Done bullet-pointing and formalizing TLP invariance
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[]$\OMS/cmsy/m/n/12 !$ \OT1/cmr/m/n/12 Val-ues in-di-cate cor-re-spon-dence be-tween a tem-plate wave-form $\OML/cmm/m/it/12 k[]\OT1/cmr/m/n/12 (\OML/cmm/m/it/12 t\OT1/cmr/m/n/12 )$ matches
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@ -309,26 +309,32 @@ $c_i(t)$ exceeds the threshold value $\thr$ during the corresponding averaging
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interval $\tlp$
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\textbf{Implication for intensity invariance:}\\
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- Convolution output $c_i(t)$ = amplitude-based quantity\\
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$\rightarrow$ Values indicate correspondence between template waveform and signal\\
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- Convolution output $c_i(t)$ quantifies temporal similarity between amplitudes of
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template waveform $k_i(t)$ and signal $\adapt(t)$ centered at time point $t$\\
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$\rightarrow$ Based on amplitudes on a graded scale
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$\rightarrow$ Values indicate correspondence between a template waveform $k_i(t)$
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matches
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the waveform of the pre-processed signal $\adapt(t)$ at a given time point $t$
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- Feature $\feat(t)$ quantifies the probability that amplitudes of $c_i(t)$
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exceed threshold value $\thr$ within interval $\tlp$ around time point $t$\\
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$\rightarrow$ Based on binned amplitudes corresponding to one of two categorical states
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$\rightarrow$ Deliberate loss of precise amplitude information\\
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$\rightarrow$ Emphasis on temporal structure (ratio of $T_1$ over $\tlp$)
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- Feature $\feat(t)$ = duty cycle-based quantity\\
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$\rightarrow$ Values indicate the ratio of two temporal quantities\\
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$\rightarrow$ Values indicate the ratio of time, or probability, that $c_i(t)$
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exceeds threshold value $\thr$ within
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- Thresholding of $c_i(t)$ and subsequent temporal averaging of $\bi(t)$ to
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obtain $\feat(t)$ constitutes a remapping of an amplitude-encoding quantity into a
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duty cycle-encoding quantity, mediated by threshold function $\nl$
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- Different scales of $c_i(t)$ can result in similar $T_1$ segments depending
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on the magnitude of the derivative of $c_i(t)$ in temporal proximity to time
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points at which $c_i(t)$ crosses threshold value $\thr$\\
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$\rightarrow$ The steeper the slope of $c_i(t)$, the less $T_1$ changes with scale variations\\
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$\rightarrow$ Extreme amplitudes of $c_i(t)$ (peaks/troughs)
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$\rightarrow$ Only amplitudes of \\
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$\rightarrow$ Absolute amplitudes of peaks/troughs of $c_i(t)$ \\
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$\rightarrow$ Acuity of peaks/troughs in $c_i(t)$ matters, not their absolute amplitude
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- From graded stimulus to categorical behavioral decision:\\
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- Feature $\feat(t)$ = duty cycle-based quantity\\
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$\rightarrow$ Values indicate how often $c_i(t)$ exceeds threshold value $\thr$
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- Thresholding of $c_i(t)$ and subsequent temporal averaging of $\bi(t)$ to obtain $\feat(t)$
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constitutes a remapping of an amplitude-based quantity (values indicating the match between) into a duty cycle-based quantity\\
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\section{Discriminating species-specific song\\patterns in feature space}
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