Started writing the discussion.

main.tex

@@ -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
+Accordingly, the signal-to-noise ratio (SNR) of input $x(t)$ in the noisy case
-$\sca$ value:
+equals the squared $\sca$ value:
-% Make sure that SNR = signal-to-noise ratio is introduced somewhere!
 \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?
+\subsection{Leveraging functional modelling to investigate sensory systems}
-% Our scientific understanding of sensory processing systems is based on the
+
-% distributed accumulation of specific anatomical, physiological, and
+Our understanding of sensory processing systems is based on the distributed
-% ethological evidence. This leaves us with the challenge of integrating the
+accumulation of anatomical, physiological, and ethological evidence. Functional
-% available knowledge fragments into a coherent whole in order to address more
+modelling provides a powerful tool to integrate the available knowledge
-% and more far-reaching questions, from the interaction between individual
+fragments into a coherent whole, which greatly fasciliates systematic
-% processing steps to comparisons between similar systems across different
+investigations and allows us to address questions of increasingly broader
-% species. One way to deal with this challenge is to build a unified framework
+scope. For instance, we were able to investigate the interaction between the
-% that captures the essential functional aspects of a sensory system. However,
+two mechanisms of intensity invariance because we can relate the output of the
-% building such a framework is a challenging task in itself. It requires a
+first mechanism to the input of the second mechanism, which would not be
-% wealth of existing knowledge of the system and the stimuli it operates on, a
+possible if both are treated as separate entities. We can also use the model
-% clearly defined scope, and careful abstraction of the underlying structures
+pathway as a general basis for comparing song representations across different
-% and mechanisms.
+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
+                     Same processing as in Fig.\,\ref{fig:pipeline_full}, using
-                     Fig.\,\ref{fig:pipeline_full}, using different
+                     different kernel-specific threshold values $\thr$
-                     kernel-specific threshold values $\thr$ (multiples of
+                     (multiples of pure-noise standard deviation
-                     pure-noise standard deviation $\sigma_{\eta_i}$ of
+                     $\sigma_{\eta_i}$ of $c_i(t)$ for $\sca=0$. See also
-                     $c_i(t)$ for $\sca=0$. See also appendix
+                     appendix Fig.\,\ref{fig:app_full_kern-sd}). Average value
-                     Fig.\,\ref{fig:app_full_kern-sd}). Average value $\muf$ of
+                     $\muf$ of each feature $f_i(t)$ over $\sca$.
-                     each feature $f_i(t)$ over $\sca$.
                      }
     \label{fig:app_cross_species_thresh}
 \end{figure}% Reference this one!