Added more to the results for the narrowband signals

main.tex

@@ -695,7 +695,7 @@ The results we show will include both the coding fraction ratio and the coding f
 
 The result (figure \ref{increases_broad}) is that $\sigma$ is a good predictor of the ratio between coding fraction at 64 cells and coding fraction of a single cell: With an increased noisiness (an increase in \sig) comes a larger ratio. This is exactly what we expected, due to the smaller coding fraction at small populations and the effects of SSR.
 The firing rate of the cells has only a small effect on the ratio, if any.  
-As 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. As seen in figure \ref{2_by_2_overview} some of the cells only begin to take off in coding fraction near that population size. So the absolute difference is quite small at this point. If we had population sizes larger than 64, the regression would make more sense; the less noisy cells will have similar values of $c_{64}$ and e.g. $c_{512}$, but for the noisier cells there can be a huge difference.
+As f'or 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. As seen in figure \ref{2_by_2_overview} some of the cells only begin to take off in coding fraction near that population size. So the absolute difference is quite small at this point. If we had population sizes larger than 64, the regression would make more sense; the less noisy cells will have similar values of $c_{64}$ and e.g. $c_{512}$, but for the noisier cells there can be a huge difference.
 
 
 
@@ -717,7 +717,14 @@ Bottom: Using the difference in coding fraction instead of the quotient makes th
 
 \subsubsection{Narrowband}
 
-Qualitatively we see very similar results when instead of the broadband signal we use the narrowband signal with a frequency cutoff of 50Hz (figure \ref{overview_experiment_results_narrow}. Again the cells in the first quantile interval show on average only a very slightly increasing coding fraction with increasing population size. Coding fraction for a population size of one on average decreases for the higher quartile intervals. The seperate coding fraction curves also show the typical flatness for the first quartile interval. The fourth quartile interval in particular contains several curves that are only just beginning to increase in coding fraction at a population size of 64 neurons.
+Qualitatively we see very similar results when instead of the broadband signal we use the narrowband signal with a frequency cutoff of 50Hz (figure \ref{overview_experiment_results_narrow}). Again the cells in the first quartile interval show on average only a very slightly increasing coding fraction with increasing population size. Coding fraction for a population size of one on average decreases for the higher quartile intervals. The seperate coding fraction curves also show the typical flatness for the first quartile interval. The fourth quartile interval in particular contains several curves that are only just beginning to increase in coding fraction at a population size of 64 neurons.
+There are some differences to the result of the broadband signal experiments:
+The first is that coding fractions in general are larger. For the case of a single cell population, coding fraction was never larger than 0.25 for the observed cells with the broadband signals. With the narrowband signal input, most cells show a coding fraction of larger than 0.25, even for a single cell sized population. 
+We also find a correlation between the firing rate and \sig when we use the narrowband signal as input (figure \ref{sigma_vs_firing_rate_for_narrow} a).
+Contrarily, the curves are in general flatter for the signal with narrowed bandwidth. This can be seen in the coding fraction as a function of population size figures (\ref{2_by_2_overview} and \ref{overview_experiment_results_narrow} B-E). 
+The numbers support the visual impression: Previously (broadband signal) most cells showed a difference of between 0.1 and about 0.4 for the increase in coding fraction from a single cell to a population of 64 cells. With the narrowband signal, most cells show an increase of less than 0.2, with most even less than 0.1. Only a few cells show the same increase we commonly saw for the broadband signal. Because for the narrowband signal many cells already show a large coding fraction at the N=1 population size, the ratio between $c_{64}$ and $c_1$ is mostly between 1 and 2. 
+Only the cells with a lower firing rate show a substantially larger ratio (figure \ref{increases_narrow} b) than that. Using a linear regression over the whole range as we have done before might not be the ideal way to handle the distribution of points. Analysing the data as we did shows a correlation between firing rate and the ratio $\frac{c_{64}}{c_1}$, even though the linear regression is clearly not the best model to capture it.\notedh{Do the analysis again only for < 200Hz (or other number?) where it looks like it could be linear?}  
+However, it works much better again if the ratio $\frac{c_{64}}{c_1}$ is compared again the noise strength \sig (figure \ref{increases_narrow} a). The correlation is lower than it was for the broadband signal (0.60 there vs 0.49 for the narrowband signal). With the narrowband signal we now also see a correlation with the difference of $c_{64}$ and $c_1$, which we didn't for the broadband signal.
 
