benda/susceptibility1
1
0
Fork 0
Code Issues Pull Requests Actions Packages Projects Releases Wiki Activity

figure captions

Browse Source
This commit is contained in:
Jan Benda 2024-06-07 17:12:38 +02:00
parent 7206b6be07
commit 42174e8d29
1 changed files with 11 additions and 22 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

33
susceptibility1.tex
View File

@ -405,7 +405,7 @@ Neuronal processing is inherently nonlinear --- spiking thresholds or rectificat
\begin{figure*}[t]
\includegraphics[width=\columnwidth]{plot_chi2}
\caption{\label{fig:lifresponse} First- (linear) and second-order response functions of the leaky integrate-and-fire model. \figitem{A} Magnitude of the first-order response function $|\chi_1(f_1)|$, also known as ``gain'' quantifies the response amplitude relative to the stimulus amplitude, both measured at the stimulus frequency. \figitem{B} Magnitude of the second-order response function $|\chi_2(f_, f_2)|$ quantifies the response at the sum of two stimulus frequencies. For linear systems, the second-order response functions is zero, because linear systems do not create new frequencies and thus there is no response at the summed frequencies. The plots show the analytical solutions from \cite{Lindner2001} and \cite{Voronenko2017} with $\mu = 1.1$ and $D = 0.001$.}
\caption{\label{fig:lifresponse} First- (linear) and second-order response functions of the leaky integrate-and-fire model. \figitem{A} Magnitude of the first-order response function $|\chi_1(f_1)|$, also known as ``gain'' quantifies the response amplitude relative to the stimulus amplitude, both measured at the stimulus frequency. \figitem{B} Magnitude of the second-order response function $|\chi_2(f_1, f_2)|$ quantifies the response at the sum of two stimulus frequencies. For linear systems, the second-order response functions is zero, because linear systems do not create new frequencies and thus there is no response at the summed frequencies. The plots show the analytical solutions from \cite{Lindner2001} and \cite{Voronenko2017} with $\mu = 1.1$ and $D = 0.001$.}
\end{figure*}
Nonlinear processes are key to neuronal information processing. Decision making is a fundamentally nonlinear process on the systemic level. On the cellular level, spiking neurons are inherently nonlinear: whether an action potential is elicited depends on the membrane potential to exceed a threshold\cite{Hodgkin1952,Koch1995}. Because of such nonlinearities, understanding and predicting neuronal responses to sensory stimuli is in general a difficult task.
@ -480,7 +480,7 @@ In the example recordings shown above (\figsrefb{fig:cells_suscept} and \fref{fi
\begin{figure*}[t]
\includegraphics[width=\columnwidth]{model_and_data}
\caption{\label{model_and_data} Estimation of second-order susceptibilities in the limit of weak stimuli. \figitem{A} \suscept{} estimated from $N=11$ trials of an electrophysiological recording of another low-CV P-unit (cell 2012-07-03-ak) driven with a weak RAM stimulus. \notejb{The stimulus strength was with 2.5\,\% contrast in the electrophysiologically recorded P-unit. The contrast for the P-unit model was with 0.009\,\%, thus reproducing the CV of the electrophysiolgoical recorded P-unit during the 2.5\,\% contrast stimulation (see table~\ref{modelparams} for model parameters).