Model Mathematics

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{- 1}{Cm} (i_Na + i_b_Na + i_K1 + i_K + i_to + i_b_K + i_Ca_L + i_b_Ca + i_NaCa + i_NaK + i_Stim)

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{0.5 R T}{F} \ln (\frac{Ca_o}{Ca_i}) E_mh = \frac{R T}{F} \ln (\frac{Na_o + 0.12 K_o}{Na_i + 0.12 K_i})

i_Na = g_Na m^{3} h (V - E_mh)

E0_m = V + 41 alpha_m = \{\begin{matrix} 2000 if | E0_m | < delta_m \\ \frac{200 E0_m}{1 - (ⅇ^{(- 0.1) E0_m})} otherwise \end{matrix} beta_m = 8000 ⅇ^{(- 0.056) (V + 66)} \frac{d}{d time} (m) = alpha_m (1 - m) - (beta_m m)

alpha_h = 20 ⅇ^{(- 0.125) (V + 75 - shift_h)} beta_h = \frac{2000}{1 + 320 ⅇ^{(- 0.1) (V + 75 - shift_h)}} \frac{d}{d time} (h) = alpha_h (1 - h) - (beta_h h)

i_K = \frac{i_Kmax x (K_i - (K_o ⅇ^{\frac{(- V) F}{R T}}))}{140}

alpha_x = \frac{0.5 ⅇ^{0.0826 (V + 50)}}{1 + ⅇ^{0.057 (V + 50)}} beta_x = \frac{1.3 ⅇ^{(- 0.06) (V + 20)}}{1 + ⅇ^{(- 0.04) (V + 20)}} \frac{d}{d time} (x) = alpha_x (1 - x) - (beta_x x)

i_K1 = \frac{\frac{g_K1 K_o}{K_o + K_mk1} (V - E_K)}{1 + ⅇ^{\frac{2 F (V - E_K - 10)}{R T}}}

i_to = g_to s r (V - E_K)

alpha_s = 0.033 ⅇ^{\frac{- V}{17}} beta_s = \frac{33}{1 + ⅇ^{(- 0.125) (V + 10)}} \frac{d}{d time} (s) = alpha_s (1 - s) - (beta_s s)

\frac{d}{d time} (r) = 333 (\frac{1}{1 + ⅇ^{\frac{- (V + 4)}{5}}} - r)

i_Ca_L_Ca = \frac{4 d f P_Ca_L_Ca (V - 50) F}{R T (1 - (ⅇ^{\frac{(- 2) (V - 50) F}{R T}}))} (Ca_i ⅇ^{\frac{100 F}{R T}} - (Ca_o ⅇ^{\frac{(- 2) (V - 50) F}{R T}})) i_Ca_L_K = \frac{0.002 d f P_Ca_L_Ca (V - 50) F}{R T (1 - (ⅇ^{\frac{(- (V - 50)) F}{R T}}))} (K_i ⅇ^{\frac{50 F}{R T}} - (K_o ⅇ^{\frac{(- (V - 50)) F}{R T}})) i_Ca_L_Na = \frac{0.01 d f P_Ca_L_Ca (V - 50) F}{R T (1 - (ⅇ^{\frac{(- (V - 50)) F}{R T}}))} (Na_i ⅇ^{\frac{50 F}{R T}} - (Na_o ⅇ^{\frac{(- (V - 50)) F}{R T}})) i_Ca_L = i_Ca_L_Ca + i_Ca_L_K + i_Ca_L_Na

E0_d = V + 24 - 5 alpha_d = \{\begin{matrix} speed_d 120 if | E0_d | < 0.00001 \\ \frac{speed_d 30 E0_d}{1 - (ⅇ^{\frac{- E0_d}{4}})} otherwise \end{matrix} beta_d = \{\begin{matrix} speed_d 120 if | E0_d | < 0.00001 \\ \frac{speed_d (- 12) E0_d}{1 - (ⅇ^{\frac{E0_d}{10}})} otherwise \end{matrix} \frac{d}{d time} (d) = alpha_d (1 - d) - (beta_d d)

E0_f = V + 34 alpha_f = \{\begin{matrix} speed_f 25 if | E0_f | < 0.00001 \\ \frac{speed_f 6.25 E0_f}{(- 1) + ⅇ^{\frac{E0_f}{4}}} otherwise \end{matrix} beta_f = \frac{speed_f 50}{1 + ⅇ^{\frac{- E0_f}{4}}} \frac{d}{d time} (f) = alpha_f (1 - f) - (beta_f f)

i_NaCa = \frac{i_NaCa_max (ⅇ^{\frac{gamma V F}{R T}} {Na_i}^{3} Ca_o - (ⅇ^{\frac{(gamma - 1) V F}{R T}} {Na_o}^{3} Ca_i))}{1 + \frac{Ca_i}{0.0069}}

i_NaK = \frac{\frac{\frac{i_NaK_max K_o}{K_mK + K_o} Na_i}{K_mNa + Na_i} 1}{1 + 0.1245 ⅇ^{\frac{(- 0.1) V F}{R T}} + 0.0353 ⅇ^{\frac{(- V) F}{R T}}}

i_b_Ca = g_b_Ca (V - E_Ca)

i_b_K = g_b_K (V - E_K)

i_b_Na = g_b_Na (V - E_Na)

Inf_CaMK = \frac{Cmdn_Ca}{0.00005} \frac{d}{d time} (F_CaMK) = \frac{Inf_CaMK - F_CaMK}{Tau_CaMK}

