\section{Introduction}
The vertex algebra $V$ is called {\em lisse}, or {\em $C_2$-cofinite},
if the dimension of the associated variety $X_V$ is zero.
For instance,  a simple affine vertex algebra $V$ associated with an affine Kac-Moody algebra $\widehat{\mathfrak{g}}$
is lisse  if and only if $V$ is an integrable representation as a $\widehat{\mathfrak{g}}$-module.
Thus, the lisse property generalizes the integrability condition to an arbitrary vertex algebra.

It is known that a lisse vertex operator algebra $V$ has nice properties, such as
the modular invariance of characters \cite{Z,Miy04},
and most
theories of vertex operator algebras have been  build under this finiteness condition
(see e.g.\  \cite{DLM00,Hua08}).
However, there do exist significant vertex algebras that do not satisfy the lisse condition.
For instance,
admissible affine vertex algebras
do not satisfy the lisse condition unless they are integrable,
but nevertheless their representations are semisimple in category ${\mathcal O}$ (\cite{AdaMil95,A12-2})
and have the modular invariance property (\cite{KacWak89,AEkeren}).
Moreover, there are a huge number of
vertex algebras
constructed in \cite{BeeLemLie15} from
 four dimensional $N=2$ superconformal field theories (SCFTs),  whose character coincides with  the Schur limit of the superconformal index of the
 corresponding
 four dimensional theories.
These vertex algebras do not
  satisfy the lisse property in general either.

 In this paper we propose the {\em quasi-lisse} condition
 that generalizes the lisse condition.
More precisely,
we call a conformal vertex algebra $V$ quasi-lisse
if
its associated variety $X_V$ has finitely many symplectic leaves.
For instance,
a simple affine vertex algebra $V$ associated with  $\widehat{\mathfrak{g}}$ is quasi-lisse if and only if $X_V$ is contained in the nilpotent cone $\mathcal{N}$ of $\mathfrak{g}$.
Therefore,
by \cite{FeiMal97,Ara09b},
all the admissible affine vertex algebras
are quasi-lisse.
Moreover,
the $W$-algebras obtained by quasi-lisse affine vertex algebras by the quantized Drinfeld-Sokolov  reduction (\cite{FF90,KacRoaWak03}) is quasi-lisse as well.
The vertex algebras
 constructed from
 4d SCFTs are also expected to be quasi-lisse,
 since their associated varieties conjecturally
 coincide with the Higgs branches of the corresponding four dimensional theories (\cite{R}).

We show that the normalized character of an ordinary representation of a  quasi-lisse vertex operator algebra
has a modular invariance property,
in the sense that it
satisfies  a {\em modular linear differential equation} (MLDE) (cf.~\cite{MMS}, \cite{KZ},
\cite{Mas07}, {\cite{Mil}, \cite{KNS} and~\cite{AKNS}).
This seems to be new even for an admissible affine vertex algebra.
Moreover,
using MLDE,
we obtain the explicit character formulas
of simple affine vertex algebras associated with the Deligne exceptional series
$
A_1\subset A_2 \subset G_2 \subset D_4 \subset F_4 \subset E_6 \subset E_7 \subset E_8
$ (\cite{D})
at level $-h^{\vee}/6-1$.
These vertex algebras arose
in \cite{BeeLemLie15}
as $2d$ chiral algebras
constructed from
4d SCFTs\footnote{For types $G_2$ and $F_4$ the connection with
4d SCFTs is conjectural.}.
Thus \cite{BeeLemLie15}, our result expresses
 the homogeneous Schur indices of
 the corresponding 4d SCFTs as  {\em (quasi)modular forms}.
This result is rather surprising
especially for types $D_4$,
$E_6$,
$E_7$ and $E_8$ (non-admissible casees), since the characters of these vertex algebras
are
written \cite{KasTan00} in terms of {\em non-trivial} Kazhdan-Lusztig polynomials
as their highest weights are not regular dominant.

We note that in \cite{CorSha16}
the authors have obtained a  conjectural expression
of Schur indices in terms of
Kontsevich-Soibelman
wall-crossing invariants,
which
we hope to investigate  in future works.
\smallskip
\subsection*{Acknowledgments}
The first named author  thanks
Victor Kac,
Anne Moreau,
Hiraku Nakajima,
Takahiro Nishinaka,
Leonardo
  Rastelli,
  Shu-Heng Shao,
 Yuji Tachikawa
  and Dan Xie
  for  valuable discussion.
  He thanks
  Christopher Beem
  for pointing out an error
  in the first version of this article.
  Some part of this work was done while he was visiting
  Academia Sinica, Taiwan, in August 2016, for
  ^^ ^^ Conference in Finite Groups and Vertex Algebras".
  He thanks  the organizers of the conference.
  He is partially  supported
by JSPS KAKENHI Grant Number
No.\ 20340007 and No.\ 23654006.
The second named author would like to thank Hiroshi Yamauchi for the helpful advice.
He was partially supported by
 JSPS KAKENHI Grant Number No.\ 14J09236.

This research was supported in part by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Economic Development \& Innovation.
\section{Quasi-lisse vertex algebras}
Let $V$ be a conformal vertex algebra,
$R_V=V/C_2(V)$
the Zhu's $C_2$-algebra of $V$ (\cite{Z}),
where
 $C_2(V)={\langle} a_{(-2)}b\mid a,b\in V{\rangle}_{\mathbb{C}}$.
 The space $R_V$ is a Poisson algebra
 by
 \begin{align*}
\bar a\cdot \bar b=\overline{a_{(-1)}b},
\quad
\{\bar a,\bar b\}=\overline{a_{(0)}b}.
\end{align*}
Here $\bar a$ denotes the image of $a\in V$ in $R_V$,
and $a(z)=\sum_{n\in \mathbb{Z}}a_{(n)}z^{-n-1}\in (\End V)[[z,z^{-1}]]$
is the quantum field corresponding to $a\in V$.
In this paper we assume $V$ is finitely strongly generated, that is,
$R_V$ is finitely generated.

The associated variety \cite{Ara12} of a vertex algebra $V$ is the finite-dimensional
algebraic variety defined
by
$$X_V=\operatorname{Specm}(R_V).$$
Since $R_V$ is a Poisson algebra,
we have a finite partition $$X_V=\bigsqcup_{k=0}^rX_k,$$
where $X_k$ are smooth analytic Poisson varieties (see e.g. \cite{Brown:2003kq}).
Thus, for any point $x\in X_k$ there is a well defined symplectic leaf $\mathscr{L}_x\subset X_k$
through it.
\begin{Def}
A  finitely strongly generated  vertex algebra $V$ is called {\em quasi-lisse}
if $X_V$ has only finitely many symplectic leaves.
\end{Def}
Let $V$ be a quasi-lisse vertex algebra.
The finiteness of the symplectic leaves implies \cite{Brown:2003kq}
 that
 the symplectic leaf $\mathscr{L}_x$
 at $x\in X_V$
coincides with the regular locus of the zero variety of the maximal Poisson ideal
contained in
the maximal ideal corresponding to $x$.
Thus,  each leaf $\mathscr{L}_x$ is a smooth connected locally-closed {\em algebraic} subvariety in $X_V$.
In particular,
every irreducible component of $X_V$ is the closure of a symplectic leaf (\cite[Corollary 3.3]{Gin03}).

For us,  the importance  of the finiteness of the symplectic leaves
 is in the following fact that has been established by Etingof and Schedler.
 \begin{Th}[\cite{EtiSch10}]\label{Th:ES}
Let $R$ be a finitely generated Poisson algebra,
and suppose that $\operatorname{Specm}(R)$ has finitely many symplectic leaves.
Then
$$\dim R/\{R,R\}<\infty.$$
\end{Th}

\section{A necessary condition for the quasi-lisse property}

A finitely strongly generated vertex algebra $V$ is called {\em conical}
if it is conformal,
and
$L_0$ gives a
$\frac{1}{m}\mathbb{Z}_{\geq 0}$-grading
$$V=\bigoplus_{\Delta\in \frac{1}{m}\mathbb{Z}_{\geq 0}}V_{\Delta}$$ on
$V$
for some $m\in \mathbb{N}$,
$\dim V_{\Delta}<\infty$ for all $\Delta$,
and $V_0=\mathbb{C}$,
where
$V_{\Delta}=\{v\in V\mid L_0v=\Delta v\}$.
Note that if  $V$ is a vertex operator algebra,
that is, if $V$ is integer-graded,
then a conical vertex operator algebra is the same as a vertex operator algebra of CFT type.

Let $V$ be a conical vertex algebra.
The $L_0$-grading induces the grading
$$R_V=\bigoplus_{\Delta\in \frac{1}{m}\mathbb{Z}_{\geq 0}}(R_V)_{\Delta},\quad (R_V)_{0}=\mathbb{C},$$
on $R_V$.
In other words,
the $L_0$-grading induces a contracting $\mathbb{C}^*$-action on
the associated variety $X_V$.

\begin{Rem}
The
associated variety of a simple conical quasi-lisse vertex algebra is conjecturally irreducible (\cite{AraMor16}).
The validity
of this conjecture implies that
the associated variety of a quasi-lisse vertex algebra is
actually {\em symplectic},
that is,
$X_V$ is the closure of a symplectic leaf.
\end{Rem}

\begin{Rem}
Conical lisse ($C_2$-cofinite) conformal vertex algebras are quasi-lisse,
since $X_V$ is a point in this case.
\end{Rem}

\begin{Pro}\label{Th:nilp1}
Let $V$ be a conical quasi-lisse vertex algebra.
Then the image $[\omega]$ of the conformal vector $\omega$ of $V$ is nilpotent in the Zhu's $C_2$-algebra $R_V$
of $V$.
\end{Pro}
\begin{proof}
Since $V$ is conical,
the $\mathbb{C}^*$-action $\rho$ on $X_V$
induced by the conformal grading
contracts to a point, say $0$,
that is, $\lim\limits_{t\rightarrow 0}\rho(t)x=0$ for all $x\in X_V$.

 Set $z=[\omega]$. It is sufficient  to show that the value of $z$ at any closed point $x$ is zero.
Pick an irreducible component  $Y$
of $X_V$ containing $x$.
Note that $Y$ is $\mathbb{C}^*$-invariant, and hence, $0\in Y$.
On the other hand,  there exists a symplectic leaf $\mathscr{L}\subset X_V$ such that
 $Y=\overline{\mathscr{L}}$, the Zariski closure of  $\mathscr{L}$.
 Since it belongs to the Poisson center of $R_V$,
 $z$ belongs to the Poisson center of $\mathcal{O}(\mathscr{L})$.
Hence $z$ is constant on $\mathscr{L}$ as  $\mathscr{L}$ is symplectic,
and thus,
so is
on $Y$.
Therefore
the value of
$z$  at $x$ is  the same as the one at
$0$,
which is clearly zero.
\end{proof}

\section{Finiteness of ordinary representations}

Recall that
a~weak $V$-module $(M,Y_M)$ is called {\it ordinary}
if $L_0$ acts semi-simply on $M$, any $L_0$-eigenspace  $M_\Delta$ of $M$ of eigenvalue $\Delta\in\mathbb{C}$ is finite-dimensional,
and for any $\Delta\in \mathbb{C}$, we have $M_{\Delta-n}=0$ for all sufficiently large $n\in\mathbb{Z}$.
\begin{Th}
Let $V$ be a quasi-lisse conformal vertex algebra.
Then the number of the simple ordinary $V$-modules is finite.
\end{Th}
\begin{proof}
Let $A(V)$ be the Zhu's algebra of $V$.
By the Zhu's theorem \cite{Z},
it is sufficient to show that the number of the simple finite-dimensional $A(V)$-modules is finite.

The algebra $A(V)$ is naturally filtered:
There is a natural filtration of $G_{\bullet}A(V)$ induced by the
filtration $\bigoplus\limits_{\Delta\leq p}V_\Delta$ of $V$ (\cite{Z})
that makes  the associated graded  $\gr _GA(V) $
a
Poisson algebra.
Moreover,
there is  a  surjection map
$$R_V\rightarrow \gr A(V)$$
of Poisson algebras \cite{De-Kac06,ALY}.
Therefore,
$\operatorname{Specm}(\gr_G A(V))$  is a Poisson subvariety of $X_V$,
and hence has a finitely many symplectic leaves.
Hence, thanks to
 Theorem 1.4 of \cite{EtiSch10} that follows from Theorem \ref{Th:ES},
 we conclude that $A(V)$ has only finitely many simple finite-dimensional representations.
\end{proof}

\section{Modular linear differential equations}
Let $\vartheta_k$ denote the Serre derivation of weight $k$
\[
\vartheta_k(f)=q\frac{d}{dq}f-\frac{k}{12}E_2 f,
\]
where $E_n(\tau)$ is the normalized Eisenstein series of weight $n\geq 2$.
Let $\vartheta_k^i=\vartheta_{k+2(i-1)}\circ \cdots \circ \vartheta_{k+2}\circ \vartheta_k$ be the $i$-th iterated Serre derivation of weight $k$ with $\vartheta^0_k=1$.
Recall that a~modular linear differential equation (MLDE) of weight $k$ is a~linear
differential equation $$\vartheta_k^n f+\sum_{j=0}^{n-1}P_{j}\vartheta_k^jf=0$$
with a~classical modular function $P_{j}$ of weight $2n-2j$ for each $0\leq j\leq n-1$.

In this section, we prove the following theorem.
Let $\mathbb{H}$ denote the complex upper-half plane.

\begin{Th}\label{Th:main}
Let $V$ be a quasi-lisse vertex operator algebra,
$c\in \mathbb{C}$ the central charge of $V$.
Then the normalized character
$$\chi_V(\tau)=\operatorname{tr}_V (e^{2\pi i\tau(L_0-c/24)})\qquad (\tau\in\mathbb{H})$$
satisfies a modular linear differential equation of weight $0$.
\end{Th}

Let $(V, Y(\cdot,z),\omega)$ be a~quasi-lisse conformal vertex operator algebra
with the weight grading $V=\bigoplus_{\Delta=0}^\infty V_\Delta$.  Y.~Zhu introduced a~second vertex operator algebra $(V,Y[\cdot,z],\tilde{\omega})$
associated to $V$ (\cite{Z}), where
 the vertex operator $Y[\cdot,z]$ is defined by
linearly extending the assignment
\[
Y[v,z]=Y(v,e^z-1)e^{z\Delta }=\sum_{n\in\mathbb{Z}}v[n]z^{-n-1}, \quad (v\in V_\Delta,\ \Delta\geq 0),
\]
and
$\tilde{\omega}=\omega-c/24$.
We write $Y[\tilde{\omega},z]=\sum_{n\in\mathbb{Z}}L[n]z^{-n-2}$ and
$V_{[\Delta]}=\{v\in V\,|\,L[0]v=\Delta v\}$ for every $\Delta\in \mathbb{Z}_{\geq 0}$.

