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Moved around and expanded experimental results/coherence measurements

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Dennis Huben 2024-09-04 16:32:06 +02:00
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main.tex
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@ -99,7 +99,7 @@ If noise is weak, the neurons behave regularly and similar to each other (A). Fo
\label{example_spiketrains}
\end{figure}
\subsection{Analysis}
\subsection{Analysis} \label{Analysis}
For each combination of parameters, a histogram of the output spikes from all neurons or a subset of the neurons was created.
The coherence $C(f)$ was calculated \citep{lindner2016mechanisms} in frequency space as the fraction between the squared cross-spectral density $|S_{sx}^2|$ of input signal $s(t) = \sigma I_{t}$ and output spikes x(t), $S_{sx}(f) = \mathcal{F}\{ s(t)*x(t) \}(f) $, divided by the product of the power spectral densities of input ($S_{ss}(f) = |\mathcal{F}\{s(t)\}(f)|^2 $) and output ($S_{xx}(f) = |\mathcal{F}\{x(t)\}(f)|^2$), where $\mathcal{F}\{ g(t) \}(f)$ is the Fourier transform of g(t).
\begin{equation}C(f) = \frac{|S_{sx}(f)|^2}{S_{ss}(f) S_{xx}(f)}\label{coherence}\end{equation}
@ -555,8 +555,33 @@ stimulus generation and attenuation, as well as preanalysis of the
data were performed online during the experiment within the
RELACS software version 0.9.7 using the efish plugin-set (by J.
Benda: http://www.relacs.net).\footnote{From St\"ockl et al. 2014}
Recordings were carried out by \notedh{insert names here}
\textit{Stimulation.} White noise stimuli with a cutoff frequency of 300{\hertz} defined an AM of the fish's signal. The stimulus was combined with the fish's own EOD in a way that the desired AM could be measured near the fish. Amplitude of the AM was 10\% (?) of the amplitude of the EOD. Stimulus duration was between 2s and 10s, with a time resolution of X.
We also used 5 narrowband stimuli in the frequency ranges 0-50Hz, 50-100Hz, 150-200Hz, 250-300Hz and 350-400Hz. Parameters for these were ... \notedh{fill information}
\subsection{Analysis}
To analyse the data we went ahead the same way we did for the simulations. For more information see section \ref{Analysis}.
We created populations out of each cell. For each p-unit, we took the different trials and added the spikes in each time bin the same way we did it for the simulated neurons. \notedh{Did I do something to build averages for smaller population sizes?} For most of the analysed cells there were between X and Y \notedh{fill information in} trials.
\subsection{Frequency dependence of sensory cells in \lepto}
\begin{figure}
\centering
\includegraphics[width=0.49\linewidth]{img/fish/coherence_example.pdf}
\includegraphics[width=0.49\linewidth]{img/fish/coherence_example_narrow.pdf}
\label{fig:ex_data}
\caption{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.
