From 073b7f8b61118722b3e0e12da46a8f09efd51992 Mon Sep 17 00:00:00 2001 From: saschuta <56550328+saschuta@users.noreply.github.com> Date: Mon, 4 Mar 2024 14:47:08 +0100 Subject: [PATCH] updated questions --- susceptibility1.tex | 9 ++++----- 1 file changed, 4 insertions(+), 5 deletions(-) diff --git a/susceptibility1.tex b/susceptibility1.tex index fb0a965..0128b69 100644 --- a/susceptibility1.tex +++ b/susceptibility1.tex @@ -477,7 +477,7 @@ Irrespective of the CV, neither cell shows the complete proposed structure of no % \subsection{Full nonlinear structure visible only in P-unit models} \subsection{Internal noise hides parts of the nonlinearity structure} -Traces of the proposed structure of second-order susceptibility are found in both ampullary and p-type electrosensory afferents. The nonlinerarity seems to depend on the CV, i.e. the level of intrinsic noise in the cells and in the next step we address whether the level of intrinsic masks the expected structures. One option to reduce the impact of this intrinsic noise is to average it out over many repetitions of the same stimulus. In the electrophysiological experiments we only have a limited recording duration and hence the number of stimulus repetitions is limited. To overcome these limitations we compared between an electrophysiologically recorded low-CV P-unit and its P-unit LIF model counterpart fitted to reproduce the P-unit behavior faithfully (see \figref{flowchart})\citealp{Barayeu2023}. In the recording depicted in \subfigrefb{model_and_data}{A} the cell was stimulated with a weak RAM stimulus with the contrast of 1\,$\%$ (red, \subfigrefb{model_and_data}\,\panel[i]{B}). The across trial average ($\n{}=11$) shows the diagonal band in second-order susceptibility at \fsumb{} (yellow diagonal in pink edges) and an additional nonlinear band appeared at \fsumehalf{} (\subfigrefb{model_and_data}{A}). The matched model reproduces the same diagonal at \fsumb{} (\subfigrefb{model_and_data}\,\panel[ii]{B}) but not the diagonal at \fsumehalf. By increasing the number of trials in the simulation, the signal-to-noise ratio and thus the estimation of the nonlinearity structures can be improved. Still, even with 1 million repetitions, no changes are observable in the nonlinearity structures (\subfigrefb{model_and_data}\,\panel[iii]{B}). The increased trial count, however goes along with generally smaller second-order susceptibility values (\subfigrefb{model_and_data}\,\panel[iii]{B}).\notejg{a problem, that we use a new noise for each trial?}\notesr{Using a new noise for each trial, is the way this method is defined. When using the same noise for one million repetiotions we will not see the triangular shape at any time. I tried this and Benjamin confirmed that this would be not possible.} +Traces of the proposed structure of second-order susceptibility are found in both ampullary and p-type electrosensory afferents. The nonlinerarity seems to depend on the CV, i.e. the level of intrinsic noise in the cells and in the next step we address whether the level of intrinsic masks the expected structures. One option to reduce the impact of this intrinsic noise is to average it out over many repetitions of the same stimulus. In the electrophysiological experiments we only have a limited recording duration and hence the number of stimulus repetitions is limited. To overcome these limitations we compared between an electrophysiologically recorded low-CV P-unit and its P-unit LIF model counterpart fitted to reproduce the P-unit behavior faithfully (see \figref{flowchart})\citealp{Barayeu2023}. In the recording depicted in \subfigrefb{model_and_data}{A} the cell was stimulated with a weak RAM stimulus with the contrast of 1\,$\%$ (red, \subfigrefb{model_and_data}\,\panel[i]{B}). The across trial average ($\n{}=11$) shows the diagonal band in second-order susceptibility at \fsumb{} (yellow diagonal in pink edges) and an additional nonlinear band appeared at \fsumehalf{} (\subfigrefb{model_and_data}{A}). The matched model reproduces the same diagonal at \fsumb{} (\subfigrefb{model_and_data}\,\panel[ii]{B}) but not the diagonal at \fsumehalf. By increasing