Managed 1st half of results text, began with 2nd half.
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j-hartling
205
main.tex
205
main.tex
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mincitenames=1
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]{biblatex}
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\addbibresource{cite.bib}
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%\bibdata
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%\bibstyle
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%\citation
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\title{Emergent intensity invariance in a physiologically inspired model of the grasshopper auditory system}
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\author{Jona Hartling, Jan Benda}
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@@ -82,7 +85,7 @@
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\newcommand{\fwrh}{\text{FWRH}} % Gaussian full-width at relative height
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\newcommand{\off}{\beta_0} % Offset for linear frequency approximation
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% Math shorthands - Threshold nonlinearity:
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% Math shorthands - Thresholding nonlinearity:
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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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@@ -90,6 +93,7 @@
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\newcommand{\soc}{s} % Song component of synthetic mixture
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\newcommand{\noc}{\eta} % Noise component of synthetic mixture
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\newcommand{\sca}{\alpha} % Multiplicative scale of song component
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\newcommand{\xvar}{\sigma_{x}^{2}} % Variance of synthetic mixture
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\newcommand{\svar}{\sigma_{\text{s}}^{2}} % Song component variance
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\newcommand{\nvar}{\sigma_{\eta}^{2}} % Noise component variance
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\newcommand{\pc}{p(c_i,\,T)} % Probability density (general interval)
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@@ -509,7 +513,7 @@ threshold value $\thr$ to obtain a binary response
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\label{eq:binary}
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\end{equation}
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which can be thought of as a categorization into "relevant" and "irrelevant"
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response values. In the grasshopper, these threshold nonlinearities might
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response values. In the grasshopper, these thresholding nonlinearities might
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either be part of the processing within the ascending neurons or take place
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further downstream~(SOURCE). Finally, the responses of the ascending neurons
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are assumed to be integrated somewhere in the supraesophageal
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@@ -543,47 +547,38 @@ can be read out by a simple linear classifier.
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\end{figure}
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\FloatBarrier
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\section{Two mechanisms driving the emergence of intensity-invariant song representations}
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\section{Two mechanisms driving the emergence of intensity-invariant song representation}
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% Still missing the SNR analysis. Should be able to write around it for now.
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The robustness of song recognition is tied to the degree of intensity
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invariance of the finalized feature representation. Ideally, the values of each
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feature should depend only on the relative amplitude dynamics of the song
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pattern but not on the overall intensity level of the song. In the grasshopper,
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the emergence of intensity-invariant representations along the song recognition
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pattern but not on the overall intensity of the song. In the grasshopper, the
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emergence of intensity-invariant representations along the song recognition
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pathway likely is a distributed process that involves different neuronal
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populations, which raises the question of what the essential computational
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mechanisms are that drive this process. Within the model pathway, we identified
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two key mechanisms that render the song representation more invariant to
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variations in baseline intensity. The two mechanisms each comprise a nonlinear
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signal transformation followed by a linear signal transformation but differ in
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the specific operations and the neural substrate involved, as outlined in the
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following sections.
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intensity variations. The two mechanisms each comprise a nonlinear signal
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transformation followed by a linear signal transformation but differ in the
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specific operations involved, as outlined in the following sections.
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\subsection{Logarithmic compression \& spike-frequency adaptation}
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The first emergence of intensity invariance along the model pathway occurs
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during the preprocessing stage, in the transition from the signal envelope
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$\env(t)$ to the logarithmically scaled envelope $\db(t)$ and then to the
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intensity-adapted envelope $\adapt(t)$. In order to disentangle the interplay
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of logarithmic compression and adaptation, we can rewrite
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$\env(t)$~(Eq.\,\ref{eq:env}) as synthetic mixture
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The first notable emergence of intensity invariance along the model pathway
