This Demonstration simulates the twodimensional persistent sodium plus potassium () neuron model described in [1].
I+I
Na,p
K
C=Ig(VE)gm(V)(VE)gn(VE)
.
V
L
L
Na
∞
Na
K
K
.
n
∞
n(V)=1/1+expKVV/(Kk)
∞
+
1/2
+
m(V)=1/1+expNaVV/(Nak)
∞
+
1/2
+
where is the capacitance density (for the squid axon, , so is fixed and set to one); is the voltage across a cell membrane in mV; is the total current density flowing across a patch of the cell membrane in ; , , are the equilibrium potentials (reverse potentials) in mV; , , are conductance densities in ; and are the activation variables (probabilities of activation gates to be open) and when with the subscript , they are voltagesensitive steadystate activation functions; and is the time constant. Note that stands for leak (ohmic leak current density is given by and it is mostly carried by ions); and are sodium and potassium ions, respectively. Finally, the Boltzmann functions and each have two parameters: and , where satisfies and similarly for sodium. The , are slope factors; that is, they adjust the slope of the Boltzmann functions at (smaller values of result in a steeper Boltzmann function).
C
C≈1μF/cm
2
C
V
I
μA/cm
2
E
L
E
Na
E
K
g
L
g
Na
g
K
mS/cm
2
m
n
∞
τ
L
g(VE)
L
L
Cl

Na
+
K
+
n(V)
∞
m(V)
∞
V
1/2
k
KV
+
1/2
nKV=0.5
∞
+
1/2
Kk
+
Nak
+
V
1/2
k
We plot four graphs. The main graph is the phase plane of the potassium conductance versus membrane voltage with the vector field (black), nullclines (orange and green), and voltage trajectory (red) superimposed. We also plot the membrane voltage versus time, sodium () and potassium () conductances versus time, and membrane currents versus time. Blue corresponds to sodium and green to potassium; nullcline is green as well.
m(V)
∞
n
There are four bookmarks, one for each bifurcation of the resting state. The parameter values for each bifurcation are listed below.
BifurcationType  E L  g L  E Na  g Na  E K  g K  Na + 1/2  Na +  τ  K + 1/2  K +  BifurcationCurrent 
SaddleNodeonInvariant Circle  80  8  60  20  90  10  20  15  1  25  5  4.51 
RegularSaddleNode  80  8  60  20  90  10  20  15  0.152  25  5  4.51 
SubcriticalAndronov Hopf  78  1  60  4  90  4  30  7  1  45  5  48.75 
SupercriticalAndronov Hopf  78  8  60  20  90  10  20  15  1  45  5  14.66 