diff --git a/main.pdf b/main.pdf index 1a9c36c..fc10545 100644 Binary files a/main.pdf and b/main.pdf differ diff --git a/main.tex b/main.tex index 4a064d2..ed566f8 100644 --- a/main.tex +++ b/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 -$\sca$ 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!