Time to sleep for now.

.gitignore

@@ -29,3 +29,4 @@ data/*
 *.synctex.gz
 *.synctex.gz(busy)
 *.pdfsync
+main.pdf

main.tex

@@ -1,5 +1,17 @@
 \documentclass[a4paper, 12pt]{article}
 
+\title{Emergent intensity invariance vs. signal-to-noise ratio at three consecutive processing stages along the grasshopper song recognition pathway}
+
+\author{Jona Hartling\textsuperscript{1},
+  Ale\v{s} \v{S}korjanc\textsuperscript{2},
+  Jan Benda\textsuperscript{1,3}}
+
+\date{\normalsize
+  \textsuperscript{1} Institute for Neurobiology, Eberhard Karls Universität, 72076 Tübingen, Germany \\ 
+  \textsuperscript{2} Department of Biology, Biotechnical Faculty, University of Ljubljana, Ve\v{c}na pot 111, 1000 Ljubljana, Slovenia\\
+  \textsuperscript{3} Bernstein Center for Computational Neuroscience Tübingen, 72076 Tübingen, Germany}
+
+
 \usepackage[left=2cm,right=2cm,top=2cm,bottom=2cm,includeheadfoot]{geometry}
 % \usepackage[onehalfspacing]{setspace}
 \usepackage{graphicx}
@@ -17,30 +29,38 @@
 \addto\captionsenglish{\renewcommand{\tablename}{Tab.}}
 \usepackage[separate-uncertainty=true, locale=DE]{siunitx}
 \sisetup{output-exponent-marker=\ensuremath{\mathrm{e}}}
+
+%%%%% section style %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\usepackage[sf,bf,it,big,clearempty]{titlesec}
+\usepackage{titling}
+\renewcommand{\maketitlehooka}{\sffamily\bfseries}
+\renewcommand{\maketitlehookb}{\rmfamily\mdseries}
+\setcounter{secnumdepth}{-1}
+
+%%%%% bibliography %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\usepackage[round,colon]{natbib}
+\renewcommand{\bibsection}{\section{References}}
+\setlength{\bibsep}{0pt}
+\setlength{\bibhang}{1.5em}
+\bibliographystyle{jneurosci}
 % \usepackage[capitalize]{cleveref}
 % \crefname{figure}{Fig.}{Figs.}
 % \crefname{equation}{Eq.}{Eqs.}
 % \creflabelformat{equation}{#2#1#3}
-\usepackage[
-    backend=bibtex,
-    style=authoryear,
-    pluralothers=true,
-    maxcitenames=1,
-    mincitenames=1
-    ]{biblatex}
-\addbibresource{cite.bib}
+%\usepackage[
+%    backend=bibtex,
+%    style=authoryear,
+%    pluralothers=true,
+%    maxcitenames=1,
+%    mincitenames=1
+%    ]{biblatex}
+%\addbibresource{cite.bib}
 %\bibdata
 %\bibstyle
 %\citation
-
-\title{Emergent intensity invariance vs. signal-to-noise ratio at three consecutive processing stages along the grasshopper song recognition pathway}
-\author{Jona Hartling$^1$, Ale\v{s} \v{S}korjanc$^2$, Jan Benda$^{1,3}$}
-\date{$^1$ Institute for Neurobiology, Eberhard Karls Universität, 72076 Tübingen, Germany \\ 
-  $^2$ Department of Biology, Biotechnical Faculty, University of Ljubljana, Ve\v{c}na pot 111, 1000 Ljubljana, Slovenia\\
-  $^3$ Bernstein Center for Computational Neuroscience Tübingen, 72076 Tübingen, Germany}
-
-\begin{document}
-\maketitle{}
+ 
+%%%%% hyperref %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\usepackage[breaklinks=true,colorlinks=true,citecolor=blue!30!black,urlcolor=blue!30!black,linkcolor=blue!30!black]{hyperref}
 
 % Text references and citations:
 \newcommand{\bcite}[1]{\cite{#1}}
@@ -110,6 +130,19 @@
 \newcommand{\tstat}{T_{\text{total}}} % Time interval where c(t) is stationary
 \newcommand{\muf}{\mu_{f_i}} % Average feature value
 
