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\journal{Brain Research} 

\makeatother

\long\def\comment#1{}

\begin{document}

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

\author[IPath,FBMI]{Petr Marsalek\corref{CorA}}
\ead{Petr.Marsalek@lf1.cuni.cz}
\author[Poly,IEM]{Zbynek Bures}
% \author[IPhys]{Pavel Sanda}
% references: {../sound_loc}

\address[IPath]{Institute of Pathological Physiology, First Medical
  Faculty, Charles University of Prague, U~Nemocnice 5, 128~53, Praha
  2, Czech Republic}

\address[FBMI]{Faculty of Biomedical Engineering, Czech Technical
  University of Prague, Nam. Sitna 3105, 272~01, Kladno, Czech
  Republic}

\address[Poly]{College of Polytechnics, Tolsteho 16, 586 01,
  Jihlava, Czech Republic}

\address[IEM]{Institute of Experimental Medicine, Academy of
  Sciences of the Czech Republic, Videnska 1083, 142 20, Praha 4,
  Czech Republic}

%\address[IPhys]{Institute of Physiology, Academy of Sciences of the
%  Czech Republic, Videnska 1083, 142~20, Praha 4, Czech Republic }

\cortext[CorA]{Corresponding author. Telephone +420 721 864 385}

\begin{abstract}
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}
\begin{keyword}
binaural hearing, coincidence detection, ergodic hypothesis, ideal
observer, interaural time difference, just noticeable difference,
medial superior olive
\end{keyword}
\maketitle

\subsection*{Abbreviations and symbols}

We use following notation, conforming with \citet{Bures2013} and
\citet{Sanda2012}.

\global\long\def\srom#1{_{{\rm #1}}}
 % my TeX definitions...
CV, coefficient of variation;
$f$, $f\srom S$, \textit{sound frequency, in the LSO paper - other meaning};
ILD, interaural level difference;
%IPD - interaural phase difference,
ITD, interaural time difference;
ISI, inter-spike interval; 
JND, just noticeable difference;
LSO, lateral superior olive;
MSO, medial superior olive;
$r$, sound level;
$R$, firing rate; 
$\sigma$, standard deviation/ timing jitter;
$t$, time;
$T$, sound period. %$T\srom{S}$

\section{Introduction}

Most of the Introduction section is to be written later.

Outline:\begin{enumerate}

\item Content and results of this paper = analytical investigations of
  our model we explored before mostly by simulation.

\item Relation to data

\item Relation of MSO and LSO

\item alternative theories of LSO/ MSO functioning

\end{enumerate}

In \citet{Sanda2012} we presented mostly 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.

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

%The neural circuit in the MSO 

The neural computation implemented by the MSO neural circuit is not
entirely described to date.

The range of sound frequencies \cite{Marsalek2004, Grothe2010}
processed in the MSO circuit is limited to the low frequency band. In
man, 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, determination of sound azimuth is
below the average taken through all these low frequencies. 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}.

\subsection{Things to do}

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:FMJND}.

Second: Present the curve from the right panel of Fig.4 from Sanda P.,
Jitter Effect on the Performance..., 
% EJBI - 1:1, 2011.
as part of Fig. \ref{fig:FM3V1}. %{fig:FMJITT}.

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)?

\section{Results}

We investigate the dependence of the circuit output on sound frequency
and sound intensity.

The degree of synchronization is measured by the vector strength
measure, \eqref{eq:M-VS}.

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 $r$ is:
%
\par\noindent
$r = (1 + \exp(k\srom S f\srom S - k \srom C f\srom C))^{-1},$
\par\noindent
%
where $k\srom S$ and $k\srom C$ are parameters, proportionality
constants, $f\srom S$ is sound frequency and $f\srom C$ is another
parameter, critical half frequency.

Figure \ref{fig:FM2V1} shows the best 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:
%
\par\noindent
$\Delta t\srom{JND} = A (f\srom S - f\srom C)^2 + B,$
\par\noindent
%
where $A$ and $B$ are fitted constants and the other parameters and
variables are as above.

Figure \ref{fig:FM3V1} shows existence of both a critical and an
optimal values of timing jitter in respect to the attaining of the JND
and output firing rate. Simulations show that with lowering
timing jitter the circuit output is virtually more and more precise.
%
The curve fits are:

