More formalizing (WIP).
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j-hartling
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This is pdfTeX, Version 3.141592653-2.6-1.40.25 (TeX Live 2023/Debian) (preloaded format=pdflatex 2025.10.28) 6 NOV 2025 09:26
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This is pdfTeX, Version 3.141592653-2.6-1.40.25 (TeX Live 2023/Debian) (preloaded format=pdflatex 2025.10.28) 6 NOV 2025 12:00
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entering extended mode
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restricted \write18 enabled.
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file:line:error style messages enabled.
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@ -376,7 +376,7 @@ Package biblatex Info: ... file 'main.bbl' found.
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Package biblatex Info: Reference section=0 on input line 15.
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Package biblatex Info: Reference segment=0 on input line 15.
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Overfull \hbox (3.81674pt too wide) in paragraph at lines 50--57
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Overfull \hbox (3.81674pt too wide) in paragraph at lines 55--62
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\OT1/cmr/m/n/12 2)...compensate for be-hav-iorally non-informative song vari-abil-ity (in-vari-ances)
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[]
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main.tex
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main.tex
@ -20,8 +20,13 @@ style=authoryear,
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\newcommand{\hp}{h_{\text{HP}}(t)} % Highpass filter function
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\newcommand{\fc}{f_{\text{cut}}} % Filter cutoff frequency
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\newcommand{\infint}{\int_{-\infty}^{\infty}} % Indefinite integral
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\newcommand{\bi}{b_\theta}
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\newcommand{\feat}{f_\theta}
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\newcommand{\thr}{\Theta_i} % Step function threshold value
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\newcommand{\nl}{H(c_i\,-\,\thr)} % Shifted Heaviside step function
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\newcommand{\bi}{b_{i,\Theta}} % Single binary response full shorthand
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\newcommand{\feat}{f_{i,\Theta}} % Single feature full shorthand
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\newcommand{\tlp}{T_{\text{LP}}} % Lowpass filter averaging interval
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\newcommand{\pc}{p(c_i,T)} % Probability density (general interval)
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\newcommand{\pclp}{p(c_i, \tlp)} % Probability density (lowpass interval)
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\section{The sensory world of a grasshopper}
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@ -86,25 +91,25 @@ Initial: Continuous acoustic input signal $x(t)$
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Filtering of behaviorally relevant frequencies by tympanal membrane\\
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$\rightarrow$ Bandpass filter 5-30 kHz
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\begin{equation}
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x(t)\,*\,\bp; \quad\quad \fc\,=\,5\,\text{kHz},\,30\,\text{kHz}
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x(t)\,*\,\bp, \quad\quad \fc\,=\,5\,\text{kHz},\,30\,\text{kHz}
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\end{equation}
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Extraction of signal envelope (AM encoding) by receptor population\\
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$\rightarrow$ Full-wave rectification, then lowpass filter 500 Hz
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\begin{equation}
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|x(t)|\,*\,\lp; \quad\quad \fc\,=\,500\,\text{Hz}
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|x(t)|\,*\,\lp, \quad\quad \fc\,=\,500\,\text{Hz}
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\end{equation}
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Logarithmically compressed intensity tuning curve of receptors\\
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$\rightarrow$ Decibel transformation
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\begin{equation}
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20\,\cdot\,\log_{10} \frac{x(t)}{x_{\text{max}}}
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10\,\cdot\,\log_{10} \frac{x(t)}{x_{\text{max}}}
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\end{equation}
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Spike-frequency adaptation in receptor and interneuron populations\\
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$\rightarrow$ Highpass filter 10 Hz
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\begin{equation}
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x(t)\,*\,\hp; \quad\quad \fc\,=\,10\,\text{Hz}
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x(t)\,*\,\hp, \quad\quad \fc\,=\,10\,\text{Hz}
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\end{equation}
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@ -134,11 +139,11 @@ $\rightarrow$ Separate convolution with each member of the kernel set
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Thresholding nonlinearity in ascending neurons (or further downstream)\\
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- Binarization of AN response traces into "relevant" vs. "irrelevant"\\
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$\rightarrow$ Heaviside step-function $H(c\,-\,\theta)$ (or steep sigmoid threshold?)
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$\rightarrow$ Shifted Heaviside step-function $\nl$ (or steep sigmoid threshold?)
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\begin{equation}
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\bi(t)\,=\,\begin{cases}
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\;1, \quad c(t)\,\geq\,\theta\\
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\;0, \quad c(t)\,<\,\theta
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\;1, \quad c_i(t)\,>\,\thr\\
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\;0, \quad c_i(t)\,\leq\,\thr
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\end{cases}
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\end{equation}
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@ -148,7 +153,7 @@ Temporal averaging by neurons of the central brain\\
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of feature values $\rightarrow$ Clusters in high-dimensional feature space\\
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$\rightarrow$ Lowpass filter 1 Hz
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\begin{equation}
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\feat(t)\,=\,\bi(t)\,*\,\lp; \quad\quad \fc\,=\,1\,\text{Hz}
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\feat(t)\,=\,\bi(t)\,*\,\lp, \quad\quad \fc\,=\,1\,\text{Hz}
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\end{equation}
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@ -165,6 +170,23 @@ Song signal $s(t)$ with variable scale $\alpha$ and fixed-scale additive noise $
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\subsection{Threshold nonlinearity \& temporal averaging}
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Convolution output $c_i(t)$ $\xrightarrow{\thr}$ Thresholded response $\bi(t)$ $\rightarrow$ Feature $\feat(t)$
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- Convolution output $c_i(t)$ has distribution $\pc$ over time interval $T$\\
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- Within $T$, $c_i(t)$ exceeds the threshold value $\thr$ for time $T_1$ ($T_1+T_0=T$)\\
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$\rightarrow$ Step-function $\nl$ bipartitions distribution $\pc$ around $\thr$
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\begin{equation}
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\int_{\thr}^{+\infty} p(c_i,T)\,dc_i\,=\,1\,-\,\int_{-\infty}^{\thr} p(c_i,T)\,dc_i\,=\,\frac{T_1}{T}
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\end{equation}
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- Ratio of time above threshold $T_1$ to total time $T$ because
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\begin{equation}
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\infint \pc\,dc_i\,=\,1
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\end{equation}
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Approximate lowpass filter as moving average over time interval $\tlp$
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\begin{equation}
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\feat(t)\,\approx\,\frac{1}{\tlp} \int_{t}^{t\,+\,\tlp} \bi(\tau)\,d\tau\,=\,\frac{T_1}{\tlp}
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\end{equation}
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\section{Discriminating species-specific song\\patterns in feature space}
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