diff --git a/cite.bib b/cite.bib
index 13dcdef..bfc05a6 100644
--- a/cite.bib
+++ b/cite.bib
@@ -133,7 +133,8 @@
volume={29},
pages={2626--2636},
year={2009},
-}
+}# Cited
+
@article{hildebrandt2015neural,
title={A neural mechanism for time-window separation resolves ambiguity of adaptive coding},
author={Hildebrandt, K Jannis and Ronacher, Bernhard and Hennig, R Matthias and Benda, Jan},
diff --git a/main.aux b/main.aux
index 97d0c19..9ab6ae2 100644
--- a/main.aux
+++ b/main.aux
@@ -22,10 +22,6 @@
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diff --git a/main.bbl b/main.bbl
index 8e8148b..8eb6fbe 100644
--- a/main.bbl
+++ b/main.bbl
@@ -20,7 +20,7 @@
\refsection{0}
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\entry{clemens2013computational}{article}{}
- \name{author}{2}{}{%
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{{un=0,uniquepart=base,hash=a2d5d9f296e8da1c1447cc2d86f121ea}{%
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\entry{clemens2013feature}{article}{}
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family={Clemens},
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\field{pages}{10434\bibrangedash 10448}
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+ {{un=0,uniquepart=base,hash=f0049b5905ec2af7caf4cdf6b716a30a}{%
+ family={Benda},
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+ {{un=0,uniquepart=base,hash=1215182dcee84bc670d002887895824e}{%
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+ \field{labeltitlesource}{title}
+ \field{journaltitle}{J Neurosci}
+ \field{title}{The origin of adaptation in the auditory pathway of locusts is specific to cell type and function}
+ \field{volume}{29}
+ \field{year}{2009}
+ \field{pages}{2626\bibrangedash 2636}
+ \range{pages}{11}
+ \endentry
\entry{machens2001discrimination}{article}{}
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diff --git a/main.bbl-SAVE-ERROR b/main.bbl-SAVE-ERROR
index 66c316f..8e8148b 100644
--- a/main.bbl-SAVE-ERROR
+++ b/main.bbl-SAVE-ERROR
@@ -139,6 +139,52 @@
\field{pages}{12136\bibrangedash 12145}
\range{pages}{10}
\endentry
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+ \field{journaltitle}{J Comp Physiol A}
+ \field{title}{Intensity invariance properties of auditory neurons compared to the statistics of relevant natural signals in grasshoppers}
+ \field{volume}{196}
+ \field{year}{2010}
+ \field{pages}{285\bibrangedash 297}
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diff --git a/main.bcf b/main.bcf
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suga1960peripheral
gollisch2002energy
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clemens2010intensity
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+ fisch2012channel
clemens2011efficient
+ clemens2011efficient
diff --git a/main.blg b/main.blg
new file mode 100644
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diff --git a/main.fdb_latexmk b/main.fdb_latexmk
index 790eb06..f9a5fe5 100644
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@@ -610,7 +612,7 @@ File: epstopdf-sys.cfg 2010/07/13 v1.3 Configuration of (r)epstopdf for TeX Live
)) (/usr/share/texlive/texmf-dist/tex/latex/translations/translations-basic-dictionary-english.trsl
File: translations-basic-dictionary-english.trsl (english translation file `translations-basic-dictionary')
)
-Package translations Info: loading dictionary `translations-basic-dictionary' for `english'. on input line 26.
+Package translations Info: loading dictionary `translations-basic-dictionary' for `english'. on input line 32.
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Package biblatex Info: ... file 'english.lbx' found.
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@@ -624,13 +626,13 @@ Package biblatex Info: Automatic encoding selection.
