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Managed 1st half of results text, began with 2nd half.

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j-hartling
2026-02-25 16:53:49 +01:00
parent c700e1723c
commit cc701a09f8
8 changed files with 216 additions and 2423 deletions
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15
main.aux
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@@ -242,17 +242,16 @@
\newlabel{eq:lowpass}{{10}{9}{}{}{}} \newlabel{eq:lowpass}{{10}{9}{}{}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces \textbf {Representations of a song of \textit {O. rufipes} during the feature extraction stage.} Different colors indicate Gabor kernels with different lobe number $n$ and sign, with lighter colors for higher $n$~($1\,\leq \,n\,\leq \,4$; both $+$ and $-$ per $n$; two kernel widths $\sigma $ of $4\,$ms and $32\,$ms per sign). \textbf {a}:~Kernel-specific filter responses. \textbf {b}:~Binary responses. \textbf {c}:~Finalized features. }}{10}{}\protected@file@percent } \@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces \textbf {Representations of a song of \textit {O. rufipes} during the feature extraction stage.} Different colors indicate Gabor kernels with different lobe number $n$ and sign, with lighter colors for higher $n$~($1\,\leq \,n\,\leq \,4$; both $+$ and $-$ per $n$; two kernel widths $\sigma $ of $4\,$ms and $32\,$ms per sign). \textbf {a}:~Kernel-specific filter responses. \textbf {b}:~Binary responses. \textbf {c}:~Finalized features. }}{10}{}\protected@file@percent }
\newlabel{fig:stages_feat}{{3}{10}{}{}{}} \newlabel{fig:stages_feat}{{3}{10}{}{}{}}
\@writefile{toc}{\contentsline {section}{\numberline {3}Two mechanisms driving the emergence of intensity-invariant song representations}{10}{}\protected@file@percent } \@writefile{toc}{\contentsline {section}{\numberline {3}Two mechanisms driving the emergence of intensity-invariant song representation}{10}{}\protected@file@percent }
\@writefile{toc}{\contentsline {subsection}{\numberline {3.1}Logarithmic compression \& spike-frequency adaptation}{10}{}\protected@file@percent } \@writefile{toc}{\contentsline {subsection}{\numberline {3.1}Logarithmic compression \& spike-frequency adaptation}{10}{}\protected@file@percent }
\newlabel{eq:toy_env}{{11}{10}{}{}{}} \newlabel{eq:toy_env}{{11}{10}{}{}{}}
\newlabel{eq:toy_snr}{{12}{11}{}{}{}} \newlabel{eq:toy_log}{{12}{11}{}{}{}}
\newlabel{eq:toy_log}{{13}{11}{}{}{}} \newlabel{eq:toy_highpass}{{13}{11}{}{}{}}
\newlabel{eq:toy_highpass}{{14}{11}{}{}{}} \newlabel{eq:toy_snr}{{14}{11}{}{}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.2}Threshold nonlinearity \& temporal averaging}{12}{}\protected@file@percent } \@writefile{toc}{\contentsline {subsection}{\numberline {3.2}Thresholding nonlinearity \& temporal averaging}{11}{}\protected@file@percent }
\newlabel{eq:pdf_split}{{15}{12}{}{}{}} \newlabel{eq:pdf_split}{{15}{12}{}{}{}}
\newlabel{eq:pdf}{{16}{12}{}{}{}} \newlabel{eq:feat_avg}{{16}{12}{}{}{}}
\newlabel{eq:feat_avg}{{17}{12}{}{}{}} \newlabel{eq:feat_prop}{{17}{12}{}{}{}}
\newlabel{eq:feat_prop}{{18}{13}{}{}{}}
\abx@aux@cite{0}{stumpner1991auditory} \abx@aux@cite{0}{stumpner1991auditory}
\abx@aux@segm{0}{0}{stumpner1991auditory} \abx@aux@segm{0}{0}{stumpner1991auditory}
\@writefile{toc}{\contentsline {section}{\numberline {4}Discriminating species-specific song\\patterns in feature space}{14}{}\protected@file@percent } \@writefile{toc}{\contentsline {section}{\numberline {4}Discriminating species-specific song\\patterns in feature space}{14}{}\protected@file@percent }

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[0] Config.pm:307> INFO - This is Biber 2.19 [0] Config.pm:307> INFO - This is Biber 2.19
[0] Config.pm:310> INFO - Logfile is 'main.blg' [0] Config.pm:310> INFO - Logfile is 'main.blg'
[37] biber:340> INFO - === Mo Feb 23, 2026, 16:48:10 [36] biber:340> INFO - === Mi Feb 25, 2026, 16:44:24
[44] Biber.pm:419> INFO - Reading 'main.bcf' [44] Biber.pm:419> INFO - Reading 'main.bcf'
[72] Biber.pm:979> INFO - Found 55 citekeys in bib section 0 [72] Biber.pm:979> INFO - Found 55 citekeys in bib section 0
[78] Biber.pm:4419> INFO - Processing section 0 [78] Biber.pm:4419> INFO - Processing section 0
[82] Biber.pm:4610> INFO - Looking for bibtex file 'cite.bib' for section 0 [83] Biber.pm:4610> INFO - Looking for bibtex file 'cite.bib' for section 0
[84] bibtex.pm:1713> INFO - LaTeX decoding ... [84] bibtex.pm:1713> INFO - LaTeX decoding ...
