More formalizing (WIP).

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}