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+ {Society for Neuroscience}% + } + \strng{namehash}{4a3a321b17a743e61e1cb58aeb8dee08} + \strng{fullhash}{4a3a321b17a743e61e1cb58aeb8dee08} + \strng{bibnamehash}{4a3a321b17a743e61e1cb58aeb8dee08} + \strng{authorbibnamehash}{4a3a321b17a743e61e1cb58aeb8dee08} + \strng{authornamehash}{4a3a321b17a743e61e1cb58aeb8dee08} + \strng{authorfullhash}{4a3a321b17a743e61e1cb58aeb8dee08} + \field{sortinit}{G} + \field{sortinithash}{32d67eca0634bf53703493fb1090a2e8} + \field{extradatescope}{labelyear} + \field{labeldatesource}{} + \field{labelnamesource}{author} + \field{labeltitlesource}{title} + \field{journaltitle}{Journal of Neuroscience} + \field{number}{34} + \field{title}{Input-driven components of spike-frequency adaptation can be unmasked in vivo} + \field{volume}{24} + \field{year}{2004} + \field{pages}{7435\bibrangedash 7444} + \range{pages}{10} + \endentry \entry{gollisch2002energy}{article}{} - \name{author}{4}{}{% + \name{author}{4}{ul=2}{% 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[] -Overfull \hbox (3.45023pt too wide) in paragraph at lines 105--136 +Overfull \hbox (3.45023pt too wide) in paragraph at lines 107--138 \OT1/cmr/m/n/12 di-rec-tional hear-ing ([]; []; [] [1 {/var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map}] -Overfull \hbox (0.45996pt too wide) in paragraph at lines 140--197 +Overfull \hbox (0.45996pt too wide) in paragraph at lines 142--199 \OT1/cmr/m/n/12 ab-so-lute time in-ter-vals ([]; []), [] -Overfull \hbox (81.98225pt too wide) in paragraph at lines 140--197 +Overfull \hbox (81.98225pt too wide) in paragraph at lines 142--199 \OT1/cmr/m/n/12 cess, in which dif-fer-ent neu-ronal pop-u-la-tions con-tribute to vary-ing de-grees ([]) [] -Overfull \hbox (0.27292pt too wide) in paragraph at lines 140--197 +Overfull \hbox (0.27292pt too wide) in paragraph at lines 142--199 \OT1/cmr/m/n/12 and by dif-fer-ent mech-a-nisms ([]). 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[] -Overfull \hbox (4.40714pt too wide) in paragraph at lines 384--393 +Overfull \hbox (4.40714pt too wide) in paragraph at lines 386--395 \OT1/cmr/m/n/12 the sub-se-quent lo-cal in-terneu-rons ([]; []) [] [7] [8] -Overfull \hbox (9.43962pt too wide) in paragraph at lines 484--484 +Overfull \hbox (9.43962pt too wide) in paragraph at lines 495--495 []\OT1/cmr/bx/n/17.28 Two mech-a-nisms driv-ing the emer-gence of intensity- [] @@ -749,15 +749,15 @@ Package logreq Info: Writing requests to 'main.run.xml'. ) Here is how much of TeX's memory you used: - 19298 strings out of 474222 - 412416 string characters out of 5748732 + 19301 strings out of 474222 + 412437 string characters out of 5748732 1937975 words of memory out of 5000000 - 41317 multiletter control sequences out of 15000+600000 + 41320 multiletter control sequences out of 15000+600000 567787 words of font info for 73 fonts, out of 8000000 for 9000 1141 hyphenation exceptions out of 8191 94i,19n,93p,955b,992s stack positions out of 10000i,1000n,20000p,200000b,200000s -Output written on main.pdf (13 pages, 1439309 bytes). +Output written on main.pdf (13 pages, 1440223 bytes). PDF statistics: 1103 PDF objects out of 1200 (max. 8388607) 699 compressed objects within 7 object streams diff --git a/main.pdf b/main.pdf index 47b3f6f..8124a60 100644 Binary files a/main.pdf and b/main.pdf differ diff --git a/main.synctex.gz b/main.synctex.gz index 1d594c3..cf48772 100644 Binary files a/main.synctex.gz and b/main.synctex.gz differ diff --git