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This happens at a point where $\sigma$ becomes large. Therefore, we excluded all values of the unit-less \(\sigma\) larger than two from the following analyses. - - -\subsection*{Methodology} - -The signal was binned according to its amplitude. The result is a discrete Gaussian distribution around 0mV, the mean of the signal, as is expected from the way the signal was created. -After accounting for time delays in signal processing, we make a histogram which contains the distribution of spikes according to signal amplitude. - This histogram is then normalized by the distribution of the signal. -The result is another histogram, where values are firing frequencies for each signal value. Because those frequencies are just firing probabilities multiplied with equal time steps, we can fit a Gaussian error function to those probabilities. - -\subsection*{Simulation} - -To confirm that the $\sigma$ parameter estimated from the fit is indeed a good measure for the noisiness, we validated it against D, the noise parameter from the simulations. We find that there is a strictly monotonous relationship between the two for different sets of simulation parameters. -%Other parameters often used to determine noisiness (citations) such as the variance of the spike PSTH, the coefficient of variation (CV) of the interspike interval are not as useful. In figure \ref{noiseparameters} we see why. The variance of the psth is not always monotonous in D and is very flat for low values of D. -%describe what happens to the others -%check Fano-factor maybe? - - - -We tried several different bin sizes (30 to 300 bins) and spike widths. There was little difference between the different parameters (see appendix). - - -\section*{-----------------------} -%We can use $\sigma$ instead of D*firing_rate: $\sigma$ makes it ind. of fr!! - \subsection*{Electrophysiology} -We find that the fits match the experimental data very well (figure \ref{sigmafits}). For very weak and very strong inputs, the firing rates themselves become noisy, because the signal only assumes those values rarely. This is especially noticeable for strong inputs, as there are more spikes there, and therefore large fluctuations, while there is very little spiking anyway for weak inputs. +We find that the fitted plots match the experimental data very well (figure \ref{sigmafits}). For very weak and very strong inputs, the firing rates themselves become noisy, because the signal only assumes those values rarely. This is especially noticeable for strong inputs, as there are more spikes there, and therefore large fluctuations, while there is very little spiking anyway for weak inputs. % fish_raster.py on oilbird for the eventplot % instructions.txt enth\"alt python-Befehle um Verteilungen und scatter zu rekonstruieren @@ -747,17 +720,121 @@ Indeed, if results are not averaged and single cells are considered, we find tha \label{ephys_sigma} \end{figure} +%The value of $\sigma$ is not signal independent. The same cell can have different values for $\sigma$ for different input signals. - -%The value of $\sigma$ is not signal independent. The same cell can have different values for $\sigma$ for different input signals. +\subsection*{Results} + +Figure \ref{fig:ex_data} A,B and C show three examples for coherence from intracellular +measurements in \lepto\. Each cell was exposed to up to 128 repetitions of the +same signal. The response was then averaged over different numbers of trials to +simulate different population sizes of homogeneous cells. We can see that an increase +in population size leads to higher coherence. Similar to what we saw in the simulations, +around the average firing rate of the cell (marked by the red vertical lines), coherence +decreases sharply. We then aggregated the results for 31 different cells (50 experiments total, +as some cells were presented with the stimulus more than once). +Figure \ref{ex_data} D shows that the increase is largest inside the +high frequency intervals. As we could see in our simulations (figures \ref{fig:popsizenarrow15} C +and \ref{fig:popsizenarrow10} C), the ratio of coding fraction in a large population +to the coding fraction in a single cell is larger for higher frequencies. +%simulation plots are from 200hz/nice coherence curves.ipynb +\begin{figure} + \centering + \includegraphics[width=0.49\linewidth]{img/fish/coherence_example.pdf} + \includegraphics[width=0.49\linewidth]{img/fish/coherence_example_narrow.pdf} + \includegraphics[width=0.49\linewidth]{{img/coherence/broad_coherence_15.0_1.0_different_popsizes_0.001}.pdf} + \includegraphics[width=0.49\linewidth]{{img/coherence/coherence_15.0_0.5_narrow_both_different_popsizes_0.001}.pdf} + \label{fig:ex_data} + \caption{A,B,C: