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main.tex

@@ -984,35 +984,39 @@ around $\Theta$:
 The right-sided part of the split $\pc$ corresponds to time $T_1$ where
 $c(t)>\Theta$, while the left-sided part corresponds to time $T_0=T-T_1$ where
 $c(t)\leq\Theta$. The semi-definite integral over the right-sided part of $\pc$
-represents the ratio of time $T_1$ to total time $T$ because the indefinite
-integral of a probability density is normalized to 1. The lowpass filtering of
-$b(t)$ can be approximated as temporal averaging over a suitable time interval
-$\tlp>\frac{1}{\fc}$ in order to express $f(t)$ as a similar temporal ratio
+represents the ratio of $T_1$ relative to $T$ because the indefinite integral
+of a probability density is normalized to 1. The lowpass filtering of $b(t)$
+can be approximated as temporal averaging over a suitable averaging interval
+$\tlp>\frac{1}{\fc}$. This allows us to express $f(t)$ as the ratio of time
+$T_1$ where $b(t)=1$ relative to $\tlp$:
 \begin{equation}
     f(t)\,\approx\,\frac{1}{\tlp} \int_{t}^{t\,+\,\tlp} b(\tau)\,d\tau\,=\,\frac{T_1}{\tlp}, \qquad b(t)\,\in\,\{0,\,1\}
     \label{eq:feat_avg}
 \end{equation}
-of time $T_1$ during which $b(t)$ is 1 within the averaging interval $\tlp$.
 Therefore, the value of $f(t)$ at every time point $t$ approximately signifies
-the cumulative probability that $c(t)$ exceeds $\Theta$ during the
-corresponding averaging interval $\tlp$:
+the cumulative probability that $c(t)$ exceeds $\Theta$ within the
+corresponding $\tlp$:
 \begin{equation}
     f(t)\,\approx\,\int_{\Theta}^{+\infty} \pclp\,dc\,=\,P(c\,>\,\Theta,\,\tlp)
     \label{eq:feat_prop}
 \end{equation}
 In a sense, $f(t)$ can be interpreted as some sort of duty cycle of $c(t)$ with
 respect to $\Theta$. For example, a feature value of $f(t)=0.4$ means that
-$c(t)$ exceeds $\Theta$ for approximately 40\,\% of the time within $\tlp$
-around $t$. In the most extreme cases, $\Theta$ lays either above the maximum
-of $c(t)$ or below the minimum of $c(t)$, which results in a minimum or maximum
-possible feature value of $f(t)=0$~(Fig.\,\ref{fig:thresh-lp_single}d, left
-column) or $f(t)=1$, respectively.
+$c(t)$ exceeds $\Theta$ for approximately 40\,\% of the time within $\tlp$. In
+the most extreme cases, $\Theta$ lays either above the maximum of $c(t)$ or
+below the minimum of $c(t)$, which results in a minimum or maximum possible
+feature value of $f(t)=0$~(Fig.\,\ref{fig:thresh-lp_single}d, left column) or
+$f(t)=1$, respectively. Furthermore, if $c(t)$ is stationary --- so that its
+statistics do not change substantially over time --- and if $\tlp$ is much
+longer than the relevant time scales of $c(t)$, then $\pclp$ is largely
+independent of $t$. In this case, $f(t)$ is approximately constant across
+$t$~(Fig.\,\ref{fig:stages_feat}c).
 
 Importantly, $f(t)$ neither retains information about the timing of individual
 threshold crossings nor the precise values of $c(t)$ apart from their relation
 to $\Theta$. Different $\sca$ can hence result in similar feature values by
 producing similar $T_1$ segments. The most reliable way of exploiting this
-invariant porperty of $f(t)$ is to set $\Theta$ to a value near 0, because
+invariant property of $f(t)$ is to set $\Theta$ to a value near 0, because
 these values are least affected by different scales of $c(t)$. For sufficiently
 large $\sca$, $f(t)$ then approaches the same constant $\mu_f$ in both the
 noiseless and the noisy case~(Fig.\,\ref{fig:thresh-lp_single}e, saturation
@@ -1571,35 +1575,108 @@ subject to decades of study will likely not be suitable for this approach yet.
 \subsection{Repetitive song patterns as design principle for robust features}
 \label{sec:constant_feat}
 
