Wrote results rect-lp and log-hp :)

Finished some more figure captions.
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j-hartling
2026-05-04 19:50:04 +02:00
parent 69f172ff2c
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15 changed files with 1376 additions and 1232 deletions

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\newlabel{fig:pathway}{{1}{6}{\textbf {Schematic organisation of the grasshopper song recognition pathway and structure of the functional model pathway.} \textbf {a}:~Simplified course of the pathway in the grasshopper, from the tympanal membrane over receptor neurons, local interneurons, and ascending neurons further towards the supraesophageal ganglion. \textbf {b}:~Schematic of synaptic connections between the three neuronal populations within the metathoracic ganglion. \textbf {c}:~Network representation of neuronal connectivity. \textbf {d}:~Flow diagram of consecutive signal representations~(boxes) and transformations~(arrows) along the model pathway. All representations are time-varying. 1st half: Preprocessing stage~(one-dimensional representation). 2nd half: Feature extraction stage~(high-dimensional representation). }{}{}}
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\newlabel{fig:rect-lp}{{4}{12}{\textbf {Rectification and lowpass filtering improves SNR but does not contribute to intensity invariance.} Input $x_{\text {raw}}(t)$ consists of song component $s(t)$ scaled by $\alpha $ with optional noise component $\eta (t)$ and is successively transformed into tympanal signal $x_{\text {filt}}(t)$ and envelope $x_{\text {env}}(t)$. Different line styles indicate different cutoff frequencies $f_{\text {cut}}$ of the lowpass filter extracting $x_{\text {env}}(t)$. \textbf {Top}:~Example representations of $x_{\text {filt}}(t)$ and $x_{\text {env}}(t)$ for different $\alpha $. \textbf {a}:~Noiseless case. \textbf {b}:~Noisy case. \textbf {Bottom}:~Intensity metrics over a range of $\alpha $. \textbf {c}:~Noiseless case: Standard deviations $\sigma _x$ of $x_{\text {filt}}(t)$ and $x_{\text {env}}(t)$. \textbf {d}:~Noisy case: Ratios of $\sigma _x$ of $x_{\text {filt}}(t)$ and $x_{\text {env}}(t)$ to the respective reference standard deviation $\sigma _{\eta }$ for input $x_{\text {raw}}(t)=\eta (t)$. \textbf {e}:~Ratios of $\sigma _x$ to $\sigma _{\eta }$ of $x_{\text {env}}(t)$ as in \textbf {d} for different species (averaged over songs and recordings, see appendix Fig.\,\ref {fig:app_rect-lp}). }{}{}}
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\newlabel{fig:log-hp}{{5}{14}{\textbf {Intensity invariance through logarithmic compression and adaptation is restricted by the noise floor and decreases SNR.} Input $x_{\text {filt}}(t)$ consists of song component $s(t)$ scaled by $\alpha $ with optional noise component $\eta (t)$ and is successively transformed into envelope $x_{\text {env}}(t)$, logarithmically compressed envelope $x_{\text {log}}(t)$, and intensity-adapted envelope $x_{\text {adapt}}(t)$. \textbf {Top}:~Example representations of $x_{\text {env}}(t)$, $x_{\text {log}}(t)$, and $x_{\text {adapt}}(t)$ for different $\alpha $. \textbf {a}:~Noiseless case. \textbf {b}:~Noisy case. \textbf {Bottom}:~Intensity metrics over a range of $\alpha $. \textbf {c}:~Noiseless case: Standard deviations $\sigma _x$ of $x_{\text {env}}(t)$, $x_{\text {log}}(t)$, and $x_{\text {adapt}}(t)$. \textbf {d}:~Noisy case: Ratios of $\sigma _x$ of $x_{\text {env}}(t)$, $x_{\text {log}}(t)$, and $x_{\text {adapt}}(t)$ to the respective reference standard deviation $\sigma _{\eta }$ for input $x_{\text {filt}}(t)=\eta (t)$. Shaded areas indicate $5\,\%$ (dark grey) and $95\,\%$ (light grey) curve span for $x_{\text {adapt}}(t)$. \textbf {e}:~Ratios of $\sigma _x$ to $\sigma _{\eta }$ of $x_{\text {adapt}}(t)$ as in \textbf {d} for different species (averaged over songs and recordings, see appendix Fig\,\ref {fig:app_log-hp_curves}). Dots indicate $95\,\%$ curve span per species. }{}{}}
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\newlabel{fig:thresh-lp_single}{{6}{16}{\textbf {Intensity invariance through thresholding and temporal averaging is mediated by the interaction of threshold value and noise floor.} Input $x_{\text {adapt}}(t)$ consists of song component $s(t)$ scaled by $\alpha $ with optional noise component $\eta (t)$ and is transformed into single kernel response $c(t)$, binary response $b(t)$, and feature $f(t)$. Different color shades indicate different threshold values $\Theta $ (multiples of reference standard deviation $\sigma _{\eta }$ of $c(t)$ for input $x_{\text {adapt}}(t)=\eta (t)$, with darker colors for higher $\Theta $). \textbf {Left}:~Noisy case: Example representations of $x_{\text {adapt}}(t)$ as well as $c(t)$, $b(t)$, and $f(t)$ for different $\alpha $. \textbf {a}:~$x_{\text {adapt}}(t)$ with kernel $k(t)$ in black. \textbf {b\,-\,d}: $c(t)$, $b(t)$, and $f(t)$ based on the same $x_{\text {adapt}}(t)$ from \textbf {a} but with different $\Theta $. \textbf {Right}:~Average value $\mu _f$ of $f(t)$ for each $\Theta $ from \textbf {b\,-\,d}, once for the noisy case (solid lines) and once for the noiseless case (dotted lines). Dots indicate $95\,\%$ curve span (noisy case). \textbf {e}:~$\mu _f$ over a range of $\alpha $. \textbf {f}:~$\mu _f$ over the standard deviation of noisy input $x_{\text {adapt}}$ corresponding to the values of $\alpha $ shown in \textbf {e}. }{}{}}
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\@writefile{lof}{\contentsline {figure}{\numberline {9}{\ignorespaces \textbf {Step-wise emergence of intensity invariant song representation along the model pathway without logarithmic compression.} Input $x_{\text {raw}}(t)$ consists of song component $s(t)$ scaled by $\alpha $ with added noise component $\eta (t)$ and is processed up to the feature set $f_i(t)$, skipping $x_{\text {log}}(t)$. Different color shades indicate different types of Gabor kernels with specific lobe number $n$ and either $+$ or $-$ sign, sorted (dark to light) first by increasing $n$ and then by sign~($1\,\leq \,n\,\leq \,4$; first $+$, then $-$ for each $n$; five kernel widths $\sigma $ of 1, 2, 4, 8, and $16\,$ms per type; 8 types, 40 kernels in total). \textbf {a}:~Example representations of $x_{\text {filt}}(t)$, $x_{\text {env}}(t)$, $x_{\text {adapt}}(t)$, $c_i(t)$, and $f_i(t)$ for different $\alpha $. \textbf {b}:~Intensity metrics over $\alpha $. For $c_i(t)$ and $f_i(t)$, the median over kernels is shown. Dots indicate $95\,\%$ curve span for $f_i(t)$. \textbf {c}:~Average value $\mu _{f_i}$ of each feature $f_i(t)$ over $\alpha $. \textbf {d}:~Ratios of intensity metrics to the respective reference value for input $x_{\text {raw}}(t)=\eta (t)$. For $c_i(t)$ and $f_i(t)$, the median over kernel-specific ratios is shown. \textbf {e}:~Ratios of $\mu _{f_i}$. \textbf {f}:~Distribution of kernel-specific $\alpha $ that correspond to $95\,\%$ curve span for $f_i(t)$. Dots indicate the value from \textbf {b}. }}{20}{}\protected@file@percent }
