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add the text about the main result

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Dennis Huben 2024-09-26 18:07:34 +02:00
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@ -721,24 +721,52 @@ The curves from which the averages were created can be seen in figure \ref{2_by_
\end{figure}
%
% \begin{figure}
% \centering
% \includegraphics[width=0.4\linewidth]{img/sigma/example_spikes_sigma_with_input.pdf}
% \includegraphics[width=0.28\linewidth]{img/ordnung/cropped_fitcurve_0_2010-08-11-aq-invivo-1_0.pdf}
% \includegraphics[width=0.28\linewidth]{img/sigma/cropped_fitcurve_0_2010-08-31-aj-invivo-1_0.pdf}
% \includegraphics[width=0.28\linewidth]{img/ordnung/cropped_fitcurve_0_2010-08-31-ad-invivo-1_0.pdf}
% \includegraphics[width=0.4\linewidth]{img/fish/dataframe_scatter_sigma_cv.pdf}
% \includegraphics[width=0.4\linewidth]{img/fish/dataframe_scatter_sigma_firing_rate.pdf}
% \includegraphics[width=0.32\linewidth]{img/fish/sigma_distribution.pdf}
% \includegraphics[width=0.32\linewidth]{img/fish/cv_distribution.pdf}
% \includegraphics[width=0.32\linewidth]{img/fish/fr_distribution.pdf}
% cropped_fitcurve_0_2010-08-31-ad-invivo-1_0.pdf: 0x0 px, 300dpi, 0.00x0.00 cm, bb=
% \caption{Histogram of spike count distribution (firing rate) and errorfunction fits. 50 bins represent different values of the Gaussian distributed input signal [maybe histogram in background again]. The value of each of those bins is the number of spikes during the times the signal was in that bin. Each of the values was normalized by the signal distribution. For very weak and very strong inputs, the firing rates themselves become noisy, because the signal only assumes those values rarely. To account for delay, we first calculated the cross-correlation of signal and spike train and took its peak as the delay. The lines show fits according to equation \eqref{errorfct}. Left and right plots show two different cells, one with a relatively narrow distribution (left) and one with a distribution that is more broad (right), as indicated by the parameter \(\sigma\). An increase of $\sigma$ is equivalent to an broader distribution. Cells with broader distributions are assumed to be noisier, as their thresholding is less sharp than those with narrow distributions. Different amounts of bins (30 and 100) showed no difference in resulting parameters.}
% \label{sigmafits}
% \end{figure}
Figure \ref{coding_fraction_n_1} shows the link between noisiness and coding fraction very apparently. There is a strong correlation between the coding fraction calculated from the response of a single neuron and the neuron's noisiness. This intuitively makes sense, because the SSR advantage noisiness offers that we discussed earlier only appears for populations. There is a smaller, but still obvious, correlation between the coding fraction and the cell's firing rate: An increase in firing rate increases the coding fraction.
Between the firing rate and the cell and its noisiness is only a very weak correlation and they appear mostly independent of each other (figure \ref{fr_sigma}.
\begin{figure}
\centering
\includegraphics[width=0.4\linewidth]{img/sigma/example_spikes_sigma_with_input.pdf}
\includegraphics[width=0.28\linewidth]{img/ordnung/cropped_fitcurve_0_2010-08-11-aq-invivo-1_0.pdf}
\includegraphics[width=0.28\linewidth]{img/sigma/cropped_fitcurve_0_2010-08-31-aj-invivo-1_0.pdf}
% \includegraphics[width=0.28\linewidth]{img/ordnung/cropped_fitcurve_0_2010-08-31-ad-invivo-1_0.pdf}
\includegraphics[width=0.4\linewidth]{img/fish/dataframe_scatter_sigma_cv.pdf}
\includegraphics[width=0.4\linewidth]{img/fish/dataframe_scatter_sigma_firing_rate.pdf}
\includegraphics[width=0.32\linewidth]{img/fish/sigma_distribution.pdf}
\includegraphics[width=0.32\linewidth]{img/fish/cv_distribution.pdf}
\includegraphics[width=0.32\linewidth]{img/fish/fr_distribution.pdf}
% cropped_fitcurve_0_2010-08-31-ad-invivo-1_0.pdf: 0x0 px, 300dpi, 0.00x0.00 cm, bb=
\caption{Histogram of spike count distribution (firing rate) and errorfunction fits. 50 bins represent different values of the Gaussian distributed input signal [maybe histogram in background again]. The value of each of those bins is the number of spikes during the times the signal was in that bin. Each of the values was normalized by the signal distribution. For very weak and very strong inputs, the firing rates themselves become noisy, because the signal only assumes those values rarely. To account for delay, we first calculated the cross-correlation of signal and spike train and took its peak as the delay. The lines show fits according to equation \eqref{errorfct}. Left and right plots show two different cells, one with a relatively narrow distribution (left) and one with a distribution that is more broad (right), as indicated by the parameter \(\sigma\). An increase of $\sigma$ is equivalent to an broader distribution. Cells with broader distributions are assumed to be noisier, as their thresholding is less sharp than those with narrow distributions. Different amounts of bins (30 and 100) showed no difference in resulting parameters.}
