# Model Mathematics

### Component: membrane

$I_star = g_Na ⁢ E_Na + g_Ca ⁢ E_Ca + g_K ⁢ E_K + g_L ⁢ E_L + g_KCa ⁢ E_K - g_inhib ⁢ E_inhib + g_ex ⁢ E_ex$$R_star = 1.0 g_Na + g_K + g_L + g_Ca + g_KCa - g_inhib + g_ex$$fr = -1.0 tau_m ⁢ln⁡ theta - R_star ⁢ I_star V_reset - R_star ⁢ I_star$$tau_m = Cm ⁢ R_star$$V_rest = R ⁢ T + T_abs F ⁢ln⁡ psi ⁢ 1000.0$$theta = V_rest + V_theta$$V_reset = V_rest + 4.0$$psi = beta_a 2 - 4 ⁢ alpha ⁢ c - beta_a 2 ⁢ alpha$$alpha = 4 ⁢ P_Ca ⁢ Ca_in ⁢ 0.001+ P_K ⁢ K_in + P_Na ⁢ Na_in + P_Cl ⁢ Cl_ex$$beta_a = P_K ⁢ K_in + P_Na ⁢ Na_in + P_Cl ⁢ Cl_ex - P_K ⁢ K_ex + P_Na ⁢ Na_ex + P_Cl ⁢ Cl_in$$c =- P_K ⁢ K_ex + 4 ⁢ P_Ca ⁢ Ca_ex ⁢ 0.001+ P_Na ⁢ Na_ex + P_Cl ⁢ Cl_in$$P_K = v_PK ⁢ BC npk K_PK + BC npk$$Res = V_R ⁢ V_rest K_R + V_rest$$K_in = K_ex theta_K$$Na_in = Na_ex theta_Na$$theta_K =ⅇ E_K k_q ⁢ T + T_abs ⁢ 1000.0$$theta_Na =ⅇ E_Na k_q ⁢ T + T_abs ⁢ 1000.0$$E_K = E_K_0 ⁢ T + T_abs T_room + T_abs$$E_Na = E_Na_0 ⁢ T + T_abs T_room + T_abs$$E_L = E_L_0 ⁢ T + T_abs T_room + T_abs$$E_Ca = k_q ⁢ T + T_abs 2.0 ⁢ln⁡ Ca_ex Ca_in ⁢ 1000.0$$E_inhib =- k_q ⁢ T + T_abs ⁢ln⁡ Cl_ex Cl_in ⁢ 1000.0$$i_Na = g_Na ⁢ V_rest - E_Na$$i_Na_abs = i_Na 2$$g_K = g_K0 + v_gk ⁢ MP K_gk + MP$$i_K = g_K ⁢ V_rest - E_K$$g_Ca = v_Ca ⁢ MP nca K_Ca + MP nca$$i_Ca = g_Ca ⁢ V_rest - E_Ca$$g_KCa = v_KCa ⁢ CC nkca K_KCa + CC nkca$$i_KCa = g_KCa ⁢ V_rest - E_K$$g_L = 1 Res$$i_L = g_L ⁢ V_rest - E_L$$g_ex = v_ex1 ⁢ i_Na_abs nex1 K_ex1 + i_Na_abs nex1 + v_ex2 ⁢ Ca_in nex2 K_ex2 + Ca_in nex2$$i_ex = g_ex ⁢ V_rest - E_ex$$i_inhib = g_inhib ⁢ V_rest - E_inhib$$GABA = GABA0 + v_GABA ⁢ VIP K_GABA + VIP$$beta = VIP VIP + K_D$$v_sPC = v_sP0 + C_T ⁢ CB K_C + CB$$Cl_in = Cl0 + v_Cl1 ⁢ MP K_Cl1 + MP + v_Cl2 ⁢ GABA nCl K_Cl2 + GABA nCl$$V_K = V_MK ⁢ Ca_in k_MK + Ca_in + V_b ⁢ beta k_b + beta$

### Component: Ca_in

$dd time Ca_in = v0 + v1 + v3 + kf ⁢ Ca_store - v2 + k ⁢ Ca_in v$$dd time Ca_store = v2 - v3 + kf ⁢ Ca_store$$k = v_kk ⁢ CC nkk K_kk + CC nkk$$v0 = v_v0 ⁢ BC nv0 K_v0 + BC nv0$$v1 = v_M1 ⁢ beta_IP3$$v2 = v_M2 ⁢ Ca_in n K_2 n + Ca_in n$$v3 = v_M3 ⁢ Ca_store m K_R m + Ca_store m ⁢ Ca_in p K_A p + Ca_in p$

