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\journal{Brain Research} % = Preprint submitted to Brain Research
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\long\def\comment#1{}
\def\citen#1{#1}
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\begin{document}
% \listoftables*
\renewcommand*\familydefault{\sfdefault}
\normalfont

\large
\title{On the precision of neural computation with interaural time
differences in the medial superior olive}
% {is in the order of tens of microseconds}

\author[Pathologic,Radio,Contact]{Petr Marsalek} %{\corref{Contact}}
\author[Pathologic,Contact]{(Marek Hajny)} %{\corref{Contact}}
\author[Robotics,Poly]{\citen{Zbynek} Bures}

\address[Pathologic]{Institute of Pathological Physiology, First Medical
 Faculty, Charles University in Prague, U~Nemocnice~5/478, 128~53, Praha~2,
 Czech Republic} %a

\address[Radio]{Department of Radioelectronics, Faculty of Electrical
 Engineering, Czech Technical University in Prague,
 Technic{k}\'a~2/1902, 16627, Praha~6, Czech Republic} %b

\address[Robotics]{Czech Institute of Informatics, Robotics and Cybernetics,
 Czech Technical University in Prague,\\
 \citen{Jugosl\'{a}v{s}k\'y{c}h partyz\'an\accent23u~4/1903}, 166~36, Praha~6,
 Czech Republic}
 \comment{Zikova~1903/~4, 166~36, Praha~6, Czech Republic} %c

\address[Poly]{College of Polytechnics,
 \citen{T{o}l{s}t\'e{h}o}~16/1556, 586~01, Jihlava, Czech Republic} %d

\address[Contact]{Corresponding authors: Petr.Marsalek@LF1.CUni.CZ,
 Marek.Hajny@LF1.CUni.CZ} %e

\begin{keyword}
binaural hearing, cochlear implant,
coincidence detection, ergodic hypothesis, ideal
observer, interaural time difference, just noticeable difference,
lateral and medial superior olive, neuronal arithmetic, 
psychophysics,
sound localization.
\end{keyword}
\maketitle

%\begin{abstract}
\subsubsection*{Abstract}
\textbf{Abstract will be rewritten after the manuscript
completion. Paper should explain anylytical MSO model, all figures
and tables.}
%
We study a model of mammalian sound source localization in horizontal
plane. Experiments on small rodents indicate that mammals use broadly
tuned channels of azimuth for localization in horizontal plane. In
mammals this neural computation is implemented by medial superior
olive for low frequency sounds up to 1500~Hz.
%
It has been shown previously that spike timing jitter, coincidence
detection window length, sound frequency, among other input parameters,
influence the output precision, measured by the just noticeable
difference in output of the circuit.
%
\comment{It is more difficult, if not impossible, to investigate the effect of
jitter and other parameters mentioned above in electrophysiological
experiment. Therefore we use previously published}
We use stochastic model with spiking neurons. 
We explore variations of the parameters mentioned above.
We calculate properties of the model with the
use of analytical methods.
% TO DO: add more text on results.
Predictions of this model have straightforward
applications in testing and designing stimulation protocols used in
cochlear implants.
\comment{
Abstract must be rewritten.
Interaural level difference (ITD) is one of basic binaural clues supporting
the spatial localization of a sound source. The rest of the abstract
will be written later. 
}
%\end{abstract}

\global\long\def\srom#1{_{{\mathrm #1}}}%
\def\cv{C\srom V} %$\cv$,~coefficient of variation;
\def\fsound{f\srom S} %$\fsound$,~$f$~sound frequency; 
\def\ILD{\citen{ILD}}
\def\IPD{\citen{IPD}}
\def\ITD{\citen{ITD}}
\def\JND{\citen{JND}}

