Subsections

• Equations
  • Delayed rectifier K⁺ current
  • Simple fast Na⁺ current
  • Hodgkin-Huxley fast Na⁺ current
  • AHP K⁺ current
  • A K⁺ current
  • m K⁺ current
  • C K⁺ current
  • h K⁺ current
  • Persistent Na⁺ current
  • High Voltage Activated Ca^{+ +} current
  • Low Voltage Activated Ca^{+ +} current
  • NMDA current
• Default values

   
CBM model equations

Equations

The following equations are used, where Vis the instantaneous membrane potential and E the voltage for which the sigmoid has its inflection point (see below):

The A coefficient is the slope of the sigmoid of the voltage versus the probability of open or closed channels. probability is 0.5 (thus it define the position of the courve on the xaxis).

In the following equations, m is associated with activation, while h is associated with inactivation.

p is the exponant for activation and q is the exponant for inactivation

A_m is the slope for activation and A_h is the slope for inactivation.

v_m is the position for activation and v_h is the position for inactivation.

Delayed rectifier K⁺ current

g = G x m^p

$${\frac{dm}{dt}}$$ = $${\frac{m_\infty - m}{\tau_m}}$$

$$\alpha_{m}^{}$$ = e^{A_m(V - E)}

$$\beta_{m}^{}$$ = e^{-A_h(V - E)}

$$\tau_{m}^{}$$ = $${\frac{1}{\lambda (\alpha_m + \beta_m)}}$$

m_$\infty$ = $${\frac{\alpha_m}{\alpha_m + \beta_m}}$$

Simple fast Na⁺ current

It is implemented using the Michaelis-Menton method.

g = G x m_$\infty$^p x (1 - m_K)

