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\@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces \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 measures 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. }}{19}{}\protected@file@percent }
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\@writefile{lof}{\contentsline {figure}{\numberline {6}{\ignorespaces \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}. 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 }
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\@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. Shaded areas indicate the average minimum $\mu _{f_i}$ across all species-specific trajectories. }}{25}{}\protected@file@percent }
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\@writefile{lof}{\contentsline {figure}{\numberline {8}{\ignorespaces \textbf {Step-wise emergence of intensity-invariant song representations along the model pathway.} 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 $f_i(t)$. 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 $x_{\text {raw}}(t)=\eta (t)$. 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 $\alpha $ that correspond to $95\,\%$ curve span for $c_i(t)$ and $f_i(t)$. Dots indicate the values from \textbf {b}. }}{28}{}\protected@file@percent }
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\@writefile{lof}{\contentsline {figure}{\numberline {9}{\ignorespaces \textbf {Effects of disabling logarithmic compression on intensity invariance along the model pathway.} 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 measures 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 measures 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}. }}{30}{}\protected@file@percent }
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\@writefile{lof}{\contentsline {figure}{\numberline {10}{\ignorespaces \textbf {Intensity invariance along the model pathway in a naturalistic setting.} Input $x_{\text {raw}}(t)$ consists of a song of \textit {P. parallelus} recorded in the field at eight different distances $d$ and is processed up to the feature set $f_i(t)$. 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). \textbf {a}:~$x_{\text {filt}}(t)$, $x_{\text {env}}(t)$, $x_{\text {log}}(t)$, $x_{\text {adapt}}(t)$, $c_i(t)$, and $f_i(t)$ at each $d$. A noise segment from the same recording is shown for reference. \textbf {b}:~Intensity measures over $d$. For $c_i(t)$ and $f_i(t)$, the median over kernels is shown. \textbf {c}:~Average value $\mu _{f_i}$ of each feature $f_i(t)$ over $d$. \textbf {d}:~Ratios of intensity measures to the respective value obtained from the noise reference. 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 $\mu _{f_i}$. }}{32}{}\protected@file@percent }
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\@writefile{lof}{\contentsline {figure}{\numberline {11}{\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 right}:~Distribution of correlation coefficients $\rho $ for each interspecific and intraspecific comparison. Dots indicate single $\rho $ values. }}{34}{}\protected@file@percent }
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[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 characters of junk seen at toplevel
|
||||
[323] Biber.pm:131> WARN - BibTeX subsystem: /tmp/biber_tmp_2sRz/347c261ec4135a5723bef5c751f5078f_24870.utf8, line 817, warning: 6 characters of junk seen at toplevel
|
||||
[323] Biber.pm:133> INFO - WARNINGS: 55
|
||||
191
main.fdb_latexmk
191
main.fdb_latexmk
@@ -1,191 +0,0 @@
|
||||
# Fdb version 4
|
||||
["biber main"] 1778854270.16835 "main.bcf" "main.bbl" "main" 1778855452.74854 0
|
||||
"cite.bib" 1770904753.08918 27483 4290db0c91f7b5055e25472ef913f6b4 ""
|
||||
"main.bcf" 1778855452.67652 112931 2a478116d80ebb1ada7083a24facd6e3 "pdflatex"
|
||||
(generated)
|
||||
"main.bbl"
|
||||
"main.blg"
|
||||
(rewritten before read)
|
||||
["pdflatex"] 1778855451.65459 "/home/hartling/phd/paper/paper_2025/main.tex" "main.pdf" "main" 1778855452.74875 0
|
||||
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|
||||
"/home/hartling/phd/paper/paper_2025/main.tex" 1778855451.52153 102661 7cb6fc878c1e09edca0dac40f59ca377 ""
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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main.run.xml
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main.run.xml
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BIN
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69
main.tex
69
main.tex
@@ -458,14 +458,15 @@ the following feature extraction stage.
|
||||
|
||||
\subsubsection{Feature extraction by individual neurons}
|
||||
|
||||
The ascending neurons extract and encode a number of different features of the
|
||||
preprocessed signal, and hence represent the signal in a higher-dimensional
|
||||
space than the preceding receptor neurons and local interneurons. 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}
|
||||
|
||||
Reference in New Issue
Block a user