From 53e43f61f59d5c737f9ca9d14f5642f191f72da4 Mon Sep 17 00:00:00 2001 From: j-hartling Date: Tue, 11 Nov 2025 15:52:07 +0100 Subject: [PATCH] Currently formalizing log-invariance (WIP). --- main.aux | 2 +- main.fdb_latexmk | 17 +++++++------- main.fls | 1 + main.log | 27 ++++++++++------------- main.pdf | Bin 142014 -> 146175 bytes main.synctex.gz | Bin 32063 -> 33969 bytes main.tex | 56 +++++++++++++++++++---------------------------- 7 files changed, 45 insertions(+), 58 deletions(-) diff --git a/main.aux b/main.aux index 0d467a3..46425a5 100644 --- a/main.aux +++ b/main.aux @@ -23,7 +23,7 @@ \newlabel{eq:toy_env}{{9}{4}{}{}{}} \newlabel{eq:toy_log}{{10}{4}{}{}{}} \newlabel{eq:toy_highpass}{{11}{4}{}{}{}} -\@writefile{toc}{\contentsline {subsection}{\numberline {3.2}Threshold nonlinearity \& temporal averaging}{4}{}\protected@file@percent } +\@writefile{toc}{\contentsline {subsection}{\numberline {3.2}Threshold nonlinearity \& temporal averaging}{5}{}\protected@file@percent } \newlabel{eq:pdf_split}{{12}{5}{}{}{}} \newlabel{eq:pdf}{{13}{5}{}{}{}} \newlabel{eq:feat_avg}{{14}{5}{}{}{}} diff --git a/main.fdb_latexmk b/main.fdb_latexmk index f30296c..afa36c3 100644 --- a/main.fdb_latexmk +++ b/main.fdb_latexmk @@ -1,14 +1,14 @@ # Fdb version 4 -["biber main"] 0 "main.bcf" "main.bbl" "main" 1762786327.42255 -1 +["biber main"] 0 "main.bcf" "main.bbl" "main" 1762872619.20857 -1 "cite.bib" 1762424399.63955 252 7fc49109df64576e199ce34111f01817 "" - "main.bcf" 1762786327.38979 107531 bb4289418b1a6604c7c781cbf02f8242 "pdflatex" + "main.bcf" 1762872619.1766 107531 bb4289418b1a6604c7c781cbf02f8242 "pdflatex" (generated) "main.bbl" "main.blg" (rewritten before read) -["pdflatex"] 1762786327.00202 "/home/hartling/phd/paper/paper_2025/main.tex" "main.pdf" "main" 1762786327.42275 0 +["pdflatex"] 1762872618.80069 "/home/hartling/phd/paper/paper_2025/main.tex" "main.pdf" "main" 1762872619.20877 0 "/etc/texmf/web2c/texmf.cnf" 1761560044.43676 475 c0e671620eb5563b2130f56340a5fde8 "" - "/home/hartling/phd/paper/paper_2025/main.tex" 1762786326.9298 12339 6ad6c5cdd4ca9344a52e34ce092ad875 "" + "/home/hartling/phd/paper/paper_2025/main.tex" 1762872618.31261 12714 174d8417d0722be3e50c61997690350b "" "/usr/share/texlive/texmf-dist/fonts/map/fontname/texfonts.map" 1577235249 3524 cb3e574dea2d1052e39280babc910dc8 "" "/usr/share/texlive/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex7.tfm" 1246382020 1004 54797486969f23fa377b128694d548df "" "/usr/share/texlive/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex8.tfm" 1246382020 988 bdf658c3bfc2d96d3c8b02cfc1c94c20 "" @@ -39,6 +39,7 @@ "/usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmr8.pfb" 1248133631 32726 0a1aea6fcd6468ee2cf64d891f5c43c8 "" "/usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy10.pfb" 1248133631 32569 5e5ddc8df908dea60932f3c484a54c0d "" "/usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy8.pfb" 1248133631 32626 4f5c1b83753b1dd3a97d1b399a005b4b "" + "/usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/symbols/msbm10.pfb" 1248133631 34694 ad62b13721ee8eda1dcc8993c8bd7041 "" "/usr/share/texlive/texmf-dist/tex/generic/iftex/iftex.sty" 1644112042 7237 bdd120a32c8fdb4b433cf9ca2e7cd98a "" "/usr/share/texlive/texmf-dist/tex/generic/infwarerr/infwarerr.sty" 1575499628 8356 7bbb2c2373aa810be568c29e333da8ed "" "/usr/share/texlive/texmf-dist/tex/generic/ltxcmds/ltxcmds.sty" 1701727651 17865 1a9bd36b4f98178fa551aca822290953 "" @@ -80,10 +81,10 @@ "/usr/share/texmf/web2c/texmf.cnf" 