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@@ -576,9 +576,8 @@ $\noc(t)$ with $\nsig=1$:
x(t)\,=\,\sca\,\cdot\,\soc(t)\,+\,\noc(t), \qquad \sca\,\geq\,0
\label{eq:noisy}
\end{equation}
Accordingly, the SNR of input $x(t)$ in the noisy case equals the squared
$\sca$ value:
% Make sure that SNR = signal-to-noise ratio is introduced somewhere!
Accordingly, the signal-to-noise ratio (SNR) of input $x(t)$ in the noisy case
equals the squared $\sca$ value:
\begin{equation}
\text{SNR}_x(\sca)\,=\,\frac{(\sca\,\cdot\,\ssig)^2}{\nsig^2}\,=\,\sca^2, \qquad \ssig\,=\,\nsig\,=\,1
\label{eq:input_snr}
@@ -1509,7 +1508,7 @@ natural song variation.
\textbf{Lower right}:~Distribution of correlation
coefficients $\rho$ for each interspecific and
intraspecific comparison. Dots indicate single $\rho$
values.
values.\\
}
\label{fig:feat_cross_species}
\end{figure}
@@ -1518,21 +1517,87 @@ natural song variation.
\newpage
\section{Discussion}
% RIPPED FROM INTRODUCTION:
In the current study, we have established a physiologically inspired functional
model of the grasshopper song recognition pathway. The model pathway covers the
entire auditory processing stream, from the sound reception at the tympanal
membrane over peripheral receptor neurons and local interneurons up to the
generation of a high-dimensional feature representation at the level of the
ascending neurons and beyond in the SEG. Using this model pathway, we have
identified two computational key mechanisms for the emergence of
intensity-invariant song representations. Each mechanism comprises a nonlinear
transformation and a subsequent linear transformation. The first mechanism
consists of logarithmic compression and adaptation, which takes place at the
level of the receptor neurons and local interneurons. The second mechanism
consists of thresholding and temporal averaging, which takes place either at
the level of the ascending neurons or further downstream in the SEG. Systematic
investigation of both mechanisms revealed a persistent trade-off between the
intensity invariance and the SNR of the song representations along the pathway.
In the following, we discuss the capabilities and limitations of our model
approach as well as the implications of our findings for the design of the
grasshopper auditory system, the evolution of species-specific grasshopper
songs, and the ethological relevance of intensity invariance in a natural
acoustic environment.
% Why functional models of sensory systems?
% Our scientific understanding of sensory processing systems is based on the
% distributed accumulation of specific anatomical, physiological, and
% ethological evidence. This leaves us with the challenge of integrating the
% available knowledge fragments into a coherent whole in order to address more
% and more far-reaching questions, from the interaction between individual
% processing steps to comparisons between similar systems across different
% species. One way to deal with this challenge is to build a unified framework
% that captures the essential functional aspects of a sensory system. However,
% building such a framework is a challenging task in itself. It requires a
% wealth of existing knowledge of the system and the stimuli it operates on, a
% clearly defined scope, and careful abstraction of the underlying structures
% and mechanisms.
\subsection{Leveraging functional modelling to investigate sensory systems}
Our understanding of sensory processing systems is based on the distributed
accumulation of anatomical, physiological, and ethological evidence. Functional
modelling provides a powerful tool to integrate the available knowledge
fragments into a coherent whole, which greatly fasciliates systematic
investigations and allows us to address questions of increasingly broader
scope. For instance, we were able to investigate the interaction between the
two mechanisms of intensity invariance because we can relate the output of the
first mechanism to the input of the second mechanism, which would not be
possible if both are treated as separate entities. We can also use the model
pathway as a general basis for comparing song representations across different
species without building a specific model for each species. However, the
potential of a functional modelling approach also depends directly on the
amount of available knowledge on the sensory system and the stimuli it operates
on. The grasshopper auditory system is a comparably simple and well-understood
system and is therefore a particularly suitable candidate for functional
modelling.
that has been studied extensively over the past decades. This makes it
a particularly suitable candidate for functional modelling.
functional
modelling is not without limitations.
However, building a
framework that captures the essential functional aspects of a sensory system is
a challenging task.
