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Meddled with "Constant features" discussion section (semi-successful).

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@@ -301,6 +301,15 @@
year={1993},
}# Cited
@article{helversen1994forces,
title={Forces driving coevolution of song and song recognition in grasshoppers},
author={Von Helversen, Otto and Von Helversen, Dagmar},
journal={Fortschritte der Zoologie},
pages={253--253},
year={1994},
publisher={Gustav Fischer Verlag}
}# Cited
@article{helversen1997recognition,
title={Recognition of sex in the acoustic communication of the grasshopper Chorthippus biguttulus (Orthoptera, Acrididae)},
author={von Helversen, Dagmar and von Helversen, Otto},

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@@ -1581,118 +1581,83 @@ system and is therefore a particularly suitable candidate for functional
modelling. Other sensory systems that are either more complex or have not been
subject to decades of study will likely not be suitable for this approach yet.
\subsection{Repetitive song patterns as design principle for robust features}
\subsection{Repetitive song structure and temporal averaging as\\design principle for a robust feature representation}
\label{sec:constant_feat}
% Theoretical constraints for constant features:
The feature set is the final song representation along the model pathway and
constitutes the basis for song recognition. The songs of different species are
represented by specific combinations of feature values, which should be as
constant as possible for the duration of a song to fasciliate recognition. The
fundamental requirement for a constant feature $f_i(t)$ is that the time where
kernel response $c_i(t)$ exceeds the threshold value $\thr$ is approximately
The songs of different species are represented by specific combinations of
feature values, which should be as constant as possible for the duration of a
song to fasciliate recognition. Feature $f_i(t)$ is constant if the time where
kernel response $c_i(t)$ exceeds the threshold value $\thr$ within the
averaging interval $\tlp$ is the same at each time point $t\in\tstat>\tlp$.
This is fulfilled if $c_i(t)$ is stationary, so that its distribution $\pci$
does not change substantially within $\tstat$.
Each
feature $f_i(t)$ approximately quantifies the proportion of time where kernel
response $c_i(t)$ exceeds the threshold value $\thr$ within the averaging
interval $\tlp$. The value of $f_i(t)$ at time point $t$ is hence determined by
the distribution $\pci$ of $c_i(t)$ around $t$.
Accordingly, if $c_i(t)$ is
stationary within some time interval $T>\tlp$ --- so that $\pci$ does not
change substantially with $t$ --- then the value of $f_i(t)$ is approximately
constant across $t$.
If the time $T_1$ where $c(t)>\Theta$ within $\tlp$ is approximately constant
across $t$ for some time interval $\tstat>\tlp$, then $f(t)$ is approximately
constant across $t\in\tstat$ as well~(Fig.\,\ref{fig:stages_feat}c). This is
fulfilled if $c(t)$ is stationary in the sense that its distribution $\pclp$
does not change substantially within $\tstat$, which requires that $\tlp$ is
much longer than the relevant time scales of $c(t)$. However, stationarity of
$c(t)$ is not a necessary condition for $f(t)$ to be constant because $f(t)$
depends only on the total $T_1$ --- irrespective of the timing of individual
threshold crossings --- and different $\pclp$ can, in principle, still result
in similar $T_1$.
Most song-evoked $c_i(t)$ are indeed highly repetitive, albeit not perfectly
periodic, which is largely an inherited property of the song itself.
Grasshopper songs are produced by stridulation, which refers to the pulling of
the serrated stridulatory file on the hindlegs across a resonating vein on the
forewings~(\bcite{helversen1977stridulatory}; \bcite{stumpner1994song};
\bcite{helversen1997recognition}). Every peg that strikes the vein generates a
brief sound pulse; multiple pulses make up a syllable; and the repetition of
syllables and pauses results in a pattern with a high degree of temporal
regularity. This temporal regularity
, which is then reflected in $c_i(t)$. A repetitive motor pattern
during stridulation hence lays the basis for constant $f_i(t)$.
% Evolutionary implications:
If constant $f_i(t)$ rely on a repetitive song pattern and are benefitial for
song recognition, then one would expect that grasshopper songs are
evolutionarily constrained towards such a repetitive temporal structure.