 
 \begin{figure}
@@ -734,7 +741,7 @@ Qualitatively we see very similar results when instead of the broadband signal w
  \includegraphics[width=0.45\linewidth]{img/sigma/narrow_0_50/scatter_and_fits_firing_rate_coding_fractions_sigma.pdf}
  \caption{Coding fraction for a single cell as a function of $\sigma$ (left) and firing rate (right). The signal used was a 0-50Hz narrowband signal. 
  Similar to what we have seen for the broadband signal (figure \ref{coding_fraction_n_1}), cells for which $\sigma$ is large, i.e. noisier cells, have a lower single cell coding fraction than cells with a smaller $\sigma$. The correlation appears to be a bit weaker though. The reverse is true for single-cell coding fraction as a function of firing rate: here, the correlation is stronger that it was for the broadband signal; it is still weaker than the correlation for the noise. Notably, there are a few cells with rather low firing rates for which the single-cell coding fraction is very close to 0. This was not the case for any of the other input signals we used, neither broadband nor higher frequency narrowband.}
- \label{coding_fraction_n_1_narrow_0_50}
+\label{coding_fraction_n_1_narrow_0_50}
 \end{figure}
 
 \begin{figure}
@@ -745,11 +752,6 @@ Qualitatively we see very similar results when instead of the broadband signal w
  \label{sigma_vs_firing_rate_for_narrow}
 \end{figure}
 
-
-
-Figures \ref{increases_narrow} and \ref{increases_narrow_high} both show that the results with regards to the increase of coding fraction for different population sizes seen for the broadband signal also appear when we use narrowband signals. For the high frequency signal (250Hz to 300Hz, figure \ref{increases_narrow_high} the correlation between the coding fraction increase and the firing rate is higher than the correlation between the coding fraction increase and $\sigma$. As seen before, taking the difference between the coding fraction of a population size of 64 ($c_{64}$) and at a population of 1 ($c_1$) might not work out. With the narrowband signal even more than with the broadband signal some of the cells only begin to take off in coding fraction near that population size of 64 \ref{overview_experiment_results_narrow}. So the absolute difference is quite small at this point. If we had population sizes larger than 64, the regression would make more sense; the less noisy cells will have similar values of $c_{64}$ and e.g. $c_{512}$, but for the noisier cells there can be a huge difference.
-
-
 %figures created with result_fits.py
 \begin{figure}
 \centering
@@ -766,24 +768,31 @@ Figures \ref{increases_narrow} and \ref{increases_narrow_high} both show that th
 \label{increases_narrow}
 \end{figure}
 
-
-
+We used other narrowband signals with a frequency width of 50Hz as well. For those, the power of the signal was not in the 0-50Hz spectrum, but in higher frequencies, e.g. 50-100Hz, 150-200Hz, 250-300Hz and 350-400Hz.
+In general, those results are comparable to the results we get with the 0-50Hz input signal. For example, we again see a correlation between the firing rate of the cells and their noisiness \sig (figure \ref{sigma_vs_firing_rate_for_narrow} b for the 250-300Hz signal).
+Taking the 250Hz-300Hz input signal as an example for the higher frequency signals, the positive correlations between \sig and the increase in coding fraction for increasing population size are quite clear, for both the ratio and the difference (figure \ref{increases_narrow_high}). Now we also see a sharp decline in the improvement for increasing firing rate, i.e. cells that fire more slowly have a much larger coding fraction increase than cells that fire more quickly. The correlation between the firing rate and the increase appears to be stronger that the correlation between \sig and the increase.
+Detailed results for all frequency ranges are shown in the appendix (figures \ref{sigma_narrow_50_100} to \ref{sigma_narrow_350_400}.
 
