% Template for PLoS % Version 1.0 January 2009 % % To compile to pdf, run: % latex plos.template % bibtex plos.template % latex plos.template % latex plos.template % dvipdf plos.template \documentclass[10pt]{article} % amsmath package, useful for mathematical formulas \usepackage{amsmath} % amssymb package, useful for mathematical symbols \usepackage{amssymb} % graphicx package, useful for including eps and pdf graphics % include graphics with the command \includegraphics \usepackage{graphicx} % cite package, to clean up citations in the main text. Do not remove. \usepackage{cite} \usepackage{color} % Use doublespacing - comment out for single spacing %\usepackage{setspace} %\doublespacing % Text layout \topmargin 0.0cm \oddsidemargin 0.5cm \evensidemargin 0.5cm \textwidth 16cm \textheight 21cm % Bold the 'Figure #' in the caption and separate it with a period % Captions will be left justified \usepackage[labelfont=bf,labelsep=period,justification=raggedright]{caption} % Use the PLoS provided bibtex style \bibliographystyle{plos2009} % Remove brackets from numbering in List of References \makeatletter \renewcommand{\@biblabel}[1]{\quad#1.} \makeatother % Leave date blank \date{} \pagestyle{myheadings} %% ** EDIT HERE ** \usepackage{hyperref} \hypersetup{ colorlinks=true, urlcolor=red, linkcolor=black, pdftoolbar=true, pdfmenubar=true, citecolor=black, filecolor=blue, pdfauthor={Marco Arieli Herrera-Valdez}, pdftitle={Herrera. Supplement. Excitability driven by drift and diffusion.} } %% ** EDIT HERE ** %% PLEASE INCLUDE ALL MACROS BELOW \usepackage{multirow} \usepackage{array} %\newcolumntype{x}[1]{{\raggedright}p{#1}} \newcolumntype{x}[1]{>{\raggedright}p{#1}} \newcommand{\tn}{\tabularnewline} \newcommand{\rr}{\raggedright} \newcommand{\icyc}{{$I_{cyc}$}} \newcommand{\calcium}{{Ca$^{2+}$}} \newcommand{\sodium}{{Na$^+$}} \newcommand{\natrium}{{Na$^+$}} \newcommand{\kalium}{{K$^+$}} \newcommand{\potassium}{{K$^+$}} \newcommand{\rfig}[1]{{Fig.~\ref{#1}}} \newcommand{\rfigs}[2]{{Figs.~\ref{#1}-\ref{#2}}} \newcommand{\rtwofigs}[2]{{Figs.~\ref{#1}~and~\ref{#2}}} \newcommand{\req}[1]{{Eq.~\ref{#1}}} \newcommand{\reqs}[2]{{Eqs.~\ref{#1}-\ref{#2}}} \newcommand{\rtwoeqs}[2]{{Eqs.~\ref{#1}~and~\ref{#2}}} \newcommand{\aquimequede}{{ \vspace{0.5cm} \textcolor{red}{AQUI ME QUEDE} \vspace{0.5cm} }} \newcommand{\stimtop}{{\textit{Top}}} \newcommand{\stimup}{{\textit{Up}}} \newcommand{\stimdown}{{\textit{Down}}} \newcommand{\mhrevone}[1]{{\textcolor{black}{#1}}} \newcommand{\mhrevtwo}[1]{{\textcolor{blue}{#1}}} % New environment for supplementary figures % \usepackage{float} \floatstyle{plain} %\floatstyle{ruled} \newfloat{suppfig}{htp}{lop} \floatname{suppfig}{\textbf{Figure S}} % \newcounter{suppfigcounter} \setcounter{suppfigcounter}{10} \newenvironment{suppfigure} { \begin{suppfig} }{ \end{suppfig} \addtocounter{suppfigcounter}{-1} } %% END MACROS SECTION \begin{document} % Title must be 150 characters or less \begin{flushleft} {\Large \textbf{Text S1. Membranes with the same ion channel populations but different excitabilities.} } % Insert Author names, affiliations and corresponding author email. \\ Marco Arieli Herrera-Valdez$^{1,2,3,4,\ast}$ \\ \bf{1} Department of Mathematics, University of Arizona, Tucson, AZ, USA \\ \bf{2} Evelyn F. McKnight Brain Institute, University of Arizona, Tucson, AZ, USA \\ \bf{3} Department of Mathematics and Physics, University of Puerto Rico in Cayey, Cayey, PR \\ \bf{4} Institute for Interdisciplinary Research, University of Puerto Rico in Cayey, Cayey, PR \\ $\ast$ E-mail: marco.herrera@upr.edu \end{flushleft} \subsubsection*{Derivation of electrodiffusion expressions for transmembrane currents} Consider a region in space that resembles one transmembrane pore within a channel together with the immediate intra and extracellular regions. Assume that any ionic species $s$ with valence $z_s$ has an electrical mobility $\mu_s$, and a smoothly varying concentration profile $C_s$. Assume further that ions in this region are influenced by an also smoothly varying electric field $U$. The flux of $s$-ions across the open channel is the sum of the fluxes caused by diffusion and electrical drift, described by the Nernst-Planck equation (NPE) \cite{weiss1996cellular1,weiss1996cellular2}: \begin{eqnarray} \vec{J}_s &=& -\mu_s \left( k_B T \nabla C_s + q_e z_s C_s \nabla U\right), \label{eq:NPE} \end{eqnarray} As in earlier sections, $k_B$, $q_e$, and $T$ in Eq.