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\large
%\title
{{\bf On the precision of neural computation with interaural time \hfil\break
differences in the medial superior olive}} 

\def\aPathologic{a}
\def\bRadio{b}
\def\eContact{e}
\def\cRobotics{c}
\def\dPoly{d}

{\small
%\author[Pathologic,Radio,Contact]
{Petr Marsalek}$^{\aPathologic,\bRadio,\eContact}$,
%\author[Pathologic,Contact]
{Marek Hajny}$^{\aPathologic,\eContact}$,
%\author[Robotics,Poly]
{Zbynek Bures}$^{\cRobotics,\dPoly}$

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

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

%\address
$^{\cRobotics}${Czech Institute of Informatics, Robotics and Cybernetics,
 Czech Technical University in Prague,
 {Jugosl\'{a}v{s}k\'y{c}h partyz\'an\r{u}~4/1903}, 166~36, Praha~6,
 Czech Republic}
% \accent23u

%\address
$^{\dPoly}${College of Polytechnics,
 {T{o}l{s}t\'e{h}o}~16/1556, 586~01, Jihlava, Czech Republic}

%\address
$^{\eContact}${Corresponding authors: Petr.Marsalek@LF1.CUni.CZ,
 Marek.Hajny@LF1.CUni.CZ}
}

%\begin{keyword}
\emph{Keywords:}
\hrule
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.
\hrule
%\end{keyword}
%\maketitle

\subsubsection*{Abstract}

{\small \textbf{Abstract will be rewritten after the manuscript
completion. Paper should explain analytical MSO model, all figures
and tables.}}

{\small
We study a model of mammalian sound source localization in horizontal
plane. (Experiments on small rodents indicate that some 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.
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.
Predictions of this model have straightforward
applications in testing and designing stimulation protocols used in
cochlear implants.
} %E O small

\subsection*{Abbreviations and symbols}

$F$, $F_{\mathrm{S}}$,~sound frequency;
$F_{\mathrm{C}}$,~critical sound frequency value;
$\varphi$,~sound phase;
ILD,~interaural level difference;
ITD,~interaural time difference;
{ISI},~inter-spike interval; 
JND,~just noticeable difference;
{LSO},~lateral superior olive;
$K_{\mathrm{C}}$, $K_S$, $A$, $B$, $C$,~proportionality constants; 
$l$,~sound level;
{MSO},~medial superior olive;
$r(V)$, $r_V$,~VS,~vector strength;
$R_F$,~firing rate;
$\sigma$,~standard deviation;
$T_J$,~timing jitter;
$t$,~time;
$T$,~$T_{\mathrm{S}}$, sound period;

\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 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 maximum spike timing precision
attained by the following two respective modalities (low and high
frequency sound localization) of natural and electrical hearing.

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}.

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, {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. 
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. \cite{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 frequencies should be
different.

In extant human, the MSO is the larger of the two nuclei and contains
approximately 10000 - 11000 neurons and the human LSO contains 5600
neurons \cite{Moore2000}.  As loudnes is implemented simpler that
microsecond time delay, most of currend auditory technology works as
if the more important of the two nuclei is the LSO in
%
\textit{Homo sapiens}.

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.

\comment{(
{\tt\bf Note:}
First look at the data enables 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.
Table \verb|~\ref{tab:speed-up}| has been deleted.
)}

The range of sound frequencies
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}

\subsection{Preliminaries}

Our procedures are mostly in using computational models, experimental
data and statistical methods. Here we list several assumptions used in
the paper.
In any model of ITD based azimuth discrimination, time differences
have to be computed with high accuracy (in the range of tens of
microseconds).

\begin{inparaenum}[(1)]
\item For single MSO neuron constants, see \cite{Toth2015}.
Membrane time constants are lower than 1~$\mu$s.
\item For the synaptic machinery, see \cite{Toth2015}.
\item For the ergodic assumption see \cite{Sanda2012}, \cite{Toth2018}.
\end{inparaenum}
The basic set of parameters is 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, 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 (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 these neurons is the azimuth signal
encoded in a spike train. Basic set of parameters is shown in
Table~\ref{tab:parameters}.