[113] bibtex.pm:1519> INFO - Found BibTeX data source 'cite.bib' [120] bibtex.pm:1519> INFO - Found BibTeX data source 'cite.bib'
[278] UCollate.pm:68> INFO - Overriding locale 'en-US' defaults 'normalization = NFD' with 'normalization = prenormalized' [297] UCollate.pm:68> INFO - Overriding locale 'en-US' defaults 'variable = shifted' with 'variable = non-ignorable'
[278] UCollate.pm:68> INFO - Overriding locale 'en-US' defaults 'variable = shifted' with 'variable = non-ignorable' [297] UCollate.pm:68> INFO - Overriding locale 'en-US' defaults 'normalization = NFD' with 'normalization = prenormalized'
[278] Biber.pm:4239> INFO - Sorting list 'nyt/global//global/global' of type 'entry' with template 'nyt' and locale 'en-US' [297] Biber.pm:4239> INFO - Sorting list 'nyt/global//global/global' of type 'entry' with template 'nyt' and locale 'en-US'
[278] Biber.pm:4245> INFO - No sort tailoring available for locale 'en-US' [297] Biber.pm:4245> INFO - No sort tailoring available for locale 'en-US'
[300] bbl.pm:660> INFO - Writing 'main.bbl' with encoding 'UTF-8' [322] bbl.pm:660> INFO - Writing 'main.bbl' with encoding 'UTF-8'
[309] bbl.pm:763> INFO - Output to main.bbl [333] bbl.pm:763> INFO - Output to main.bbl
[310] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_DBXa/347c261ec4135a5723bef5c751f5078f_38197.utf8, line 10, warning: 6 characters of junk seen at toplevel [333] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_E030/347c261ec4135a5723bef5c751f5078f_66584.utf8, line 10, warning: 6 characters of junk seen at toplevel
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[310] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_DBXa/347c261ec4135a5723bef5c751f5078f_38197.utf8, line 38, warning: 6 characters of junk seen at toplevel [333] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_E030/347c261ec4135a5723bef5c751f5078f_66584.utf8, line 38, warning: 6 characters of junk seen at toplevel
[310] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_DBXa/347c261ec4135a5723bef5c751f5078f_38197.utf8, line 49, warning: 6 characters of junk seen at toplevel [333] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_E030/347c261ec4135a5723bef5c751f5078f_66584.utf8, line 49, warning: 6 characters of junk seen at toplevel
[310] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_DBXa/347c261ec4135a5723bef5c751f5078f_38197.utf8, line 58, warning: 6 characters of junk seen at toplevel [334] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_E030/347c261ec4135a5723bef5c751f5078f_66584.utf8, line 58, warning: 6 characters of junk seen at toplevel
[310] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_DBXa/347c261ec4135a5723bef5c751f5078f_38197.utf8, line 73, warning: 6 characters of junk seen at toplevel [334] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_E030/347c261ec4135a5723bef5c751f5078f_66584.utf8, line 73, warning: 6 characters of junk seen at toplevel
[310] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_DBXa/347c261ec4135a5723bef5c751f5078f_38197.utf8, line 82, warning: 6 characters of junk seen at toplevel [334] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_E030/347c261ec4135a5723bef5c751f5078f_66584.utf8, line 82, warning: 6 characters of junk seen at toplevel
[310] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_DBXa/347c261ec4135a5723bef5c751f5078f_38197.utf8, line 91, warning: 6 characters of junk seen at toplevel [334] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_E030/347c261ec4135a5723bef5c751f5078f_66584.utf8, line 91, warning: 6 characters of junk seen at toplevel
[310] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_DBXa/347c261ec4135a5723bef5c751f5078f_38197.utf8, line 100, warning: 6 characters of junk seen at toplevel [334] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_E030/347c261ec4135a5723bef5c751f5078f_66584.utf8, line 100, warning: 6 characters of junk seen at toplevel
[310] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_DBXa/347c261ec4135a5723bef5c751f5078f_38197.utf8, line 109, warning: 6 characters of junk seen at toplevel [334] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_E030/347c261ec4135a5723bef5c751f5078f_66584.utf8, line 109, warning: 6 characters of junk seen at toplevel
[310] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_DBXa/347c261ec4135a5723bef5c751f5078f_38197.utf8, line 118, warning: 6 characters of junk seen at toplevel [334] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_E030/347c261ec4135a5723bef5c751f5078f_66584.utf8, line 118, warning: 6 characters of junk seen at toplevel
[310] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_DBXa/347c261ec4135a5723bef5c751f5078f_38197.utf8, line 127, warning: 6 characters of junk seen at toplevel [334] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_E030/347c261ec4135a5723bef5c751f5078f_66584.utf8, line 127, warning: 6 characters of junk seen at toplevel