1707919699 40399 f2c302f7d2af602abb742093540a5834 "" "/var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map" 1761648465.84635 5472669 54eaf61a88b6b7896ebd0dac973cb29c "" "/var/lib/texmf/web2c/pdftex/pdflatex.fmt" 1761648508 8213325 7fd20752ab46ff9aa583e4973d7433df "" - "main.aux" 1762786327.38879 2095 14794d615d49fbf91a4e5fcc2bcea8b4 "pdflatex" - "main.bbl" 1762785994.90795 2032 d9585839a1478809d4c6e9e5bffc8fe4 "biber main" - "main.run.xml" 1762786327.38979 2305 d0067550f22ade5b49f22d40cea22838 "pdflatex" - "main.tex" 1762786326.9298 12339 6ad6c5cdd4ca9344a52e34ce092ad875 "" + "main.aux" 1762872619.1756 2095 e1147be531e57b160e5a1f27b419d862 "pdflatex" + "main.bbl" 1762858512.77528 2032 d9585839a1478809d4c6e9e5bffc8fe4 "biber main" + "main.run.xml" 1762872619.1776 2305 d0067550f22ade5b49f22d40cea22838 "pdflatex" + "main.tex" 1762872618.31261 12714 174d8417d0722be3e50c61997690350b "" (generated) "main.aux" "main.bcf" diff --git a/main.fls b/main.fls index 84e3925..a1ee165 100644 --- a/main.fls +++ b/main.fls @@ -152,3 +152,4 @@ INPUT /usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmr6.pfb INPUT /usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmr8.pfb INPUT /usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy10.pfb INPUT /usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy8.pfb +INPUT /usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/symbols/msbm10.pfb diff --git a/main.log b/main.log index 1692bb3..b4e785d 100644 --- a/main.log +++ b/main.log @@ -1,4 +1,4 @@ -This is pdfTeX, Version 3.141592653-2.6-1.40.25 (TeX Live 2023/Debian) (preloaded format=pdflatex 2025.10.28) 10 NOV 2025 15:52 +This is pdfTeX, Version 3.141592653-2.6-1.40.25 (TeX Live 2023/Debian) (preloaded format=pdflatex 2025.10.28) 11 NOV 2025 15:50 entering extended mode restricted \write18 enabled. file:line:error style messages enabled. @@ -392,24 +392,19 @@ LaTeX Font Info: Trying to load font information for U+msb on input line 17. 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signal \newcommand{\dec}{\log_{10}} % Logarithm base 10 @@ -37,6 +38,7 @@ style=authoryear, \newcommand{\bi}{b_{i,\Theta}} % Single threshold-constrained binary response \newcommand{\feat}{f_{i,\Theta}} % Single threshold-constrained feature +\newcommand{\thp}{T_{\text{HP}}} % Highpass filter adaptation interval \newcommand{\tlp}{T_{\text{LP}}} % Lowpass filter averaging interval \newcommand{\pc}{p(c_i,\,T)} % Probability density (general interval) \newcommand{\pclp}{p(c_i,\,\tlp)} % Probability density (lowpass interval) @@ -104,28 +106,28 @@ Initial: Continuous acoustic input signal $x(t)$ Filtering of behaviorally relevant frequencies by tympanal membrane\\ $\rightarrow$ Bandpass filter 5-30 kHz \begin{equation} - \filt(t)\,=\,\raw(t)\,*\,\bp, \quad\quad \fc\,=\,5\,\text{kHz},\,30\,\text{kHz} + \filt(t)\,=\,\raw(t)\,*\,\bp, \qquad \fc\,=\,5\,\text{kHz},\,30\,\text{kHz} \label{eq:bandpass} \end{equation} Extraction of signal envelope (AM encoding) by receptor population\\ $\rightarrow$ Full-wave rectification, then lowpass filter 500 Hz \begin{equation} - \env(t)\,=\,|\filt(t)|\,*\,\lp, \quad\quad \fc\,=\,500\,\text{Hz} + \env(t)\,=\,|\filt(t)|\,*\,\lp, \qquad \fc\,=\,500\,\text{Hz} \label{eq:env} \end{equation} Logarithmically compressed intensity tuning curve of receptors\\ $\rightarrow$ Decibel transformation \begin{equation} - \db(t)\,=\,10\,\cdot\,\dec \frac{\env(t)}{\max[\env(t)]} + \db(t)\,=\,10\,\cdot\,\dec \frac{\env(t)}{\dbref}, \qquad \dbref\,=\,\max[\env(t)] \label{eq:log} \end{equation} Spike-frequency adaptation in receptor and interneuron populations\\ $\rightarrow$ Highpass filter 10 Hz \begin{equation} - \adapt(t)\,=\,\db(t)\,*\,\hp, \quad\quad \fc\,=\,10\,\text{Hz} + \adapt(t)\,=\,\db(t)\,*\,\hp, \qquad \fc\,=\,10\,\text{Hz} \label{eq:highpass} \end{equation} @@ -173,62 +175,50 @@ Temporal averaging by neurons of the central brain\\ of feature values $\rightarrow$ Clusters in high-dimensional feature space\\ $\rightarrow$ Lowpass filter 1 Hz \begin{equation} - \feat(t)\,=\,\bi(t)\,*\,\lp, \quad\quad \fc\,=\,1\,\text{Hz} + \feat(t)\,=\,\bi(t)\,*\,\lp, \qquad \fc\,=\,1\,\text{Hz} \label{eq:lowpass} \end{equation} - \section{Two mechanisms driving the emergence of intensity-invariant song representation} - \subsection{Logarithmic scaling \& spike-frequency adaptation} Envelope $\env(t)$ $\xrightarrow{\text{dB}}$ Logarithmic $\db(t)$ $\xrightarrow{\hp}$ Adapted $\adapt(t)$ -Example signal envelope $\env(t)$ ($\env(t)>0$ for all $t\in T$):\\ -- Song signal $s(t)$ with $\sigs=1$\\ -- Variable multiplicative song scale $\alpha\geq0$\\ -- Fixed-scale additive noise $\eta(t)$ with $\sign=1$\\ -- Suitable observed time interval $T$\\ -- Decibel reference intensity $m\,=\,\max[\env(t)]$ +- Rewrite signal envelope $\env(t)$ (Eq.\,\ref{eq:env}) as a synthetic mixture:\\ +1) Song signal $s(t)$ ($\sigs=1$) with variable multiplicative scale $\alpha\geq0$\\ +2) Fixed-scale additive noise $\eta(t)$ ($\sign=1$) \begin{equation} - \env(t)\,=\,\alpha\,\cdot\,s(t)\,+\,\eta(t),\quad\quad x:T\to(0,\infty) + \env(t)\,=\,\alpha\,\cdot\,s(t)\,+\,\eta(t),\qquad \env(t)\,>\,0\enspace\forall\enspace t\,\in\,\mathbb{R} \label{eq:toy_env} \end{equation} \textbf{Logarithmic component:}\\ +- Apply decibel transformation (Eq.\,\ref{eq:log}) to synthetic $\env(t)$\\ +- Isolate scale $\alpha$ and reference $\dbref$ using logarithm product/quotient laws \begin{equation} \begin{split} - \db(t)\,&=\,10\,\cdot\,\dec \frac{\alpha\,\cdot\,s(t)\,+\,\eta(t)}{m}\\ - &=\,10\,\cdot\,\big(\dec \frac{\alpha}{m}\,+\,\dec[s(t)\,+\,\frac{\eta(t)}{\alpha}]\big) + \db(t)\,&=\,10\,\cdot\,\dec \frac{\alpha\,\cdot\,s(t)\,+\,\eta(t)}{\dbref}\\ + &=\,10\,\cdot\,\big(\dec \frac{\alpha}{\dbref}\,+\,\dec[s(t)\,+\,\frac{\eta(t)}{\alpha}]\big) \end{split} \label{eq:toy_log} \end{equation} - -% \begin{equation} -% \begin{split} -% \db(t)\,&=\,\log{[\alpha\,\cdot\,s(t)\,+\,\eta(t)]}\\ -% &=\,\log{\alpha}\,+\,\log{[s(t)\,+\,\frac{\eta(t)}{\alpha}]} -% \end{split} -% \label{eq:toy_log} -% \end{equation} +$\rightarrow$ 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 logarithmically scaled $\db(t)$ (Eq.\,\ref{eq:highpass}) can +be approximated as subtraction of the signal offset (DC removal) within a suitable +time interval $\thp$ ($0 < \thp < \frac{1}{\fc}$)\\ \begin{equation} \begin{split} - \adapt(t)\,\approx\,\db(t)\,-\,\dec \frac{\alpha}{m}\,=\,\dec{[s(t)\,+\,\frac{\eta(t)}{\alpha}]} + \adapt(t)\,\approx\,\db(t)\,-\,\dec \frac{\alpha}{\dbref}\,=\,\dec{[s(t)\,+\,\frac{\eta(t)}{\alpha}]} \end{split} \label{eq:toy_highpass} \end{equation} -% \textbf{Adaptation component:}\\ -% \begin{equation} -% \begin{split} -% \adapt(t)\,\approx\,\db(t)\,-\,\log{\alpha}\,=\,\log{[s(t)\,+\,\frac{\eta(t)}{\alpha}]} -% \end{split} -% \label{eq:toy_highpass} -% \end{equation} - \subsection{Threshold nonlinearity \& temporal averaging} Convolved $c_i(t)$ $\xrightarrow{\nl}$ Binary $\bi(t)$ $\xrightarrow{\lp}$ Feature $\feat(t)$