~\eqref{eq:NPE} are, respectively, the electrical potential, Boltzmann's constant (mJ/K), the elementary charge, and absolute temperature (K). The diffusion of ions inside an open channel can be assumed to occur along one of three spatial dimensions \cite{hille1978potassium,eisenberg1998ionic,NonnerEisenberg1998,eisenberg1999structure}. As a consequence, the motion of ions in the other two dimensions is assumed from here on to be negligible. Recall $v_B=Tk_B/q_e$. Using the integrating factor ${\exp\left(~zU/v_B~\right)}$, a one-dimensional version of Eq.~\eqref{eq:NPE} can then be written as: \begin{eqnarray} J_s&=& - \mu_s k T \exp \left( -\frac{z_s U}{v_B}\right) \frac{d}{dx} \left[ C_s \exp \left(\frac{z_s U}{v_B} \right) \right], \label{eq:NPE1} \end{eqnarray} where $x$ is a one-dimensional variable between 0 and $L$ representing the length of the channel pore across the membrane. The values 0 and $L$ represent the extracellular and intracellular ends of the channel, respectively. Taking a macroscopic perspective, assume the current flow through the open pore is a constant $i$ \cite{Endresenetal2000}. The current density between 0 and $L$ is then \begin{equation} \frac{i}{A(x)} = q z_s J_s(x), \label{eq:ix} \end{equation} where$A$ is the cross sectional area of the channel, assumed to be a smooth and slowly varying function on the interval [0,L]. This macroscopic description of the flux across the channel is reasonable \textit{at least} for channels permeable to potasssium (\kalium) and calcium (\calcium), which display single-file fluxes \cite{hille1978potassium,almers1984non}. As a consequence, \begin{equation} \frac{i}{A(x)} \exp \left( \frac{z_s U}{v_B}\right) = - z_s \mu_s q k T \frac{d}{dx} \left[ C_s \exp \left(\frac{z_s U}{v_B} \right) \right], \label{eq:NPE2} \end{equation} For notation purposes, let $U_i =U(0)$, $U_e =U(L)$, $S_i~=~C(0)$, and ${S_e = C(L)}$. The idea is to use Eqs.~\eqref{eq:NPE1} to write the total current in the open channel as a function of the membrane potential $v=U_i-U_e$. Eq.~\eqref{eq:NPE2} can be integrated between 0 and $L$ to obtain expressions in terms of the intra- and extracellular electrical potentials \begin{small} \begin{equation} i = \frac{ \mu z_s q k T}{\int_0^L \frac{dx}{A(x)} \exp\left( \frac{z_sU}{v_B} \right) } \left[ S_i \exp \left( \frac{z_s U_i }{v_B}\right) - S_e \exp \left( \frac{z_s U_e }{v_B}\right) \right] \label{eq:transmcurrent} \end{equation} \end{small} The integral in the denominator of the right hand side of Eq.~\eqref{eq:transmcurrent} describes the dependence of the current on the shape of the channel. %An alternative way to this assumption is to replace the number of ions %per unit time between 0 and $L$ by an intermediate value that %represents the average flux across the length of the pore. %As a consequence, the flux through the channel is inversely proportional to the %cross-sectional area, and the current density can then be written as %\begin{equation} % \frac{i}{A(x)} = z q J(x) \label{eq:consI} %\end{equation} %with $i$ representing the constant current. Assume also that the %distance between intra- \emck{and} extracellular compartments is $L$, %with $x=0$ and $x=L$ representing, respectively, the locations where %the intra- and extracellular domains begin in a direction %perpendicular to the membrane. Next, we rewrite Eq.