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occurs during the transformation of the signal envelope $\env(t)$ into the
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logarithmically scaled envelope $\db(t)$ and then into the intensity-adapted
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envelope $\adapt(t)$. In order to disentangle the interplay of logarithmic
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compression and adaptation, $\env(t)$ can be rewritten as a synthetic mixture
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\begin{equation}
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\env(t)\,=\,\sca\,\cdot\,\soc(t)\,+\,\noc(t), \qquad \env(t)\,>\,0\enspace\forall\enspace t\,\in\,\mathbb{R}
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\label{eq:toy_env}
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\end{equation}
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of a song component $\soc(t)$ with variable multiplicative scale $\sca\geq0$
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and a fixed-scale noise component $\noc(t)$. Both $\soc(t)$ and $\noc(t)$ are
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assumed to have unit variance~($\svar=\nvar=1$). If $\soc(t)$ and $\noc(t)$ are
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uncorrelated~($\soc(t)\perp\noc(t)$), the signal-to-noise ratio (SNR) of the
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synthetic $\env(t)$ with ($\sca>0$) and without ($\sca=0$) song component
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$\soc(t)$ is given by
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\begin{equation}
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\text{SNR}\,=\,\frac{\sigma_{s+\eta}^{2}}{\nvar}\,=\,\frac{\alpha^{2}\,\cdot\,\svar\,+\,\nvar}{\nvar}\,=\,\alpha^{2}\,+\,1
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\label{eq:toy_snr}
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\end{equation}
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When simplifying the decibel transformation~(Eq.\,\ref{eq:log}), the logarithmically
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scaled envelope $\db(t)$ can be expressed as a sum of two logarithmic terms
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assumed to have unit variance. By conversion of $\env(t)$ to decibel
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scale~(Eq.\,\ref{eq:log}), $\sca$ turns from a multiplicative scale in linear
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space into an additive term, or offset, in logarithmic space
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\begin{equation}
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\begin{split}
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\db(t)\,&=\,\log \frac{\alpha\,\cdot\,s(t)\,+\,\eta(t)}{\dbref}\\
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@@ -591,99 +586,90 @@ scaled envelope $\db(t)$ can be expressed as a sum of two logarithmic terms
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\end{split}
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\label{eq:toy_log}
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\end{equation}
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\textbf{Logarithmic component:}\\
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- Simplify decibel transformation (Eq.\,\ref{eq:log}) and apply to synthetic $\env(t)$\\
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- Isolate scale $\alpha$ and reference $\dbref$ using logarithm product/quotient laws
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$\rightarrow$ In log-space, a multiplicative scaling factor becomes additive\\
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$\rightarrow$ Allows for the separation of song signal $s(t)$ and its scale $\alpha$\\
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$\rightarrow$ Introduces scaling of noise term $\eta(t)$ by the inverse of $\alpha$\\
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$\rightarrow$ Normalization by $\dbref$ applies equally to all terms (no individual effects)
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\textbf{Adaptation component:}\\
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- Highpass filter over $\db(t)$ (Eq.\,\ref{eq:highpass}) can
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be approximated as subtraction of the local signal offset within a suitable time
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interval $\thp$ ($0 \ll \thp < \frac{1}{\fc}$)
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%
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which allows for its separation from $\soc(t)$ but introduces a scaling of
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$\noc(t)$ by the inverse of $\sca$. The subsequent
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highpass-filtering~(Eq.\,\ref{eq:highpass}) of $\db(t)$ can then be
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approximated as a subtraction of the local offset within a suitable time
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interval $0 \ll \thp < \frac{1}{\fc}$:
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\begin{equation}
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\begin{split}
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\adapt(t)\,\approx\,\db(t)\,-\,\log \frac{\alpha}{\dbref}\,=\,\log\left[s(t)\,+\,\frac{\eta(t)}{\alpha}\right]
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\end{split}
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\label{eq:toy_highpass}
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\end{equation}
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%
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\textbf{Implication for intensity invariance:}\\
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- Logarithmic scaling is essential for equalizing different song intensities\\
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$\rightarrow$ Intensity information can be manipulated more easily when in form
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of a signal offset in log-space than a multiplicative scale in linear space
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- Scale $\alpha$ can only be redistributed, not entirely eliminated from $\adapt(t)$\\
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$\rightarrow$ Turn initial scaling of song $s(t)$ by $\alpha$ into scaling of noise $\eta(t)$ by $\frac{1}{\alpha}$
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- Capability to compensate for intensity variations, i.e. selective amplification
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of output $\adapt(t)$ relative to input $\env(t)$, is limited by input SNR (Eq.\,\ref{eq:toy_snr}):\\
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$\alpha\gg1$: Attenuation of $\eta(t)$ term $\rightarrow$ $s(t)$ dominates $\adapt(t)$\\
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$\alpha\approx1$ Negligible effect on $\eta(t)$ term $\rightarrow$ $\adapt(t)=\log[s(t)+\eta(t)]$\\