} Pink edges mark baseline firing rate where enhanced nonlinear responses are expected. \figitem[i]{B} \textit{Standard condition} of model simulations with intrinsic noise (bottom) and a RAM stimulus (top). \figitem[ii]{B} \suscept{} estimated from simulations of the cell's LIF model counterpart (cell 2012-07-03-ak, table~\ref{modelparams}) based on the same number of trials as in the electrophysiological recording. \figitem[iii]{B} Same as \panel[ii]{B} but using $10^6$ stimulus repetitions. \figitem[i-iii]{C} Same as in \panel[i-iii]{B} but in the \textit{noise split} condition: there is no external RAM signal driving the model. Instead, a large part (90\,\%) of the total intrinsic noise is treated as signal and is presented as an equivalent amplitude modulation (\signalnoise, center), while the intrinsic noise is reduced to 10\,\% of its original strength (see methods for details). In addition to one million trials, this reveals the full expected structure of the second-order susceptibility.}
\caption{\label{model_and_data} Estimation of second-order susceptibilities in the limit of weak stimuli. \figitem{A} \suscept{} estimated from $N=11$ trials \notejb{of duration XXX?} of an electrophysiological recording of another low-CV P-unit (cell 2012-07-03-ak, $\fbase=120$\,Hz, CV=0.20) driven with a weak RAM stimulus with contrast 2.5\,\%. Pink edges mark baseline firing rate where enhanced nonlinear responses are expected. \figitem[i]{B} \textit{Standard condition} of model simulations with intrinsic noise (bottom) and a RAM stimulus (top). \notejb{Since the model overestimated the sensitivity of the real P-unit, we adjusted the RAM contrast to 0.009\,\%, such that the resulting spike trains had the same CV as the electrophysiolgical recorded P-unit during the 2.5\,\% contrast stimulation (see table~\ref{modelparams} for model parameters).}\notejb{Specify fbase and baseline CV} \figitem[ii]{B} \suscept{} estimated from simulations of the cell's LIF model counterpart (cell 2012-07-03-ak, table~\ref{modelparams}) based on the same number of trials as in the electrophysiological recording. \figitem[iii]{B} Same as \panel[ii]{B} but using $10^6$ stimulus repetitions. \figitem[i-iii]{C} Same as in \panel[i-iii]{B} but in the \textit{noise split} condition: there is no external RAM signal driving the model. Instead, a large part (90\,\%) of the total intrinsic noise is treated as signal and is presented as an equivalent amplitude modulation (\signalnoise, center), while the intrinsic noise is reduced to 10\,\% of its original strength (see methods for details). In addition to one million trials, this reveals the full expected structure of the second-order susceptibility.}
\end{figure*}
One reason could be simply too little data for a good estimate of the second-order susceptibility. Electrophysiological recordings are limited in time, and therefore responses to only a limited number of trials, i.e. repetitions of the same RAM stimulus, are available. As a consequence, the cross-spectra, \Eqnref{eq:crosshigh}, are insufficiently averaged and the full structure of the second-order susceptibility might be hidden in finite-data noise. This experimental limitation can be overcome by using a computational model for the P-unit, a stochastic leaky integrate-and-fire model with adaptation current and dendritic preprocessing, and parameters fitted to the experimentally recorded P-unit (\figrefb{flowchart}) \cite{Barayeu2023}. The model faithfully reproduces the second-order susceptibility of a low-CV cell estimated from the same low number of trials as in the experiment ($\n{}=11$, compare \panel{A} and \panel[ii]{B} in \figrefb{model_and_data}).
