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@ -497,70 +497,6 @@ We also find all other results replicated even with refractory periods of 1ms or
We recorded electrophysiological data from X cells from Y different fish.
\textit{Surgery}. Twenty-two E. virescens (10 to 21 cm) were used for
single-unit recordings. Recordings of electroreceptors were made
from the anterior part of the lateral line nerve.
Fish were initially anesthetized with 150 mg/l MS-222 (PharmaQ,
Fordingbridge, UK) until gill movements ceased and were then
respirated with a constant flow of water through a mouth tube,
containing 120 mg/l MS-222 during the surgery to sustain anesthesia.
The lateral line nerve was exposed dorsal to the operculum. Fish were
fixed in the setup with a plastic rod glued to the exposed skull bone.
The wounds were locally anesthetized with Lidocainehydrochloride
2\% (bela-pharm, Vechta, Germany) before the nerve was exposed.
Local anesthesia was renewed every 2 h by careful application of
Lidocaine to the skin surrounding the wound.
After surgery, fish were immobilized with 0.05 ml 5 mg/ml tubocurarine (Sigma-Aldrich, Steinheim, Germany) injected into the trunk
muscles.
\sout{Since tubocurarine suppresses all muscular activity, it also
suppresses the activity of the electrocytes of the electric organ and thus
strongly reduces the EOD of the fish. We therefore mimicked the EOD
by a sinusoidal signal provided by a sine-wave generator (Hameg HMF
2525; Hameg Instruments, Mainhausen, Germany) via silver electrodes
in the mouth tube and at the tail. The amplitude and frequency of the
artificial field were adjusted to the fish’s own field as measured before
surgery.} After surgery, fish were transferred into the recording tank of the
setup filled with water from the fish’s housing tank not containing
MS-222. Respiration was continued without anesthesia. The animals
were submerged into the water so that the exposed nerve was just above
the water surface. Electroreceptors located on the parts above water
surface did not respond to the stimulus and were excluded from analysis.
Water temperature was kept at 26°C.\footnote{From St\"ockl et al. 2014}
\textit{Recording. }Action potentials from electroreceptor afferents were
recorded intracellularly with sharp borosilicate microelectrodes
(GB150F-8P; Science Products, Hofheim, Germany), pulled to a resistance between 20 and 100 M and filled with a 1 M KCl solution.
Electrodes were positioned by microdrives (Luigs-Neumann, Ratingen,
Germany). As a reference, glass microelectrodes were used. They were
placed in the tissue surrounding the nerve, adjusted to the isopotential line
of the recording electrode. The potential between the micropipette and the
reference electrode was amplified (SEC-05X; npi electronic) and lowpass filtered at 10 kHz. Signals were digitized by a data acquisition board
(PCI-6229; National Instruments) at a sampling rate of 20 kHz. Spikes
were detected and identified online based on the peak-detection algorithm
proposed by Todd and Andrews (1999).
The EOD of the fish was measured between the head and tail via
two carbon rod electrodes (11 cm long, 8-mm diameter). The potential
at the skin of the fish was recorded by a pair of silver wires, spaced
1 cm apart, which were placed orthogonal to the side of the fish at
two-thirds body length. The residual EOD potentials were recorded
and monitored with a pair of silver wire electrodes placed in a piece
of tube that was put over the tip of the tail. These EOD voltages were
amplified by a factor of 1,000 and band-pass filtered between 3 Hz and
1.5 kHz (DPA-2FXM; npi electronics).
Stimuli were attenuated (ATN-01M; npi electronics), isolated from
ground (ISO-02V; npi electronics), and delivered by two carbon rod
electrodes (30-cm length, 8-mm diameter) placed on either side of the
fish parallel to its longitudinal axis. Stimuli were calibrated to evoke
defined AM measured close to the fish. Spike and EOD detection,
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}.
@ -581,14 +517,6 @@ We created populations out of each cell. For each p-unit, we took the different
}
\end{figure}
\begin{figure}
\includegraphics[width=0.65\linewidth]{img/sigma/cf_N_sigma.pdf}
\caption{Coding fraction as a function of population size. Each line represents one cell. Populations were created by taking separate trials from each cell. Line color indicates the firing rate of the cell. \notedh{What are they sorted/divided by? $\sigma$?}}
\label{Curve_examples}
\end{figure}
\subsection{How to determine noisiness}
@ -608,7 +536,7 @@ Stocks (2000) uses one such function to simulate groups of noisy spiking neurons
where $\sigma$ is the parameter quantifying the noise (figure \ref{Curve_examples}). $\sigma$ determines the steepness of the curve.
A neuron with a $\sigma$ of 0 would be a perfect thresholding mechanism. Firing probability for all inputs below the threshold is 0, and firing probability for all inputs above is 1.
Larger values mean a flatter activation curve. Neurons with such an activation curve can sometimes fire even for signals below the firing threshold, while it will sometimes not fire for inputs above the firing threshold. Its firing behaviour is influenced less by the signal and more affected by noise.
We also tried different other methods of quantifying noise commonly used (citations), but none of them worked as well as the errorfunction fit (fig. \ref{noiseparameters} and \ref{noiseparameters2}).
We also tried different other methods of quantifying noise commonly used (citations), but none of them worked as well as the errorfunction fit (fig. \ref{noiseparameters}. For example, the coefficient of variation (cv) which is commonly \notedh{add citations} used as a measure of noisiness does not only depend on the noisiness of the cell, but also on other cell parameters like the membrane constant or refractory periods. As
@ -623,19 +551,10 @@ This histogram is then normalized by the distribution of the signal. The result
\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.
