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Formalizing intensity invariances (WIP).

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j-hartling
2025-11-10 15:53:47 +01:00
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main.tex
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@@ -2,6 +2,7 @@
\usepackage{parskip}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage[
backend=biber,
style=authoryear,
@@ -19,14 +20,26 @@ style=authoryear,
\newcommand{\lp}{h_{\text{LP}}(t)} % Lowpass filter function
\newcommand{\hp}{h_{\text{HP}}(t)} % Highpass filter function
\newcommand{\fc}{f_{\text{cut}}} % Filter cutoff frequency
\newcommand{\infint}{\int_{-\infty}^{\infty}} % Indefinite integral
\newcommand{\raw}{x} % Placeholder input signal
\newcommand{\filt}{\raw_{\text{filt}}} % Bandpass-filtered signal
\newcommand{\env}{\raw_{\text{env}}} % Signal envelope
\newcommand{\db}{\raw_{\text{dB}}} % Logarithmically scaled signal
\newcommand{\adapt}{\raw_{\text{adapt}}} % Adapted signal
\newcommand{\dec}{\log_{10}} % Logarithm base 10
\newcommand{\sigs}{\sigma_{\text{s}}} % Song standard deviation
\newcommand{\sign}{\sigma_{\eta}} % Noise standard deviation
\newcommand{\infint}{\int_{-\infty}^{+\infty}} % Indefinite integral
\newcommand{\thr}{\Theta_i} % Step function threshold value
\newcommand{\nl}{H(c_i\,-\,\thr)} % Shifted Heaviside step function
\newcommand{\bi}{b_{i,\Theta}} % Single binary response full shorthand
\newcommand{\feat}{f_{i,\Theta}} % Single feature full shorthand
\newcommand{\bi}{b_{i,\Theta}} % Single threshold-constrained binary response
\newcommand{\feat}{f_{i,\Theta}} % Single threshold-constrained feature
\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)
\newcommand{\pc}{p(c_i,\,T)} % Probability density (general interval)
\newcommand{\pclp}{p(c_i,\,\tlp)} % Probability density (lowpass interval)
\section{The sensory world of a grasshopper}
@@ -91,25 +104,29 @@ Initial: Continuous acoustic input signal $x(t)$
Filtering of behaviorally relevant frequencies by tympanal membrane\\
$\rightarrow$ Bandpass filter 5-30 kHz
\begin{equation}
x(t)\,*\,\bp, \quad\quad \fc\,=\,5\,\text{kHz},\,30\,\text{kHz}
\filt(t)\,=\,\raw(t)\,*\,\bp, \quad\quad \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}
|x(t)|\,*\,\lp, \quad\quad \fc\,=\,500\,\text{Hz}
\env(t)\,=\,|\filt(t)|\,*\,\lp, \quad\quad \fc\,=\,500\,\text{Hz}
\label{eq:env}
\end{equation}
Logarithmically compressed intensity tuning curve of receptors\\
$\rightarrow$ Decibel transformation
\begin{equation}
10\,\cdot\,\log_{10} \frac{x(t)}{x_{\text{max}}}
\db(t)\,=\,10\,\cdot\,\dec \frac{\env(t)}{\max[\env(t)]}
\label{eq:log}
\end{equation}
Spike-frequency adaptation in receptor and interneuron populations\\
$\rightarrow$ Highpass filter 10 Hz
\begin{equation}
x(t)\,*\,\hp, \quad\quad \fc\,=\,10\,\text{Hz}
\adapt(t)\,=\,\db(t)\,*\,\hp, \quad\quad \fc\,=\,10\,\text{Hz}
\label{eq:highpass}
\end{equation}
@@ -130,11 +147,13 @@ Template matching by individual ANs\\
- Gabor parameters: $\sigma, \phi, f$ $\rightarrow$ Determines kernel sign and lobe number
\begin{equation}
k(t)\,=\,e^{-\frac{t^{2}}{2\sigma^{2}}}\,\cdot\,\sin(2\pi f t\,+\,\phi)
\label{eq:gabor}
\end{equation}
$\rightarrow$ Separate convolution with each member of the kernel set
\begin{equation}
c_i(t)\,=\,x(t)\,*\,k_i(t)
= \infint x(\tau)\,\cdot\,k_i(t\,-\,\tau)\,d\tau
c_i(t)\,=\,\adapt(t)\,*\,k_i(t)
= \infint \adapt(\tau)\,\cdot\,k_i(t\,-\,\tau)\,d\tau
\label{eq:conv}
\end{equation}
Thresholding nonlinearity in ascending neurons (or further downstream)\\
@@ -145,6 +164,7 @@ $\rightarrow$ Shifted Heaviside step-function $\nl$ (or steep sigmoid threshold?