%%% \comment{
\subsection*{Abbreviations and symbols}

% \def\cv{C\srom V}$\cv$,~coefficient of variation;
\def\fsound{F\srom S}$\fsound$,%~$f$~
sound frequency; 
$F\srom C$,~critical sound frequency value;
$\varphi$,~sound phase;
% Note: sound frequency, in the LSO paper - other symbol;
\ILD,~interaural level difference;
% \citen{IPD} - interaural phase difference, not used
\ITD,~interaural time difference;
\citen{ISI},~inter-spike interval; 
\JND,~just noticeable difference;
\citen{LSO},~lateral superior olive;
$K\srom C$, $K\srom S$, $A$, $B$, $C$,~proportionality constants; 
$l$,~sound level;
\citen{MSO},~medial superior olive;
$r(V)$, $r\srom V$,~VS,~vector strength;
$R\srom F$,~firing rate;
$\sigma$,~standard deviation;
$T\srom J$,~timing jitter;
$t$,~time;
$T$,~$T\srom{S}$, sound period;
%%% } % End Comment

\section{Introduction}

Mammalian sound localization circuits contain two nuclei in the
auditory brainstem, the medial and the lateral superior olive (MSO and
LSO). Neurons in these nuclei are the first binaural neurons in the
auditory pathway, they are connected to both ears.

This article presents description of information encoding and neural
computation in the MSO obtained mostly with analytical
computations. Using the analytical tools we extend quantitative results
obtained by \citet{Sanda2012} only in simulations and connect them to
\citet{Bures2013} to arrive to unified description of neural circuits
in the superior olive.
%
We use this description to find maximum spike timing precision
attained by the following two respective modalities (low and high
frequency sound localization) of natural and electrical hearing.

\comment{
After a simulation study of the MSO in
\cite{Sanda2012} and an analytical theory of the LSO in
\cite{Bures2013}, we present an analytical and quantitative
calculation of the MSO workings in this paper.
... \citet{Bures2013} ...
\citet{Sanda2012} presented simulation model of the stochastic
computation of the sound azimuth from the \ITD\ in the MSO neural
circuit.
%
In this paper we present some theoretical and analytical calculations
together with the simulations. They are related to the dependence of
the circuit output on sound frequency and sound intensity.
} %\EOC

Due to the physical nature of the binaural sound, the MSO neurons
process spike timing differences in the range of tens of microseconds,
$\mu$s.
%
The MSO processes low frequency sounds, in human this is from 20~Hz to
not more than 2~kHz.
%
The LSO processes high frequency sounds, in human this is from 1~kHz
up to 20~kHz.
%
The overlapping region is known to have a sensitivity drop at 1.5~kHz
\cite{Mills1958}.

% Neural coding/ computation
Firing rate, first spike latency and individual spike timings are used
in neural system coding, especially in the auditory pathway. The
highest known spike timing precision that has been proven to be
utilized in the mammalian auditory system is the motor response in a bat.
In behavioral experiment,
the big brown bat,
\textit{Eptesicus fuscus}
catches tidbit larvae of the
meal-worm beetle
\textit{Tenebrio molitor}.
The motor precision is in
the range of hundreds of nanoseconds, \citen{ns}, and auditory
separation of ultrasound echoes is in the order of microsecond units,
1~$\mu$s \cite{Simmons1998}.
%
Comparable performance of the human MSO is in the range of
microseconds, 10~$\mu$s and can be improved after several hours of
training.

Different neural mechanisms are employed in
the two nuclei. 
% \continue

It has been reported
that computation in the MSO is independent on sound intensity
\cite{Grothe2010}. With higher sound intensity, first spike latency is
shortening. Relation of this dependence to ITD and ILD has been
described, yet it is difficult to interpret. \citet{Michelet2012} have
shown the latencies in experiment on the domestic cat,
%
\textit{Felis silvestris}.

The LSO and MSO extract location information with the use of different
physical cues. The sensitivity of the system in dependence on the main
sound frequency to sounds of different main frequencies should be
different.
%
In human, MSO is the larger of the two nuclei and contains
approximately 15500 neurons, human LSO contains 5600 neurons.
% {\bf In practice}

We have previously studied parameter free models of human sound
localization. Current development of binaural cochlear implant
technology enables to stimulate auditory periphery with rich
repertoire of patterns. Better understanding to the MSO mechanisms
will help to improve electrical hearing.

