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j-hartling
172
main.tex
172
main.tex
@@ -612,109 +612,114 @@ without ($\sca=0$) song component $\soc(t)$
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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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which depends quadratically on $\sca$ if $\soc(t)\perp\noc(t)$. Overall, the
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combination of logarithmic compression and adaptation allows for the
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equalization of different sufficiently large song scales, which is essential
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for intensity-invariant song representation. However, this mechanism is unable
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to recover songs that have already sunken below the noise floor, which
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emphasizes the importance of a sufficiently high SNR at the intial reception of
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the signal for reliable song 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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$f_i(t)$. Kernel response $c_i(t)$ quantifies the degree of similarity between
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kernel $k_i(t)$ and the preprocessed signal $\adapt(t)$. The thresholding
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nonlinearity $\nl$ categorizes the value of $c_i(t)$ at every time point $t$
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into "relevant" ($c_i(t)>\thr$, $b_i(t)=1$) and "irrelevant" ($c_i(t)\leq\thr$,
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$b_i(t)=0$) response values
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By passing $c_i(t)$ through the thresholding
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nonlinearity $\nl$, its amplitude values are binned
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into one of two categories~(Eq.\,\ref{eq:binary}).
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: $c_i(t)>\thr$
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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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its probability density $\pc$ within some observed time interval $T$ is split
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around 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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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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The right-sided part of the split $\pc$ corresponds to time $T_1$ where
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$c_i(t)>\thr$, while the left-sided part corresponds to time $T_0=T-T_1$ where
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$c_i(t)\leq\thr$. The semi-definite integral over the right-sided part of $\pc$
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represents the ratio of time $T_1$ to total time $T$ because the indefinite
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integral of a probability density is normalized to 1. Following the
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thresholding nonlinearity, the resulting binary responses $b_i(t)$ are
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lowpass-filtered~(Eq.\,\ref{eq:lowpass}) to obtain $f_i(t)$, which can be
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approximated as temporal averaging over a suitable time interval
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$\tlp>\frac{1}{\fc}$
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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}, \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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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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Feature $f_i(t)$
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If the 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 $\tlp>\frac{1}{\fc}$, then $f_i(t)$ can
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be linked to a similar temporal ratio
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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}, \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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of time $T_1$ during which $b_i(t)$ is 1 within the total averaging interval
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$\tlp$. 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 $\tlp$:
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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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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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In a sense, $f_i(t)$ resembles a duty cycle of some sort, which quantifies
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purely temporal relations in the structure of $c_i(t)$ with no regard for
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precise amplitude values apart from their relation 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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Accordingly, a substantial amount of information about the degree of similarity
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between signal $\adapt(t)$ and kernel $k_i(t)$ that is contained in $c_i(t)$ is
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lost during its transformation into $f_i(t)$. Instead, $f_i(t)$ only retains
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information about the temporal relation of $c_i(t)$ relative to $\thr$
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This loss of amplitude information is the key to the intensity
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invariance of $f_i(t)$: For a given $\thr$, different scales of $c_i(t)$ can
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still result in similar $T_1$ segments depending on the magnitude of the
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derivative of $c_i(t)$ in temporal proximity to time points at which $c_i(t)$
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crosses $\thr$. The steeper the slope of $c_i(t)$ around the threshold
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crossings, the less $T_1$ changes with scale variations.
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In a sense, $f_i(t)$ resembles a duty
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cycle of some sort, as it quantifies purely temporal relations in the structure
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of $c_i(t)$ with no regard for precise amplitude values apart from their
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relation to $\thr$. This near-complete loss of amplitude information is the key
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to the intensity invariance of $f_i(t)$: For a given $\thr$, different scales
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of $c_i(t)$ can still result in similar $T_1$ segments depending on the
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magnitude of the derivative of $c_i(t)$ in temporal proximity to time points at
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which $c_i(t)$ crosses $\thr$. The steeper the slope of $c_i(t)$ around the
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threshold crossings, the less $T_1$ changes with scale variations.
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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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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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- Feature $f_i(t)$ 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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- Thresholding of $c_i(t)$ and subsequent temporal averaging of $b_i(t)$ to
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obtain $f_i(t)$ 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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$\rightarrow$ If $T_1$ is invariant to scale variation in $c_i(t)$, then so is $f_i(t)$
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- Suggests a relatively simple rule for optimal choice of threshold value $\thr$:\\
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$\rightarrow$ Find amplitude $c_i$ that maximizes absolute derivative of $c_i(t)$ over time\\
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$\rightarrow$ Optimal with respect to intensity invariance of $f_i(t)$, not necessarily for
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other criteria such as song-noise separation or diversity between features
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- Nonlinear operations can be used to detach representations from graded physical
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stimulus (to fasciliate categorical behavioral decision-making?):\\
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1) Capture sufficiently precise amplitude information: $\env(t)$, $\adapt(t)$\\
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$\rightarrow$ Closely following the AM of the acoustic stimulus\\
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2) Quantify relevant stimulus properties on a graded scale: $c_i(t)$\\
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$\rightarrow$ More decorrelated representation, compared to prior stages\\
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3) Nonlinearity: Distinguish between "relevant vs irrelevant" values: $b_i(t)$\\
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$\rightarrow$ Trading a graded scale for two or more categorical states\\
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4) Represent stimulus properties under relevance constraint: $f_i(t)$\\
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$\rightarrow$ Graded again but highly decorrelated from the acoustic stimulus\\
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5) Categorical behavioral decision-making requires further nonlinearities\\
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$\rightarrow$ Parameters of a behavioral response may be graded (e.g. approach speed),
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initiation of one behavior over another is categorical (e.g. approach/stay)
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\section{Discriminating species-specific song\\patterns in feature space}
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\section{Conclusions \& outlook}
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\textbf{Song recognition pathway: Grasshopper vs. model:}\\
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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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\textbf{Definition of invariance (general, systemic):}\\
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Invariance = Property of a system to maintain a stable output with respect to a
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set of relevant input parameters (variation to be represented) but irrespective
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@@ -743,9 +748,26 @@ $\rightarrow$ Inability to recover $s(t)$ when initially masked by noise floor $
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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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\textbf{Thresh-LP: Implication for intensity invariance:}\\
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- Role of song periodicity for feature representation!
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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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- Suggests a relatively simple rule for optimal choice of threshold value $\thr$:\\
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$\rightarrow$ Find amplitude $c_i$ that maximizes absolute derivative of $c_i(t)$ over time\\
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$\rightarrow$ Optimal with respect to intensity invariance of $f_i(t)$, not necessarily for
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other criteria such as song-noise separation or diversity between features
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- Nonlinear operations can be used to detach representations from graded physical
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stimulus (to fasciliate categorical behavioral decision-making?):\\
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1) Capture sufficiently precise amplitude information: $\env(t)$, $\adapt(t)$\\
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$\rightarrow$ Closely following the AM of the acoustic stimulus\\
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2) Quantify relevant stimulus properties on a graded scale: $c_i(t)$\\
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$\rightarrow$ More decorrelated representation, compared to prior stages\\
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3) Nonlinearity: Distinguish between "relevant vs irrelevant" values: $b_i(t)$\\
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$\rightarrow$ Trading a graded scale for two or more categorical states\\
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4) Represent stimulus properties under relevance constraint: $f_i(t)$\\
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$\rightarrow$ Graded again but highly decorrelated from the acoustic stimulus\\
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5) Categorical behavioral decision-making requires further nonlinearities\\
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$\rightarrow$ Parameters of a behavioral response may be graded (e.g. approach speed),
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initiation of one behavior over another is categorical (e.g. approach/stay)
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\end{document}
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