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\def\keyFont{\fontsize{8}{11}\helveticabold }
\def\firstAuthorLast{Marsalek {et~al.}} %use et al only if is more than 1 author
\def\Authors{Petr Marsalek\,$^{1,2,*}$, Pavel Sanda\,$^{3}$ and Zbynek Bures\,$^{2,4}$}
% Affiliations should be keyed to the author's name with superscript numbers and be listed as follows: Laboratory, Institute, Department, Organization, City, State abbreviation (USA, Canada, Australia), and Country (without detailed address information such as city zip codes or street names).
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\def\Address{$^{1}$ Institute of Pathological Physiology, First Medical
 Faculty, Charles University in Prague, U~Nemocnice~5/478,
 128~53, Praha~2, Czech Republic\\
  $^{2}$Department of Natural Sciences, Faculty of Biomedical
  Engineering, Czech Technical University in Prague,
  n\'{a}m. S\'{\i}tn\'{a} 3105, 272~01 Kladno, Czech Republic\\
$^{3}$Institute of Computer Science, Czech Academy
 of Sciences, Pod Vod\'{a}renskou v\v{e}\v{z}\'{\i}~2/271,
 182~07, Praha~8, Czech Republic\\
$^{4}$Czech Institute of Informatics, Robotics and Cybernetics,
 Czech Technical University in Prague,
 Jugosl\'{a}v{s}k\'y{c}h partyz\'an{\accent23u}~4/1903,
 166~36, Praha~6, Czech Republic}

% The Corresponding Author should be marked with an asterisk
% Provide the exact contact address (this time including street name and city zip code) and email of the corresponding author

\def\corrAuthor{Petr Marsalek}

\def\corrEmail{Petr.Marsalek@cvut.cz}

\begin{document}

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\title[Precision of neural computation with ITD in MSO]{On the precision of neural computation with interaural time differences in the medial superior olive} 

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%\extraAuth{corresponding Author2 \\ Laboratory X2, Institute X2, Department X2, Organization X2, Street X2, City X2 , State XX2 (only USA, Canada and Australia), Zip Code2, X2 Country X2, email2@uni2.edu}

\maketitle

\begin{abstract}

%%% Leave the Abstract empty if your article does not require one, please see the Summary Table for full details.

In the auditory nerve and the following auditory pathway, incoming
sound is encoded into spike trains -- series of neural action potentials.
At the third neuron of the auditory pathway, spike trains of the left
and right sides converge and are processed to yield sound localization
information. Two different localization encoding mechanisms are employed
for low and high sound frequencies in two dedicated nuclei in the
brainstem: the medial and lateral superior olivary nuclei. Building
upon our previous computational model of medial superior olive (MSO),
this paper brings analytical estimates of parameters needed to describe
auditory coding in the MSO circuit. We arrive to best estimates for
neuronal signaling with the use of just noticeable difference and
the ideal observer concepts. We describe spike timing jitter and its
role in the spike train processing. We study the dependence of sound
localization precision on the sound frequency. All parameters are
accompanied with detailed estimates of their values and variability.
Intervals bounding all the parameters from lower and higher values
are discussed.

%\tiny
\small
\keyFont
{ \uppercase{Keywords:} %\subsection*{Keywords:}
binaural hearing, coincidence detection, ideal observer,
interaural time difference, just noticeable difference,
lateral and medial superior olive, sound localization,
spike timing jitter}
%All article types: you may provide up to 8 keywords; at least 5 are mandatory.
\end{abstract}

%There are 5 heading levels
% \subsection{Level 2} \subsubsection{Level 3}
% \paragraph{Level 4} \subparagraph{Level 5}

\subsection*{Abbreviations and symbols}

$f_{\mathrm{S}}$,~sound frequency; $F_{\mathrm{C}}$,~critical
sound frequency value; $\varphi$,~sound phase; ILD,~interaural
level difference; IPD,~interaural phase difference; ITD,~interaural
time difference; JND,~just noticeable difference, also difference
limen, or difference threshold; $K(.)$, $K_{\mathrm{C}}$, $K_{S}$,
$K_{X}$; $A$, $B$, $C$, $\dots$~proportionality constants; $l$,~sound
level, also rate level of point process; LSO,~lateral superior olive;
MSO, medial superior olive; $R(.)$, $R_{\mathrm{VS}}$,~VS,~vector
strength; $R_{F}$,~firing rate; $\sigma$,~standard deviation;
\global\long\def\timijitt{t_{\mathrm{J}}}%
 $\timijitt$~timing jitter; $t$, $\Delta t$,~time, time difference;
$T$,~$T_{\mathrm{X}}$, ~$T_{\mathrm{(.)}}$, sound periods, time
constants.