\subsection{ITD readout curve}

Let us have
a monotonous function with firing rate as an input, which outputs
azimuth. We will call it the ITD readout curve. In the paper by
\cite{Sanda2012} this curve is constructed by curve fitting to
simulated points. Here we construct the curve based on assumption
that the main 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 is defined as follows, by
\cite{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.)
(Discrete sum) vector strength of sample
\( \varphi_1, \dots, \varphi_N \) is defined as

\begin{equation}
r_{\mathrm{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{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 rate code JND
(in other words, the precision of rate coding)
using theoretical tools. 

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 equal standard
deviations $\sigma$. 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 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 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.
\label{R_SDT2}
\end{equation}

\section{Results}

We investigate how circuit output and overall performance depend on
sound frequency and sound intensity.
All Figures show sample data of numerical simulation as black line,
lower estimate of data as blue line and upper estimate of data as red
line.
Figure \ref{fig:FM1V1} shows an example of typical vector strength
(which cannot be larger than 1) lowering towards higher frequencies,
as it can be observed in the numerical module of the auditory
periphery (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, simulated data
({\tt\bf Note:} recorded by \cite{Joris1996} from the MSO in 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 $r_{\mathrm{V}}$ in dependence on sound
frequency $F_\mathrm{S}$ is:

\begin{equation}
r_{\mathrm{V}} = (1 + \exp(K_{\mathrm{S}} F_{\mathrm{S}} -
K_{\mathrm{C}} F_{\mathrm{C1/2}} ))^{-1},
\end{equation} %(4)
%
where $K_{\mathrm{S}}$ (sound coefficient) and $K_{\mathrm{C}}$
(critical coefficient), $F_{\mathrm{C1/2}}$ (critical half frequency)
are proportionality constants.

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}

where upper and lower bound parameters $A$ and $B$ are fitted
constants and the other parameters and variables are as above. See
also \cite{Zwislocki1956}.

Figure \ref{fig:FM3V1} shows how the
% just noticeable difference
(JND) of ITD depends on timing jitter $\tau_{\mathrm{J}}$.
There is one
%
{\it critical} $\tau_{\mathrm{JC}}~=~0.2$ and one {\it optimal} value
(normalized to $\tau_{\mathrm{JO}}~=~1$) 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.
%
Yet, when the jitter is lower than critical value $\tau_{\mathrm{JC}}$,
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:

1. fit of exponential function to simulations (red curve):
\begin{equation}
\Delta t_{\mathrm{JND}}= \exp(A_1(\tau_{\mathrm{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 (blue curve):

\begin{equation}
\Delta t_{\mathrm{JND}} = A_2(\tau_{\mathrm{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.
({\tt\bf Note:} this must be explained
- that this fit is after normalization!).

Logarithm of the simulated JND lowers with the exponential curve
\eqref{eq:exponential}, which is concave function of jitter
$\tau_{\mathrm{J}}$ as the jitter gets lower. The trend towards higher
accuracy diverges from the parabolic fit \eqref{eq:parabolic},
when jitter reaches critical value between
%
$\tau_{\mathrm{JC}}$ and $\tau_{\mathrm{JO}}$.

\begin{equation}
\tau_{\mathrm{JC}}=0.2\lessapprox
\tau_{\mathrm{J}}\lessapprox 0.5~\lessapprox
\tau_{\mathrm{JO}}=1~\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.

({\tt\bf Note:} The other (optimal) jitter value notable in this
Figure~\ref{fig:FM3V1} is firing rate in dependency on the jitter
values.)

({\tt\bf Note:} Check this.)
This is an analytical dependency obtained in \cite{Salinas2000}
% [Salinas and Sejnowski, J Neuroscience, 2000]
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 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.

\begin{table}
\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. 
({\tt\bf Note:} the table should be here, in analogy to the tables in \cite{Sanda2012}.)
}
\end{table}

\section{Discussion}

\subsection{Things to do}
(First/ we may skip this, since this adds no explanatory value to the text:
Reproduce the curve in panel with sound frequency on x-axis and
JND on the y-axis in Fig. \ref{fig:FM2V1}.)
\par Second: Present the curve from the right panel of Fig.4 from Sanda P.,
Jitter Effect on the Performance..., 
as part of Fig. \ref{fig:FM3V1}.
\par
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) in the ITD = 0 direction?
(This we will do later: We should also discuss the relation of the MSO
neural circuit to its neighboring nucleus, the LSO, and its output.)