Set
$A=\mathbb{C}[\tilde{G}_4(q),\tilde{G}_6(q)]$, $V_A=V\otimes A$.
Here, $q$ is a~formal variable and $\tilde{G}_n(q)=\sum_{j=0}^\infty a_{n,j}q^j$,
where $\{a_{n,j}\}_{j\geq 0}$ ($n\geq 2$) are the Fourier coefficients of the Eisenstein series $G_n(\tau)$ of weight $n$, that is,
$G_n(\tau)=\sum_{j=0}^\infty a_{n,j} e^{2\pi i\tau j}$.
Let
 $O_q(V)$ be the
$A$-submodule of $V_A$
generated by
\[
a[-2]b+\sum_{k=2}^\infty (2k-1)\tilde{G}_{2k}(q)a[2k-2]b\qquad (a,b\in V).
\]
Let $[V_A,V_A]$ denote
the $A$-span of elements $a[0]b$, $a,b\in V$, in $V_A$.

\begin{Pro}\label{Pro:key}
The $A$-module
$$V_A/([V_A,V_A]+O_q(V))$$ is finitely generated.
\end{Pro}
\begin{proof}
We set
$$U:=[V_A,V_A]+O_q(V).$$
Also,
put
$R=R_V=V/C_2(V)$,  $C_2(V)_A=C_2(V)\otimes_\mathbb{C} A$, and
\begin{align*}
R_A=R\*_\mathbb{C} A=V_A/C_2(V)_A.
\end{align*}

Define an increasing filtration $G_{\bullet}V_A$ of the $A$-module $V_A$ by
\begin{align*}
G_p V_A=\bigoplus_{\Delta\leq p}V_\Delta\* A.
\end{align*}
This induces the filtration on $U$ and $V_A/U$:
 $G_p U=U\cap G_p V_A$,
$\gr_G U=\bigoplus_p G_p U/G_{p-1}U$,
$G_p (V_A/U)=G_p V_A/G_p U$,
and
$$\gr_G (V_A/U)=\bigoplus_p G_p (V_A/U)/G_{p-1} (V_A/U)
= V_A/\gr_G U.$$

Since
$$\gr_G U\supset \{R_A,R_A\}+C_2(V)_A,$$
we have a surjective map of $A$-modules
$$R_A/\{R_A,R_A\}\twoheadrightarrow
V_A/\gr_G U=\gr_G
(V_A/U).$$
As
$R_A/\{R_A,R_A\}=(R/\{R,R\})\otimes_\mathbb{C} A$,
the assertion follows from
Theorem \ref{Th:ES}.
\end{proof}

Since $A$ is a~Noetherian ring,
it follows from  Proposition \ref{Pro:key} that
$V_A/([V_A,V_A]+O_q(V))$ is a~Noetherian $A$-module.
Hence,
 we have the following lemma.
\begin{Lem}\label{Th:kankei1}
For an element  $a$ of $V$,
there exist $s\in\mathbb{Z}_{\geq 0}$ and $g_i(q)\in A$
{\rm(}$0\leq i\leq s-1${\rm)} such that
\[
L[-2]^sa+\sum_{i=0}^{s-1} g_i(q) L[-2]^ia\in [V_A,V_A]+O_q(V).
\]
\end{Lem}

Let $M$ be an~ordinary $V$-module.
Define the zero-mode action $o(\cdot):V\rightarrow \mathrm{End}(M)$
by linearly extending the assignment
\[
o(a)=\mathrm{Res}_z Y^M(a,z)z^{\Delta-1}dz:M\rightarrow M, \quad (a\in V_\Delta,\ \Delta\geq 0),
\]
For any $a\in V$,  define the formal $1$-point function $\tilde{\chi}_M(a,q)$ by
 $$\tilde{\chi}_M(a,q)=\mathrm{tr}|_M o(a)q^{L_0-c/24}.$$
For each $q$-series $f(q)$ and $k\geq 0$,
define the formal Serre derivation $\partial_k$ of weight $k$ by
\[
\partial_k f=q\frac{d}{dq}f(q)+k\,\tilde{G}_2(q)f(q).
\]
Let $\Delta$ and $\ell$ be non-negative integers.
For each $a\in V_{[\Delta]}$ and $f(q)\in  \mathbb{C}[\tilde{G}_2(q),\tilde{G}_4(q),\tilde{G}_6(q)]$ of weight $n$, define the formal iterated Serre derivation $\partial^i$ by
\begin{align*}
\partial^i (f(q) \tilde{\chi}_M(a,q))=\partial_{\Delta+\ell+2i-2}(\partial^{i-1}(f(q) \tilde{\chi}_M(a,q))) \quad (i\geq 1),
\end{align*}
and $\partial^0=\mathrm{id}$.
Here, $f(q)$ is said to be of weight $n$
if $f(q)$ is a~homogeneous element of weight $n$ of the graded algebra $\mathbb{C}[\tilde{G}_2(q),\tilde{G}_4(q),\tilde{G}_6(q)]$,
where the weight of $\tilde{G}_k(q)$ is $k$ for each $k=2,4,6$.

\begin{Lem}[{cf.~\cite[(5.9)]{DLM00}}]\label{Th:vir1}
Let $v$ be an~element of $V$ and $M$ an~ordinary $V$-module.
We have
\[
\tilde{\chi}_M(L[-2]v,q)=\partial \tilde{\chi}_M(v,q)+\sum_{\ell=2}^\infty \tilde{G}_{2\ell}(q)\tilde{\chi}_M(L[2\ell-2]v,q).
\]
\end{Lem}

\begin{proof}
The assertion follows by~\cite[Proposition~4.3.5]{Z} with $a=\omega$ and $b=v$.
\end{proof}

Let $a$ be a primiary vector of $V$ of weight $\Delta$,
that is, $L[0]a=\Delta a$ and
$L[n]a=0$ for $n\geq 1$.

\begin{Lem}[{cf.~\cite[Lemma~6.2]{DLM00}}]\label{Th:vir2}
For each $i\geq 1$, there exist elements $f_j(q)\in A$ {\rm(}$0\leq j\leq i-1${\rm)} such that for any ordinary $V$-module $M$,
\begin{equation}\label{eqn:serre}
\tilde{\chi}_M(L[-2]^ia,q)=\partial^i\tilde{\chi}_M(a,q)+
\sum_{j=0}^{i-1}f_j(\tau)\partial^j\tilde{\chi}_M(a,q).
\end{equation}
\end{Lem}

\begin{proof}
The proof is similar to that of~\cite[Lemma~6.2]{DLM00}.
We prove the assertion by induction on $i$.
When $i=1$, it follows from Lemma~\ref{Th:vir1} that
$\tilde{\chi}_M(L[-2]a,q)=\partial\tilde{\chi}_M(a,q)$, and therefore~\eqref{eqn:serre} follows.
Suppose that $i\geq 2$.
Then by~ Lemma \ref{Th:vir1}, we see that
\begin{align*}
&\tilde{\chi}_M(L[-2]^i a,q)\\&=\partial \tilde{\chi}_M(L[-2]^{i-1}a,q)+\sum_{k=2}^\infty
\tilde{G}_{2k}(q) \tilde{\chi}_M(L[2k-2]L[-2]^{i-1}a,q).
\end{align*}
By using the relation of the Virasoro algebra, we have
$$L[2k-2]L[-2]^{i-1}a=c_k \cdot L[-2]^{i-k}a$$
with  a~scalar $c_k$ for each
$2\leq k\leq i$ and $L[2k-2]L[-2]^{i-1}a=0$ if $k\geq i+1$.
 Therefore,
 \begin{align*}
\tilde{\chi}_M(L[-2]^i a,q)=\partial \tilde{\chi}_M(L[-2]^{i-1}a,q)+\sum_{k=2}^{i}
c_k \tilde{G}_{2k}(q) \tilde{\chi}_M(L[-2]^{i-k}a,q).
\end{align*}
By the induction hypothesis, we have~\eqref{eqn:serre}, which completes the proof.
\end{proof}

Let $u$ and $v$ be elements of $V$.

\begin{Lem}[{\cite[Proposition 4.3.6]{Z}}]\label{Th:kankei3}
For every ordinary $V$-module $M$,
\[
\tilde{\chi}_M(u[0]v,q)=0,
\]
\[
\tilde{\chi}_M(u[-2]v,q)+\sum_{k=2}^\infty (2k-1)\tilde{G}_{2k}(q)
\tilde{\chi}_M(u[2k-2]v,q)=0.
\]
\end{Lem}

\begin{Th}\label{Th:abs}
Let $V$ be a quasi-lisse vertex operator algebra,
$a\in V$ primary with $L[0]a=\Delta a$.
For each ordinary $V$-module $M$, the series $\tilde{\chi}_M(a,q)$ converges absolutely and
uniformly in every closed subset of the domain $\{q\,|\,|q|<1\}$, and the limit function $\chi_M(a,q)$
has the form $q^hf(q)$ with some analytic function $f(q)$ in  $\{q\,|\,|q|<1\}$.
Moreover, the space spanned by $\chi_M(a,q)$ for all ordinary $V$-module $M$ is a~subspace of the space of the solutions of a~modular linear differential equation
of weight $\Delta$.
\end{Th}

\begin{proof}
The proof is similar to those of~\cite[Theorem 4.4.1]{Z} and~\cite[Lemma 6.3]{DLM00}.
By Lemma~\ref{Th:kankei1}, we have $L[-s]^sa+\sum_{i=0}^{s-1}g_i(q)L[-2]^ia\in O_q(V)$ where $s\in \mathbb{Z}_{\geq 0}$ and
$g_i(q)\in A$  for each $0\leq i\leq s-1$.
It then follows by the definition of $O_q(V)$ and Lemma~\ref{Th:kankei3}
that
$\tilde{\chi}_M(L[-2]^sa+\sum_{i=0}^{s-1} g_i(q)L[-2]^ia,q)=0$.
By Lemma~\ref{Th:vir2}, we obtain a~differential equation
\begin{equation}\label{eq:diffproof}
\partial^s \tilde{\chi}_M(a,q)+\sum_{i=0}^{s-1}h_i(q)\partial^i\tilde{\chi}_M(a,q)=0.
\end{equation}
for the formal series
$\tilde{\chi}_M(a,q)$ with $h_i(q)\in A$.
Since $h_i(q)$ converges absolutely and uniformly
on every closed subset of  $\{q\,|\,|q|<1\}$, and~\eqref{eq:diffproof} is
regular, it follows that $\tilde{\chi}_M(a,q)$ converges uniformly
on every closed subset of $\{q\,|\,|q|<1\}$.
By using~\eqref{eq:diffproof} again, we see that
 the space spanned by $\chi_M(a,q)$ for all ordinary $V$-module $M$ is a~subset of the space of the solutions of the MLDE
$$
\vartheta^s_{\Delta} \chi_M(a,q)+\sum_{i=0}^{s-1}p_i(q)\vartheta^i_{\Delta}\chi_M(a,q)=0,
$$
where  $p_i(q)\in\mathbb{C}[G_4(\tau),G_6(\tau)]$ is the limit function of $h_i(q)$ with $q=e^{2\pi i\tau}$ and $\tau\in\mathbb{H}$.
The remainder of the theorem is clear.
\end{proof}

Theorem~\ref{Th:main} follows from Theorem~\ref{Th:abs} with $a=|0\rangle$, the vacuum vector of $V$.

\section{Examples of quasi-lisse vertex algebras}
 Let
 $V^k(\mathfrak{g})$ be the universal affine vertex algebra associated with a simple Lie algebra
 $\mathfrak{g}$ at level $k\in \mathbb{C}$,
 and let
 $V_k(\mathfrak{g})$ be the unique simple graded quotient of $V^k(\mathfrak{g})$.
 We have $X_{V^k(\mathfrak{g})}=\mathfrak{g}^*$,
 where $\mathfrak{g}^*$ is equipped with the Kirillov-Kostant-Souriau Poisson structure,
 and
  $X_{V_k(\mathfrak{g})}$ is a conic, $G$-invariant, Poisson subvariety of $X_{V^k(\mathfrak{g})}=\mathfrak{g}^*$,
 where $G$ is the adjoint group of $\mathfrak{g}$ (see \cite{Ara12}).

 Let $\mathcal{N}=\{x\in\mathfrak{g}\mid \ad x \text{ is nilpotent}\}$,
 the nilpotent cone of $\mathfrak{g}$,
 which is identified with the zero locus
 of the argumentation ideal $\mathbb{C}[\mathfrak{g}^*]^G_+$ of
 the invariant ring $\mathbb{C}[\mathfrak{g}^*]^G$ via the identification $\mathfrak{g}=\mathfrak{g}^*$.
 It is well-known since Kostant \cite{Kos63}
 that
 the number of $G$-orbits in  $\mathcal{N}$
 is finite.
 \begin{Lem}
The affine vertex algebra $V_k(\mathfrak{g})$ is quasi-lisse if and only if
the associated variety
$X_{V_k(\mathfrak{g})}\subset \mathcal{N}$.
\end{Lem}
\begin{proof}
The ^^ ^^ if" part
is clear
since the symplectic leaves in $\mathfrak{g}^*$
 are the coadjoint  orbits of $G$.
 Conversely,
 suppose that $X_{V_k(\mathfrak{g})}\not \subset\mathcal{N} $.
 Since  $X_{V_k(\mathfrak{g})}$ is closed,
there exists a nonzero semisimple element $s$ in $X_{V_k(\mathfrak{g})}$.
As it  is conic,
$X_{V_k(\mathfrak{g})}$ contains  infinitely many orbits of the form
$G.\lambda s$, $\lambda\in \mathbb{C}^*$.
\end{proof}

Recall that
$V_k(\mathfrak{g})$ is called {\em admissible} if it is an admissible representation (\cite{KacWak89})
as a module over the affine Kac-Moody algebra $\widehat{\mathfrak{g}}$
associated with $\mathfrak{g}$.
All the admissible affine vertex algebras are quasi-lisse,
since
their associated varieties are  contained in $\mathcal{N}$
 (\cite{FeiMal97,Ara09b}).
In fact,
 the associated variety of an admissible affine vertex algebra $V_k(\mathfrak{g})$
is irreducible, that is,
$X_{V_k(\mathfrak{g})}=\overline{\mathbb{O}}$ for some nilpotent orbit $\mathbb{O}$
of $\mathfrak{g}$ (see \cite{Ara09b} for the explicit description of the orbit $\mathbb{O}$).