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\@writefile{lof}{\contentsline {figure}{\numberline {10}{\ignorespaces \textbf {Interspecific and intraspecific feature variability.} Average value $\mu _{f_i}$ of each feature $f_i(t)$ against its counterpart from a 2nd feature set based on a different input $x_{\text {raw}}(t)$. Each dot within a subplot represents a single feature $f_i(t)$. Different color shades indicate different types of Gabor kernels with specific lobe number $n$ and either $+$ or $-$ sign, sorted (dark to light) first by increasing $n$ and then by sign~($1\,\leq \,n\,\leq \,4$; first $+$, then $-$ for each $n$; five kernel widths $\sigma $ of 1, 2, 4, 8, and $16\,$ms per type; 8 types, 40 kernels in total). Data is based on the analysis underlying Fig\,\ref {fig:pipeline_full}. \textbf {Lower triangular}:~Interspecific comparisons between single songs of different species. \textbf {Upper triangular}:~Intraspecific comparisons between different songs of a single species (\textit {O. rufipes}). \textbf {Lower left}:~Distribution of correlation coefficients $\rho $ for each interspecific and intraspecific comparison. Dots indicate single $\rho $ values. }}{21}{}\protected@file@percent }
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[1620] Biber.pm:131> WARN - BibTeX subsystem: C:\Users\jonah\AppData\Local\Temp\biber_tmp_n7RG\347c261ec4135a5723bef5c751f5078f_20112.utf8, line 729, warning: 6 characters of junk seen at toplevel
[1621] Biber.pm:131> WARN - BibTeX subsystem: C:\Users\jonah\AppData\Local\Temp\biber_tmp_n7RG\347c261ec4135a5723bef5c751f5078f_20112.utf8, line 749, warning: 6 characters of junk seen at toplevel
[1621] Biber.pm:131> WARN - BibTeX subsystem: C:\Users\jonah\AppData\Local\Temp\biber_tmp_n7RG\347c261ec4135a5723bef5c751f5078f_20112.utf8, line 766, warning: 6 characters of junk seen at toplevel
[1621] Biber.pm:131> WARN - BibTeX subsystem: C:\Users\jonah\AppData\Local\Temp\biber_tmp_n7RG\347c261ec4135a5723bef5c751f5078f_20112.utf8, line 775, warning: 6 characters of junk seen at toplevel
[1621] Biber.pm:131> WARN - BibTeX subsystem: C:\Users\jonah\AppData\Local\Temp\biber_tmp_n7RG\347c261ec4135a5723bef5c751f5078f_20112.utf8, line 800, warning: 6 characters of junk seen at toplevel
[1621] Biber.pm:131> WARN - BibTeX subsystem: C:\Users\jonah\AppData\Local\Temp\biber_tmp_n7RG\347c261ec4135a5723bef5c751f5078f_20112.utf8, line 817, warning: 6 characters of junk seen at toplevel
[1621] Biber.pm:133> INFO - WARNINGS: 55
[37] biber:340> INFO - === Mo Mai 4, 2026, 19:48:34
[45] Biber.pm:419> INFO - Reading 'main.bcf'
[75] Biber.pm:979> INFO - Found 55 citekeys in bib section 0
[80] Biber.pm:4419> INFO - Processing section 0
[85] Biber.pm:4610> INFO - Looking for bibtex file 'cite.bib' for section 0
[87] bibtex.pm:1713> INFO - LaTeX decoding ...
[117] bibtex.pm:1519> INFO - Found BibTeX data source 'cite.bib'
[296] UCollate.pm:68> INFO - Overriding locale 'en-US' defaults 'variable = shifted' with 'variable = non-ignorable'
[296] UCollate.pm:68> INFO - Overriding locale 'en-US' defaults 'normalization = NFD' with 'normalization = prenormalized'
[296] Biber.pm:4239> INFO - Sorting list 'nyt/global//global/global' of type 'entry' with template 'nyt' and locale 'en-US'
[296] Biber.pm:4245> INFO - No sort tailoring available for locale 'en-US'
[320] bbl.pm:660> INFO - Writing 'main.bbl' with encoding 'UTF-8'
[331] bbl.pm:763> INFO - Output to main.bbl
[331] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_vMHw/347c261ec4135a5723bef5c751f5078f_55959.utf8, line 10, warning: 6 characters of junk seen at toplevel
[331] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_vMHw/347c261ec4135a5723bef5c751f5078f_55959.utf8, line 21, warning: 6 characters of junk seen at toplevel
[331] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_vMHw/347c261ec4135a5723bef5c751f5078f_55959.utf8, line 38, warning: 6 characters of junk seen at toplevel
[331] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_vMHw/347c261ec4135a5723bef5c751f5078f_55959.utf8, line 49, warning: 6 characters of junk seen at toplevel
[331] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_vMHw/347c261ec4135a5723bef5c751f5078f_55959.utf8, line 58, warning: 6 characters of junk seen at toplevel
[331] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_vMHw/347c261ec4135a5723bef5c751f5078f_55959.utf8, line 73, warning: 6 characters of junk seen at toplevel
[331] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_vMHw/347c261ec4135a5723bef5c751f5078f_55959.utf8, line 82, warning: 6 characters of junk seen at toplevel
[331] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_vMHw/347c261ec4135a5723bef5c751f5078f_55959.utf8, line 91, warning: 6 characters of junk seen at toplevel
[331] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_vMHw/347c261ec4135a5723bef5c751f5078f_55959.utf8, line 100, warning: 6 characters of junk seen at toplevel
[331] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_vMHw/347c261ec4135a5723bef5c751f5078f_55959.utf8, line 109, warning: 6 characters of junk seen at toplevel
[331] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_vMHw/347c261ec4135a5723bef5c751f5078f_55959.utf8, line 118, warning: 6 characters of junk seen at toplevel
[331] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_vMHw/347c261ec4135a5723bef5c751f5078f_55959.utf8, line 127, warning: 6 characters of junk seen at toplevel
[331] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_vMHw/347c261ec4135a5723bef5c751f5078f_55959.utf8, line 136, warning: 6 characters of junk seen at toplevel
[331] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_vMHw/347c261ec4135a5723bef5c751f5078f_55959.utf8, line 157, warning: 6 characters of junk seen at toplevel
[331] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_vMHw/347c261ec4135a5723bef5c751f5078f_55959.utf8, line 178, warning: 6 characters of junk seen at toplevel
[331] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_vMHw/347c261ec4135a5723bef5c751f5078f_55959.utf8, line 187, warning: 6 characters of junk seen at toplevel
[331] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_vMHw/347c261ec4135a5723bef5c751f5078f_55959.utf8, line 196, warning: 6 characters of junk seen at toplevel
[331] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_vMHw/347c261ec4135a5723bef5c751f5078f_55959.utf8, line 207, warning: 6 characters of junk seen at toplevel