~\eqref{eq:transmcurrent} as a function of the membrane potential. In normal resting conditions, the membrane potential of a cell is $v = U_i - U_e<0$, so $U_i < U_e$. Since the electric field is continuous, there is a location $x_m$ along the length of the pore such that the corresponding voltage $U_m=U(x_m)$ satisfies \begin{equation} U_i -U_m = \frac{v}{2} = U_m-U_e . \end{equation} Introducing $U_m$ into the exponential terms of Eq.~\eqref{eq:transmcurrent} (multiplying and dividing by $\exp\left(-\frac{zU_m}{v_B}\right)$), and factoring out the square roots of $S_e$ and $S_i$, the current is \begin{small} \begin{eqnarray} i &=& %- \frac{ \mu z q k T}{\int_0^L \frac{dx}{A(x)} \exp\left( \frac{zU}{v_B}\right) } \left[ S_i \exp \left( \frac{z U_i}{v_B}\right)-S_e \exp \left( \frac{z U_e }{v_B}\right) \right] \nonumber \\ &=& \frac{\mu z q k T \exp\left( \frac{z_SU_m}{v_B} \right)}{\int_0^L \frac{dx}{A(x)} \exp\left( \frac{z_S U}{v_B} \right) } \left[ S_i \exp \left( \frac{z_S (U_i-U_m)}{v_B}\right) -S_e \exp \left( \frac{z_S (U_e-U_m) }{v_B}\right) \right] \label{eq:DDUm} \nonumber \\ &=& % \frac{ \mu z q k T \exp\left( \frac{zU_m}{v_B} \right)}{\int_0^L % \frac{dx}{A(x)} \exp\left( \frac{zU}{v_B} \right) } \sqrt{S_e S_i} \nonumber \\ % && \left[ % \sqrt{\frac{S_e}{S_i}} \exp \left( \frac{z (U_e-U_m) }{v_B}\right) - \sqrt{ \frac{S_i}{S_e}} \exp \left( \frac{z (U_i-U_m) }{v_B}\right) \right] % \nonumber \\ &=& \frac{\mu_s z_s q k T \exp\left( \frac{z_sU_m}{v_B} \right)}{\int_0^L \frac{dx}{A(x)} \exp\left( \frac{z_sU}{v_B} \right) } \sqrt{S_e S_i} \nonumber \\ &&\times \left[ \sqrt{\frac{S_i}{S_e}} \exp \left( \frac{zv }{2v_B}\right) - \sqrt{ \frac{S_e}{S_i}} \exp \left( \frac{-z v }{2v_B}\right) \right] \label{eq:DDv} \end{eqnarray} \end{small} The expression in the right hand side Eq.~\eqref{eq:DDv} can be simplified to give a convenient functional form in terms of the Nernst potential $v_s$. To do so, integrate the zero-flux version of the NPE between 0 and $L$ and rewrite in terms of the intra- and extracellular concentrations for $S$: %and solving for the membrane potential: %\begin{equation} % v_S= \frac{v_B}{z_S} \ln\left( \frac{S_e}{S_i} \right). \label{eq:Nernst} %\end{equation} %rewritten as \begin{equation} \frac{S_e}{S_i}= \exp\left( \frac{z_S}{v_B} v_S \right). \label{eq:NernstConc} \end{equation} Substituting the right hand side of Eq.~\eqref{eq:NernstConc} into Eq.~\eqref{eq:DDv}, the current through the open channel can be rewritten as \begin{small} %\begin{eqnarray} \begin{equation} i = \frac{ \mu z q k T \exp\left( \frac{zU_m} {v_B} \right)}{\int_0^L \frac{dx}{A(x)} \exp\left( \frac{zU}{v_B} \right) } \sqrt{S_e S_i} \left\{ \exp \left[ \frac{z_S(v-v_S)}{2v_B} \right] - \exp \left[ \frac{z_S (v-v_S)}{2v_B} \right] \right\} \label{eq:DDv} %\end{eqnarray} \end{equation} \end{small} which can be simplified into \begin{eqnarray} i &=& \tilde{a}_s T \sqrt{ S_e S_i} \sinh \left[ \frac{z_s (v-v_S) }{2v_B} \right]. % - \frac{ \mu z q k T \exp\left( \frac{zU_m}{v_B} \right)}{\int_0^L % \frac{dx}{A(x)} \exp\left( \frac{zU}{v_B} \right) } % \sqrt{ S_e S_i} \left[ \exp\left( \frac{z (v_s-v) }{2v_B}\right) - \exp \left( \frac{z(v-v_s)}{2v_B}\right) \right] % \nonumber \\ &=& \label{eq:isinh} \end{eqnarray} with $\bar{a}$ representing a maximum whole-membrane current amplitude of the form \begin{equation} \tilde{a}_s = 2 \frac{ \mu_s z_s q k \exp\left( \frac{z_sU_m}{v_B} \right)}{\int_0^L \frac{dx}{A(x)} \exp\left( \frac{z_sU}{v_B} \right) }. \label{eq:tildea} \end{equation} The term $v_S$ %is called the \textit{Nernst} potential of $S$-ions, and can be though of as the membrane potential for which the net transmembrane flux of $S$-ions is zero. Therefore, the current is zero when $v=v_S$. Eq.~\eqref{eq:isinh} can be regarded as a macroscopic description of transmembrane current driven by electric \textit{drift and diffusion} through an open channel that will be herein referred to as a DD current. The integral in the denominator of Eq.