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Model Mathematics

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Component: environment

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 = \frac{1.0}{g_Na + g_K + g_L + g_Ca + g_KCa - (g_inhib + g_ex)}

fr = \frac{-1.0}{tau_m \ln (\frac{theta - (R_star I_star)}{V_reset - (R_star I_star)})}

tau_m = Cm R_star

V_rest = \frac{R (T + T_abs)}{F} \ln psi 1000.0

theta = V_rest + V_theta

V_reset = V_rest + 4.0

psi = \frac{\sqrt{{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 \frac{{BC}^{npk}}{K_PK + {BC}^{npk}}

Res = V_R \frac{V_rest}{K_R + V_rest}

K_in = \frac{K_ex}{theta_K}

Na_in = \frac{Na_ex}{theta_Na}

theta_K = ⅇ^{\frac{E_K}{k_q (T + T_abs) 1000.0}}

theta_Na = ⅇ^{\frac{E_Na}{k_q (T + T_abs) 1000.0}}

E_K = \frac{E_K_0 (T + T_abs)}{T_room + T_abs}

E_Na = \frac{E_Na_0 (T + T_abs)}{T_room + T_abs}

E_L = \frac{E_L_0 (T + T_abs)}{T_room + T_abs}

E_Ca = \frac{k_q (T + T_abs)}{2.0} \ln (\frac{Ca_ex}{Ca_in}) 1000.0

E_inhib = (- k_q) (T + T_abs) \ln (\frac{Cl_ex}{Cl_in}) 1000.0

i_Na = g_Na (V_rest - E_Na)

i_Na_abs = \sqrt{{i_Na}^{2}}

g_K = g_K0 + v_gk \frac{MP}{K_gk + MP}

i_K = g_K (V_rest - E_K)

g_Ca = v_Ca \frac{{MP}^{nca}}{K_Ca + {MP}^{nca}}

i_Ca = g_Ca (V_rest - E_Ca)

g_KCa = v_KCa \frac{{CC}^{nkca}}{K_KCa + {CC}^{nkca}}

i_KCa = g_KCa (V_rest - E_K)

g_L = \frac{1}{Res}

i_L = g_L (V_rest - E_L)

g_ex = v_ex1 \frac{{i_Na_abs}^{nex1}}{K_ex1 + {i_Na_abs}^{nex1}} + v_ex2 \frac{{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 + \frac{v_GABA VIP}{K_GABA + VIP}

beta = \frac{VIP}{VIP + K_D}

v_sPC = v_sP0 + \frac{C_T CB}{K_C + CB}

Cl_in = Cl0 + v_Cl1 \frac{MP}{K_Cl1 + MP} + v_Cl2 \frac{{GABA}^{nCl}}{K_Cl2 + {GABA}^{nCl}}

V_K = \frac{V_MK Ca_in}{k_MK + Ca_in} + \frac{V_b beta}{k_b + beta}

Component: Ca_in

\frac{d}{d time} (Ca_in) = v0 + v1 + v3 + kf Ca_store - (v2 + k {Ca_in}^{v})

\frac{d}{d time} (Ca_store) = v2 - (v3 + kf Ca_store)

k = v_kk \frac{{CC}^{nkk}}{K_kk + {CC}^{nkk}}

v0 = v_v0 \frac{{BC}^{nv0}}{K_v0 + {BC}^{nv0}}

v1 = v_M1 beta_IP3

v2 = v_M2 \frac{{Ca_in}^{n}}{{K_2}^{n} + {Ca_in}^{n}}

v3 = v_M3 \frac{{Ca_store}^{m}}{{K_R}^{m} + {Ca_store}^{m}} \frac{{Ca_in}^{p}}{{K_A}^{p} + {Ca_in}^{p}}

Component: VIP

\frac{d}{d time} (VIP) = v_VIP \frac{{fr}^{n_VIP}}{K_VIP + {fr}^{n_VIP}} - (kd_VIP {VIP}^{nd_VIP})

Component: MP

\frac{d}{d time} (MP) = v_sPC \frac{{BN}^{n}}{{KAP}^{n} + {BN}^{n}} - (vmP \frac{MP}{KmP + MP} + kdmp MP)