[331] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_vMHw/347c261ec4135a5723bef5c751f5078f_55959.utf8, line 218, warning: 6 characters of junk seen at toplevel
[331] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_vMHw/347c261ec4135a5723bef5c751f5078f_55959.utf8, line 229, warning: 6 characters of junk seen at toplevel
[331] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_vMHw/347c261ec4135a5723bef5c751f5078f_55959.utf8, line 240, warning: 6 characters of junk seen at toplevel
[331] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_vMHw/347c261ec4135a5723bef5c751f5078f_55959.utf8, line 249, warning: 6 characters of junk seen at toplevel
[331] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_vMHw/347c261ec4135a5723bef5c751f5078f_55959.utf8, line 258, warning: 6 characters of junk seen at toplevel
[331] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_vMHw/347c261ec4135a5723bef5c751f5078f_55959.utf8, line 269, warning: 6 characters of junk seen at toplevel
[331] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_vMHw/347c261ec4135a5723bef5c751f5078f_55959.utf8, line 278, warning: 6 characters of junk seen at toplevel
[331] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_vMHw/347c261ec4135a5723bef5c751f5078f_55959.utf8, line 289, warning: 6 characters of junk seen at toplevel
[331] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_vMHw/347c261ec4135a5723bef5c751f5078f_55959.utf8, line 300, warning: 6 characters of junk seen at toplevel
[331] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_vMHw/347c261ec4135a5723bef5c751f5078f_55959.utf8, line 309, warning: 6 characters of junk seen at toplevel
[331] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_vMHw/347c261ec4135a5723bef5c751f5078f_55959.utf8, line 328, warning: 6 characters of junk seen at toplevel
[331] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_vMHw/347c261ec4135a5723bef5c751f5078f_55959.utf8, line 337, warning: 6 characters of junk seen at toplevel
[331] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_vMHw/347c261ec4135a5723bef5c751f5078f_55959.utf8, line 400, warning: 6 characters of junk seen at toplevel
[331] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_vMHw/347c261ec4135a5723bef5c751f5078f_55959.utf8, line 419, warning: 6 characters of junk seen at toplevel
[331] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_vMHw/347c261ec4135a5723bef5c751f5078f_55959.utf8, line 428, warning: 6 characters of junk seen at toplevel
[331] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_vMHw/347c261ec4135a5723bef5c751f5078f_55959.utf8, line 437, warning: 6 characters of junk seen at toplevel
[331] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_vMHw/347c261ec4135a5723bef5c751f5078f_55959.utf8, line 456, warning: 6 characters of junk seen at toplevel
[331] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_vMHw/347c261ec4135a5723bef5c751f5078f_55959.utf8, line 491, warning: 6 characters of junk seen at toplevel
[331] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_vMHw/347c261ec4135a5723bef5c751f5078f_55959.utf8, line 526, warning: 6 characters of junk seen at toplevel
[331] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_vMHw/347c261ec4135a5723bef5c751f5078f_55959.utf8, line 535, warning: 6 characters of junk seen at toplevel
[332] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_vMHw/347c261ec4135a5723bef5c751f5078f_55959.utf8, line 556, warning: 6 characters of junk seen at toplevel
[332] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_vMHw/347c261ec4135a5723bef5c751f5078f_55959.utf8, line 565, warning: 6 characters of junk seen at toplevel
[332] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_vMHw/347c261ec4135a5723bef5c751f5078f_55959.utf8, line 576, warning: 6 characters of junk seen at toplevel
[332] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_vMHw/347c261ec4135a5723bef5c751f5078f_55959.utf8, line 587, warning: 6 characters of junk seen at toplevel
[332] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_vMHw/347c261ec4135a5723bef5c751f5078f_55959.utf8, line 619, warning: 6 characters of junk seen at toplevel
[332] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_vMHw/347c261ec4135a5723bef5c751f5078f_55959.utf8, line 648, warning: 6 characters of junk seen at toplevel
[332] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_vMHw/347c261ec4135a5723bef5c751f5078f_55959.utf8, line 658, warning: 6 characters of junk seen at toplevel
[332] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_vMHw/347c261ec4135a5723bef5c751f5078f_55959.utf8, line 667, warning: 6 characters of junk seen at toplevel
[332] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_vMHw/347c261ec4135a5723bef5c751f5078f_55959.utf8, line 688, warning: 6 characters of junk seen at toplevel
[332] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_vMHw/347c261ec4135a5723bef5c751f5078f_55959.utf8, line 709, warning: 6 characters of junk seen at toplevel
[332] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_vMHw/347c261ec4135a5723bef5c751f5078f_55959.utf8, line 720, warning: 6 characters of junk seen at toplevel
[332] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_vMHw/347c261ec4135a5723bef5c751f5078f_55959.utf8, line 729, warning: 6 characters of junk seen at toplevel
[332] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_vMHw/347c261ec4135a5723bef5c751f5078f_55959.utf8, line 749, warning: 6 characters of junk seen at toplevel
[332] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_vMHw/347c261ec4135a5723bef5c751f5078f_55959.utf8, line 766, warning: 6 characters of junk seen at toplevel
[332] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_vMHw/347c261ec4135a5723bef5c751f5078f_55959.utf8, line 775, warning: 6 characters of junk seen at toplevel
[332] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_vMHw/347c261ec4135a5723bef5c751f5078f_55959.utf8, line 800, warning: 6 characters of junk seen at toplevel
[332] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_vMHw/347c261ec4135a5723bef5c751f5078f_55959.utf8, line 817, warning: 6 characters of junk seen at toplevel
[332] Biber.pm:133> INFO - WARNINGS: 55

View File

@@ -1,180 +1,184 @@
# Fdb version 4
["biber main"] 1777829759 "main.bcf" "main.bbl" "main" 1777829958 0
"./cite.bib" 1771664051 28308 4290db0c91f7b5055e25472ef913f6b4 ""
"main.bcf" 1777829957 112931 2a478116d80ebb1ada7083a24facd6e3 "pdflatex"
["biber main"] 1777916914.1072 "main.bcf" "main.bbl" "main" 1777916953.0495 0
"cite.bib" 1770904753.08918 27483 4290db0c91f7b5055e25472ef913f6b4 ""
"main.bcf" 1777916952.98174 112931 2a478116d80ebb1ada7083a24facd6e3 "pdflatex"
(generated)
"main.bbl"
"main.blg"
(rewritten before read)
["pdflatex"] 1777829955 "c:/Users/jonah/Desktop/phd/paper/paper_2025/main.tex" "main.pdf" "main" 1777829958 0
