Done bullet-pointing and formalizing TLP invariance

main.tex

@@ -309,26 +309,32 @@ $c_i(t)$ 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 correspondence between template waveform and signal\\
+- Convolution output $c_i(t)$ quantifies temporal similarity between amplitudes of
+template waveform $k_i(t)$ and signal $\adapt(t)$ centered at time point $t$\\
+$\rightarrow$ Based on amplitudes on a graded scale
 
-$\rightarrow$ Values indicate correspondence between a template waveform $k_i(t)$
-matches
-the waveform of the pre-processed signal $\adapt(t)$ at a given time point $t$
+- Feature $\feat(t)$ quantifies the probability that amplitudes of $c_i(t)$
+exceed threshold value $\thr$ within interval $\tlp$ around time point $t$\\
+$\rightarrow$ Based on binned amplitudes corresponding to one of two categorical states
+$\rightarrow$ Deliberate loss of precise amplitude information\\
+$\rightarrow$ Emphasis on temporal structure (ratio of $T_1$ over $\tlp$)
 
-- Feature $\feat(t)$ = duty cycle-based quantity\\
-$\rightarrow$ Values indicate the ratio of two temporal quantities\\
-$\rightarrow$ Values indicate the ratio of time, or probability, that $c_i(t)$
-exceeds threshold value $\thr$ within
+- Thresholding of $c_i(t)$ and subsequent temporal averaging of $\bi(t)$ to
+obtain $\feat(t)$ constitutes a remapping of an amplitude-encoding quantity into a
+duty cycle-encoding quantity, mediated by threshold function $\nl$
 
+- Different scales of $c_i(t)$ can result in similar $T_1$ segments depending
+on the magnitude of the derivative of $c_i(t)$ in temporal proximity to time
+points at which $c_i(t)$ crosses threshold value $\thr$\\
+$\rightarrow$ The steeper the slope of $c_i(t)$, the less $T_1$ changes with scale variations\\
+$\rightarrow$ Extreme amplitudes of $c_i(t)$ (peaks/troughs)
 
+$\rightarrow$ Only amplitudes of \\
+$\rightarrow$ Absolute amplitudes of peaks/troughs of $c_i(t)$ \\
+$\rightarrow$ Acuity of peaks/troughs in $c_i(t)$ matters, not their absolute amplitude
 
+- From graded stimulus to categorical behavioral decision:\\
 
-- 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}