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(./main.bbl)
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@@ -642,16 +644,21 @@ File: figures/fig_auditory_pathway.pdf Graphic file (type pdf)
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+Overfull \hbox (81.90907pt too wide) in paragraph at lines 178--187
+\OT1/cmr/m/n/12 terneu-rons (Fig.1[]b). Both the lo-cal in-terneu-rons ([]; [])
+ []
+
+
+Overfull \hbox (13.04964pt too wide) in paragraph at lines 192--194
[]\OT1/cmr/m/n/12 Similar re-sponse/filter prop-er-ties within re-cep-tor/interneuron pop-u-la-tions ([]Clemens, Kutzki,
[]
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[]\OT1/cmr/bx/n/17.28 Two mech-a-nisms driv-ing the emer-gence of intensity-
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diff --git a/main.pdf b/main.pdf
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diff --git a/main.tex b/main.tex
index 1a0f35a..0f32f7c 100644
--- a/main.tex
+++ b/main.tex
@@ -1,6 +1,7 @@
\documentclass[a4paper, 12pt]{article}
\usepackage[left=2.5cm,right=2.5cm,top=2cm,bottom=2cm,includeheadfoot]{geometry}
+\usepackage[onehalfspacing]{setspace}
\usepackage{graphicx}
\usepackage{svg}
\usepackage{import}
@@ -11,11 +12,16 @@
\usepackage{amssymb}
\usepackage[separate-uncertainty=true, locale=DE]{siunitx}
\sisetup{output-exponent-marker=\ensuremath{\mathrm{e}}}
+% \usepackage[capitalize]{cleveref}
+% \crefname{figure}{Fig.}{Figs.}
+% \crefname{equation}{Eq.}{Eqs.}
+% \creflabelformat{equation}{#2#1#3}
\usepackage[
backend=biber,
style=authoryear,
- mincitenames=1,
- maxcitenames=2
+ pluralothers=true,
+ maxcitenames=1,
+ mincitenames=1
]{biblatex}
\addbibresource{cite.bib}
@@ -26,11 +32,26 @@
\begin{document}
\maketitle{}
+% Text references and citations:
+\newcommand{\bcite}[1]{\mbox{\cite{#1}}}
+% \newcommand{\fref}[1]{\mbox{\cref{#1}}}
+% \newcommand{\fref}[1]{\mbox{Fig.\,\ref{#1}}}
+% \newcommand{\eref}[1]{\mbox{\cref{#1}}}
+% \newcommand{\eref}[1]{\mbox{Eq.\,\ref{#1}}}
+
+% Math shorthands - Standard symbols:
+\newcommand{\dec}{\log_{10}} % Logarithm base 10
+\newcommand{\infint}{\int_{-\infty}^{+\infty}} % Indefinite integral
+
+% Math shorthands - Spectral filtering:
\newcommand{\bp}{h_{\text{BP}}(t)} % Bandpass filter function
\newcommand{\lp}{h_{\text{LP}}(t)} % Lowpass filter function
\newcommand{\hp}{h_{\text{HP}}(t)} % Highpass filter function
\newcommand{\fc}{f_{\text{cut}}} % Filter cutoff frequency
+\newcommand{\tlp}{T_{\text{LP}}} % Lowpass filter averaging interval
+\newcommand{\thp}{T_{\text{HP}}} % Highpass filter adaptation interval
+% Math shorthands - Early representations:
\newcommand{\raw}{x} % Placeholder input signal
\newcommand{\filt}{\raw_{\text{filt}}} % Bandpass-filtered signal
\newcommand{\env}{\raw_{\text{env}}} % Signal envelope
@@ -38,18 +59,18 @@
\newcommand{\dbref}{\raw_{\text{ref}}} % Decibel reference intensity
\newcommand{\adapt}{\raw_{\text{adapt}}} % Adapted signal
-\newcommand{\dec}{\log_{10}} % Logarithm base 10
-\newcommand{\svar}{\sigma_{\text{s}}^{2}} % Song signal variance
-\newcommand{\nvar}{\sigma_{\eta}^{2}} % Noise signal variance
-\newcommand{\infint}{\int_{-\infty}^{+\infty}} % Indefinite integral
+% Math shorthands - Kernel parameters:
+\newcommand{\ks}{\sigma_i} % Gabor kernel width
+\newcommand{\kf}{f_i} % Gabor kernel frequency
+\newcommand{\kp}{\phi_i} % Gabor kernel phase
+
+% Math shorthands - Threshold nonlinearity:
\newcommand{\thr}{\Theta_i} % Step function threshold value
