diff --git a/susceptibility1.tex b/susceptibility1.tex index 7a1137a..7735616 100644 --- a/susceptibility1.tex +++ b/susceptibility1.tex @@ -558,14 +558,7 @@ Traces of the expected structure of second-order susceptibility are found in bot In the electrophysiological experiments we only have a limited number of trials and this insufficient averaging may occlude the full nonlinear structure. This 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, with 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 repetitions ($\n{}=11$, compare \panel{A} and \panel[ii]{B} in \figrefb{model_and_data}). In the model we can increase the number of repetitions substantially but still do not observe the full nonlinear structure ($\n{}=10^6$, \subfigrefb{model_and_data}\,\panel[iii]{B}). A possible reason for this could be that by applying a broadband stimulus the effective input-noise level is increased and this may linearize the signal transmission \cite{Longtin1993, Chialvo1997, Roddey2000, Voronenko2017}. Assuming that the intrinsic noise level in this P-unit is small enough, the full nonlinear structure should appear in the limit of weak AMs. Again, this cannot be done experimentally, because the problem of insufficient averaging becomes even more severe for weak AMs (low contrast). In the model, however, we know the time course of the intrinsic noise and can use this knowledge to determine the susceptibilities by input-output correlations via the Furutsu-Novikov theorem \cite{Furutsu1963, Novikov1965}. This theorem, in its simplest form, states that the cross-spectrum $S_{x\eta}(\omega)$ of a Gaussian noise $\eta(t)$ driving a nonlinear system and the system's output $x(t)$ is proportional to the linear susceptibility $S_{x\eta}(\omega)=\chi(\omega)S_{\eta\eta}(\omega)$. Here $\chi(\omega)$ characterizes the linear response to an infinitely weak signal $\varepsilon s(t)$ in the presence of the background noise $\eta(t)$. Likewise, the nonlinear susceptibility can be determined in an analogous fashion from higher-order input-output cross-spectra (see methods \eqref{eq:crosshigh}, \eqref{eq:susceptibility}) \cite{Egerland2020}. In line with an alternative derivation of the Furutsu-Novikov theorem \cite{Lindner2022}, we can split the total noise and consider a fraction of it as stimulus. This allows to calculate the susceptibility from the cross-spectrum between the output and this stimulus fraction of the noise. Adapting this approach to our P-unit model (see methods), we replace the intrinsic noise by an approximately equivalent RAM stimulus $s_\xi(t)$ and a weak remaining intrinsic noise ($\sqrt{2D \, c_{noise}} \cdot \xi(t)$, with $c_\text{noise} = 0.1$, see methods \eqref{eq:ram_split}, \eqref{eq:Noise_split_intrinsic}, \eqref{eq:Noise_split_intrinsic_dendrite}, \subfigrefb{model_and_data}\,\panel[i]{C}). We tune the amplitude / standard deviation of the RAM stimulus $s_\xi(t)$ such that the output firing rate and variability (CV) are the same as in the baseline activity (i.e. full intrinsic noise $\sqrt{2D}\xi(t)$ in the voltage equation but no RAM) and compute the second- and third-order cross-spectra between the RAM part of the noise $s_\xi(t)$ and the output spike train. This procedure has two consequences: (i) by means of the cross-spectrum between the output and $s_\xi(t)$, which is a large fraction of the noise, the signal-to-noise ratio of the measured susceptibilities is drastically improved; (ii) the total noise in the system has been reduced (by what was before the external RAM stimulus $s(t)$), which makes the system more nonlinear. For both reasons we now see the expected nonlinear features in the second-order susceptibility for a sufficient number of trials (\subfigrefb{model_and_data}\,\panel[iii]{B}), but not for a number of trials comparable to the experiment (\subfigrefb{model_and_data}\,\panel[ii]{B}). In addition to the strong response for \fsumb{}, we now also observe pronounced nonlinear responses at \foneb{} and \ftwob{} (vertical and horizontal lines, \subfigrefb{model_and_data}\,\panel[iii]{C}). Note, that the increased number of repetitions goes along with a substantial reduction of second-order susceptibility values (\subfigrefb{model_and_data}\,\panel[iii]{B}), that saturate in its peak values for $N>10^5$ (\figrefb{trialnr}). This demonstrates the limited reliability of the statistics that is based on 11 trials. However, we would like to point out that already the limited number of trials as used in the experiments reveals key features of the nonlinear response. - -%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 -%Adding an additional, independent, RAM stimulus to the simulation does not heavily influence the qualitative observations but the nonlinearity becomes weaker (compare \subfigrefb{model_and_data}\,\panel[iii]{C} and \panel[iv]{C}) - -% 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 \cite{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 \cite{Voronenko2017}. +In the 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 \cite{Voronenko2017}. \begin{figure*}[!hb] \includegraphics[width=\columnwidth]{model_and_data} diff --git a/susceptibility1.tex.bak b/susceptibility1.tex.bak index 0bf847f..7735616 100644 --- a/susceptibility1.tex.bak +++ b/susceptibility1.tex.bak @@ -556,19 +556,9 @@ Irrespective of the CV, neither cell shows the complete proposed structure of no Traces of the expected structure of second-order susceptibility are found in both ampullary and p-type