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{- (i_Na + i_K1 + i_to + i_Kur + i_Kr + i_Ks + i_B_Na + i_B_Ca + i_NaK + i_CaP + i_NaCa + i_Ca_L + i_st)}{Cm}

Component: fast_sodium_current

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

Component: fast_sodium_current_m_gate

alpha_m = \{\begin{matrix} 3.2 if V = - 47.13 \\ \frac{0.32 (V + 47.13)}{1 - (ⅇ^{(- 0.1) (V + 47.13)})} otherwise \end{matrix} beta_m = 0.08 ⅇ^{\frac{- V}{11}} m_inf = \frac{alpha_m}{alpha_m + beta_m} tau_m = \frac{1}{alpha_m + beta_m} \frac{d}{d time} (m) = \frac{m_inf - m}{tau_m}

Component: fast_sodium_current_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} + 3.1 e 5 ⅇ^{0.35 V} if V < - 40 \\ \frac{1}{0.13 (1 + ⅇ^{\frac{V + 10.66}{- 11.1}})} otherwise \end{matrix} h_inf = \frac{alpha_h}{alpha_h + beta_h} tau_h = \frac{1}{alpha_h + beta_h} \frac{d}{d time} (h) = \frac{h_inf - h}{tau_h}

Component: fast_sodium_current_j_gate

alpha_j = \{\begin{matrix} \frac{((- 1.2714 e 5) ⅇ^{0.2444 V} - (3.474 e -5 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^{(- 2.535 e -7) V}}{1 + ⅇ^{(- 0.1) (V + 32)}} otherwise \end{matrix} j_inf = \frac{alpha_j}{alpha_j + beta_j} tau_j = \frac{1}{alpha_j + beta_j} \frac{d}{d time} (j) = \frac{j_inf - j}{tau_j}

Component: time_independent_potassium_current

E_K = \frac{R T}{F} \ln (\frac{K_o}{K_i}) i_K1 = \frac{Cm g_K1 (V - E_K)}{1 + ⅇ^{0.07 (V + 80)}}

Component: transient_outward_K_current

i_to = Cm g_to {oa}^{3} oi (V - E_K)

Component: transient_outward_K_current_oa_gate

alpha_oa = 0.65 {(ⅇ^{\frac{V - (- 10)}{- 8.5}} + ⅇ^{\frac{V - (- 10) - 40}{- 59}})}^{- 1} beta_oa = 0.65 {(2.5 + ⅇ^{\frac{(V - (- 10)) + 72}{17}})}^{- 1} tau_oa = \frac{{(alpha_oa + beta_oa)}^{- 1}}{K_Q10} oa_infinity = {(1 + ⅇ^{\frac{(V - (- 10)) + 10.47}{- 17.54}})}^{- 1} \frac{d}{d time} (oa) = \frac{oa_infinity - oa}{tau_oa}

Component: transient_outward_K_current_oi_gate

alpha_oi = {(18.53 + 1 ⅇ^{\frac{(V - (- 10)) + 103.7}{10.95}})}^{- 1} beta_oi = {(35.56 + 1 ⅇ^{\frac{V - (- 10) - 8.74}{- 7.44}})}^{- 1} tau_oi = \frac{{(alpha_oi + beta_oi)}^{- 1}}{K_Q10} oi_infinity = {(1 + ⅇ^{\frac{(V - (- 10)) + 33.1}{5.3}})}^{- 1} \frac{d}{d time} (oi) = \frac{oi_infinity - oi}{tau_oi}

Component: ultrarapid_delayed_rectifier_K_current

g_Kur = 0.005 + \frac{0.05}{1 + ⅇ^{\frac{V - 15}{- 13}}} i_Kur = Cm g_Kur {ua}^{3} ui (V - E_K)

Component: ultrarapid_delayed_rectifier_K_current_ua_gate

alpha_ua = 0.65 {(ⅇ^{\frac{V - (- 10)}{- 8.5}} + ⅇ^{\frac{V - (- 10) - 40}{- 59}})}^{- 1} beta_ua = 0.65 {(2.5 + ⅇ^{\frac{(V - (- 10)) + 72}{17}})}^{- 1} tau_ua = \frac{{(alpha_ua + beta_ua)}^{- 1}}{K_Q10} ua_infinity = {(1 + ⅇ^{\frac{(V - (- 10)) + 20.3}{- 9.6}})}^{- 1} \frac{d}{d time} (ua) = \frac{ua_infinity - ua}{tau_ua}

Component: ultrarapid_delayed_rectifier_K_current_ui_gate

alpha_ui = {(21 + 1 ⅇ^{\frac{V - (- 10) - 195}{- 28}})}^{- 1} beta_ui = \frac{1}{ⅇ^{\frac{V - (- 10) - 168}{- 16}}} tau_ui = \frac{{(alpha_ui + beta_ui)}^{- 1}}{K_Q10} ui_infinity = {(1 + ⅇ^{\frac{V - (- 10) - 109.45}{27.48}})}^{- 1} \frac{d}{d time} (ui) = \frac{ui_infinity - ui}{tau_ui}