Left: One signal with a maximum frequency of 300Hz. Right: Three different signals (0-50Hz, 150-200Hz and 350-400Hz). With increasing population size the increase in coherence was especially noticeable for the higher frequency ranges. See also figure \ref{fish_result_summary_yue} b). \notedh{Show a different cell with all five narrowband signals?}
}
\end{figure}
\subsection{How to determine noisiness}
@ -607,9 +632,6 @@ To confirm that the $\sigma$ parameter estimated from the fit is indeed a good m
\begin{figure}
\centering
%\includegraphics[width=0.45\linewidth]{img/ordnung/base_D_sigma}\\
\includegraphics[width=0.23\linewidth]{img/simulation_sigma_examples/fitcurve_50hz_0.0002noi500s_0_capped.pdf}
\includegraphics[width=0.23\linewidth]{img/simulation_sigma_examples/fitcurve_50hz_0.001noi500s_0_capped.pdf}
\includegraphics[width=0.23\linewidth]{img/simulation_sigma_examples/fitcurve_50hz_0.1noi500s_0_capped.pdf}
\includegraphics[width=0.23\linewidth]{img/ISI_explanation.pdf}
\includegraphics[width=0.23\linewidth]{img/cv_psth_sigma_compare/dataframe_scatter_labels_D_sigma_membrane_50.pdf}
\includegraphics[width=0.23\linewidth]{img/cv_psth_sigma_compare/dataframe_scatter_labels_D_cv_membrane_50.pdf}
@ -650,6 +672,9 @@ We can see from figure \ref{sigmafits_example} that the fits look very close to
\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}
\includegraphics[width=0.23\linewidth]{img/simulation_sigma_examples/fitcurve_50hz_0.0002noi500s_0_capped.pdf}
\includegraphics[width=0.23\linewidth]{img/simulation_sigma_examples/fitcurve_50hz_0.001noi500s_0_capped.pdf}
\includegraphics[width=0.23\linewidth]{img/simulation_sigma_examples/fitcurve_50hz_0.1noi500s_0_capped.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. 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 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?}}
\label{sigmafits_example}
@ -658,16 +683,12 @@ We can see from figure \ref{sigmafits_example} that the fits look very close to
\begin{figure}
\includegraphics[width=0.65\linewidth]{img/sigma/cf_N_sigma.pdf}
\caption{Firing rate as a function of signal strength. Examples from experimental data.}
\caption{Coding fraction as a function of population size. Each line represents one cell. Populations were created by taking seperate trials from each cell. \notedh{What are they sorted/divided by?}}
\label{Curve_examples}
\end{figure}
%TODO insert plot with sigma x-axis and delta_cf on y-axis here; also, plot with sigma as function of firing rate, also absoulte cf for different population size as function of sigma.
When we group neurons by their noise and plot coding fraction as a function of population size for averages of the groups, we see results similar to what we see for simulations.
Noisier cells have a lower coding fraction for small populations. For increasing population size, coding fraction increases for all groups, but the increase is much larger for noisy cells. For large population sizes the noisy cells show a better linear encoding of the signal than the more regular cells.
\begin{figure}
\includegraphics[width=0.45\linewidth]{img/ordnung/sigma_popsize_curves_0to300}
\caption{Left: Coding fraction as a function of population size for all recorded neurons. Color are by \(\sigma\) from the fit of the function in equation \ref{errorfct}, so that there are roughly an equal number of neurons in each category. Red: \(\sigma = \) 0 to 0.5, pink: 0.5 to 1.0, purple: 1.0 to 1.5, blue: 1.5 and above. Thick colored lines are average of the neurons in each group. For a population size of 1, coding fraction descreases on average with increasing \(\sigma\). As population sizes increase, coding fraction for weak noise neurons quickly stops increasing. Strong noise neurons show better coding performance for larger popuation sizes (about 8 to 32 neurons). Right [missing]: Increase in coding as a function of sigma. y-axis shows the difference in coding fraction between N=1 and N=32,}
@ -675,13 +696,6 @@ Noisier cells have a lower coding fraction for small populations. For increasing
\end{figure}
\subsection*{Electrophysiology}
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
\begin{figure}
\centering
\includegraphics[width=0.4\linewidth]{img/sigma/example_spikes_sigma.pdf}
@ -725,7 +739,7 @@ Indeed, if results are not averaged and single cells are considered, we find tha
\subsection*{Results}
Figure \ref{fig:ex_data} A,B and C show three examples for coherence from intracellular
Figure \ref{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
@ -739,24 +753,7 @@ and \ref{fig:popsizenarrow10} C), the ratio of coding fraction in a large popula
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}
@ -813,13 +810,14 @@ to the coding fraction in a single cell is larger for higher frequencies.
\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}
\label{fish_result_summary_yue}
\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 $quot_cf$.}
\caption{This is about frequency and how it determines $delta_cf$. In other paper I have used $quot_cf$. \notedh{The x-axis labels don't make sense to me.}}
\end{figure}
\subsection{Discussion}