the number of trials in the simulation, the signal-to-noise ratio and thus the estimation of the nonlinearity structures can be improved. Still, even with 1 million repetitions, no changes are observable in the nonlinearity structures (\subfigrefb{model_and_data}\,\panel[iii]{B}). The increased trial count, however goes along with generally smaller second-order susceptibility values (\subfigrefb{model_and_data}\,\panel[iii]{B}). %Note that this line doesn't appear in the susceptibility matrix of the model (\subfigrefb{model_and_data}{C})Each cell has an intrinsic noise level (\subfigrefb{model_and_data}{A}, bottom). %The signal component (purple) compensates for this total noise reduction, is not a weak signal anymore and @@ -487,7 +487,7 @@ Based on the Novikov-Furutsu Theorem \citealp{Novikov1965, Furutsu1963} the intr % Based on this a weak RAM signal as the input to the P-unit model (red in \subfigrefb{model_and_data}\,\panel[i]{A}) can be approximated by a model where no RAM stimulus is present (red) but instead the total noise of the model is split up into a reduced intrinsic noise component (gray) and a signal component (purple), maintaining the CV and \fbase{} as during baseline (\subfigrefb{model_and_data}\,\panel[i]{B}, see methods section \ref{intrinsicsplit_methods} for more details). This signal component (purple) can be used for the calculation of the second-order susceptibility. With the reduced noise component the signal-to-noise ratio increases and the number of stimulus realizations can be reduced. This noise split cannot be applied in experimentally measured cells. If this noise split is applied in the model with $\n{}=11$ stimulus realizations the nonlinearity at \fsumb{} is still present (\subfigrefb{model_and_data}\,\panel[iii]{B}). If instead, the RAM stimulus is drawn 1 million times the diagonal nonlinearity band is complemented by vertical and horizontal lines appearing at \foneb{} and \ftwob{} (\subfigrefb{model_and_data}\,\panel[iv]{B}). These nonlinear structures correspond to the ones observed in previous works \citealp{Voronenko2017}. If now a weak RAM stimulus is added (\subfigrefb{model_and_data}\,\panel[i]{C}, red), simultaneously the noise is split up into a noise and signal component (gray and purple) and the calculation is performed on the sum of the signal component and the RAM (red plus purple) only the diagonal band is present with $\n{}=11$ (\subfigrefb{model_and_data}\,\panel[iii]{C}) but also with 1\,million stimulus realizations (\subfigrefb{model_and_data}\,\panel[iv]{C}). -In a high-CV P-unit we could not find such nonlinear structures, neither in the electrophysiologically recorded data nor in the respective model (not shown), corresponding to the theoretical predictions \citealp{Voronenko2017}.\notejg{How can we understand this? If we take a larger chunk of the intrinsic noise should it not help? Is there more than just noise that changes the structure?}\notejg{No this would not help. Splitting up the noise doesn't change the total noise. Nonlinear effects are expected only until a certain amount of noise. Once this noise is surpassed a better signal-to-noise ratio will not help to see that nonlinear effects that are not there. In the low CV cells the noise split changes the signal-to-noise ratio, not the total noise.} +In a high-CV P-unit we could not find such nonlinear structures, neither in the electrophysiologically recorded data nor in the respective model (not shown), corresponding to the theoretical predictions \citealp{Voronenko2017}. \begin{figure*}[hb!] \includegraphics[width=\columnwidth]{model_and_data} @@ -548,6 +548,8 @@ In this chapter, the nonlinearity of P-units and ampullary cells was retrieved b Here it was demonstrated that the second-order susceptibility for the two RAM noise input frequencies \fone{} and \ftwo{} can approximate a three-fish setting, where the driving force for the P-unit are two beats with frequencies \bone{} and \btwo{}. This was confirmed by a low-CV P-unit, where nonlinearities in the P-unit response occurred at the sum and difference of the beat frequencies for pure sine-wave stimulation (\figrefb{motivation}). In this P-unit the nonlinearity appeared in a three-wave setting with \bone{} being close to \fbase{}, corresponding to a frequency combination on the vertical line at \foneb{} in the second-order susceptibility (\subfigrefb{model_full}{B}). This implies that even if only the diagonal structure can be accessed with noise stimulation in the second-order susceptibility in an experiment (\subfigrefb{model_full}{A}), it can be taken as an indicator that the whole nonlinear structure should be present during pure sine-wave stimulation in the same cell. With this RAM stimulation was demonstrated to be an effective method to scan the three-fish or two-beat plane and estimate the whole theoretically predicted nonlinear structure in experimentally recorded cells. +%\notejg{a problem, that we use a new noise for each trial?}\notesr{Using a new noise for each trial, is the way this method is defined. When using the same noise for one million repetiotions we will not see the triangular shape at any time. I tried this and Benjamin confirmed that this would be not possible.} + %the power spectrum of the firing rate %The existence of the extended nonlinearity structure (\figrefb{model_full}) was %\bsum{} and \bdiff{} @@ -727,13 +729,11 @@ The second-order susceptibility was calculated by dividing the higher-order cros The absolute value of a second-order susceptibility matrix is visualized in \figrefb{model_full}. There the upper right and the lower left quadrants characterize the nonlinearity in the response $r(t)$ at the sum frequency of the two input frequencies. The lower right and upper left quadrants characterize the nonlinearity in the response $r(t)$ at the difference of the input frequencies. \paragraph{Nonlinearity index}\label{projected_method} -\notejg{use of $f_{Base}$ or $f_{base}$ or $f_0$ should be consistent throughout the manuscript.} We expect to see nonlinear susceptibility when $\omega_1 + \omega_2 = \fbase{}$. To characterize this we calculated the nonlinearity index (NLI) as \begin{equation} \label{eq:nli_equation} NLI(\fbase{}) = \frac{\max_{\fbase{}-5\,\rm{Hz} \leq f \leq \fbase{}+5\,\rm{Hz}} D(f)}{\mathrm{med}(D(f))} \end{equation} -\notejg{sollte es $D(\omega)$ sein?} For this index, the second-order susceptibility matrix was projected onto the diagonal $D(f)$, by taking the mean of the anti-diagonals. The peakedness at the frequency $\fbase{}$ in $D(f)$ was quantified by finding the maximum of $D(f)$ in the range $\fbase{} \pm 5$\,Hz (\subfigrefb{cells_suscept}{G}) and dividing it by the median of $D(f)$. If the same frozen noise was recorded several times in a cell, each noise repetition resulted in a separate second-order susceptibility matrix. The mean of the corresponding \nli{} values was used for the population statistics in \figref{data_overview_mod}. @@ -848,7 +848,6 @@ with $\rho$ a scaling factor that compensates (see below) for the signal transfo A big portion of the total noise was assigned to the signal component ($c_{signal} = 0.9$) and the remaining part to the noise component ($c_{noise} = 0.1$, \subfigrefb{flowchart}\,\panel[iii]{C}). For the application of the Novikov-Furutsu Theorem \citealp{Novikov1965, Furutsu1963} it is critical that both components add up to the initial 100\,$\%$ of the total noise. This is easily achieved in a model without a carrier if the condition $c_{signal}+c_{noise}=1$ is satisfied. The situation here is more complicated. To compensate for the transformations the signal undergoes before it enters the LIF core, $\xi_{signal}(t)$ was scaled up by the factor $\rho$ (\eqnref{eq:ram_split}, red in \subfigrefb{flowchart}\,\panel[i]{D}). $\rho$ was found by bisecting the space of possible $\rho$ scaling factors by minimizing the error between the baseline CV (only carrier) and the CV during stimulation with noise split. -\notejg{Etwas kurz, die Tabelle. Oder benutzen wir hier tatsaechlich nur die zwei Modelle?} % See section \ref{lifmethods} for model and parameter description. \begin{table*}[hp!] \caption{\label{modelparams} Model parameters of LIF models, fitted to 2 electrophysiologically recorded P-units (\citealp{Ott2020}).}