@ -769,7 +769,7 @@ Whenever the membrane voltage $V_m(t)$ crossed the spiking threshold $\theta=1$
\begin{figure*}[t]
\includegraphics[width=\columnwidth]{flowchart}
\caption{\label{flowchart}
Flowchart of a LIF P-unit model with EOD carrier. The main steps of the model are depicted in the left model column (Model cell identifier 2012-07-03-ak, see table~\ref{modelparams} for model parameters). The three other columns show the relevant signals in three different settings. (i) the baseline situation, no external stimulus, only the animal's self-generated EOD (i.e. the carrier) is present (ii) RAM stimulation, the carrier is amplitude modulated with a weak (2\,\% contrast) stimulus, (iii) Noise split condition in which 90\,\% of the internal noise is used as a driving stimulus scaled with the correction factor $\rho$ (see text). Note: that the firing rate and the CV of the ISI distribution is the same in this and the baseline condition. \figitem{A} Thresholding: a simple linear threshold was applied to the EOD carrier, \Eqnref{eq:eod}. The red line on top depicts the amplitude modulation (AM). \figitem{B} Dendritic low-pass filtering attenuates the carrier. \figitem{C} A Gaussian noise is added to the signal in B. Note the reduced internal noise amplitude in the noise split (iii) condition. \figitem{D} Spiking output of the LIF model in response to the addition of B and C. \figitem{E} Power spectra of the LIF neuron's spiking activity. Under the baseline condition (\panel[i]{E}) there are several peaks, from left to right, at the baseline firing rate $\fbase{}$, $f_{EOD} - \fbase{}$ $f_{EOD}$, and $f_{EOD} + \fbase{}$. In the stimulus driven regime (\panel[ii]{E}), there is only a peak at \feod, while under the noise split condition (\panel[iii]{E}) again all peaks are present.}
Components of the P-unit model. The main steps of the model are illustrated in the left column. The three other columns show the corresponding signals in three different settings: (i) the baseline situation, no external stimulus, only the fish's self-generated EOD (i.e. the carrier) is present. (ii) RAM stimulation, the carrier is amplitude modulated with a weak (2\,\% contrast) nad-limited white-noise stimulus. (iii) Noise split condition in which 90\,\% of the internal noise is used as a driving RAM stimulus scaled with the correction factor $\rho$ (see text). Note that the mean firing rate and the CV of the ISI distribution is the same in this and the baseline condition. As an example, simulations of the model for cell ``2012-07-03-ak'' are shown (see table~\ref{modelparams} for model parameters). \figitem{A} Thresholding: a simple linear threshold was applied to the EOD carrier, \Eqnref{eq:eod}. \figitem{B} Subsequent dendritic low-pass filtering attenuates the carrier and carves out the AM signal. \figitem{C} Gaussian white-noise is added to the signal in \panel{B}. Note the reduced internal noise amplitude in the noise split (iii) condition. \figitem{D} Spiking output of the LIF model in response to the sum of B and C. \figitem{E} Power spectra of the LIF neuron's spiking activity. Under the baseline condition (\panel[i]{E}) there are several peaks, from left to right, at the baseline firing rate $\fbase{}$, $f_{EOD} - \fbase{}$, $f_{EOD}$, and $f_{EOD} + \fbase{}$. In the stimulus driven regime (\panel[ii]{E}), there is only a peak at \feod, while under the noise split condition (\panel[iii]{E}) the same peaks as in the baseline condition are present.}
\end{figure*}
\subsection{Numerical implementation}
@ -873,37 +873,26 @@ In the here used model a small portion of the original noise was assigned to the
\section{Supporting information}
%\subsection{High-CV P-units do not exhibit increased nonlinear interactions at \fsum}
\paragraph*{S1 Second-order susceptibility of high-CV P-unit}
\subsection{S1 Second-order susceptibility of high-CV P-unit}
CVs in P-units can range up to 1.5 \cite{Grewe2017, Hladnik2023}. We show the same analysis as in \figrefb{fig:cells_suscept} for an example higher-CV P-unit. Similar to the low-CV cell, high-CV P-units fire at multiples of the EOD period (\subfigrefb{fig:cells_suscept_high_CV}{A}). In contrast to low-CV P-units, however, the higher CV characterizes the noisier baseline firing pattern and the peak at \fbase{} is less pronounced in the power spectrum of the baseline activity (\subfigrefb{fig:cells_suscept_high_CV}{B}). High-CV P-units do not exhibit a clear nonlinear structure related to \fbase{} neither in the second-order susceptibility matrices (\subfigrefb{fig:cells_suscept_high_CV}{E--F}), nor in the projected diagonals (\subfigrefb{fig:cells_suscept_high_CV}{G}). The overall level of nonlinearity, however shows the same dependence on the stimulus contrast. It is much reduced for high-contrast stimuli that drive the neuron much stronger (\subfigrefb{fig:cells_suscept_high_CV}{F}).
