Merge branch 'main' of https://whale.am28.uni-tuebingen.de/git/j_hartling/paper_2025

main.tex

@@ -1691,7 +1691,7 @@ additional certainty.
 \subsection{Invariant processing in the grasshopper auditory system}
 \label{sec:general_inv}
 
-% Invariance in the general (systemic) sense:
+% Invariance in the general (systemic) sense (could be skipped if too much):
 The notion of invariance is fundamental for sensory processing systems.
 Invariance, in the general sense, can be described as the property of a
 transformation to maintain variation across certain meaningful input parameters
@@ -1785,52 +1785,61 @@ that are robust to noise masking~(\bcite{einhaupl2011attractiveness}).
 % Trading SNR for log-HP intensity invariance (+variability, +general principle):
 The SNR of each song representation prior to $\adapt(t)$ increases
 monotonically with $\sca$~(excluding $0<\sca\ll1$, noise regime). These
-representations maintain and improve the initial SNR of $\raw(t)$ and hence
-never achieve intensity invariance. In contrast, the SNR of the
-intensity-invariant $\adapt(t)$ never exceeds its saturation level even for
-arbitrarily high $\sca$. The saturation level of $\adapt(t)$ varies across
-species and songs. This variability is likely rooted in the way in which
-logarithmic compression acts on the specific distribution of $\env(t)$, which
-depends on the $\fc$ of the lowpass filter as well as the temporal structure
-and frequency spectrum of the rectified $\filt(t)$. Overall, $\adapt(t)$ has
-never been observed to exceed a SNR of around~10 across all songs. The low SNR
-of $\adapt(t)$ partially results from the amplification of smaller values of
-$\env(t)$ by the logarithm, which raises the noise floor of $\adapt(t)$. Still,
-the reduction in SNR is substantial --- considering that the SNR of preceeding
-song representations has been orders of magnitude higher --- but is likely a
-necessary price to pay for the intensity invariance of $\adapt(t)$. After all,
-a transformation cannot compress a range of different input intensities into a
+representations maintain the full extent of the initial SNR of $\raw(t)$ and
+hence never achieve intensity invariance. In contrast, the SNR of $\adapt(t)$
+saturates for sufficiently high $\sca$. Accordingly, $\adapt(t)$ is
+intensity-invariant but cannot have a higher SNR than the saturation level,
+which indicates a fundamental trade-off. The saturation level of $\adapt(t)$
+varies across species and songs. This variability is likely rooted in the way
+in which logarithmic compression acts on the specific $\env(t)$, which depends
+on the $\fc$ of the lowpass filter as well as the temporal structure and
+frequency spectrum of the rectified $\filt(t)$. Across all songs, the
+saturation level of $\adapt(t)$ has never been observed to exceed a SNR of
+around~10. This is a substantial reduction in SNR, considering that the SNR of
+preceeding representations had been orders of magnitude higher. Part of this
+reduction stems from the amplification of smaller values of $\env(t)$ by
+logarithmic compression, which raises the noise floor of $\adapt(t)$ relative
+to the song. Accordingly, the low SNR of $\adapt(t)$ appears to be a necessary
+price to pay for its intensity invariance through logarithmic compression and
+adaptation. But the trade-off between intensity invariance and SNR likely goes
+beyond the particular mechanisms along the pathway. After all, a transformation
+is not expected to compress a range of different input intensities into a
 constant output intensity without sacrificing some of the corresponding input
-SNR. Accordingly, the trade-off between intensity invariance and SNR is not
-expected to be specific to the particular mechanisms along the pathway but
-presumably applies to any transformation that achieves or improves intensity
-invariance.
+SNR. This suggests that the trade-off is a more general principle that applies
+to any transformation that achieves or improves intensity invariance.
 
-Thresholding and temporal averaging renders feature $f_i(t)$
-intensity-invariant for sufficiently large $\sca$. The trade-off between
-intensity invariance and SNR is mediated by threshold value $\thr$. A lower
-$\thr$ ($\thr\to0$) improves intensity invariance by shifting the saturation
-point towards lower $\sca$ but also decreases the SNR of $f_i(t)$. The
-saturation level of $f_i(t)$ is independent of $\thr$ as long as the intensity
-invariance by the previous mechanism is neglected. The SNR of $f_i(t)$ is
-therefore determined solely by the pure-noise response of $f_i(t)$. The
-distribution $\pci$ of the pure-noise kernel response $c_i(t)$ is largely a
-normal distribution with mean $\mu\approx0$ for all kernels $k_i(t)$. The value
-of the pure-noise $f_i(t)$ is hence 0.5 for $\thr=0$ and decreases for higher
-$\thr$. If $\thr$ is set above the maximum of $c_i(t)$, the pure-noise feature
-value is 0, which results in an "unlimited" SNR of $f_i(t)$. In this case, any
-non-zero feature value that is sustained for a sufficient duration could serve
-as indicator for the presence of $\soc(t)$, although at the cost of a higher
-saturation point. The maximum of the pure-noise $c_i(t)$ is assumed to be very
+The second mechanism of intensity invariance consists of thresholding and
+temporal averaging of $c_i(t)$ into $f_i(t)$. Here, the trade-off between
+intensity invariance and SNR is mediated by the threshold value $\thr$. A lower
+$\thr$~($\thr\to0$) improves the intensity invariance of $f_i(t)$ by shifting
+the saturation point towards lower $\sca$. However, a lower $\thr$ also raises
+the noise floor of $f_i(t)$ by including more of the pure-noise $c_i(t)$, which
+decreases the SNR of $f_i(t)$. The distribution $\pci$ of the pure-noise
+$c_i(t)$ is very close to a normal distribution with mean $\mu\approx0$ for all
+kernels $k_i(t)$. The value of the pure-noise $f_i(t)$ is hence 0.5 for
+$\thr=0$ and decreases for higher $\thr$. If $\thr$ is set above the maximum of
+$c_i(t)$, the pure-noise feature value is 0, which results in an "unlimited"
+SNR of $f_i(t)$. In this case, any non-zero feature value that is sustained for
+a sufficient duration could serve as indicator for the presence of $\soc(t)$ in
+$\raw(t)$, although at the cost of a higher saturation point. Of course, this
+would require a fine evolutionary tuning of $\thr$ to the properties of the
+natural noise in a certain habitat to avoid false positives.
+
+The saturation level of $f_i(t)$ is independent of $\thr$ as long as the
+intensity invariance by the previous mechanism is neglected.
+
+
+If $f_i(t)$ can achieve an arbitrarily high SNR, it can counteract the
+comparably low SNR of $\adapt(t)$ 
+
+The maximum of the pure-noise $c_i(t)$ is assumed to be very
 small due to the various SNR improvements along the pathway, so that the
 required increase in $\thr$ and hence the saturation point of $f_i(t)$ is not
-expected to be substantial. However, exploiting the capacity of $f_i(t)$ for
-arbitrarily high SNR would certainly require a fine evolutionary tuning of
-$\thr$ to the properties of both the species-specific song and the natural
-noise in a certain habitat.
+expected to be substantial.
 
-\newpage
-\subsection{Intensity invariance versus intensity invariance}
+
+% \newpage
+% \subsection{Intensity invariance versus intensity invariance}
 
 Two consecutive mechanisms of intensity invariance do not necessarily add up to
 a stronger overall intensity invariance. If the first mechanism results in a