) if Version = 1 \\ g_to_Centre_1DCapable + FCell (g_to_Periphery_1DCapable - g_to_Centre_1DCapable) otherwise \end{matrix} g_sus = \{\begin{matrix} g_sus_Centre_Published + FCell (g_sus_Periphery_Published - g_sus_Centre_Published) if Version = 0 \\ g_sus_Centre_0DCapable + FCell (g_sus_Periphery_0DCapable - g_sus_Centre_0DCapable) if Version = 1 \\ g_sus_Centre_1DCapable + FCell (g_sus_Periphery_1DCapable - g_sus_Centre_1DCapable) otherwise \end{matrix} i_to = g_to q r (V - E_K) i_sus = g_sus r (V - E_K)

Component: four_AP_sensitive_currents_q_gate

\frac{d}{d time} (q) = \frac{q_infinity - q}{tau_q} q_infinity = \frac{1}{1 + ⅇ^{\frac{V + 59.37}{13.1}}} tau_q = \{\begin{matrix} 0.0101 + \frac{0.06517}{0.57 ⅇ^{(- 0.08) (V + 49)}} + 2.4 e -5 ⅇ^{0.1 (V + 50.93)} if Version = 0 \\ \frac{0.001}{3} (30.31 + \frac{195.5}{0.5686 ⅇ^{(- 0.08161) (V + 39 + 10 FCell)} + 0.7174 ⅇ^{(0.2719 - (0.1719 FCell)) 1 (V + 40.93 + 10 FCell)}}) if Version = 1 \\ 0.0101 + \frac{0.06517}{0.5686 ⅇ^{(- 0.08161) (V + 39)} + 0.7174 ⅇ^{0.2719 (V + 40.93)}} otherwise \end{matrix}

Component: four_AP_sensitive_currents_r_gate

\frac{d}{d time} (r) = \frac{r_infinity - r}{tau_r} r_infinity = \frac{1}{1 + ⅇ^{\frac{- (V - 10.93)}{19.7}}} tau_r = \{\begin{matrix} 0.001 (2.98 + \frac{15.59}{1.037 ⅇ^{0.09 (V + 30.61)} + 0.369 ⅇ^{(- 0.12) (V + 23.84)}}) if Version = 0 \\ 0.0025 (1.191 + \frac{7.838}{1.037 ⅇ^{0.09012 (V + 30.61)} + 0.369 ⅇ^{(- 0.119) (V + 23.84)}}) if Version = 1 \\ 0.001 (2.98 + \frac{19.59}{1.037 ⅇ^{0.09012 (V + 30.61)} + 0.369 ⅇ^{(- 0.119) (V + 23.84)}}) otherwise \end{matrix}

Component: rapid_delayed_rectifying_potassium_current

g_K_r = \{\begin{matrix} g_K_r_Centre_Published + FCell (g_K_r_Periphery_Published - g_K_r_Centre_Published) if Version = 0 \\ g_K_r_Centre_0DCapable + FCell (g_K_r_Periphery_0DCapable - g_K_r_Centre_0DCapable) if Version = 1 \\ g_K_r_Centre_1DCapable + FCell (g_K_r_Periphery_1DCapable - g_K_r_Centre_1DCapable) otherwise \end{matrix} i_K_r = g_K_r P_a P_i (V - E_K) P_a = 0.6 P_af + 0.4 P_as

Component: rapid_delayed_rectifying_potassium_current_P_af_gate

\frac{d}{d time} (P_af) = \frac{P_af_infinity - P_af}{tau_P_af} P_af_infinity = \{\begin{matrix} \frac{1}{1 + ⅇ^{\frac{- (V + 14.2)}{10.6}}} if Version \neq 2 \\ \frac{1}{1 + ⅇ^{\frac{- (V + 13.2)}{10.6}}} otherwise \end{matrix} tau_P_af = \{\begin{matrix} \frac{1}{37.2 ⅇ^{\frac{V - 9}{15.9}} + 0.96 ⅇ^{\frac{- (V - 9)}{22.5}}} if Version \neq 2 \\ \frac{1}{37.2 ⅇ^{\frac{V - 10}{15.9}} + 0.96 ⅇ^{\frac{- (V - 10)}{22.5}}} otherwise \end{matrix}