m_$\infty$ = $${\frac{1}{1+ \mbox{e}^{-2 A_m (V-E)}}}$$

Hodgkin-Huxley fast Na⁺ current

g = G x m^p x h^q

$${\frac{dh}{dt}}$$ = $${\frac{h_\infty - h}{\tau_h}}$$

m_$\infty$ = $${\frac{1}{1+ \mbox{e}^{-2 A_m (V-E)}}}$$

h_$\infty$ = $${\frac{1}{1+ \mbox{e}^{-2 A_h (V-E)}}}$$

$$\tau_{h}^{}$$ = constant

AHP K⁺ current

g = G x m_$\infty$

m_$\infty$ = $${\frac{[\mbox{Ca}^{++}]_i}{[\mbox{Ca}^{++}]_i + K}}$$

A K⁺ current

g = G x m_$\infty$^p x h^q

$${\frac{dh}{dt}}$$ = $${\frac{h_\infty - h}{\tau_h}}$$

m_$\infty$ = $${\frac{1}{1+ \mbox{e}^{-2 A_m (V-E)}}}$$

h_$\infty$ = $${\frac{1}{1+ \mbox{e}^{-2 A_h (V-E)}}}$$

$$\tau_{h}^{}$$ = constant

m K⁺ current

g = G x m

$${\frac{dm}{dt}}$$ = $${\frac{m_\infty - m}{\tau_m}}$$

$$\alpha_{m}^{}$$ = e^{A(V - E)}

$$\beta_{m}^{}$$ = e^{-A(V - E)}

$$\tau_{m}^{}$$ = $${\frac{1}{\lambda (\alpha_m + \beta_m)}}$$

m_$\infty$ = $${\frac{\alpha_m}{\alpha_m + \beta_m}}$$

C K⁺ current

g = G x m

$${\frac{dm}{dt}}$$ = $${\frac{m_\infty - m}{\tau_m}}$$

$$\alpha_{m}^{}$$ = [Ca^{+ +}]e^{A(V - E)}

$$\beta_{m}^{}$$ = e^{-A(V - E)}

$$\tau_{m}^{}$$ = $${\frac{1}{\lambda (\alpha_m + \beta_m)}}$$

m_$\infty$ = $${\frac{\alpha_m}{\alpha_m + \beta_m}}$$

h K⁺ current

g = G x m

$${\frac{dm}{dt}}$$ = $${\frac{m_\infty - m}{\tau_m}}$$

$$\alpha_{m}^{}$$ = e^{A(V - E)}

$$\beta_{m}^{}$$ = e^{-A(V - E)}

$$\tau_{m}^{}$$ = $${\frac{1}{\lambda (\alpha_m + \beta_m)}}$$

m_$\infty$ = $${\frac{\alpha_m}{\alpha_m + \beta_m}}$$

Persistent Na⁺ current

g = G x m

$${\frac{dm}{dt}}$$ = $${\frac{m_\infty - m}{\tau_m}}$$

$$\alpha_{m}^{}$$ = e^{A(V - E)}

$$\beta_{m}^{}$$ = e^{-A(V - E)}

$$\tau_{m}^{}$$ = $${\frac{1}{\lambda (\alpha_m + \beta_m)}}$$

m_$\infty$ = $${\frac{\alpha_m}{\alpha_m + \beta_m}}$$

High Voltage Activated Ca^{+ +} current

g = G x m^p x h

$${\frac{dm}{dt}}$$ = $${\frac{m_\infty - m}{\tau_m}}$$

m_$\infty$ = $${\frac{1}{1+ \mbox{e}^{-2A (V-E)}}}$$

h = $${\frac{K}{K + [\mbox{Ca}^{++}]_i}}$$

Low Voltage Activated Ca^{+ +} current

g = G x m^p x h

$${\frac{dm}{dt}}$$ = $${\frac{m_\infty - m}{\tau_m}}$$

$${\frac{dh}{dt}}$$ = $${\frac{h_\infty - h}{\tau_h}}$$

$$\alpha_{m}^{}$$ = e^{A(V - E)}

$$\beta_{m}^{}$$ = e^{-A(V - E)}

$$\tau_{m}^{}$$ = constant

m_$\infty$ = $${\frac{\alpha_m}{\alpha_m + \beta_m}}$$

$$\alpha_{h}^{}$$ = e^{-A(V - E)}

$$\beta_{h}^{}$$ = e^{A(V - E)}

$$\tau_{h}^{}$$ = $${\frac{1}{\lambda}}$$

h_$\infty$ = $${\frac{\alpha_h}{\alpha_h + \beta_h}}$$

NMDA current

g = $${\frac{G}{1+\eta [\mbox{Mg}^{++}] \mbox{e}^{-\gamma V}}}$$

Default values

These are the default values of a Conductance Based Model as stored in the .G_unit file:

diametre: 20.00000000000000
TauDecayCa: 20.00000000000000
spike_slope:       10000.000
TimeStep: 0.10000000149012
Temp: 20.00000000000000
Na_ext: 140.00000000000000
Na_int: 14.00000000000000
K_ext: 4.00000000000000
K_int: 140.00000000000000
E_Ca: 124.00000000000000
E_Cl: -89.00000000000000
E_H: -43.00000000000000
PosReversalPot: -10.00000000000000
NegReversalPot: -90.00000000000000
E_fuite: -50.00000000000000
g_Na: 120.00000000000000
g_K: 36.00000000000000
g_Ca: 1.00000000000000
g_Cl: 0.00000000000000
g_A: 0.00000000000000
g_M: 0.00000000000000
g_C: 0.00000000000000
g_ahp: 0.00000000000000
g_Nap: 0.00000000000000
g_Cab: 0.00000000000000
g_H: 0.00000000000000
g_fuite: 0.30000001192093
g_Syn_pos: 0.05000000074506
g_Syn_neg: 0.05000000074506
capacite: 1.00000000000000
input_fact: 0.00000000000000
tau_attack_epsp: 2.50000000000000
tau_decay_epsp: 10.00000000000000
tau_attack_ipsp: 2.50000000000000
tau_decay_ipsp: 10.00000000000000
Ca_i: 0.00999999977648
K_ext: 4.00000000000000
NoiseMean: 0.00000000000000
NoiseSd: 0.00000000000000
magnesium: 1.00000000000000
g_NMDA: 1.79999995231628
rev_pot_NMDA: 0.00000000000000
eta_NMDA: 0.33000001311302
gamma_NMDA: 0.05999999865890
tau_attack_NMDA: 80.00000000000000
tau_decay_NMDA: 0.66000002622604
Ca_Influx:    0.0001500000
Ca_Removal:    0.0049999999
Ca_Kd:    1.0000000000
K_Ca_Kd:    0.5000000000
lambda_Ik:    0.0810000002
V_Na:  -31.0000000000
A_Na:    0.0650999993
Pow_Na:    3.0000000000
V_K:  -46.0000000000
A_K:    0.0551000014
Pow_K:    4.0000000000
V_Ca:  -40.0000000000
A_Ca:    0.2000000030
Pow_Ca:    2.0000000000
Tau_Ca:   50.0000000000
V_Ia_A:  -20.0000000000
A_Ia_A:    0.0209999997
Pow_Ia_A:    1.0000000000
V_Ia_B:  -70.0000000000
A_Ia_B:   -0.1000000015
Pow_Ia_B:    1.0000000000
Tau_Ia_B:   10.0000000000
V_Ic:  -90.0000000000
A_Ic:    0.0500000007
V_Im:  -35.0000000000
A_Im:    0.0500000007
V_Inap:  -56.0000000000
A_Inap:    0.0700000003
Vm_Icab:  -45.0000000000
Am_Icab:    0.1000000015
Pow_m_cab:    3.0000000000
Vh_Icab:  -70.0000000000
Ah_Icab:    0.1000000015
V_Ih:  -75.0000000000
A_Ih:   -0.0900000036
lambda_Ic:    5.0000000000
lambda_Im:    0.0030000000
lambda_Inap:    0.1000000015
lambda_Icab:    0.0049999999
lambda_Ih:    0.0005000000
Tau_m_Cab:   10.0000000000
K_rest:    2.5000000000
 m_ahp_init:    0.0000000000
m _K_init:    0.1599999964
l e_potentiel_init:  -60.0000000000
le_potentiel_E_Na_init:   55.0000000000
le_potentiel_E_K_init:  -72.0000000000
m_Ca_init:    0.0010000000
Ca_i_init:    0.0099999998
h_A_init:    0.0500000007
m_Nap_init:    0.0000000000
m_Cab_init:    0.0000000000
m_Ih_init:    0.0000000000
h_Cab_init:    0.0000000000

If modified with a text editor, remind that the labels are case sensitive.