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$\alpha\ll1$: Amplification of $\eta(t)$ term $\rightarrow$ $\eta(t)$ dominates $\adapt(t)$\\
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$\rightarrow$ Ability to equalize between different sufficiently large scales of $s(t)$\\
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$\rightarrow$ Inability to recover $s(t)$ when initially masked by noise floor $\eta(t)$
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- Logarithmic scaling emphasizes small amplitudes (song onsets, noise floor) \\
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$\rightarrow$ Recurring trade-off: Equalizing signal intensity vs preserving initial SNR
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\subsection{Threshold nonlinearity \& temporal averaging}
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Convolved $c_i(t)$ $\xrightarrow{\nl}$ Binary $b_i(t)$ $\xrightarrow{\lp}$ Feature $f_i(t)$
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\textbf{Thresholding component:}\\
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- Within an observed time interval $T$, $c_i(t)$ follows probability density $\pc$\\
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- Within $T$, $c_i(t)$ exceeds threshold value $\thr$ for time $T_1$ ($T_1+T_0=T$)\\
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- Threshold $\nl$ splits $\pc$ around $\thr$ in two complementary parts
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%
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This means that $\sca$ cannot be entirely eliminated from $\adapt(t)$, only
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redistributed between $\soc(t)$ and $\noc(t)$. In consequence, if $\sca$ is
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sufficiently large ($\sca\gg1$), $\noc(t)$ is attenuated to the point of being
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negligible, so that $\adapt(t)$ represents $\soc(t)$ in a scale-free manner. If
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$\soc(t)$ and $\noc(t)$ are at similar scales ($\sca\approx1$), $\adapt(t)$
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largely resembles $\db(t)$. However, if $\sca$ is sufficiently small
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($\sca\ll1$), $\noc(t)$ masks $\soc(t)$ even after the intensity adaptation.
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Therefore, the effective intensity invariance of $\adapt(t)$ relative to
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$\env(t)$ is limited by the initial scaling of $\soc(t)$ relative to $\noc(t)$;
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that is, the signal-to-noise ratio (SNR) of $\env(t)$ with ($\sca>0$) and
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without ($\sca=0$) song component $\soc(t)$
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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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\text{SNR}(\sca)\,=\,\frac{\xvar}{\nvar}\,=\,\frac{\alpha^{2}\,\cdot\,\svar\,+\,\nvar}{\nvar}\,=\,\alpha^{2}\,+\,1, \qquad \svar\,=\,\nvar\,=\,1
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\label{eq:toy_snr}
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\end{equation}
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which depends quadratically on $\sca$ if $\soc(t)$ and $\noc(t)$ are
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uncorrelated~($\soc(t)\perp\noc(t)$). In summary, the combination of
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logarithmic compression and adaptation allows for the equalization of different
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sufficiently large song scales, which is essential for intensity-invariant song
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representation. However, this mechanism is unable to recover songs that have
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already sunken below the noise floor, which emphasizes the importance of a
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sufficiently high SNR at the intial reception of the signal for reliable song
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recognition.
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\subsection{Thresholding nonlinearity \& temporal averaging}
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The second key mechanism for the emergence of intensity invariance along the
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model pathway takes place during the transformation of the kernel responses
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$c_i(t)$ over the binary responses $b_i(t)$ into the finalized features
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$f_i(t)$. This mechanism is mediated by the thresholding nonlinearity $\nl$. By
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passing $c_i(t)$ through the thresholding nonlinearity~(Eq.\,\ref{eq:binary}),
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its probability density within some observed time interval $T$ is split around
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threshold value $\thr$ into two complementary parts:
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\begin{equation}
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\int_{\thr}^{+\infty} \pc\,dc_i\,=\,1\,-\,\int_{-\infty}^{\thr} \pc\,dc_i\,=\,\frac{T_1}{T}, \qquad \infint \pc\,dc_i\,=\,1
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\label{eq:pdf_split}
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\end{equation}
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%
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$\rightarrow$ Semi-definite integral over right-sided portion of split $\pc$ gives ratio
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of time $T_1$ where $c_i(t)>\thr$ to total time $T$ due to normalization of $\pc$
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%
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Due to the normalization of $\pc$, the semi-definite integral over the
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right-sided part of the split $\pc$ is the ratio of time $T_1$ during which
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$c_i(t)$ exceeds $\thr$ within the total time $T$. If the subsequent lowpass
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filter~(Eq.\,\ref{eq:lowpass}) over $b_i(t)$ is approximated as temporal
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averaging over a suitable time interval
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$\tlp>\frac{1}{\fc}$
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\begin{equation}
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\infint \pc\,dc_i\,=\,1
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\label{eq:pdf}
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\end{equation}
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%
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\textbf{Averaging component:}\\
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- Lowpass filter over binary response $b_i(t)$ (Eq.\,\ref{eq:lowpass}) can be