Component: rapid_delayed_rectifier_K_current

i_Kr = \frac{Cm g_Kr xr (V - E_K)}{1 + ⅇ^{\frac{V + 15}{22.4}}}

Component: rapid_delayed_rectifier_K_current_xr_gate

alpha_xr = \{\begin{matrix} 0.0015 if | V + 14.1 | < 1 e -10 \\ \frac{0.0003 (V + 14.1)}{1 - (ⅇ^{\frac{V + 14.1}{- 5}})} otherwise \end{matrix} beta_xr = \{\begin{matrix} 3.7836118 e -4 if | V - 3.3328 | < 1 e -10 \\ \frac{0.000073898 (V - 3.3328)}{ⅇ^{\frac{V - 3.3328}{5.1237}} - 1} otherwise \end{matrix} tau_xr = {(alpha_xr + beta_xr)}^{- 1} xr_infinity = {(1 + ⅇ^{\frac{V + 14.1}{- 6.5}})}^{- 1} \frac{d}{d time} (xr) = \frac{xr_infinity - xr}{tau_xr}

Component: slow_delayed_rectifier_K_current

i_Ks = Cm g_Ks {xs}^{2} (V - E_K)

Component: slow_delayed_rectifier_K_current_xs_gate

alpha_xs = \{\begin{matrix} 0.00068 if | V - 19.9 | < 1 e -10 \\ \frac{0.00004 (V - 19.9)}{1 - (ⅇ^{\frac{V - 19.9}{- 17}})} otherwise \end{matrix} beta_xs = \{\begin{matrix} 0.000315 if | V - 19.9 | < 1 e -10 \\ \frac{0.000035 (V - 19.9)}{ⅇ^{\frac{V - 19.9}{9}} - 1} otherwise \end{matrix} tau_xs = 0.5 {(alpha_xs + beta_xs)}^{- 1} xs_infinity = {(1 + ⅇ^{\frac{V - 19.9}{- 12.7}})}^{- 0.5} \frac{d}{d time} (xs) = \frac{xs_infinity - xs}{tau_xs}

Component: L_type_Ca_channel

i_Ca_L = Cm g_Ca_L d f f_Ca (V - 65)

Component: L_type_Ca_channel_d_gate

d_infinity = {(1 + ⅇ^{\frac{V + 10}{- 8}})}^{- 1} tau_d = \{\begin{matrix} \frac{4.579}{1 + ⅇ^{\frac{V + 10}{- 6.24}}} if | V + 10 | < 1 e -10 \\ \frac{1 - (ⅇ^{\frac{V + 10}{- 6.24}})}{0.035 (V + 10) (1 + ⅇ^{\frac{V + 10}{- 6.24}})} otherwise \end{matrix} \frac{d}{d time} (d) = \frac{d_infinity - d}{tau_d}

Component: L_type_Ca_channel_f_gate

f_infinity = \frac{ⅇ^{\frac{- (V + 28)}{6.9}}}{1 + ⅇ^{\frac{- (V + 28)}{6.9}}} tau_f = 9 {(0.0197 ⅇ^{(- ({0.0337}^{2})) {(V + 10)}^{2}} + 0.02)}^{- 1} \frac{d}{d time} (f) = \frac{f_infinity - f}{tau_f}

Component: L_type_Ca_channel_f_Ca_gate

f_Ca_infinity = {(1 + \frac{Ca_i}{0.00035})}^{- 1} tau_f_Ca = 2 \frac{d}{d time} (f_Ca) = \frac{f_Ca_infinity - f_Ca}{tau_f_Ca}

Component: sodium_potassium_pump

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

Component: background_currents

E_Ca = \frac{R T}{2 F} \ln (\frac{Ca_o}{Ca_i}) i_B_Na = Cm g_B_Na (V - E_Na) i_B_Ca = Cm g_B_Ca (V - E_Ca) i_B_K = Cm g_B_K (V - E_K)

Component: Na_Ca_exchanger_current

i_NaCa = \frac{Cm I_NaCa_max (ⅇ^{\frac{gamma F V}{R T}} {Na_i}^{3} Ca_o - (ⅇ^{\frac{(gamma - 1) F V}{R T}} {Na_o}^{3} Ca_i))}{({K_mNa}^{3} + {Na_o}^{3}) (K_mCa + Ca_o) (1 + K_sat ⅇ^{\frac{(gamma - 1) V F}{R T}})}

Component: sarcolemmal_calcium_pump_current

i_CaP = \frac{Cm i_CaP_max Ca_i}{0.0005 + Ca_i}

Component: Ca_release_current_from_JSR

Fn = 1 e 3 (1 e -15 V_rel i_rel - (\frac{1 e -15}{2 F} (0.5 i_Ca_L - (0.2 i_NaCa)))) i_rel = K_rel u^{2} v w (Ca_rel - Ca_i)