\newcommand{\nl}{H(c_i\,-\,\thr)} % Shifted Heaviside step function
-\newcommand{\bi}{b_{i,\Theta}} % Single threshold-constrained binary response
-\newcommand{\feat}{f_{i,\Theta}} % Single threshold-constrained feature
-
-\newcommand{\thp}{T_{\text{HP}}} % Highpass filter adaptation interval
-\newcommand{\tlp}{T_{\text{LP}}} % Lowpass filter averaging interval
+% Math shorthands - Minor symbols and helpers:
+\newcommand{\svar}{\sigma_{\text{s}}^{2}} % Song signal variance
+\newcommand{\nvar}{\sigma_{\eta}^{2}} % Noise signal variance
\newcommand{\pc}{p(c_i,\,T)} % Probability density (general interval)
\newcommand{\pclp}{p(c_i,\,\tlp)} % Probability density (lowpass interval)
@@ -126,42 +147,43 @@ $\rightarrow$ More general, simpler, unfitted formalized Gabor filter bank
\subsection{Population-driven signal pre-processing}
Grasshoppers receive airborne sound waves by a tympanal organ at each side of
-the thorax~(Fig.\,\ref{fig:pathway}a). The tympanal membrane acts as a mechanical resonance filter:
-Vibrations of specific frequencies are focused on different membrane areas,
-while other frequencies are attenuated~(\mbox{\cite{michelsen1971frequency}};
-\mbox{\cite{windmill2008time}}; \mbox{\cite{malkin2014energy}}). This
-processing step can be approximated by an initial bandpass filter
+the thorax~(Fig.\,\ref{fig:pathway}a). The tympanal membrane acts as a
+mechanical resonance filter, that focuses vibrations of specific frequencies on
+different membrane areas while attenuating
+others~(\bcite{michelsen1971frequency}; \bcite{windmill2008time};
+\bcite{malkin2014energy}). This processing step can be approximated by an
+initial bandpass filter
\begin{equation}
\filt(t)\,=\,\raw(t)\,*\,\bp, \qquad \fc\,=\,5\,\text{kHz},\,30\,\text{kHz}
\label{eq:bandpass}
\end{equation}
applied to the acoustic input signal $\raw(t)$. The auditory receptor neurons
-connect directly to the tympanal membrane and transduce mechanical vibrations
-into electro-chemical potentials. The receptor population is substrate to
-several known signal processing steps. First, the receptors extract
-the signal envelope~(\mbox{\cite{machens2001discrimination}}), which likely
-involves a rectifying nonlinearity~(\mbox{\cite{machens2001representation}}).
-This can be modelled as full-wave rectification followed by lowpass filtering
+connect directly to the tympanal membrane. Besides performing the
+mechano-electrical transduction, the receptor population further is substrate
+to several known processing steps. First, the receptors extract the signal
+envelope~(\bcite{machens2001discrimination}), which likely involves a
+rectifying nonlinearity~(\bcite{machens2001representation}). This can be
+modelled as full-wave rectification followed by lowpass filtering
\begin{equation}
\env(t)\,=\,|\filt(t)|\,*\,\lp, \qquad \fc\,=\,500\,\text{Hz}
\label{eq:env}
\end{equation}
of the tympanal signal $\filt(t)$. Furthermore, the receptors exhibit a
sigmoidal response curve over logarithmically compressed intensity
-levels~(\mbox{\cite{suga1960peripheral}}; \mbox{\cite{gollisch2002energy}}). In
-the model, logarithmic compression is achieved by conversion to decibel scale
+levels~(\bcite{suga1960peripheral}; \bcite{gollisch2002energy}). In the model,
+logarithmic compression is achieved by conversion to decibel scale
\begin{equation}
\db(t)\,=\,10\,\cdot\,\dec \frac{\env(t)}{\dbref}, \qquad \dbref\,=\,\max[\env(t)]
\label{eq:log}
\end{equation}
relative to the maximum intensity $\dbref$ of the signal envelope $\env(t)$.