Highest weight representations
of an admissible affine vertex algebra $V_k(\mathfrak{g})$
are exactly the admissible representations $L(\lambda)$ of $\widehat{\mathfrak{g}}$ of level $k$
whose integral Weyl groups are
obtained from  that of $V_k(\mathfrak{g})$
by an element of the extended affine Weyl group
 (\cite{AdaMil95, A12-2}).
Let  $\mathfrak{h}$ be the Cartan subalgebra of $\mathfrak{g}$.
The modular invariance of the normalized {\em full} characters
$$e^{2\pi kt}\operatorname{tr}_{L(\lambda)}(q^{L_0-c/24}e^{2\pi i x}),\quad (\tau,x,t)\in Y,
$$
of those representations,
where $Y$ is some domain in $\mathbb{H}\times \mathfrak{h}\times \mathbb{C}$,
 has been known since
Kac and Wakimoto
 \cite{KacWak88,KacWak89},
 and was extended in \cite{AEkeren} to
 that of the general (full) trace functions.
  Here it is essential to consider their full characters,
 since an admissible representation is not an ordinary representation in general,
 and thus,
the normalized character
 $\operatorname{tr}_V (e^{2\pi i\tau(L_0-c/24)})$ is not always well-defined.

Theorem \ref{Th:abs}
states the modular invariance of
the normalized character
(instead of the normalized full character)
of
an admissible representation that is ordinary.
As far as the authors know, this fact is new.

\smallskip

Here are more examples of quasi-lisse affine vertex algebras.
\begin{Th}[\cite{AM15}]\label{Th:AM}
Assume that $\mathfrak{g}$ belongs to the Deligne exceptional  series
$$A_1\subset A_2\subset G_2\subset D_4\subset F_4\subset E_6\subset E_7\subset E_8,$$
and let $k=-h^{\vee}/6-1$.
Then $X_{V_k(\mathfrak{g})}\cong \overline{\mathbb{O}_{min}}$,
where $\mathbb{O}_{min}$ is the minimal nilpotent orbit of $\mathfrak{g}$.
\end{Th}

In Theorem \ref{Th:AM},
the affine vertex algebra $V_{-h^{\vee}/6-1}(\mathfrak{g})$ is admissible
 for types $A_1$,  $A_2$, $G_2$, $ F_4$,
and so the statement is contained in \cite{Ara09b}.
However, $V_{-h^{\vee}/6-1}(\mathfrak{g})$  is not admissible for types $D_4$, $E_6$, $E_7$, $E_8$.
These non-admissible quasi-lisse affine vertex algebras have appeared in \cite{BeeLemLie15}
as main examples of chiral algebras coming from 4d SCFTs.
In fact, the labels $D_4$, $E_6$, $E_7$, $E_8$ also appear in Kodaira's classification of
isotrivial elliptic fibrations,
and the corresponding 4d SCFTs are obtained by applying the $F$-theory to these isotrivial elliptic fibrations.
By construction \cite{BeeLemLie15},
 the character of the above non-admissible quasi-lisse affine vertex algebras
are the (homogeneous) Schur indices of these 4d SCFTs obtained from elliptic fibrations.
In mathematics,
such a non-admissible affine vertex algebra was first extensively studied in \cite{Per13}.

In the next section
we derive the explicit form of the characters of these non-admissible quasi-lisse affine vertex algebras.

\smallskip

Now let us give examples of quasi-lisse vertex algebras
outside affine vertex algebras.
Let $\mathscr{W}^k(\mathfrak{g},f)$ be the $W$-algebra associated with $\mathfrak{g}$ and a nilpotent element $f\in \mathfrak{g}$ at level $k$,
defined by the quantized Drinfeld-Sokolov reduction
\begin{align*}
\mathscr{W}^k(\mathfrak{g},f)=H^0_{DS,f}(V^k(\mathfrak{g})),
\end{align*}
where $H^\bullet_{DS,f}(M)$ denotes the cohomology of the BRST complex with coefficient $M$ associated with
the Drinfeld-Sokolov reduction with respect to $f$.
This definition  was discovered
by Feigin and Frenkel \cite{FF90} in the case that $f$ is principal
as a generalization of Kostant's Whittaker model of the center of $U(\mathfrak{g})$
(\cite{Kos78}),
and was generalized to an arbitrary $f$  by Kac and Wakimoto
 (\cite{KacRoaWak03}).

By \cite{Ara09b},
the natural surjection $V^k(\mathfrak{g})\rightarrow V_k(\mathfrak{g})$
induces a surjective homomorphism
$\mathscr{W}^k(\mathfrak{g},f)\twoheadrightarrow H^0_{DS,f}(V_k(\mathfrak{g}))$ of vertex algebras,
and moreover,
\begin{align*}
X_{H^0_{DS,f}(V_k(\mathfrak{g}))}\cong X_{V_k(\mathfrak{g})}\cap \mathcal{S}_f,
\end{align*}
where $\mathcal{S}_f$ is the Slodowy slice at $f$,
that is,
$\mathcal{S}_f=f+\mathfrak{g}^e$.
Here $\{e,f,h\}$ is an $\mathfrak{sl}_2$-triple and
$\mathfrak{g}^e$ is the centralizer of $e$ in $\mathfrak{g}$.
Therefore, we have the following assertion.
\begin{Lem}\label{Lem:W}
Let $k$ be non-critical
and suppose that $V_k(\mathfrak{g})$ is quasi-lisse,
that is, $X_{V_k(\mathfrak{g})}\subset \mathcal{N}$.
For any $f\in X_{V_k(\mathfrak{g})}$,
$H^0_{DS,f}(V_k(\mathfrak{g}))$ is quasi-lisse,
and hence, so is the simple quotient $\mathscr{W}_k(\mathfrak{g},f)$ of $\mathscr{W}^k(\mathfrak{g},f)$.
\end{Lem}
We note that
$H^0_{DS,f}(V_k(\mathfrak{g}))\cong \mathscr{W}_k(\mathfrak{g},f)$
if $G.f\subset X_{V_k(\mathfrak{g})}$
for type $A$ (\cite{Ara08-a})
and this conjecturally holds for any $\mathfrak{g}$
(\cite{KacWak08}).

Lemma \ref{Lem:W}
implies
admissible affine vertex algebras produce many quasi-lisse $W$-algebras
by applying the Drinfeld-Sokolov reduction.
For instance,
if $k$ is a non-degenerate admissible number (see \cite{A2012Dec}),
then $X_{V_k(\mathfrak{g})}=\mathcal{N}$, and hence,
\begin{align*}
X_{H^0_{DS,f}(V_k(\mathfrak{g}))}\cong \mathcal{N}\cap \mathcal{S}_f,
\end{align*}
which is  irreducible and therefore symplectic (\cite{Pre02}).
In particular, if $f$ is a subregular nilpotent element in types $ADE$,
$ X_{H^0_{DS,f}(V_k(\mathfrak{g}))}$ has the simple singularity of the same type as $\mathfrak{g}$ (\cite{Slo80}).
In type $A$,
it has been recently shown by Genra \cite{Genra} that
the subregular $W$-algebra $\mathscr{W}^k(\mathfrak{sl}_n,f_{subreg})$
is isomorphic to   Feigin-Semikhatov's $W_n^{(2)}$-algebra (\cite{FeiSem04}) at level $k$ .

\smallskip

See \cite{AM15,AraMor16b} for more examples of quasi-lisse vertex algebras,
and see e.g. \cite{Dan} for more examples of vertex algebras obtained from 4d SCFTs.

\section{The characters of affine vertex algebras associated with the Deligne exceptional series}\label{section:DLE}

In this section, we give the explicit character formulas of the quasi-lisse affine vertex algebras associated with the Deligne exceptional series
 appeared in Theorem \ref{Th:AM}
by using MLDEs.

Let $\mathfrak{g}$ be a~Lie algebra in the Deligne exceptional series
and $V$ the simple affine vertex algebra $V_k(\mathfrak{g})$ with $k=-h^\vee/6-1$.
The Deligne dimension formula~\eqref{eqn:deligne1} below
implies that the central charge $c$ of $V$ is
given by $$c=-2h^\vee-2.$$

\begin{Lem}
 The square of the Virasoro element $\omega$ of $V$ is $0$ in the Zhu's $C_2$-algebra $R_V$.
\end{Lem}
\begin{proof}
Let $I$ be the ideal of $R_{V^k(\mathfrak{g})}=S(\mathfrak{g})$
generated by the image of the maximal submodule of $V^k(\mathfrak{g})$,
so that
$R_{V_k(\mathfrak{g})}=S(\mathfrak{g})/I$.
We need to show that $\Omega^2\in I$,
where $\Omega$ is the Casimir element   of $S(\mathfrak{g})$.

If $\mathfrak{g}$ is not of  type $A$,
then this result has been already stated in  Lemma 2.1 of \cite{AM15},
see (the proof of) Theorem 3.1 of  \cite{AM15}.
So let $\mathfrak{g}$ be of type $A$,
in which case
the maximal submodule of $V^k(\mathfrak{g})$
is generated by a singular vector, say $v$ (\cite{KacWak88}).
For
$\mathfrak{g}=\mathfrak{sl}_2$,
 the assertion follows immediately from a result in \cite{FeiMal97},
 which says that the  image of $v$ in $I$
coincides with $\Omega e$ up to nonzero constant multiplication,
see  the proof of Theorem 4.2.1 of \cite{FeiMal97}.
Finally let $\mathfrak{g}=\mathfrak{sl}_3$.
Then
the vector $v$ has degree $2$,
cf. ~ \cite{Per08}.
Let $V$ be the $\mathfrak{g}$-submodule of $S^2(\mathfrak{g})$ generated by the image $[v]$ of $v$ in $I$.
Proposition 3.3 of  \cite{GanSav04} (which is valid for type $A$ cases as well)
says that $\mathfrak{g} \cdot V\subset S^3(\mathfrak{g})$ contains a submodule isomorphic to $\mathfrak{g}$.
On the other hand,
Kostant's Separation Theorem  (\cite{Kos63}, cf. Proposition 3.2 of \cite{GanSav04}) implies that
$\mathfrak{g} \cdot \Omega$ is the unique submodule of $ S^3(\mathfrak{g})$ isomorphic to $\mathfrak{g}$.
Thus,
$\mathfrak{g} \cdot \Omega\subset  \mathfrak{g}  \cdot V$,
and the assertion follows.
\end{proof}
As $(V,Y[\cdot,z])$ is isomorphic to $(V,Y(\cdot,z))$,
it follows that
$$L[-2]^2|0\rangle =
\sum_{i=1}^\ell b_i[-2]c_i $$
for some
$\ell\geq 0$,
where $b_i$ and $c_i$, $1\leq i\leq \ell$, are $L[0]$-homogeneous elements of $V$
such that $L[0](b_i[-2]c_i)=4b_i[-2]c_i$.
On the other hand,
it  follows from the definition of $O_q(V)$ that
$$b_i[-2]c_i\equiv -\sum_{k=2}^\infty f_k(q) b_i[2k-2]c_i\pmod{O_q(V)}$$
for  $1\leq i\leq \ell$.
However,
 $b_i[2]c_i\in\mathbb{C}|0\rangle$ and  $b_i[2k-2]c_i=0$ for $k\geq 3$
 as the $L[0]$-weight of $b_i[-2]c_i$ is $4$.
Therefore,
we get that $$L[-2]^2|0\rangle+g(q)|0\rangle \in O_q(V)$$ with
$g(q)\in A$.
By using Lemma~\ref{Th:vir2},
we see that the formal characters $\tilde{\chi}_M(|0\rangle,q)$ of all ordinary $V$-modules $M$ satisfy
a~second order differential equation of the form
$\partial^2\tilde{\chi}_M(|0\rangle ,q)+f_1(q)\partial \tilde{\chi}_M(|0\rangle ,q)+
f_2(q)\tilde{\chi}_M(|0\rangle ,q)=0$ with $f_1(q),f_2(q)\in A$.
Hence, the characters $\chi_M(\tau)$ of the ordinary $V$-modules $M$
satisfy a~second order MLDE
$L(f)=0$ of weight 0.

The second order MLDEs (of weight 0) have the form
\begin{equation}\label{eqn:kz}
f''(\tau)-\frac{1}{6}E_2(\tau)f'(\tau)-\frac{k(k+2)}{144}E_4(\tau)f(\tau)=0
\end{equation}
with $k\in \mathbb{C}$.
Here,
\[
'=q\frac{d}{dq}=\frac{1}{2\pi i}\frac{d}{d\tau}.
\]
A~function $f(\tau)$ is called of {\it vacuum type} if $f$ has the form
$f(\tau)=q^{-\alpha/24}(1+\sum_{n=1}^\infty a_nq^n)$ with $\alpha\in \mathbb{Q}$ and $a_n\in \mathbb{Z}_{\geq 0}$ for each $n\geq 1$, where $q=e^{2\pi i \tau}$.
Let $f$ be a~solution of~\eqref{eqn:kz} of the form
$f(\tau)=q^{-\alpha/24}(1+O(q))$ with $\alpha\in \mathbb{Q}$.
Then by substituting $f$ into~\eqref{eqn:kz}, we see that
$\alpha=-k/12$ or $(k+2)/12$.
If $\alpha=-k/12$, it follows that $k$ is one of the following numbers~\cite[(3.12)]{KNS}:
\begin{equation}\label{cand1}
k=\frac{1}{5},\frac{1}{2},1,\frac{7}{5},2,\frac{13}{4},3,\frac{7}{2},\frac{19}{5},4.
\end{equation}
On the other hand, if $\alpha=(k+2)/12$, we have~\cite[(3.16)]{KNS}
\begin{equation}\label{cand2}
k=\frac{1}{5},\frac{1}{2},1,2,3,5,8,11,17,23,29,53.
\end{equation}

Since $V$ is of CFT-type with the central charge $-h^\vee-2$, the character $\chi_V(\tau)$ is of vacuum type and has the form $\chi_V(\tau)=q^{(h^\vee+1)/12}(1+O(q))$.
Therefore,~\eqref{cand1} and~\eqref{cand2}
imply that the MLDE $L(f)=0$ must be
the following one:
\begin{equation}\label{eqn:diff1}
f''(\tau)-\frac{1}{6}E_2(\tau)f'(\tau)-\frac{(h^\vee-1)(h^\vee+1)}{144}E_4(\tau)f(\tau)=0.
\end{equation}

The vacuum type solutions of~\eqref{eqn:diff1} are also given in~\cite{KK} and~\cite{KNS}.
As a result, we conclude that
\begin{align*}
&\chi_{V_{-4/3}(A_1)}=\frac{\eta(3\tau)^3}{\eta(\tau)^3},\quad
\chi_{V_{-3/2}(A_2)}=\frac{\eta(2\tau)^8}{\eta(\tau)^8},\quad
\chi_{V_{-5/3}(G_2)}=\frac{E_1^{(3)}(\tau)\eta(3\tau)^6}{\eta(\tau)^8},\\
&\chi_{V_{-2}(D_4)}=\frac{E_4'(\tau)}{240\eta(\tau)^{10}},\quad
\chi_{V_{-5/2}(F_4)}=\frac{E_2^{(2)}(\tau)\eta(2\tau)^{24}}{\eta(\tau)^{28}},\\
&\chi_{V_{-3}(E_6)}=
-\frac{1}{462}\left(\frac{E_6(\tau)E_4'(\tau)}{240\eta(\tau)^{22}}
-\eta(\tau)^2\right),\\
&\chi_{V_{-4}(E_7)}=\frac{1}{204204}\Biggl(
\Delta(\tau)P_2\left(\frac{E_6(\tau)}{\sqrt{\Delta(\tau)}}\right)
\frac{E_4'(\tau)}{240\eta(\tau)^{34}}-\Delta(\tau)\frac{E_6(\tau)}{\eta(\tau)^{34}}
\Biggr),\\
&\chi_{V_{-6}(E_8)}=\frac{1}{38818159380}\Biggl(
\Delta(\tau)^2P_4\left(\frac{E_6(\tau)}{\sqrt{\Delta(\tau)}}\right)
\frac{E_4'(\tau)}{240\eta(\tau)^{58}}-\Delta(\tau)^{5/2}
Q_4\left(\frac{E_6(\tau)}{\sqrt{\Delta(\tau)}}\right)\frac{1}{\eta(\tau)^{58}}
\Biggr).
\end{align*}
Here,
$\eta(\tau)=q^{1/24}\prod_{n=1}^\infty (1-q^n)$,
$\Delta(\tau)=\eta(\tau)^{24}$,
$E_2^{(2)}(\tau)=2E_2(2\tau)-E_2(\tau)$,
\[
E_1^{(3)}(\tau)=1+6\sum_{n=1}^\infty \Bigl(\sum_{d|n}\left(\dfrac{d}{3}\right)(n/d)^2\Bigr)q^n
\]
with the Legendre symbol $\left(\dfrac{\cdot}{\cdot}\right)$,
$P_2(x)=x^2+462$, $P_4(x)=x^4+1341x^2+201894$, and $Q_4(x)=x^3+879x$.