\label{sigmafits}
\includegraphics[width=0.45\linewidth]{img/sigma/0_300/scatter_and_fits_sigma_coding_fractions_firing_rate.pdf}
\includegraphics[width=0.45\linewidth]{img/sigma/0_300/scatter_and_fits_firing_rate_coding_fractions_sigma.pdf}
\caption{Low firing rate and strong noise bith lead a a small coding fraction for single neurons.}
\label{coding_fraction_n_1}
\end{figure}
%figures created with box_script.py
\begin{figure}
\includegraphics[width=0.45\linewidth]{img/sigma/0_300/scatter_and_fits_sigma_firing_rate_contrast.pdf}
\includegraphics[width=0.45\linewidth]{img/sigma/0_300/scatter_and_fits_contrast_firing_rate_sigma.pdf}
\caption{Relationship between firing rate and $\sigma$ and cv respectively. Noisier tend to be cells that fire slower, but the relationship is very slight.}
\label{fr_sigma}
\end{figure}
We can further quantify the effect of SSR on the encoding by studying the difference in coding fraction for populations of different sizes. There are two different ways to do this. The first is to take the coding fraction at a large population size (here: 64 neurons) and divide it by the coding fraction for a single neuron. It is important to note that a large gain does not necessarily mean a good performance: a neuron that starts with a coding fraction of 0.01 for a population size of 1 can have a gain of 10. It would still perform worse for a population of 64 neurons than a cell that starts with a coding fraction of 0.11 even though that cell will certainly have a gain lower than 10, as coding fraction is limited at 1.
The alternative is taking the difference between the two coding fraction values for a large population and a single neuron. However, this might not be ideal in case of cells which need a population size larger than the 64 neurons observed here; the coding fraction increase from 1 to 64 neurons might then look small, even though the cells actually fit our model very well. Examples for neurons like this are in figure \ref{2_by_2_overview}: in the bottom right panel there are the bottom two lines with only a small increase in coding fraction, but both lines appear to become steeper with rising population size, so it is not unthinkable that they would rise much further for very large populations. It's a limitation of the current experiments that we can only record a finite amount of trials from each neuron.\notedh{Discussion??}
The result shown in figure \ref{increases_broad} is that $\sigma$ is a good predictor of the gain (quotient between coding fraction at 64 cells and coding fraction at 1 cell). Additionally, the firing rate negatively correlates with the gain, but more weakly.
For the difference between coding fraction of a single neuron and a population we do not see any correlation neither with $\sigma$ nor with the firing rate.
%figures created with result_fits.py
\begin{figure}
%\includegraphics[width=0.45\linewidth]{img/ordnung/sigma_popsize_curves_0to300}
\centering
@ -748,7 +776,8 @@ The curves from which the averages were created can be seen in figure \ref{2_by_
\includegraphics[width=0.45\linewidth]{img/sigma/0_300/scatter_and_fits_sigma_diff_firing_rate}%
\includegraphics[width=0.45\linewidth]{img/sigma/0_300/scatter_and_fits_firing_rate_diff_contrast}%
\caption{Top: The relative increase in coding fraction for population sizes 64 and 1. Note that the y-axis scales logarithmically Left: As a function of $\sigma$. Red curve shows a regression between $\sigma$ and $\log(c_{64}/c_{1})$. There is a strong relationship between the noisiness and the increase. Noisier cells (larger $\sigma$) have generally lower coding fractions for a single neuron, so they have a bigger potential for gain. Right: As a function of cell firing rate. The relationship is much weaker, but still there. }
\caption{Top: The relative increase in coding fraction for population sizes 64 and 1. Note that the y-axis scales logarithmically Left: As a function of $\sigma$. Red curve shows a regression between $\sigma$ and $\log(c_{64}/c_{1})$. There is a strong relationship between the noisiness and the increase. Noisier cells (larger $\sigma$) have generally lower coding fractions for a single neuron, so they have a bigger potential for gain. Right: As a function of cell firing rate. The relationship is much weaker, but still there.
Bottom: Using the difference in coding fraction instead of the quotient makes the relationship between the increase in coding fraction and the two parameters $\sigma$ and firing rate disappear. This might be different for larger population sizes.}
\label{increases_broad}
\end{figure}