# Model Mathematics

### Component: membrane

$dd time V =- i_sp + i_Na + i_K + i_l + i_Ca + i_SK + i_ex Cm$

### Component: excitatory_synapse

$i_sp = G ⁢ V - E_rev dd time G = L ⁢ 1 tau1 - D ⁢ 1 tau2$

### Component: sodium_channel

$i_Na = g_Na ⁢ m 3.0 ⁢ h ⁢ V - E_Na$

### Component: sodium_channel_m_gate

$alpha_m = 0.109 ⁢ V + 38.0 1.0 -ⅇ- V + 38.0 5.0 beta_m = 0.0744 ⁢ V + 38.0 ⅇ V + 38.0 5.0 - 1.0 dd time m = alpha_m ⁢ 1.0 - m - beta_m ⁢ m$

### Component: sodium_channel_h_gate

$alpha_h = 0.0192 ⁢ⅇ- 55.0 + V 15.0 beta_h = 2.48 1.0 +ⅇ 17.0 - V 21.0 dd time h = alpha_h ⁢ 1.0 - h - beta_h ⁢ h$

### Component: potassium_channel

$i_K = g_K ⁢ n 4.0 ⁢ V - E_K$

### Component: potassium_channel_n_gate

$alpha_n = 0.012 ⁢ V + 45.0 1.0 -ⅇ- 45.0 + V 5.0 beta_n = 0.204 ⁢ⅇ- V + 50.0 40.0 dd time n = alpha_n ⁢ 1.0 - n - beta_n ⁢ n$

### Component: leak_channel

$i_l = g_l ⁢ V - E_l$

### Component: calcium_channel

$i_Ca = PCa ⁢ m 2.0 ⁢ h ⁢ ghk ghk = V ⁢ 2.0 ⁢ F 2.0 R ⁢ T ⁢ Cao - Ca0 ⁢ⅇ 2.0 ⁢ V ⁢ F R ⁢ T 1.0 -ⅇ 2.0 ⁢ V ⁢ F R ⁢ T$

### Component: calcium_channel_m_gate

$alpha_m = 5.0 1.0 +ⅇ -0.075 ⁢ V - 5.0 beta_m = 0.05 ⁢ 2.0 - V 1.0 -ⅇ V - 2.0 8.0 dd time m = alpha_m ⁢ 1.0 - m - beta_m ⁢ m$

### Component: calcium_channel_h_gate

$h = Ff ⁢ⅇ- time tau_f + Fm ⁢ⅇ- time tau_m + Fs ⁢ⅇ- time tau_s + Fr$

### Component: calcium_dependent_potassium_channel

$i_SK = g_SK ⁢ m 2.0 ⁢ V - E_K$

### Component: calcium_dependent_potassium_channel_m_gate

$m_infinity = Car 1.0 + Car tau_m = 1.0 beta_k ⁢ 1.0 + Car Car = Ca0 Cac n_SK dd time m =- m - m_infinity tau_m$

### Component: buffered_calcium_diffusion

$Cai = CaBu ⁢ k2b - Cai ⁢ Bu ⁢ k1b Bu = CaBu ⁢ k2b - Cai ⁢ Bu ⁢ k1b CaBu = Cai ⁢ Bu ⁢ k1b - CaBu ⁢ k2b kfc = DCa ⁢ Ai2 delta_r ⁢ vol_i kbc = DCa ⁢ Ai2 delta_r ⁢ vol_i$

### Component: calcium_extrusion

$i_ex = Vmax ⁢ Ca0 Kex + Ca0 - Vmax ⁢ Ca_rest Kex + Ca_rest$

### Component: calcium_triggered_exocytosis

$k1f = a1 ⁢ Ca0 b1 + Ca0 k1b = a2 k2 = a3 ⁢ Ca0 3.0 B = A ⁢ k1f - B ⁢ k1b + B ⁢ k2 C = B ⁢ k2 SR = k2 ⁢ B$

### Component: calcium_indicator

$Dy = Dyi2 ⁢ kbd - Dy ⁢ kfd kfd = Ddy ⁢ Ai2 delta_r ⁢ vol_i kbd = Ddy ⁢ Ai2 delta_r ⁢ vol_i CaDy = Dy ⁢ Cai ⁢ k1d - CaDy ⁢ k2d FDy = 1.0 - F0 ⁢ FCaDy + F0$

### Component: calcium_concentration

$dd time Ca0 = kbc ⁢ Ca1 + k2b ⁢ CaBu + k2d ⁢ CaDy - i_Ca + i_ex ⁢ A0 2.0 ⁢ F ⁢ vol0 + kfc ⁢ Ca0 + k1b ⁢ Bu ⁢ Ca0 + k1d ⁢ Dy ⁢ Ca0$

### Component: model_parameters

$delta_r = diameter 2.0 ⁢ Nc - 1.0$
Source
Derived from workspace Warashina, Ogura, 2004 at changeset 17f4aef262d8.
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