\label{S1:highcvpunit}
\begin{figure*}[!ht]
\includegraphics[width=\columnwidth]{cells_suscept_high_CV}
\caption{\label{fig:cells_suscept_high_CV} Response of experimentally measured P-units (cell identifier ``2018-08-24-af") to RAM stimuli. Light purple -- low RAM contrast. Dark purple -- high RAM contrast. Noisy high-CV P-Unit. \figitem{A} Interspike intervals (ISI) distribution during baseline. \figitem{B} Baseline power spectrum of the firing rate. \figitem{C} Top: EOD carrier (gray) with RAM (red). Middle: Spike trains in response to a low RAM contrast. Bottom: Spike trains in response to a high RAM contrast. \figitem{D} First-order susceptibility (see \Eqnref{linearencoding_methods}). \figitem{E} Absolute value of the second-order susceptibility, \Eqnref{eq:susceptibility}, for the low RAM contrast.
Pink lines -- edges of the structure when \fone, \ftwo{} or \fsum{} are equal to \fbase{}. \figitem{F} Absolute value of the second-order susceptibility for the higher RAM contrast. \figitem{G} Projected diagonals, calculated as the mean of the anti-diagonals of the matrices in \panel{E--F}. Gray dots: \fbase{}. Dashed lines: Medians of the projected diagonals.
}
\caption{\label{fig:cells_suscept_high_CV} Response of experimentally measured noisy P-units (cell identifier ``2018-08-24-af") with a relatively high CV of 0.34 to RAM stimuli with two different contrasts. \figitem{A} Interspike intervals (ISI) distribution during baseline. \figitem{B} Baseline power spectrum. \figitem{C} Top: EOD carrier (gray) with RAM (red). Center: Spike trains in response to the 5\,\% RAM contrast. Bottom: Spike trains in response to the 10\,\% RAM contrast. \figitem{D} First-order susceptibility (\Eqnref{linearencoding_methods}). \figitem{E} Absolute value $|\chi_2(f_1, f_2)|$ of the second-order susceptibility, \Eqnref{eq:susceptibility}, for the 5\,\% RAM contrast. Pink lines -- edges of the structure when \fone, \ftwo{} or \fsum{} are equal to \fbase{}. \figitem{F} $|\chi_2(f_1, f_2)|$ for the 10\,\% RAM contrast. \figitem{G} Projected diagonals, calculated as the mean of the anti-diagonals of the matrices in \panel{E--F}. Gray dots: \fbase{}. Dashed lines: Medians of the projected diagonals.}
\end{figure*}
\begin{figure*}[hp]%hp!
\includegraphics[width=\columnwidth]{trialnr}
\caption{\label{fig:trialnr} Saturation of the second-order susceptibility depending on the stimulus repetition number $\n{}$. \figitem{A} Gray line -- 99.9th percentile of the second-order susceptibility matrix. \figitem{B} Gray line -- 10th percentile of the second-order susceptibility matrix.
\includegraphics[width=\columnwidth]{trialnr}
\notejb{ylabel: absolute value of chi_2!}
\notejb{sort the legend labels!}
\notejb{one of the 99.99\% percentiles is enough}
\notejb{add 10^7 tick on x-axis}
\caption{\label{fig:trialnr} Dependence of the estimate of the second-order susceptibility on the number of trials $\n$. While the estimate of the noise floor (10th and 90th percentile) of the $|\chi_2(f_1, f_2)|$ matrix does not saturate yet, the estimates of the high values in the matrix that make up the characteristic ridges saturate for $N>10^6$.
}
\end{figure*}
% \bibliographystyle{iscience}
%\bibliographystyle{apalike}%alpha}%}%alpha}%apalike}
%\bibliographystyle{elsarticle-num-names}%elsarticle-num-names}
%\ExecuteBibliographyOptions{sorting=nty}
%\bibliographystyle{authordate2}
%\bibliography{journalsabbrv,references}
%\begin{thebibliography}{00}
\end{document}