"C:/Users/jonah/AppData/Local/MiKTeX/fonts/map/pdftex/pdftex.map" 1770058784 279778 aa8c8b08a6e6f5b912fcdb1120e47ad8 ""
"C:/Users/jonah/AppData/Local/MiKTeX/miktex/data/le/pdftex/pdflatex.fmt" 1692800637 24120620 9856d41a93bb8d2530a663260e901d35 ""
"C:/Users/jonah/AppData/Local/Programs/MiKTeX/fonts/tfm/public/amsfonts/cmextra/cmex7.tfm" 1233951848 1004 54797486969f23fa377b128694d548df ""
"C:/Users/jonah/AppData/Local/Programs/MiKTeX/fonts/tfm/public/amsfonts/cmextra/cmex8.tfm" 1233951848 988 bdf658c3bfc2d96d3c8b02cfc1c94c20 ""
"C:/Users/jonah/AppData/Local/Programs/MiKTeX/fonts/tfm/public/amsfonts/symbols/msam10.tfm" 1233951854 916 f87d7c45f9c908e672703b83b72241a3 ""
"C:/Users/jonah/AppData/Local/Programs/MiKTeX/fonts/tfm/public/amsfonts/symbols/msam7.tfm" 1233951854 928 2dc8d444221b7a635bb58038579b861a ""
"C:/Users/jonah/AppData/Local/Programs/MiKTeX/fonts/tfm/public/amsfonts/symbols/msbm10.tfm" 1233951854 908 2921f8a10601f252058503cc6570e581 ""
"C:/Users/jonah/AppData/Local/Programs/MiKTeX/fonts/tfm/public/amsfonts/symbols/msbm7.tfm" 1233951854 940 228d6584342e91276bf566bcf9716b83 ""
"C:/Users/jonah/AppData/Local/Programs/MiKTeX/fonts/tfm/public/cm/cmbx10.tfm" 1136765053 1328 c834bbb027764024c09d3d2bf908b5f0 ""
"C:/Users/jonah/AppData/Local/Programs/MiKTeX/fonts/tfm/public/cm/cmbx12.tfm" 1136765053 1324 c910af8c371558dc20f2d7822f66fe64 ""
"C:/Users/jonah/AppData/Local/Programs/MiKTeX/fonts/tfm/public/cm/cmbxti10.tfm" 1136765053 1532 9162035f4e7176612125649e348e2195 ""
"C:/Users/jonah/AppData/Local/Programs/MiKTeX/fonts/tfm/public/cm/cmex10.tfm" 1136765053 992 662f679a0b3d2d53c1b94050fdaa3f50 ""
"C:/Users/jonah/AppData/Local/Programs/MiKTeX/fonts/tfm/public/cm/cmmi10.tfm" 1136765053 1528 abec98dbc43e172678c11b3b9031252a ""
"C:/Users/jonah/AppData/Local/Programs/MiKTeX/fonts/tfm/public/cm/cmmi12.tfm" 1136765053 1524 4414a8315f39513458b80dfc63bff03a ""
"C:/Users/jonah/AppData/Local/Programs/MiKTeX/fonts/tfm/public/cm/cmmi6.tfm" 1136765053 1512 f21f83efb36853c0b70002322c1ab3ad ""
"C:/Users/jonah/AppData/Local/Programs/MiKTeX/fonts/tfm/public/cm/cmmi8.tfm" 1136765053 1520 eccf95517727cb11801f4f1aee3a21b4 ""
"C:/Users/jonah/AppData/Local/Programs/MiKTeX/fonts/tfm/public/cm/cmr10.tfm" 1136765053 1296 45809c5a464d5f32c8f98ba97c1bb47f ""
"C:/Users/jonah/AppData/Local/Programs/MiKTeX/fonts/tfm/public/cm/cmr12.tfm" 1136765053 1288 655e228510b4c2a1abe905c368440826 ""
"C:/Users/jonah/AppData/Local/Programs/MiKTeX/fonts/tfm/public/cm/cmr17.tfm" 1136765053 1292 296a67155bdbfc32aa9c636f21e91433 ""
"C:/Users/jonah/AppData/Local/Programs/MiKTeX/fonts/tfm/public/cm/cmr6.tfm" 1136765053 1300 b62933e007d01cfd073f79b963c01526 ""
"C:/Users/jonah/AppData/Local/Programs/MiKTeX/fonts/tfm/public/cm/cmr8.tfm" 1136765053 1292 21c1c5bfeaebccffdb478fd231a0997d ""
"C:/Users/jonah/AppData/Local/Programs/MiKTeX/fonts/tfm/public/cm/cmsy10.tfm" 1136765053 1124 6c73e740cf17375f03eec0ee63599741 ""
"C:/Users/jonah/AppData/Local/Programs/MiKTeX/fonts/tfm/public/cm/cmsy6.tfm" 1136765053 1116 933a60c408fc0a863a92debe84b2d294 ""
"C:/Users/jonah/AppData/Local/Programs/MiKTeX/fonts/tfm/public/cm/cmsy8.tfm" 1136765053 1120 8b7d695260f3cff42e636090a8002094 ""
"C:/Users/jonah/AppData/Local/Programs/MiKTeX/fonts/tfm/public/cm/cmti12.tfm" 1136765053 1484 ed72f8f5cf654cda15ecc8e32bfcbee5 ""
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"C:/Users/jonah/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmbx12.pfb" 1247596666 32080 340ef9bf63678554ee606688e7b5339d ""
"C:/Users/jonah/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmbxti10.pfb" 1247596667 36554 b67dc2cfa451409e100b3fcf5f506509 ""
"C:/Users/jonah/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmex10.pfb" 1247596667 30251 6afa5cb1d0204815a708a080681d4674 ""
"C:/Users/jonah/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmmi10.pfb" 1247596667 36299 5f9df58c2139e7edcf37c8fca4bd384d ""
"C:/Users/jonah/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmmi12.pfb" 1247596667 36741 0ee9e374ec3e30da87cdfb0ea3575226 ""
"C:/Users/jonah/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmmi6.pfb" 1247596666 37166 8ab3487cbe3ab49ebce74c29ea2418db ""
"C:/Users/jonah/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmmi8.pfb" 1247596666 35469 dcf3a5f2fc1862f5952e3ee5eb1d98c4 ""
"C:/Users/jonah/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmr10.pfb" 1247596667 35752 024fb6c41858982481f6968b5fc26508 ""
"C:/Users/jonah/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmr12.pfb" 1247596667 32722 d7379af29a190c3f453aba36302ff5a9 ""
"C:/Users/jonah/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmr17.pfb" 1247596666 32362 bc3f3eec7ab7d65fe700963d4017d32c ""
"C:/Users/jonah/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmr6.pfb" 1247596667 32734 69e00a6b65cedb993666e42eedb3d48f ""
"C:/Users/jonah/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmr8.pfb" 1247596667 32726 39f0f9e62e84beb801509898a605dbd5 ""
"C:/Users/jonah/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmsy10.pfb" 1247596667 32569 5e5ddc8df908dea60932f3c484a54c0d ""
"C:/Users/jonah/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmsy8.pfb" 1247596667 32626 5abc8bb2f28aa647d4c70f8ea38cc0d3 ""
"C:/Users/jonah/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/cm/cmti12.pfb" 1247596667 36118 fad905eba93cff5bce1e185fe980a177 ""
"C:/Users/jonah/AppData/Local/Programs/MiKTeX/fonts/type1/public/amsfonts/symbols/msbm10.pfb" 1247596673 34694 870c211f62cb72718a00e353f14f254d ""
"C:/Users/jonah/AppData/Local/Programs/MiKTeX/tex/context/base/mkii/supp-pdf.mkii" 1580390158 71627 94eb9990bed73c364d7f53f960cc8c5b ""
"C:/Users/jonah/AppData/Local/Programs/MiKTeX/tex/generic/babel/babel.sty" 1689436349 147701 c67babac98fcd1d4a6956667b84c7fca ""
"C:/Users/jonah/AppData/Local/Programs/MiKTeX/tex/generic/babel/locale/de/babel-de.ini" 1689436351 4096 fa1802f91144b2c083d61dddf097a6d2 ""
"C:/Users/jonah/AppData/Local/Programs/MiKTeX/tex/generic/babel/locale/de/babel-german.tex" 1689436351 699 565e921a0b7b8366adf163b0ecc00e44 ""
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@@ -33,7 +33,7 @@
%\bibstyle
%\citation
\title{Emergent intensity invariance in a physiologically inspired model of the grasshopper auditory system}
\title{Emergent intensity invariance vs. signal-to-noise ratio at three consecutive processing stages along the grasshopper song recognition pathway}
\author{Jona Hartling, Jan Benda}
\date{}
@@ -403,7 +403,7 @@ pathway, logarithmic compression is achieved by conversion to decibel scale
\db(t)\,=\,20\,\cdot\,\dec \frac{\env(t)}{\dbref}, \qquad \dbref\,=\,1
\label{eq:log}
\end{equation}
relative to the maximum intensity $\dbref$ of the signal envelope $\env(t)$.
relative to the common reference intensity $\dbref$.
