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

main.tex

@@ -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,