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approximated as temporal averaging over a suitable time interval $\tlp$ ($\tlp > \frac{1}{\fc}$)\\
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- Within $\tlp$, $b_i(t)$ takes a value of 1 ($c_i(t)>\thr$) for time $T_1$ ($T_1+T_0=\tlp$)
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%
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\begin{equation}
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f_i(t)\,\approx\,\frac{1}{\tlp} \int_{t}^{t\,+\,\tlp} b_i(\tau)\,d\tau\,=\,\frac{T_1}{\tlp}
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f_i(t)\,\approx\,\frac{1}{\tlp} \int_{t}^{t\,+\,\tlp} b_i(\tau)\,d\tau\,=\,\frac{T_1}{\tlp}, \qquad b_i(t)\,\in\,\{0,\,1\}
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\label{eq:feat_avg}
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\end{equation}
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%
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$\rightarrow$ Temporal averaging over $b_i(t)\in[0,1]$ (Eq.\,\ref{eq:binary}) gives
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ratio of time $T_1$ where $c_i(t)>\thr$ to total averaging interval $\tlp$\\
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$\rightarrow$ Feature $f_i(t)$ approximately represents supra-threshold fraction of $\tlp$
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\textbf{Combined result:}\\
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- Feature $f_i(t)$ can be linked to the distribution of $c_i(t)$ using Eqs.\,\ref{eq:pdf_split} \& \ref{eq:feat_avg}
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%
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feature $f_i(t)$ likewise represents a ratio of time $T_1$ during which
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$b_i(t)$ is 1 within the total averaging interval $\tlp$. Since $b_i(t)$ is 1
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where $c_i(t)>\thr$, $f_i(t)$ relates to the probability density of $c_i(t)$ by
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\begin{equation}
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f_i(t)\,\approx\,\int_{\thr}^{+\infty} \pclp\,dc_i\,=\,P(c_i\,>\,\thr,\,\tlp)
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\label{eq:feat_prop}
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\end{equation}
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%
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$\rightarrow$ Because the integral over a probability density is a cumulative
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probability, the value of feature $f_i(t)$ (temporal compression of $b_i(t)$)
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at every time point $t$ signifies the probability that convolution output
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$c_i(t)$ exceeds the threshold value $\thr$ during the corresponding averaging
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interval $\tlp$
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Therefore, the value of $f_i(t)$ at every time point $t$ approximately
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signifies the cumulative probability that $c_i(t)$ exceeds $\thr$ during the
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corresponding averaging interval $\tlp$. Accordingly, the combination of
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thresholding nonlinearity and temporal averaging constitutes a remapping of a
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quantity that encodes temporal similarity between signal $\adapt(t)$ and kernel
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$k_i(t)$ into a quantity that encodes a duty cycle with respect to $\thr$.
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Accordingly, the combination of
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thresholding nonlinearity and temporal averaging constitutes a remapping of the
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amplitude-encoding quantity $c_i(t)$ into the duty cycle-encoding quantity
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$f_i(t)$ by binning graded amplitude values into one of two categorical states.
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This deliberate loss of precise amplitude information is the key to intensity
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invariance of the finalized features, as different scales of $c_i(t)$ can
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result in similar $T_1$ segments depending on the magnitude of the derivative
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of $c_i(t)$ in temporal proximity to time points at which $c_i(t)$ crosses
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$\thr$.
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\textbf{Implication for intensity invariance:}\\
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- Convolution output $c_i(t)$ quantifies temporal similarity between amplitudes of
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@@ -743,6 +729,21 @@ large-scale AM, current overall intensity level)\\
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$\rightarrow$ Without time scale selectivity, any fully intensity-invariant
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output will be a flat line
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\textbf{Log-HP: Implication for intensity invariance:}\\
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- Logarithmic scaling is essential for equalizing different song intensities\\
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$\rightarrow$ Intensity information can be manipulated more easily when in form
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of a signal offset in log-space than a multiplicative scale in linear space
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- Capability to compensate for intensity variations, i.e. selective amplification
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of output $\adapt(t)$ relative to input $\env(t)$, is limited by input SNR (Eq.\,\ref{eq:toy_snr}):\\
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$\rightarrow$ Ability to equalize between different sufficiently large scales of $s(t)$\\
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$\rightarrow$ Inability to recover $s(t)$ when initially masked by noise floor $\eta(t)$
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- Logarithmic scaling emphasizes small amplitudes (song onsets, noise floor) \\
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$\rightarrow$ Recurring trade-off: Equalizing signal intensity vs preserving initial SNR
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The model pathway includes a rather large number of Gabor kernels compared to
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the 15 to 20 ascending neurons in the grasshopper auditory
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system~(\bcite{stumpner1991auditory}).
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