 %figures created with result_fits.py
 \begin{figure}
-\centering
+ \centering 
+\includegraphics[width=0.45\linewidth]{img/sigma/narrow_250_300/scatter_and_fits_sigma_coding_fractions_firing_rate.pdf}
+\includegraphics[width=0.45\linewidth]{img/sigma/narrow_250_300/scatter_and_fits_firing_rate_coding_fractions_sigma.pdf}
+
+ \centering
 \includegraphics[width=0.45\linewidth]{img/sigma/narrow_250_300/scatter_and_fits_sigma_quot_firing_rate}%
 \includegraphics[width=0.45\linewidth]{img/sigma/narrow_250_300/scatter_and_fits_firing_rate_quot_contrast}%
 
 \centering
 \includegraphics[width=0.45\linewidth]{img/sigma/narrow_250_300/scatter_and_fits_sigma_diff_firing_rate}%
 \includegraphics[width=0.45\linewidth]{img/sigma/narrow_250_300/scatter_and_fits_firing_rate_diff_contrast}%
- \caption{Same graphs as in figure \ref{increases_narrow}, but with a narrowband signal with 250Hz-300Hz. 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 $\log(c_{64}/c_{1})$.  Right: As a function of cell firing rate. Both cell firing rate and $sigma$ determine the increase, even though firing rate and $sigma$ themselves are almost independent of each other (figure \ref{fr_sigma})
+ \caption{Same graphs as in figure \ref{increases_narrow}, but with a narrowband signal with 250Hz-300Hz.
+ Top row: Coding fraction for a single cell as a function of $\sigma$ (left) and firing rate (right). Points show individual trials and the red line shows a linear regression between the points. On the left, as a function of \sig, on the right as a function of the firing rate.
+ Middle row: 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 $\log(c_{64}/c_{1})$.  Right: As a function of cell firing rate. Both cell firing rate and $sigma$ determine the increase, even though firing rate and $sigma$ themselves are almost independent of each other (figure \ref{fr_sigma})
 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_narrow_high}
 \end{figure}
 
-The results (figure \ref{overview_fits}) show \todo{write}
 
 \begin{figure}
 \centering
@@ -799,6 +808,10 @@ The results (figure \ref{overview_fits}) show \todo{write}
 \label{overview_fits_narrow}
 \end{figure}
 
+Figure \ref{overview_fits_narrow} compiles the results for the narrowband signal in all the frequency ranges. For three of the ranges (50-100Hz, 150-200Hz and 350-400Hz) there were few trials available for analysis (six, four and five respectively). Therefore, those results are less reliable, which is also shown in the respective p-values in the figure. For the cases for which we have good data, we see the same trend we saw for the broadband signal in the relationship between the coding fraction increase from increasing population size and \sig still exists. Other than what we saw for the broadband signal, we can also observe a correlation between the coding fraction increase and the firing rate of the cell.
+\todo{Das in der Diskussion einbringen!}
+
+
 \notedh{link to the appropriate chapter from theory results}
 In addition to the ``pure'' narrowband signals, I also analysed the coding fraction change for a smaller part of the spectrum in the experiments using the broadband signal. Figure \ref{increases_narow_in_broad} shows part of the results and again we see the strong correlation between $\sigma$ and the gain and a lesser correlation between the firing rate and the gain. In this case we see the same correlation also for the coding fraction difference.
 Similar results can be observed for the other frequency bands. \notedh{Images to the appendix? The sigma/gain of all in one plot?}
@@ -831,6 +844,8 @@ Left columns shows results for the difference in coding fraction, right column s
 \label{overview_fits_broad}
 \end{figure}
 
+\section*{Appendix}
+
 %compare_params_300.py auf oilbird
 \begin{figure}
  \includegraphics[width=0.30\linewidth]{img/sigma/parameter_assessment/bins100v300.pdf}
@@ -1025,7 +1040,7 @@ Bottom: Same as above, but using the difference in coding fraction instead of th
 
 
 
-\section{Literature}
+%\section{Literature}
 
 
 \clearpage