It requires comprehensive information on the system and the
stimuli it operates on as well as careful abstraction of the underlying
structures and mechanisms. The grasshopper auditory system is a comparably
simple, well-understood system that has been studied extensively over the past
decades.
and is therefore a
particularly suitable candidate for functional modelling. Many other sensory
systems
\textbf{Song recognition pathway: Grasshopper vs. model:}\\
The model pathway includes a rather large number of Gabor kernels compared to
the 15 to 20 ascending neurons in the grasshopper auditory
system~(\bcite{stumpner1991auditory}).
\subsection{Interplay of song representation and song design}
\textbf{The role of repetitive songs for the feature representation:}
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 forewings~(\bcite{helversen1977stridulatory};
\bcite{stumpner1994song}; \bcite{helversen1997recognition}). Every "tooth" 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
characteristic amplitude-modulated waveform pattern.
\subsection{Intensity invariance versus SNR along the auditory pathway}
\subsection{Behavior in a natural acoustic environment}
% RIPPED FROM INTRODUCTION:
% Multi-species, multi-individual communally inhabited environments\\
% - Temporal overlap: Simultaneous singing across individuals/species common\\
@@ -1597,15 +1662,6 @@ operate on unmodified recordings of natural grasshopper songs instead of
condensed pulse train approximations, which widens its scope towards more
realistic, ecologically relevant scenarios.
\textbf{The role of repetitive songs for the feature representation:}
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 forewings~(\bcite{helversen1977stridulatory};
\bcite{stumpner1994song}; \bcite{helversen1997recognition}). Every "tooth" 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
characteristic amplitude-modulated waveform pattern.
\textbf{Excursion into time-warp invariance:}
For instance, the temporal structure of grasshopper songs warps with
temperature~(\bcite{skovmand1983song}). The auditory system can compensate for
@@ -1614,11 +1670,6 @@ absolute time intervals~(\bcite{creutzig2009timescale};
\bcite{creutzig2010timescale}), as those remain relatively constant across
different temperatures~(\bcite{helversen1972gesang}).
\textbf{Song recognition pathway: Grasshopper vs. model:}\\
The model pathway includes a rather large number of Gabor kernels compared to
the 15 to 20 ascending neurons in the grasshopper auditory
system~(\bcite{stumpner1991auditory}).
\textbf{Definition of invariance (general, systemic):}\\
Invariance = Property of a system to maintain a stable output with respect to a
set of relevant input parameters (variation to be represented) but irrespective
@@ -1633,7 +1684,6 @@ large-scale AM, current overall intensity level)\\
$\rightarrow$ Without time scale selectivity, any fully intensity-invariant
output will be a flat line
\textbf{Log-HP: Implication for intensity invariance:}\\
- Logarithmic scaling is essential for equalizing different song intensities\\
$\rightarrow$ Intensity information can be manipulated more easily when in form
@@ -1683,7 +1733,7 @@ initiation of one behavior over another is categorical (e.g. approach/stay)
$\noc(t)$ within the signal envelope $\env(t)$ over scale
$\sca$. Based on input $\raw(t)$ with $\sigma_{\eta}=1$
(corresponding to the analysis underlying
Fig.\,\ref{fig:rect-lp}), using 100 realizations of
Fig.\,\ref{fig:rect-lp}), using random 100 realizations of
$\noc(t)$.}
\label{fig:app_env-sd}
\end{figure}% Referenced.
@@ -1842,13 +1892,12 @@ initiation of one behavior over another is categorical (e.g. approach/stay)
\includegraphics[width=\textwidth]{figures/fig_invariance_cross_species_thresh_appendix.pdf}
\caption{\textbf{Threshold-dependent intensity invariance of
species-specific feature sets.}
Same input and processing as in
Fig.\,\ref{fig:pipeline_full}, using different
kernel-specific threshold values $\thr$ (multiples of
pure-noise standard deviation $\sigma_{\eta_i}$ of
$c_i(t)$ for $\sca=0$. See also appendix
Fig.\,\ref{fig:app_full_kern-sd}). Average value $\muf$ of
each feature $f_i(t)$ over $\sca$.
Same processing as in Fig.\,\ref{fig:pipeline_full}, using
different kernel-specific threshold values $\thr$
(multiples of pure-noise standard deviation
$\sigma_{\eta_i}$ of $c_i(t)$ for $\sca=0$. See also
appendix Fig.\,\ref{fig:app_full_kern-sd}). Average value
$\muf$ of each feature $f_i(t)$ over $\sca$.
}
\label{fig:app_cross_species_thresh}
\end{figure}% Reference this one!