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={-∞
(2.12)
where $Δk$ is the $k$-dimensional simplex in $Rk+1$ given by $y1+…+yk+1=1,yi≥0.$ The 4D HH model $dxdt=F(x)$, equations 2.1 to 2.4, is defined for $x∈X$, and the 14D HH model $dydt=G(y)$, equations 2.7 and 2.9, is defined for $y∈Y$. We introduce a dimension-reducing mapping $R:Y→X$ as in Table 1 and a mapping from lower to higher dimension, $H:X→Y$, as in Table 2. We construct $R$ and $H$ in such a way that $R∘H$ acts as the identity on $X$, that is, for all $x∈X$, $x=R(H(x))$. 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
$(v,m,h,n)$ $(v,m00,…,m31,n0,…,n4)$
$v$ $v$
$(1-n)4$ $n0$
$4(1-n)3n$ $n1$
$6(1-n)2n2$ $n2$
$4(1-n)n3$ $n3$
$n4$ $n4$
$(1-m)3(1-h)$ $m00$
$3(1-m)2m(1-h)$ $m10$
$3(1-m)m2(1-h)$ $m20$
$m3(1-h)$ $m30$
$(1-m)3h$ $m01$
$3(1-m)2mh$ $m11$
$3(1-m)m2h$ $m21$
$m3h$ $m31$
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