In particular, it follows
that the characters
of $V_{-4/3}(A_1)$,
$V_{-3/2}(A_2)$,
$V_{-5/3}(G_2)$, and
$V_{-5/2}(F_4)$  are modular forms,
while those of
 $V_{-2}(D_4)$, $V_{-3}(E_6)$, $V_{-4}(E_7)$ and $V_{-6}(E_8)$ are quasimodular forms of positive depths
(\cite[pp.450]{KNS}).
Moreover, if $h^\vee$ is the dual Coxeter number of $D_4$, $E_6$, $E_7$ or $E_8$, then MLDE~\eqref{eqn:diff1}
has a~solution with a~logarithmic term (see \cite[section~5]{KK} and~\cite[Remark~3.8]{KNS}).
Note that the  above formula for $A_1$ and $A_2$
follows also from  the recent remark \cite{KW4} by Kac and Wakimoto.

\begin{Rem}\label{Th:remk1}
It should be notable that
the coefficient of $E_4(\tau)f(\tau)$ in~\eqref{eqn:diff1} is a~non-constant rational function in $h^\vee$,
as
such phenomena are
 often observed for the Deligne exceptional series.
In fact,
the dimensions of specific modules over any Lie algebra in the Deligne exceptional series satisfy
the so-called {\it Deligne dimension formulas}, which are rational functions in $h^\vee$. For example,
\begin{equation}\label{eqn:deligne1}
\dim \mathfrak{g} = \frac{2(h^\vee +1)(5h^\vee -6)}{h^\vee +6},
\end{equation}
and
$
\dim L(2\theta) = 5(h^\vee)^2(2h^\vee+3)(5h^\vee-6)/(h^\vee+12)(h^\vee+6)
$
(\cite{CdM}, \cite{D} and \cite{LM02}).
Here,
$L(2\theta)$ is the irreducible highest weight module of weight $2\theta$ over $\mathfrak{g}$.
In the vertex algebra setting,
the first example of such phenomena
was first observed
in \cite{T}, where the coefficients of the MLDEs which the characters of the affine vertex algebras $V_1(\mathfrak{g})$ at level~$1$
associated with the Deligne exceptional series satisfy are expressed as rational functions in  $h^\vee$.
The second example of such phenomena was found
in \cite{K}, where
the minimal affine $\mathcal{W}$-algebras associated with the Deligne exceptional series at level $-h^\vee/6$ were shown to be lisse and rational.
\end{Rem}

\begin{Rem}
It follows from a result of~\cite{KNS} that
there is a~vacuum type solution of~\eqref{eqn:diff1} with $h^\vee=24$.
Although the Coxeter number $h^\vee$ of any Lie algebra in the Deligne exceptional series does not coincide with $24$,
the number ``$h^\vee=24$" appears in many studies of the Deligne exceptional series (see e.g.~\cite{CdM} and \cite{K14}).
\end{Rem}