\section{Introduction}

Mammalian sound localization circuits contain two nuclei in the
auditory brainstem - the medial and the lateral superior olive, MSO
and LSO, respectively. Neurons in these nuclei are the first binaural
neurons in the auditory pathway, processing the information arriving
from both ears. The MSO processes low frequency sounds, in human this
is from 20~Hz to not more than 2~kHz, and the LSO processes high
frequency sounds, in human this is from 1~kHz up to 20~kHz
\cite{Middlebrooks1991}. Due to the physical nature of the binaural
stimulus, the MSO neurons process spike timing differences, Interaural
Time Differences, ITDs.  The ITDs emerge as a result of different
distance from the sound source to each ear. The ITDs take values given
by the distance between the ears
% \textmu -> \mu
(in human this is circa 16 cm/ 466
\emph{$\mu$}s sound takes to travel this distance in the air)
and are detected with precision of tens of microseconds,
%
10 \emph{$\mu$}s \cite{Laback2008}. The LSO detects
the Interaural Level Difference, ILD, given by the acoustic shadow
of the head. The frequency region, where these two mechanisms overlap
(around 1.5~kHz) is known to have a drop in localization sensitivity
\cite{Mills1958}.

Firing rate, first spike latency and individual spike timings are
used in neural system coding, especially in the auditory pathway.
Human MSO can detect binaural spike timing shifts as small as
%
10~\emph{$\mu$}s
%
and this ability is reported to be improved two- to five-fold after
several hours of training \cite{Middlebrooks1991}.
Considering that duration of a neuronal action potential is approximately
1 ms, such performance is striking. Furthermore, to be etologically
useful, the MSO must provide stable operation over a wide range of
sound frequencies and intensities and listening conditions. While
it has been reported that computation in the MSO is independent on
sound intensity \cite{Grothe2010},
it is known that its performance drops as the stimulus frequency increases..
Relation of this dependence to binaural cues has been described, yet
it is difficult to interpret.

It is generally agreed that the main reason, why the precision of
the MSO circuit deteriorates towards higher frequency is lowering
of the synchronization of spike trains with the sound phase. However,
exact mechanisms are not completely understood. One of our aims is
thus to explore the conditions and limits of MSO circuit operation
using the computational modeling approach.

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 numerical computations in \cite{Sanda2012}.
We compare this analytical MSO description to the LSO description
in \cite{Bures2013} to arrive to
unified description of neural circuits in the superior olive. We use
this description to find the performance limits of the MSO circuit
in dependence on sound frequency and intensity. Apparently, low and
high frequency sound localization use different neuronal mechanisms.
Sound features in low and high sound frequencies are encoded by distinctive
codes and this multitude of codes also affects binaural processing.
\section{Methods}

\subsection{Preliminaries}

{In the neural circuit model used here spikes, or action
potentials, fired in the arbitrarily precise time are individual
events of neural computation. Arbitrary precise timing would imply
arbitrary high information content. In the model, this is limited
by assumptions of intrinsic noise content. Numerical implementation
has been described in \cite{Sanda2012}. Here we develop combined
stochastic and analytical description of the model. Our aim is to
arrive at parameters and constants useful in the MSO description.}

The neural circuit consists of neurons, functional units
exchanging spikes. Incoming sound is sequentially processed in the
auditory periphery. All the processing stages, including cochlea,
are modeled as all-or-none units with various degree of biological
realism. After cochlea, locations of individual neurons in model
correspond nearly to anatomical neuron numbering \cite{Kulesza2007}.
The zero order neuron is the whole mechanical-to-electrical cochlear
mechanism, neurons of order 1 are in auditory nerve, order 2 are
neurons in the cochlear nuclei, order two and a half is the medial
nucleus of trapezoid body, we regard this nucleus as a ''polarity
inverter'', and order 3 are the neurons of the MSO itself. The order 3
neurons are the binaural part of the circuit.

