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PDF statistics: - 72 PDF objects out of 1000 (max. 8388607) - 43 compressed objects within 1 object stream + 78 PDF objects out of 1000 (max. 8388607) + 47 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 117c143..fc53b66 100644 Binary files a/main.pdf and b/main.pdf differ diff --git a/main.synctex.gz b/main.synctex.gz index bf0cab1..0400d91 100644 Binary files a/main.synctex.gz and b/main.synctex.gz differ diff --git a/main.tex b/main.tex index fcb411d..019ad51 100644 --- a/main.tex +++ b/main.tex @@ -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}