\section*{Electric fish as a real world model system}

To put the results from our simulations into a real world context, we chose the 
weakly electric fish \textit{Apteronotus leptorhynchus} as a model system.
\lepto\ uses an electric organ to produce electric fields which it
uses for orientation, prey detection and communication. Distributed over the skin
of \lepto\ are electroreceptors which produce action potentials
in response to electric signals. 

These receptor cells ("p-units") are analogous to the 
simulated neurons we used in our simulations because they do not receive any
input other than the signal they are encoding. Individual cells fire independently
of each other and there is no feedback. 




\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}


\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}

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.