Added/changed some of the results text

main.tex

@@ -607,23 +607,26 @@ of each other and there is no feedback.
 
 \subsection*{Electrophysiology}
 
-We can see from figure \ref{sigmafits_example} that the fits look very close to the data. Due to the gaussian signal distribution there are fewer samples for very weak and very strong inputs. In these regions the firing rates become somewhat noisy.  This is especially noticeable for strong inputs, as there are more spikes there, and therefore large fluctuations. Fluctuations are less visible for weak inputs where there is very little spiking anyway.
+After correcting for the time delay between the emission of the signal and the recording of the electrophysiological data, we apply a timeshift to the signal, so it matches up better with the data (figure \ref{timeshift}).
+Next we fit the curve from equation \ref{errorfct} to the recorded data. We find that that the fits look very close to the data (figure \ref{sigmafits_example}). Due to the gaussian signal distribution there are fewer samples for very weak and very strong inputs. In these regions the firing rates become somewhat noisy.  This is especially noticeable for strong inputs, as there are more spikes there than for weaker inputs. Fluctuations are less visible for weak inputs where there is very little spiking anyway.
 
 \begin{figure}
  \includegraphics[width=0.4\linewidth]{img/ordnung/cropped_fitcurve_0_2010-08-31-ad-invivo-1_0.pdf}
  \includegraphics[width=0.4\linewidth]{img/ordnung/cropped_fitcurve_0_2010-08-11-aq-invivo-1_0.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 amplitude of the Gaussian distributed input signal \notedh{[maybe histogram in background again] - or better: One plot where I show the raw data - histrogram in background, number of spikes as dots.}. 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. To account for delay, we first calculated the cross-correlation of signal and spike train and took its peak as the delay. This is necessary because there are delays between the signal being emitted and being registered by the cell. For an explanation, see figure \ref{timeshift}. The lines show fits according to equation \eqref{errorfct}. 
- Below are some examples of different cells, with differently wide distributions. one with a relatively narrow distribution and one with a distribution that is more broad, as indicated by the parameter \(\sigma\). Different amounts of bins (30 and 100) showed no difference in resulting parameters. \notedh{Show a plot.} \notedh{Show more than two plots?}}
+ \caption{Histogram of spike count distribution (firing rate) and errorfunction fits. 50  bins represent different values of the amplitude of the Gaussian distributed input signal \notedh{[maybe histogram in background again] - or better: One plot where I show the raw data - histrogram in background, number of spikes as dots.}. 
+ 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. 
+ To account for delay, we first calculated the cross-correlation of signal and spike train and took its peak as the delay. This is necessary because there are delays between the signal being emitted and being registered by the cell (see figure \ref{timeshift}). The lines show fits according to equation \eqref{errorfct}. 
+ Below are some examples of different cells, with differently wide distributions. one with a relatively narrow distribution and one with a distribution that is more broad, as indicated by the parameter \(\sigma\). Different amounts of bins (30 and 100) showed no difference in resulting parameters (figure \ref{sigma_bins}). \notedh{Show more than two plots?}}
  \label{sigmafits_example}
 \end{figure}
 
-Figure \ref{fr_sigma} shows that 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} shows that between the firing rate and the cell and its noisiness is only a very weak correlation and they appear mostly independent of each other. This matches the observations we had from the analysis of the simulated data.
 
 \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_cv_firing_rate_contrast.pdf}
- \caption{Relationship between firing rate and $\sigma$ and firing rate and CV respectively. Noisier cells might overall be cells that fire slower, but the relationship is very weak, if present at all. The color of the markers indicates the contrast, a measure for the strength of the signal. For the values used in the experiments, there doesn't appear to be much of a correlation between firing rate and the contrast (figure \ref{contrast_firing_rate} in the appendix.}
+ \caption{Relationship between firing rate and $\sigma$ and firing rate and CV respectively. Noisier cells (as measured by \sig) might overall be cells that fire less frequently, but the relationship is very weak, if present at all. The color of the markers indicates the contrast, a measure for the strength of the signal. For the values used in the experiments, there doesn't appear to be much of a correlation between firing rate and the contrast (figure \ref{contrast_firing_rate}).}
  \label{fr_sigma}
 \end{figure}
 
@@ -670,8 +673,9 @@ The curves from which the averages were created can be seen in figure \ref{2_by_
 %  \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.
-
+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 and matches what we observed in the simulations, because a single cell simply becomes less reliable by additional noise. The advantage of SSR only comes into play at larger population sizes. 
+There is a smaller correlation between the coding fraction and the cell's firing rate: An increase in firing rate increases the single-cell coding fraction.
 
 
 
@@ -684,11 +688,14 @@ Figure \ref{coding_fraction_n_1} shows the link between noisiness and coding fra
 
 
 
-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??}
+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 could have a gain of 10. It would still perform worse for a population of 64 neurons than a cell that shows a coding fraction of 0.11 at N=1. 
+The alternative to taking the ratio is to take 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 large population sizes to show a good performance. 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 results we show will include both the coding fraction ratio and the coding fraction difference at the different population sizes.
 
-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.   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.
+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.
 
 
 
@@ -871,7 +878,7 @@ Left columns shows results for the difference in coding fraction, right column s
 \begin{figure}
 \centering
  \includegraphics[width=0.49\linewidth]{img/sigma/0_300/scatter_and_fits_contrast_firing_rate_contrast}
- \caption{No correlation between firing rate and contrast.}
+ \caption{Firing rate of the recorded cells as a function of signal contrast. The signal used was the broandband signal with 300Hz cutoff frequency. The red line shows a linear regression through the points. No correlation between firing rate and contrast is observed.}
  \label{contrast_firing_rate}
 \end{figure}