+%%%%% notes %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\newcommand{\note}[2][]{\textcolor{red}{[#1: #2]}}
+%\newcommand{\note}[2][]{}
+\newcommand{\notejh}[1]{\note[JH]{#1}}
+\newcommand{\notejb}[1]{\note[JB]{#1}}
+\newcommand{\noteas}[1]{\note[AS]{#1}}
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5
+\begin{document}
+
+\maketitle
+
 \section{Introduction}
 % % Drosophila/visual/article:
 % \bcite{ketkar2023multifaceted}
@@ -130,22 +163,16 @@
 % \bcite{bolding2018recurrent}
 
 % Introduction to intensity invariance:
-Intensity invariance is a fundamental property of sensory systems across
-modalities and species, from fruit flies~(\bcite{ozeri2018fast};
-\bcite{ketkar2023multifaceted}) over crickets~(\bcite{benda2008spike}) and
-grasshoppers~(\bcite{clemens2010intensity}) to
-rodents~(\bcite{bolding2018recurrent}) and
-primates~(\bcite{barbour2011intensity}). It allows for the robust recognition
+Intensity invariance is a fundamental property of sensory systems across different modalities. For example, it has been shown in auditory systems of drosophila \citep{ozeri2018fast}, crickets \citep{benda2008spike}, grasshoppers \citep{clemens2010intensity}, and primates \citep{barbour2011intensity}, visual systems in drosophila \citep{ketkar2023multifaceted}, and olfactory systems of rodents \citep{bolding2018recurrent}. It allows for the robust recognition
 of behaviorally relevant stimuli despite variations in stimulus intensity.
 However, the computational mechanisms underlying intensity invariance are often
 difficult to disentangle. Here, we use a physiologically inspired functional
-model of the grasshopper song recognition pathway to investigate the emergence
-of intensity invariance throughout the auditory processing stream.
+model of the grasshopper (\textit{Acrididae}) song recognition pathway to investigate the emergence
+of intensity invariance at different levels of the auditory processing stream, which has been studied extensively.
 
 % Why the grasshopper auditory system?
 % Why focus on song recognition among other auditory functions?
-The auditory system of grasshoppers~(\textit{Acrididae}) has been studied
-extensively over the years. Grasshoppers rely on their sense of hearing for
+Grasshoppers rely on their hearing for
 intraspecific communication --- including mate
 attraction~(\bcite{helversen1972gesang}) and
 evaluation~(\bcite{stange2012grasshopper}), sender
@@ -157,7 +184,7 @@ contexts~(\bcite{otte1970comparative}). The most conspicuous acoustic signals
 of grasshoppers are their species-specific calling songs, which broadcast the
 presence of the singing individual to potential mates within range. These songs
 are usually more characteristic of a species than morphological
-traits~(\bcite{tishechkin2016acoustic}; \bcite{tarasova2021eurasius}), which
+traits (\bcite{tishechkin2016acoustic}; \bcite{tarasova2021eurasius}), which
 can vary greatly within species~(\bcite{rowell1972variable};
 \bcite{kohler2017morphological}). The reliance on songs to mediate reproduction
 represents a strong evolutionary driving force that resulted in a massive
@@ -182,7 +209,7 @@ Grasshopper songs, like all acoustic signals, are subject to sound attenuation,
 which depends on the distance from the sound source, the frequency content of
 the signal, and the vegetation of the habitat~(\bcite{michelsen1978sound}).
 Sound attenuation has two major consequences for song recognition. First, the
-amplitude dynamics of the song pattern degrade with increasing distance to the
+amplitude dynamics of the song pattern degrades with increasing distance to the
 sender, which limits the effective communication range of grasshoppers
 to~\mbox{1\,--\,2\,m} in their typical grassland
 habitats~(\bcite{lang2000acoustic}). Second, the intensity of a song at the
@@ -203,9 +230,9 @@ degrees~(\bcite{clemens2010intensity}) and by different
 mechanisms~(\bcite{hildebrandt2009origin}).
 
