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More formalizing (WIP).

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@ -376,7 +376,7 @@ Package biblatex Info: ... file 'main.bbl' found.
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\OT1/cmr/m/n/12 2)...compensate for be-hav-iorally non-informative song vari-abil-ity (in-vari-ances) \OT1/cmr/m/n/12 2)...compensate for be-hav-iorally non-informative song vari-abil-ity (in-vari-ances)
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@ -20,8 +20,13 @@ style=authoryear,
\newcommand{\hp}{h_{\text{HP}}(t)} % Highpass filter function \newcommand{\hp}{h_{\text{HP}}(t)} % Highpass filter function
\newcommand{\fc}{f_{\text{cut}}} % Filter cutoff frequency \newcommand{\fc}{f_{\text{cut}}} % Filter cutoff frequency
\newcommand{\infint}{\int_{-\infty}^{\infty}} % Indefinite integral \newcommand{\infint}{\int_{-\infty}^{\infty}} % Indefinite integral
\newcommand{\bi}{b_\theta} \newcommand{\thr}{\Theta_i} % Step function threshold value
\newcommand{\feat}{f_\theta} \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{\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)
\section{The sensory world of a grasshopper} \section{The sensory world of a grasshopper}
@ -86,25 +91,25 @@ Initial: Continuous acoustic input signal $x(t)$
Filtering of behaviorally relevant frequencies by tympanal membrane\\ Filtering of behaviorally relevant frequencies by tympanal membrane\\
$\rightarrow$ Bandpass filter 5-30 kHz $\rightarrow$ Bandpass filter 5-30 kHz
\begin{equation} \begin{equation}
x(t)\,*\,\bp; \quad\quad \fc\,=\,5\,\text{kHz},\,30\,\text{kHz} x(t)\,*\,\bp, \quad\quad \fc\,=\,5\,\text{kHz},\,30\,\text{kHz}
\end{equation} \end{equation}
Extraction of signal envelope (AM encoding) by receptor population\\ Extraction of signal envelope (AM encoding) by receptor population\\
$\rightarrow$ Full-wave rectification, then lowpass filter 500 Hz $\rightarrow$ Full-wave rectification, then lowpass filter 500 Hz
\begin{equation} \begin{equation}
|x(t)|\,*\,\lp; \quad\quad \fc\,=\,500\,\text{Hz} |x(t)|\,*\,\lp, \quad\quad \fc\,=\,500\,\text{Hz}
\end{equation} \end{equation}
Logarithmically compressed intensity tuning curve of receptors\\ Logarithmically compressed intensity tuning curve of receptors\\
$\rightarrow$ Decibel transformation $\rightarrow$ Decibel transformation
\begin{equation} \begin{equation}
20\,\cdot\,\log_{10} \frac{x(t)}{x_{\text{max}}} 10\,\cdot\,\log_{10} \frac{x(t)}{x_{\text{max}}}
\end{equation} \end{equation}
Spike-frequency adaptation in receptor and interneuron populations\\ Spike-frequency adaptation in receptor and interneuron populations\\
$\rightarrow$ Highpass filter 10 Hz $\rightarrow$ Highpass filter 10 Hz
\begin{equation} \begin{equation}
x(t)\,*\,\hp; \quad\quad \fc\,=\,10\,\text{Hz} x(t)\,*\,\hp, \quad\quad \fc\,=\,10\,\text{Hz}
\end{equation} \end{equation}
@ -134,11 +139,11 @@ $\rightarrow$ Separate convolution with each member of the kernel set
Thresholding nonlinearity in ascending neurons (or further downstream)\\ Thresholding nonlinearity in ascending neurons (or further downstream)\\
- Binarization of AN response traces into "relevant" vs. "irrelevant"\\ - Binarization of AN response traces into "relevant" vs. "irrelevant"\\
$\rightarrow$ Heaviside step-function $H(c\,-\,\theta)$ (or steep sigmoid threshold?) $\rightarrow$ Shifted Heaviside step-function $\nl$ (or steep sigmoid threshold?)
\begin{equation} \begin{equation}
\bi(t)\,=\,\begin{cases} \bi(t)\,=\,\begin{cases}
\;1, \quad c(t)\,\geq\,\theta\\ \;1, \quad c_i(t)\,>\,\thr\\
\;0, \quad c(t)\,<\,\theta \;0, \quad c_i(t)\,\leq\,\thr
\end{cases} \end{cases}
\end{equation} \end{equation}
@ -148,7 +153,7 @@ Temporal averaging by neurons of the central brain\\
of feature values $\rightarrow$ Clusters in high-dimensional feature space\\ of feature values $\rightarrow$ Clusters in high-dimensional feature space\\
$\rightarrow$ Lowpass filter 1 Hz $\rightarrow$ Lowpass filter 1 Hz
\begin{equation} \begin{equation}
\feat(t)\,=\,\bi(t)\,*\,\lp; \quad\quad \fc\,=\,1\,\text{Hz} \feat(t)\,=\,\bi(t)\,*\,\lp, \quad\quad \fc\,=\,1\,\text{Hz}
\end{equation} \end{equation}
@ -165,6 +170,23 @@ Song signal $s(t)$ with variable scale $\alpha$ and fixed-scale additive noise $
\subsection{Threshold nonlinearity \& temporal averaging} \subsection{Threshold nonlinearity \& temporal averaging}
Convolution output $c_i(t)$ $\xrightarrow{\thr}$ Thresholded response $\bi(t)$ $\rightarrow$ 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 $\thr$
\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}
\end{equation}
- Ratio of time above threshold $T_1$ to total time $T$ because
\begin{equation}
\infint \pc\,dc_i\,=\,1
\end{equation}
Approximate lowpass filter as moving average over time interval $\tlp$
\begin{equation}
\feat(t)\,\approx\,\frac{1}{\tlp} \int_{t}^{t\,+\,\tlp} \bi(\tau)\,d\tau\,=\,\frac{T_1}{\tlp}
\end{equation}
\section{Discriminating species-specific song\\patterns in feature space} \section{Discriminating species-specific song\\patterns in feature space}