From 915b8ea4ffef090869f60234abae9efb6cadbbb9 Mon Sep 17 00:00:00 2001 From: j-hartling <129943012+j-hartling@users.noreply.github.com> Date: Sat, 16 May 2026 14:43:13 +0200 Subject: [PATCH] Finished methods. 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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}. Dots indicate $95\,\%$ curve span (noisy case). \textbf {e}:~$\mu _f$ over a range of $\alpha $, once for the noisy case (solid lines) and once for the noiseless case (dotted lines). \textbf {f}:~Noisy case: $\mu _f$ over the standard deviation of input $x_{\text {adapt}}$ corresponding to the values of $\alpha $ shown in \textbf {e}. Shaded area indicates standard deviations that would be capped in the output $x_{\text {adapt}}(t)$ of the previous transformation pair (see Fig.\,\ref {fig:log-hp}cd). }}{22}{}\protected@file@percent } -\newlabel{fig:thresh-lp_single}{{6}{22}{}{}{}} -\@writefile{toc}{\contentsline {subsection}{\numberline {3.2}Intensity invariance of species-specific feature representations}{23}{}\protected@file@percent } -\@writefile{lof}{\contentsline {figure}{\numberline {7}{\ignorespaces \textbf {Feature representation of different species-specific songs saturates at different points in feature space.} Same input and processing as in Fig.\,\ref {fig:thresh-lp_single} but with three different kernels $k_i$, each with a single kernel-specific threshold value $\Theta _i=0.5\cdot \sigma _{\eta _i}$. \textbf {a}:~Examples of species-specific grasshopper songs. \textbf {Middle}:~Average value $\mu _{f_i}$ of each feature $f_i(t)$ over $\alpha $ per species (averaged over songs and recordings, see appendix Figs.\,\ref {fig:app_thresh-lp_pure} and \ref {fig:app_thresh-lp_noise}). Different color shades indicate different kernels $k_i$. Dots indicate $95\,\%$ curve span per $k_i$. \textbf {b}:~Noiseless case. \textbf {c}:~Noisy case. \textbf {Bottom}:~2D feature spaces spanned by each pair of $f_i(t)$. Each trajectory corresponds to a species-specific combination of $\mu _{f_i}$ that develops with $\alpha $ (colorbars). Horizontal dashes in the colorbar indicate $5\,\%$ (dark grey) and $95\,\%$ (light grey) curve span of the norm across all three $\mu _{f_i}$ per species. \textbf {d}:~Noiseless case. \textbf {e}:~Noisy case. 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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 $x_{\text {filt}}(t)$, $x_{\text {env}}(t)$, $x_{\text {log}}(t)$, $x_{\text {adapt}}(t)$, $c_i(t)$, and $f_i(t)$ for different $\alpha $. \textbf {b}:~Intensity measures over $\alpha $. For $c_i(t)$ and $f_i(t)$, the median over kernels is shown. Dots indicate $95\,\%$ curve span for $x_{\text {log}}(t)$, $x_{\text {adapt}}(t)$, $c_i(t)$, and $f_i(t)$. \textbf {c}:~Average value $\mu _{f_i}$ of each feature $f_i(t)$ over $\alpha $. \textbf {d}:~Ratios of intensity measures to the respective reference value for input $x_{\text {raw}}(t)=\eta (t)$. 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- helversen1977stridulatory - stumpner1994song - helversen1997recognition - helversen1972gesang - helversen1993absolute - balakrishnan2001song - helversen2004acoustic - helversen1997recognition - skovmand1983song - creutzig2009timescale - creutzig2010timescale - helversen1972gesang - michelsen1978sound - lang2000acoustic - benda2008spike - barbour2011intensity - ozeri2018fast - benda2021neural - romer1976informationsverarbeitung - gollisch2004input - hildebrandt2009origin - clemens2010intensity - fisch2012channel - clemens2010intensity - hildebrandt2009origin - clemens2013computational - hennig2014time - clemens2013feature - ronacher2015computational - clemens2013feature - clemens2013computational - rokem2006spike - clemens2011efficient - clemens2012nonlinear diff --git a/main.blg b/main.blg deleted file mode 100644 index 94af8be..0000000 --- a/main.blg +++ /dev/null @@ -1,71 +0,0 @@ -[0] Config.pm:307> INFO - This is Biber 2.19 -[0] Config.pm:310> INFO - Logfile is 'main.blg' -[36] biber:340> INFO - === Fr Mai 15, 2026, 16:11:10 -[45] Biber.pm:419> INFO - Reading 'main.bcf' -[73] Biber.pm:979> INFO - Found 55 citekeys in bib section 0 -[79] Biber.pm:4419> INFO - Processing section 0 -[83] Biber.pm:4610> INFO - Looking for bibtex file 'cite.bib' for section 0 -[85] bibtex.pm:1713> INFO - LaTeX decoding ... -[116] bibtex.pm:1519> INFO - Found BibTeX data source 'cite.bib' -[289] UCollate.pm:68> INFO - Overriding locale 'en-US' defaults 'variable = shifted' with 'variable = non-ignorable' -[289] UCollate.pm:68> INFO - Overriding locale 'en-US' defaults 'normalization = NFD' with 'normalization = prenormalized' -[289] Biber.pm:4239> INFO - Sorting list 'nyt/global//global/global' of type 'entry' with template 'nyt' and locale 'en-US' -[289] Biber.pm:4245> INFO - No sort tailoring available for locale 'en-US' -[312] bbl.pm:660> INFO - Writing 'main.bbl' with encoding 'UTF-8' -[322] bbl.pm:763> INFO - Output to main.bbl -[322] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_2sRz/347c261ec4135a5723bef5c751f5078f_24870.utf8, line 10, warning: 6 characters of junk seen at toplevel -[322] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_2sRz/347c261ec4135a5723bef5c751f5078f_24870.utf8, line 21, warning: 6 characters of junk seen at toplevel -[322] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_2sRz/347c261ec4135a5723bef5c751f5078f_24870.utf8, line 38, warning: 6 characters of junk seen at toplevel -[322] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_2sRz/347c261ec4135a5723bef5c751f5078f_24870.utf8, line 49, warning: 6 characters of junk seen at toplevel -[322] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_2sRz/347c261ec4135a5723bef5c751f5078f_24870.utf8, line 58, warning: 6 characters of junk seen at toplevel -[322] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_2sRz/347c261ec4135a5723bef5c751f5078f_24870.utf8, line 73, warning: 6 characters of junk seen at toplevel -[322] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_2sRz/347c261ec4135a5723bef5c751f5078f_24870.utf8, line 82, warning: 6 characters of junk seen at toplevel -[322] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_2sRz/347c261ec4135a5723bef5c751f5078f_24870.utf8, line 91, warning: 6 characters of junk seen at toplevel -[322] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_2sRz/347c261ec4135a5723bef5c751f5078f_24870.utf8, line 100, warning: 6 characters of junk seen at toplevel -[322] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_2sRz/347c261ec4135a5723bef5c751f5078f_24870.utf8, line 109, warning: 6 characters of junk seen at toplevel -[322] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_2sRz/347c261ec4135a5723bef5c751f5078f_24870.utf8, line 118, warning: 6 characters of junk seen at toplevel -[322] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_2sRz/347c261ec4135a5723bef5c751f5078f_24870.utf8, line 127, warning: 6 characters of junk seen at toplevel -[322] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_2sRz/347c261ec4135a5723bef5c751f5078f_24870.utf8, line 136, warning: 6 characters of junk seen at toplevel -[322] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_2sRz/347c261ec4135a5723bef5c751f5078f_24870.utf8, line 157, warning: 6 characters of junk seen at toplevel -[322] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_2sRz/347c261ec4135a5723bef5c751f5078f_24870.utf8, line 178, warning: 6 characters of junk seen at toplevel -[322] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_2sRz/347c261ec4135a5723bef5c751f5078f_24870.utf8, line 187, warning: 6 characters of junk seen at toplevel -[322] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_2sRz/347c261ec4135a5723bef5c751f5078f_24870.utf8, line 196, warning: 6 characters of junk seen at toplevel -[322] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_2sRz/347c261ec4135a5723bef5c751f5078f_24870.utf8, line 207, warning: 6 characters of junk seen at toplevel -[322] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_2sRz/347c261ec4135a5723bef5c751f5078f_24870.utf8, line 218, warning: 6 characters