Before they converge on the MSO the two, left and right,
branches process sound from left and right ear. After the sound is
encoded by cochleas into spike trains, the rest is the processing
of spike trains by neurons. The spike trains are subject to delays
and synaptic relying. A remarkable property of the auditory pathway
is that both synapses and neurons have the shortest response times
and highest time precision in the mammalian brain. If the neurons
were represented by R{C} circuit, or similar equivalent biophysical
models, their time constants would be comparable to, or lower than
1~ms. Due to vernier mechanisms known from various parts of peripheral
sensory pathways, they can capture time events in the range of tens
of microseconds. This capability has been described in human,
\cite{Mills1958}.
In some animal species -- localization specialists, responses are in
the range of tens of nanoseconds, as it has been shown in experiment
on bats by \cite{Simmons1998}.
Several other time constants and frequencies are characteristic for
this neural circuit. They are shown in Table~\ref{tab:parameters}.

\subsection{Model of the MSO neural circuit}

Our model MSO circuit is based on connected phenomenological neurons.
Input sound to left and right sides is transformed by the auditory
periphery module into spike trains. Spikes in these trains are point
events, where only spike times matter and the details of spike numerical
implementation do not make any differences in model output. These
spike trains converge and diverge into higher order neurons. They
are relayed from the auditory nerve and cochlear nucleus through the
medial and lateral nuclei of trapezoid body up to the neurons of medial
superior olive, which are first binaural neurons. Output of the binaural
neurons is the azimuth signal encoded in a spike train. However, components
of our computational model can not be identified with individual neurons
and their anatomical connectivity. Rather, the model consists of functional
block units representing the crucial properties of the explored neural
mechanism, such as intrinsic noise, temporal delay and coincidence
detection. This functional block model layout provides access to descriptive
parameters, which are setting the circuit's performance limits, see
Figure~\ref{fig:circuit0}. In our MSO model, the
interaural time delay (ITD) present at the MSO input is represented by
firing rate at the MSO output. The relationship between the input
ITD and output spike rate is called ITD readout curve, see the next
section.

\begin{figure}[h!] %[ht]
\begin{centering}
{\includegraphics[width=0.75\textwidth]{../figs/circuit_small} }
\par\end{centering}
\noindent
{\caption{ \textbf{Schematic MSO circuit with anatomic and connection layouts.}
This Figure with modifications is reprinted with permission from the
manuscript of \cite{Sanda2012} to show the numerical model key features.}
\label{fig:circuit0} }
\end{figure}

\subsection{ITD readout curve}

Let us have a monotonous function with firing rate as an input, which
outputs azimuth. We call it the ITD readout curve. In the paper by
\cite{Sanda2012} this curve was constructed by curve fitting to
simulated points. Several assumptions about the sound objects had to
be fulfilled to assure the existence of the readout curve. Here we
simplified therefore improved construction of this curve. Now it is
based on assumption that the main (i.e. dominant) frequency of sound
input exist, is unique and is known. In addition to this known
frequency, other parameters of the readout curve are set to make the
fitting well posed and to obtain correct position of the curve
maximum.

\subsection{Vector strength}

The vector strength \cite{Kessler2021} has been first used in the
context of sound localization by \cite{Goldberg1969}. It is commonly
used in auditory neuroscience to quantify how well a spike train phase-locks,
or synchronizes with periodic stimulus phases. Its definition follows.
Let us have sample spike phases $\varphi_{i}$, $i=1,2,\dots,N$
relative to phases of a given input master periodic function. Only
the phases enter the formula. The periodicity of vowel tones making
up speech is a perfect example of such stimulus. Discrete sum vector
strength of samples {$\varphi_{1},\dots,\varphi_{N}$} attains
values from 0 to 1 and is defined as
%
\begin{equation}
  R_{\mathrm{VS}}(\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}}.%1
\label{eq:M-VS}
\end{equation}
%
Here we use the vector strength to analyze temporal acuity of inputs
to the MSO model and the consequences how the degree of synchronization
affects the MSO precision.

\subsection{JND and ideal observer}

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 different spike count, then to distinguish
between two different stimuli, the associated spike count change must
be larger than the spike count variability. This way we determine
the Just Noticeable Difference (JND) of the rate code. In other words,
this is the precision of rate coding.