\newcommand{\etalchar}[1]{$^{#1}$}


\section{Introduction}
\label{sec:theIntro}
While finite difference (FD), Runge-Kutta (RK) and exponential-time-differencing (ETD) methods have numerically integrated Hodgkin-Huxley (HH)
type neural models for over fifty years, it could be argued a measure of uncertainty remains regarding the differences one can expect by choosing
one numerical method or integration step-size over another. Such uncertainties include: How does simulation inaccuracy present? How do action
potential (AP) waveforms change as integration step size increases? Do ETD methods obviate the need for implicit integration methods \cite{Borgers}?
Does oscillation onset appear only on the rising edge of spiking membrane voltage waveforms \cite{Borgers}? If the Hines-Crank-Nicolson (HCN)
method allows decaying oscillations as step sizes grow large, when and where do these oscillations present and to what extent
\cite[ch.4]{CarnevaleHines}? We address these questions as they relate to integrating a branched, $74$ compartment, single cell, neuronal model
over integration step sizes ranging from $1$--$100\mu$s. \\ \\
Rallpacks formalized the measurement of similar data to answer the question, ``How does one simulator's performance compare to others in
terms of accuracy, model size, simulation speed, and core memory needed?''\cite{BBB_Rall}. In contrast this study graphically illustrates how
performance varies for different numerical methods as integration step size increases. Rallpacks' first two benchmarks were passive models
and the third added two active ion membrane channels. Each compartment in our study's model has eleven active membrane ion channels
and includes calcium dynamics.\\ \\
Section \ref{sec:TestProbAndNumMethods} begins by defining our test problem and listing the numerical methods we test. Section \ref{sec:AccStab}
reviews the predicted accuracy and stability of the integration methods evaluated in this study. Here we show that the HH partial differential cable
equation renders multiple order explicit RK methods mere first-order methods. Section \ref{sec:AnalysisMethods} describes data analysis techniques. \\ \\
The literature suggests significant AP waveform structures and measurements include action-potential (AP) spikes and bursts \cite{Feld12, Gol06,
Hage15, Kepecs00, Kuznet10, Miller81, Pinsky94, Shad98, T2005}, spiking multiplicities, amplitudes and spike rise times \cite{Varela2012},
after-depolarization-pulses (ADP) \cite{Chu06, Kepecs00, Metz07, Varela2012}, and power spectrums \cite{Bahr13}. Results presented in Section
\ref{sec:Results} show numerical integration method and step size combinations capable of inducing inaccuracies able to either masquerade as or
distort these same membrane voltage waveform measures. We show that while ETD method stability makes it an excellent choice when integrating
HH equations for single compartment models, the first-order exponential Euler method is not suited for multi-compartment models. We then identify
integration step sizes and locations where oscillations appear in AP cycle waveforms integrated by the HCN method as well as examine their magnitude
and duration. Finally, Section \ref{sec:SumAndDiscuss} discusses our results and how they relate to other studies.
\section{Test Problem and Numerical Methods}
\label{sec:TestProbAndNumMethods}
\subsection{Governing Equations}
The membrane voltage of compartmental excitable nerve cell models with varying cross-section diameters depends on neighboring compartments'
membrane voltages as well as their own membrane ion channel states as described by the partial differential cable equation \cite[ch.6]{DayanAbbott}
\begin{equation}
\label{CableEquationRoot}
c_m\frac{\partial V}{\partial t} + \frac{(V-E_L)}{r_m} + \sum_i g_i(V-E_i) - \frac{1}{2ar_L}\frac{\partial}{\partial x}\bigg(a^2\frac{\partial V}{\partial x}\bigg)=0.
\end{equation}
Parameters of equation (\ref{CableEquationRoot}) are defined in Table \ref{tab:cableEqnParms}.
\begin{table}[h!]\small
\caption{Cable Equation Parameters}
\ \\
\centering
\setlength{\tabcolsep}{1.0mm}
\begin{tabular}{l l l}
\toprule
Symbol & Parameter & Units   \\
\midrule \\[-2.0ex]
$a$ & compartment radius & $m$ \\
$c_m$  & specific membrane capacitance  &  $F/m^2$ \\
$C_m$  & compartment membrane capacitance  &  $F$ \\
$E_i$ & membrane ion channel $i$ Nernst Potential & $volts$ \\
$E_L$ & membrane leakage current Reversal Potential & $volts$ \\
$g_i$ & membrane ion channel $i$ specific conductance & $S/m^2~=~(\Omega\cdot m^2)^{-1}$ \\
$r_L$  & specific axial resistance  &  $\Omega \cdot m$ \\
$R_a$  & compartment axial resistance  &  $\Omega$ \\
$r_m$  & specific membrane resistance  &  $\Omega \cdot m^2$ \\
$V$ & membrane voltage & $volts$ \\
\bottomrule
\end{tabular}
\label{tab:cableEqnParms}
\end{table}
\\
Membrane ion channel $i$ specific conductance,
\begin{equation}
\label{ionChanSpecCond}
g_i=\overline{g}_if_i(m_i,h_i),~\text{for}~i=1,2,...,11,
\end{equation}
is the product of channel specific concentration, $\overline{g}_i$ and the channel specific gate states function, $f_i(m_i,h_i)$, for each membrane
ion channel. Activation, $m_i$, and inactivation, $h_i$, states of channel constituent gates are governed by
\begin{equation}
\label{eqn:gateStateODE}
\frac{dy}{dt}=\frac{y_{\infty}(V)-y}{\tau_y(V)}~\text{for }y=m~\text{or}~h,
\end{equation}
where gate steady state, $y_{\infty}(V)$, and gate time constant, $\tau_y(V)$, are functions of membrane voltage.
\subsection{Neural Model}
\label{sec:TraubL23rsModel}
Our model is Traub's cortical L2/3 regular spiking pyramidal neuron whose parameters were defined in \cite{T2005}.  It is deterministic and
without hyper/de-polarizing or synaptic currents.
\subsection{Numerical Methods}
Numerical integration methods are listed in Table \ref{tab:schemeTruncError}. Forward-time central-space, backward-time central-space and
Hines-Crank-Nicolson, are numerical PDE methods. Traditional ODE methods, exponential Euler and RK numerical integrators, were included
to demonstrate how not accounting for a PDE's changing spatial derivative affects an integrator's truncation error and our ability to predict step
size upper limits guaranteeing stability. HCN is the only method to integrate the membrane voltage and gate states in a staggered fashion,
temporally centering the gate state integration between membrane voltage time-steps as described in \cite{CarnevaleHines,Hines}.
\subsection{Simulation Specification}
Simulation parameters are listed in Table \ref{tab:simSpec}.
\begin{table}[h!]\small
\caption{Simulation Specification}
\ \\
\centering
\begin{tabular}{l | l}
\toprule
Duration & $3.0$ seconds ($\sim40$ AP cycles) \\
\hline \\[-2.5ex]
Step Sizes & $1,~2,~3,~...~,~99~~\mu$s (given stability)\\
\hline \\[-2.0ex]
Initial Values & Compartment membrane voltage, $V=-0.07$ volts\\
& Calcium ion concentration, [Ca$^{2+}]=0.0$ \\
& Gate states: steady-state, $y_{\infty}\big(V=-0.07\big)$ \\
\bottomrule
\end{tabular}
\label{tab:simSpec}
\end{table}
\section{Accuracy and Stability}
\label{sec:AccStab}
\subsection{Accuracy}
\label{subsec:Acc}
Integration methods listed in Table \ref{tab:schemeTruncError} are intended for different contexts. At the same time there exist myriad examples
of these methods being used to integrate generic HH models \cite{GEN, Borgers, CarnevaleHines, CooleyDodge66, MooreRamon74, T2005}.
To be clear, we integrated ion channel states represented by equation (\ref{eqn:gateStateODE}) with ODE methods forward Euler, backward
Euler and trapezoidal rule in place of PDE methods forward-time central-space (FTCS), backward-time central-space (BTCS) and HCN, respectively.
\begin{table}[h!]\small
\caption{Membrane Voltage Integration Truncation Error, $1^{st}$ and $2^{nd}$ Order Terms}
\ \\
\centering
\setlength{\tabcolsep}{1.0mm}
\begin{tabular}{l | l}
\toprule
Method & $P_{k,h}(\cdot)-P(\cdot)$  \\
\midrule \\[-2.0ex]
\parbox{2.5cm}{Forward-Time \\ Central-Space}
& $\displaystyle ~~\frac{V_{tt}}{2}k + \frac{V_{ttt}}{6}k^2 -\gamma\bigg(\overline{a}^2V_{xxxx}+2\big(\overline{a}^2\big)_xV_{xxx}\bigg)\frac{h^2}{12}$
\\[2.0ex] \hline \\[-2.0ex]
\parbox{2.5cm}{Backward-Time Central-Space}
& $\displaystyle -\frac{V_{tt}}{2}k-\frac{V_{ttt}}{3}k^2-\gamma\bigg(\overline{a}^2V_{xxxx}+2\big(\overline{a}^2\big)_xV_{xxx}\bigg)\frac{h^2}{12}$
\\ [1.0ex] \hline \\[-2.0ex]
\parbox{2.5cm}{Exponential \\ Euler}
& $\displaystyle ~~\frac{V_{tt}}{2}k+\frac{V_{ttt}}{6}k^2-\gamma\bigg(\overline{a}^2V_{xxxx}+2\big(\overline{a}^2\big)_xV_{xxx}\bigg)\frac{h^2}{12}
+\frac{BV_t}{2}k - \frac{BV_t}{6}k^2$ \\[2.0ex] \hline \\[-2.0ex]
\parbox{2.5cm}{Hines-Crank-Nicolson}
& $\displaystyle \qquad\qquad \frac{V_{ttt}}{24}k^2-\gamma\bigg(\overline{a}^2V_{xxxx}+2\big(\overline{a}^2\big)_xV_{xxx}\bigg)\frac{h^2}{12}$
\\ [1.0ex] \hline \\[-2.0ex]
\parbox{2.5cm}{RK21}
& $\displaystyle ~~\frac{V_{tt}}{2}k + \frac{V_{ttt}}{6}k^2 -\gamma\bigg(\overline{a}^2V_{xxxx}+2\big(\overline{a}^2\big)_xV_{xxx}\bigg)\frac{h^2}{12}
+\frac{BV_t}{2}k$\\ [1.0ex] \hline \\[-2.0ex]
\parbox{2.5cm}{RK41}
& $\displaystyle ~~\frac{V_{tt}}{2}k + \frac{V_{ttt}}{6}k^2 -\gamma\bigg(\overline{a}^2V_{xxxx}+2\big(\overline{a}^2\big)_xV_{xxx}\bigg)\frac{h^2}{12}
+\frac{BV_t}{2}k$ \\ [1.0ex] \hline \\[-2.0ex]
Definitions & $\displaystyle \alpha=\frac{1}{r_mc_m}, \quad \beta_i=\frac{g_i}{c_m} \quad \text{and} \quad \gamma=\frac{1}{2ar_Lc_m},$ \\ \\[-1.0ex]
& $\displaystyle B=\Big(\alpha+\sum_i\beta_i\Big)+\bigg(\frac{1}{R_aC_m}+\frac{1}{R'_aC_m}\bigg)$ \\ \\[-1.0ex]
& $\overline{a}:$ Average compartment radius, $~~ \big(\overline{a}^2\big)_x\equiv\big(a(x)^2\big)_x|_{x=\overline{a}}$ \\[1.0ex]
\bottomrule
\end{tabular}
\label{tab:schemeTruncError}
\end{table}
\\ \\
For integrating equation (\ref{CableEquationRoot}) we normalized all our methods by recasting traditional ODE methods as quasi-FD methods to
assess their accuracy and stability on an equal footing. Truncation errors are summarized in Table \ref{tab:schemeTruncError} not just to point out
the order of a scheme's time and space error, but also to show differences between similarly ordered methods. Table \ref{tab:schemeTruncError}
includes the claim that HH partial differential cable equations render multi-order explicit RK integrators mere first-order methods. The following
derivation of RK21 truncation error describes the basis for this claim.
\subsubsection{RK21 Truncation Error}
\label{subsubsec:RK21truncErr}
Using process and notation from \cite{Str} we define truncation error as the difference between the cable equation's differential operator and,
what we define below, RK21's quasi-FD operator.  Modifying derivative notation in equation (\ref{CableEquationRoot}) lets us write the cable
equation's differential operator as
\begin{multline}
\label{txtCableEquationOperator}
P(V) = V_t + \Big(\alpha+\sum_i\beta_i\Big)V -\Big(\alpha E_L+\sum_i\beta_iE_i\Big) -\gamma\big(a^2V_x\big)_x  = 0,
\end{multline}
where \[ \alpha=\frac{1}{r_mc_m}, \quad \beta_i=\frac{g_i}{c_m} \quad \text{and} \quad \gamma=\frac{1}{2ar_Lc_m}. \] \\
Relationships between specific properties $r_L$, $c_m$ and a compartment's actual physical properties are
\begin{equation}
\label{txtSpecificAxialResDefs}
\frac{1}{R_aC_m}=\frac{a^2/a}{2r_Lc_mh^2}~~\text{and}~~\frac{1}{R'_aC_m}=\frac{a'^2/a}{2r_Lc_mh^2},
\end{equation}
\\
where $a$ is the radius of a given compartment with subscript $j$, $a'$ the radius of neighboring compartment, subscripted $j+1$, and $h$
is the compartmentally defined spatial resolution. Relationships between $a$, $a'$ and the diffusion term's spatial derivatives in equation  (\ref{txtCableEquationOperator}) needed below include
\begin{multline}
\label{txtIdentitiesForTE}
\big(a^2V_x\big)_x\equiv\overline{a}^2V_{xx}+\big(\overline{a}^2\big)_xV_x, \\ \\[-1.5ex]
\quad \text{where} \quad\overline{a}^2\equiv\frac{a'^2+a^2}{2}, \quad \text{and}
\quad\big(\overline{a}^2\big)_x\equiv\displaystyle\lim_{h\rightarrow 0}\frac{a'^2-a^2}{h}.
\end{multline}
\\
Next, we spatially discretize equation (\ref{txtCableEquationOperator}) and rewrite its diffusion term as a function of the compartment's
actual physical properties giving us the ODE
\begin{multline}
\label{cableODEtxt}
\big(V_t\big)_j + \Big(\alpha+\sum_i\beta_i\Big)V_j -\Big(\alpha E_L+\sum_i\beta_iE_i\Big) ~ ... \hfill \\ \\[-2.0ex]
-\Bigg(\frac{V_{j-1}-V_j}{R_aC_m}+\frac{V_{j+1}-V_j}{R'_aC_m}\Bigg)=0
\end{multline}
\begin{multline}
\label{simpleODEtxt}
\text{which we may abreviate as } \quad \big(V_t\big)_j=A-BV_j, \hfill
\end{multline}
\begin{multline}
\label{simpleODEtxtDefs}
\text{where } A=\alpha E_L +\sum_i \beta_i E_i +\bigg(\frac{V_{j-1}}{R_aC_m} +\frac{V_{j+1}}{R'_aC_m}\bigg), \\
\text{ and } \quad B=\alpha +\sum_i \beta_i +\bigg(\frac{1}{R_aC_m} +\frac{1}{R'_aC_m}\bigg).
\end{multline}
For the context of a generic HH multi-compartment model, we define the format of an RK method as
\begin{multline}
\label{rk21TruncErr0text}
V_j^{n+1}=V_j^n+kF(t_n,V_j^n,k;f) ~~\text{where}~~F(t_n,V_j^n,k;f)=\sum_{i=1}^s b_iK_i, \\ \\[-2.0ex]
K_i=f\big(t_n+c_ik,V_j^n+k\sum_{j=1}^sa_{ij}K_j\big), ~~ f(t_n,V_j^n) = \big(V_t\big)_j^n = A-BV_j^n,
\end{multline}
$k$ is the integration step size and terms $A,B$ are defined in equation (\ref{simpleODEtxtDefs}).
\begin{table}[h!]\small
\caption{RK21 Tableau}
\ \\
\centering
\begin{tabular}{l l}
\toprule \\[-2.0ex]
\begin{tabular}{l | l l}
$\displaystyle c_1$ & $a_{11}$  & \\
\hline \\[-2.0ex]
 & $b_1$ & $b_2$
\end{tabular}
&
$~ \Rightarrow ~$
\begin{tabular}{l | l l}
$1$ & $1$ & \\
\hline \\[-2.0ex]
 & $\frac{1}{2}$ &  $\frac{1}{2}$
\end{tabular}
\\ \\[-2.0ex]
\bottomrule
\end{tabular}
\label{tabInText:RK21}
\end{table}
RK21 rendered a quasi-FD method then becomes
\begin{multline}
\label{rk21TruncErr1text}
V_j^{n+1}=V_j^n+\frac{k}{2}\Big( K_1+K_2 \Big), ~~ \text{where} \hfill \\ \\[-2.0ex]
K_1=f\big(t_n,V_j^n\big)= A-BV_j^n, ~~ \text{and} \hfill \\ \\[-2.0ex]
K_2= f\big(t_n,V_j^n+kK_1\big)  = A-B\Big(V_j^n+k\big(A-BV_j^n\big)\Big). \hfill
\end{multline}
\\[-1.0ex]
After combining terms we may express the RK21 quasi-FD operator as
\begin{multline}
\label{rk21TruncErr2text}
P_{h,k}(V) = V_j^{n+1}-V_j^n-k\Big(A-BV_j^n \Big) + \frac{k^2}{2}\Big(B\big(A-BV_j^n\big)\Big)=0. \hfill
\end{multline}
\\
Taylor expanding the operator, simplifying, dividing by $k$ and substituting the first occurrence of $A$ and $B$ with definitions in
equation (\ref{simpleODEtxtDefs}) leads to
\begin{multline}
\label{rk21TruncErr3text}
V_t+\frac{k}{2}V_{tt}+\frac{k^2}{3!}V_{ttt}+O(k^3)
-\Bigg(\bigg[\alpha E_L+\sum_i\beta_iE_i +\bigg(\frac{V^n_{j-1}}{R_aC_m} +\frac{V^n_{j+1}}{R'_aC_m}\bigg)\bigg]\Bigg. ~...\\
\Bigg. - \bigg[\alpha +\sum_i \beta_i +\bigg(\frac{1}{R_aC_m} +\frac{1}{R'_aC_m}\bigg) \bigg] V_j^n \Bigg) +\frac{Bk}{2}\Big( A-BV_j^n \Big)=0.
\end{multline}
\\
Taylor expanding neighboring compartment voltage terms, simplifying and applying definitions in equation (\ref{txtIdentitiesForTE}), brings us to
\begin{multline}
\label{rk21TruncErr4text}
V_t+\frac{k}{2}V_{tt}+\frac{k^2}{6}V_{ttt}
-\Bigg(\Big(\alpha E_L+\sum_i\beta_iE_i\Big)-\Big(\alpha+\sum_i\beta_i\Big)V^n_j\Bigg. ~... \hfill  \\
+\frac{a^2/a}{2r_Lc_mh^2}\bigg(-hV_x +\frac{h^2}{2}V_{xx}-\frac{h^3}{6}V_{xxx}+\frac{h^4}{24}V_{xxxx} \bigg)~ ... \hfill \\
\Bigg. +\frac{a'^2/a}{2r_Lc_mh^2}\bigg(hV_x+\frac{h^2}{2}V_{xx}+\frac{h^3}{3!}V_{xxx}+\frac{h^4}{4!}V_{xxxx}\bigg) \Bigg
+\frac{Bk}{2}\Big( A-BV_j^n \Big)=O(k^3,h^3).
\end{multline}
Rearranging, again applying definitions in equation (\ref{txtIdentitiesForTE}) and eliminating third-order terms gives us
\begin{multline}
\label{rk21TruncErr5text}
V_t+\frac{V_{tt}}{2}k+\frac{V_{ttt}}{6}k^2 -\Bigg(\Big(\alpha E_L+\sum_i\beta_iE_i\Big)- \Big(\alpha+\sum_i\beta_i\Big)V^n_j  \Bigg.~... \hfill \\
\Bigg. +\gamma\bigg[\Big(\overline{a}^2V_{xx}+\big(\overline{a}^2\big)_xV_x\Big)+\Big(\overline{a}^2V_{xxxx}
+2\big(\overline{a}^2\big)_xV_{xxx}\Big)\frac{h^2}{12} \bigg] \Bigg)
+\frac{Bk}{2}\Big( A-BV_j^n \Big) =0.
\end{multline}
After replacing the remaining occurrence of $A-BV^n_j$ with what the first occurrence in equation (\ref{rk21TruncErr2text}) eventually
became in equation \ref{rk21TruncErr5text} we have
\begin{multline}
\label{rk21TruncErr6text}
V_t+\frac{V_{tt}}{2}k+\frac{V_{ttt}}{6}k^2
-\Bigg(\Big(\alpha E_L+\sum_i\beta_iE_i\Big) - \Big(\alpha+\sum_i\beta_i\Big)V^n_j \Bigg.~... \hfill \\
\hfill \Bigg. +\gamma\bigg[\big(\overline{a}^2V_x\big)_x+\Big(\overline{a}^2V_{xxxx} +2\big(\overline{a}^2\big)_xV_{xxx}\Big)\frac{h^2}{12} \bigg] \Bigg) ~ ... \\
+\frac{Bk}{2}\Bigg(\Big(\alpha E_L+\sum_i\beta_iE_i\Big) - \Big(\alpha+\sum_i\beta_i\Big)V^n_j \Bigg.~... \hfill \\
\Bigg. +\gamma\bigg[\big(\overline{a}^2V_x\big)_x+\Big(\overline{a}^2V_{xxxx} +2\big(\overline{a}^2\big)_xV_{xxx}\Big)\frac{h^2}{12} \bigg] \Bigg) =0.
\end{multline}
\begin{multline}
\label{rk21TruncErr7text}
\Rightarrow V_t+\Big(\alpha+\sum_i\beta_i\Big)V^n_j - \Big(\alpha E_L+\sum_i\beta_iE_i\Big)-\gamma\big(\overline{a}^2V_x\big)_x~... \hfill \\
+\frac{V_{tt}}{2}k+\frac{V_{ttt}}{6}k^2-\gamma\Big(\overline{a}^2V_{xxxx} +2\big(\overline{a}^2\big)_xV_{xxx}\Big)\frac{h^2}{12} ~ ... \\
+\frac{Bk}{2}\bigg(V_t+\gamma\Big(\overline{a}^2V_{xxxx} +2\big(\overline{a}^2\big)_xV_{xxx}\Big)\frac{h^2}{12}\bigg) =0,
\end{multline}
\\ \\[-2.0ex]