 % How did we expand on the previous framework (feat. Clemens et al.)?
-In the current study, we leverage functional modelling to trace the emergence
-of intensity invariance through individual processing steps of the grasshopper
-song recognition pathway. The model pathway we propose here is based on a
+In the current study, we use functional modelling of the grasshopper
+song recognition pathway to identify individual processing steps that
+contribute to intensity invariance of the auditory system. The model pathway we propose here is based on a
 previous functional model framework for song recognition in both
 crickets~(\bcite{clemens2013computational}; \bcite{hennig2014time}) and
 grasshoppers~(\bcite{clemens2013feature}; review on
@@ -216,14 +243,14 @@ It includes feature extraction by a bank of linear-nonlinear feature detectors,
 evidence accumulation by temporal averaging of each feature, and categorical
 decision making by a weighted linear combination of feature values. We adopted
 the general structure of the existing framework and extended it by a
-physiologically plausible preprocessing stage --- including spectral filtering,
+physiologically plausible preprocessing stage --- spectral filtering,
 envelope extraction, logarithmic compression, and intensity adaptation ---
 which allows the model to operate on unmodified recordings of natural
 grasshopper songs. The resulting model pathway thus covers the entire auditory
 processing stream from the initial reception of airborne sound waves to the
 generation of a high-dimensional feature representation that allows for the
 categorical recognition of conspecific songs. It incorporates anatomical,
-physiological, and ethological evidence from several decades of research on the
+physiological, and ethological evidence from research on the
 grasshopper auditory system. In the following, we provide a side-by-side
 account of the known physiological processing steps along the song recognition
 pathway and their functional approximations in the model pathway. We then
@@ -1664,7 +1691,7 @@ additional certainty.
 \subsection{Invariant processing in the grasshopper auditory system}
 \label{sec:general_inv}
 
-% Invariance in the general (systemic) sense:
+% Invariance in the general (systemic) sense (could be skipped if too much):
 The notion of invariance is fundamental for sensory processing systems.
 Invariance, in the general sense, can be described as the property of a
 transformation to maintain variation across certain meaningful input parameters
@@ -1758,71 +1785,130 @@ that are robust to noise masking~(\bcite{einhaupl2011attractiveness}).
 % Trading SNR for log-HP intensity invariance (+variability, +general principle):
 The SNR of each song representation prior to $\adapt(t)$ increases
 monotonically with $\sca$~(excluding $0<\sca\ll1$, noise regime). These
-representations maintain and improve the initial SNR of $\raw(t)$ and hence
-never achieve intensity invariance. In contrast, the SNR of the
-intensity-invariant $\adapt(t)$ never exceeds its saturation level even for
-arbitrarily high $\sca$. The saturation level of $\adapt(t)$ varies across
-species and songs. This variability is likely rooted in the way in which
-logarithmic compression acts on the specific distribution of $\env(t)$, which
-depends on the $\fc$ of the lowpass filter as well as the temporal structure
-and frequency spectrum of the rectified $\filt(t)$. Overall, $\adapt(t)$ has
-never been observed to exceed a SNR of around~10 across all songs. The low SNR
-of $\adapt(t)$ partially results from the amplification of smaller values of
-$\env(t)$ by the logarithm, which raises the noise floor of $\adapt(t)$. Still,
-the reduction in SNR is substantial --- considering that the SNR of preceeding
-song representations has been orders of magnitude higher --- but is likely a
-necessary price to pay for the intensity invariance of $\adapt(t)$. After all,
-a transformation cannot compress a range of different input intensities into a
+representations maintain the full extent of the initial SNR of $\raw(t)$ and
+hence never achieve intensity invariance. In contrast, the SNR of $\adapt(t)$
+saturates for sufficiently high $\sca$. Accordingly, $\adapt(t)$ is
+intensity-invariant but cannot have a higher SNR than the saturation level,
+which indicates a fundamental trade-off. The saturation level of $\adapt(t)$
+varies across species and songs. This variability is likely rooted in the way
+in which logarithmic compression acts on the specific $\env(t)$, which depends
+on the $\fc$ of the lowpass filter as well as the temporal structure and
+frequency spectrum of the rectified $\filt(t)$. Across all songs, the
+saturation level of $\adapt(t)$ has never been observed to exceed a SNR of
+around~10. This is a substantial reduction in SNR, considering that the SNR of
+preceeding representations had been orders of magnitude higher. Part of this
+reduction stems from the amplification of smaller values of $\env(t)$ by
+logarithmic compression, which raises the noise floor of $\adapt(t)$ relative
+to the song. Accordingly, the low SNR of $\adapt(t)$ appears to be a necessary
+price to pay for its intensity invariance through logarithmic compression and
+adaptation. But the trade-off between intensity invariance and SNR likely goes
+beyond the particular mechanisms along the pathway. After all, a transformation
+is not expected to compress a range of different input intensities into a
 constant output intensity without sacrificing some of the corresponding input
-SNR. Accordingly, the trade-off between intensity invariance and SNR is not
-expected to be specific to the particular mechanisms along the pathway but
-presumably applies to any transformation that achieves or improves intensity
-invariance.
+SNR. Accordingly, the trade-off likely is a more general principle that might
+apply to any transformation that achieves or improves intensity invariance.
 