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 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
us assume that the random variables $n_{i}$, $i=1,2,\dots$, have
probability distributions $p(n_{i})$ with means $\mu_{i}$ and equal
standard deviation $\sigma$. A detection distance is then defined
\cite{Tanner1961} as 
%
\begin{equation}
d'=\frac{\mu_{2}-\mu_{1}}{\sigma}.%2
\label{R_SDT1}
\end{equation}

This definition expresses the fact that the larger is the
variance of the spike count, the worse is 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 (normal)
distributed, 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_{\mathrm{JND}}=1/\delta'=\frac{l_{2}-l_{1}}{\mu_{2}-\mu_{1}}\sigma.%3
\label{R_SDT2}
\end{equation}
%
We apply these tools to the MSO output to study the precision of the
sound localization circuit to discuss its dependence on input sound
frequency.

\section{Results}

We investigate how the MSO circuit output and overall performance
depend on sound frequency and sound intensity. Figures from{\ref{fig:FM1V1}}
to{\ref{fig:FM4V1}} show results of numerical simulations
obtained using the computational model from \cite{Sanda2012} side to
side to experimental data as black lines and points, analytically
described lower estimates as blue lines, upper estimates as red lines
and standard errors of measurement and variations, which are shown in
green.

Successive figures show individual steps of sound localization signal
processing. Fig.~{\ref{fig:FM1V1}} shows how vector strength
in individual units lowers towards high sound frequencies.
Fig.~{\ref{fig:FM2V1}}
shows the range of ITDs in sound localization precision.
Fig.~{\ref{fig:FM3V1}}
shows the JND of the neural circuit in the dependence on the spike
timing jitter. Fig.~{\ref{fig:FM4V1}} shows synchronization
to main sound frequency (in the case when it exists).

Figure{\ref{fig:FM1V1}} shows how vector strength \emph{R}\textsubscript{VS}
lowers towards higher frequencies, as it can be observed in the module
of the auditory periphery consisting of auditory nerve and cochlear
nucleus. The prevailing majority of neurons in the auditory pathway
has vector strength spike train statistics sigmoidally dropping towards
higher sound frequencies as it is in this example. In this figure,
\nocite{Joris1995}
data originally recorded by \cite{Joris1996}
\nocite{Joris1998_envelope}
at the MSO of domestic cat, were fitted to the sigmoidal curve with
the general formula of the Boltzmann function used in \cite{Marsalek2005}.
The curve fit of vector strength $R_{\mathrm{VS}}$ in dependence
on sound frequency $F_{\mathrm{S}}$ is:
%
\begin{equation}
  R_{\mathrm{VS}}=
  1/(1+\exp(K_{\mathrm{S}}f_{\mathrm{S}}-K_{\mathrm{C}}F_{\mathrm{C1/2}})),
\end{equation}
%
where named parameters with values are $K_{\mathrm{S}}=2/0.75=2.666$~ms~
(kHz$^{-1}$), sound frequency coefficient;
$K_{\mathrm{C}}=4/0.75=5.333$~ms~(kHz$^{-1}$),
critical coefficient; and $F_{\mathrm{C1/2}}=0.75$~kHz, critical
half frequency. The numerical values are the proportionality constants
of the $R_{\mathrm{VS}}$ upper bound curve, denoted red. Note that
at sound frequencies from 20 to 100 Hz, there are two branches reflecting
the existence of two alternative ways how to compute the lower limit,
which is 90 \% of the upper limit. The first, blue curve corresponds
to quantile between 80 and 90 \% of the upper limit. The second, green,
lower curve is based on estimate of probability of spike coming from
both sides, which progressively lowers towards lower frequencies,
calculated as in \cite{Marsalek2005}, see Discussion section.

Figure \ref{fig:FM2V1} shows the curves limiting the ITD
obtained with the basic parameter set in dependence on the sound frequency.
The quadratic curve fit of the JND denoted $\Delta t_{\mathrm{JND}}$
is:
%
\begin{equation}
\Delta t_{\mathrm{JND}}=A(f_{\mathrm{S}}-F_{\mathrm{C1/2}})^{2}+B.
\end{equation}
%
Values of these parameters are $A=10^{-5}$, $B=0.05$,
$F_{\mathrm{C1/2}}=$~1~kHz.
The parameters in the Figures were constructed as follows: firstly,
splines were fitted to experimental data, published by {[}Mills 1958{]}
and reproduced by his followers. Splines parameters were allowed to
vary within the 10 \% of their original value and were approximated
by the rounded off values. These procedures were also used to get
parameter values below. For ranges of the audiogram parameters, see
also \cite{Zwislocki1956}. Analogously to Figure \ref{fig:FM1V1},
sound frequencies from 20 to 100 Hz exhibit higher spread between
lower and upper limits, as the fitting method used, quadratic fit,
is the same for both limits.