where the term $V_t$ in the last line of equation (\ref{rk21TruncErr7text}) came by using equation (\ref{txtCableEquationOperator}) to eliminate
terms in the last two lines of equation (\ref{rk21TruncErr6text}). Then after discarding the third-order $kh^2$ term we are left with
\begin{multline}
\label{rk21TruncErr8text}
V_t+\Big(\alpha+\sum_i\beta_i\Big)V^n_j - \Big(\alpha E_L+\sum_i\beta_iE_i\Big)-\gamma\big(\overline{a}^2V_x\big)_x~... \hfill \\
+\Big(BV_t+V_{tt}\Big)\frac{k}{2}+\frac{V_{ttt}}{6}k^2-\gamma\Big(\overline{a}^2V_{xxxx} +2\big(\overline{a}^2\big)_xV_{xxx}\Big)\frac{h^2}{12}=0.
\end{multline}
\\
Terms $A$ and $B$ in equation (\ref{simpleODEtxt}) are both functions of the compartment's membrane voltage, $V^n_j$, through ion channel
conductance term $\sum\beta_i$, and, of course, $V^n_j$ changes over time. Hence, $V_{tt}=A_t-(BV_t+B_tV)$. Clearly the first-order term is
not eliminated and truncation error for RK21 is
\begin{multline}
\label{rk21TruncErr9text}
P_{k,h}(V)-P(V) = ~ ... \hfill \\
\Big(BV_t+V_{tt}\Big)\frac{k}{2}+\frac{V_{ttt}}{6}k^2-\gamma\Big(\overline{a}^2V_{xxxx} +2\big(\overline{a}^2\big)_xV_{xxx}\Big)\frac{h^2}{12},
\end{multline}
which is only first-order in time, second in space.
\subsection{Stability}
\label{ssec:SchemeStab}
Results of Von Neumann stability analyses \cite[ch.2]{Str} for this study's six membrane voltage integration methods on an unbounded
domain are listed in Table \ref{tab:schemeStability}. The bounded stability analysis presented in \cite[ch.4]{CarnevaleHines}, confirmed the
same results shown in Table \ref{tab:schemeStability} for FTCS, BTCS and HCN.
\begin{table}[h!]\small
\caption{Integration Method Stability}
\ \\
\centering
\setlength{\tabcolsep}{1.0mm}
\begin{tabular}{l | l}
\toprule
Method & Condition for Growth Factor Magnitude, $|g(\theta,k)|^2<1$ \\
\midrule \\[-3.0ex]
\parbox{2.5cm}{Forward-Time Central-Space} & $\displaystyle k<\min_{\text{AP cycle}}\bigg(\frac{2}{K+2L}\bigg)=7\mu$s.
\\[2.0ex] \hline \\[-2.0ex]
\parbox{2.5cm}{Exponential \\ Euler} & Unconditionally Stable.
\\[2.0ex] \hline \\[-2.0ex]
\parbox{2.5cm}{Backward-Time Central-Space} & Unconditionally Stable.
\\ [2.0ex] \hline \\[-2.0ex]
\parbox{2.5cm}{Hines-Crank-Nicolson} & Unconditionally Stable, but $\displaystyle k >\min_{\text{AP cycle}}\bigg(\frac{2}{K+2L}\bigg)$  \\ \\[-2.0ex]
 & $\Rightarrow-1<g(k,\theta)<0\Rightarrow$ sol'n oscillates with decreasing amplitude. \\ [1.0ex] \hline \\[-2.0ex]
\parbox{2.5cm}{RK21} & $\displaystyle k<\min_{\text{AP cycle}}\bigg(\frac{2}{B}\bigg)=14\mu$s.$~~$ \\ [2.0ex] \hline \\[-2.0ex]
\parbox{2.5cm}{RK41} & $\displaystyle k<\min_{\text{AP cycle}}\bigg(\frac{\sim 2.7853}{B}\bigg)=20\mu$s. \\ [2.0ex] \hline \\[-2.0ex]
Definitions & $\displaystyle K=\Big(\alpha+\sum_i\beta_i\Big),~L=\bigg(\frac{1}{R_aC_m}+\frac{1}{R'_aC_m}\bigg)\sin^2{\frac{\theta}{2}},
~B$: see Table \ref{tab:schemeTruncError} \\
\bottomrule
\end{tabular}
\label{tab:schemeStability}
\end{table}
\section{Analysis Methods}
\label{sec:AnalysisMethods}
While our model neuron is composed of $74$ compartments, we assess only somatic compartment data as in \cite{T2005}.
\subsection{Waveform Definitions}
\label{sec:waveformDefs}
We categorize AP spiking waveforms according to two distinct features. {\it Spikes} refer to pulse peaks that rise monotonically from
less than $-0.04$ volts to over $-0.01$ volts.  {\it ADP} refers to an after-depolarization-pulse, a modest local maximum rarely rising above
$-0.055$ volts following the last {\it Spike}.  AP waveforms are classified, as in Figures \ref{fig:p23rsAPclassFE_2OT_RK21RK41} and
\ref{fig:p23rsAPclassBE_HCN}, using $Spikes$ and $ADP$ numbers as follows.
\begin{equation}
\label{eqn:classificationFormat}
\text{(number of $Spikes$ - number of $ADP$)}
\end{equation}
\subsection{Accuracy}
\label{sec:AccMeasure}
The correct solution for the active Rallpacks $3$ model could not be found analytically. Instead, Rallpacks provided simulator code ``used
to generate the results in NEURON \cite{CarnevaleHines} and GENESIS \cite{GEN}. These models were'' integrated with the HCN method
using a time-step of $1\mu$s for which correspondence between the two was better than $1\%$ \cite{BBB_Rall}. To assess this correspondence
``... spike peaks [were] aligned for the voltage calculations, and interval differences [were] separately added to the root mean squared total
\cite{BBB_Rall}.'' \\ \\
Likewise, our model's exact waveform solution cannot be determined analytically. Furthermore, Rallpacks' choice of an AP cycle waveform
integrated by HCN using a $1\mu$s time-step as the standard of accuracy works well only if different step size waveforms compared
to it are also integrated by HCN. This because each integration method generates different AP spiking waveform phases as shown in Figure
\ref{fig:SpikePhaseComp}, even for $1\mu$s step sizes. For this reason we have chosen the standard of accuracy for each method as the
AP cycle waveform integrated by the same method when using $1\mu$s step sizes. \\ \\
Note that because AP cycle absolute minimums and periods do not stabilize until the twentieth cycle, as shown in Figures
\ref{fig:minsVapidx}-\ref{fig:periodsVapidx}, model statistics in Figure \ref{fig:somaStatsWithStd} were only based on cycles following the nineteenth AP.
Likewise, accuracy of simulation waveforms, shown in Figure \ref{fig:methodErrVsts}, was determined by comparing the synchronized twentieth
AP cycle from each step size's simulation to the reference waveform, that being the twentieth AP cycle generated by the same integration
method using a step size of $1\mu$s.
\subsection{Spectral Analysis}
\label{subsection:SpecAnalysis}
Power spectral density (PSD) graphs in Figures \ref{fig:HCNandBEPSD}-\ref{fig:HCNandExpoEPSD} are periodograms. Integration step sizes,
$1$ to $100\mu$sec, correspond to, respectively, sampling frequencies of $1$ MHz and $10$ KHz. But the latter is higher than necessary since
neuroscience power spectrum interests coincide with e.e.g. bands, all less than $64$ Hz.  Simulation voltages were therefore downsampled to
achieve a sampling frequency of $250$ Hz allowing us to see the distribution of signal power as high as $125$ Hz. Power spectral densities were
then computed with the Welch periodogram in MATLAB; sampling frequency was $250$ Hz, Hamming window and DFT widths were $250$
samples with $150$ sample overlaps. \\ \\
We define the spectral centroid of an FFT computed from compartmentally sampled membrane voltages, used to compute step size upper limits
shown in Figures \ref{fig:fEstepSizeLimit}-\ref{fig:RKstepsizeLimits}, by writing the magnitude of a plane wave's spatial frequency spectrum as
$|H(\omega)|$, a function of frequency in radians where
\begin{equation}
\label{eqn:omegaSpan}
\omega \equiv \omega_i=\frac{i\pi}{16}\quad\text{for } i = 0, 1, 2, ... , 16.
\end{equation}
The spectral centroid is then
\begin{equation}
\label{eqn:spectralCentroid}
\omega_c=\frac{\sum_i \omega_i|H(\omega_i)|}{\sum_i |H(\omega_i)|}.
\end{equation}
\subsection{Oscillation Magnitude and Duration}
\label{sec:OscDet}
Our oscillation search method looks for concavities with alternating polarities. Two consecutive nonzero concavities with alternate polarities
are however common in AP cycle waveforms near membrane voltage spiking swings, especially as integration step size grows larger than
$50\mu$s. So, we require greater than two consecutive nonzero concavities with alternate polarities for a membrane voltage waveform
to be considered an oscillation. \\ \\
Next we consider how to best approximate concavity. Divided differences, the discrete approximation of derivatives, represent the estimate of
an instantaneous rate of change. In contrast, undivided differences are an approximation of a differential, the estimate of an amount, not a rate.
Since we are most interested in the magnitude an oscillating membrane voltage swings to reach the opposite polar extreme in one integration
step, regardless of step size, we will estimate concavity with the second undivided difference. \\ \\
The shortest oscillation is a sequence of membrane voltages whose second undivided differences change sign three or more consecutive
integration steps. Figure \ref{fig:oscMagMeasure} illustrates a sequence of four concavities with alternating polarity to demonstrate that amplitude
magnitude is one half the peak-to-peak voltage, or identically one quarter the second undivided difference.
\begin{figure}[h!]
\centering
\includegraphics[height=2.0in,width=4.0in]{oscMagMeasure.pdf}
\caption{Oscillation amplitude is one quarter the second undivided difference.}
\label{fig:oscMagMeasure}
\end{figure}
Figure \ref{fig:exOsc} shows an example of membrane voltage oscillation induced by FTCS using a step size of $7\mu$s. The magnitude of
second undivided differences, in red, is the basis for computing the RMS of oscillations shown in Figures
\ref{fig:fE2OToscVsTimestep}-\ref{fig:hCNoscVsTimestep}.
\begin{figure}[h!]
\centering
\includegraphics[height=4.5in,width=4.0in]{exampleOscillations.pdf}
\caption{The measure of oscillation amplitude is plotted in red. The lower pane is a closeup of the first AP spike in the upper frame.
Note that, contrary to observations described in \cite{Borgers}, instability can present on the descending slopes of an AP spike.}
\label{fig:exOsc}
\end{figure}
\section{Results}
\label{sec:Results}
Simulation software was developed by the author in MATLAB. Table \ref{tab:SomaWaveSummary} lists graphs of results and their
corresponding figure numbers.
\begin{table}[h!]\footnotesize
\caption{P23RS Somatic Waveform Performance Figures}
\ \\
\centering
\renewcommand{\arraystretch}{0.5
\begin{tabular}{l|c}
\toprule
Figure Summary & Figure \\
\midrule
\parbox{10.5cm}{Stable step size intervals versus method} &  \ref{fig:successfulStepSizes} \\ \\
\parbox{10.5cm}{Ideal AP cycle, integrated by HCN, step size $1\mu$s} & \ref{fig:idealAPcyclesHCNsts1} \\ \\
\parbox{12.5cm}{Absolute minimums between AP cycles reach steady state by $20$ AP cycles} & \ref{fig:minsVapidx} \\ \\
\parbox{12.5cm}{AP periods reach steady state by $20$ AP cycles}  & \ref{fig:periodsVapidx} \\ \\
\parbox{10.5cm}{Integration method accuracy comparison} & \ref{fig:methodErrVsts} \\ \\
\parbox{10.5cm}{Integration method statistics with standard deviations} &  \ref{fig:somaStatsWithStd} \\ \\
\parbox{10.5cm}{Spiking classifications for FTCS, RK21 and RK41 methods} & \ref{fig:p23rsAPclassFE_2OT_RK21RK41} \\ \\
\parbox{10.5cm}{Spiking classifications for BTCS and HCN versus step size} & \ref{fig:p23rsAPclassBE_HCN} \\ \\
\parbox{10.5cm}{Spiking classifications for exponential Euler versus step size} & \ref{fig:p23rsAPclassExpoE} \\ \\
\parbox{10.5cm}{Examples of exponential Euler AP spiking distortions}  &  \ref{fig:corruptExpoEulerAPs} \\ \\
\parbox{10.5cm}{Integration method AP spike phases at step sizes $1,9,50,80$ and $99\mu$s} & \ref{fig:SpikePhaseComp} \\ \\
\parbox{12.5cm}{Comparison of power spectral density (PSD) for HCN and BTCS methods} & \ref{fig:HCNandBEPSD} \\ \\
\parbox{12.5cm}{Comparison of PSD for HCN and exponential Euler methods} & \ref{fig:HCNandExpoEPSD} \\ \\
\parbox{12.5cm}{Maximum step size for FTCS versus AP cycle time} &  \ref{fig:fEstepSizeLimit} \\ \\
\parbox{12.5cm}{RK step size limits with/without consideration of plane wave phase angle, $\theta$} &  \ref{fig:RKstepsizeLimits} \\ \\
\parbox{10.5cm}{RMS of FTCS, RK21 and RK41 oscillation magnitudes} & \ref{fig:fE2OToscVsTimestep} \\ \\
\parbox{10.5cm}{RMS of exponential Euler and BTCS oscillation magnitudes vs step size} & \ref{fig:eE_BTCSoscVsTimestep} \\ \\
\parbox{10.5cm}{RMS of HCN oscillation versus step size} & \ref{fig:hCNoscVsTimestep} \\ \\
\parbox{10.5cm}{HCN Growth factor span versus step size} & \ref{fig:hCNGrwthFctrVsTimestep} \\ \\
\parbox{12.5cm}{HCN oscillations during Spiking phase} & \ref{fig:Spiking} \\ \\
\parbox{10.5cm}{HCN oscillations during ADP phase} & \ref{fig:ADP} \\ \\
\parbox{10.5cm}{HCN oscillations during Max Polarization phase} & \ref{fig:maxPolarization} \\ \\
\parbox{10.5cm}{HCN oscillation magnitude increases with step size and decays before the next AP cycle begins} & \ref{fig:hCNOscDecay} \\
\bottomrule
\end{tabular}
\label{tab:SomaWaveSummary}
\end{table}
\subsection{Accuracy}
\label{sec:Accuracy}
Figure \ref{fig:idealAPcyclesHCNsts1} shows the ideal AP cycle and a closer view of its three spikes as generated by the HCN method with
a step size of $1\mu$s. AP cycle waveform minimum's and periods for all methods required almost twenty AP cycles to reach macro view
equilibriums, regardless of step size, exemplified by three methods shown in Figures \ref{fig:minsVapidx}-\ref{fig:periodsVapidx}. Membrane ion
gate phase portraits included in the supplemental resources also support this observation.  AP waveform statistics shown in \mbox{Figure
\ref{fig:somaStatsWithStd}} were therefore based on APs following the nineteenth cycle. \\ \\
Accuracy comparisons, as described in Section \ref{sec:AccMeasure}, appear in Figure \ref{fig:methodErrVsts}. For every integration method,
spiking variations increased as step size increased.  For example, the center pane in Figure \ref{fig:somaStatsWithStd} shows HCN's maximum
spike height stable until step sizes reached $53\mu$s, after which both mean and standard variation increased. In contrast, maximum spike
heights for every other integration method decreased steadily as step size grew larger than $1\mu$s. Increasing step sizes led to a varying
number of spikes in each AP cycle as shown in Figures \ref{fig:p23rsAPclassFE_2OT_RK21RK41}--\ref{fig:p23rsAPclassExpoE}. Prominent among
them, waveforms generated by the exponential Euler method, as shown in Figure \ref{fig:corruptExpoEulerAPs}, were the most variable and
inconsistent. Artifacts of accuracy loss also include changing AP spike phase presented by each method, except HCN, as shown in Figure
\ref{fig:SpikePhaseComp}. PSDs of the exponential Euler and BTCS methods were each compared to the PSD of HCN in Figures
\ref{fig:HCNandBEPSD}-\ref{fig:HCNandExpoEPSD}.
\subsection{Stability}
\label{sec:Stability}
Figure \ref{fig:successfulStepSizes} illustrates stable integration step size intervals observed for each method. While the exponential Euler
method was always stable, its waveforms were sufficiently inconsistent to prevent comparison with waveforms presented by other methods.
So this method was only exercised for step sizes from $1-50\mu$s. \\ \\
Figure \ref{fig:fEstepSizeLimit} shows the correspondence between membrane voltage, the model's plane wave's phase angle and the Von Neumann
step size limit for FTCS. Spectral centroid of the phase angle was computed, as described in Section \ref{subsection:SpecAnalysis}, from the FFT of
membrane voltages of compartments from the tip of any distal dendrite to the tip of the axon. These thirteen values were then padded with nineteen
zeros to increase low frequency resolution. \\ \\
Figure \ref{fig:RKstepsizeLimits} shows integration step size limits for RK21 and RK41 predicted two ways. The first using RK stability analysis from
\cite{Butcher} for a true ODE, not a spatially discretized PDE, does not consider the plane wave's phase angle. The second, Von Neumann
stability analysis applied to RK methods cast as a quasi-FD methods, is in terms of the phase angle. Figure \ref{fig:successfulStepSizes} and Figure
\ref{fig:RKstepsizeLimits} show that step size limits from Von Neumann analysis match those we observed for RK21 and RK41, $14\mu$sec
and $20\mu$sec respectively. \\ \\
Magnitude and location of oscillations presented by each integration method, as defined in Section \ref{sec:OscDet}, are shown in Figures
\ref{fig:fE2OToscVsTimestep}--\ref{fig:hCNoscVsTimestep}. Note that FTCS method oscillation magnitudes grow significantly one or two
microseconds in step size before stability is lost. Exponential Euler's oscillations approach $10^{-2}m$volts shortly after integration step size
increases beyond $30\mu$s. \\ \\
While HCN was stable for all step sizes from $1$ to $99\mu$s, small amplitude decaying oscillations grew in duration and magnitude as step
sizes grew larger than $50\mu$s as illustrated in Figure \ref{fig:hCNoscVsTimestep}. HCN's Von Neumann growth factor, spanning one AP
cycle, is plotted versus integration step size in Figure \ref{fig:hCNGrwthFctrVsTimestep}. Key oscillation locations are illustrated in Figures
\ref{fig:ADP}--\ref{fig:maxPolarization}. Figure \ref{fig:hCNOscDecay} illustrates that while oscillations are decaying, their magnitude increases
with step size. Videos in the supplemental resources show HCN's oscillations evolving for step sizes $1-99\mu$s on three intervals, AP spikes,
ADP pulse and maximum membrane polarization separating AP cycles.
\begin{figure}[h!]
\centering
\begin{tabular}{l}
\includegraphics[height=1.75in,width=4.0in]{successfulStepSizes.pdf}
\end{tabular}
\caption{Stable step size intervals integrated by each numerical method.}
\label{fig:successfulStepSizes}
\end{figure}
\begin{figure}[h!]
\centering
\begin{tabular}{l}
\includegraphics[height=3.5in,width=3.5in]{idealAPcyclesHCNsts1.pdf}
\end{tabular}
\caption{The ideal waveform, a triplette, whose spike heights descend in order, and one modest after-depolarization-pulse (ADP).}
\label{fig:idealAPcyclesHCNsts1}
\end{figure}

