generally, in the Hodgkin-Huxley style model, the Na/K ion-pump is not explicitly modeled, instead, it is assumed that the passive conductance restores the resting membrane potential.

I want to model the explicit effect that the Na/K pump has on the Na and K concentrations and then on membrane potential.

I considered a single compartment. I inserted the following HH mechanism which contains the voltage-gated K and Na channel mechanisms, the Na/K pump and ion accumulation mechanisms:

Code: Select all

```
TITLE HH channels
:
: Fast Na+ and K+ currents responsible for action potentials
: Iterative equations
:
: Equations modified by Traub, for Hippocampal Pyramidal cells, in:
: Traub & Miles, Neuronal Networks of the Hippocampus, Cambridge, 1991
:
: range variable vtraub adjust threshold
:
: Written by Alain Destexhe, Salk Institute, Aug 1992
:
: Modifications by Arthur Houweling for use in MyFirstNEURON
: Modifications by Paulo Aguiar: vh changed from 5 to 6 - NOT ANYMORE: vh=5 as originally set
INDEPENDENT {t FROM 0 TO 1 WITH 1 (ms)}
NEURON {
SUFFIX HH2
USEION na READ ena WRITE ina
USEION k READ ek WRITE ik
RANGE gnabar, gkbar, vtraub
RANGE m_inf, h_inf, n_inf
RANGE tau_m, tau_h, tau_n
RANGE m_exp, h_exp, n_exp
RANGE ik, ina
}
UNITS {
(mA) = (milliamp)
(mV) = (millivolt)
}
PARAMETER {
gnabar = .1 (mho/cm2)
gkbar = .06 (mho/cm2)
ena (mV)
ek (mV)
celsius (degC)
dt (ms)
v (mV)
vtraub = -55 (mV) : adjusts threshold
}
STATE {
m h n
}
ASSIGNED {
ina (mA/cm2)
ik (mA/cm2)
il (mA/cm2)
m_inf
h_inf
n_inf
tau_m
tau_h
tau_n
m_exp
h_exp
n_exp
tadj
}
BREAKPOINT {
SOLVE states
ina = gnabar * m*m*m*h * (v - ena)
ik = gkbar * n*n*n*n * (v - ek)
}
:DERIVATIVE states { : use this for exact Hodgkin-Huxley equations
: evaluate_fct(v)
: m' = (m_inf - m) / tau_m
: h' = (h_inf - h) / tau_h
: n' = (n_inf - n) / tau_n
:}
PROCEDURE states() { : this discretized form is more stable
evaluate_fct(v)
m = m + m_exp * (m_inf - m)
h = h + h_exp * (h_inf - h)
n = n + n_exp * (n_inf - n)
VERBATIM
return 0;
ENDVERBATIM
}
UNITSOFF
INITIAL {
:
: Q10 was assumed to be 3 for both currents
:
tadj = 3.0 ^ ((celsius-36)/ 10 )
evaluate_fct(v)
m= m_inf
h= h_inf
n= n_inf
}
PROCEDURE evaluate_fct(v(mV)) { LOCAL a,b,v2,vh
v2 = v - vtraub : convert to traub convention
vh = 5
a = 0.32 * (13-v2) / ( exp((13-v2)/4) - 1)
b = 0.28 * (v2-40) / ( exp((v2-40)/5) - 1)
tau_m = 1 / (a + b) / tadj
m_inf = a / (a + b)
:a = 0.128 * exp((17-v2)/18)
:b = 4 / ( 1 + exp((40-v2)/5) )
:tau_h = 1 / (a + b) / tadj
:h_inf = a / (a + b)
a = 0.128 * exp((17-v2-vh)/18)
b = 4 / ( 1 + exp((40-v2-vh)/5) )
tau_h = 1 / (a + b) / tadj
h_inf = a / (a + b)
a = 0.032 * (15-v2) / ( exp((15-v2)/5) - 1)
b = 0.5 * exp((10-v2)/40)
tau_n = 1 / (a + b) / tadj
n_inf = a / (a + b)
m_exp = 1 - exp(-dt/tau_m)
h_exp = 1 - exp(-dt/tau_h)
n_exp = 1 - exp(-dt/tau_n)
}
UNITSON
```

Code: Select all

```
TITLE sodium potassium pump
: from Lindblad et al Am J Physiol 1996 275:H1666
NEURON {
SUFFIX nkpump
USEION k READ ko, ki WRITE ik
USEION na READ nao, nai WRITE ina
RANGE ik, ina , INaKmax, ink, Kmko, Kmnai,inapump
GLOBAL dummy : prevent vectorization for use with CVODE
}
UNITS {
(mA) = (milliamp)
(mV) = (millivolt)
}
PARAMETER {
INaKmax = 2.18660e-3 (mA/cm2) <0,1e6>
Kmnai = 10 (mM) <0,1e6>
Kmko = 1.5 (mM) <0,1e6>
}
ASSIGNED {
celsius (degC)
v (mV)
ik (mA/cm2)
ina (mA/cm2)
ko (mM)
ki (mM)
nao (mM)
nai (mM)
ink (mA/cm2)
dummy
}
BREAKPOINT {
ink= INaKmax/((1 + (Kmnai/nai)^1.5)*(1 + Kmko/ko))
ina = 3*ink
ik = -2*ink
}
```

Code: Select all

```
NEURON {
SUFFIX nacumst
USEION na READ ina WRITE nai
RANGE tau, C0
}
UNITS {
(mA) = (milliamp)
(mol) = (1)
(molar) = (mol/liter)
(mM) = (millimolar)
(uM) = (micromolar)
(um) = (micrometer)
FARADAY = (faraday) (coulombs)
}
PARAMETER {
tau = 5 (ms) : AB soma; 300 ms for PD soma
C0 = 10 (mM)
}
ASSIGNED {
ina (mA/cm2)
diam (um)
}
STATE {
nai (mM)
}
INITIAL {
nai = C0
}
BREAKPOINT {
SOLVE states METHOD cnexp
}
DERIVATIVE states {
nai' = -(1e4)*2*ina/(FARADAY*diam) + (C0 - nai)/tau
}
```

At the moment, of course, I have that the v is going below the resting potential because the currents generated by the pump is not balancing the current genereted by the the channels.

Here figures for the Na:

The ink is dependent on the change of ionic concentration (nai and ko). The change of ionic concentrations is own to the activated currents.

I think would be useful to keep track of the numbers of ions are moving in or out the cell, but how?

I think I should modify something in

ink= INaKmax/((1 + (Kmnai/nai)^1.5)*(1 + Kmko/ko))

but, in which direction to go?

Or I can use persistent Na and K channels inseatd?

Best

Menica