[310] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_DBXa/347c261ec4135a5723bef5c751f5078f_38197.utf8, line 136, warning: 6 characters of junk seen at toplevel [334] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_E030/347c261ec4135a5723bef5c751f5078f_66584.utf8, line 136, warning: 6 characters of junk seen at toplevel
[310] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_DBXa/347c261ec4135a5723bef5c751f5078f_38197.utf8, line 157, warning: 6 characters of junk seen at toplevel [334] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_E030/347c261ec4135a5723bef5c751f5078f_66584.utf8, line 157, warning: 6 characters of junk seen at toplevel
[310] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_DBXa/347c261ec4135a5723bef5c751f5078f_38197.utf8, line 178, warning: 6 characters of junk seen at toplevel [334] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_E030/347c261ec4135a5723bef5c751f5078f_66584.utf8, line 178, warning: 6 characters of junk seen at toplevel
[310] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_DBXa/347c261ec4135a5723bef5c751f5078f_38197.utf8, line 187, warning: 6 characters of junk seen at toplevel [334] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_E030/347c261ec4135a5723bef5c751f5078f_66584.utf8, line 187, warning: 6 characters of junk seen at toplevel
[310] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_DBXa/347c261ec4135a5723bef5c751f5078f_38197.utf8, line 196, warning: 6 characters of junk seen at toplevel [334] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_E030/347c261ec4135a5723bef5c751f5078f_66584.utf8, line 196, warning: 6 characters of junk seen at toplevel
[310] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_DBXa/347c261ec4135a5723bef5c751f5078f_38197.utf8, line 207, warning: 6 characters of junk seen at toplevel [334] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_E030/347c261ec4135a5723bef5c751f5078f_66584.utf8, line 207, warning: 6 characters of junk seen at toplevel
[310] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_DBXa/347c261ec4135a5723bef5c751f5078f_38197.utf8, line 218, warning: 6 characters of junk seen at toplevel [334] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_E030/347c261ec4135a5723bef5c751f5078f_66584.utf8, line 218, warning: 6 characters of junk seen at toplevel
[310] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_DBXa/347c261ec4135a5723bef5c751f5078f_38197.utf8, line 229, warning: 6 characters of junk seen at toplevel [334] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_E030/347c261ec4135a5723bef5c751f5078f_66584.utf8, line 229, warning: 6 characters of junk seen at toplevel
[310] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_DBXa/347c261ec4135a5723bef5c751f5078f_38197.utf8, line 240, warning: 6 characters of junk seen at toplevel [334] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_E030/347c261ec4135a5723bef5c751f5078f_66584.utf8, line 240, warning: 6 characters of junk seen at toplevel
[310] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_DBXa/347c261ec4135a5723bef5c751f5078f_38197.utf8, line 249, warning: 6 characters of junk seen at toplevel [334] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_E030/347c261ec4135a5723bef5c751f5078f_66584.utf8, line 249, warning: 6 characters of junk seen at toplevel
[310] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_DBXa/347c261ec4135a5723bef5c751f5078f_38197.utf8, line 258, warning: 6 characters of junk seen at toplevel [334] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_E030/347c261ec4135a5723bef5c751f5078f_66584.utf8, line 258, warning: 6 characters of junk seen at toplevel
[310] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_DBXa/347c261ec4135a5723bef5c751f5078f_38197.utf8, line 269, warning: 6 characters of junk seen at toplevel [334] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_E030/347c261ec4135a5723bef5c751f5078f_66584.utf8, line 269, warning: 6 characters of junk seen at toplevel
[310] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_DBXa/347c261ec4135a5723bef5c751f5078f_38197.utf8, line 278, warning: 6 characters of junk seen at toplevel [334] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_E030/347c261ec4135a5723bef5c751f5078f_66584.utf8, line 278, warning: 6 characters of junk seen at toplevel
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[310] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_DBXa/347c261ec4135a5723bef5c751f5078f_38197.utf8, line 328, warning: 6 characters of junk seen at toplevel [334] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_E030/347c261ec4135a5723bef5c751f5078f_66584.utf8, line 328, warning: 6 characters of junk seen at toplevel