-Thresholding and temporal averaging renders feature $f_i(t)$
-intensity-invariant for sufficiently large $\sca$. The trade-off between
-intensity invariance and SNR is mediated by threshold value $\thr$. A lower
-$\thr$ ($\thr\to0$) improves intensity invariance by shifting the saturation
-point towards lower $\sca$ but also decreases the SNR of $f_i(t)$. The
-saturation level of $f_i(t)$ is independent of $\thr$ as long as the intensity
-invariance by the previous mechanism is neglected. The SNR of $f_i(t)$ is
-therefore determined solely by the pure-noise response of $f_i(t)$. The
-distribution $\pci$ of the pure-noise kernel response $c_i(t)$ is largely a
-normal distribution with mean $\mu\approx0$ for all kernels $k_i(t)$. The value
-of the pure-noise $f_i(t)$ is hence 0.5 for $\thr=0$ and decreases for higher
+% Dependence of thresh-LP intensity invariance on threshold value (+unlimited SNR):
+The second mechanism of intensity invariance consists of thresholding and
+temporal averaging of $c_i(t)$ into $f_i(t)$. Here, the trade-off between
+intensity invariance and SNR is mediated by the threshold value $\thr$. The
+effects of $\thr$ on the intensity invariance and SNR of $f_i(t)$ are best
+assessed if the mechanism is viewed in isolation. A lower $\thr$~($\thr\to0$)
+improves the intensity invariance of $f_i(t)$ by shifting the saturation point
+towards lower $\sca$. The saturation level of $f_i(t)$ is mostly independent of
+$\thr$, assuming that $\sca$ is sufficiently large. However, the lower $\thr$,
+the more of the pure-noise $c_i(t)$ is included in $f_i(t)$ and hence the
+higher the noise floor of $f_i(t)$, which decreases the SNR of $f_i(t)$. The
+distribution $\pci$ of the pure-noise $c_i(t)$ is very close to a normal
+distribution with mean $\mu\approx0$ for all kernels in the set. The value of
+the pure-noise $f_i(t)$ is hence 0.5 for $\thr=0$ and decreases for higher
 $\thr$. If $\thr$ is set above the maximum of $c_i(t)$, the pure-noise feature
 value is 0, which results in an "unlimited" SNR of $f_i(t)$. In this case, any
 non-zero feature value that is sustained for a sufficient duration could serve
-as indicator for the presence of $\soc(t)$, although at the cost of a higher
-saturation point. The maximum of the pure-noise $c_i(t)$ is assumed to be very
-small due to the various SNR improvements along the pathway, so that the
-required increase in $\thr$ and hence the saturation point of $f_i(t)$ is not
-expected to be substantial. However, exploiting the capacity of $f_i(t)$ for
-arbitrarily high SNR would certainly require a fine evolutionary tuning of
-$\thr$ to the properties of both the species-specific song and the natural
-noise in a certain habitat.
+as indicator for the presence of $\soc(t)$ in $\raw(t)$. Of course, this would
+require a fine evolutionary tuning of $\thr$ to the properties of the natural
+noise in a certain habitat in order to avoid false positives.
 
