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j-hartling
2025-12-08 16:42:30 +01:00
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@@ -126,18 +126,40 @@ $\rightarrow$ More general, simpler, unfitted formalized Gabor filter bank
\subsection{Population-driven signal pre-processing}
Grasshoppers receive airborne sound waves by a tympanal organ at each side of
the thorax. The tympanal membrane~(Fig.\,\ref{fig:pathway}) vibrates in
response to incoming sound waves in a frequency-dependent manner: Vibrations of
specific frequencies are focused on different membrane areas, while other
frequencies are attenuated~(\mbox{\cite{michelsen1971frequency}};
the thorax~(Fig.\,\ref{fig:pathway}a). The tympanal membrane acts as a mechanical resonance filter:
Vibrations of specific frequencies are focused on different membrane areas,
while other frequencies are attenuated~(\mbox{\cite{michelsen1971frequency}};
\mbox{\cite{windmill2008time}}; \mbox{\cite{malkin2014energy}}). This
mechanical resonance filter can be modelled by an initial bandpass filter
processing step can be approximated by an initial bandpass filter
\begin{equation}
\filt(t)\,=\,\raw(t)\,*\,\bp, \qquad \fc\,=\,5\,\text{kHz},\,30\,\text{kHz}
\label{eq:bandpass}
\end{equation}
applied to the acoustic input signal $\raw(t)$.
applied to the acoustic input signal $\raw(t)$. The auditory receptor neurons
connect directly to the tympanal membrane and transduce mechanical vibrations
into electro-chemical potentials. The receptor population is substrate to
several known signal processing steps. First, the receptors extract
the signal envelope~(\mbox{\cite{machens2001discrimination}}), which likely
involves a rectifying nonlinearity~(\mbox{\cite{machens2001representation}}).
This can be modelled as full-wave rectification followed by lowpass filtering
\begin{equation}
\env(t)\,=\,|\filt(t)|\,*\,\lp, \qquad \fc\,=\,500\,\text{Hz}
\label{eq:env}
\end{equation}
of the tympanal signal $\filt(t)$. Furthermore, the receptors exhibit a
sigmoidal response curve over logarithmically compressed intensity
levels~(\mbox{\cite{suga1960peripheral}}; \mbox{\cite{gollisch2002energy}}). In
the model, logarithmic compression is achieved by conversion to decibel scale
\begin{equation}
\db(t)\,=\,10\,\cdot\,\dec \frac{\env(t)}{\dbref}, \qquad \dbref\,=\,\max[\env(t)]
\label{eq:log}
\end{equation}
relative to the maximum intensity $\dbref$ of the signal envelope $\env(t)$.
Next, the axons of the receptor neurons project into the metathoracic ganglion,
where they synapse onto local interneurons~(Fig.\,\ref{fig:pathway}b). Both the
auditory receptors~(\mbox{\cite{fisch2012channel}}) and the subsequent
interneurons~(\mbox{\cite{clemens2010intensity}}) display spike-frequency
adaptation.
@@ -154,22 +176,13 @@ Initial: Continuous acoustic input signal $x(t)$
Filtering of behaviorally relevant frequencies by tympanal membrane\\
$\rightarrow$ Bandpass filter 5-30 kHz
Extraction of signal envelope (AM encoding) by receptor population\\
$\rightarrow$ Full-wave rectification, then lowpass filter 500 Hz
%
\begin{equation}
\env(t)\,=\,|\filt(t)|\,*\,\lp, \qquad \fc\,=\,500\,\text{Hz}
\label{eq:env}
\end{equation}
%
Logarithmically compressed intensity tuning curve of receptors\\
$\rightarrow$ Decibel transformation
%
\begin{equation}
\db(t)\,=\,10\,\cdot\,\dec \frac{\env(t)}{\dbref}, \qquad \dbref\,=\,\max[\env(t)]
\label{eq:log}
\end{equation}
%
Spike-frequency adaptation in receptor and interneuron populations\\
$\rightarrow$ Highpass filter 10 Hz
%