Cellular information dynamics through transmembrane flow of ions

The average such excess is assumed to be minimized, as an expression of natural selection (see above). The average is evaluated by simply weighting the r.h.s. of (3) with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${p}_{i}(t)$$\end{document}pi(t) and integrating over all t, as4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\int }_{0}^{\infty }{p}_{i}(t)\mathrm{ln}(({p}_{i}(t)/{q}_{i})dt\equiv {H}_{KL}({p}_{i}||{q}_{i})=minimum,\,i=1,\ldots ,N.$$\end{document}∫0∞pi(t)ln((pi(t)/qi)dt≡HKL(pi||qi)=minimum,i=1,…,N.

In words, Eq. (4) states the average amount of wasted code length over all ion occurrences i equals the relative entropy \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${H}_{KL}(p||q)$$\end{document}HKL(p||q) or the ‘Kullback–Leibler divergence22. This is effectively a distance between probabilities \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{p}}_{{i}}({t})\,\,$$\end{document}pi(t)and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${q}_{i}$$\end{document}qi, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${i}={1},\ldots ,{N}$$\end{document}i=1,…,N.

For example, at any i and t for which probability \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{p}}_{{i}}({t})={{q}}_{{i}}$$\end{document}pi(t)=qi, there is contribution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\rm{l}}{\rm{n}}({1})={0}$$\end{document}ln(1)=0 to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${H}_{KL}({p}_{i}||{q}_{i})$$\end{document}HKL(pi||qi) in Eq. (4). There is thus zero distance between such pairs. This implies that if probabilities \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${p}_{i}(t)$$\end{document}pi(t) are formed, coded, and decoded at the membrane according to principle (4) the external environmental events will be identified by the cell membrane with minimum error and maximum speed. We propose that these optimization principles, because the allow maximally fast and accurate responses to external perturbations, are fundamental Darwinian selection forces in the evolution of all cells.

Quite aside from the above requirement by natural evolution, the minimization requirement in Eq. (4) is also suggested by the dual facts that (i) the Kullback–Leibler divergence Eq. (4) has the form of the Shannon information 12, and (ii) the transmission of the ion through the membrane is an irreversible thermodynamic process which, therefore, obeys coarse graining23–26. Such processes can only lose information, and so we again arrive at principle (4) which states that optimal function requires this loss of information to be a minimum27.

We note that the Kullback–Leibler principle Eq. (4) is being used as a principle of thermodynamics. Such use allows microscale effects, for example occurrence rates \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{p}}_{{i}}({t})$$\end{document}pi(t), to be found without detailed knowledge of that microscale, such as the mathematical dependences of the fields of potential (e.g., inverse square) that are provided by ions. Thermodynamics is thereby seen to provide a basis for the phenomenon of natural selection.

The minimum in principle (4) is always an extreme value since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{H}}_{{K}{L}}({{p}}_{{A}}||{{q}}_{{A}})$$\end{document}HKL(pA||qA) is a convex functional. This is fortunate, as finding its extreme value is then elementary (see examples below).

In summary to this point, the solution to principle (4) is found under the special condition Eq. (2) of a constant \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{q}}_{{i}}({t})\equiv {{q}}_{{i}}$$\end{document}qi(t)≡qi. Also the solution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{p}}_{{i}}({t})$$\end{document}pi(t) obeys:

(i) minimum departure from that of the constant ion rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{q}}_{{i}}$$\end{document}qi (of Eq. (2)) exterior to the cell, in the sense of carrying a level of Shannon information that departs minimally from that of the constant \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{q}}_{{i}}$$\end{document}qi, and

(ii) optimally efficient information transfer of environmental messages (such that they are decoded with maximum speed within the membrane).

Note that in Eq. (4) that at all t for which \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{p}}_{{i}}({t})\,\approx \,{{q}}_{{i}}({t}),\,\,$$\end{document}pi(t)≈qi(t),since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\rm{l}}{\rm{n}}({1})={0}$$\end{document}ln(1)=0 there is a very small contribution to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${H}_{KL}$$\end{document}HKL. Conversely, for all t for which \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{p}}_{{i}}({t})\,\gg \,{{q}}_{{i}}({t})$$\end{document}pi(t)≫qi(t) there is large contribution to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${H}_{KL}$$\end{document}HKL.