Both the receptor neurons~(\bcite{romer1976informationsverarbeitung};
\bcite{gollisch2004input}; \bcite{fisch2012channel}) and, on a larger scale,
the subsequent local interneurons~(\bcite{hildebrandt2009origin};
@@ -555,7 +555,7 @@ can be read out by a simple linear classifier.
\end{figure}
\FloatBarrier
\section{Two mechanisms driving the emergence of intensity-invariant song representation}
\section{Mechanisms driving the emergence of\\intensity-invariant song representation}
% Still missing the SNR analysis. Should be able to write around it for now.
The robustness of song recognition is tied to the degree of intensity
@@ -573,6 +573,54 @@ specific operations involved, as outlined in the following sections.
\subsection{Full-wave rectification \& lowpass filtering}
The first nonlinear transformation along the model pathway is the full-wave
rectification of the tympanal signal $\filt(t)$ during the extraction of the
signal envelope (Eq.\,\ref{eq:env}). Rectification transforms the distribution
of $\filt(t)$ from an approximately zero-centered distribution with both
positive and negative values into a strictly non-negative distribution. Signal
envelope $\env(t)$ is then obtained by lowpass filtering the rectified
$\filt(t)$. The effects of this transformation pair on SNR and potential
intensity invariance were analyzed by rescaling and processing the input signal
$\raw(t)$ and comparing standard deviations between the resulting $\filt(t)$
and $\env(t)$, once for the noiseless case~(Fig.\,\ref{fig:rect-lp}a) and once
for the noisy case~(Fig.\,\ref{fig:rect-lp}b). In addition, the cutoff
frequency $\fc$ of the lowpass filter was varied to investigate the influence
of different filter bandwidths. In the noiseless case, the standard deviations
of $\filt(t)$ and $\env(t)$ are each reduced compared to the input $\raw(t)$ by
a multiplicative factor. These factors are constant across all $\sca$, which
results in a downward shift of the respective curve on a double-logarithmic
scale, away from the diagonal~(Fig.\,\ref{fig:rect-lp}c). For $\filt(t)$, the
reduction is a consequence of the bandpass filtering~(Eq.\,\ref{eq:bandpass})
of $\raw(t)$. For $\env(t)$, the standard deviation is further reduced compared
to $\filt(t)$. Rectification contributes much less to this reduction than
lowpass filtering. The degree of reduction by lowpass filtering depends on the
cutoff frequency $\fc$, with lower $\fc$ (narrow bandwidth) resulting in a
stronger reduction. In the noisy case, the standard deviations of $\filt(t)$
and $\env(t)$ can be related to the respective pure-noise reference standard
deviation~(Fig.\,\ref{fig:rect-lp}d). This causes each curve to start with a
constant regime of SNR values near 1 for smaller $\sca$, which reflects the
dominance of the noise component $\noc(t)$ over the song component $\soc(t)$ in
the input $\raw(t)$. For larger $\sca$, all curves transition into a regime of
linearly increasing SNR on a double-logarithmic scale. For $\filt(t)$, the
linear part of the curve deviates only slightly from the diagonal. For
$\env(t)$, however, the transition occurs at lower $\sca$ compared to
$\filt(t)$, and the linear part of the curve is shifted leftward away from the
diagonal, which means that higher SNR values are achieved for the same $\sca$.
This effect is more pronounced for lower $\fc$ of the lowpass filter and is
presumably caused by the attenuation of high-frequency components in the
signal, which are more prominent in the noise component $\noc(t)$ than in the
song component $\soc(t)$. The effect also appears relatively consistent across
different species, although small variations based on different song structures
and distributions exist~(Fig.\,\ref{fig:rect-lp}e). In summary, the standard
deviation of $\env(t)$ has never been observed to transition into a saturation
regime for larger $\sca$ but rather continues to increase proportionally to
$\sca$ for all tested $\fc$, in both the noiseless and the noisy case and
across different species. Consequently, the combination of rectification and
lowpass filtering does not contribute to intensity invariance. However, this
transformation pair does improve the SNR of $\env(t)$ relative to $\filt(t)$
and thus provides subsequent processing stages with a more robust input
representation and higher input SNR.
\begin{figure}[!ht]
\centering
\includegraphics[width=\textwidth]{figures/fig_invariance_rect_lp.pdf}
@@ -605,73 +653,113 @@ specific operations involved, as outlined in the following sections.
\subsection{Logarithmic compression \& spike-frequency adaptation}
The first notable emergence of intensity invariance along the model pathway
occurs during the transformation of the signal envelope $\env(t)$ into the
logarithmically scaled envelope $\db(t)$ and then into the intensity-adapted
envelope $\adapt(t)$. In order to disentangle the interplay of logarithmic
compression and adaptation, $\env(t)$ can be rewritten as a synthetic mixture
The second nonlinear transformation along the model pathway is the logarithmic
compression of the signal envelope $\env(t)$ into $\db(t)$, Eq.\,\ref{eq:log},
which is then followed by the highpass filtering of $\db(t)$,
Eq.\,\ref{eq:highpass}, to obtain the intensity-adapted envelope $\adapt(t)$.
The interplay of this transformation pair was analyzed by rescaling and
processing the input signal $\filt(t)$ and comparing standard deviations
between the resulting $\env(t)$, $\db(t)$, and $\adapt(t)$. It is necessary to
use $\filt(t)$ as input for this analysis instead of $\env(t)$, because
$\env(t)$ results from a nonlinear transformation and hence cannot be
synthesized as an additive mixture of song component $\soc(t)$ and noise
component $\noc(t)$. % <-- Sentence may be methods section material.