a/main.tex b/main.tex index 4869943..e450dfb 100644 --- a/main.tex +++ b/main.tex @@ -72,6 +72,8 @@ \newcommand{\kpi}{\kp_i} % Specific Gabor kernel phase \newcommand{\kni}{\kn_i} % Specific Gabor kernel lobe number \newcommand{\ksi}{\ks_i} % Specific Gabor kernel sign +\newcommand{\rh}{\text{RH}} % Relative Gaussian height for FWRH +\newcommand{\fwrh}{\text{FWRH}} % Gaussian full-width at relative height % Math shorthands - Threshold nonlinearity: \newcommand{\thr}{\Theta_i} % Step function threshold value @@ -415,25 +417,38 @@ matching, can be modelled as a convolution \label{eq:conv} \end{equation} of the intensity-adapted envelope $\adapt(t)$ with a kernel $k_i(t)$ per -ascending neuron. We used Gabor kernels as basis functions for creating -different preferred patterns. An arbitrary one-dimensional, real Gabor kernel -is generated by multiplication of a Gaussian envelope and a sinusoidal carrier +ascending neuron. We use Gabor kernels as basis functions for creating +different template patterns. An arbitrary one-dimensional, real Gabor kernel is +generated by multiplication of a Gaussian envelope and a sinusoidal carrier \begin{equation} k_i(t,\,\kwi,\,\kfi,\,\kpi)\,=\,e^{-\frac{t^{2}}{2{\kwi}^{2}}}\,\cdot\,\sin(\kfi\,t\,+\,\kpi), \qquad \kfi\,=\,2\pi f_{sin} \label{eq:gabor} \end{equation} with Gaussian standard deviation or kernel width $\kwi$, carrier frequency -$\kfi$, and carrier phase $\kpi$. Different combinations of $\kwi$, $\kfi$, and -$\kpi$ result in Gabor kernels with different lobe number $\kni$ and sign -$\ksi$. If the function space is constrained to only include mirror- or -point-symmetric Gabor kernels, frequency $\kf$ is related to lobe number $\kn$ -by +$\kfi$, and carrier phase $\kpi$. Different combinations of $\kw$, $\kf$, and +$\kp$ result in Gabor kernels with different lobe number $\kn$, which is the +number of half-periods of the carrier that fit under the Gaussian envelope +within reasonable limits of attenuation. These limits are a matter of +definition, since the Gaussian function never fully decays to zero. A good +measure is the Gaussian full-width at relative height, which can be calculated +as +\begin{equation} + \fwrh(\kw,\,\rh)\,=\,2\,\cdot\,\sqrt{-2\,\ln \rh}\cdot\,\kw, \qquad \rh\,\in\,(0,\,1] +\end{equation} +With this, an appropriate carrier frequency $\kf$ for obtaining a Gabor kernel +with width $\kw$ and a desired lobe number $\kn$ can be approximated as +\begin{equation} + \kf(\kn,\,\fwrh)\,=\,\frac{0.5\,\cdot\,\kn\,+\,0.5}{\fwrh} +\end{equation} +We restrict the Gabor kernels to be either even functions~(mirror-symmetric, +uneven $\kn$) or odd functions~(point-symmetric, even $\kn$). Under this +condition, phase $\kp$ is related to lobe number $\kn$ by \begin{equation} \kp(\kn,\,\ks)\,=\,0.5\,\cdot\,(1\,-\,\text{mod}[\kn,\,2]\,+\,\ks) + \label{eq:gabor_phase} \end{equation} which results in the specific phase values shown in -Table\,\mbox{\ref{tab:gabor_phases}}. - +Table\,\mbox{\ref{tab:gabor_phase}}. \FloatBarrier \begin{table}[!ht] \centering @@ -447,14 +462,10 @@ Table\,\mbox{\ref{tab:gabor_phases}}. -1 & $-\pi\,/\,2$ & $0$\\ \hline \end{tabular} - \label{tab:gabor_phases} + \label{tab:gabor_phase} \end{table} \FloatBarrier -In order to create a Gabor kernel with a specific lobe number $\kn$ and kernel -width $\kw$, frequency $\kf$ has to be set to -\begin{equation} - \kf(\kn,\,\kw)\,=\,\frac{\kn}{2\pi\,\kw} -\end{equation} + \textbf{Stage-specific processing steps and functional approximations:}