[310] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_DBXa/347c261ec4135a5723bef5c751f5078f_38197.utf8, line 337, warning: 6 characters of junk seen at toplevel [334] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_E030/347c261ec4135a5723bef5c751f5078f_66584.utf8, line 337, warning: 6 characters of junk seen at toplevel
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[310] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_DBXa/347c261ec4135a5723bef5c751f5078f_38197.utf8, line 419, warning: 6 characters of junk seen at toplevel [334] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_E030/347c261ec4135a5723bef5c751f5078f_66584.utf8, line 419, warning: 6 characters of junk seen at toplevel
[310] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_DBXa/347c261ec4135a5723bef5c751f5078f_38197.utf8, line 428, warning: 6 characters of junk seen at toplevel [334] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_E030/347c261ec4135a5723bef5c751f5078f_66584.utf8, line 428, warning: 6 characters of junk seen at toplevel
[310] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_DBXa/347c261ec4135a5723bef5c751f5078f_38197.utf8, line 437, warning: 6 characters of junk seen at toplevel [334] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_E030/347c261ec4135a5723bef5c751f5078f_66584.utf8, line 437, warning: 6 characters of junk seen at toplevel
[310] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_DBXa/347c261ec4135a5723bef5c751f5078f_38197.utf8, line 456, warning: 6 characters of junk seen at toplevel [334] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_E030/347c261ec4135a5723bef5c751f5078f_66584.utf8, line 456, warning: 6 characters of junk seen at toplevel
[310] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_DBXa/347c261ec4135a5723bef5c751f5078f_38197.utf8, line 491, warning: 6 characters of junk seen at toplevel [334] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_E030/347c261ec4135a5723bef5c751f5078f_66584.utf8, line 491, warning: 6 characters of junk seen at toplevel
[310] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_DBXa/347c261ec4135a5723bef5c751f5078f_38197.utf8, line 526, warning: 6 characters of junk seen at toplevel [334] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_E030/347c261ec4135a5723bef5c751f5078f_66584.utf8, line 526, warning: 6 characters of junk seen at toplevel
[310] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_DBXa/347c261ec4135a5723bef5c751f5078f_38197.utf8, line 535, warning: 6 characters of junk seen at toplevel [334] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_E030/347c261ec4135a5723bef5c751f5078f_66584.utf8, line 535, warning: 6 characters of junk seen at toplevel
[310] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_DBXa/347c261ec4135a5723bef5c751f5078f_38197.utf8, line 556, warning: 6 characters of junk seen at toplevel [334] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_E030/347c261ec4135a5723bef5c751f5078f_66584.utf8, line 556, warning: 6 characters of junk seen at toplevel
[310] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_DBXa/347c261ec4135a5723bef5c751f5078f_38197.utf8, line 565, warning: 6 characters of junk seen at toplevel [334] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_E030/347c261ec4135a5723bef5c751f5078f_66584.utf8, line 565, warning: 6 characters of junk seen at toplevel
[310] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_DBXa/347c261ec4135a5723bef5c751f5078f_38197.utf8, line 576, warning: 6 characters of junk seen at toplevel [334] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_E030/347c261ec4135a5723bef5c751f5078f_66584.utf8, line 576, warning: 6 characters of junk seen at toplevel
[310] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_DBXa/347c261ec4135a5723bef5c751f5078f_38197.utf8, line 587, warning: 6 characters of junk seen at toplevel [334] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_E030/347c261ec4135a5723bef5c751f5078f_66584.utf8, line 587, warning: 6 characters of junk seen at toplevel
[310] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_DBXa/347c261ec4135a5723bef5c751f5078f_38197.utf8, line 619, warning: 6 characters of junk seen at toplevel [334] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_E030/347c261ec4135a5723bef5c751f5078f_66584.utf8, line 619, warning: 6 characters of junk seen at toplevel
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@@ -29,6 +29,9 @@
mincitenames=1 mincitenames=1
]{biblatex} ]{biblatex}
\addbibresource{cite.bib} \addbibresource{cite.bib}
%\bibdata
%\bibstyle
%\citation
\title{Emergent intensity invariance in a physiologically inspired model of the grasshopper auditory system} \title{Emergent intensity invariance in a physiologically inspired model of the grasshopper auditory system}
\author{Jona Hartling, Jan Benda} \author{Jona Hartling, Jan Benda}
@@ -82,7 +85,7 @@