Moving sound objects are localized using the ITD and ILD cues and in
addition to this, Doppler effect pitch changes signal approaching and
retreating objects, even though Doppler cues are less salient. Due to
these phenomena, neural circuits are robust to pitch variation. It is
therefore interesting to pose a question, whether wider pitch changes
in a thought experiment in rescaling time will affect these neural
mechanisms.
% First look at the data could enable us to generalize when we ...
Speeding up, or slowing down the system time can be extended to two
octaves up and down the regular time, see Table~\ref{tab:speed-up}.

The range of sound frequencies
%\cite{Marsalek2004, Grothe2010}
processed in the MSO circuit is limited to the low frequency band. In
human, where audible sound frequencies range from 20~Hz to 20~kHz,
this low frequency band spans from 20~Hz to 1.5~kHz. Moreover, in the
marginal values of this interval, performance of sound azimuth
discrimination is worse.
%
It is generally agreed that the main reason, why the workings of the
MSO circuit deteriorate towards higher frequency is lowering of the
synchronization of spike trains in the circuit with the sound source
phase. The synchronization between two corresponding series of point
events can be expressed as a (discrete version of) vector strength,
defined below in equation \eqref{eq:M-VS}.

\section{Methods}
%\section{Experimental Procedures}

\subsection{Preliminaries}

Our procedures are mostly in using computational models, experimental
data and statistical methods. Here we list several assumptions used in
the paper.

\begin{inparaenum}[(1)]
\item Single MSO neuron constants, see \cite{Toth2015}.
\item Synaptic machinery, see \cite{Toth2015}.
\item Ergodic assumption see \cite{Sanda2012}, \cite{Toth2018}.
\end{inparaenum}

% keywords: Vector strength, ideal observer, phenomenological spike,
% just noticeable difference

\subsection{Vector strength}

The vector strength is defined as it is used since
\citet{Goldberg1969}.
%
Let us have sample (spike) phases 
%
$\varphi_i$, $i=1, 2, \dots, N$ 
%
relative to phases of a given input master periodic function, which
does not enter the formula. (Such function can be for example sound
stimulus eliciting the spike train as a response.)
%
(Discrete sum) vector strength of sample
%
\( \varphi_1, \dots, \varphi_N \) is defined as
%
\begin{equation}
r\srom V(\varphi_i) = \frac{1}{N}
\sqrt{\left( \sum_{i=1}^N \cos\varphi_i \right)^2
+ \left( \sum_{i=1}^N\sin\varphi_i \right)^2}.
\label{eq:M-VS}
\end{equation}

\subsection{\label{Meth_JND}Just noticeable differences of \ITD}

This subsection of Methods should be analogous to the corresponding
section in \citet{Bures2013}...

\comment{
A higher variability of firing leads to a lower precision of the rate
code. Intuitively, if a repeated presentation of the same stimulus
evokes each time a different spike count, then to distinguish between
two different stimuli, the associated spike count change must be larger
than the spike count variability. Mathematically, the current work
determines the \JND\ of \ILD\
(in other words, the precision of \ILD\ coding)
using theoretical tools. 
% derived in \citen{Green}1966, \citen{Teich}1979, \citen{Colburn}2003).
We will ask whether it is possible to distinguish between two random
processes with rates $L_{1}$ and $L_{2}$, $L_{2}>L_{1}$. If we
count events in a given counting window, we get counts $n_{1}$ and
$n_{2}$. The probability that the observer obtains a result that
$L_{2}>L_{1}$ equals to the probability that $n_{2}>n_{1}$. Let's
assume that the random variables $n_{i}$, $i=1,2$, have probability
distribution $p(n_{i})$ with means $\mu_{i}$ and roughly equal standard
deviation $\sigma$. A detection distance is then defined as 
\begin{equation}
d'=\frac{\mu_{2}-\mu_{1}}{\sigma}\label{R_SDT1}
\end{equation}
This definition expresses the fact that the larger the variance of
the spike count, the worse the detection capability.

