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PDF statistics: - 78 PDF objects out of 1000 (max. 8388607) - 47 compressed objects within 1 object stream + 83 PDF objects out of 1000 (max. 8388607) + 50 compressed objects within 1 object stream 0 named destinations out of 1000 (max. 500000) 1 words of extra memory for PDF output out of 10000 (max. 10000000) diff --git a/main.pdf b/main.pdf index fc53b66..761a502 100644 Binary files a/main.pdf and b/main.pdf differ diff --git a/main.synctex.gz b/main.synctex.gz index 0400d91..f5944b5 100644 Binary files a/main.synctex.gz and b/main.synctex.gz differ diff --git a/main.tex b/main.tex index 019ad51..c14e171 100644 --- a/main.tex +++ b/main.tex @@ -25,6 +25,7 @@ style=authoryear, \newcommand{\filt}{\raw_{\text{filt}}} % Bandpass-filtered signal \newcommand{\env}{\raw_{\text{env}}} % Signal envelope \newcommand{\db}{\raw_{\text{dB}}} % Logarithmically scaled signal +\newcommand{\dbref}{\raw_{\text{ref}}} % Decibel reference intensity \newcommand{\adapt}{\raw_{\text{adapt}}} % Adapted 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 $\feat(t)$