Added .gitignore that currently only excludes contents of ./data/.
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j-hartling
112
main.tex
112
main.tex
@@ -10,8 +10,11 @@
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\usepackage{parskip}
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\usepackage{amsmath}
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\usepackage{amssymb}
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\usepackage{subcaption}
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\usepackage{subcaption}
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\usepackage[labelfont=bf, textfont=small]{caption}
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\usepackage[german,english]{babel}
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\addto\captionsenglish{\renewcommand{\figurename}{Fig.}}
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\addto\captionsenglish{\renewcommand{\tablename}{Tab.}}
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\usepackage[separate-uncertainty=true, locale=DE]{siunitx}
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\sisetup{output-exponent-marker=\ensuremath{\mathrm{e}}}
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% \usepackage[capitalize]{cleveref}
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@@ -66,14 +69,18 @@
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\newcommand{\kf}{\omega} % Unspecific Gabor kernel frequency
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\newcommand{\kp}{\phi} % Unspecific Gabor kernel phase
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\newcommand{\kn}{n} % Unspecific Gabor kernel lobe number
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\newcommand{\ks}{s} % Unspecific Gabor kernel sign
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% \newcommand{\ks}{s} % Unspecific Gabor kernel sign
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\newcommand{\kwi}{\kw_i} % Specific Gabor kernel width
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\newcommand{\kfi}{\kf_i} % Specific Gabor kernel frequency
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\newcommand{\kpi}{\kp_i} % Specific Gabor kernel phase
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\newcommand{\kni}{\kn_i} % Specific Gabor kernel lobe number
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\newcommand{\ksi}{\ks_i} % Specific Gabor kernel sign
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\newcommand{\rh}{\text{RH}} % Relative Gaussian height for FWRH
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% \newcommand{\ksi}{\ks_i} % Specific Gabor kernel sign
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% Math shorthands - Auxiliary kernel parameters:
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\newcommand{\fsin}{f_{\text{sin}}} % Carrier frequency
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\newcommand{\rh}{h_{\text{rel}}} % Relative Gaussian height for FWRH
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\newcommand{\fwrh}{\text{FWRH}} % Gaussian full-width at relative height
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\newcommand{\off}{\beta_0} % Offset for linear frequency approximation
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% Math shorthands - Threshold nonlinearity:
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\newcommand{\thr}{\Theta_i} % Step function threshold value
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@@ -421,58 +428,62 @@ ascending neuron. We use Gabor kernels as basis functions for creating
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different template patterns. An arbitrary one-dimensional, real Gabor kernel is
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generated by multiplication of a Gaussian envelope and a sinusoidal carrier
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\begin{equation}
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k_i(t,\,\kwi,\,\kfi,\,\kpi)\,=\,e^{-\frac{t^{2}}{2{\kwi}^{2}}}\,\cdot\,\sin(\kfi\,t\,+\,\kpi), \qquad \kfi\,=\,2\pi f_{sin}
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k_i(t,\,\kwi,\,\kfi,\,\kpi)\,=\,e^{-\frac{t^{2}}{2{\kwi}^{2}}}\,\cdot\,\sin(\kfi\,t\,+\,\kpi), \qquad \kfi\,=\,2\pi\fsin
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\label{eq:gabor}
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\end{equation}
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with Gaussian standard deviation or kernel width $\kwi$, carrier frequency
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$\kfi$, and carrier phase $\kpi$. Different combinations of $\kw$, $\kf$, and
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$\kp$ result in Gabor kernels with different lobe number $\kn$, which is the
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number of half-periods of the carrier that fit under the Gaussian envelope
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within reasonable limits of attenuation. These limits are a matter of
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definition, since the Gaussian function never fully decays to zero. A good
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measure is the Gaussian full-width at relative height, which can be calculated
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as
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$\kfi$, and carrier phase $\kpi$. Different combinations of $\kw$ and $\kf$
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result in Gabor kernels with different lobe number $\kn$, which is the number
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of half-periods of the carrier that fit under the Gaussian envelope within
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reasonable limits of attenuation. The interval under the Gaussian envelope that
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contains the relevant lobes of the kernel can be defined as Gaussian full-width
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measured at relative peak height $\rh$
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\begin{equation}
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\fwrh(\kw,\,\rh)\,=\,2\,\cdot\,\sqrt{-2\,\ln \rh}\cdot\,\kw, \qquad \rh\,\in\,(0,\,1]
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\fwrh(\kw,\,\rh)\,=\,2\,\cdot\,\sqrt{-2\,\cdot\,\ln \rh}\cdot\,\kw, \qquad \rh\,\in\,(0,\,1]
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\end{equation}
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With this, an appropriate carrier frequency $\kf$ for obtaining a Gabor kernel
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with width $\kw$ and a desired lobe number $\kn$ can be approximated as
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with width $\kw$ and desired lobe number $\kn$ can be approximated as
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% \begin{equation}
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% \kf(\kn,\,\fwrh)\,=\,\frac{0.5\,\cdot\,\kn\,+\,\off}{\fwrh}, \qquad \kn\,\geq\,2\enspace\forall\enspace \kn\,\in\,\mathbb{Z}
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% \end{equation}
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\begin{equation}
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\kf(\kn,\,\fwrh)\,=\,\frac{0.5\,\cdot\,\kn\,+\,0.5}{\fwrh}
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\kf(\kn,\,\kw,\,\rh)\,=\,\frac{\kn\,+\,\off}{4\,\cdot\,\sqrt{-2\,\cdot\,\ln \rh}}, \qquad \kn\,\geq\,2\enspace\forall\enspace \kn\,\in\,\mathbb{Z}
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\end{equation}
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We restrict the Gabor kernels to be either even functions~(mirror-symmetric,
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uneven $\kn$) or odd functions~(point-symmetric, even $\kn$). Under this
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condition, phase $\kp$ is related to lobe number $\kn$ by
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\begin{equation}
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\kp(\kn,\,\ks)\,=\,0.5\,\cdot\,(1\,-\,\text{mod}[\kn,\,2]\,+\,\ks)
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\label{eq:gabor_phase}
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\end{equation}
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which results in the specific phase values shown in
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Table\,\mbox{\ref{tab:gabor_phase}}.