Component: rapid_delayed_rectifying_potassium_current_P_as_gate

\frac{d}{d time} (P_as) = \frac{P_as_infinity - P_as}{tau_P_as} P_as_infinity = P_af_infinity tau_P_as = \{\begin{matrix} \frac{1}{4.2 ⅇ^{\frac{V - 9}{17}} + 0.15 ⅇ^{\frac{- (V - 9)}{21.6}}} if Version \neq 2 \\ \frac{1}{4.2 ⅇ^{\frac{V - 10}{17}} + 0.15 ⅇ^{\frac{- (V - 10)}{21.6}}} otherwise \end{matrix}

Component: rapid_delayed_rectifying_potassium_current_P_i_gate

tau_P_i = \{\begin{matrix} 0.002 if Version = 0 \\ 0.002 if Version = 1 \\ 0.006 otherwise \end{matrix} \frac{d}{d time} (P_i) = \frac{P_i_infinity - P_i}{tau_P_i} P_i_infinity = \frac{1}{1 + ⅇ^{\frac{V + 18.6}{10.1}}}

Component: slow_delayed_rectifying_potassium_current

g_K_s = \{\begin{matrix} g_K_s_Centre_Published + FCell (g_K_s_Periphery_Published - g_K_s_Centre_Published) if Version = 0 \\ g_K_s_Centre_0DCapable + FCell (g_K_s_Periphery_0DCapable - g_K_s_Centre_0DCapable) if Version = 1 \\ g_K_s_Centre_1DCapable + FCell (g_K_s_Periphery_1DCapable - g_K_s_Centre_1DCapable) otherwise \end{matrix} i_K_s = g_K_s {xs}^{2} (V - E_K_s)

Component: slow_delayed_rectifying_potassium_current_xs_gate

\frac{d}{d time} (xs) = alpha_xs (1 - xs) - (beta_xs xs) alpha_xs = \frac{14}{1 + ⅇ^{\frac{- (V - 40)}{9}}} beta_xs = 1 ⅇ^{\frac{- V}{45}}

Component: hyperpolarisation_activated_current

g_f_Na = \{\begin{matrix} g_f_Na_Centre_Published + FCell (g_f_Na_Periphery_Published - g_f_Na_Centre_Published) if Version = 0 \\ g_f_Na_Centre_0DCapable + FCell (g_f_Na_Periphery_0DCapable - g_f_Na_Centre_0DCapable) if Version = 1 \\ g_f_Na_Centre_1DCapable + FCell (g_f_Na_Periphery_1DCapable - g_f_Na_Centre_1DCapable) otherwise \end{matrix} i_f_Na = \{\begin{matrix} g_f_Na y (V - E_Na) if Version \neq 2 \\ g_f_Na y (V - 77.6) otherwise \end{matrix} g_f_K = \{\begin{matrix} g_f_K_Centre_Published + FCell (g_f_K_Periphery_Published - g_f_K_Centre_Published) if Version = 0 \\ g_f_K_Centre_0DCapable + FCell (g_f_K_Periphery_0DCapable - g_f_K_Centre_0DCapable) if Version = 1 \\ g_f_K_Centre_1DCapable + FCell (g_f_K_Periphery_1DCapable - g_f_K_Centre_1DCapable) otherwise \end{matrix} i_f_K = \{\begin{matrix} g_f_K y (V - E_K) if Version \neq 2 \\ g_f_K y (V + 102) otherwise \end{matrix}

Component: hyperpolarisation_activated_current_y_gate

\frac{d}{d time} (y) = alpha_y (1 - y) - (beta_y y) alpha_y = \{\begin{matrix} 1 ⅇ^{\frac{- (V + 78.91)}{26.62}} if Version = 0 \\ 1 ⅇ^{\frac{- (V + 78.91)}{26.63}} otherwise \end{matrix} beta_y = 1 ⅇ^{\frac{V + 75.13}{21.25}}

Component: sodium_background_current

g_b_Na = \{\begin{matrix} g_b_Na_Centre_Published + FCell (g_b_Na_Periphery_Published - g_b_Na_Centre_Published) if Version = 0 \\ g_b_Na_Centre_0DCapable + FCell (g_b_Na_Periphery_0DCapable - g_b_Na_Centre_0DCapable) if Version = 1 \\ g_b_Na_Centre_1DCapable + FCell (g_b_Na_Periphery_1DCapable - g_b_Na_Centre_1DCapable) otherwise \end{matrix} i_b_Na = g_b_Na (V - E_Na)