In psychophysics, a threshold value is commonly defined as that value
for which the percentage of correct answers equals 75. In our case,
the examined value is the just-noticeable change of firing rate,
$\Delta L=L_{2}-L_{1}$.
%
Assuming that both $p(n_{1})$ and $p(n_{2})$ are Gaussian, the 75\%
probability of $n_{2}-n_{1}>0$ corresponds to $d'=1$. To obtain
the \JND\ of firing rate, we scale the detection distance with $\Delta L$
and put $\delta'=d'/\Delta L$. Then, the \JND\ of firing rate is 

\begin{equation}
\Delta L_{\text{\JND}}=1/\delta'=
\frac{L_{2}-L_{1}}{\mu_{2}-\mu_{1}}\sigma\label{R_SDT2}
\end{equation}
} %end - comment

\subsection{Model of the MSO neural circuit}

To do: Describe model and calculations.

% \nocite us/ mutual citations:
% Author(s): \citen{Koutsou}; \citen{Christodoulou}; \citen{Bugmann};
% and others
% Source: NEURAL COMPUTATION, Volume: 24, Issue: 9, Pages: 2318-2345 

\section{Results}

We investigate how circuit output and overall performance depend on
sound frequency and sound intensity.
Figure \ref{fig:FM1V1} shows an example of typical vector strength
lowering towards higher frequencies, as observed in the auditory nerve
(AN). The prevailing majority of neurons in the auditory pathway has a
vector strength spike train statistics sigmoidally dropping towards
higher sound frequencies as it is in this example.
%
In this figure, data recorded by \citet{Joris1996} from the MSO in cat
were fitted to the sigmoidal curve with the general formula of the
Boltzmann function used in \citet{Marsalek2005}.
%
The curve fit of vector strength $r\srom V$ in dependence on sound
frequency $\fsound$ is:
%
\begin{equation}
r\srom V = (1 + \exp(K\srom S \fsound - K\srom C F\srom C))^{-1},
\end{equation}
%
where $K\srom S$ (sound coefficient) and $K\srom C$ (critical
coefficient), $F\srom C$ (critical half frequency) are proportionality
constants.

Figure \ref{fig:FM2V1} shows the best just noticeable difference
(\JND) obtained with the basic parameter set in dependence on the
sound frequency.
%
The quadratic curve fit of the \JND\ denoted $\Delta t\srom{\JND}$ is:
%
\begin{equation}
\Delta t\srom{\JND} = A (f\srom S - F\srom C)^2 + B,
\end{equation}
%
where $A$ and $B$ are fitted constants and the other parameters and
variables are as above. See also \citet{Zwislocki1956}.

Figure \ref{fig:FM3V1} shows how ITD JND depends on timing jitter
$\tau\srom{J}$.
%
There is one critical $\tau\srom{J}~=~0.2$ and one optimal value
(normalized to $\tau\srom{J}~=~1$) of timing jitter related to the
output \JND\ and firing rate. Simulations show that with lowering
timing jitter the circuit output is virtually more and more precise.
%
The two curves fitted to the simulation are:

1. fit of exponential function to simulations:
\begin{equation}
\Delta t\srom{\JND}= \exp(A_1(\tau\srom{J}-B_1)) - C_1,
\label{eq:exponential}
\end{equation}
%
where $A_1=1.9$, $B_1=1.25$ and $C_1=0.2$ are fitted parameters. This
is shown conveniently by the logarithmic y-axis in this figure.

2. Another fit, which also takes into account shot noise in lower
jitter values, is to a quadratic function:
%
\begin{equation}
\Delta t\srom{\JND} = A_2(\tau\srom{J}-B_2)^2 + C_2,
\label{eq:parabolic}
\end{equation}
%
where $A_2=2.5$, $B_2=1$, $C_2=1$. There is only one parameter
($A$) fitted to data, as the point $(x,y) = (B_2,C_2)$ has been chosen
to be a unit, this fit is a normalized fit of the model.
%{this must be explained - that this fit is after normalization!}.
% \par\indent