\newcommand{\fwrh}{\text{FWRH}} % Gaussian full-width at relative height \newcommand{\fwrh}{\text{FWRH}} % Gaussian full-width at relative height
\newcommand{\off}{\beta_0} % Offset for linear frequency approximation \newcommand{\off}{\beta_0} % Offset for linear frequency approximation
% Math shorthands - Threshold nonlinearity: % Math shorthands - Thresholding nonlinearity:
\newcommand{\thr}{\Theta_i} % Step function threshold value \newcommand{\thr}{\Theta_i} % Step function threshold value
\newcommand{\nl}{H(c_i\,-\,\thr)} % Shifted Heaviside step function \newcommand{\nl}{H(c_i\,-\,\thr)} % Shifted Heaviside step function
@@ -90,6 +93,7 @@
\newcommand{\soc}{s} % Song component of synthetic mixture \newcommand{\soc}{s} % Song component of synthetic mixture
\newcommand{\noc}{\eta} % Noise component of synthetic mixture \newcommand{\noc}{\eta} % Noise component of synthetic mixture
\newcommand{\sca}{\alpha} % Multiplicative scale of song component \newcommand{\sca}{\alpha} % Multiplicative scale of song component
\newcommand{\xvar}{\sigma_{x}^{2}} % Variance of synthetic mixture
\newcommand{\svar}{\sigma_{\text{s}}^{2}} % Song component variance \newcommand{\svar}{\sigma_{\text{s}}^{2}} % Song component variance
\newcommand{\nvar}{\sigma_{\eta}^{2}} % Noise component variance \newcommand{\nvar}{\sigma_{\eta}^{2}} % Noise component variance
\newcommand{\pc}{p(c_i,\,T)} % Probability density (general interval) \newcommand{\pc}{p(c_i,\,T)} % Probability density (general interval)
@@ -509,7 +513,7 @@ threshold value $\thr$ to obtain a binary response
\label{eq:binary} \label{eq:binary}
\end{equation} \end{equation}
which can be thought of as a categorization into "relevant" and "irrelevant" which can be thought of as a categorization into "relevant" and "irrelevant"
response values. In the grasshopper, these threshold nonlinearities might response values. In the grasshopper, these thresholding nonlinearities might
either be part of the processing within the ascending neurons or take place either be part of the processing within the ascending neurons or take place
further downstream~(SOURCE). Finally, the responses of the ascending neurons further downstream~(SOURCE). Finally, the responses of the ascending neurons
are assumed to be integrated somewhere in the supraesophageal are assumed to be integrated somewhere in the supraesophageal
@@ -543,47 +547,38 @@ can be read out by a simple linear classifier.
\end{figure} \end{figure}
\FloatBarrier \FloatBarrier
\section{Two mechanisms driving the emergence of intensity-invariant song representations} \section{Two mechanisms driving the emergence of intensity-invariant song representation}
% Still missing the SNR analysis. Should be able to write around it for now. % Still missing the SNR analysis. Should be able to write around it for now.
The robustness of song recognition is tied to the degree of intensity The robustness of song recognition is tied to the degree of intensity
invariance of the finalized feature representation. Ideally, the values of each invariance of the finalized feature representation. Ideally, the values of each
feature should depend only on the relative amplitude dynamics of the song feature should depend only on the relative amplitude dynamics of the song
pattern but not on the overall intensity level of the song. In the grasshopper, pattern but not on the overall intensity of the song. In the grasshopper, the
the emergence of intensity-invariant representations along the song recognition emergence of intensity-invariant representations along the song recognition
pathway likely is a distributed process that involves different neuronal pathway likely is a distributed process that involves different neuronal
populations, which raises the question of what the essential computational populations, which raises the question of what the essential computational
mechanisms are that drive this process. Within the model pathway, we identified mechanisms are that drive this process. Within the model pathway, we identified
two key mechanisms that render the song representation more invariant to two key mechanisms that render the song representation more invariant to
variations in baseline intensity. The two mechanisms each comprise a nonlinear intensity variations. The two mechanisms each comprise a nonlinear signal
signal transformation followed by a linear signal transformation but differ in transformation followed by a linear signal transformation but differ in the
the specific operations and the neural substrate involved, as outlined in the specific operations involved, as outlined in the following sections.
following sections.
\subsection{Logarithmic compression \& spike-frequency adaptation} \subsection{Logarithmic compression \& spike-frequency adaptation}
The first emergence of intensity invariance along the model pathway occurs The first notable emergence of intensity invariance along the model pathway