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where $\off$ is a small positive offset to the near-linear relationship between
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$\kf$ and $\kn$ to balance the amplitude of the $\kn$ desired lobes of the
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kernel --- which should be maximized --- against the amplitude of the
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next-outer lobes, which should not exceed the threshold value determined by
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$\rh$. For $\kn=1$, carrier frequency $\kf$ is set to zero, which results in a
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simple Gaussian kernel. Carrier phase $\kp$ determines the position of the
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kernel lobes relative to the kernel center. By setting $\kp$ to one of only
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four specific phase values~(Tab.\,\ref{tab:gabor_phases}), we restrict the
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Gabor kernels to be either even functions~(mirror-symmetric, uneven $\kn$) or
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odd functions~(point-symmetric, even $\kn$) with either positive or negative
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sign, which refers to the sign of the kernel's central lobe (even kernels) or
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the left of the two central lobes (odd kernels).
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\FloatBarrier
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\begin{table}[!ht]
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\centering
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\captionsetup{width=.55\textwidth}
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\caption{}
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\captionsetup{width=.46\textwidth}
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\caption{Values of phase $\kp$ that are specific for the four major groups
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of Gabor kernels.}
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\begin{tabular}{|ccc|}
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\hline
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sign $\ks$ & even $\kn$ & odd $\kn$\\
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sign & even kernels & odd kernels\\
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\hline
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+1 & $+\pi\,/\,2$ & $\pi$\\
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-1 & $-\pi\,/\,2$ & $0$\\
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$+$ & $+\pi\,/\,2$ & $\pi$\\
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$-$ & $-\pi\,/\,2$ & $0$\\
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\hline
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\end{tabular}
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\label{tab:gabor_phase}
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\label{tab:gabor_phases}
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\end{table}
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\FloatBarrier
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\textbf{Stage-specific processing steps and functional approximations:}
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Thresholding nonlinearity in ascending neurons (or further downstream)\\
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- Binarization of AN response traces into "relevant" vs. "irrelevant"\\
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$\rightarrow$ Shifted Heaviside step-function $\nl$ (or steep sigmoid threshold?)
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%
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These four groups of Gabor kernels allow for the extraction of different types
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of signal features, such as the presence of peaks (even, $+$), troughs (even,
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$-$), onsets (odd, $+$), and offsets (odd, $-$) at various time scales.
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Following the convolutional template matching, each kernel-specific response
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$c_i(t)$ is passed through a shifted Heaviside step-function $\nl$ with threshold
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value $\thr$ to obtain a binary response
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\begin{equation}
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b_i(t,\,\thr)\,=\,\begin{cases}
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\;1, \quad c_i(t)\,>\,\thr\\
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@@ -480,18 +491,23 @@ $\rightarrow$ Shifted Heaviside step-function $\nl$ (or steep sigmoid threshold?
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\end{cases}
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\label{eq:binary}
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\end{equation}
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%
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Temporal averaging by neurons of the central brain\\
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- Finalized set of slowly changing kernel-specific features (one per AN)\\
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- Different species-specific song patterns are characterized by a distinct combination
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of feature values $\rightarrow$ Clusters in high-dimensional feature space\\
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$\rightarrow$ Lowpass filter 1 Hz
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%
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which can be thought of as a categorization into "relevant" and "irrelevant"
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response values. In the grasshopper, these threshold nonlinearities might
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either be part of the processing within the ascending neurons or take place
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further downstream~(SOURCE). Finally, the responses of the ascending neurons
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are assumed to be integrated somewhere in the supraesophageal
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ganglion~(\bcite{ronacher1986routes}; \bcite{bauer1987separate};
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\bcite{bhavsar2017brain}). This processing step can be approximated as temporal
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averaging of the binary responses $b_i(t)$ by a lowpass filter
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\begin{equation}
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f_i(t)\,=\,b_i(t)\,*\,\lp, \qquad \fc\,=\,1\,\text{Hz}
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\label{eq:lowpass}
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\end{equation}
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%
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to obtain a final set of slowly changing kernel-specific features $f_i(t)$. In
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the resulting high-dimensional feature space, different species-specific song
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patterns are characterized by a distinct combination of feature values, which
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can be read out by a simple linear classifier.
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\section{Two mechanisms driving the emergence of intensity-invariant song representation}
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\textbf{Definition of invariance (general, systemic):}\\
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@@ -671,4 +687,8 @@ initiation of one behavior over another is categorical (e.g. approach/stay)
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\section{Conclusions \& outlook}
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The model pathway includes a rather large number of Gabor kernels compared to
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the 15 to 20 ascending neurons in the grasshopper auditory
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system~(\bcite{stumpner1991auditory}).
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\end{document}
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