\begin{figure}[h!]
\centering
\setlength{\tabcolsep}{0.7mm}
\begin{tabular}{l}
\includegraphics[height=1.8in,width=4.5in]{fEminsVapidx.pdf} \\
\includegraphics[height=1.8in,width=4.5in]{bEminsVapidx.pdf} \\
\includegraphics[height=1.8in,width=4.5in]{hCNminsVapidx.pdf} \\
\end{tabular}
\caption{Relative minimums between AP cycles stabilize by the twentieth AP cycle.}
\label{fig:minsVapidx}
\end{figure}

\begin{figure}[h!]
\centering
\setlength{\tabcolsep}{0.7mm}
\begin{tabular}{l}
\includegraphics[height=1.8in,width=4.5in]{fEperiodVapidx.pdf} \\
\includegraphics[height=1.8in,width=4.5in]{bEperiodVapidx.pdf} \\
\includegraphics[height=1.8in,width=4.5in]{hCNperiodVapidx.pdf} \\
\end{tabular}
\caption{Action potential (AP) periods stabilize by the twentieth AP cycle.}
\label{fig:periodsVapidx}
\end{figure}

\begin{figure}[h!]
\centering
\setlength{\tabcolsep}{0.7mm}
\begin{tabular}{l}
\includegraphics[height=6.0in,width=4.5in]{methodErrVsts.pdf} \\
\end{tabular}
\caption{Accuracy of membrane voltage integration found by comparing the twentieth AP cycle at each step size to the same AP cycle,
integrated by the same method with a $1\mu$s step size. FTCS data presents typical first-order, straight line, growth of error, and
is hidden behind BTCS data. BTCS data presents square-root-like, $\frac{1}{2}$-order, error growth. HCN presents typical second-order,
concave-up, growth of error. As predicted, both RK methods present straight line, first-order, error growth. Of particular interest are shocks
in BTCS, RK21 and RK41 data that indicate step sizes beyond which loss of accuracy for each first order method leads to spiking morphology
changes. The expanded view on the bottom supports approximately $\frac{1}{2}$-Order growth of error for BTCS as well as RK41.}
\label{fig:methodErrVsts}
\end{figure}

\begin{figure}[h!]
\centering
\setlength{\tabcolsep}{0.7mm}
\begin{tabular}{l}
\includegraphics[height=2.25in,width=4.4in]{p23rsSomaMinWithStd.pdf} \\
\includegraphics[height=2.25in,width=4.4in]{p23rsSomaMaxWithStd.pdf} \\
\includegraphics[height=2.25in,width=4.4in]{p23rsSomaPeriodWithStd.pdf}
\end{tabular}
\caption{Comparison of somatic compartment statistical means and standard deviations based on \mbox{AP cycles $20-$end}.}
\label{fig:somaStatsWithStd}
\end{figure}

\clearpage
\begin{figure}[h!]
\centering
\begin{tabular}{l}
\includegraphics[height=2.7in,width=3.5in]{p23rsAPperSimFE-RK21-RK41.pdf}
\end{tabular}
\caption{FTCS spiking morphology deteriorates just $1\mu$s prior to the onset of instability. RK21 and RK41, both rendered first-order
methods, present deteriorating spiking morphologies for step sizes larger than $8\mu$s and $9-10\mu$s, respectively.}
\label{fig:p23rsAPclassFE_2OT_RK21RK41}
\end{figure}
\begin{figure}[h!]
\centering
\begin{tabular}{l}
\includegraphics[height=2.8in,width=2.4in]{p23rsAPperSimBackE.pdf}
\includegraphics[height=2.8in,width=2.4in]{p23rsAPperSimHCN.pdf}
\end{tabular}
\caption{BTCS presents deteriorating spiking morphologies for step sizes $>78\mu$s. HCN's class $3$-$0$ waveforms, i.e
apparent loss of ADPs, are an inaccurate artifact. They are in fact class $3$-$1$ whose oscillations prevented software from detecting
the ADP local maximum. This was intentionally left to visually corroborate oscillations in \mbox{Figure \ref{fig:hCNoscVsTimestep}.}}
\label{fig:p23rsAPclassBE_HCN}
\end{figure}

\begin{figure}[h!]
\centering
\begin{tabular}{l}
\includegraphics[height=3.5in,width=3.8in]{p23rsAPperSimExpoE.pdf}
\end{tabular}
\caption{AP spiking classification map for exponential Euler integration method .}
\label{fig:p23rsAPclassExpoE}
\end{figure}
\begin{figure}[h!]
\centering
\begin{tabular}{l}
\includegraphics[height=3.0in,width=4.5in]{p23rsSomaSampleExpoEulerSims.pdf}
\end{tabular}
\caption{Corrupted AP cycle waveforms integrated by the exponential Euler method.}
\label{fig:corruptExpoEulerAPs}
\end{figure}

\begin{figure}[h!]
\centering
\setlength{\tabcolsep}{0.7mm}
\begin{tabular}{l}
\includegraphics[height=7.0in,width=5.0in]{SpikePhaseComp.pdf}
\end{tabular}
\caption{AP cycle 22 spike phase comparisons at step sizes of $1,9,50,80$ and $99\mu$s.}
\label{fig:SpikePhaseComp}
\end{figure}

\clearpage

\begin{figure}[h!]
\centering
\setlength{\tabcolsep}{0.7mm}
\begin{tabular}{l}
\includegraphics[height=5.9in,width=4.4in]{p23rsSomaPSDHCNandBE.pdf}
\end{tabular}
\caption{Spectral density comparison showing the power distribution for HCN remaining largely fixed regardless of integration step size.
BTCS is just as reliable up until $80\mu$sec step sizes where a likely bifurcation incursion changes spiking multiplicities for all AP cycles as
shown in Figure \ref{fig:p23rsAPclassBE_HCN}. For both methods, the lowest frequency spectral peaks, those near $15$ Hz correspond
to each numerical method's respective fundamental AP frequency. All other peaks are mere harmonics. }
\label{fig:HCNandBEPSD}
\end{figure}

\begin{figure}[h!]
\centering
\setlength{\tabcolsep}{0.7mm}
\begin{tabular}{l}
\includegraphics[height=5.9in,width=4.4in]{p23rsSomaPSDHCNandExpoE.pdf}
\end{tabular}
\caption{Spectral density comparison showing the frequency of exponential Euler's power peaks decrease proportionally as integration
step size increases.  This frequency shift is also corroborated by the decreasing AP cycle frequency, or increasing AP cycle period shown
in Figures \ref{fig:somaStatsWithStd} and \ref{fig:p23rsAPclassExpoE}.}
\label{fig:HCNandExpoEPSD}
\end{figure}

\begin{figure}[h!]
\centering
\setlength{\tabcolsep}{0.7mm}
\begin{tabular}{l}
\includegraphics[height=4.5in,width=3.9in]{fEstepSizeLimit.pdf}
\end{tabular}
\caption{Graphs above illustrate correspondence between membrane voltage, phase angle of the model's plane wave from tip of
distal dendrites to axon tip, and the integration step size limit for FTCS. The step size limit shown here for FTCS of $6\mu$sec matches
simulation data shown in Figure \ref{fig:successfulStepSizes}.}
\label{fig:fEstepSizeLimit}
\end{figure}

\begin{figure}[h!]
\centering
\setlength{\tabcolsep}{0.7mm}
\begin{tabular}{l}
\includegraphics[height=1.6in,width=3.7in]{RK21stsLimWithoutTheta.pdf} \\
\includegraphics[height=1.6in,width=3.7in]{RK21stepsizeLimit.pdf} \\
\includegraphics[height=1.6in,width=3.7in]{RK41stsLimWithoutTheta.pdf} \\
\includegraphics[height=1.6in,width=3.7in]{RK41stepsizeLimit.pdf}
\end{tabular}
\caption{Integration step size limits shown above for RK21 and RK41 predicted two ways. The first using RK stability analysis from \cite{Butcher}
for a true ODE, not a spatially discretized PDE, does not consider the plane wave's phase angle. The second, Von Neumann stability analysis
applied to RK methods cast as a quasi-FD methods, is in terms of the phase angle. As shown above, and in Figure \ref{fig:successfulStepSizes},
step size limits from Von Neumann analysis match those observed for RK21 and RK41, $14\mu$sec and
$20\mu$sec respectively.}
\label{fig:RKstepsizeLimits}
\end{figure}

\begin{figure}[h!]
\centering
\setlength{\tabcolsep}{0.7mm}
\begin{tabular}{l}
\includegraphics[height=3.1in,width=4.0in]{fE-RK21-RK41oscVsTimestep.pdf}
\end{tabular}
\caption{Oscillation magnitudes versus step size for FTCS, RK21 and RK41. Each data point represents one of, as many as, $42$ AP cycles.}
\label{fig:fE2OToscVsTimestep}
\end{figure}

\begin{figure}[h!]
\centering
\setlength{\tabcolsep}{0.7mm}
\begin{tabular}{l}
\includegraphics[height=3.2in,width=2.5in]{eEoscVsTimestep.pdf}
\includegraphics[height=3.2in,width=2.5in]{bEoscVsTimestep.pdf}
\end{tabular}
\caption{Oscillation magnitudes versus step size for exponential Euler and BTCS. Each data point represents one of, as many as, $42$ AP cycles.}
\label{fig:eE_BTCSoscVsTimestep}
\end{figure}

\begin{figure}[h!]
\centering
\setlength{\tabcolsep}{0.7mm}
\begin{tabular}{l}
\includegraphics[height=3.2in,width=3.3in]{hCNoscVsTimestep.pdf}
\end{tabular}
\caption{Oscillation magnitudes versus step size for HCN. Each data point represents one of $42$ AP cycles.}
\label{fig:hCNoscVsTimestep}
\end{figure}

\begin{figure}[h!]
\centering
\setlength{\tabcolsep}{0.7mm}
\begin{tabular}{l}
\includegraphics[height=3.2in,width=3.3in]{hCNGrwthFctrVsTimestep.pdf} \\
\end{tabular}
\caption{Span of Von Neumann growth factor vs step size for HCN.}
\label{fig:hCNGrwthFctrVsTimestep}
\end{figure}

\clearpage

\begin{figure}[h!]
\centering
\setlength{\tabcolsep}{0.7mm}
\begin{tabular}{l}
\includegraphics[height=6.0in,width=4.5in]{hCNAP26Spkg.pdf}
\end{tabular}
\caption{Comparison of spiking phase membrane voltage, $V$, the first and second undivided differences of membrane voltage, and
the Von Neumann growth factor for integration step sizes $1$ and $51\mu$sec. The onset of oscillation by the time step size has increased
to $51\mu$sec is also demonstrated by HCN's AP maximums shown in Figure \ref{fig:somaStatsWithStd}, by RMS oscillation magnitudes
in Figure \ref{fig:hCNoscVsTimestep} and justified by the trend of growth factor illustrated in Figure \ref{fig:hCNGrwthFctrVsTimestep}.}
\label{fig:Spiking}
\end{figure}

\begin{figure}[h!]
\centering
\setlength{\tabcolsep}{0.7mm}
\begin{tabular}{l}
\includegraphics[height=6.0in,width=4.5in]{hCNAP26ADP.pdf}
\end{tabular}
\caption{Comparison of the ADP phase membrane voltage, $V$, the first and second undivided differences of membrane voltage, and
the Von Neumann growth factor for integration step sizes $1$ and $64\mu$s. The onset of oscillation by the time step size has
increased to $64\mu$s is also demonstrated by evidence of ADP oscillation in the lower right graph of Figure \ref{fig:p23rsAPclassBE_HCN},
by RMS oscillation magnitudes in Figure \ref{fig:hCNoscVsTimestep} and corroborated by the trend of growth factor illustrated in Figure
\ref{fig:hCNGrwthFctrVsTimestep}.}
\label{fig:ADP}
\end{figure}

\begin{figure}[h!]
\centering
\setlength{\tabcolsep}{0.7mm}
\begin{tabular}{l}
\includegraphics[height=6.0in,width=4.5in]{hCNAP26MxPol.pdf}
\end{tabular}
\caption{Comparison at maximum polarization of membrane voltage, $V$, the first and second undivided differences of membrane voltage,
and the Von Neumann growth factor for integration step sizes $1$ and $64\mu$s. The onset of oscillation by the time step size has increased
to $64\mu$s is demonstrated by RMS oscillation magnitudes in Figure \ref{fig:hCNoscVsTimestep} and corroborated by the trend of growth factor
illustrated in Figure \ref{fig:hCNGrwthFctrVsTimestep}.}
\label{fig:maxPolarization}
\end{figure}

\begin{figure}[h!]
\centering
\setlength{\tabcolsep}{0.7mm}
\begin{tabular}{l}
\includegraphics[height=6.0in,width=4.5in]{hCNOscDecay.pdf}
\end{tabular}
\caption{HCN oscillation magnitude increases with step size, as predicted in \cite[ch.4]{CarnevaleHines}, and decays before the next AP cycle begins.}
\label{fig:hCNOscDecay}
\end{figure}