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 (see figure \ref{noiseparameters}) 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?
\begin{figure}
\includegraphics[width=0.45\linewidth]{img/ordnung/base_D_sigma}
\includegraphics[width=0.45\linewidth]{img/dataframe_scatter_D_normalized_psth_1ms_test_tau}
\includegraphics[width=0.45\linewidth]{img/dataframe_scatter_D_psth_5ms_test}
% \includegraphics[width=0.45\linewidth]{img/dataframe_scatter_D_cv_test}
\caption{a)The parameter \(\sigma\) as a function of the noise parameter D in LIF-simulations. There is a strictly monotonous relationship between the two, which allows us to use \(\sigma\) as a susbtitute for D in the analysis of electrophysiological experiments. b-d) different other parameters commonly used to quantify noise. None of these functions is stricly monotonous and therefore none is useful as a substitute for D. b) Peri-stimulus time histogram (PSTH) of the spikes with a bin width of 1ms, normalized by c) PSTH of the spikes with a bin width of 5ms. d) coefficient of variation (cv) of the interspike-intervals.}
\label{noiseparameters}
\end{figure}
\begin{figure}
\centering
%\includegraphics[width=0.45\linewidth]{img/ordnung/base_D_sigma}\\
@ -651,7 +570,7 @@ To confirm that the $\sigma$ parameter estimated from the fit is indeed a good m
b-e) Left to right: $\sigma$, CV and standard deviation of the psth with two diffrent kernel widths as a function of D for different membrane constants (4ms, 10ms and 16ms). The membrane constant $\tau$ determines how quickly the voltage of a LIF-neuron changes, with lower constants meaning faster changes. Only $\sigma$ does not change its values with different $\tau$. The CV (c)) is not even monotonous in the case of a timeconstant of 4ms, ruling out any potential usefulness.
f-i) Left to right: $\sigma$, CV and standard deviation of the psth with two diffrent kernel widths as a function of D for different refractory periods (0ms, 1ms and 5ms). Only $\sigma$ does not change with different refractory periods.
}
\label{noiseparameters2}
\label{noiseparameters}
\end{figure}
We tried several different bin sizes (30 to 300 bins) and spike widths. There was little difference between the different parameters (see figure \ref{sigma_bins} in appendix).
@ -698,7 +617,7 @@ Figure \ref{fr_sigma} shows that between the firing rate and the cell and its no
\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_contrast_firing_rate_sigma.pdf}
\caption{Relationship between firing rate and $\sigma$ and cv respectively. Noisier tend to be cells that fire slower, but the relationship is very slight.}
\caption{Relationship between firing rate and $\sigma$ and cv respectively. Noisier tend to be cells that fire slower, but the relationship is very weak.}
\label{fr_sigma}
\end{figure}
@ -793,7 +712,7 @@ Qualitatively we see very similar results when instead of the broadband signal w
\label{overview_experiment_results_narrow}
\end{figure}
Figures \ref{increases_narow} and \ref{increases_narow_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.
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 $\sigma$.
%figures created with result_fits.py
\begin{figure}
@ -803,9 +722,9 @@ Figures \ref{increases_narow} and \ref{increases_narow_high} both show that the
\includegraphics[width=0.45\linewidth]{img/sigma/narrow_0_50/scatter_and_fits_sigma_diff_firing_rate}%
\includegraphics[width=0.45\linewidth]{img/sigma/narrow_0_50/scatter_and_fits_firing_rate_diff_contrast}%
\caption{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.
\caption{Narrowband signal with 0Hz-50Hz. 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.
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_narow}
\label{increases_narrow}
\end{figure}
@ -815,11 +734,12 @@ Bottom: Using the difference in coding fraction instead of the quotient makes th
\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{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.
\caption{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.
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_narow_high}
\label{increases_narrow_high}
\end{figure}
\notedh{link to the appropriate chapter from theory results}
@ -834,40 +754,12 @@ Similar results can be observed for the other frequency bands. \notedh{Images to
\includegraphics[width=0.45\linewidth]{img/sigma/0_50/scatter_and_fits_sigma_diff_firing_rate}%
\includegraphics[width=0.45\linewidth]{img/sigma/0_50/scatter_and_fits_firing_rate_diff_contrast}%
\caption{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.
\caption{Broadband signal, analyzed in the range 0Hz-50Hz. 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.
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.}
\label{increases_narow_in_broad}
\end{figure}
\subsection*{Results}
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
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
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}
%compare_params_300.py auf oilbird
\begin{figure}
\includegraphics[width=0.30\linewidth]{img/sigma/parameter_assessment/bins100v300.pdf}
@ -886,6 +778,7 @@ to the coding fraction in a single cell is larger for higher frequencies.
%compare_params.py auf oilbird
\begin{figure}
\centering
\includegraphics[width=0.45\linewidth]{img/sigma/parameter_assessment/gauss1v5_30.pdf}
\includegraphics[width=0.45\linewidth]{img/sigma/parameter_assessment/gauss1v5_50.pdf}
\includegraphics[width=0.45\linewidth]{img/sigma/parameter_assessment/gauss1v5_100.pdf}
@ -895,44 +788,6 @@ to the coding fraction in a single cell is larger for higher frequencies.
\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}
@ -963,8 +818,6 @@ 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}
\section{Literature}
\clearpage