If constant $f_i(t)$ rely on a repetitive song pattern and are benefitial for
reliable song recognition, one would expect that repetitiveness is a common
design principle of species-specific grasshopper songs.
This is true for many species-specific calling songs but less for
courtship songs, which tend to have a more complex structure~()
If constant $f_i(t)$ rely on a repetitive song pattern and are benefitial for
song recognition, then one would expect that grasshopper songs are
evolutionarily constrained to have such a repetitive temporal structure.
From an evolutionary perspective, one would then expect that grasshopper songs
are evolutionarily constrained to have a repetitive temporal structure in order
to elicit a robust feature representation.
Certain grasshopper species like \textit{Chorthippus dorsatus} are known to
switch their stridulation pattern in the middle of a
song~(\bcite{stumpner1994song}). \textit{C. dorsatus} starts stridulating with
both hindlegs in synchrony and thereby generates a pronounced syllable-pause
pattern similar to that of \textit{P. parallelus}. For the last part of its
song, however, \textit{C. dorsatus} switches to an alternating leg movement,
which results in a more continuous but not entirely unstructured rattling
sound. It is unclear what this composite design means for the feature
representation of \textit{C. dorsatus} songs. In principle, both parts of the
song could result in similar $\pci$ despite their different temporal structure,
which would allow for consistent $f_i(t)$ across the entire song. However, it
appears more likely that only one part of the song encodes species identity,
while the other part serves a different purpose such as fitness
advertisement~(\bcite{stumpner1992recognition}).
% Constraints on the song structure:
% Also: Constant model features vs. actual grasshopper (calling) songs:
% (Also: Third revision and this section still doesn't sound good)
Grasshoppers sing by pulling the stridulatory file on the hindlegs across a
resonating vein on the forewings~(\bcite{helversen1977stridulatory};
\bcite{stumpner1994song}; \bcite{helversen1997recognition}). Different
stridulatory motor patterns allow for the production of vastly different song
patterns without modifying the overall stridulation
apparatus~(\bcite{stumpner1994song}). The song pattern could hence be changed
frequently and substantially throughout the song; yet many species resort to
songs with a regular, highly repetitive
structure~(\bcite{tishechkin2009acoustic}). From the perspective of the model
pathway, a repetitive song pattern is necessary because it translates into a
repetitive structure of $c_i(t)$. If the song pattern is sufficiently regular,
$c_i(t)$ is assumed to be stationary, which lays the basis for constant
$f_i(t)$ and hence reliable recognition throughout the song. This is
particularly relevant for the calling songs --- whose primary function is the
broadcasting of species identity --- whereas the courtship songs tend towards a
slightly more complex structure~(\bcite{vedenina2003complex};
\bcite{vedenina2011speciation}; \bcite{vedenina2014stable}). Different accounts
of how the ancestral calling song could have looked like agree that it likely
possessed a repetitive structure~(\bcite{helversen1994forces};
\bcite{vedenina2011speciation}; \bcite{sevastianov2023evolution}). This
suggests that the song recognition pathway has long been evolving around
repetitive song patterns. The calling songs of many extant species --- while
extremely diverse in their details~(\bcite{tishechkin2009acoustic}) --- might
still have to conform to this design principle in order to be recognizable.
However, there are exceptions: For example, \textit{Chorthippus dorsatus}
produces a composite calling song that consists of a repetitive syllable-pause
pattern followed by a noisy rattling sound~(\bcite{stumpner1992recognition};
\bcite{stumpner1994song}). Females respond to the song only if both parts are
present~(\bcite{stumpner1992recognition}). It has been suggested that the
second part of the song instead serves as a fitness signal, so that females
might recognize the song based on the first part but choose not to respond if
the second part is missing.