~\eqref{eq:tildea} can be approximated by a constant \citep{NonnerEisenberg1998}. If the absolute temperature and the transmembrane concentrations of $S$ can be assumed to be constant, then $\tilde{a} T \sqrt{S_e S_i}$ can be replaced by a constant representing the maximum amplitude of the current through a single open channel. Note this approximation does not necessarily hold in the case of {\calcium} channels. \subsubsection*{Notes about steady state currents, fixed points, and saddle-node bifurcations} Consider the sum of all currents except $I_S$ \begin{equation} I(v,\alpha) = I_N(v,w,\alpha) + I_K(v,w,\alpha) + I_L(v,\alpha) \label{eq:Iinfty} \end{equation} where $\alpha$ is a vector containing all the parameters of the system. The \textit{steady state currents} are obtained after replacing $w$ by $w_{\infty}(v)= \left\{ 1 + \exp \left[ \eta_w (v_w-v) / v_B \right] \right\}^{-1}$ in the currents on the right hand side of \req{eq:Iinfty}: \begin{eqnarray} I_{N \infty}(v,\alpha) &=& I_N(v,w_{\infty},\alpha) \\ I_{K \infty}(v,\alpha) &=& I_K(v,w_{\infty},\alpha) \end{eqnarray} The \textit{total steady state current} without stimulation is the sum of the steady state currents \begin{equation} I_{\infty}(v) = I_{N \infty}(v,\alpha) + I_{K \infty}(v,\alpha) + I_L(v,\alpha) \end{equation} It is useful to recall at this point that the steady states of the system are points of the form $(v_*,w_*)$ where $w_*=w_{\infty}(v_*)$. As a consequence, the solutions of the equation $I_S-I_{\infty}(v_*)=0$ for constant $I_S$ can be used to determine the fixed points of the system by substitution into the steady state $w_{\infty}(v)$. Recall that a \textit{bifurcation} occurs when either the number, the type, or the stability of the fixed points or cycles of the system change \cite{strogatz1994nonlinear}. Non-monotonic curves $I_{\infty}$ are indicative of at least one of the bifurcations induced by current injection, namely, saddle-node bifurcations in which the number of fixed points changes between 3 and 1. To find the fixed points of the system in the presence of current stimulation, consider the zeros of the curve $I_S-I_{\infty}(v)$, which can be studied by observing the graph of $I_{\infty}(v)$ while moving a vertical line from left to right (in the direction of $I_S$). Bifurcations can be detected this way, as the number of fixed points may change for different values of $I_S$. If the curve is non-monotonic, the number of zero crosses would change from 1 to 3 (saddle-node bifurcation, or SN bifurcation for short), and then from 3 to 1 as the vertical line moves rightward. The $I_S-I_{\infty}(v)$ curves can be used to find the fixed points but do not help to find all the equilibria (\textit{e.g.} cycles) in these kinds of systems, nor do they help to find the nature of all the bifurcations induced by injection of current or changes in other parameters. In particular, as shown by examples in the following paragraphs, the transitions between rest and repetitive oscillations (bifurcations of the steady state) are not always characterized by the monotonicity, or lack thereof, in the $I_S-I_{\infty}(v)$ curve. For instance, the curve may not be monotonic, in which case current injection could induce a SN bifurcation. However, the transition between rest and sustained oscillations may not occur at the SN point. In cases like these, the system is bistable (one stable fixed point and one limit cycle coexist) within a small interval of $I_S$ values, with the SN bifurcation somewhere in the right end of the interval. This means that when the stimulus amplitude is within the interval of bistability, the membrane could be placed within the basin of attraction of the limit cycle. This happens with square pulse stimulation but may not happen with ramp stimuli \cite{rinzel1988threshold,baer1989slow,rinzel1989analysis,IzhikevichBook}. The $I_{\infty}(v)$ in the DD and CB versions of the model changes from non-monotonic to monotonic as $a_K$ increases. The eigenvalues of the Jacobian matrix of the system evaluated at a fixed point determine the type and stability of the fixed point. Node fixed