Syncing to home.

main.tex

@@ -1568,10 +1568,10 @@ 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{Song design, temporal averaging, and feature representation}
+\subsection{Repetitive song patterns as design principle for robust features}
 \label{sec:constant_feat}
 
-% Recap of feature theory and relevant parameters:
+% Recap of feature theory and relevant variables:
 The feature set is the final song representation along the model pathway and
 constitutes the basis for song recognition. Each feature $f_i(t)$ results from
 the thresholding of the respective kernel response $c_i(t)$ by $\nl$ and the
@@ -1582,7 +1582,7 @@ the threshold value $\thr$ within the averaging interval $\tlp$ specified by
 $\fc$. The value of $f_i(t)$ is hence determined by $\thr$ with respect to the
 distribution $\pci$ of $c_i(t)$ and is restricted to the interval $[0,1]$.
 
-% Feature representation and the constraint of repetitive song structure:
+% Constant features and the constraint of repetitive song structure:
 Different species-specific songs are represented by different combinations of
 feature values, which should preferably be constant for the duration of a song
 to fasciliate recognition. The fundamental requirement for constant $f_i(t)$ is
@@ -1601,43 +1601,15 @@ syllables and pauses results in a pattern with a high degree of temporal
 regularity. A repetitive motor pattern during stridulation hence lays the basis
 for constant $f_i(t)$.
 
-The second requirement for constant $f_i(t)$ is a suitable averaging interval
+% Evolutionary implications:
-$\tlp$. The minimum $\tlp$ should encompass at least a few cycles of $c_i(t)$
+If constant $f_i(t)$ rely on a repetitive song pattern and are benefitial for
-to ensure a stable $\pci$. The maximum $\tlp$ should not exceed the duration of
+song recognition, then one would expect that grasshopper songs are
-the song to avoid the inclusion of noise. The duration of species-specific
+evolutionarily constrained towards such a repetitive temporal structure.
-grasshopper songs can range from a few hundred milliseconds~(e\,.g
-\textit{Stethophyma grossum}) to well over a minute~(e\,.g. \textit{C.
-mollis}), so that the optimal $\tlp$ likely differs between species. The longer
-$\tlp$, the longer $f_i(t)$ takes to stabilize after the onset of the song,
-which narrows the time window for reliable recognition.
-\\What about \bcite{ronacher1998song}??
-\\ $\rightarrow$ Answer might be \bcite{clemens2021sex}
-
-
-
-
-If the basis for constant $f_i(t)$ is already laid 
-
-The basis for constant $f_i(t)$ is hence already 
-
-The basis for a robust feature representation in the sense
-of constant $f_i(t)$ is hence already laid during the song production.
-
-If the feature representation relies on a repetitive song pattern, one would
-expect that grasshopper songs are evolutionary constrained to include such a
-pattern.
 
 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.
 
-, and if constant
-$f_i(t)$ are required for reliable song recognition, then one would expect that
-
-
-grasshopper songs are evolutionarily constrained to have such a repetitive
-structure.
-
 This is true for many species-specific calling songs but less for
 courtship songs, which tend to have a more complex structure~()
 
@@ -1649,20 +1621,6 @@ 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.
 
-
-Various grasshopper species, especially those with longer songs like \textit{C.
-mollis}, \textit{G. rufus}, or \textit{O. rufipes}, tend to stridulate softly
-at first and then continuously increase the amplitude of their song over time.
-This slow "ramping" amplitude modulation makes the overall song less periodic
-despite its temporal regularity. The "ramping" appears more pronounced in
-$\env(t)$ compared to $\adapt(t)$, which suggests that the logarithmic
-compression and adaptation during the preprocessing stage might be at least
-partially beneficial for mitigating the effect of this amplitude modulation on
-later representations. However, the adaptation of $\adapt(t)$ can only act on
-certain time scales --- depending on the cutoff frequency of the underlying
-highpass filter --- and is hence not able to compensate for "ramping" across
-the entire duration of a song.
-
 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
@@ -1678,6 +1636,37 @@ 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 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
+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.
+
 \subsection{Invariant processing in the grasshopper auditory system}
 
 % Invariance in the general (systemic) sense:
@@ -1823,6 +1812,19 @@ marginally lower saturation points. This raises the question whether two
 consecutive mechanisms of intensity invariance are actually beneficial for the
 overall system.
 
+Various grasshopper species, especially those with longer songs like \textit{C.
+mollis}, \textit{G. rufus}, or \textit{O. rufipes}, tend to stridulate softly
+at first and then continuously increase the amplitude of their song over time.
+This slow "ramping" amplitude modulation makes the overall song less periodic
+despite its temporal regularity. The "ramping" appears more pronounced in
+$\env(t)$ compared to $\adapt(t)$, which suggests that the logarithmic
+compression and adaptation during the preprocessing stage might be at least
+partially beneficial for mitigating the effect of this amplitude modulation on
+later representations. However, the adaptation of $\adapt(t)$ can only act on
+certain time scales --- depending on the cutoff frequency of the underlying
+highpass filter --- and is hence not able to compensate for "ramping" across
+the entire duration of a song.
+
 From a purely functional perspective, the answer could be that logarithmic
 compression and adaptation is a necessary preprocessing step towards a robust
 feature representation, even if thresholding and temporal averaging alone would