\clearpage
\section{Summary and Discussion}
\label{sec:SumAndDiscuss}
Our model's computational sequence is composed of two sub-steps, the first to integrate each compartment's eleven ion channel state ODEs,
represented in equation (\ref{eqn:gateStateODE}), and second to integrate each compartment's membrane voltage PDE, in equation
(\ref{CableEquationRoot}), as a function of the updated channel states. The solution to our model's dynamical system is periodic, a sequence of AP
cycles, each with its own sub-sequence of three states, Spiking, Re-polarization and a long slow Depolarization, as shown in Figure
\ref{fig:idealAPcyclesHCNsts1}. We expect numerical methods integrating our model's dynamical system to maintain periodicity and synchronization
of all channel state cycles in order to ensure consistent AP cycle waveforms and statistics such as relative minimums and maximums, period lengths,
spike phases and multiplicities. \\ \\
One of our first observations is that roughly twenty AP cycles were required before state statistics reached a mature equilibrium, regardless of
integration method, as shown in Figures \ref{fig:minsVapidx}-\ref{fig:periodsVapidx}. Membrane ion gate phase portraits included in the supplemental
resources also support this observation. These are examples of common characteristics that combine to delay the onset of a mature dynamical system
state. Such 'warm-up' time could be reduced by saving a model's state variables at the end of a deterministic run-up to a mature state for use as initial
values prior to varying conditions and introducing stochastic regimes. For this reason research articles would be more helpful if they indicate how far
into a simulation their data was collected \cite{Ascoli10,Sousa15,Kuznet10}. \\ \\
The purpose of our study, a comparison of numerical integration methods, prompts the question, ``Why should we expect any method-dependent
integration differences at all?'' While an integration method's truncation error is not the same as the inaccuracy of the solution it generates, terms in
the former help show why the later is different for each integration method. The first and second terms of truncation errors, listed in Table
\ref{tab:schemeTruncError} for the membrane voltage integration, are second and third derivatives of the membrane voltage which change throughout
the AP cycle and ensure error differences between methods. The following two sections describe integration method-specific differences in accuracy
and stability.
\subsection{Accuracy}
\label{subsec:AccuracyDiscussion}
Perhaps the simplest generalization worth making is that every explicit ODE method integrating a multi-compartment neural model sacrifices accuracy
by failing to account for the changing spatial derivative which is no different than assuming it remains constant for the duration of each step. Any hope
of recovering this loss by replacing one explicit integrator with another of higher order, given our selection of integrators, is lost either because instability
takes over at such small step sizes, as shown in Figure \ref{fig:successfulStepSizes}, or the cable equation renders RK methods first-order, a matter we
discuss below. \\ \\
Our study assessed integration method accuracy in the following ways. First, we computed the RMS difference of AP waveforms relative to the
reference waveform integrated by the same method at $1\mu$s step size, as described in Section \ref{sec:AccMeasure}. These measurements
of difference, plotted versus step size for each method in Figure \ref{fig:methodErrVsts} corroborate most truncation error orders presented in Table \ref{tab:schemeTruncError}. The abrupt shocks in the same figure presented by BTCS, RK21 and RK41
are discussed below.  \\ \\
As summarized in Table \ref{tab:schemeTruncError} and confirmed in Figure \ref{fig:methodErrVsts}, RK21 and RK41 both present, at first glance,
linearly increasing first-order error. Of course, RK21 and RK41 are indeed second- and fourth-order methods, respectively, when integrating each
compartment's ion channel state ODEs. It is when integrating the discretized cable equation, as shown in Section \ref{subsubsec:RK21truncErr},
that truncation errors become first-order. In contrast, HCN presents second-order error growth, concave up as step size increases. \\ \\
There remains one unusual nonlinear anomaly in Figure \ref{fig:methodErrVsts} affecting BTCS and RK41. If FTCS and RK21 had remained stable
long enough their data may also have shown a similar trend. The anomaly is that their error plots present an error order less than unity evidenced
by the error curve's negative concavity. Perhaps BTCS was most affected as it was the only first-order method to remain stable. The second most
significantly affected was RK41, also a, albeit situationally rendered, first-order method. We suggest the most likely reason for this anomaly is the
same reason HCN does {\it{not}} present it. That is, to minimize error accrued by integrating highly nonlinear HH type cable equations, HCN has
always prescribed staggering gate state integrations by one half-step relative to the membrane voltage integration. By not staggering channel state
updates between membrane voltage updates, all other methods were more likely to accrue more integration error than did HCN. \\ \\
The second way we assessed accuracy of numerical integration methods was with AP waveform statistics. These statistics are a measure of an
integration method's ability to consistently control ion channel constituent gate states periodicity over broad step size intervals. Without such
consistency the reliability of an adaptive step method would be uncertain. Figure \ref{fig:somaStatsWithStd} illustrates two method-specific
inconsistencies which present alone and together. We first thought that the meandering mean and discontinuous variation of AP maximums
presented by HCN after $50\mu$s step sizes was one of these inconsistencies. In fact, the source of this artifact is a decaying oscillation and
is discussed below. \\ \\
The first method-specific inconsistency seen in Figure \ref{fig:somaStatsWithStd} is a smooth nonzero rate-of-change over increasing integration
step size, error characteristic of first order methods. With the exception of erratic AP maximums after $50\mu$s step sizes, HCN presents perfect
consistency without any change to AP minimums, maximums and periods. While not as perfectly controlled as HCN, BTCS statistics present a very
small rate-of-change over the same interval and also present control of AP maximums after $50\mu$s step sizes superior to HCN. Remaining
integrators, all explicit methods, present significant and sometimes discontinuous change to AP minimums, maximums and periods as a function
of integration step size. \\ \\
A second type of inconsistency seen in the top and bottom panes of Figure \ref{fig:somaStatsWithStd} presents as significant jumps in the data
and affects all methods but HCN. RK21 and RK41 both present these data shocks from step sizes $9-11\mu$s, and $12-15\mu$s respectively.
While BTCS data shocks appear limited to $81-90\mu$s step sizes, the exponential Euler method is affected throughout much of its step size
testing interval, $1-50\mu$s. Notable for exponential Euler are perhaps the step size intervals where its waveform inconsistencies are not affected,
$7-8\mu$s and $13-19\mu$s.  These same AP spiking multiplicity variations are illustrated from different vantage points in Figure \ref{fig:methodErrVsts},
Figures \ref{fig:p23rsAPclassFE_2OT_RK21RK41}--\ref{fig:p23rsAPclassExpoE} and exponential Euler's waveform plots in Figure
\ref{fig:corruptExpoEulerAPs}. Simulation videos showing the full simulation train of spiking waveforms for each method over all step sizes are
included with the supplemental materials. These videos provide a more global perspective of sudden significant changes to AP waveforms as
step sizes increase. \\ \\
The cause for these waveform anomalies becomes apparent by watching the evolution of membrane voltage, axial and membrane ion channel
currents as step size increases with $1\mu$s resolution. As step size increases, integration of first order linear membrane channel ODEs with a
first order numerical method leads to decreasing rise times and increasing widths of pulses with diminished magnitudes. The result in our case
is for compartment membrane voltage to be pushed just above an unstable equilibrium of the dynamical system at a time that delayed membrane
ion channels have not yet returned to resting states and results in another AP spike. While we are not hypothesizing the presence of a bifurcation,
otherwise similar mechanics are described in \cite[ch.3]{Izh}. A video showing the onset of spiking multiplicity variation for one AP cycle integrated
by RK21 is included in the supplemental materials. \\ \\
The third way we assess the accuracy of a numerical integration method is by observing AP spike phase changes as step size increases. Note that
each method's AP spike phases are different from each other even when the step size is $1\mu$s.  We also see that every method except HCN
allows its spike phases to change with increasing step size as shown in Figure \ref{fig:SpikePhaseComp}. \\ \\
Performance of a network model is often evaluated by its power spectrum. So, the fourth way we assessed the accuracy of a numerical integration
method was by observing three method's power spectrums. Figures \ref{fig:HCNandBEPSD}-\ref{fig:HCNandExpoEPSD} show HCN's power spectral
densities (PSD) consistent for step sizes $1-99\mu$s. BTCS PSDs were consistent for step sizes up to $80\mu$s when spiking waveform anomalies
developed. HCN and BTCS PSDs were also consistent with each other. Exponential Euler's power spectral peaks varied between $5$ and $10$ Hz
as step size grew from $1\mu$s to $50\mu$s.
\subsection{Stability}
\label{subsec:Stability}
Figure \ref{fig:successfulStepSizes} illustrates step size intervals successfully integrated without oscillation by each numerical method. As
predicted in Table \ref{tab:schemeStability}, the implicit first-order method BTCS was stable over all step sizes. Exponential Euler was
also predicted, and found, to be stable for all step sizes, but was only exercised up to a step size of $50\mu$s
due to inconsistent waveform morphology. \\ \\
Von Neumann stability analysis \cite[pg.126]{Str} of first-order FTCS provided the step size upper limit shown in Table \ref{tab:schemeStability}
and confirmed in Figure \ref{fig:successfulStepSizes}. When FTCS step size increased to $7\mu$s, oscillations appeared on the trailing edges
of AP spikes as shown in the second pane of Figure \ref{fig:exOsc}, an outcome different than predicted in \cite{Borgers}. \\ \\
RK methods are intended to integrate native ODEs and not spatially discretized PDEs. RK method predictions of stable step size upper limits,
described by \cite[pg.100]{Butcher}, shown in Figure \ref{fig:RKstepsizeLimits}, do not match the accurate predictions made by Von Neumann
stability analysis of the quasi-FD RK methods, derived in \ref{subsubsec:RK21stability} and \ref{subsubsec:RK41stability}, listed in Table
\ref{tab:schemeStability} and also shown in Figure \ref{fig:RKstepsizeLimits}, because their upper step size limit expressions do not consider
the model's spatial dimension, in particular the model's plane wave's phase angle. \\ \\
Figures \ref{fig:fE2OToscVsTimestep}-\ref{fig:hCNoscVsTimestep} illustrate the RMS of oscillation magnitude, as defined in Section
\ref{sec:OscDet}, for all integration methods. Except for the last step size prior to expected instability, magnitudes are far below $10^{-6}$mv.
That each method would have any oscillation is not unexpected given the neural model's governing DE changes after every step. \\ \\
While HCN is formally unconditionally stable \cite[pg.62]{CarnevaleHines}, Figure \ref{fig:hCNGrwthFctrVsTimestep} shows its growth factor,
$g(k,\theta)\Rightarrow-1$ from above as step size increases. Associated oscillation amplitude increases shown in Figures
\ref{fig:hCNoscVsTimestep},  \ref{fig:ADP}--\ref{fig:hCNOscDecay} are the source of widely varying AP cycle maximums in Figure
\ref{fig:somaStatsWithStd} which indicate oscillation amplitudes can be as high as $10$ millivolts. Amplitude magnitudes outside the spiking
interval only rise to a few tenths of a millivolt as shown in Figure \ref{fig:hCNOscDecay}. But given such small oscillation magnitudes it remains
to be seen what, if any, untoward outcomes are likely. A clearer picture of relative oscillation magnitudes is presented by videos included in the
supplementary material that show the data presented in Figure \ref{fig:SpikePhaseComp} and \ref{fig:ADP}--\ref{fig:hCNOscDecay} as step size
changes from $1$ to $99\mu$s. \\ \\
Finally, while our model was deterministic, varying oscillation amplitudes of just one AP cycle for each step size shown in Figures
\ref{fig:fE2OToscVsTimestep}--\ref{fig:hCNoscVsTimestep} clearly indicate the model's state space is large enough that AP cycles are unlikely
to ever be identical.
\subsection{Conclusions}
\label{subsec:Conclusions}
The traditional advantage of explicit methods, ease of implementation, did not make up for their short span of stable step sizes
(Figure \ref{fig:successfulStepSizes}), low order error growth \mbox{(Figure \ref{fig:methodErrVsts})}, and variation in waveform statistics
\mbox{(Figure \ref{fig:somaStatsWithStd})}. \\ \\
We can only cite two potentially unwanted outcomes of HCN's decaying oscillations. Fluctuation of maximums would matter if, during a
network simulation, the magnitude of APs reaching a synapse through an axon are proportional to the source neuron's maximum spiking
magnitude and its attenuation over randomly determined distances. Furthermore, ADP oscillations would matter if their presence misleads
software intended to detect the ADP's peak just as we experienced (see lower right pane of Figure \ref{fig:p23rsAPclassBE_HCN}).
Otherwise, HCN's accuracy, stability and immunity to the data shocks experienced by the first order methods, are unmatched in our study. \\ \\
Our results also demonstrate that whenever neural model simulation studies name their simulation platform but not the numerical integration
method or merely describe their integration step size as ``variable'' without further details or not mention elapsed simulation time prior to
data collection, repeatability of a study's results is no longer a certainty \cite{MBL}.
\section{Future Work}
A similar examination of the implicit integrators' performance after hyper/depolarizing and synaptic currents have been added would be instructive.
HCN's method of staggering gate state updates should be applied to all other methods to see if their less than unity error order in Figure
\ref{fig:methodErrVsts} will vanish. The simulations run in this study should also be repeated after substituting symmetric axial resistances for the
asymmetrically modeled resistances used here.
\section{Acknowledgements}
This research did not receive any grants from funding agencies in the public, commercial, or not-for-profit sectors.