Logarithm of the simulated \JND\ lowers with the exponential curve
\eqref{eq:exponential}, which is concave function of jitter
$\tau\srom{J}$ as the jitter gets lower. The trend towards higher
accuracy diverges from the parabolic fit \eqref{eq:parabolic},
when jitter reaches critical value between
% \lessapprox and \gtrapprox.
$0.2\lessapprox \tau\srom{J}\lessapprox 0.5$~ms. Beyond that point
towards the lower jitter values, the neural circuit cannot function
properly, as too low jitter prevents the interaction of spikes from
the left and right side within the coincidence detection
mechanism.
% The other (optimal) jitter value notable in this Figure~
% \ref{fig:FM3V1} is firing rate in dependency on the jitter values.
\comment{
This is an analytical dependency obtained in \cite{Salinas2000} for a
perfect integrator model with several inputs. The mechanism studied
thereof is close to the MSO neural mechanism studied here.
%
The firing rate changes in dependency on the input spike timing
variability of {\it partially correlated} definition?/ description?
input spike trains.}

% ---------

Figure \ref{fig:FM4V1} shows the calibration curve. The rising slope
of this curve is used as a read{out} function yielding th firing rate
in dependency on the \ITD\, which in turn signals the sound azimuth to
the next nuclei up to the auditory pathway.

\begin{table} % Table 1
\begin{centering}
\begin{tabular}{|l |l | r| c | c |}
\hline 
Parameter & Symbol & Units & Typical Value & Ranges\tabularnewline
\hline 
\hline 
Jitter & $T_{J}=\sigma$ & ms & 1 & 0.125 - 8 \tabularnewline
\hline 
Window & $w_{CD}$ & ms & 0.6 & 0.15 - 1.5 \tabularnewline
\hline 
Sound Frequency & $f$ & Hz & 200 & 40-1600 \tabularnewline
\hline 
 & & & & \tabularnewline
\hline 
\end{tabular}
\par\end{centering}

\caption{\label{tab:parameters}
The basic set of parameters. 
%should be here, in analogy to the tables in \citet{Sanda2012}.
}
\end{table}

Comment
\comment{

\begin{table} % B O table box commented
\begin{centering}
\small
\vskip 0.25 cm
%
\begin{tabular}{|l||l|l|l|l|} %[!t]
\hline
Animal, its head & width $W$, & ITD latency $l_{\rm ITD}$,
& cochlear latency $l_{\max}$
& and basilar membrane length $L$.\\
\hline
% \hline
% elephant & 1.00~m & 2940~$\mu$s & ? (1000-2000~$\mu$s) & 
% 60~mm \cite{bekesy60XX}\\ %(100-200~mm) \\
\hline
human & 0.16~m & 470~$\mu$s & 640~$\mu$s &
 35~mm \citen{bekesy60XX}\\
% \hline
% dolphin & 0.14~m & 90~$\mu$s (in sea water) & 640~$\mu$s &
% 35~mm \cite{wever71XX}\\ %\cite{branstetter03XX}
\hline
cat & 0.08~m & 235~$\mu$s & 400~$\mu$s &
 22~mm (cat) \citen{ehret75XX}\\
\hline
mouse / or gerbil & 0.03~m& 90~$\mu$s & 128~$\mu$s &
 7~mm (mouse) \citen{ehret75XX}\\
\hline
\end{tabular}
\end{centering}
% E O \rm
%
\normalfont
\vskip 0.25 cm
%
\caption{\label{tab:parameters}
%\centerline{
Table 1: \bf Maximum interaural time
 $l_{\rm ITD}$ and maximum cochlear latency $l_{\max}$
\noindent
Table compares head width $W$, maximal interaural time $l_{\rm ITD}$
($l = W/c$) and maximal cochlear delay $l_{\max}$ proportional to
basilar membrane length $L$.}
% {table:animals}

\end{table} % E O table box

} % E O C ---


TO-DO - This Table X should be extended: more parameters and range of
parameters...