Other periodic functions can be used as the ITD readout curves. In
\cite{Toth2018} we have compared the Sine function with the circular
Beta density, and as other alternatives, regular Normal density and
circular Normal 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 testing differences between two
probability densities, ibidem.

({\tt\bf Note:}
Below are suggestions what papers to cite in Discussion. )

Cite: \cite{Mlynarski2014}, this is a paper about statistics of
natural environmental sounds in relation to low frequency sound
localization...

Cite: \cite{Bures2019}, this is a paper submitted this spring to
Biosystems, about encoding of low frequency sounds...

\begin{figure}[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 a 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 and blue curve shows lower
limit. Black points are data simulated with the use of
point-process-spikes train generation with the use of the dead time
Poisson process.}
\label{fig:FM1V1} 
\end{figure}

({\tt\bf Note:} Should we have here a schema showing 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}
\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 simulations.
% figure 2
}
\label{fig:FM2V1} 
\end{figure}

\begin{figure}[ht]
\begin{centering}
\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,
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}
\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 sine
function fir and blue curve is fit by the Normal density (Gauss
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 \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 \cite{Smith1991} and \cite{Smith1993}
(papers from the Joris group) are relevant.

The ergodic paper of \cite{Toth2018} is already cited, but in other
context it may also go to Discussion...

Another MSO model, already studied in 2005 is: \cite{Zhou2005}, this
is a great example of simplistic model, I (Petr) am much inspired by
this model...

Cochlear implants and MSO, see \cite{Kerber2012}...

(No)Cite? Check \cite{Wu1992}: I thought we should cite this because
of high precision of some synapses, but maybe this is not necessary...

(No)Cite? Here are the three LSO experimental papers, which have detailed
methodology applicable to overlap of sound frequency ranges with the MSO;
maybe none of these is needed...
\cite{Joris1995};
\cite{Joris1996};
\cite{Joris1998_envelope}.

Cite \cite{Jeffress1962} should go here instead of Jeffress, 1948, which
is a bit too notorious...

(No)Cite? \cite{Bures2012} is neuronal arithmetic in {Biological}
{Cybernetics}.

(No)Cite? \cite{Bures2013} is ``dual/ LSO'' paper in Brain
{Research}

(No)Cite? Do we still need to cite this review \cite{Grothe2010}?...

(No)Cite? There are the classics: \cite{Goldberg1969}...

Cite: Cochlear delays, very important: \cite{Joris2006}.

Yet another neuronal arithmetic: \cite{Koutsou2012}.

(No)Cite? This is about other dimension; if we cite this, then we must
discuss all the three, \cite{Kopco2012}. Shall we do that?

(No)Cite? Jitter and variability by \cite{Kostal2010}...

(No)Cite? \cite{Krips2009}...

(No)Cite? \cite{Laback2008}...

(No)Cite? This is a chronological record of the neuronal arithmetic ideas,
starting from:
\cite{Marsalek2000},
through
\cite{Marsalek2001}
through
\cite{Marsalek2004} 
to the
\cite{Marsalek2005_Kofranek};
watch the years ...

No(Cite)? Some more detailed account of the diversity of cell
functions and shapes: \cite{Reed2002}...

Next possible citations: \cite{Cariani2011}, \cite{Bouse2019},
\cite{Bures2014},
\cite{Michelet2012}, \cite{Srinivasan2018}.

({\tt\bf Note:}
Other notes: Figure 4 should correct the omission in the
paper by Sanda and Marsalek. When we cover everything else, we should
add this: 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.

\subsubsection*{Other Notes}
Things to do to co-authors.

Petr Marsalek. Clean up and finish first pass writing.

Marek Hajny is joining the author team as: middle position author with
tasks: 1) to compile Discussion and write and 2) to rewrite parts of
the manuscript.

Zbynek Bures. Should do final reading of the manuscript before
submission to the journal.

\subsection*{Acknowledgments}

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

\subsection*{Compilation notes}

\noindent\LaTeX\ compilation date is: \today.

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