\;1, \quad c_i(t)\,>\,\thr\\
\;0, \quad c_i(t)\,\leq\,\thr
\end{cases}
\label{eq:binary}
\end{equation}
Temporal averaging by neurons of the central brain\\
@@ -154,6 +174,7 @@ 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}
\label{eq:lowpass}
\end{equation}
@@ -162,31 +183,103 @@ $\rightarrow$ Lowpass filter 1 Hz
\subsection{Logarithmic scaling \& spike-frequency adaptation}
Song signal $s(t)$ with variable scale $\alpha$ and fixed-scale additive noise $\eta(t)$
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 $m\,=\,\max[\env(t)]$
\begin{equation}
\alpha\,\cdot\,s(t)\,+\,\eta(t)
\env(t)\,=\,\alpha\,\cdot\,s(t)\,+\,\eta(t),\quad\quad x:T\to(0,\infty)
\label{eq:toy_env}
\end{equation}
\textbf{Logarithmic component:}\\
\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)
\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}
\textbf{Adaptation component:}\\
\begin{equation}
\begin{split}
\adapt(t)\,\approx\,\db(t)\,-\,\dec \frac{\alpha}{m}\,=\,\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}
Convolution output $c_i(t)$ $\xrightarrow{\thr}$ Thresholded response $\bi(t)$ $\rightarrow$ Feature $\feat(t)$
Convolved $c_i(t)$ $\xrightarrow{\nl}$ Binary $\bi(t)$ $\xrightarrow{\lp}$ Feature $\feat(t)$
- Convolution output $c_i(t)$ has distribution $\pc$ over time interval $T$\\
- Within $T$, $c_i(t)$ exceeds the threshold value $\thr$ for time $T_1$ ($T_1+T_0=T$)\\
$\rightarrow$ Step-function $\nl$ bipartitions distribution $\pc$ around $\thr$
\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}
\int_{\thr}^{+\infty} p(c_i,T)\,dc_i\,=\,1\,-\,\int_{-\infty}^{\thr} p(c_i,T)\,dc_i\,=\,\frac{T_1}{T}
\label{eq:pdf_split}
\end{equation}
- Ratio of time above threshold $T_1$ to total time $T$ because
$\rightarrow$ Semi-definite integral over right-sided portion of split $\pc$ gives ratio
of time $T_1$ where $c_i(t)>\thr$ to total time $T$ due to normalization of $\pc$
\begin{equation}
\infint \pc\,dc_i\,=\,1
\label{eq:pdf}
\end{equation}
Approximate lowpass filter as moving average over time interval $\tlp$
\textbf{Averaging component:}\\
- Lowpass filter over binary response $\bi(t)$ (Eq.\,\ref{eq:lowpass}) can be
approximated as temporal averaging over a suitable time interval $\tlp$ ($\tlp > \frac{1}{\fc}$)\\
- Within $\tlp$, $\bi(t)$ takes a value of 1 ($c_i(t)>\thr$) for time $T_1$ ($T_1+T_0=\tlp$)
\begin{equation}
\feat(t)\,\approx\,\frac{1}{\tlp} \int_{t}^{t\,+\,\tlp} \bi(\tau)\,d\tau\,=\,\frac{T_1}{\tlp}
\label{eq:feat_avg}
\end{equation}
$\rightarrow$ Temporal averaging over $\bi(t)\in[0,1]$ (Eq.\ref{eq:binary}) gives
ratio of time $T_1$ where $c_i(t)>\thr$ to total averaging interval $\tlp$\\
$\rightarrow$ Feature $\feat(t)$ approximately represents supra-threshold fraction of $\tlp$
\textbf{Combined result:}\\
- Feature $\feat(t)$ can be linked to the distribution of $c_i(t)$ using Eqs.\,\ref{eq:pdf_split} \& \ref{eq:feat_avg}
\begin{equation}
\feat(t)\,\approx\,\int_{\thr}^{+\infty} \pclp\,dc_i\,=\,P(c_i\,>\,\thr,\,\tlp)
\label{eq:feat_prop}
\end{equation}
$\rightarrow$ Because the integral over a probability density is a cumulative
probability, the value of feature $\feat(t)$ (temporal compression of $\bi(t)$)
at every time point $t$ signifies the probability that convolution output
$c_i(t)$ exceeds the threshold value $\thr$ during the corresponding averaging
interval $\tlp$
\textbf{Implication for intensity invariance:}\\
- Convolution output $c_i(t)$ = amplitude-based quantity\\
$\rightarrow$ Values indicate how well template waveform $k_i(t)$ matches signal $x(t)$\\
- Feature $\feat(t)$ = duty cycle-based quantity\\
$\rightarrow$ Values indicate how often $c_i(t)$ exceeds threshold value $\thr$
- Thresholding of $c_i(t)$ and subsequent temporal averaging of $\bi(t)$ to obtain $\feat(t)$
constitutes a remapping of an amplitude-based quantity (values indicating the match between) into a duty cycle-based quantity\\
\section{Discriminating species-specific song\\patterns in feature space}