-\newpage
-\subsection{Intensity invariance versus intensity invariance}
+% Interaction between the two mechanisms of intensity invariance (expectations):
+% (Also: Extremely important, but maybe too wordy?)
+The intensity invariance of $f_i(t)$ is not only determined by the second
+mechanism but by the interaction between the two consecutive mechanisms along
+the pathway. This interaction is difficult to assess systematically due to the
+multitude of involved parameters. A basic expectation is that the combined
+effects of the two mechanisms mostly depend on which mechanism achieves a lower
+saturation point, assuming that $f_i(t)$ is always intensity-invariant if
+$\adapt(t)$ is already intensity-invariant. Furthermore, it is necessary to
+distinguish between the intrinsic saturation point of $f_i(t)$ --- the
+saturation point that the second mechanism can achieve in isolation --- and its
+actual saturation point including the effects of the first mechanism. The same
+distinction applies to the saturation level of $f_i(t)$. If the intrinsic
+saturation point of $f_i(t)$ is lower than the saturation point of $\adapt(t)$,
+$f_i(t)$ is expected to reach the intrinsic saturation level at the intrinsic
+saturation point. In contrast, if the intrinsic saturation point of $f_i(t)$ is
+higher than the saturation point of $\adapt(t)$, $f_i(t)$ is expected to
+saturate at the lower saturation point of $\adapt(t)$ instead. This has no
+detrimental effect on the intensity invariance of $f_i(t)$. However, $f_i(t)$
+is then also expected to saturate below its intrinsic saturation level.
+Moreover, the saturation level of $f_i(t)$ is not independent of $\thr$ anymore
+but decreases with increasing $\thr$. A lower saturation level of $f_i(t)$ is
+not necessarily detrimental to the SNR of $f_i(t)$ --- $f_i(t)$ can still
+achieve an arbitrarily high SNR by setting $\thr$ just above the maximum
+pure-noise $c_i(t)$. More importantly, a lower saturation level of $f_i(t)$
+also means that the range of possible feature values that $f_i(t)$ can take on
+is limited compared to the case where $f_i(t)$ can reach its intrinsic
+saturation level. This effectively restricts the part of the feature space that
+is available for species-specific song representation. The interaction between
+the two mechanisms of intensity invariance could therefore have unfavorable
+consequences if the first mechanism results in a lower saturation point than
+the second mechanism.
 
-Two consecutive mechanisms of intensity invariance do not necessarily add up to
-a stronger overall intensity invariance. If the first mechanism results in a
-lower saturation point than the second mechanism by itself, the saturation
-point of feature $f_i(t)$ will be determined solely by the first mechanism. In
-this case, the saturation level of $f_i(t)$ will conform to the intensity that
-$f_i(t)$ can reach for the given saturation point rather than the intrinsic
-saturation level of $f_i(t)$. Conversely, if the second mechanism results in a
-lower saturation point than the first mechanism, both the saturation point and
-the saturation level of $f_i(t)$ will be determined by the second mechanism.
-The saturation points of $f_i(t)$ across the set are distributed over a much
-wider range than those of the preceeding kernel responses $c_i(t)$, which
-suggests that the interaction between the two mechanisms is specific to
-individual kernels. A number of $f_i(t)$ achieve a lower saturation point than
-the respective $c_i(t)$, whereas some $f_i(t)$ exhibit similar or only
-marginally lower saturation points. In these cases, the question arises to what
+% Interaction between the two mechanisms of intensity invariance (current results):
+The saturation point and saturation level of a feature in the set varies with
+the specific kernel.
+
+The combined effects of the two mechanisms on the intensity invariance of a
+specific feature in the set vary between different kernels
+
+Based on the current results, it is difficult to assess which of the two
+mechanisms has a stronger effect on the intensity invariance of a specific
+feature in the set.
+
+The combined effects of the two mechanisms on the intensity invariance of a
+specific feature in the set vary between different kernels. It is difficult to
+assess which of the two mechanisms achieves a lower saturation point for a
+specific feature. On the one hand, the distribution of saturation levels across
+the feature set is not symmetric around a feature value of 0.5, which is the
+case if the logarithmic compression along the pathway is disabled. This result
+indicates that a number of features does not reach the intrinsic saturation
+level, which suggests that the intensity invariance of these features is
+determined by the first mechanism rather than the second mechanism. One the
+other hand, the distribution of saturation points across the feature set
+indicates that a number of features does indeed achieve a lower saturation
+point than the preceeding representations. This result suggests that the
+intensity invariance of these features is determined by the second mechanism
+rather than the first mechanism. In either case, the question arises to what
 extent two consecutive mechanisms of intensity invariance are actually
 beneficial for the overall system.
 