Component: Ca_release_current_from_JSR_u_gate

tau_u = 8 u_infinity = {(1 + ⅇ^{\frac{- (Fn - 3.4175 e -13)}{13.67 e -16}})}^{- 1} \frac{d}{d time} (u) = \frac{u_infinity - u}{tau_u}

Component: Ca_release_current_from_JSR_v_gate

tau_v = 1.91 + 2.09 {(1 + ⅇ^{\frac{- (Fn - 3.4175 e -13)}{13.67 e -16}})}^{- 1} v_infinity = 1 - ({(1 + ⅇ^{\frac{- (Fn - 6.835 e -14)}{13.67 e -16}})}^{- 1}) \frac{d}{d time} (v) = \frac{v_infinity - v}{tau_v}

Component: Ca_release_current_from_JSR_w_gate

tau_w = \{\begin{matrix} \frac{6 0.2}{1.3} if | V - 7.9 | < 1 e -10 \\ \frac{6 (1 - (ⅇ^{\frac{- (V - 7.9)}{5}}))}{(1 + 0.3 ⅇ^{\frac{- (V - 7.9)}{5}}) 1 (V - 7.9)} otherwise \end{matrix} w_infinity = 1 - ({(1 + ⅇ^{\frac{- (V - 40)}{17}})}^{- 1}) \frac{d}{d time} (w) = \frac{w_infinity - w}{tau_w}

Component: transfer_current_from_NSR_to_JSR

i_tr = \frac{Ca_up - Ca_rel}{tau_tr}

Component: Ca_uptake_current_by_the_NSR

i_up = \frac{I_up_max}{1 + \frac{K_up}{Ca_i}}

Component: Ca_leak_current_by_the_NSR

i_up_leak = \frac{I_up_max Ca_up}{Ca_up_max}

Component: Ca_buffers

Ca_CMDN = \frac{CMDN_max Ca_i}{Ca_i + Km_CMDN} Ca_TRPN = \frac{TRPN_max Ca_i}{Ca_i + Km_TRPN} Ca_CSQN = \frac{CSQN_max Ca_rel}{Ca_rel + Km_CSQN}

Component: intracellular_ion_concentrations

V_i = V_cell 0.68 V_rel = 0.0048 V_cell V_up = 0.0552 V_cell \frac{d}{d time} (Na_i) = \frac{(- 3) i_NaK - (3 i_NaCa + i_B_Na + i_Na)}{V_i F} \frac{d}{d time} (K_i) = \frac{2 i_NaK - (i_K1 + i_to + i_Kur + i_Kr + i_Ks + i_B_K)}{V_i F} \frac{d}{d time} (Ca_i) = \frac{B1}{B2} B1 = \frac{2 i_NaCa - (i_CaP + i_Ca_L + i_B_Ca)}{2 V_i F} + \frac{V_up (i_up_leak - i_up) + i_rel V_rel}{V_i} B2 = 1 + \frac{TRPN_max Km_TRPN}{{(Ca_i + Km_TRPN)}^{2}} + \frac{CMDN_max Km_CMDN}{{(Ca_i + Km_CMDN)}^{2}} \frac{d}{d time} (Ca_up) = i_up - (i_up_leak + \frac{i_tr V_rel}{V_up}) \frac{d}{d time} (Ca_rel) = (i_tr - i_rel) {(1 + \frac{CSQN_max Km_CSQN}{{(Ca_rel + Km_CSQN)}^{2}})}^{- 1}

Model Mathematics

Component: environment

Component: membrane

Component: fast_sodium_current

Component: fast_sodium_current_m_gate

Component: fast_sodium_current_h_gate

Component: fast_sodium_current_j_gate

Component: time_independent_potassium_current

Component: transient_outward_K_current

Component: transient_outward_K_current_oa_gate

Component: transient_outward_K_current_oi_gate

Component: ultrarapid_delayed_rectifier_K_current

Component: ultrarapid_delayed_rectifier_K_current_ua_gate

Component: ultrarapid_delayed_rectifier_K_current_ui_gate

Component: rapid_delayed_rectifier_K_current

Component: rapid_delayed_rectifier_K_current_xr_gate

Component: slow_delayed_rectifier_K_current

Component: slow_delayed_rectifier_K_current_xs_gate

Component: L_type_Ca_channel

Component: L_type_Ca_channel_d_gate

Component: L_type_Ca_channel_f_gate

Component: L_type_Ca_channel_f_Ca_gate

Component: sodium_potassium_pump

Component: background_currents

Component: Na_Ca_exchanger_current

Component: sarcolemmal_calcium_pump_current

Component: Ca_release_current_from_JSR

Component: Ca_release_current_from_JSR_u_gate

Component: Ca_release_current_from_JSR_v_gate

Component: Ca_release_current_from_JSR_w_gate

Component: transfer_current_from_NSR_to_JSR

Component: Ca_uptake_current_by_the_NSR

Component: Ca_leak_current_by_the_NSR

Component: Ca_buffers

Component: intracellular_ion_concentrations

Component: standard_ionic_concentrations