-% Recap of feature theory and relevant variables:
+% 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 $\thr$ within averaging
+interval $\tlp$ is the same for all time points $t$. This is fulfilled if
+$c_i(t)$ is stationary within a certain time window and $\tlp$ is much longer
+than the relevant time scales of $c_i(t)$, so that the distribution $\pci$ of
+$c_i(t)$ and hence the value of $f_i(t)$ remain stable across $t$.
+
+% Practical cases that allow for approximately constant features:
+There are two very different practical cases in which $c_i(t)$ could fulfill
+the stationarity requirement. First, $c_i(t)$ is 
+
+can be assumed to be
+stationary: Either $c_i(t)$ is entirely unstructured on most time scales, or
+$c_i(t)$ is periodic.
+
+
+First, $c_i(t)$ is entirely unstructured on most time scales, which
+
+
+
+The structure of noise-evoked $c_i(t)$ is largely random with an
+approximately normal $\pci$ with constant mean and variance across $t$.
+
+Either the structure of $c_i(t)$ is largely random, or 
+
+Noise-evoked $c_i(t)$ are 
+
+Noise-evoked $c_i(t)$ are largely unstructured and follow a roughly
+normal $\pci$ with constant mean and variance across $t$. In contrast,
+song-evoked $c_i(t)$ are highly 
+
+Noise-evoked $c_i(t)$ fulfill the stationary requirement because their $\pci$
+is approximately a normal distribution with constant mean and variance across
+$t$.
+
+Each feature $f_i(t)$ approximately
+quantifies the proportion of time where the respective kernel response $c_i(t)$
+exceeds the threshold value $\thr$ within the averaging interval $\tlp$. The
+songs of different species are represented by specific combinations of feature
+values, which should preferably be as constant as possible during a song to
+fasciliate recognition. The fundamental requirement for constant $f_i(t)$ is
+that the time where $c_i(t)>\thr$ within $\tlp$ is the same for all $t$, which
+is fulfilled if the distribution $\pci$ of 
+
+
+The
+value of $f_i(t)$ is hence determined by $\thr$ with respect to the
+distribution $\pci$ of $c_i(t)$.
+
+
+$c_i(t)>\thr$.
+
 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
 subsequent temporal averaging of binary response $b_i(t)$ by a lowpass filter
-with extremely low cutoff frequency $\fc$. At a given time point $t$, $f_i(t)$
-approximately quantifies the proportion of time during which $c_i(t)$ exceeds
-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 $[0,1]$.
+with cutoff frequency $\fc$, which specifies the averaging interval $\tlp$.
+Feature $f_i(t)$ approximately quantifies the proportion of time during which
+$c_i(t)$ exceeds the threshold value $\thr$ within $\tlp$. The value of
+$f_i(t)$ at time point $t$ is hence determined by $\thr$ with respect to the
+distribution $\pci$ of $c_i(t)$ around $t$ and restricted to the interval
+$[0,1]$.
 
-% 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
-that the time where $c_i(t)>\thr$ during $\tlp$ is the same for all $t$, which
-is fulfilled if $\pci$ is stable across $t$. The most straightforward way to
-achieve a stable $\pci$ is that $c_i(t)$ is periodic and $\tlp$ is sufficiently
-long to average over multiple cycles of $c_i(t)$. Most song-evoked $c_i(t)$ are
-indeed highly repetitive, albeit not perfectly periodic, which is largely an
-inherited property of the song itself. Most 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
+% Theoretical constraints for constant features:
+The songs of different species are represented by specific combinations of
+values across the feature set, which should preferably be constant for the
+duration of a song to fasciliate recognition. The fundamental requirement for
+constant $f_i(t)$ is that the time where $c_i(t)>\thr$ within $\tlp$ is the
+same for all $t$.
+
+This is fulfilled if $c_i(t)$ is stationary across $t$ and
+$\tlp$ is much longer than the relevant time scales of $c_i(t)$, so that $\pci$
+is independent of $t$.
+
+
+, which is fulfilled if $\pci$ is stable across $t$.
+
+The most
+straightforward way to achieve a stable $\pci$ is that $c_i(t)$ is stationary
+and $\tlp$ is sufficiently long to average over the stationary distribution of
+
+The most
+straightforward way to achieve a stable $\pci$ is that $c_i(t)$ is periodic and
+$\tlp$ is sufficiently long to average over multiple cycles of $c_i(t)$.
+
+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
+\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. A repetitive motor pattern during stridulation hence lays the basis
-for constant $f_i(t)$.
+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