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Finished methods.

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Attempting to clean up tex files.
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2026-05-16 14:43:13 +02:00
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@@ -458,14 +458,15 @@ the following feature extraction stage.
\subsubsection{Feature extraction by individual neurons}
The ascending neurons extract and encode a number of different features of the
preprocessed signal, and hence represent the signal in a higher-dimensional
space than the preceding receptor neurons and local interneurons. Each
ascending neuron is assumed to scan the signal for a specific template pattern,
which can be thought of as a kernel of a particular structure and on a
particular time scale. This process, known as template matching, can be
modelled as a convolution of the intensity-adapted envelope $\adapt(t)$ with a
kernel $k_i(t)$ specific to the $i$-th ascending neuron:
The population of ascending neurons extracts and encodes a number of different
features of the preprocessed signal, and hence represents the signal in a
higher-dimensional space than the preceding receptor neurons and local
interneurons~(\bcite{clemens2011efficient}). Each ascending neuron is assumed
to scan the signal for a specific template pattern, which can be thought of as
a kernel of a particular structure and on a particular time scale. This
process, known as template matching, can be modelled as a convolution of the
intensity-adapted envelope $\adapt(t)$ with a kernel $k_i(t)$ specific to the
$i$-th ascending neuron:
\begin{equation}
c_i(t)\,=\,\adapt(t)\,*\,k_i(t)
= \infint \adapt(\tau)\,\cdot\,k_i(t\,-\,\tau)\,d\tau
@@ -495,14 +496,14 @@ $\rh$ relative to the maximum of the Gaussian:
With this, an appropriate carrier frequency $\kfi$ for obtaining a Gabor kernel
with width $\kwi$ and desired lobe number $\kni$ can be approximated as
\begin{equation}
\kfi(\kni,\,\kwi,\,\rh)\,=\,\frac{0.5\,\cdot\,(\kni\,+\,\beta_0)}{\fdrm(\kwi,\,\rh)}, \qquad \kni\,\geq\,2\enspace\forall\enspace \kni\,\in\,\mathbb{Z}
\kfi(\kni,\,\kwi,\,\rh)\,=\,\frac{0.5\,\cdot\,\kni\,+\,\beta_0}{\fdrm(\kwi,\,\rh)}, \qquad \kni\,\geq\,2\enspace\forall\enspace \kni\,\in\,\mathbb{Z}
\label{eq:gabor_freq}
\end{equation}
% \begin{equation}
% \kfi(\kni,\,\kwi,\,\rh)\,=\,\frac{0.5\,\cdot\,(\kni\,+\,\beta_0)}{2\,\cdot\,\sqrt{-2\,\cdot\,\ln \rh}\cdot\kwi}, \qquad \kni\,\geq\,2\enspace\forall\enspace \kni\,\in\,\mathbb{Z}
% \kfi(\kni,\,\kwi,\,\rh)\,=\,\frac{0.5\,\cdot\,\kni\,+\,\beta_0}{2\,\cdot\,\sqrt{-2\,\cdot\,\ln \rh}\cdot\kwi}, \qquad \kni\,\geq\,2\enspace\forall\enspace \kni\,\in\,\mathbb{Z}
% \end{equation}
The relationship between $\kfi$ and $\kni$ is approximately linear except for
small $\kni$. The offset term $\beta_0\approx0.5$ was added to balance the
small $\kni$. The offset term $\beta_0\approx0.26$ was added to balance the
amplitudes of the $\kni$ desired lobes of the kernel --- which should be
maximized --- against the amplitudes of the next-outer lobes, which should not
exceed the threshold value determined by $\rh$. Note that simple Gaussian
@@ -549,23 +550,30 @@ response~(Fig.\,\ref{fig:stages_feat}b):
\label{eq:binary}
\end{equation}
The thresholding of $c_i(t)$ into $b_i(t)$ can be thought of as a
categorization into "relevant" and "irrelevant" response values.
% It is unclear whether such a thresholding nonlinearity is actually implemented
% either by the ascending neurons or at some point further downstream in the SEG.
Finally, the responses of the ascending neurons are assumed to be integrated
somewhere in the SEG~(\bcite{ronacher1986routes}; \bcite{bauer1987separate};
\bcite{bhavsar2017brain}). This processing step can be approximated as temporal
averaging of the binary responses $b_i(t)$ by a lowpass filter
categorization into "relevant" and "irrelevant" response values. Similar
thresholding nonlinearities have been a crucial processing step in previous
models that deal with the extraction of behaviorally relevant song features in
insects~(\bcite{clemens2013computational}; \bcite{clemens2013feature};
\bcite{hennig2014time}; \bcite{ronacher2015computational}).
% However, there is no direct physiological evidence that would allow to
% determine the exact location or underlying mechanism of such a nonlinearity in
% either the ascending neurons or at some point further downstream in the SEG.
In the grasshopper, the responses of the ascending neurons are assumed to be
integrated somewhere in the SEG~(\bcite{ronacher1986routes};
\bcite{bauer1987separate}; \bcite{bhavsar2017brain}). In the model pathway,
temporal integration is implemented as temporal averaging of the binary
responses $b_i(t)$ by a lowpass filter with extremely low cutoff frequency:
\begin{equation}
f_i(t)\,=\,b_i(t)\,*\,\lp, \qquad \fc\,=\,1\,\text{Hz}
\label{eq:lowpass}
\end{equation}
to obtain a final set of slowly changing kernel-specific features $f_i(t)$. In
the resulting high-dimensional feature space, different species-specific song
patterns are characterized by a distinct combination of feature values, which
can be read out by a simple linear classifier.
% Cite somewhere:
This processing step results in a set of slowly changing kernel-specific
features $f_i(t)$, which is the final representation along the model
pathway~(Fig.\,\ref{fig:stages_feat}c). In the resulting high-dimensional
feature space, different species-specific song patterns can be distinguished by
their distinct combination of feature values, e.\,g. using Euclidian geometry
or a simple linear classifier.
\begin{figure}[!ht]
\centering
\includegraphics[width=\textwidth]{figures/fig_feat_stages.pdf}
@@ -770,6 +778,19 @@ stable position and height of the microphone array during recording. The
resulting recordings were then processed through the model pathway and analyzed
according to the procedure described in Section~\ref{sec:intensity_measures}.
\subsection{Determining kernel-specific threshold values}
Different kernels $k_i(t)$ result in specific kernel responses $c_i(t)$,
Eq.\,\ref{eq:conv}, which are then transformed further into binary responses
$b_i(t)$, Eq.\,\ref{eq:binary}, by thresholding nonlinearity $\nl$. The
threshold value $\thr$ is specific to each $k_i(t)$. Across all analyses,
$\thr$ has been specified as a multiple of the pure-noise reference standard
deviation $\sigma_{c_i}$ for input $x(t)=\noc(t)$. This ensures that $\thr$ as
well as the resulting $b_i(t)$ and $f_i(t)$ are comparable across different
$k_i(t)$ because each pure-noise $c_i(t)$ approximately follows a normal
distribution~(see appendix
Figs.\,\ref{fig:app_thresh-lp_kern-sd}-\ref{fig:app_field_kern-sd}).
\section{Results}
\subsection{Mechanisms driving the emergence of intensity invariance}