However, it is much easier to conceive a mathematical description of the
effects of logarithmic compression and adaptation if $\env(t)$ itself is
assumed to be composed of $\soc(t)$ and $\noc(t)$. In the noiseless
case~(Fig.\,\ref{fig:log-hp}a), $\env(t)$ takes the form of
\begin{equation}
\env(t)\,=\,\sca\,\cdot\,\soc(t), \qquad \env(t)\,>\,0\enspace\forall\enspace t\,\in\,\mathbb{R}
\label{eq:toy_env_pure}
\end{equation}
The standard deviation of $\env(t)$ increases linearly with $\sca$ on a
double-logarithmic scale and is slightly reduced~(Fig.\,\ref{fig:log-hp}c)
compared to the input $\filt(t)$, which is consistent with the results of the
previous analysis~(Fig.\,\ref{fig:rect-lp}c). By conversion of $\env(t)$ to
decibel scale, $\sca$ turns from a multiplicative scale in linear space into an
additive term, or offset, in logarithmic space:
\begin{equation}
\db(t)\,=\,20\,\cdot\,\dec \left[\,\sca\,\cdot\,s(t)\,\right]\,=\,20\,\cdot\,\left[\dec \sca\,+\,\dec s(t)\right], \qquad \sca\,>\,0
\label{eq:toy_log_pure}
\end{equation}
The highpass filtering of $\db(t)$ can be approximated as a subtraction of the
local signal offset within a suitable time interval $0 \ll \thp <
\frac{1}{\fc}$:
\begin{equation}
\begin{split}
\adapt(t)\,\approx\,\db(t)\,-\,20\,\cdot\,\dec \sca\,=\,20\,\cdot\,\dec s(t)
\end{split}
\label{eq:toy_highpass_pure}
\end{equation}
This eliminates $\sca$ from $\adapt(t)$ and thus renders it perfectly
intensity-invariant, with a constant standard deviation of around 10\,dB across
all $\sca>0$~(Fig.\,\ref{fig:log-hp}c). In contrast, in the noisy
case~(Fig.\,\ref{fig:log-hp}b), $\env(t)$ takes the form of
\begin{equation}
\env(t)\,=\,\sca\,\cdot\,\soc(t)\,+\,\noc(t), \qquad \env(t)\,>\,0\enspace\forall\enspace t\,\in\,\mathbb{R}
\label{eq:toy_env}
\label{eq:toy_env_noise}
\end{equation}
of a song component $\soc(t)$ with variable multiplicative scale $\sca\geq0$
and a fixed-scale noise component $\noc(t)$. Both $\soc(t)$ and $\noc(t)$ are
assumed to have unit variance. By conversion of $\env(t)$ to decibel
scale~(Eq.\,\ref{eq:log}), $\sca$ turns from a multiplicative scale in linear
space into an additive term, or offset, in logarithmic space
Similar to the previous analysis~(Fig.\,\ref{fig:rect-lp}d), the ratio of the
standard deviation of $\env(t)$ to its pure-noise reference standard deviation
on a double-logarithmic scale follows a constant regime for small $\sca$ and a
linearly increasing regime for larger $\sca$~(Fig.\,\ref{fig:log-hp}d). Decibel
conversion of $\env(t)$
% \begin{equation}
% \begin{split}
% \db(t)\,&=\,\dec \frac{\alpha\,\cdot\,s(t)\,+\,\eta(t)}{\dbref}\\
% &=\,\dec \frac{\alpha}{\dbref}\,+\,\dec \left[s(t)\,+\,\frac{\eta(t)}{\alpha}\right], \qquad \sca\,>\,0
% \db(t)\,&=\,20\,\cdot\,\dec \left[\,\sca\,\cdot\,s(t)\,+\,\eta(t)\,\right]\\
% &=\,20\,\cdot\,\left(\dec \sca\,+\,\dec \left[s(t)\,+\,\frac{\eta(t)}{\sca}\right]\right), \qquad \sca\,>\,0
% \end{split}
% \label{eq:toy_log}
% \label{eq:toy_log_noise}
% \end{equation}
\begin{equation}
\begin{split}
\db(t)\,&=\,20\,\cdot\,\dec \left[\,\sca\,\cdot\,s(t)\,+\,\eta(t)\,\right]\\
&=\,20\,\cdot\,\left(\dec \sca\,+\,\dec \left[s(t)\,+\,\frac{\eta(t)}{\sca}\right]\right), \qquad \sca\,>\,0
\end{split}
\label{eq:toy_log}
\db(t)\,=\,20\,\cdot\,\left(\dec \sca\,+\,\dec \left[s(t)\,+\,\frac{\eta(t)}{\sca}\right]\right), \qquad \sca\,>\,0
\label{eq:toy_log_noise}
\end{equation}
allows for the separation of $\sca$ from $\soc(t)$ but introduces a scaling of
$\noc(t)$ by the inverse of $\sca$, which remains present even after the offset
subtraction:
\begin{equation}
\begin{split}
\adapt(t)\,\approx\,20\,\cdot\,\dec\left[s(t)\,+\,\frac{\eta(t)}{\sca}\right]
\end{split}
\label{eq:toy_highpass_noise}
\end{equation}
which allows for its separation from $\soc(t)$ but introduces a scaling of
$\noc(t)$ by the inverse of $\sca$. The subsequent
highpass filtering~(Eq.\,\ref{eq:highpass}) of $\db(t)$ can then be
approximated as a subtraction of the local offset within a suitable time
interval $0 \ll \thp < \frac{1}{\fc}$:
% \begin{equation}
% \begin{split}
% \adapt(t)\,\approx\,\db(t)\,-\,\dec \frac{\sca}{\dbref}\,=\,\dec\left[s(t)\,+\,\frac{\eta(t)}{\sca}\right], \qquad \sca\,>\,0
% \adapt(t)\,\approx\,\db(t)\,-\,20\,\cdot\,\dec \sca\,=\,20\,\cdot\,\dec\left[s(t)\,+\,\frac{\eta(t)}{\sca}\right]
% \end{split}
% \label{eq:toy_highpass}
% \label{eq:toy_highpass_noise}
% \end{equation}
\begin{equation}
\begin{split}
\adapt(t)\,\approx\,\db(t)\,-\,20\,\cdot\,\dec \sca\,=\,20\,\cdot\,\dec\left[s(t)\,+\,\frac{\eta(t)}{\sca}\right], \qquad \sca\,>\,0
\end{split}
\label{eq:toy_highpass}
\end{equation}
This means that $\sca$ cannot be entirely eliminated from $\adapt(t)$, only
redistributed between $\soc(t)$ and $\noc(t)$. In consequence, if $\sca$ is
sufficiently large ($\sca\gg1$), $\noc(t)$ is attenuated to the point of being
negligible, so that $\adapt(t)$ represents $\soc(t)$ in a scale-free manner. If
$\soc(t)$ and $\noc(t)$ are at similar scales ($\sca\approx1$), $\adapt(t)$
largely resembles $\db(t)$. However, if $\sca$ is sufficiently small
($\sca\ll1$), $\noc(t)$ masks $\soc(t)$ even after the intensity adaptation.
Therefore, the effective intensity invariance of $\adapt(t)$ relative to
$\env(t)$ is limited by the initial scaling of $\soc(t)$ relative to $\noc(t)$;
that is, the signal-to-noise ratio (SNR) of $\env(t)$ with ($\sca>0$) and
without ($\sca=0$) song component $\soc(t)$
\begin{equation}
\text{SNR}(\sca)\,=\,\frac{\xvar}{\nvar}\,=\,\frac{\alpha^{2}\,\cdot\,\svar\,+\,\nvar}{\nvar}\,=\,\alpha^{2}\,+\,1, \qquad \svar\,=\,\nvar\,=\,1
\label{eq:toy_snr}
\end{equation}
which depends quadratically on $\sca$ if $\soc(t)\perp\noc(t)$. Overall, the
combination of logarithmic compression and adaptation allows for the
equalization of different sufficiently large song scales, which is essential
for intensity-invariant song representation. However, this mechanism is unable
to recover songs that have already sunken below the noise floor, which
emphasizes the importance of a sufficiently high SNR at the intial reception of
the signal for reliable song recognition.
This means that, in the noisy case, $\sca$ cannot be entirely eliminated from
$\adapt(t)$, only redistributed between $\soc(t)$ and $\noc(t)$. If $\sca$ is
sufficiently large ($\sca\gg1$, saturation regime), $\noc(t)$ is attenuated to
the point of being negligible, so that $\adapt(t)$ is a scale-free
representation of $\soc(t)$. If $\sca$ and $\noc(t)$ are at similar scales
($\sca\approx1$, transient regime), $\adapt(t)$ largely resembles $\db(t)$.