-Next, the axons of the receptor neurons project into the metathoracic ganglion,
-where they synapse onto local interneurons~(Fig.\,\ref{fig:pathway}b). Both the
-auditory receptors~(\mbox{\cite{fisch2012channel}}) and the subsequent
-interneurons~(\mbox{\cite{clemens2010intensity}}) display spike-frequency
-adaptation.
-
-
+The axons of the receptor neurons project into the metathoracic ganglion, where
+they synapse onto local interneurons~(Fig.\,\ref{fig:pathway}b). Both the local
+interneurons~(\bcite{hildebrandt2009origin}; \bcite{clemens2010intensity}) and,
+to a lesser extent, the receptors themselves~(\bcite{fisch2012channel}) display
+spike-frequency adaptation in response to sustained stimulation.
+This behavior is crucial to render subsequent signal representations invariant
+to variations in sound intensity.
"Pre-split portion" of the auditory pathway:\\
@@ -205,10 +227,10 @@ $\rightarrow$ Individual neuron-specific response traces from this stage onwards
Template matching by individual ANs\\
- Filter base (STA approximations): Set of Gabor kernels\\
-- Gabor parameters: $\sigma, \phi, f$ $\rightarrow$ Determines kernel sign and lobe number
+- Gabor parameters: $\ks, \kp, \kf$ $\rightarrow$ Determines kernel sign and lobe number
%
\begin{equation}
- k(t)\,=\,e^{-\frac{t^{2}}{2\sigma^{2}}}\,\cdot\,\sin(2\pi f t\,+\,\phi)
+ k_i(t,\,\ks,\,\kf,\,\kp)\,=\,e^{-\frac{t^{2}}{2{\ks}^{2}}}\,\cdot\,\sin(2\pi\kf\,\cdot\,t\,+\,\phi_i)
\label{eq:gabor}
\end{equation}
%
@@ -225,7 +247,7 @@ Thresholding nonlinearity in ascending neurons (or further downstream)\\
$\rightarrow$ Shifted Heaviside step-function $\nl$ (or steep sigmoid threshold?)
%
\begin{equation}
- \bi(t)\,=\,\begin{cases}
+ b_i(t,\,\thr)\,=\,\begin{cases}
\;1, \quad c_i(t)\,>\,\thr\\
\;0, \quad c_i(t)\,\leq\,\thr
\end{cases}
@@ -239,7 +261,7 @@ of feature values $\rightarrow$ Clusters in high-dimensional feature space\\
$\rightarrow$ Lowpass filter 1 Hz
%
\begin{equation}
- \feat(t)\,=\,\bi(t)\,*\,\lp, \qquad \fc\,=\,1\,\text{Hz}
+ f_i(t)\,=\,b_i(t)\,*\,\lp, \qquad \fc\,=\,1\,\text{Hz}
\label{eq:lowpass}
\end{equation}
%
@@ -273,7 +295,7 @@ $\env(t)$ with ($\alpha>0$) and without ($\alpha=0$) song signal $s(t)$, assumin
\begin{equation}
\begin{split}
\db(t)\,&=\,\log \frac{\alpha\,\cdot\,s(t)\,+\,\eta(t)}{\dbref}\\
- &=\,\log \frac{\alpha}{\dbref}\,+\,\log \big[s(t)\,+\,\frac{\eta(t)}{\alpha}\big]
+ &=\,\log \frac{\alpha}{\dbref}\,+\,\log b_ig[s(t)\,+\,\frac{\eta(t)}{\alpha}b_ig]
\end{split}
\label{eq:toy_log}
\end{equation}
@@ -290,7 +312,7 @@ interval $\thp$ ($0 \ll \thp < \frac{1}{\fc}$)
%
\begin{equation}
\begin{split}
- \adapt(t)\,\approx\,\db(t)\,-\,\log \frac{\alpha}{\dbref}\,=\,\log \big[s(t)\,+\,\frac{\eta(t)}{\alpha}\big]
+ \adapt(t)\,\approx\,\db(t)\,-\,\log \frac{\alpha}{\dbref}\,=\,\log b_ig[s(t)\,+\,\frac{\eta(t)}{\alpha}b_ig]