### Component: VIP

$dd time VIP = v_VIP ⁢ fr n_VIP K_VIP + fr n_VIP - kd_VIP ⁢ VIP nd_VIP$

### Component: MP

$ddtime MP = v_sPC ⁢ BN n KAP n + BN n - vmP ⁢ MP KmP + MP + kdmp ⁢ MP$

### Component: MC

$ddtime MC = vsC ⁢ BN n KAC n + BN n - vmC ⁢ MC KmC + MC + kdmc ⁢ MC$

### Component: MB

$ddtime MB = vsB ⁢ KIB m KIB m + BN m - vmB ⁢ MB KmB + MB + kdmb ⁢ MB$

### Component: PC

$ddtime PC = ksP ⁢ MP + V2P ⁢ PCP Kdp + PCP + k4 ⁢ PCC - V1P ⁢ PC Kp + PC + k3 ⁢ PC ⁢ CC + kdn ⁢ PC$

### Component: CC

$ddtime CC = ksC ⁢ MC + V2C ⁢ CCP Kdp + CCP + k4 ⁢ PCC - V1C ⁢ CC Kp + CC + k3 ⁢ PC ⁢ CC + kdnc ⁢ CC$

### Component: PCP

$ddtime PCP = V1P ⁢ PC Kp + PC - V2P ⁢ PCP Kdp + PCP + vdPC ⁢ PCP Kd + PCP + kdn ⁢ PCP$

### Component: CCP

$ddtime CCP = V1C ⁢ CC Kp + CC - V2C ⁢ CCP Kdp + CCP + vdCC ⁢ CCP Kd + CCP + kdn ⁢ CCP$

### Component: PCC

$ddtime PCC = V2PC ⁢ PCCP Kdp + PCCP + k3 ⁢ PC ⁢ CC + k2 ⁢ PCN - V1PC ⁢ PCC Kp + PCC + k4 ⁢ PCC + k1 ⁢ PCC + kdn ⁢ PCC$

### Component: PCN

$ddtime PCN = V4PC ⁢ PCNP Kdp + PCNP + k1 ⁢ PCC + k8 ⁢ IN - V3PC ⁢ PCN Kp + PCN + k2 ⁢ PCN + k7 ⁢ BN ⁢ PCN + kdn ⁢ PCN$

### Component: PCCP

$ddtime PCCP = V1PC ⁢ PCC Kp + PCC - V2PC ⁢ PCCP Kdp + PCCP + vdPCC ⁢ PCCP Kd + PCCP + kdn ⁢ PCCP$

### Component: PCNP

$ddtime PCNP = V3PC ⁢ PCN Kp + PCN - V4PC ⁢ PCNP Kdp + PCNP + vdPCN ⁢ PCNP Kd + PCNP + kdn ⁢ PCNP$

### Component: BC

$ddtime BC = V2B ⁢ BCP Kdp + BCP + k6 ⁢ BN + ksB ⁢ MB - V1B ⁢ BC Kp + BC + k5 ⁢ BC + kdn ⁢ BC$

### Component: BCP

$ddtime BCP = V1B ⁢ BC Kp + BC - V2B ⁢ BCP Kdp + BCP + vdBC ⁢ BCP Kd + BCP + kdn ⁢ BCP$

### Component: BN

$ddtime BN = V4B ⁢ BNP Kdp + BNP + k5 ⁢ BC + k8 ⁢ IN - V3B ⁢ BN Kp + BN + k6 ⁢ BN + k7 ⁢ BN ⁢ PCN + kdn ⁢ BN$

### Component: BNP

$ddtime BNP = V3B ⁢ BN Kp + BN - V4B ⁢ BNP Kdp + BNP + vdBN ⁢ BNP Kd + BNP + kdn ⁢ BNP$

### Component: IN

$ddtime IN = k7 ⁢ BN ⁢ PCN - k8 ⁢ IN + vdIN ⁢ IN Kd + IN + kdn ⁢ IN$

### Component: CB

$ddtime CB = V_K ⁢ 1.0 - CB K1CB + 1.0 - CB - vP ⁢ CB K2CB + CB WT$

### Component: model_parameters

Source
Derived from workspace Vasalou and Henson 2010 at changeset b73ad1c71089.
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