Currently formalizing log-invariance (WIP).

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 $\feat(t)$