of junk seen at toplevel -[322] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_2sRz/347c261ec4135a5723bef5c751f5078f_24870.utf8, line 229, warning: 6 characters of junk seen at toplevel -[322] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_2sRz/347c261ec4135a5723bef5c751f5078f_24870.utf8, line 240, warning: 6 characters of junk seen at toplevel -[322] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_2sRz/347c261ec4135a5723bef5c751f5078f_24870.utf8, line 249, warning: 6 characters of junk seen at toplevel -[322] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_2sRz/347c261ec4135a5723bef5c751f5078f_24870.utf8, line 258, warning: 6 characters of junk seen at toplevel -[322] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_2sRz/347c261ec4135a5723bef5c751f5078f_24870.utf8, line 269, warning: 6 characters of junk seen at toplevel -[322] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_2sRz/347c261ec4135a5723bef5c751f5078f_24870.utf8, line 278, warning: 6 characters of junk seen at toplevel -[322] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_2sRz/347c261ec4135a5723bef5c751f5078f_24870.utf8, line 289, warning: 6 characters of junk seen at toplevel -[322] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_2sRz/347c261ec4135a5723bef5c751f5078f_24870.utf8, line 300, warning: 6 characters of junk seen at toplevel -[322] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_2sRz/347c261ec4135a5723bef5c751f5078f_24870.utf8, line 309, warning: 6 characters of junk seen at toplevel -[322] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_2sRz/347c261ec4135a5723bef5c751f5078f_24870.utf8, line 328, warning: 6 characters of junk seen at toplevel -[322] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_2sRz/347c261ec4135a5723bef5c751f5078f_24870.utf8, line 337, warning: 6 characters of junk seen at toplevel -[322] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_2sRz/347c261ec4135a5723bef5c751f5078f_24870.utf8, line 400, warning: 6 characters of junk seen at toplevel -[322] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_2sRz/347c261ec4135a5723bef5c751f5078f_24870.utf8, line 419, warning: 6 characters of junk seen at toplevel -[322] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_2sRz/347c261ec4135a5723bef5c751f5078f_24870.utf8, line 428, warning: 6 characters of junk seen at toplevel -[322] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_2sRz/347c261ec4135a5723bef5c751f5078f_24870.utf8, line 437, warning: 6 characters of junk seen at toplevel -[322] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_2sRz/347c261ec4135a5723bef5c751f5078f_24870.utf8, line 456, warning: 6 characters of junk seen at toplevel -[322] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_2sRz/347c261ec4135a5723bef5c751f5078f_24870.utf8, line 491, warning: 6 characters of junk seen at toplevel -[322] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_2sRz/347c261ec4135a5723bef5c751f5078f_24870.utf8, line 526, warning: 6 characters of junk seen at toplevel -[322] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_2sRz/347c261ec4135a5723bef5c751f5078f_24870.utf8, line 535, warning: 6 characters of junk seen at toplevel -[322] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_2sRz/347c261ec4135a5723bef5c751f5078f_24870.utf8, line 556, warning: 6 characters of junk seen at toplevel -[322] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_2sRz/347c261ec4135a5723bef5c751f5078f_24870.utf8, line 565, warning: 6 characters of junk seen at toplevel -[322] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_2sRz/347c261ec4135a5723bef5c751f5078f_24870.utf8, line 576, warning: 6 characters of junk seen at toplevel -[322] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_2sRz/347c261ec4135a5723bef5c751f5078f_24870.utf8, line 587, warning: 6 characters of junk seen at toplevel -[322] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_2sRz/347c261ec4135a5723bef5c751f5078f_24870.utf8, line 619, warning: 6 characters of junk seen at toplevel -[322] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_2sRz/347c261ec4135a5723bef5c751f5078f_24870.utf8, line 648, warning: 6 characters of junk seen at toplevel -[322] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_2sRz/347c261ec4135a5723bef5c751f5078f_24870.utf8, line 658, warning: 6 characters of junk seen at toplevel -[322] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_2sRz/347c261ec4135a5723bef5c751f5078f_24870.utf8, line 667, warning: 6 characters of junk seen at toplevel -[323] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_2sRz/347c261ec4135a5723bef5c751f5078f_24870.utf8, line 688, warning: 6 characters of junk seen at toplevel -[323] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_2sRz/347c261ec4135a5723bef5c751f5078f_24870.utf8, line 709, warning: 6 characters of junk seen at toplevel -[323] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_2sRz/347c261ec4135a5723bef5c751f5078f_24870.utf8, line 720, warning: 6 characters of junk seen at toplevel -[323] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_2sRz/347c261ec4135a5723bef5c751f5078f_24870.utf8, line 729, warning: 6 characters of junk seen at toplevel -[323] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_2sRz/347c261ec4135a5723bef5c751f5078f_24870.utf8, line 749, warning: 6 characters of junk seen at toplevel -[323] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_2sRz/347c261ec4135a5723bef5c751f5078f_24870.utf8, line 766, warning: 6 characters of junk seen at toplevel -[323] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_2sRz/347c261ec4135a5723bef5c751f5078f_24870.utf8, line 775, warning: 6 characters of junk seen at toplevel -[323] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_2sRz/347c261ec4135a5723bef5c751f5078f_24870.utf8, line 800, warning: 6 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Each -ascending neuron is assumed to scan the signal for a specific template pattern, -which can be thought of as a kernel of a particular structure and on a -particular time scale. This process, known as template matching, can be -modelled as a convolution of the intensity-adapted envelope $\adapt(t)$ with a -kernel $k_i(t)$ specific to the $i$-th ascending neuron: +The population of ascending neurons extracts and encodes a number of different +features of the preprocessed signal, and hence represents the signal in a +higher-dimensional space than the preceding receptor neurons and local +interneurons~(\bcite{clemens2011efficient}). Each ascending neuron is assumed +to scan the signal for a specific template pattern, which can be thought of as +a kernel of a particular structure and on a particular time scale. This +process, known as template matching, can be modelled as a convolution of the +intensity-adapted envelope $\adapt(t)$ with a kernel $k_i(t)$ specific to the +$i$-th ascending neuron: \begin{equation} c_i(t)\,=\,\adapt(t)\,*\,k_i(t) = \infint \adapt(\tau)\,\cdot\,k_i(t\,-\,\tau)\,d\tau @@ -495,14 +496,14 @@ $\rh$ relative to the maximum of the Gaussian: With this, an appropriate carrier frequency $\kfi$ for obtaining a Gabor kernel with width $\kwi$ and desired lobe number $\kni$ can be approximated as \begin{equation} - \kfi(\kni,\,\kwi,\,\rh)\,=\,\frac{0.5\,\cdot\,(\kni\,+\,\beta_0)}{\fdrm(\kwi,\,\rh)}, \qquad \kni\,\geq\,2\enspace\forall\enspace \kni\,\in\,\mathbb{Z} + \kfi(\kni,\,\kwi,\,\rh)\,=\,\frac{0.5\,\cdot\,\kni\,+\,\beta_0}{\fdrm(\kwi,\,\rh)}, \qquad \kni\,\geq\,2\enspace\forall\enspace \kni\,\in\,\mathbb{Z} \label{eq:gabor_freq} \end{equation} % \begin{equation} -% \kfi(\kni,\,\kwi,\,\rh)\,=\,\frac{0.5\,\cdot\,(\kni\,+\,\beta_0)}{2\,\cdot\,\sqrt{-2\,\cdot\,\ln \rh}\cdot\kwi}, \qquad \kni\,\geq\,2\enspace\forall\enspace \kni\,\in\,\mathbb{Z} +% \kfi(\kni,\,\kwi,\,\rh)\,=\,\frac{0.5\,\cdot\,\kni\,+\,\beta_0}{2\,\cdot\,\sqrt{-2\,\cdot\,\ln \rh}\cdot\kwi}, \qquad \kni\,\geq\,2\enspace\forall\enspace \kni\,\in\,\mathbb{Z} % \end{equation} The relationship between $\kfi$ and $\kni$ is approximately linear except for -small $\kni$. The offset term $\beta_0\approx0.5$ was added to balance the +small $\kni$. The offset term $\beta_0\approx0.26$ was added to balance the amplitudes of the $\kni$ desired lobes of the kernel --- which should be maximized --- against the amplitudes of the next-outer lobes, which should not exceed the threshold value determined by $\rh$. Note that simple Gaussian @@ -549,23 +550,30 @@ response~(Fig.