Component: MC

\frac{d}{d time} (MC) = vsC \frac{{BN}^{n}}{{KAC}^{n} + {BN}^{n}} - (vmC \frac{MC}{KmC + MC} + kdmc MC)

Component: MB

\frac{d}{d time} (MB) = vsB \frac{{KIB}^{m}}{{KIB}^{m} + {BN}^{m}} - (vmB \frac{MB}{KmB + MB} + kdmb MB)

Component: PC

\frac{d}{d time} (PC) = ksP MP + V2P \frac{PCP}{Kdp + PCP} + k4 PCC - (V1P \frac{PC}{Kp + PC} + k3 PC CC + kdn PC)

Component: CC

\frac{d}{d time} (CC) = ksC MC + V2C \frac{CCP}{Kdp + CCP} + k4 PCC - (V1C \frac{CC}{Kp + CC} + k3 PC CC + kdnc CC)

Component: PCP

\frac{d}{d time} (PCP) = V1P \frac{PC}{Kp + PC} - (V2P \frac{PCP}{Kdp + PCP} + vdPC \frac{PCP}{Kd + PCP} + kdn PCP)

Component: CCP

\frac{d}{d time} (CCP) = V1C \frac{CC}{Kp + CC} - (V2C \frac{CCP}{Kdp + CCP} + vdCC \frac{CCP}{Kd + CCP} + kdn CCP)

Component: PCC

\frac{d}{d time} (PCC) = V2PC \frac{PCCP}{Kdp + PCCP} + k3 PC CC + k2 PCN - (V1PC \frac{PCC}{Kp + PCC} + k4 PCC + k1 PCC + kdn PCC)

Component: PCN

\frac{d}{d time} (PCN) = V4PC \frac{PCNP}{Kdp + PCNP} + k1 PCC + k8 IN - (V3PC \frac{PCN}{Kp + PCN} + k2 PCN + k7 BN PCN + kdn PCN)

Component: PCCP

\frac{d}{d time} (PCCP) = V1PC \frac{PCC}{Kp + PCC} - (V2PC \frac{PCCP}{Kdp + PCCP} + vdPCC \frac{PCCP}{Kd + PCCP} + kdn PCCP)

Component: PCNP

\frac{d}{d time} (PCNP) = V3PC \frac{PCN}{Kp + PCN} - (V4PC \frac{PCNP}{Kdp + PCNP} + vdPCN \frac{PCNP}{Kd + PCNP} + kdn PCNP)

Component: BC

\frac{d}{d time} (BC) = V2B \frac{BCP}{Kdp + BCP} + k6 BN + ksB MB - (V1B \frac{BC}{Kp + BC} + k5 BC + kdn BC)

Component: BCP

\frac{d}{d time} (BCP) = V1B \frac{BC}{Kp + BC} - (V2B \frac{BCP}{Kdp + BCP} + vdBC \frac{BCP}{Kd + BCP} + kdn BCP)

Component: BN

\frac{d}{d time} (BN) = V4B \frac{BNP}{Kdp + BNP} + k5 BC + k8 IN - (V3B \frac{BN}{Kp + BN} + k6 BN + k7 BN PCN + kdn BN)

Component: BNP

\frac{d}{d time} (BNP) = V3B \frac{BN}{Kp + BN} - (V4B \frac{BNP}{Kdp + BNP} + vdBN \frac{BNP}{Kd + BNP} + kdn BNP)

Component: IN

\frac{d}{d time} (IN) = k7 BN PCN - (k8 IN + vdIN \frac{IN}{Kd + IN} + kdn IN)

Component: CB

\frac{d}{d time} (CB) = \frac{\frac{V_K (1.0 - CB)}{K1CB + 1.0 - CB} - (\frac{vP CB}{K2CB + CB})}{WT}

Component: model_parameters

Source: Derived from workspace Vasalou and Henson 2010 at changeset 48e0e8d07104.

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A multiscale model to investigate circadian rhythmicity of pacemaker neurons in the suprachiasmatic nucleus