N_CaMK = {(\frac{F_CaMK}{0.7})}^{2} k_1 = gain_k1 (30625000 {Ca_i}^{2} - (245 i_Ca_L)) k_2 = \frac{gain_k2 450}{1 + \frac{0.36}{Ca_SR}} k_3 = gain_k3 1.885 {(\frac{F_SRCa_RyR}{0.22})}^{N_CaMK} k_4 = gain_k4 1.8 \frac{d}{d time} (F_1) = k_3 F_3 - (k_4 F_1) - (k_1 F_1) \frac{d}{d time} (F_2) = k_1 F_1 - (k_2 F_2) F_3 = 1 - (F_1 + F_2) F_rel = {(\frac{F_2}{F_2 + 0.25})}^{2} \frac{d}{d time} (F_SRCa_RyR) = \frac{Ca_SR - F_SRCa_RyR}{Tau_SRCa_RyR} K_rel = \frac{K_rel_max F_SRCa_RyR}{F_SRCa_RyR + 0.2} j_rel = (K_rel F_rel + K_leak_rate) (Ca_SR - Ca_i)

f_b = {(\frac{Ca_i}{0.00024})}^{2} r_b = {(\frac{Ca_SR}{1.64})}^{2} j_up = \frac{F_CaMK V_max_f f_b - (V_max_r r_b)}{1 + f_b + r_b}

dCmdn_Ca_dtime = alpha_cmdn (Cmdn_tot - Cmdn_Ca) Ca_i - (beta_cmdn Cmdn_Ca) \frac{d}{d time} (Cmdn_Ca) = alpha_cmdn (Cmdn_tot - Cmdn_Ca) Ca_i - (beta_cmdn Cmdn_Ca)

dTrpn_Ca_dtime = alpha_trpn (Trpn_tot - Trpn_Ca) Ca_i - (\frac{beta_trpn (1 + 2 (1 - Force_norm))}{3} Trpn_Ca) \frac{d}{d time} (Trpn_Ca) = alpha_trpn (Trpn_tot - Trpn_Ca) Ca_i - (\frac{beta_trpn (1 + 2 (1 - Force_norm))}{3} Trpn_Ca)

\frac{d}{d time} (Ca_i) = (\frac{- (i_Ca_L_Ca + i_b_Ca - (2 i_NaCa))}{2 v_i F} - j_up) + \frac{j_rel v_SR}{v_i} - dCmdn_Ca_dtime - dTrpn_Ca_dtime

\frac{d}{d time} (Ca_SR) = \frac{j_up v_i}{v_SR} - j_rel

\frac{d}{d time} (Na_i) = \frac{- (i_Na + i_b_Na + i_Ca_L_Na + 3 i_NaCa + 3 i_NaK)}{v_i F}

\frac{d}{d time} (K_i) = \frac{- (i_K1 + i_K + i_to + i_b_K + i_Ca_L_K - (2 i_NaK))}{v_i F}

f_01 = 3 f_XB f_12 = 10 f_XB f_23 = 7 f_XB g_01 = g_XB (2 - SL_norm) g_12 = 2 g_XB (2 - SL_norm) g_23 = 3 g_XB (2 - SL_norm) SL_norm = \frac{SL - 1.7}{0.7} N_tm = 3.5 + 2.5 SL_norm K_tm = \frac{1}{1 + \frac{\frac{beta_trpn}{alpha_trpn}}{0.0017 - (0.0009 SL_norm)}} alpha_tm = beta_tm {(\frac{Trpn_Ca}{Trpn_tot K_tm})}^{N_tm} \frac{d}{d time} (N_0) = (beta_tm P_0 - (alpha_tm N_0)) + g_01 N_1 \frac{d}{d time} (P_0) = (- (beta_tm + f_01)) P_0 + alpha_tm N_0 + g_01 P_1 \frac{d}{d time} (P_1) = (- (beta_tm + f_12 + g_01)) P_1 + alpha_tm N_1 + f_01 P_0 + g_12 P_2 \frac{d}{d time} (P_2) = (- (f_23 + g_12)) P_2 + f_12 P_1 + g_23 P_3 \frac{d}{d time} (P_3) = (- g_23) P_3 + f_23 P_2 N_1 = 1 - (N_0 + P_0 + P_1 + P_2 + P_3) sigma_paths = 1 g_XB 2 g_XB 3 g_XB + 1 f_01 2 g_XB 3 g_XB + 1 f_01 1 f_12 3 g_XB + 1 f_01 1 f_12 1 f_23 P_1_max = \frac{1 f_01 2 g_XB 3 g_XB}{sigma_paths} P_2_max = \frac{1 f_01 1 f_12 3 g_XB}{sigma_paths} P_3_max = \frac{1 f_01 1 f_12 1 f_23}{sigma_paths} Force_max = P_1_max + 2 P_2_max + 3 P_3_max phi_SL = \{\begin{matrix} \frac{SL - 0.6}{1.4} if SL \geq 1.7 \land SL ≦ 2 \\ 1 if SL > 2 \land SL ≦ 2.2 \\ \frac{3.6 - SL}{1.4} if SL > 2.2 \land SL ≦ 2.3 \end{matrix} Force_norm = \frac{phi_SL (P_1 + N_1 + 2 P_2 + 3 P_3)}{Force_max} Force = zeta Force_norm