\,\ref{fig:stages_feat}b): \label{eq:binary} \end{equation} The thresholding of $c_i(t)$ into $b_i(t)$ can be thought of as a -categorization into "relevant" and "irrelevant" response values. -% It is unclear whether such a thresholding nonlinearity is actually implemented -% either by the ascending neurons or at some point further downstream in the SEG. -Finally, the responses of the ascending neurons are assumed to be integrated -somewhere in the SEG~(\bcite{ronacher1986routes}; \bcite{bauer1987separate}; -\bcite{bhavsar2017brain}). This processing step can be approximated as temporal -averaging of the binary responses $b_i(t)$ by a lowpass filter +categorization into "relevant" and "irrelevant" response values. Similar +thresholding nonlinearities have been a crucial processing step in previous +models that deal with the extraction of behaviorally relevant song features in +insects~(\bcite{clemens2013computational}; \bcite{clemens2013feature}; +\bcite{hennig2014time}; \bcite{ronacher2015computational}). +% However, there is no direct physiological evidence that would allow to +% determine the exact location or underlying mechanism of such a nonlinearity in +% either the ascending neurons or at some point further downstream in the SEG. + +In the grasshopper, the responses of the ascending neurons are assumed to be +integrated somewhere in the SEG~(\bcite{ronacher1986routes}; +\bcite{bauer1987separate}; \bcite{bhavsar2017brain}). In the model pathway, +temporal integration is implemented as temporal averaging of the binary +responses $b_i(t)$ by a lowpass filter with extremely low cutoff frequency: \begin{equation} f_i(t)\,=\,b_i(t)\,*\,\lp, \qquad \fc\,=\,1\,\text{Hz} \label{eq:lowpass} \end{equation} -to obtain a final set of slowly changing kernel-specific features $f_i(t)$. In -the resulting high-dimensional feature space, different species-specific song -patterns are characterized by a distinct combination of feature values, which -can be read out by a simple linear classifier. - -% Cite somewhere: +This processing step results in a set of slowly changing kernel-specific +features $f_i(t)$, which is the final representation along the model +pathway~(Fig.\,\ref{fig:stages_feat}c). In the resulting high-dimensional +feature space, different species-specific song patterns can be distinguished by +their distinct combination of feature values, e.\,g. using Euclidian geometry +or a simple linear classifier. \begin{figure}[!ht] \centering \includegraphics[width=\textwidth]{figures/fig_feat_stages.pdf} @@ -770,6 +778,19 @@ stable position and height of the microphone array during recording. The resulting recordings were then processed through the model pathway and analyzed according to the procedure described in Section~\ref{sec:intensity_measures}. +\subsection{Determining kernel-specific threshold values} + +Different kernels $k_i(t)$ result in specific kernel responses $c_i(t)$, +Eq.\,\ref{eq:conv}, which are then transformed further into binary responses +$b_i(t)$, Eq.\,\ref{eq:binary}, by thresholding nonlinearity $\nl$. The +threshold value $\thr$ is specific to each $k_i(t)$. Across all analyses, +$\thr$ has been specified as a multiple of the pure-noise reference standard +deviation $\sigma_{c_i}$ for input $x(t)=\noc(t)$. This ensures that $\thr$ as +well as the resulting $b_i(t)$ and $f_i(t)$ are comparable across different +$k_i(t)$ because each pure-noise $c_i(t)$ approximately follows a normal +distribution~(see appendix +Figs.\,\ref{fig:app_thresh-lp_kern-sd}-\ref{fig:app_field_kern-sd}). + \section{Results} \subsection{Mechanisms driving the emergence of intensity invariance}