Skip to Main Content
In order to compare the trajectories of the 14D HH equations with trajectories of the standard 4D equations, we define lower-dimensional and higher-dimensional domains X and Y, respectively, as
X={-<v<,0m1,0h1,0n1}=R×[0,1]3R4,Y={-<v<}0mij,i=03j=01mij=10ni,i=04ni=1=R×Δ7×Δ4R14,
(2.12)
where Δk is the k-dimensional simplex in Rk+1 given by y1++yk+1=1,yi0. The 4D HH model dxdt=F(x), equations 2.1 to 2.4, is defined for xX, and the 14D HH model dydt=G(y), equations 2.7 and 2.9, is defined for yY. We introduce a dimension-reducing mapping R:YX as in Table 1 and a mapping from lower to higher dimension, H:XY, as in Table 2. We construct R and H in such a way that RH acts as the identity on X, that is, for all xX, x=R(H(x)). The maps H and R are consistent with a multinomial structure for the ion channel state distribution, in the following sense. The space Y covers all possible probability distributions on the eight sodium channel states and the five potassium channel states. Those distributions, which are products of one multinomial distribution on the K+-channel1and a second multinomial distribution on the Na+-channel,2 form a submanifold of Δ7×Δ4. In this way we define a submanifold, denoted M=H(X), the image of X under H.
Table 1:
R: Map from the 14D HH Model (m00,,m31,n0,,n4) to the 4D HH Model (m,h,n).
14D Model4D Model
(v,m00,,m31,n0,,n4) (v,m,h,n) 
v v 
13(m11+m10)+23(m21+m20)+m31+m30 m 
m01+m11+m21+m31 h 
n1/4+n2/2+3n3/4+n4 n‘ 
14D Model4D Model
(v,m00,,m31,n0,,n4) (v,m,h,n) 
v v 
13(m11+m10)+23(m21+m20)+m31+m30 m 
m01+m11+m21+m31 h 
n1/4+n2/2+3n3/4+n4 n‘ 

Note: Note that both {m00,,m31} and {n0,,n4} follow multinomial distributions.

Table 2:
H: Map from the 4D HH Model (m,h,n) and the 14D HH Model (m00,,m31,n0,,n4).
4D Model14D Model
(v,m,h,n) (v,m00,,m31,n0,,n4) 
v v 
(1-n)4 n0 
4(1-n)3n n1 
6(1-n)2n2 n2 
4(1-n)n3 n3 
n4 n4 
(1-m)3(1-h) m00 
3(1-m)2m(1-h) m10 
3(1-m)m2(1-h) m20 
m3(1-h) m30 
(1-m)3h m01 
3(1-m)2mh m11 
3(1-m)m2h m21 
m3h m31 
4D Model14D Model
(v,m,h,n) (v,m00,,m31,n0,,n4) 
v v 
(1-n)4 n0 
4(1-n)3n n1 
6(1-n)2n2 n2 
4(1-n)n3 n3 
n4 n4 
(1-m)3(1-h) m00 
3(1-m)2m(1-h) m10 
3(1-m)m2(1-h) m20 
m3(1-h) m30 
(1-m)3h m01 
3(1-m)2mh m11 
3(1-m)m2h m21 
m3h m31 
Close Modal

or Create an Account

Close Modal
Close Modal