\end{split}
\label{eq:toy_highpass}
\end{equation}
@@ -316,7 +338,7 @@ $\rightarrow$ Recurring trade-off: Equalizing signal intensity vs preserving ini
\subsection{Threshold nonlinearity \& temporal averaging}
-Convolved $c_i(t)$ $\xrightarrow{\nl}$ Binary $\bi(t)$ $\xrightarrow{\lp}$ Feature $\feat(t)$
+Convolved $c_i(t)$ $\xrightarrow{\nl}$ Binary $b_i(t)$ $\xrightarrow{\lp}$ Feature $f_i(t)$
\textbf{Thresholding component:}\\
- Within an observed time interval $T$, $c_i(t)$ follows probability density $\pc$\\
@@ -337,29 +359,29 @@ of time $T_1$ where $c_i(t)>\thr$ to total time $T$ due to normalization of $\pc
\end{equation}
%
\textbf{Averaging component:}\\
-- Lowpass filter over binary response $\bi(t)$ (Eq.\,\ref{eq:lowpass}) can be
+- Lowpass filter over binary response $b_i(t)$ (Eq.\,\ref{eq:lowpass}) can be
approximated as temporal averaging over a suitable time interval $\tlp$ ($\tlp > \frac{1}{\fc}$)\\
-- Within $\tlp$, $\bi(t)$ takes a value of 1 ($c_i(t)>\thr$) for time $T_1$ ($T_1+T_0=\tlp$)
+- Within $\tlp$, $b_i(t)$ takes a value of 1 ($c_i(t)>\thr$) for time $T_1$ ($T_1+T_0=\tlp$)
%
\begin{equation}
- \feat(t)\,\approx\,\frac{1}{\tlp} \int_{t}^{t\,+\,\tlp} \bi(\tau)\,d\tau\,=\,\frac{T_1}{\tlp}
+ f_i(t)\,\approx\,\frac{1}{\tlp} \int_{t}^{t\,+\,\tlp} b_i(\tau)\,d\tau\,=\,\frac{T_1}{\tlp}
\label{eq:feat_avg}
\end{equation}
%
-$\rightarrow$ Temporal averaging over $\bi(t)\in[0,1]$ (Eq.\,\ref{eq:binary}) gives
+$\rightarrow$ Temporal averaging over $b_i(t)\in[0,1]$ (Eq.\,\ref{eq:binary}) gives
ratio of time $T_1$ where $c_i(t)>\thr$ to total averaging interval $\tlp$\\
-$\rightarrow$ Feature $\feat(t)$ approximately represents supra-threshold fraction of $\tlp$
+$\rightarrow$ Feature $f_i(t)$ approximately represents supra-threshold fraction of $\tlp$
\textbf{Combined result:}\\
-- Feature $\feat(t)$ can be linked to the distribution of $c_i(t)$ using Eqs.\,\ref{eq:pdf_split} \& \ref{eq:feat_avg}
+- Feature $f_i(t)$ can be linked to the distribution of $c_i(t)$ using Eqs.\,\ref{eq:pdf_split} \& \ref{eq:feat_avg}
%
\begin{equation}
- \feat(t)\,\approx\,\int_{\thr}^{+\infty} \pclp\,dc_i\,=\,P(c_i\,>\,\thr,\,\tlp)
+ f_i(t)\,\approx\,\int_{\thr}^{+\infty} \pclp\,dc_i\,=\,P(c_i\,>\,\thr,\,\tlp)
\label{eq:feat_prop}
\end{equation}
%
$\rightarrow$ Because the integral over a probability density is a cumulative
-probability, the value of feature $\feat(t)$ (temporal compression of $\bi(t)$)
+probability, the value of feature $f_i(t)$ (temporal compression of $b_i(t)$)
at every time point $t$ signifies the probability that convolution output
$c_i(t)$ exceeds the threshold value $\thr$ during the corresponding averaging
interval $\tlp$
@@ -369,25 +391,25 @@ interval $\tlp$
template waveform $k_i(t)$ and signal $\adapt(t)$ centered at time point $t$\\