\comment{
Table number One-and-one-half:
Top description (x-axis)
a 1) normal hearing
b 2) hearing impaired
c 3) electrical - no jitter
d 4) artificial jitter
e 5) (sub)optimal jitter

Side description (y-axis)
1) 40 - 160 Hz
2) 160 - 640 Hz
3) 640 - 2,560 Hz
4) 2,560 - 10,240 Hz}

\comment{
\begin{table} % Table~2 & 0
\begin{centering}
\begin{tabular}{|l | l || c | c | c | c | c |}
\hline 
% octaves
[a-top] & src & MSO/ low & MSO/ low & critical & LSO/ hi & LSO/ hi 
 \cr\hline\hline 
central frequency & - & 0.250 & 0.500 & 1 & 2 & 4 \cr\hline
range & - & 0.175 - 0.350& 0.350 - 700& 0.7 - 1.4& 1.4 - 2.8& 2.8 - 5.6\cr\hline
N.H. & Bures Marsalek 2019{}& 1~dB & 1~dB & 1~dB & 1~dB & 1~dB \cr\hline
H.I. & cochlear model {}{}{}& 1~dB & 1~dB & 1~dB & 1~dB & 1~dB \cr\hline
E.H. no jitter & literature & 1~dB & 1~dB & 1~dB & 1~dB & 1~dB \cr\hline
E.H. with jitter & Laback{} & 1~dB & 1~dB & 1~dB & 1~dB & 1~dB \cr\hline
E.H. other & tuning {} {}{} & 1~dB & 1~dB & 1~dB & 1~dB & 1~dB \cr\hline
\end{tabular}
\par\end{centering}
\caption{\label{tab:jnds} Just noticeable differences in models of
 different operation regimes}
box a-top: mode/ frequency band [kHz]
\end{table}
} %end comment

%
% %%%
\begin{table} % Table~3
\begin{centering}
\begin{tabular}{|l |l | r|| c | c | c | c |}
\hline 
% octaves
Parameters (man; time warped) & Symbol & Units &
 4$\times$ Slower & Normal Speed&
 4$\times$ Faster & 16$\times$ Faster \cr\hline\hline 
Sound frequency ranges &
 $f$ & Hz/ kHz & 15-125 & 60-500 & 240-2~kHz & 960-8~kHz \cr\hline
%$f$ & Hz & 17.5 - 110 & 70 - 440 & 280 - 1760 & 1120 - 7040 Hz \cr\hline
%$f$ & Hz & 35 & 140 & 560 & 1120 \cr\hline
Window size & $w\srom{CD}$ & $\mu$s & 2400 & 600 & 150 & 37.5 \cr\hline 
Timing Jitter & $T\srom{J}$ & ms & 4 & 1 & 0.25 & 62.5 $\mu$s \cr\hline 
Predicted \JND\ & $T$ & $\mu$s &
 40 & 10 & 2.5 & N/ A \cr\hline % 0.625 $\mu$s
Predicted detection time$^{*}$ & $T$ & ms &
 2,600 & 650 & 162.5 & N/ A \cr\hline % 81.25 \cr\hline
\end{tabular}
\par\end{centering}
%
\caption{\label{tab:speed-up} TO DO add the last OVERLAPPING band 4480
 - 28160 and make all a bit narrower to start at 20 Hz and end at
 20000 Hz. This Table is direct scaling of parameters used in
 \citet{Sanda2012}. Make also MSO and LSO entries...}
%
Predicted values are: 1) just noticeable
difference (\JND) of the \ITD\
% expressed as a ratio of the \ITD\ over
2) the time of observation by an ideal observer reading the information
from one neuron.
\label{tab:frequency}
\end{table}
% %%%

\section{Discussion}

\subsection{Things to do}

\comment{Outline:\begin{enumerate}
% \hline
OK \item Content and results of this paper = analytical investigations of
our model we explored before mostly by simulation.
\item Relation to data
OK \item Relation of MSO and LSO
Discussion \item alternative theories of LSO/ MSO functioning
\end{enumerate}} %\EOC

\comment{Simulations to support/ correct the theoretical results.}

First: Reproduce the curve in panel with sound frequency on x-axis and
\JND\ on the y-axis in Fig. \ref{fig:FM2V1}. %{fig:FM-J-N-D}.