during the preprocessing stage, in the transition from the signal envelope occurs during the transformation of the signal envelope $\env(t)$ into the
$\env(t)$ to the logarithmically scaled envelope $\db(t)$ and then to the logarithmically scaled envelope $\db(t)$ and then into the intensity-adapted
intensity-adapted envelope $\adapt(t)$. In order to disentangle the interplay envelope $\adapt(t)$. In order to disentangle the interplay of logarithmic
of logarithmic compression and adaptation, we can rewrite compression and adaptation, $\env(t)$ can be rewritten as a synthetic mixture
$\env(t)$~(Eq.\,\ref{eq:env}) as synthetic mixture
\begin{equation} \begin{equation}
\env(t)\,=\,\sca\,\cdot\,\soc(t)\,+\,\noc(t), \qquad \env(t)\,>\,0\enspace\forall\enspace t\,\in\,\mathbb{R} \env(t)\,=\,\sca\,\cdot\,\soc(t)\,+\,\noc(t), \qquad \env(t)\,>\,0\enspace\forall\enspace t\,\in\,\mathbb{R}
\label{eq:toy_env} \label{eq:toy_env}
\end{equation} \end{equation}
of a song component $\soc(t)$ with variable multiplicative scale $\sca\geq0$ of a song component $\soc(t)$ with variable multiplicative scale $\sca\geq0$
and a fixed-scale noise component $\noc(t)$. Both $\soc(t)$ and $\noc(t)$ are and a fixed-scale noise component $\noc(t)$. Both $\soc(t)$ and $\noc(t)$ are
assumed to have unit variance~($\svar=\nvar=1$). If $\soc(t)$ and $\noc(t)$ are assumed to have unit variance. By conversion of $\env(t)$ to decibel
uncorrelated~($\soc(t)\perp\noc(t)$), the signal-to-noise ratio (SNR) of the scale~(Eq.\,\ref{eq:log}), $\sca$ turns from a multiplicative scale in linear
synthetic $\env(t)$ with ($\sca>0$) and without ($\sca=0$) song component space into an additive term, or offset, in logarithmic space
$\soc(t)$ is given by
\begin{equation}
\text{SNR}\,=\,\frac{\sigma_{s+\eta}^{2}}{\nvar}\,=\,\frac{\alpha^{2}\,\cdot\,\svar\,+\,\nvar}{\nvar}\,=\,\alpha^{2}\,+\,1
\label{eq:toy_snr}
\end{equation}
When simplifying the decibel transformation~(Eq.\,\ref{eq:log}), the logarithmically
scaled envelope $\db(t)$ can be expressed as a sum of two logarithmic terms
\begin{equation} \begin{equation}
\begin{split} \begin{split}
\db(t)\,&=\,\log \frac{\alpha\,\cdot\,s(t)\,+\,\eta(t)}{\dbref}\\ \db(t)\,&=\,\log \frac{\alpha\,\cdot\,s(t)\,+\,\eta(t)}{\dbref}\\
@@ -591,99 +586,90 @@ scaled envelope $\db(t)$ can be expressed as a sum of two logarithmic terms
\end{split} \end{split}
\label{eq:toy_log} \label{eq:toy_log}
\end{equation} \end{equation}
which allows for its separation from $\soc(t)$ but introduces a scaling of
$\noc(t)$ by the inverse of $\sca$. The subsequent
highpass-filtering~(Eq.\,\ref{eq:highpass}) of $\db(t)$ can then be
approximated as a subtraction of the local offset within a suitable time
\textbf{Logarithmic component:}\\ interval $0 \ll \thp < \frac{1}{\fc}$:
- Simplify decibel transformation (Eq.\,\ref{eq:log}) and apply to synthetic $\env(t)$\\
- Isolate scale $\alpha$ and reference $\dbref$ using logarithm product/quotient laws
$\rightarrow$ In log-space, a multiplicative scaling factor becomes additive\\
$\rightarrow$ Allows for the separation of song signal $s(t)$ and its scale $\alpha$\\
$\rightarrow$ Introduces scaling of noise term $\eta(t)$ by the inverse of $\alpha$\\
$\rightarrow$ Normalization by $\dbref$ applies equally to all terms (no individual effects)
\textbf{Adaptation component:}\\
- Highpass filter over $\db(t)$ (Eq.\,\ref{eq:highpass}) can
be approximated as subtraction of the local signal offset within a suitable time
interval $\thp$ ($0 \ll \thp < \frac{1}{\fc}$)
%
\begin{equation} \begin{equation}
\begin{split} \begin{split}
\adapt(t)\,\approx\,\db(t)\,-\,\log \frac{\alpha}{\dbref}\,=\,\log\left[s(t)\,+\,\frac{\eta(t)}{\alpha}\right] \adapt(t)\,\approx\,\db(t)\,-\,\log \frac{\alpha}{\dbref}\,=\,\log\left[s(t)\,+\,\frac{\eta(t)}{\alpha}\right]
\end{split} \end{split}
\label{eq:toy_highpass} \label{eq:toy_highpass}
\end{equation} \end{equation}
% This means that $\sca$ cannot be entirely eliminated from $\adapt(t)$, only
\textbf{Implication for intensity invariance:}\\ redistributed between $\soc(t)$ and $\noc(t)$. In consequence, if $\sca$ is
- Logarithmic scaling is essential for equalizing different song intensities\\ sufficiently large ($\sca\gg1$), $\noc(t)$ is attenuated to the point of being
$\rightarrow$ Intensity information can be manipulated more easily when in form negligible, so that $\adapt(t)$ represents $\soc(t)$ in a scale-free manner. If
of a signal offset in log-space than a multiplicative scale in linear space $\soc(t)$ and $\noc(t)$ are at similar scales ($\sca\approx1$), $\adapt(t)$
largely resembles $\db(t)$. However, if $\sca$ is sufficiently small
- Scale $\alpha$ can only be redistributed, not entirely eliminated from $\adapt(t)$\\ ($\sca\ll1$), $\noc(t)$ masks $\soc(t)$ even after the intensity adaptation.