$\rightarrow$ Based on amplitudes on a graded scale
-- Feature $\feat(t)$ quantifies the probability that amplitudes of $c_i(t)$
+- Feature $f_i(t)$ quantifies the probability that amplitudes of $c_i(t)$
exceed threshold value $\thr$ within interval $\tlp$ around time point $t$\\
$\rightarrow$ Based on binned amplitudes corresponding to one of two categorical states
$\rightarrow$ Deliberate loss of precise amplitude information\\
$\rightarrow$ Emphasis on temporal structure (ratio of $T_1$ over $\tlp$)
-- Thresholding of $c_i(t)$ and subsequent temporal averaging of $\bi(t)$ to
-obtain $\feat(t)$ constitutes a remapping of an amplitude-encoding quantity into a
+- Thresholding of $c_i(t)$ and subsequent temporal averaging of $b_i(t)$ to
+obtain $f_i(t)$ constitutes a remapping of an amplitude-encoding quantity into a
duty cycle-encoding quantity, mediated by threshold function $\nl$
- Different scales of $c_i(t)$ can result in similar $T_1$ segments depending
on the magnitude of the derivative of $c_i(t)$ in temporal proximity to time
points at which $c_i(t)$ crosses threshold value $\thr$\\
$\rightarrow$ The steeper the slope of $c_i(t)$, the less $T_1$ changes with scale variations\\
-$\rightarrow$ If $T_1$ is invariant to scale variation in $c_i(t)$, then so is $\feat(t)$
+$\rightarrow$ If $T_1$ is invariant to scale variation in $c_i(t)$, then so is $f_i(t)$
- Suggests a relatively simple rule for optimal choice of threshold value $\thr$:\\
$\rightarrow$ Find amplitude $c_i$ that maximizes absolute derivative of $c_i(t)$ over time\\
-$\rightarrow$ Optimal with respect to intensity invariance of $\feat(t)$, not necessarily for
+$\rightarrow$ Optimal with respect to intensity invariance of $f_i(t)$, not necessarily for
other criteria such as song-noise separation or diversity between features
- Nonlinear operations can be used to detach representations from graded physical
@@ -396,9 +418,9 @@ stimulus (to fasciliate categorical behavioral decision-making?):\\
$\rightarrow$ Closely following the AM of the acoustic stimulus\\
2) Quantify relevant stimulus properties on a graded scale: $c_i(t)$\\
$\rightarrow$ More decorrelated representation, compared to prior stages\\
-3) Nonlinearity: Distinguish between "relevant vs irrelevant" values: $\bi(t)$\\
+3) Nonlinearity: Distinguish between "relevant vs irrelevant" values: $b_i(t)$\\
$\rightarrow$ Trading a graded scale for two or more categorical states\\
-4) Represent stimulus properties under relevance constraint: $\feat(t)$\\
+4) Represent stimulus properties under relevance constraint: $f_i(t)$\\
$\rightarrow$ Graded again but highly decorrelated from the acoustic stimulus\\
5) Categorical behavioral decision-making requires further nonlinearities\\
$\rightarrow$ Parameters of a behavioral response may be graded (e.g. approach speed),