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  pdfauthor={Marco Arieli Herrera-Valdez},
  pdftitle={Herrera. Supplement. Excitability driven by drift and diffusion.}
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\newcommand{\icyc}{{$I_{cyc}$}}
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\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}