Finally, if $\sca$ is sufficiently small ($0<\sca\ll1$, noise regime),
$\noc(t)$ masks $\soc(t)$ even after the intensity adaptation. Accordingly, the
effective intensity invariance of $\adapt(t)$ through logarithmic compression
and adaptation is limited by the SNR of $\env(t)$: Songs that have already
sunken into the noise floor at the level of $\env(t)$ cannot be recovered by
subsequent processing steps, which emphasizes the importance of the SNR
improvement by rectification and lowpass filtering during the previous
processing step~(Fig.\,\ref{fig:rect-lp}d). The general pattern of noise
regime, transient regime, and saturation regime remains consistent across
different species~(Fig.\,\ref{fig:log-hp}e). However, the specific value of
$\sca$ at which the saturation regime is reached (see appendix
Fig.\,\ref{fig:app_log-hp_saturation}) and the maximum SNR value of $\adapt(t)$
within the saturation regime vary considerably between and within species. For
example, \textit{C. biguttulus} and \textit{C. mollis} display a noticably
lower maximum SNR of $\adapt(t)$ compared to other species. These differences
are not to be underestimated, since the SNR of $\adapt(t)$ within the
saturation regime determines the maximum input SNR for subsequent processing
steps. In other words, the fact that $\adapt(t)$ eventually reaches a
saturation regime is, of course, desirable in the context of intensity
invariance, but it also means to pass up on the higher SNR values that are
achieved by $\env(t)$ for the same $\sca$ (up to several orders of magnitude,
Fig.\,\ref{fig:log-hp}d). This trade-off between intensity invariance and SNR
--- and the consequences it has further downstream along the pathway --- are
adressed in the following sections.
\begin{figure}[!ht]
\centering
@@ -744,30 +832,6 @@ the signal for reliable song recognition.
\end{figure}
\FloatBarrier
% \caption{\textbf{Rectification and lowpass filtering improves SNR
% but does not contribute to intensity invariance.}
% Input $\raw(t)$ consists of song component $\soc(t)$ scaled by
% $\sca$ with optional noise component $\noc(t)$ and is
% successively transformed into tympanal signal $\filt(t)$ and
% envelope $\env(t)$. Different line styles indicate different
% cutoff frequencies $\fc$ of the lowpass filter extracting
% $\env(t)$.
% \textbf{Top}:~Example representations of $\filt(t)$ and
% $\env(t)$ for different $\sca$.
% \textbf{a}:~Noiseless case.
% \textbf{b}:~Noisy case.
% \textbf{Bottom}:~Intensity metrics over a range of $\sca$.
% \textbf{c}:~Noiseless case: Standard deviations of $\filt(t)$
% and $\env(t)$.
% \textbf{d}:~Noisy case: Ratios of standard deviations of
% $\filt(t)$ and $\env(t)$ to the respective reference standard
% deviation for input $\raw(t)=\noc(t)$.
% \textbf{e}:~Ratios of standard deviations of $\env(t)$ as in
% \textbf{b} for different species (averaged over songs and
% recordings, see appendix Fig.\,\ref{fig:app_rect-lp}).
% }
\begin{figure}[!ht]
\centering
\includegraphics[width=\textwidth]{figures/fig_invariance_thresh_lp_species.pdf}
@@ -796,17 +860,67 @@ the signal for reliable song recognition.
curve span of the norm across all three $\mu_{f_i}$ per
species.
\textbf{d}:~Noiseless case.
\textbf{e}:~Noisy case. Shaded areas
\textbf{e}:~Noisy case. Shaded areas indicate the average
minimum $\mu_{f_i}$ across all species-specific trajectories.
}
\label{fig:thresh-lp_species}
\end{figure}
\FloatBarrier
% \caption{\textbf{Rectification and lowpass filtering improves SNR
% but does not contribute to intensity invariance.}
% Input $\raw(t)$ consists of song component $\soc(t)$ scaled by
% $\sca$ with optional noise component $\noc(t)$ and is
% successively transformed into tympanal signal $\filt(t)$ and
% envelope $\env(t)$. Different line styles indicate different
% cutoff frequencies $\fc$ of the lowpass filter extracting
% $\env(t)$.
% \textbf{Top}:~Example representations of $\filt(t)$ and
% $\env(t)$ for different $\sca$.
% \textbf{a}:~Noiseless case.
% \textbf{b}:~Noisy case.
% \textbf{Bottom}:~Intensity metrics over a range of $\sca$.
% \textbf{c}:~Noiseless case: Standard deviations of $\filt(t)$
% and $\env(t)$.
% \textbf{d}:~Noisy case: Ratios of standard deviations of
% $\filt(t)$ and $\env(t)$ to the respective reference standard
% deviation for input $\raw(t)=\noc(t)$.
% \textbf{e}:~Ratios of standard deviations of $\env(t)$ as in
% \textbf{b} for different species (averaged over songs and
% recordings, see appendix Fig.\,\ref{fig:app_rect-lp}).
% }
\begin{figure}[!ht]
\centering
\includegraphics[width=\textwidth]{figures/fig_invariance_full_Omocestus_rufipes.pdf}
\caption{\textbf{Step-wise emergence of intensity invariant song
representation along the model pathway.}
\caption{\textbf{Step-wise emergence of intensity-invariant song
representation along the full model pathway.}
Input $\raw(t)$ consists of song component $\soc(t)$
scaled by $\sca$ with added noise component $\noc(t)$ and
is processed up to the feature set $f_i(t)$. Different
color shades indicate different types of Gabor kernels
with specific lobe number $\kn$ and either $+$ or $-$
sign, sorted (dark to light) first by increasing $\kn$ and
then by sign~($1\,\leq\,\kn\,\leq\,4$; first $+$, then $-$
for each $\kn$; five kernel widths $\kw$ of 1, 2, 4, 8,
and $16\,$ms per type; 8 types, 40 kernels in total).
\textbf{a}:~Example representations of $\filt(t)$,
$\env(t)$, $\db(t)$, $\adapt(t)$, $c_i(t)$, and $f_i(t)$
for different $\sca$.
\textbf{b}:~Intensity metrics over $\sca$. For $c_i(t)$
and $f_i(t)$, the median over kernels is shown. Dots
indicate $95\,\%$ curve span for $\db(t)$, $\adapt(t)$,
$c_i(t)$, and $f_i(t)$.
\textbf{c}:~Average value $\mu_{f_i}$ of each feature
$f_i(t)$ over $\sca$.
\textbf{d}:~Ratios of intensity metrics to the respective
reference value for input $\raw(t)=\noc(t)$. For $c_i(t)$
and $f_i(t)$, the median over kernel-specific ratios is
shown.
\textbf{e}:~Ratios of standard deviation $\sigma_{c_i}$ of
each $c_i(t)$.
\textbf{f}:~Ratios of $\mu_{f_i}$.
\textbf{g}:~Distributions of kernel-specific $\sca$ that
correspond to $95\,\%$ curve span for $c_i(t)$ and
$f_i(t)$. Dots indicate the values from \textbf{b}.
}
\label{fig:pipeline_full}
\end{figure}
@@ -816,7 +930,34 @@ the signal for reliable song recognition.
\centering
\includegraphics[width=\textwidth]{figures/fig_invariance_short_Omocestus_rufipes.pdf}
\caption{\textbf{Step-wise emergence of intensity invariant song
representation along the model pathway.}
representation along the model pathway without logarithmic
compression.}
Input $\raw(t)$ consists of song component $\soc(t)$
scaled by $\sca$ with added noise component $\noc(t)$ and
is processed up to the feature set $f_i(t)$, skipping
$\db(t)$. Different color shades indicate different types
of Gabor kernels with specific lobe number $\kn$ and
either $+$ or $-$ sign, sorted (dark to light) first by
increasing $\kn$ and then by
sign~($1\,\leq\,\kn\,\leq\,4$; first $+$, then $-$ for
each $\kn$; five kernel widths $\kw$ of 1, 2, 4, 8, and
$16\,$ms per type; 8 types, 40 kernels in total).