Thus, mathematically, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${H}_{KL}({p}_{i}||{q}_{i})$$\end{document}HKL(pi||qi) is a type of logarithmic measure of the ‘distance’ between corresponding \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{p}}_{{i}}$$\end{document}pi and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{q}}_{{i}}$$\end{document}qi over all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${i}={1},\ldots ,{N}$$\end{document}i=1,…,N.

(Note: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{H}}_{{K}{L}}$$\end{document}HKL isn’t a formal ‘distance’ measure as it isn’t symmetric in exchange of corresponding \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{p}}_{{i}}$$\end{document}pi and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{q}}_{{i}}$$\end{document}qi, but this is not important for our purposes, as will become apparent).

The minimization in principle (4) is always found for a given state of prior knowledge, usually in the form of equality constraints on the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${p}_{i}(t)$$\end{document}pi(t). These are implemented by the simple weighted addition of the constraints to principle (4), where each is multiplied by a corresponding constant Lagrange multiplier constant. These constants are found by requiring them to obey the known constraints (see below).

The membrane is regarded to generally have some level of transmembrane potential V that can change with time t. Hence principle (4) is applied for some arbitrary values of V t. In addition, the minimum in Eq. (4) is attained in the presence of known constraints obeyed by the unknown laws \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{p}}_{{i}}({t}),{{q}}_{{i}}$$\end{document}pi(t),qi. In general, these obey two, trivial normalization constraints, which are obeyed in all problems plus other, more substantive constraints such as mean passage time through the membrane. These effectively define the particular problem at hand.

We next compared results with known experimental observations, specifically of those of ions that pass through membrane ion channels in the membrane of a neuron – as described by the classical Hodgkin-Huxley equation. Hence, let the law \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{p}}_{{i}}({t})$$\end{document}pi(t) obey substantive constraints as follows. As with Hodgkin and Huxley10 let the ions pass through the cell membrane over fixed average times \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\tau }_{{i}}$$\end{document}τi,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\,{1},\ldots ,{N}$$\end{document}1,…,N. These \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\tau }_{{i}}$$\end{document}τi, were empirically observed by Hodgkin and Huxley.

It is then seen that the particular values of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\tau }_{{i}}$$\end{document}τi are substantive in strongly influencing the output rates \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{p}}_{{i}}({t})$$\end{document}pi(t) obeying principle (4).

Hodgkin and Huxley investigated, in particular, the case of N = 3 types of ions, specifically Na+, K+ and Cl−. They observed average path times \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\tau }_{{i}},\,{i}={1},\,{2},{3}$$\end{document}τi,i=1,2,3 for these types of ions. Hence, these are used as constraints on the time behavior determined by principle (4). As is usual, these constraints are entered via Lagrange constraint multipliers \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\lambda }_{{i}},\,{i}\,=\,{1},{2},{3}$$\end{document}λi,i=1,2,3. Constancy effect Eq. (1) and normalization Eq. (2) are also so used.

The resulting constrained principle (4) is5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\int }_{0}^{T}{p}_{i}(t)\mathrm{ln}(({p}_{i}(t)/{q}_{i})dt+{\lambda }_{1}[{\int }_{0}^{T\,}dt{p}_{i}(t)-1]+{\lambda }_{2}[{\int }_{0}^{T}dt{q}_{i}-1]+{\lambda }_{3}[{\int }_{0}^{T}dtt{p}_{i}(t)-{\tau }_{i}]=min$$\end{document}∫0Tpi(t)ln((pi(t)/qi)dt+λ1[∫0Tdtpi(t)−1]+λ2[∫0Tdtqi−1]+λ3[∫0Tdttpi(t)−τi]=minWe will see that this principle gives rise to the known dynamical equations of Hodgkin and Huxley for occurrence rates \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{p}}_{{i}}({t})$$\end{document}pi(t) at the assumed potential value V.

