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j-hartling
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@@ -105,6 +105,7 @@
\newcommand{\pc}{p(c,\,T)} % Probability density (general interval)
\newcommand{\pclp}{p(c,\,\tlp)} % Probability density (lowpass interval)
\newcommand{\pci}{p(c_i,\,\tlp)} % Kernel-specific probability density (lowpass interval)
\newcommand{\tstat}{T_{\text{total}}} % Time interval where c(t) is stationary
\newcommand{\muf}{\mu_{f_i}} % Average feature value
\section{Introduction}
@@ -827,11 +828,10 @@ between the resulting $\env(t)$, $\db(t)$, and $\adapt(t)$. It is necessary to
use $\filt(t)$ as input for this analysis instead of $\env(t)$, because
$\env(t)$ results from a nonlinear transformation and hence cannot be
synthesized as an additive mixture of song component $\soc(t)$ and noise
component $\noc(t)$. % <-- Sentence may be methods section material.
However, it is much easier to conceive a mathematical description of the
effects of logarithmic compression and adaptation if $\env(t)$ itself is
assumed to be composed of $\soc(t)$ and $\noc(t)$. In the noiseless
case~(Fig.\,\ref{fig:log-hp}a), $\env(t)$ takes the form of
component $\noc(t)$. However, it is much easier to conceive a mathematical
description of the effects of logarithmic compression and adaptation if
$\env(t)$ itself is assumed to be composed of $\soc(t)$ and $\noc(t)$. In the
noiseless case~(Fig.\,\ref{fig:log-hp}a), $\env(t)$ takes the form of
\begin{equation}
\env(t)\,=\,\sca\,\cdot\,\soc(t), \qquad \env(t)\,>\,0\enspace\forall\enspace t\,\in\,\mathbb{R}
\label{eq:toy_env_pure}
@@ -1000,35 +1000,44 @@ corresponding $\tlp$:
f(t)\,\approx\,\int_{\Theta}^{+\infty} \pclp\,dc\,=\,P(c\,>\,\Theta,\,\tlp)
\label{eq:feat_prop}
\end{equation}
% Little bit of patch-work here...
% 1) Interpretation of the feature value:
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 $\tlp$. 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).
$f(t)=1$, respectively.
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 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
regime).
% 2) Constant feature values 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 $T_1$.
The saturation level of $f(t)$ is independent of the precise value of $\Theta$,
but the saturation point decreases with
$\Theta$~(Fig.\,\ref{fig:thresh-lp_single}e). Therefore, a threshold value of
$\Theta=0$ would be the optimal choice for achieving intensity invariance at
the lowest possible $\sca$. In stark contrast, the closer $\Theta$ is to 0, the
higher $\mu_f$ in response to the pure noise component $\noc(t)$ and the lower
the resulting SNR of $f(t)$ between noise regime and saturation
% 3) Constant feature values across alpha:
Similarly, $f(t)$ retains no information about the precise values of $c(t)$
apart from their relation to $\Theta$. Different scales $\sca$ can hence result
in similar values of $f(t)$ as long as $T_1$ remains similar across $\sca$. The
most reliable way of exploiting this 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 regime). The saturation
level of $f(t)$ is independent of the precise value of $\Theta$, but the
saturation point decreases with $\Theta$~(Fig.\,\ref{fig:thresh-lp_single}e).
Therefore, a threshold value of $\Theta=0$ would be the optimal choice for
achieving intensity invariance at the lowest possible $\sca$. In stark
contrast, the closer $\Theta$ is to 0, the higher $\mu_f$ in response to the
pure noise component $\noc(t)$ and the lower the resulting SNR of $f(t)$
between noise regime and saturation
regime~(Fig.\,\ref{fig:thresh-lp_single}b-d, left column, and
Fig.\,\ref{fig:thresh-lp_single}e). This trade-off between intensity invariance
and SNR has already been observed during the previous analysis on logarithmic
@@ -1581,89 +1590,30 @@ 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.
kernel response $c_i(t)$ exceeds the threshold value $\thr$ is approximately
First, $c_i(t)$ is entirely unstructured on most time scales, which
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 $t$.
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 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
$[0,1]$.
% 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 $c_i(t)$.
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.