{Figure \ref{fig:FM3V1} shows how the JND of ITD depends
on timing jitter magnitude $\timijitt$. Figure purpose is to capture,
what is the best JND. There are several time constants, which are
defined in relation to physical properties of spatial sound processing.
To attain to rounded off parameters as in the other figures, we select
individual values of the spike timing jitter and describe their purpose
in the localization precision estimation. }
\textit{Critical}{
timing jitter is lower estimate of timing jitter captured by spike
train of typical mammalian neuron, $T_{\mathrm{JC}}~=~0.2$~ms. }
\textit{Normalized}{
value timing jitter $T_{\mathrm{JN}}~=~1$~ms is the value of timing
jitter normalized in relation to the output JND with respect to average
firing rate. }\textit{Optimal}{ value timing jitter
$T_{\mathrm{JO}}~=~1.66$~ms is result of crossing two fits described
below.}

Simulations show that with lowering timing jitter the circuit
output is virtually more and more precise. Yet, when the jitter is
lower than critical value $T_{\mathrm{JC}}$, determined by intrinsic
noise, duration of coincidence detection window, and by other time
constants, the precision lowers again. The two curves fitted to the
simulation are:
\hfill\break
\noindent
% \paragraph*{1.}
{\bf 1.} fit of exponential function to simulations, red
curve,
%
\begin{equation}
  \Delta t_{\mathrm{JND}}=\exp(A_{1}(\timijitt-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 relation is shown conveniently by the logarithmic y-axis in this
Figure.
\hfill\break
\noindent
%\paragraph*{2.}
{\bf 2.} another fit, which also takes into account shot noise
in lower jitter values, is to a quadratic function, blue curve,
%
\begin{equation}
\Delta t_{\mathrm{JND}}=A_{2}(\timijitt-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
sought by numerical simulation. This is $A$, fitted to data, as the
point $(x,y)=(B_{2},C_{2})$ has been chosen to be a unit. This fit
is the normalized fit of the model.

{Logarithm of the simulated JND lowers with the exponential
curve \eqref{eq:exponential}, which is concave function of sought
jitter $\timijitt$ as the jitter gets lower. The trend towards higher
accuracy diverges from the parabolic fit in equation \eqref{eq:parabolic},
when jitter reaches critical value between $T_{\mathrm{JC}}$ and
$T_{\mathrm{JO}}$.}
%
\begin{equation}
T_{\mathrm{JC}}=0.2\lessapprox\timijitt%\lessapprox0.5
\lessapprox T_{\mathrm{JO}}\lessapprox2\mathrm{ms}.\label{eq:jitters}
\end{equation}
%
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.

This corresponds to the 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 \textit{partially correlated}
input spike trains.

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

In the numerical model of \cite{Sanda2012} we reproduced
a procedure to obtain azimuth tuning curves based on hypotheses, how
mammalian neural circuits work \cite{Grothe2010}. In this procedure,
prior assumption of the existence of the main (dominant) sound frequency
and neural tuning to this frequency was used by experimentalists,
but not by the authors of the numerical only model in (Sanda and Marsalek,
2012). When we use this assumption, which is stronger than in the
numerical only model, in the calculations presented in this paper,
we obtain a fitting curve which is more coherent, it has the higher
vector strength value. Two estimate errors are present in this Figure.
The first is the mismatch between the use of Gaussian (normal) probability
density function, as it is used in some of the experimental literature,
and Sine function. Sine function is simpler circular statistics counterpart
of the Gaussian, where the rigorous circular statistics choice is
the von Mises distribution. More details of the circular statistics
use in the sound localization context are explained in the article
on the ergodicity assumption by \cite{Toth2018}. The difference
between the two functions is shown here as the green curve. The other
error, not shown in Figure, would arise from the not using the prior
assumption as in the numerical only model mentioned above. It appears
that the assumption of the main frequency existence leads to more
precise estimates. This should be, because the assumption adds more
information besides the fitted data.{}