+These cases raise the question whether the first mechanism is actually
+necessary for the overall system if the second mechanism can apparently achieve
+intensity invariance with a lower saturation point. These cases raise the
+question whether intensity invariance by the first mechanism --- while
+achieving a lower saturation point than the second mechanism --- is actually
+beneficial
+
+The saturation point of $f_i(t)$ varies between different kernels in the set. A
+number of $f_i(t)$ achieve a lower saturation point than $c_i(t)$ --- and hence a lower saturation point than $\adapt(t)$ --- which
+indicates that the second mechanism takes precedence over the first mechanism.
+Some $f_i(t)$ exhibit similar or only marginally lower saturation points than
+
+The saturation points of $f_i(t)$ across the set are distributed over a much
+wider range than those of the preceeding $c_i(t)$, 
+
+In these cases, the question arises to what extent two
+consecutive mechanisms of intensity invariance are actually beneficial for the
+overall system.
+
 From a computational perspective, the answer could be that logarithmic
 compression and adaptation is a necessary preprocessing step towards robust
 $f_i(t)$ because it works towards a more consistent distribution $\pci$ of
@@ -1927,7 +2013,8 @@ habitat.
 % - How to integrate the available knowledge on anatomy, physiology, ethology?\\
 % $\rightarrow$ Abstract, simplify, formalize $\rightarrow$ Functional model framework
 
-\printbibliography
+%\printbibliography
+\bibliography{cite}
 
 \newpage
 \section{Appendix}

python/fig_invariance_full.py

@@ -442,6 +442,7 @@ for stage in stages:
         
     # Indicate saturation point(s):
     if stage in ['log', 'inv', 'conv', 'feat']:
+        # Get and plot single curve saturation point:
         ind = get_saturation(curve, **plateau_settings)[1]
         crit_inds[stage] = ind
         scale = scales[ind]
@@ -452,6 +453,13 @@ for stage in stages:
                          transform=raw_axes[0].get_xaxis_transform())
         raw_axes[0].vlines(scale, raw_axes[0].get_ylim()[0], curve[ind],
                            color=color, **plateau_line_kwargs)
+        if stage in ['conv', 'feat']:
+            # Get and log distribution of swarm saturation points:
+            inds = np.array(get_saturation(measure, **plateau_settings)[1])
+            if np.isnan(inds).sum():
+                print('WARNING: Found NaN saturation point(s)!')
+                inds = inds[~np.isnan(inds)].astype(int)
+            crit_scales_swarm[stage] = scales[inds]
 
     ## NORMALIZED MEASURE:
 
@@ -476,11 +484,6 @@ for stage in stages:
         fill_kwargs = dist_fill_kwargs | dict(color=color)
         y_dist(base_insets[i1], measure[-1], nbins=100, log=True,
                line_kwargs=line_kwargs, fill_kwargs=fill_kwargs)
-        # Get and log distribution of saturation points:
-        inds = np.array(get_saturation(measure, **plateau_settings)[1])
-        if np.isnan(inds).sum():
-            inds = inds[~np.isnan(inds)].astype(int)
-        crit_scales_swarm[stage] = scales[inds]
         if stage == 'feat':
             # Plot distribution of saturation points on shared bins:
             bin_lims = [0.01, 1.1 * max([s.max() for s in crit_scales_swarm.values()])]