$\rightarrow$ Turn initial scaling of song $s(t)$ by $\alpha$ into scaling of noise $\eta(t)$ by $\frac{1}{\alpha}$ Therefore, the effective intensity invariance of $\adapt(t)$ relative to
$\env(t)$ is limited by the initial scaling of $\soc(t)$ relative to $\noc(t)$;
- Capability to compensate for intensity variations, i.e. selective amplification that is, the signal-to-noise ratio (SNR) of $\env(t)$ with ($\sca>0$) and
of output $\adapt(t)$ relative to input $\env(t)$, is limited by input SNR (Eq.\,\ref{eq:toy_snr}):\\ without ($\sca=0$) song component $\soc(t)$
$\alpha\gg1$: Attenuation of $\eta(t)$ term $\rightarrow$ $s(t)$ dominates $\adapt(t)$\\
$\alpha\approx1$ Negligible effect on $\eta(t)$ term $\rightarrow$ $\adapt(t)=\log[s(t)+\eta(t)]$\\
$\alpha\ll1$: Amplification of $\eta(t)$ term $\rightarrow$ $\eta(t)$ dominates $\adapt(t)$\\
$\rightarrow$ Ability to equalize between different sufficiently large scales of $s(t)$\\
$\rightarrow$ Inability to recover $s(t)$ when initially masked by noise floor $\eta(t)$
- Logarithmic scaling emphasizes small amplitudes (song onsets, noise floor) \\
$\rightarrow$ Recurring trade-off: Equalizing signal intensity vs preserving initial SNR
\subsection{Threshold nonlinearity \& temporal averaging}
Convolved $c_i(t)$ $\xrightarrow{\nl}$ Binary $b_i(t)$ $\xrightarrow{\lp}$ Feature $f_i(t)$
\textbf{Thresholding component:}\\
- Within an observed time interval $T$, $c_i(t)$ follows probability density $\pc$\\
- Within $T$, $c_i(t)$ exceeds threshold value $\thr$ for time $T_1$ ($T_1+T_0=T$)\\
- Threshold $\nl$ splits $\pc$ around $\thr$ in two complementary parts
%
\begin{equation} \begin{equation}
\int_{\thr}^{+\infty} p(c_i,T)\,dc_i\,=\,1\,-\,\int_{-\infty}^{\thr} p(c_i,T)\,dc_i\,=\,\frac{T_1}{T} \text{SNR}(\sca)\,=\,\frac{\xvar}{\nvar}\,=\,\frac{\alpha^{2}\,\cdot\,\svar\,+\,\nvar}{\nvar}\,=\,\alpha^{2}\,+\,1, \qquad \svar\,=\,\nvar\,=\,1
\label{eq:toy_snr}
\end{equation}
which depends quadratically on $\sca$ if $\soc(t)$ and $\noc(t)$ are
uncorrelated~($\soc(t)\perp\noc(t)$). In summary, the combination of
logarithmic compression and adaptation allows for the equalization of different
sufficiently large song scales, which is essential for intensity-invariant song
representation. However, this mechanism is unable to recover songs that have
already sunken below the noise floor, which emphasizes the importance of a
sufficiently high SNR at the intial reception of the signal for reliable song
recognition.
\subsection{Thresholding nonlinearity \& temporal averaging}
The second key mechanism for the emergence of intensity invariance along the
model pathway takes place during the transformation of the kernel responses
$c_i(t)$ over the binary responses $b_i(t)$ into the finalized features
$f_i(t)$. This mechanism is mediated by the thresholding nonlinearity $\nl$. By
passing $c_i(t)$ through the thresholding nonlinearity~(Eq.\,\ref{eq:binary}),
its probability density within some observed time interval $T$ is split around
threshold value $\thr$ into two complementary parts:
\begin{equation}
\int_{\thr}^{+\infty} \pc\,dc_i\,=\,1\,-\,\int_{-\infty}^{\thr} \pc\,dc_i\,=\,\frac{T_1}{T}, \qquad \infint \pc\,dc_i\,=\,1
\label{eq:pdf_split} \label{eq:pdf_split}
\end{equation} \end{equation}
% Due to the normalization of $\pc$, the semi-definite integral over the
$\rightarrow$ Semi-definite integral over right-sided portion of split $\pc$ gives ratio right-sided part of the split $\pc$ is the ratio of time $T_1$ during which
of time $T_1$ where $c_i(t)>\thr$ to total time $T$ due to normalization of $\pc$ $c_i(t)$ exceeds $\thr$ within the total time $T$. If the subsequent lowpass
% filter~(Eq.\,\ref{eq:lowpass}) over $b_i(t)$ is approximated as temporal
averaging over a suitable time interval
$\tlp>\frac{1}{\fc}$
\begin{equation} \begin{equation}
\infint \pc\,dc_i\,=\,1 f_i(t)\,\approx\,\frac{1}{\tlp} \int_{t}^{t\,+\,\tlp} b_i(\tau)\,d\tau\,=\,\frac{T_1}{\tlp}, \qquad b_i(t)\,\in\,\{0,\,1\}
\label{eq:pdf}
\end{equation}
%
\textbf{Averaging component:}\\
- Lowpass filter over binary response $b_i(t)$ (Eq.\,\ref{eq:lowpass}) can be