We note the absence of any derivative terms \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${d}{p}{/}{d}{t}$$\end{document}dp/dt in the integrands of Eq. (5). Then its Euler-Lagrange solution simplifies to that obtained by simply setting partial derivative \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\rm{\partial }}{/}{\rm{\partial }}{{p}}_{{i}}$$\end{document}∂/∂pi of its Lagrangian (total integrand) to zero. This gives a requirement6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1+ln{p}_{i}\,-\,ln{q}_{i}+{\lambda }_{1}+{\lambda }_{3}t=0$$\end{document}1+lnpi−lnqi+λ1+λ3t=0at the arbitrary time t.

Solving for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{p}}_{{i}}$$\end{document}pi,7\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\,{{p}}_{{i}}\,({t})={{q}}_{{i}}{e}{x}{p}(-{1}{-}\,{\lambda }_{{1}}\,{-}\,{\lambda }_{{3}}{t})$$\end{document}pi(t)=qiexp(−1−λ1−λ3t)Boundary values \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{p}}_{{i}}({0}),{{p}}_{{i}}({T})$$\end{document}pi(0),pi(T) were likewise observed by H and H. These should be obeyed by solution (7).

Enforcing these in (7) gives the following solutions.8\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${p}_{i}(t)={p}_{i}(T)\,-\,{q}_{i}exp(-1\,-\,{\lambda }_{1})\exp (-{\lambda }_{3}t),{q}_{i}=const.$$\end{document}pi(t)=pi(T)−qiexp(−1−λ1)exp(−λ3t),qi=const.Evaluating this at t = 0 gives\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${p}_{i}(0)={p}_{i}(T)-{q}_{i}exp(-1\,-\,{\lambda }_{1})$$\end{document}pi(0)=pi(T)−qiexp(−1−λ1)Solving,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${q}_{i}exp(-1\,-\,{\lambda }_{1})={p}_{i}(T)\,-\,{p}_{i}(0)$$\end{document}qiexp(−1−λ1)=pi(T)−pi(0)Putting this in (8) then gives9a\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${p}_{i}(t)={p}_{i}(T)\,-\,({p}_{i}(T)\,-\,{p}_{i}(0))\exp (-{\lambda }_{3}t)$$\end{document}pi(t)=pi(T)−(pi(T)−pi(0))exp(−λ3t)

In principle (5), the mean time constraint \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\int }_{{0}}^{{T}}{d}{t}{t}{{p}}_{{i}}({t})={\tau }_{{i}}$$\end{document}∫0Tdttpi(t)=τi and normalization constraint\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\int }_{0}^{T}{p}_{i}(t)dt=1$$\end{document}∫0Tpi(t)dt=1giving the result9b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\int }_{0}^{T}t{p}_{i}(t)dt={\lambda }_{3}=1/{\tau }_{i}$$\end{document}∫0Ttpi(t)dt=λ3=1/τiThen result (9a) becomes10\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${p}_{i}(t)={p}_{i}(T)\,-\,({p}_{i}(T)\,-\,{p}_{i}(0))\exp (-\frac{t}{{\tau }_{i}}),\,i=1,\ldots ,N.$$\end{document}pi(t)=pi(T)−(pi(T)−pi(0))exp(−tτi),i=1,…,N.These are equivalent to the Hodgkin and Huxley (H and H) equations numbered (8), (17) and (18) in10. Thus, the dynamics described by the “Hodgkin and Huxley” equations apply to a wider scope of cell membranes than merely neurons. As a check, we compare Eq. (10) with Eq. (8) of H and H.

In their notation this is11\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n={n}_{\infty }-({n}_{\infty }-{n}_{0})\exp (-\frac{t}{{\tau }_{i}})$$\end{document}n=n∞−(n∞−n0)exp(−tτi)Thus our notation for the ith rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{p}}_{{i}}\,\,$$\end{document}piin Eq. (10) becomes simply n for H and H, with their subscript denoting the time. Hodgkin and Huxley take their end time T to be very large, effectively \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\,\infty $$\end{document}∞. Thus our notation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{p}}_{{i}}({T})$$\end{document}pi(T) translates into \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{n}}_{{T}}\equiv {{n}}_{{\rm{\infty }}}$$\end{document}nT≡n∞ in their Eq. (8) [Eq. (11) preceding]. Also as in Hodgkin and Huxley14, a ‘resting state’ is defined by V = 0. Let this occur at time t = 0.