\textbf{a}:~Example representations of $\filt(t)$,
$\env(t)$, $\adapt(t)$, $c_i(t)$, and $f_i(t)$ for
different $\sca$.
\textbf{b}:~Intensity metrics over $\sca$. For $c_i(t)$
and $f_i(t)$, the median over kernels is shown. Dots
indicate $95\,\%$ curve span for $f_i(t)$.
\textbf{c}:~Average value $\mu_{f_i}$ of each feature
$f_i(t)$ over $\sca$.
\textbf{d}:~Ratios of intensity metrics to the respective
reference value for input $\raw(t)=\noc(t)$. For $c_i(t)$
and $f_i(t)$, the median over kernel-specific ratios is
shown.
\textbf{e}:~Ratios of $\mu_{f_i}$.
\textbf{f}:~Distribution of kernel-specific $\sca$ that
correspond to $95\,\%$ curve span for $f_i(t)$. Dots
indicate the value from \textbf{b}.
}
\label{fig:pipeline_short}
\end{figure}
@@ -825,7 +966,28 @@ the signal for reliable song recognition.
\begin{figure}[!ht]
\centering
\includegraphics[width=\textwidth]{figures/fig_features_cross_species.pdf}
\caption{\textbf{Inter- and intraspecific feature variability.}
\caption{\textbf{Interspecific and intraspecific feature variability.}
Average value $\mu_{f_i}$ of each feature $f_i(t)$ against
its counterpart from a 2nd feature set based on a
different input $\raw(t)$. Each dot within a subplot
represents a single feature $f_i(t)$. Different color
shades indicate different types of Gabor kernels with
specific lobe number $\kn$ and either $+$ or $-$ sign,
sorted (dark to light) first by increasing $\kn$ and then
by sign~($1\,\leq\,\kn\,\leq\,4$; first $+$, then $-$ for
each $\kn$; five kernel widths $\kw$ of 1, 2, 4, 8, and
$16\,$ms per type; 8 types, 40 kernels in total). Data is
based on the analysis underlying
Fig\,\ref{fig:pipeline_full}.
\textbf{Lower triangular}:~Interspecific comparisons
between single songs of different species.
\textbf{Upper triangular}:~Intraspecific comparisons
between different songs of a single species (\textit{O.
rufipes}).
\textbf{Lower left}:~Distribution of correlation
coefficients $\rho$ for each interspecific and
intraspecific comparison. Dots indicate single $\rho$
values.
}
\label{fig:feat_cross_species}
\end{figure}

View File

@@ -164,7 +164,7 @@ ylabels = dict(
conv='$c_i$\n$[\\text{dB}]$',
feat='$f_i$',
raw=['$m$', '$\\mu_{f_i}$'],
base=['$m\\,/\\,m_{\\eta}$', '$\\sigma_{c_i}$', '$\\mu_{f_i}$', '$\\text{PDF}_{\\alpha}$']
base=['$m\\,/\\,m_{\\eta}$', '$\\sigma_{c_i}\\,/\\,\\sigma_{\\eta_i}$', '$\\mu_{f_i}\\,/\\,\\mu_{\\eta_i}$', '$\\text{PDF}_{\\alpha}$']
)
xlab_big_kwargs = dict(
y=0,

View File

@@ -57,7 +57,7 @@ snip_cutoff = np.array([np.nan, 2500, 250, 25])[2]
# GRAPH SETTINGS:
fig_kwargs = dict(
figsize=(32/2.54, 32/2.54),
figsize=(32/2.54, 20/2.54),
)
super_grid_kwargs = dict(
nrows=3,
@@ -68,7 +68,7 @@ super_grid_kwargs = dict(
right=1,
bottom=0,
top=1,
height_ratios=[1, 1, 1]
height_ratios=[1, 1, 2]
)
subfig_specs = dict(
pure=(0, slice(None)),

View File

@@ -162,7 +162,7 @@ ylabels = dict(
conv='$c_i$\n$[\\text{dB}]$',
feat='$f_i$',
raw=['$m$', '$\\mu_{f_i}$'],
base=['$m\\,/\\,m_{\\eta}$', '$\\mu_{f_i}$', '$\\text{PDF}_{\\alpha}$']
base=['$m\\,/\\,m_{\\eta}$', '$\\mu_{f_i}\\,/\\,\\mu_{\\eta_i}$', '$\\text{PDF}_{\\alpha}$']
)
xlab_big_kwargs = dict(
y=0,

View File

@@ -1,3 +1,5 @@
import gc
import copy
import numpy as np
from thunderhopper.modeltools import load_data, save_data
from thunderhopper.filetools import search_files, crop_paths
@@ -16,7 +18,7 @@ target_species = [
# 'Gomphocerippus_rufus',
# 'Omocestus_rufipes',
# 'Pseudochorthippus_parallelus',
][1]
][0]
example_file = {
'Chorthippus_biguttulus': 'Chorthippus_biguttulus_GBC_94-17s73.1ms-19s977ms',
'Chorthippus_mollis': 'Chorthippus_mollis_DJN_41_T28C-46s4.58ms-1m15s697ms',
@@ -110,8 +112,8 @@ for data_path, name in zip(data_paths, crop_paths(data_paths)):
scaled = song * scale + noise
# Process mixture (excluding features):
signals, rates = process_signal(config, returns=pre_stages,
signal=scaled, rate=rate)
signals, _ = process_signal(config, returns=pre_stages,
signal=scaled, rate=rate)
# Store non-feature results:
for stage in pre_stages:
# Log intensity measures:
@@ -120,12 +122,16 @@ for data_path, name in zip(data_paths, crop_paths(data_paths)):
# Log optional snippet data:
if save_detailed and scale in example_scales:
scale_ind = np.nonzero(example_scales == scale)[0][0]
snippets[f'snip_{stage}'][:, ..., scale_ind] = signals[stage]
snippets[f'snip_{stage}'][:, ..., scale_ind] = copy.deepcopy(signals[stage])
conv = copy.deepcopy(signals['conv'])
del scaled, signals
gc.collect()
# Execute piecewise again:
for j, thresholds in enumerate(thresh_abs):
# Finalize processing:
feat = sosfilter((signals['conv'] > thresholds).astype(float),
feat = sosfilter((conv > thresholds).astype(float),
rate, config['feat_fcut'], 'lp',
padtype='fixed', padlen=config['padlen'])
@@ -134,19 +140,27 @@ for data_path, name in zip(data_paths, crop_paths(data_paths)):
# Log optional snippet data:
if save_detailed and scale in example_scales:
snippets['snip_feat'][:, :, scale_ind, j] = feat
snippets['snip_feat'][:, :, scale_ind, j] = copy.deepcopy(feat)
del feat
gc.collect()
# Save analysis results:
if save_path is not None:
data = dict(
archive = dict(
scales=scales,
example_scales=example_scales,
thresh_rel=thresh_rel,
thresh_abs=thresh_abs,
)
data.update(measures)
archive.update(measures)
if save_detailed:
data.update(snippets)
save_data(save_path + name, data, config, overwrite=True)
archive.update(snippets)
save_data(save_path + name, archive, config, overwrite=True)
del archive
del measures, data, config, conv
if save_detailed:
del snippets
gc.collect()
print('Done.')
embed()