% Constraints on the averaging interval:
The second requirement for constant $f_i(t)$ is a suitable averaging interval
$\tlp$. The minimum $\tlp$ should encompass at least a few cycles of $c_i(t)$
to ensure a stable $\pci$. Experiments with artificial songs have shown that
replacing every second syllable with one of different duration does not
drastically impair song recognition~(\bcite{helversen1998acoustic}). In
particular, recognition was least impaired if the average replacement duration
corresponded roughly to the original syllable duration, even though the
individual replacements were much shorter or longer. Accordingly, the more
cycles of $c_i(t)$ are included in $\tlp$, the more robust $f_i(t)$ is against
irregularities in the song pattern. However, the longer $\tlp$, the longer
$f_i(t)$ takes to stabilize after the onset of the song due to the inclusion of
noise, which narrows the time window during which $f_i(t)$ is constant. If
$\tlp$ exceeds the duration of the song, $f_i(t)$ will never be constant at
all. In the model pathway, $\tlp$ is in the range of around 1
second~($\fc=1\,$Hz), so that $f_i(t)$ takes accordingly long to stabilize. In
contrast, \textit{C. biguttulus} has been shown to respond to songs that
consist of only 3~syllable-pause cycles and are merely 250\,ms
long~(\bcite{ronacher1998song}). This suggests a shorter $\tlp$ in this species
than in the model pathway. It also appears plausible that grasshoppers
% Also: Fixed model averaging interval vs. literature:
The minimum $\tlp$ must encompass at least a few cycles of $c_i(t)$ to ensure a
stationary $\pci$. Behavioral experiments have shown that recognition of the
song pattern is not drastically impaired if the duration of some syllables is
changed~(\bcite{helversen1998acoustic}). In particular, recognition was least
impaired if the average replacement duration corresponded roughly to the
original duration, even though the individual replacements were much shorter or
longer. Accordingly, the more cycles of $c_i(t)$ are included in $\tlp$, the
more robust $f_i(t)$ is against irregularities in the song pattern. However,
the longer $\tlp$, the longer $f_i(t)$ takes to stabilize after the onset of
the song due to the inclusion of noise, which narrows the time window during
which $f_i(t)$ is constant. If $\tlp$ exceeds the duration of the song,
$f_i(t)$ will never be constant at all. The optimal $\tlp$ for a specific song
is hence determined by the duration of a typical syllable-pause cycle~(lower
bound) and the total song duration~(upper bound). Both parameters vary widely
across different grasshopper species~(\bcite{tishechkin2009acoustic}), which
suggests that the optimal $\tlp$ is likely species-specific. In the model
pathway, $\tlp$ is fixed at around 1 second~($\fc=1\,$Hz), so that $f_i(t)$
takes accordingly long to stabilize. In contrast, \textit{C. biguttulus} has
been shown to respond to songs that consist of only 3~syllable-pause cycles and
are merely 250\,ms long~(\bcite{ronacher1998song}), which suggests a much
shorter $\tlp$ in this species. It appears plausible that grasshoppers
recognize conspecific songs not by a singular combination of feature values~(a
point in feature space) but within a certain tolerance~(a region in feature
space). Song responsiveness in grasshoppers is subject to a speed-accuracy
trade-off~(\bcite{clemens2021sex}) --- a grasshopper could thus either respond
as soon as $f_i(t)$ is within tolerance or wait for $f_i(t)$ to stabilize for
additional certainty. Overall, it is difficult to assess a suitable $\tlp$ for
a specific song. However, it is known that both the song duration and the
duration of a typical syllable-pause cycle vary widely across different
grasshopper species~(\bcite{tishechkin2009acoustic}), so that the optimal
$\tlp$ is likely species-specific.
additional certainty.
\subsection{Invariant processing in the grasshopper auditory system}
@@ -1894,8 +1859,11 @@ all nearby individuals. Importantly, the limitation of intensity invariance by
SNR likely applies to all grasshoppers regardless of species, so that the
behavioral strategies could be shared among the species that coexist in a given
habitat.
\\ \bcite{stange2012grasshopper}?
\\ \bcite{kramer2018robustness}
\\ \bcite{einhaupl2011attractiveness}
\\ \bcite{snedden1998mechanisms}?
\\ \bcite{tishechkin2009acoustic}?
% Because the presumed restriction of song recognition
% by means of the noise floor applies to all grasshoppers in a certain area,