examples of coherence in the p-Units of \lepto. Each plot shows + the coherence of the response of a single cell to a stimulus for different numbers of trials. + Like in the simulations, increased population sizes lead to a higher coherence. + D: Encoding of higher frequency intervals profits more from an increase in + population size than encoding of lower frequency intervals. + The ratio of the coding fraction for the largest number of trials divided by + the coding fraction for a single trial for each of six different frequency + intervals. Shown here are the data for all 50 experiments (31 different cells). + The orange line signifies the median value for all cells. The box + extends over the 2nd and 3rd quartile. } +\end{figure} -\input{sigma} -\subsection{Results} +\begin{figure} + \centering + broad + + \includegraphics[width=0.48\linewidth]{img/fish/cf_curves/cfN_broad_0.pdf} + \includegraphics[width=0.48\linewidth]{img/fish/cf_curves/cfN_broad_1.pdf} + \includegraphics[width=0.48\linewidth]{img/fish/cf_curves/cfN_broad_2.pdf} + \includegraphics[width=0.48\linewidth]{img/fish/cf_curves/cfN_broad_3.pdf} +\end{figure} + +%box_script.py, quot_sigma() und quot_sigma_narrow() +\begin{figure} + \centering + broad + + \includegraphics[width=0.16\linewidth]{img/fish/scatter/sigma_cf_quot_broad_0_50.pdf} + \includegraphics[width=0.16\linewidth]{img/fish/scatter/sigma_cf_quot_broad_50_100.pdf} + \includegraphics[width=0.16\linewidth]{img/fish/scatter/sigma_cf_quot_broad_100_150.pdf} + \includegraphics[width=0.16\linewidth]{img/fish/scatter/sigma_cf_quot_broad_150_200.pdf} + \includegraphics[width=0.16\linewidth]{img/fish/scatter/sigma_cf_quot_broad_200_250.pdf} + \includegraphics[width=0.16\linewidth]{img/fish/scatter/sigma_cf_quot_broad_250_300.pdf} + + + \includegraphics[width=0.16\linewidth]{img/fish/scatter/check_fr_quot_broad_0_50.pdf} + \includegraphics[width=0.16\linewidth]{img/fish/scatter/check_fr_quot_broad_50_100.pdf} + \includegraphics[width=0.16\linewidth]{img/fish/scatter/check_fr_quot_broad_100_150.pdf} + \includegraphics[width=0.16\linewidth]{img/fish/scatter/check_fr_quot_broad_150_200.pdf} + \includegraphics[width=0.16\linewidth]{img/fish/scatter/check_fr_quot_broad_200_250.pdf} + \includegraphics[width=0.16\linewidth]{img/fish/scatter/check_fr_quot_broad_250_300.pdf} + + narrow + + \includegraphics[width=0.16\linewidth]{img/fish/scatter/sigma_cf_quot_narrow_0_50.pdf} + \includegraphics[width=0.16\linewidth]{img/fish/scatter/sigma_cf_quot_narrow_50_100.pdf} + \includegraphics[width=0.16\linewidth]{img/fish/scatter/sigma_cf_quot_narrow_150_200.pdf} + \includegraphics[width=0.16\linewidth]{img/fish/scatter/sigma_cf_quot_narrow_250_300.pdf} + \includegraphics[width=0.16\linewidth]{img/fish/scatter/sigma_cf_quot_narrow_350_400.pdf} + + \includegraphics[width=0.16\linewidth]{img/fish/scatter/check_fr_quot_narrow_0_50.pdf} + \includegraphics[width=0.16\linewidth]{img/fish/scatter/check_fr_quot_narrow_50_100.pdf} + \includegraphics[width=0.16\linewidth]{img/fish/scatter/check_fr_quot_narrow_150_200.pdf} + \includegraphics[width=0.16\linewidth]{img/fish/scatter/check_fr_quot_narrow_250_300.pdf} + \includegraphics[width=0.16\linewidth]{img/fish/scatter/check_fr_quot_narrow_350_400.pdf} + +\end{figure} + + + +\begin{figure} +\centering +\includegraphics[width=0.4\linewidth]{img/fish/diff_box.pdf} +\includegraphics[width=0.4\linewidth]{img/fish/diff_box_narrow.pdf} + \includegraphics[width=0.4\linewidth]{img/relative_coding_fractions_box.pdf} + \notedh{needs figure 3.6 from yue and equivalent} +\end{figure} + +\begin{figure} + \includegraphics[width=0.49\linewidth]{img/fish/ratio_narrow.pdf} + \includegraphics[width=0.49\linewidth]{img/fish/broad_ratio.pdf} + \label{freq_delta_cf} + \caption{This is about frequency and how it determines $delta_cf$. In other paper I have used $quot_cf$.} +\end{figure} + +\subsection{Discussion} -\input{fish_bands} +We also confirmed that the results from the theory part of the paper play a role in a +real world example. Inside the brain of the weakly electric fish +\textit{Apteronotus leptorhynchus} pyramidal cells in different areas +are responsible for encoding different frequencies. In each of those areas, +cells integrate over different numbers of the same receptor cells. +Artificial populations consisting of different trials of the same receptor cell +show what we have seen in our simulations: Larger populations help +especially with the encoding of high frequency signals. These results +are in line with what is known about the pyramidal cells of \lepto: +The cells which encode high frequency signals best are the cells which +integrate over the largest number of neurons. \section{Discussion: Combining experiment and simulation}