\begin{table}
\begin{centering}
{}%
\begin{tabular}{|l|l|r|c|c|}
\hline 
{Parameter } & {Symbol } & {Units } & {Typical Value } & {Ranges}\tabularnewline
\hline 
\hline 
{Timing Jitter } & {$\timijitt,\sigma$ } & {ms } & {1 } & {0.125 - 8 }\tabularnewline
\hline 
{Window of Coincidence Detection} & {$w_{CD}$} & {ms} & {0.6} & {0.15 - 1.5}\tabularnewline
\hline 
{Sound Frequency } & {$f$ } & {Hz } & {200 } & {40-1600 }\tabularnewline
\hline 
{Shortest Perceptual Time } & {$T_{PT}$ } & {ms } & {20 } & {20 - 80 }\tabularnewline
\hline 
\end{tabular}
\par\end{centering}
{\caption{\label{tab:parameters} The basic set of parameters.}
}
\end{table}

\section{Discussion}

{In this paper we have revisited numerical simulations by
\cite{Sanda2012}. }We have added analytical estimations to the description
of the MSO function, which have not been known previously. Our analytical
calculations make possible to derive time constants useful in description
of normal human hearing. Because of minimalist set of assumptions
about stimuli, we hypothesize that the descriptions are valid also
for hearing with hearing aids and cochlear implants. All Figures contain
model parameters and analytically expressed upper and lower limits
of model transfer functions.

{Figure \ref{fig:FM1V1} contains two lower limit branches
at low frequencies (shown by the blue and green curves, respectively).
The green curve uses an assumption of lower energy and lower contribution
to spike rate in neural units in lower frequencies. Limiting lower
bound by two different analytical functions (branches) can be understood
as the estimate uncertainty. A conservative estimate of the lower
bound always considers the lower of the two branches. This uncertainty
should be recognized as one of original results presented for the
first time in this paper. Its existence has been proposed in a doctoral
thesis by \cite{Bures2014}. To our knowledge this observation has
not yet been published elsewhere. }

{Figure \ref{fig:FM2V1} depicts a quadratic fit. Clearly
the data cannot be captured by the linear curve. The procedure to
obtain the fit is analogous to obtaining other parameters in this
paper. Initially splines were used and then their output was rounded
to arrive to the quadratic fit. This fit is the simplest analytical
way, how to capture nonlinear and band limited span of human hearing
range.}

{Figure \ref{fig:FM3V1} calculations use assumptions about
intrinsic noise \cite{Bures2014}. The simulation data have been obtained
by arbitrary precision calculation. Any neural data recording cannot
reach this precision due to the internal noise of both neurons and
recording electronics. In order to capture circuit noisiness, we have
used both exponential and quadratic fits. When we attempt to use them
as upper and lower bounds, we notice that they exchange their order
in region close to the optimal jitter value. In other words, at the
lower jitter values the two estimates exchange their ordering. This
is the choice of the quadratic fit to obtain a normalized bound together
with other data-points. Numerical simulation with the basic set of
parameters around the x-axis value of $\timijitt$ = 1 ms lies beyond
this point, but close to the exponential fit. }

Figure \ref{fig:FM4V1} contains better fit of the Sine function,
as compared to \cite{Sanda2012}. As the lower bound we can also use
circular normal density (von Mises) function, the difference is negligible,
not shown \cite{Toth2018}. Comparison of time constants and sound
periods in the model presented here will answer a tentative question:
What is the highest slope of the ITD interpolation curve, such that
it gives the resolution of the well known minimum audible angle in
the midline (ITD = 0), which is 4$^{\circ}$ in angular degrees? This
slope is more steep in higher frequency sounds, its maximum is attained
in maximum frequency of the MSO circuit operation, around 1~kHz
\cite{Marsalek2005}.

