Writing discussion.

main.tex

@@ -1781,56 +1781,99 @@ constant output intensity without sacrificing some of the corresponding input
 SNR. This suggests that the trade-off is a more general principle that applies
 to any transformation that achieves or improves intensity invariance.
 
+% Dependence of thresh-LP intensity invariance on threshold value (+unlimited SNR):
 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
+intensity invariance and SNR is mediated by the threshold value $\thr$. The
+effects of $\thr$ on the intensity invariance and SNR of $f_i(t)$ are best
+assessed if the mechanism is initially viewed in isolation. 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
+kernels in the set. 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.
+$\raw(t)$. 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.
+% Interaction between the two mechanisms of intensity invariance:
+% (Also: Extremely important, but maybe too wordy?)
+The combined effect of the two consecutive mechanisms of intensity invariance
+depends on which mechanism results in a lower saturation point. In case of
+$f_i(t)$, it is necessary to distinguish between its intrinsic saturation
+point~(the saturation point that the second mechanism can achieve in isolation)
+and its actual saturation point~(including the effects of the first mechanism).
+The same distinction applies to the saturation level of $f_i(t)$. The intrinsic
+saturation point of $f_i(t)$ increases with increasing $\thr$. The intrinsic
+saturation level of $f_i(t)$ is largely independent of $\thr$, assuming that
+$\thr$ is sufficiently small or $\sca$ is sufficiently large. If the intrinsic
+saturation point of $f_i(t)$ is lower than the saturation point of $\adapt(t)$,
+the second mechanism will take precedence over the first mechanism. In this
+case, $f_i(t)$ will reach the intrinsic saturation level at the intrinsic
+saturation point. In contrast, if the intrinsic saturation point of $f_i(t)$ is
+higher than the saturation point of $\adapt(t)$, the first mechanism will take
+precedence over the second mechanism. In this case, the actual saturation point
+of $f_i(t)$ will be determined by the saturation point of $\adapt(t)$ rather
+than the intrinsic saturation point of $f_i(t)$. This has no detrimental effect
+on the intensity invariance of $f_i(t)$. However, a lower saturation point of
+$f_i(t)$ means that the actual saturation level of $f_i(t)$ will be lower than
+its intrinsic saturation level. Moreover, the saturation level of $f_i(t)$ will
+not be independent of $\thr$ anymore but will decrease with increasing $\thr$.
+A lower saturation level of $f_i(t)$ does not necessarily impair the SNR of
+$f_i(t)$ --- $f_i(t)$ can still achieve an arbitrarily high SNR by setting
+$\thr$ just above the maximum pure-noise $c_i(t)$. However, a lower saturation
+level of $f_i(t)$ does mean that the range of possible feature values that
+$f_i(t)$ can take on is restricted compared to the case where $f_i(t)$ can
+reach its intrinsic saturation level. In summary, the interaction between the
+two mechanisms of intensity invariance along the pathway can have unfavorable
+consequences for the overall system if the first mechanism takes precedence
+over the second mechanism. However, this interaction does not so much affect
+the intensity invariance or the SNR of $f_i(t)$ but rather constraints the part
+of the feature space that is available for species-specific song
+representation.
+
+% Check log-axis histogram counts!
+% Why do so many features have a lower saturation point than adapt if so many
+% do not reach the intrinsic saturation level??
+Judging from the distribution of saturation points across the set of $f_i(t)$,
+both interactions between the two mechanisms appear to be present in the
+current pathway. A number of $f_i(t)$ achieve a lower saturation point than
+$\adapt(t)$, which indicates that the second mechanism takes precedence over
+the first mechanism. These cases raise the question whether the first mechanism
+is actually necessary for the overall system if the second mechanism can
+apparently achieve intensity invariance with a lower saturation point. There
+are also some $f_i(t)$ whose saturation point matches the saturation point of
+$\adapt(t)$, which indicates that the first mechanism takes precedence over the
+second mechanism. These cases raise the question whether intensity invariance
+by the first mechanism --- while achieving a lower saturation point than the
+second mechanism --- is actually beneficial
+
+The saturation point of $f_i(t)$ varies between different kernels in the set. A
+number of $f_i(t)$ achieve a lower saturation point than $c_i(t)$ --- and hence a lower saturation point than $\adapt(t)$ --- which
+indicates that the second mechanism takes precedence over the first mechanism.
+Some $f_i(t)$ exhibit similar or only marginally lower saturation points than
 
 
-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.
-
-
-% \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
-lower saturation point than the second mechanism by itself, the saturation
-point of feature $f_i(t)$ will be determined solely by the first mechanism. In
-this case, the saturation level of $f_i(t)$ will conform to the intensity that
-$f_i(t)$ can reach for the given saturation point rather than the intrinsic
-saturation level of $f_i(t)$. Conversely, if the second mechanism results in a
-lower saturation point than the first mechanism, both the saturation point and
-the saturation level of $f_i(t)$ will be determined by the second mechanism.
 The saturation points of $f_i(t)$ across the set are distributed over a much
-wider range than those of the preceeding kernel responses $c_i(t)$, which
-suggests that the interaction between the two mechanisms is specific to
-individual kernels. A number of $f_i(t)$ achieve a lower saturation point than
-the respective $c_i(t)$, whereas some $f_i(t)$ exhibit similar or only
-marginally lower saturation points. In these cases, the question arises to what
-extent two consecutive mechanisms of intensity invariance are actually
-beneficial for the overall system.
+wider range than those of the preceeding $c_i(t)$, 
+
+which suggests that the
+interaction between the two mechanisms is specific to individual kernels. A
+number of $f_i(t)$ achieve a lower saturation point than the respective
+$c_i(t)$, 
+
+whereas some $f_i(t)$ exhibit similar or only marginally lower
+saturation points. 
+
+
+In these cases, the question arises to what extent two
+consecutive mechanisms of intensity invariance are actually beneficial for the
+overall system.
 
 From a computational perspective, the answer could be that logarithmic
 compression and adaptation is a necessary preprocessing step towards robust