Second: Present the curve from the right panel of Fig.4 from Sanda P.,
Jitter Effect on the Performance..., 
% E-J-B-I - 1:1, 2011.
as part of Fig. \ref{fig:FM3V1}. %{fig:FM-J-I-T-T}.

Some other tentative questions: What is the highest slope of the \ITD\
interpolation curve, such that it gives the resolution of 4$^\circ$
(angular degrees)?

(This we will do later: We also discuss the relation of the MSO neural
circuit to its neighboring nucleus, the LSO, and its output.)

\comment{To cite: exotic media, like \citen{Scholarpedia}, or
 \citen{ArXive}, see theoretical work in \citen{ArXive} 
 \citen{Mlynarski2014}
} % \EOC 
Cite: \cite{Mlynarski2014}...
Cite: \cite{Bures2019}...

% We should further discuss \cite{Sanda2011}, etc, etc...

%%% End Discussion / Beginning Figures

%%%
\comment{
\begin{figure}[ht]
\begin{centering} % [width=0.5\textwidth]
\includegraphics[scale=0.125]{../figs/GENERIC}
\par\end{centering}
\noindent
\caption{
\textbf{Caption title.}
%
Figure description
%
}
\label{fig:FMSCHEM} 
\end{figure}
} % end comment
%%%

\begin{figure}[ht]
\begin{centering}
% [scale=0.5; 0.8]
% \includegraphics[width=13cm]{../figs/FM1V1}
\includegraphics[width=0.75\textwidth]{../figs/FM1V1}
\par\end{centering}
\noindent
\caption{ 
%
\textbf{Vector strength of auditory nerve spike trains
in dependence on sound frequency.}
%
X-axis shows fundamental sound frequency in~Hz in a logarithmic scale
and y-axis shows the vector strength.
%
Even though is some nuclei up the auditory pathway the synchronization
can be maintained towards higher frequencies than shown here, the
decrease of the vector strength towards higher frequencies is a
general property of all neurons in the auditory pathway.}
% figure 1
\label{fig:FM1V1} 
\end{figure}

\comment{
This schema shows the anatomical wiring of the mammalian brainstem
circuit together with the equivalent circuit used in the model.
%
Without the loss of generality, for the sake of simplicity, only one
half of the circuit is shown. Ipsi-lateral and contra-lateral sides
are shown as a left and right sides, respectively
}
%

\begin{figure}[ht]
\begin{centering} % scale=0.8; width=0.75\textwidth
\includegraphics[width=0.75\textwidth]{../figs/FM2V1}
\par\end{centering}
\noindent
\caption{
\textbf{The shortest \ITD\ detected in the dependence on sound frequency.}
%
X-axis shows sound frequency in~Hz in a logarithmic scale and y-axis
the shortest \JND\ in $\mu$s. This is a theoretical prediction based on
the analytical model and basic parameter set used in previous
simulations.
% figure 2
}
\label{fig:FM2V1} 
\end{figure}

\begin{figure}[ht]
\begin{centering} % scale=0.8; width=0.75\textwidth
\includegraphics[width=0.75\textwidth]{../figs/FM3V1}
\par\end{centering}
\noindent
\caption{
%
\textbf{The \JND\ values in basic parameter set in dependence on the
jitter magnitude.}
%
This plot in semi-logarithmic y-scale shows just noticeable difference
of interaural time difference depending on variation of the spike
timing jitter. Jagged line: simulated data,
% from \cite{Sanda2012},
solid line: an exponential fit to the simulations under the assumption
of arbitrary time precision in the model circuit, line with circles: a
quadratic function estimate of spike timing precision in a system with
addition of noise.
% figure 3
}
\label{fig:FM3V1}
\end{figure}

\begin{figure}[ht]
\begin{centering} % scale=0.5; width=0.75\textwidth
\includegraphics[width=0.75\textwidth]{../figs/FM4V1}
\par\end{centering}
\noindent
\caption{
\textbf{Gaussian fit of the \ITD\ to firing rate slope calibration curve.}
%
X-axis shows \ITD\ in~ms and y-axis shows corresponding firing rate in
action potentials per second. Note that the curve peak is offset from
the origin of coordinates at $t\srom{\ITD}=0$.
%
}
\label{fig:FM4V1} 
\end{figure}