Towards the analytical descriptions it is important to note
that other periodic functions can be used as the ITD readout curves.
In \cite{Toth2018} we have compared the \textit{Sine function}
with the \textit{circular beta density}, and other
alternative functions. To impose periodic and infinite boundaries
to the problem, regular normal density and circular normal (von Mises)
density have been used and tested in previous versions of our model.
No differences between these densities with proper parameters have
been shown by common statistical tests when testing differences between
two probability densities, ibidem.

\begin{figure}[h!] %[ht]
\begin{centering}
{\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 sound frequency in~Hz in logarithmic
scale and y-axis shows the vector strength. Even though in 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. Red curve shows upper theoretical limit, blue
curve shows lower limit, green curve shows limit imposed by lower
firing rate and lower energy of low frequency sound. Black points
are data simulated with the use of point-process spike train generation
with the use of the dead time Poisson process. Note that in frequency
$f_{\mathrm{S}}$ range from 20 to 200 Hz the lower limit is shown
by curve branching to two branches to the left. The upper is the Boltzmann
function fit and the lower is decrease of vector strength at low frequencies
due to stochastic response of high spontaneous rate neurons.}
\label{fig:FM1V1} }
\end{figure}

\begin{figure}[h!] %[ht]
\begin{centering}
{\includegraphics[width=0.75\textwidth]{../figs/FM2V1} }
\par\end{centering}
\noindent {\caption{ \textbf{The shortest JND of 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 of ITD in $\mu$s. This is a theoretical
prediction based on the analytical model and basic parameter set used
in simulations. As in other figures, black line is obtained by simulation
and red and blue lines are respectively upper and lower bounds obtained
by an analytic fit. }
\label{fig:FM2V1}}
\end{figure}

\begin{figure}[h!] %[ht]
\begin{centering}
{\includegraphics[width=0.75\textwidth]{../figs/FM3V1} }
\par\end{centering}
\noindent {\caption{ \textbf{JND values in basic parameter set in dependence on the spike
timing jitter magnitude.} This plot in semi-logarithmic y-scale shows
JND (just noticeable difference) of interaural time difference depending
on variation of the spike timing jitter. Jagged black line: simulated
data, solid line: an exponential fit to the simulations under the
assumption of arbitrary time precision in the model circuit, dotted
line: a quadratic function estimate of spike timing precision in a
system with addition of noise. Note that in this Figure % curves of
the exponential and quadratic fits cross at $f_{\mathrm{S}}$ = 1.66
ms. In order to correspond to other Figures showing the upper and
lower bounds of the estimate of stochastic model, the two fits are
split into two branches of the same function at this point of $f_{\mathrm{S}}$
= 1.66 ms. For lower x-axis $\protect\timijitt$ values, quadratic
fit is larger than the exponential, and vice versa. This is indicated
by distinctive data-points. (These are circles and triangles; no data-points
and squares, respectively.) Also notice that the curve of the quadratic
fit goes through the point {[}1,1{]}, this is a consequence of using
normalized parameter set. }
\label{fig:FM3V1}}
\end{figure}

\begin{figure}[h!] %[ht]
\begin{centering}
{\includegraphics[width=0.75\textwidth]{../figs/FM4V1} }
\par\end{centering}
\noindent {\caption{ \textbf{Fit of example functions to firing rate slope ITD readout
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 $t_{\mathrm{ITD}}=0$. Red
curve is the Sine density function fit and blue curve is fit by the
normal density function with the variance set to correspond to the
known sound main period. Green curve shows the difference between
the red and blue curves. }
\label{fig:FM4V1}}
\end{figure}

\cite{Franken2014} use recorded spike trains of several
nuclei in the MSO neural circuit to demonstrate that coincidence detection
is an essential part of the neuronal arithmetic executed by the neural
circuit, see also \cite{Bures2012}. These authors show simulations
combined with experimental description of MSO workings in line with
findings of this paper and with coincidence detection theories. Another
MSO model, already studied in 2005 is: \cite{Zhou2005}, this is an
example of simplistic model, motivating the neural circuit description
presented also in this paper. For discussion of neural coding in the
auditory nerve, auditory pathway, cochlear implants and brainstem
neural circuits see \cite{Kerber2012}.

{Our investigation of quantitative properties of the superior
olive neural circuit is also motivated by the three LSO experimental
papers, which have detailed methodology applicable to LSO, to the
overlap of sound frequency ranges between the LSO and the MSO; and
also to the MSO range itself; \cite{Joris1995}; \cite{Joris1996};
\cite{Joris1998_envelope} in experiments on the domestic cat.}

Following his (notorious) paper from 1948, Lloyd A. Jeffress dedicated
lots of efforts to the search of a mechanism, by which microsecond
time scale events of directional sound difference can be transformed
into a code processed and transmitted by action potentials lasting
several microseconds \cite{Jeffress1962}. Historical comments on
Jeffress papers from 1948 and 1962 are summarized by Cariani in
Scholarpedia \cite{Cariani2011}.
A plausible explanation of the microsecond precision of the MSO circuit
can be based on descriptions of the neural computation using leading
edges of action potentials and post-synaptic potentials, \cite{Marsalek2000},
\cite{Toth2018}.