approximated as temporal averaging over a suitable time interval $\tlp$ ($\tlp > \frac{1}{\fc}$)\\
- Within $\tlp$, $b_i(t)$ takes a value of 1 ($c_i(t)>\thr$) for time $T_1$ ($T_1+T_0=\tlp$)
%
\begin{equation}
f_i(t)\,\approx\,\frac{1}{\tlp} \int_{t}^{t\,+\,\tlp} b_i(\tau)\,d\tau\,=\,\frac{T_1}{\tlp}
\label{eq:feat_avg} \label{eq:feat_avg}
\end{equation} \end{equation}
% feature $f_i(t)$ likewise represents a ratio of time $T_1$ during which
$\rightarrow$ Temporal averaging over $b_i(t)\in[0,1]$ (Eq.\,\ref{eq:binary}) gives $b_i(t)$ is 1 within the total averaging interval $\tlp$. Since $b_i(t)$ is 1
ratio of time $T_1$ where $c_i(t)>\thr$ to total averaging interval $\tlp$\\ where $c_i(t)>\thr$, $f_i(t)$ relates to the probability density of $c_i(t)$ by
$\rightarrow$ Feature $f_i(t)$ approximately represents supra-threshold fraction of $\tlp$
\textbf{Combined result:}\\
- Feature $f_i(t)$ can be linked to the distribution of $c_i(t)$ using Eqs.\,\ref{eq:pdf_split} \& \ref{eq:feat_avg}
%
\begin{equation} \begin{equation}
f_i(t)\,\approx\,\int_{\thr}^{+\infty} \pclp\,dc_i\,=\,P(c_i\,>\,\thr,\,\tlp) f_i(t)\,\approx\,\int_{\thr}^{+\infty} \pclp\,dc_i\,=\,P(c_i\,>\,\thr,\,\tlp)
\label{eq:feat_prop} \label{eq:feat_prop}
\end{equation} \end{equation}
% Therefore, the value of $f_i(t)$ at every time point $t$ approximately
$\rightarrow$ Because the integral over a probability density is a cumulative signifies the cumulative probability that $c_i(t)$ exceeds $\thr$ during the
probability, the value of feature $f_i(t)$ (temporal compression of $b_i(t)$) corresponding averaging interval $\tlp$. Accordingly, the combination of
at every time point $t$ signifies the probability that convolution output thresholding nonlinearity and temporal averaging constitutes a remapping of a
$c_i(t)$ exceeds the threshold value $\thr$ during the corresponding averaging quantity that encodes temporal similarity between signal $\adapt(t)$ and kernel
interval $\tlp$ $k_i(t)$ into a quantity that encodes a duty cycle with respect to $\thr$.
Accordingly, the combination of
thresholding nonlinearity and temporal averaging constitutes a remapping of the
amplitude-encoding quantity $c_i(t)$ into the duty cycle-encoding quantity
$f_i(t)$ by binning graded amplitude values into one of two categorical states.
This deliberate loss of precise amplitude information is the key to intensity
invariance of the finalized features, as different scales of $c_i(t)$ can
result in similar $T_1$ segments depending on the magnitude of the derivative
of $c_i(t)$ in temporal proximity to time points at which $c_i(t)$ crosses
$\thr$.
\textbf{Implication for intensity invariance:}\\ \textbf{Implication for intensity invariance:}\\
- Convolution output $c_i(t)$ quantifies temporal similarity between amplitudes of - Convolution output $c_i(t)$ quantifies temporal similarity between amplitudes of
@@ -743,6 +729,21 @@ large-scale AM, current overall intensity level)\\
$\rightarrow$ Without time scale selectivity, any fully intensity-invariant $\rightarrow$ Without time scale selectivity, any fully intensity-invariant
output will be a flat line 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
of a signal offset in log-space than a multiplicative scale in linear space
- Capability to compensate for intensity variations, i.e. selective amplification
of output $\adapt(t)$ relative to input $\env(t)$, is limited by input SNR (Eq.\,\ref{eq:toy_snr}):\\
$\rightarrow$ Ability to equalize between different sufficiently large scales of $s(t)$\\
$\rightarrow$ Inability to recover $s(t)$ when initially masked by noise floor $\eta(t)$
- Logarithmic scaling emphasizes small amplitudes (song onsets, noise floor) \\
$\rightarrow$ Recurring trade-off: Equalizing signal intensity vs preserving initial SNR
The model pathway includes a rather large number of Gabor kernels compared to The model pathway includes a rather large number of Gabor kernels compared to
the 15 to 20 ascending neurons in the grasshopper auditory the 15 to 20 ascending neurons in the grasshopper auditory
system~(\bcite{stumpner1991auditory}). system~(\bcite{stumpner1991auditory}).