\subsection*{Notes}
%
% by graduate research program no. 266509/ 2013 to P.M. at the Charles
% University 
% thanks to X{Y}Z for English language copy-editing.
%%% \newpage
%%% \nocite{*} % = all records in the ../sound_local.bib file are OK
%\nocite{XX}

To Do: Add citations to discussion:

Cite \cite{Franken2014}, this is an experimental description of MSO
workings in line with our concepts and coincidence detection theories.

Cite \cite{Silver2010}, because this is a prospect of neuronal
arithmetic.

To do: check, whether both \citet{Smith1991} and \citet{Smith1993} are
relevant.

The ergodic paper of \citet{Toth2018} goes to Discussion.

Another MSO model, already in 2005 is: \citet{Zhou2005}.

Cochlear implants and MSO: \cite{Kerber2012}.

Check \citet{Wu1992}: WHY DO WE CITE THIS?

Here are the three LSO experimental papers, which have detailed
methodology applicable to overlap of sound frequency ranges with the MSO
\cite{Joris1995}; % = LSO I
\cite{Joris1996}; % = LSO II
\cite{Joris1998_envelope}. % = LSO III

\citet{Jeffress1962} goes instead of Jeffress, 1948.

% below are several no-cite items/ alphabetical order

(No)Cite: \citet{Bures2012} is neuronal arithmetic in \citen{Biological}
\citen{Cybernetics}

(No)Cite: \cite{Bures2013} is ``dual'' paper in Brain \citen{Research} 

Do we still need to cite this review: \cite{Grothe2010}?...

Thee are the classics...: \cite{Goldberg1969}.

Cochlear delays, very important: \citet{Joris2006}.

Yet another neuronal arithmetic: \cite{Koutsou2012}.

This is about other dimension; if we cite this, then we must discuss all
the three, \citet{Kopco2012}. hall we do that?

Jitter and variability... : \cite{Kostal2010}.

\nocite{Krips2009}

\nocite{Laback2008}

This is a chronological record of the neuronal arithmetic ideas,
starting from:
\citet{Marsalek2000},
through
\citet{Marsalek2001}
some more detailed account o the diversity of cell functions and shapes:
\citet{Reed2002},
through
\citet{Marsalek2004} 
to the
\citet{Marsalek2005_Kofranek};
watch the years ...

Remains to be cited: no-cites of \citet{Smith1993}, where we have
by mistake fitted a curve without a-priory entering the known sound
frequency.

% \citet{Szalisznyo2004}.

Next \citet{Cariani2011}, \citet{Bouse2019}, \citet{Bures2014},
% \citet{Bures2019},
\cite{Michelet2012}, \cite{Srinivasan2018}.

Other insider comments: Figure 4 should correct the omission in the
paper by Sanda and Marsalek. When we cover everything else, we should
add this ``weeping of Jeremiah'': we urgently need binaural data/
psychophysics/ or electrophysiology, all the same.

\section*{Conclusions}

This theoretical paper is a followup of sound localization precision
descriptions in the MSO \cite{Sanda2012} and in the LSO
\cite{Bures2013}. Novel results are two: 1) analytical estimates of
results obtained previously only by numerical simulation and 2)
comparison of natural and electrical hearing modalities and
demonstrating what is the maximum timing precision which can be
attained by the two modalities.

% \section*{Notes}
% Marek Hajny can join the author team as: middle position author with
% tasks: 1) to compile Discussion and write and 2) to rewrite parts of
% the manuscript.

\subsection*{Acknowledgments}

PM and M{H} were supported by: Charles University graduate
students research program (acronym SVV)
No. 260371/ 2017-2019.
Special thanks to Pavel Sanda for permission to use his simulations.

\subsection*{Acknowledgments}

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