Then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{p}}_{{i}}({0})$$\end{document}pi(0) in Eq. (10) becomes \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{n}}_{{0}}$$\end{document}n0 in their Eq. (8) [same as preceding Eq. (11)], although the argument 0 in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{p}}_{{i}}({0})$$\end{document}pi(0) explicitly represents their case of zero potential.

Note that if all time constants \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\tau }_{{i}}$$\end{document}τi in Eq. (10) are very large this gives \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{p}}_{{i}}({t})\approx {{p}}_{{i}}({0}),$$\end{document}pi(t)≈pi(0), no environmental information significantly enters the cell. However, such large \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\tau }_{{i}}$$\end{document}τi have their use. As noted above, passage of any ion across the membrane requires that a specific number of gates open through which the ion then passes. For example, passage of a K+ ion through a membrane channel will not occur until all 3 gates open. The latter probability is of approximate size \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{p}}_{{{K}}_{+}}{({0})}^{{3}}$$\end{document}pK+(0)3.

Alternatively, the passage of an Na+ ion requires that 2 gates in the ion channel open, which then has probability\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\,{{p}}_{{N}{a}+}{({0})}^{{2}}$$\end{document}pNa+(0)2. As the square of a probability tends to be greater than the cube of a comparable one, the Na+ ions are more probable to pass through the channel than are 3 K+ ions. Consequently, a cell tends to efflux Na+ before influxing K+. Further, there is a need for this time gap between the two since, otherwise, the passages of the two ions might so overlap in time such that they collide with one another. Thus, it is important for K+ ions to be delayed in their passage through the membrane. Collectively, the flow solutions both accelerate and slow selected ions to optimize decoding times and avoid collisions.

Probabilities \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{p}}_{{i}}({0}),{{p}}_{{i}}({\rm{\infty }}),\,{i}={1},\ldots ,{N}$$\end{document}pi(0),pi(∞),i=1,…,N must be further defined. This involves the vital role played by the membrane potential V.

Ion transitions results from opening of channel gates that are governed by a perturbation in the transmembrane voltage V. However, as in the preceding, at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${t}={0}$$\end{document}t=0 one assumes no perturbation is applied to the membrane so voltage \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${V}={0.}$$\end{document}V=0. Also, let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\alpha }_{{m}}({0}),{\beta }_{{m}}({0})$$\end{document}αm(0),βm(0) represent, respectively, the rate of ion transfer at this voltage from outside the cell membrane to inside and from inside to outside. This represents slow “leakage”10 current with a probability12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${p}_{i}(0)=\frac{{\alpha }_{i}(0)}{{\alpha }_{i}(0)+{\beta }_{i}(0)}.$$\end{document}pi(0)=αi(0)αi(0)+βi(0).Next, we have assumed that a fixed, finite voltage V is applied to the membrane due to arrival of a propagating wave down the axon of the nerve.

Then rates \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\alpha }_{{i}}={\alpha }_{{i}}({V}),{\beta }_{{i}}={\beta }_{{i}}({V})$$\end{document}αi=αi(V),βi=βi(V) instantly take on fixed values (1), appropriate to the value of V over time t that represents the course of the action potential. This allows us to suppress all V notation (as in ref.10). Thus in particular13\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${p}_{i}(\infty )=\frac{{\alpha }_{i}}{{\alpha }_{i}+{\beta }_{i}}.$$\end{document}pi(∞)=αiαi+βi.Endpoint values (12) and (13) supplement the interior H-H solution Eq. (10).

Finally, by the above definition of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\alpha }_{{i}},$$\end{document}αi, with V fixed and finite, it must be that the out-to-in ion rate obeys \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\alpha }_{{i}}={{p}}_{{i}}({\rm{\infty }}){/}{\tau }_{{i}}$$\end{document}αi=pi(∞)/τi and the in-to-out rate obeys \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\beta }_{{i}}=\frac{{1}{-}{{p}}_{{i}}({\rm{\infty }})}{{\tau }_{{i}}}.$$\end{document}βi=1−pi(∞)τi.