1. fit of exponential function to simulations:
\par\noindent
$\Delta t\srom{JND}= \exp(A(\tau\srom{J}-B)) - C,$
\par\noindent
%
where $A~=~1.9$, $B~=~1.25$ and $C~=~0.2$ are fitted parameters. 
\par\noindent
%
2. Another fit, together with noise, to a quadratic function:
%
\par\noindent
$\Delta t\srom{JND} = A(\tau\srom{J}-B)^2 + C,$
\par\noindent
%
where $A~=~2.5$, $B~=~1$, $C~=~1$.
\par\noindent
%
Logarithm of the JND lowers with a curve close to the square root of
the jitter. This trend ends up at the critical value of jitter:
$\tau\srom{J}\approx~0.5~ms$. Beyond that point towards the lower
jitter values, the neural circuit cannot function properly, as a 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 a firing rate
in dependency on the jitter values. 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 partially correlated 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}
\begin{centering}
\begin{tabular}{|c|c|}
\hline 
Parameter & Value\tabularnewline
\hline 
\hline 
$T_{J}=\sigma$ & 1 ms\tabularnewline
\hline 
$w_{CD}$ & 600 $\mu$s\tabularnewline
\hline 
$f$ & 40-1600 Hz\tabularnewline
\hline 
 & \tabularnewline
\hline 
\end{tabular}
\par\end{centering}

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

%%%%

\begin{table}
\begin{centering}
\begin{tabular}{| l| r| c | c | c |}
\hline 
Parameter & units & values &  &  \cr
\hline 
\hline 
sound frequency & Hz & 140 & 700 & 1400 \cr
\hline 
window size & $\mu$s & 600 & 120 & 60 \cr
\hline 
jitter & ms & 1 & 0.2 & 0.1 \cr
\hline 
JND/ time $^{*}$ & $\mu$s/ ms & 10/ 650 & 2/ 130 & 1/ 65 \cr
\hline 
\end{tabular}
\par\end{centering}

\caption{This Table is direct scaling of parameters used in
  \citet{Sanda2012}.}
%
$^{*}$ JND/ time is the just noticeable
difference of the ITD expressed as a ratio of the ITD over the time of
observation by an ideal observer reading the information from one
neuron.
\label{tab:frequency}
\end{table}

%%%%%%%%%%%

\section{Discussion}


\begin{figure}[ht]
\begin{centering}
\includegraphics{figs/FM1V1}
\par\end{centering}
\noindent
\caption{ 
%
\textbf{Vector strength of AN 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.}
%
\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
}
%
%%%
\comment{
\begin{figure}[ht]
\begin{centering}
\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}
\includegraphics{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.
%
}
\label{fig:FM2V1} 
\end{figure}

%%%

\begin{figure}[ht]
\begin{centering}
\includegraphics[scale=0.8]{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
realistic noise.
%
}
\label{fig:FM3V1} 
\end{figure}


\begin{figure}[ht]
\begin{centering}
\includegraphics[scale=0.8]{figs/FM4V1}
\par\end{centering}
\noindent
\caption{
\textbf{Gaussian fit of the ITD to firing rate sloping 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}

\section{Experimental Procedures}

The vector strength is defined as follows:
\cite{Goldberg1969}
%
Let us have sample (spike) phases 
%
$\varphi_i$, $i=1, 2, \dots, N$ 
%
relative to phases of a given master periodic function, which can be
for example sound stimulus eliciting the spike train as a response.
%
(Discrete) vector strength of sample \( \varphi_1, \dots, \varphi_N \)
is defined as
\begin{equation}
\rho(\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 \citep{} (Green 1966, T{e{i}c}h 1979, 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}

% below are several no-cite items
%%% \input{arch/kap3B}
% \input{arch/kap4}

\nocite{Grothe2010}

\nocite{Goldberg1969}
\nocite{Jeffress1948}

\nocite{Joris2006}

% \nocite us:

% Author(s): Koutsou, Achilleas; Christodoulou, Chris; Bugmann, Guido;
% and others
% Source: NEURAL COMPUTATION  Volume: 24   Issue: 9   Pages: 2318-2345 

\nocite{Kopco2012}

% \nocite{Szalisznyo2006} %--> citation belongs to the LSO paper!
\nocite{Szalisznyo2004}

% Citations of our materials:
% To do: include Sanda - 2012 - E{J}B{I}

\nocite{Sanda2011}
\nocite{Salinas2000}

% \nocite{Joris1996}
\nocite{Smith1993}

\nocite{Bures2012} 
\nocite{Bures2013} % submitted to Brain Research 
\nocite{Sanda2012}

\nocite{Marsalek2000, Reed2002, Marsalek2004, Marsalek2005_Kofranek}

\subsection*{Acknowledgments}

Supported by projects M00176 ``Co-operation in biomedicine and
electronics'' and ``Support of individual careers of prospective
academic workers'' to Z.B. at the College of Polytechnics Jihlava,
%
by graduate research program no. 264507/ 2012 to P.M. at the Charles
University 
%
and by the Czech Ministry of Health program N{T} 11532 - Medical
Technology Assessment to P.M. at the Czech Technical University.
%
Thanks to Pavel Sanda for help with data
%
and to
%
X.Y.
%
for English language copy-editing.

%%% \newpage

\section*{References}

\bibliographystyle{elsarticle-harv}
\bibliography{../sound_loc}

\end{document}