Component: potassium_background_current

g_b_K = \{\begin{matrix} g_b_K_Centre_Published + FCell (g_b_K_Periphery_Published - g_b_K_Centre_Published) if Version = 0 \\ g_b_K_Centre_0DCapable + FCell (g_b_K_Periphery_0DCapable - g_b_K_Centre_0DCapable) if Version = 1 \\ g_b_K_Centre_1DCapable + FCell (g_b_K_Periphery_1DCapable - g_b_K_Centre_1DCapable) otherwise \end{matrix} i_b_K = g_b_K (V - E_K)

Component: calcium_background_current

g_b_Ca = \{\begin{matrix} g_b_Ca_Centre_Published + FCell (g_b_Ca_Periphery_Published - g_b_Ca_Centre_Published) if Version = 0 \\ g_b_Ca_Centre_0DCapable + FCell (g_b_Ca_Periphery_0DCapable - g_b_Ca_Centre_0DCapable) if Version = 1 \\ g_b_Ca_Centre_1DCapable + FCell (g_b_Ca_Periphery_1DCapable - g_b_Ca_Centre_1DCapable) otherwise \end{matrix} i_b_Ca = g_b_Ca (V - E_Ca)

Component: sodium_calcium_exchanger

k_NaCa = \{\begin{matrix} k_NaCa_Centre_Published + FCell (k_NaCa_Periphery_Published - k_NaCa_Centre_Published) if Version = 0 \\ k_NaCa_Centre_0DCapable + FCell (k_NaCa_Periphery_0DCapable - k_NaCa_Centre_0DCapable) if Version = 1 \\ k_NaCa_Centre_1DCapable + FCell (k_NaCa_Periphery_1DCapable - k_NaCa_Centre_1DCapable) otherwise \end{matrix} i_NaCa = \{\begin{matrix} \frac{k_NaCa ({Na_i}^{3} Ca_o ⅇ^{0.03743 V gamma_NaCa} - ({Na_o}^{3} Ca_i ⅇ^{0.0374 V (gamma_NaCa - 1)}))}{1 + d_NaCa (Ca_i {Na_o}^{3} + Ca_o {Na_i}^{3})} if Version = 0 \\ \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})} otherwise \end{matrix}

Component: sodium_potassium_pump

i_p_max = \{\begin{matrix} i_p_max_Centre_Published + FCell (i_p_max_Periphery_Published - i_p_max_Centre_Published) if Version = 0 \\ i_p_max_Centre_0DCapable + FCell (i_p_max_Periphery_0DCapable - i_p_max_Centre_0DCapable) if Version = 1 \\ i_p_max_Centre_1DCapable + FCell (i_p_max_Periphery_1DCapable - i_p_max_Centre_1DCapable) otherwise \end{matrix} i_p = \frac{i_p_max {(\frac{Na_i}{K_m_Na + Na_i})}^{3} {(\frac{K_o}{K_m_K + K_o})}^{2} 1.6}{1.5 + ⅇ^{\frac{- (V + 60)}{40}}}

Component: persistent_calcium_current

i_Ca_p_max = \{\begin{matrix} i_Ca_p_max_Centre_Published + FCell (i_Ca_p_max_Periphery_Published - i_Ca_p_max_Centre_Published) if Version = 0 \\ i_Ca_p_max_Centre_0DCapable + FCell (i_Ca_p_max_Periphery_0DCapable - i_Ca_p_max_Centre_0DCapable) if Version = 1 \\ i_Ca_p_max_Centre_1DCapable + FCell (i_Ca_p_max_Periphery_1DCapable - i_Ca_p_max_Centre_1DCapable) otherwise \end{matrix} i_Ca_p = \frac{i_Ca_p_max Ca_i}{Ca_i + 0.0004}

Component: ionic_concentrations

Component: reversal_and_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_K_s = \{\begin{matrix} \frac{R T}{F} \ln (\frac{K_o + 0.12 Na_o}{K_i + 0.12 Na_i}) if Version = 0 \\ \frac{R T}{F} \ln (\frac{K_o + 0.03 Na_o}{K_i + 0.03 Na_i}) otherwise \end{matrix}