Comparative physiology is useful in showing how the same mechanism
works in other animals and what alternatives are found in the
phylogeny tree. Let us mention just two examples here:
\hfill\break
\noindent
{\bf 1.} \emph{Ormia
ochracea} (fly parasiting on cricket, \emph{Gryllus campestris}) has
the same precision of motor response in ITD processing
as human. The `computational unit' there is chitin lever connecting
two eardrums.
\hfill\break
\noindent
{\bf 2.} Much higher precision demonstrated by behavioral response
is described in bats. \emph{Eptesicus fuscus} (brown bat) catches
tidbit larvae of the meal-worm beetle \emph{Tenebrio molitor}. The
motor precision is in the range of hundreds of nanoseconds, 100~{n}s,
and auditory separation of ultrasound echoes in the bats sonar sense
is in the order of microsecond units,
%
1~\emph{$\mu$}s \cite{Simmons1998}.

In contemporary human, the MSO is the larger of the two nuclei and
contains approximately 10000 - 11000 neurons, while the human LSO
contains 5600 neurons \cite{Moore2000}.
To implement the loudness change is much simpler than to record and
implement microsecond time delay. Therefore in sound generation and
processing, most of current auditory technology works as if the more
important of the two localization cues in \emph{Homo sapiens} were
the sound intensity cue \cite{Vencovsky2016},
even though the evidence is far not definitive.

In \cite{Marsalek2000}, individual steps of signal processing
in the superior olive neural circuits have been investigated. Various
synaptic mechanisms have been proposed \cite{Marsalek2005_Kofranek}.
Spike timing jitter and spike variability have been systematically
analytically investigated by \cite{Kostal2010}.

{In this last paragraph of Discussion section, we should mention
briefly rest of scientific papers, where we found ideas towards the
design of the analytically tractable model presented here. Article
by \cite{Michelet2012} discusses interaural phase delays (IPDs; when
they exist, their utility is equivalent to that of the ITDs) and cochlear
delays. In the case of cochlear delays, very important is to review
the ranges of delays in comparison to sound periods and classically
described excitatory-excitatory and excitatory-inhibitory responses
to binaural inputs in \cite{Joris2006}. Paper of Srinivasan, Laback
and Majdak cites current progress of ITD encoding by binaural cochlear
implants, this is important for model validations and applications
to studies with hearing aids and electrical hearing, \cite{Srinivasan2018}.}

\section*{Conclusions}

{This theoretical paper is continuation of sound localization
precision descriptions in the MSO \cite{Sanda2012} and in the LSO
\cite{Bures2013}. Major novel results here are two:
1) analytical estimates of results obtained previously only by
numerical simulation and 2) estimates of auditory parameters and
functions bounding from the bottom and from the top known
characteristics of the human sound localization circuit. }

\section*{Conflict of Interest Statement}
%All financial, commercial or other relationships that might be perceived by the academic community as representing a potential conflict of interest must be disclosed. If no such relationship exists, authors will be asked to confirm the following statement: 

The authors declare that the research was conducted in the absence of
any commercial or financial relationships that could be construed as a
potential conflict of interest.

\section*{Author Contributions}

Model design: PM, ZB. Methodology: PM, PS. Writing original draft: PM.
Writing and editing: PM, PS, ZB.

\section*{Funding}
This project was in part funded by Charles University graduate
students research program, acronym S{V}V, No. 260 519/ 2020-2022,
to Petr Marsalek.

\section*{Acknowledgments}

We acknowledge uploading this manuscript first to
ArXiv repository prior to impacted journal submission.
Special thanks to Marek Hajny for reading and commenting on the
manuscript.

\section*{Data Availability Statement}

The datasets and source codes of programs used in this study can be
found when requested at the web pages of PM, see link
%
\texttt{http://patf-onyx.lf1.cuni.cz/mlab/ftp/}.

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