points yield eigenvalues with zero imaginary part and nonzero real parts of the \textit{same} sign. Saddle-node points yield eigenvalues have a zero imaginary part and real parts of \textit{different} signs. The eigenvalues of focus points have non-zero imaginary parts and non-zero real parts; if the real part is zero the point is called a center. The eigenvalues of asymptotically stable fixed points have negative real parts. \subsubsection*{Notes on channel biophysical properties and model validation} All ion channels considered here are assumed to have the same (macroscopic) biophysical properties if encoded by the same gene. Changes in the biophysical properties of channels can be regarded as dependent on splicing of the gene \cite{eisenberg1999structure,lin2009alternative}, or on modulation by hormones, mono-amines, and other molecules \cite{macica2003modulation}. In the absence of modulation, the biophysical properties of channels from the same splice variant can be assumed to be constrained to a narrow range. For instance, compare the biophysical properties of Shab channels reported in \cite{Covarrubiasetal1991} and \cite{TsunodaSalkoff1995b}; which are two articles reporting voltage-clamp data obtained from different expression systems. Importantly, the formulation for the gating time constant is such that the fits to currents recorded under voltage-clamp do not require exponents in the gating variables \cite{herrera-valdez2011dissertation}. Most of the parameters involved with a channel gating variable $u$ can be constrained based on biophysical experiments. The data for the channels included in the model was digitized from \cite{TsunodaSalkoff1995b}. \subsubsection*{Notes on bifurcation analysis} For $a_K$=1, the two models have three fixed points. The DD model has two stable fixed points, separated in phase space by a \textit{separatrix} defined by the stable orbit of the saddle point \cite{fitzhugh1960thresholds}. The CB model has only one stable fixed point that corresponds to the resting potential. The qualitative behavior of the DD system is different when the maximum {\kalium} current amplitude is twice as big as the maximum {\natrium} current amplitude ($a_K= \bar{a}_K/ \bar{a}_N$ = 2). In this case the number of fixed points at $I_S=0$ is also 3, but there is only one asymptotically stable fixed point (resting potential), one saddle-node, and one unstable node point. The system becomes bistable for larger $I_S$ as a stable limit cycle (in turn surrounding an unstable cycle around the stable fixed point) emerges through a fold-limit cycle (FLC) bifurcation (not shown for clarity as it would crowd the bifurcation diagrams). For larger values of $I_S$, the stable fixed point eventually becomes unstable through a (sub-critical) AH bifurcation; in this case the unstable limit cycle collapses onto the stable focus point, leaving an unstable fixed point surrounded by a stable limit cycle. If $I_S$ increases further, the system undergoes a SN bifurcation, with the number of fixed points decreasing from 3 to 1. In this case the limit cycle remains after the SN bifurcation as the only attractor of the system for a range of values of $I_S$. The limit cycle eventually shrinks to a single stable focus point with a larger $v$-value (depolarization block) through another AH bifurcation. This explains the oscillations around the depolarization block sometimes observed in experiments in which the amplitude of the injected current is too large. As current increases even further, the fixed point turns into a stable node. A distinct pattern of bifurcations occurs in the DD model when the maximum {\potassium} current amplitude is 3 times larger than the maximum {\sodium} current ($a_K=\bar{a}_K / \bar{a}_N=3$). In this case there is only one fixed point for each of the values of $I_S$ under consideration, so the only bifurcations caused by increasing $I_S$ involve a change in the type or the stability of the fixed point. As $I_S$ increases within the range considered here, the fixed point is a stable node first; then it becomes a stable focus which eventually looses its stability through an AH bifurcation. For larger values of $I_S$ the focus point eventually becomes stable again (another AH bifurcation) and later it becomes a stable node. Similar situations occur for the cases $a_K$=4 and 5, all of which can be thought of as membrane models in which the expression of potassium channels is significantly higher than the membranes with $a_K \in \{1,2\}$ (top two curves). \subsubsection*{Bifurcation sequences for $a_K$=2.5} The DD model undergoes a SN bifurcation first (from 1 to 3 fixed points near $I_S$=210 pA); followed by a fold limit cycle (FLC) bifurcation (near $I_S$=383 pA) in which two cycles emerge around the fixed point. In this case, the stable fixed point is surrounded by an unstable cycle that, in turn, is surrounded by a stable limit cycle (repetitive spiking). The unstable cycle separates the basins of attraction of the fixed point and the limit cycle, resulting in bistability that remains for a small interval with $I_S$ larger than 383~pA. This means that the system will go back to the resting potential if within the unstable cycle, or into a repetitive spiking regime if outside the unstable cycle. In contrast, all the trajectories from different initial conditions for $I_S$=383~pA in the CB model converge to the only fixed point of the system, which is asymptotically stable. For values of $I_S$ near 400 pA, the stable focus becomes unstable (as the unstable limit cycle closes into it through a subcritical AH bifurcation), leaving the limit cycle as the only attractor of the system with 3 unstable fixed points. The system then undergoes an SN bifurcation where the focus and saddle points meet and disappear, leaving only one unstable node and the limit cycle. The membrane potential trajectories of the DD model from different initial conditions and $I_S$=675 pA result in repetitive spiking dynamics around a unique unstable node. In contrast, the CB model is bistable, the two attractors being a limit cycle and a fixed point, separated by an unstable limit cycle. The CB system undergoes a FLC bifurcation near $I_S$=675 pA, then the fixed point loses its stability through another AH bifurcation, then it becomes stable again, and finally the unstable focus transforms into an unstable node. Therefore, the sequences of bifurcations displayed by the two models for $a_K$=2.5 are different as a function of $I_S$. \subsubsection*{Phase plane trajectories with UTD stimulation} In the phase plane $w-v$, the limit cycles of the DD and CB models have similar shapes, but they have different sizes. The difference between cycles in the phase plane is easier to observe in the $w$-direction: the range of $w$ values that spans a smaller interval in the DD model in comparison to CB. The location of the fixed point of the system before stimulation is very similar for both models, but during the maximal stimulation, the $v$-component of the fixed point in the DD model is more than 10 mV higher in the CB model (Fig.~S6). The phase plane also explains how the repetitive spiking started in the CB model: as the current stimulus began to increase, the system trajectory $(w,v)(t)$ slowly tracks the fixed point while it was stable. In agreement with the previous analysis, the fixed point becomes unstable at {\stimtop}, giving rise to a spiral trajectory away from the fixed point and eventually reaching the limit cycle. The differences in the cycles can be better observed in the $(dv/dt,v)$ plots. For instance, the down-stroke of the action potential in the DD model starts with an acceleration near the peak of the action potential, then it slows down shortly, but continues downward until the valley of the action potential is reached. The initial change in acceleration after the peak is absent in the CB model. \bibliography{membranebiophysics} \end{document}