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Package biblatex Info: Reference section=0 on input line 15. 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PDF statistics: 72 PDF objects out of 1000 (max. 8388607) 43 compressed objects within 1 object stream diff --git a/main.pdf b/main.pdf index f453d89..117c143 100644 Binary files a/main.pdf and b/main.pdf differ diff --git a/main.synctex.gz b/main.synctex.gz index 2ccf006..bf0cab1 100644 Binary files a/main.synctex.gz and b/main.synctex.gz differ diff --git a/main.tex b/main.tex index 9d136f7..fcb411d 100644 --- a/main.tex +++ b/main.tex @@ -20,8 +20,13 @@ style=authoryear, \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{\bi}{b_\theta} -\newcommand{\feat}{f_\theta} +\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{\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} @@ -86,25 +91,25 @@ 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} + x(t)\,*\,\bp, \quad\quad \fc\,=\,5\,\text{kHz},\,30\,\text{kHz} \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} + |x(t)|\,*\,\lp, \quad\quad \fc\,=\,500\,\text{Hz} \end{equation} Logarithmically compressed intensity tuning curve of receptors\\ $\rightarrow$ Decibel transformation \begin{equation} - 20\,\cdot\,\log_{10} \frac{x(t)}{x_{\text{max}}} + 10\,\cdot\,\log_{10} \frac{x(t)}{x_{\text{max}}} \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} + x(t)\,*\,\hp, \quad\quad \fc\,=\,10\,\text{Hz} \end{equation} @@ -134,11 +139,11 @@ $\rightarrow$ Separate convolution with each member of the kernel set Thresholding nonlinearity in ascending neurons (or further downstream)\\ - 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} \bi(t)\,=\,\begin{cases} - \;1, \quad c(t)\,\geq\,\theta\\ - \;0, \quad c(t)\,<\,\theta + \;1, \quad c_i(t)\,>\,\thr\\ + \;0, \quad c_i(t)\,\leq\,\thr \end{cases} \end{equation} @@ -148,7 +153,7 @@ 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, \quad\quad \fc\,=\,1\,\text{Hz} \end{equation} @@ -165,6 +170,23 @@ Song signal $s(t)$ with variable scale $\alpha$ and fixed-scale additive noise $ \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}