electrosensory afferents. In the recordings shown above (\figrefb{fig:cells_suscept}, \figrefb{fig:ampullary}), the nonlinear response is strong whenever the two frequencies (\fone{}, \ftwo{}) fall onto the antidiagonal \fsumb{}, which is in line with theoretical expectations \cite{Voronenko2017}. However, a pronounced nonlinear response for frequencies with \foneb{} or \ftwob{}, although predicted by theory, cannot be observed. Here we investigate how these discrepancies can be understood. In the electrophysiological experiments we only have a limited number of trials and this insufficient averaging may occlude the full nonlinear structure. This 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, with 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 repetitions ($\n{}=11$, compare \panel{A} and \panel[ii]{B} in \figrefb{model_and_data}). In the model we can increase the number of repetitions substantially but still do not observe the full nonlinear structure ($\n{}=10^6$, \subfigrefb{model_and_data}\,\panel[iii]{B}). A possible reason for this could be that by applying a broadband stimulus the effective input-noise level is increased and this may linearize the signal -transmission \cite{Longtin1993, Chialvo1997, Roddey2000, Voronenko2017}. Assuming that the intrinsic noise level in this P-unit is small enough, the full nonlinear structure should appear in the limit of weak AMs. Again, this cannot be done experimentally, because the problem of insufficient averaging becomes even more severe for weak AMs (low contrast). In the model, however, we know the time course of the intrinsic noise and can use this knowledge to determine the susceptibilities by input-output correlations via the Furutsu-Novikov theorem \cite{Furutsu1963, Novikov1965}. This theorem, in its simplest form, states that the cross-spectrum $S_{x\eta}(\omega)$ of a Gaussian noise $\eta(t)$ driving a nonlinear system and the system's output $x(t)$ is proportional to the linear susceptibility $S_{x\eta}(\omega)=\chi(\omega)S_{\eta\eta}(\omega)$. Here $\chi(\omega)$ characterizes the linear response to an infinitely weak signal $\varepsilon s(t)$ in the presence of the background noise $\eta(t)$. Likewise, the nonlinear susceptibility can be determined in an analogous fashion from higher-order input-output cross-spectra (see methods \eqref{eq:crosshigh}, \eqref{eq:susceptibility}) \cite{Egerland2020}. In line with an alternative derivation of the Furutsu-Novikov theorem \cite{Lindner2022}, we can split the total noise and consider a fraction of it as stimulus. This allows to calculate the susceptibility from the cross-spectrum between the output and this stimulus fraction of the noise. Adapting this approach to our P-unit model (see methods), we replace the intrinsic noise by an approximately equivalent RAM stimulus $s_\xi(t)$ and a weak remaining intrinsic noise ($\sqrt{2D \, c_{noise}} \cdot \xi(t)$, with $c_\text{noise} = 0.1$, see methods \eqref{eq:ram_split}, \eqref{eq:Noise_split_intrinsic}, \eqref{eq:Noise_split_intrinsic_dendrite}, \subfigrefb{model_and_data}\,\panel[i]{C}). We tune the amplitude / standard deviation of the RAM stimulus $s_\xi(t)$ such that the output firing rate and variability (CV) are the same as in the baseline activity (i.e. full intrinsic noise $\sqrt{2D}\xi(t)$ in the voltage equation but no RAM) and compute the second- and third-order cross-spectra between the RAM part of the noise $s_\xi(t)$ and the output spike train. This procedure has two consequences: (i) by means of the cross-spectrum between the output and $s_\xi(t)$, which is a large fraction of the noise, the signal-to-noise ratio of the measured susceptibilities is drastically improved; (ii) the total noise in the system has been reduced (by what was before the external RAM stimulus $s(t)$), which makes the system more nonlinear. For both reasons we now see the expected nonlinear features in the second-order susceptibility for a sufficient number of trials (\subfigrefb{model_and_data}\,\panel[iii]{B}), but not for a number of trials comparable to the experiment (\subfigrefb{model_and_data}\,\panel[ii]{B}). In addition to the strong response for \fsumb{}, we now also observe pronounced nonlinear responses at \foneb{} and \ftwob{} (vertical and horizontal lines, \subfigrefb{model_and_data}\,\panel[iii]{C}). Note, that the increased number of repetitions goes along with a substantial reduction of second-order susceptibility values (\subfigrefb{model_and_data}\,\panel[iii]{B}), that saturate in its peak values for $N>10^5$ (\figrefb{trialnr}). This demonstrates the limited reliability of the statistics that is based on 11 trials. However, we would like to point out that already the limited number of trials as used in the experiments +transmission \cite{Longtin1993, Chialvo1997, Roddey2000, Voronenko2017}. Assuming that the intrinsic noise level in this P-unit is small enough, the full nonlinear structure should appear in the limit of weak AMs. Again, this cannot be done experimentally, because the problem of insufficient averaging becomes even more severe for weak AMs (low contrast). In the model, however, we know the time course of the intrinsic noise and can use this knowledge to determine the susceptibilities by input-output correlations via the Furutsu-Novikov theorem \cite{Furutsu1963, Novikov1965}. This theorem, in its simplest form, states that the cross-spectrum $S_{x\eta}(\omega)$ of a Gaussian noise $\eta(t)$ driving a nonlinear system and the system's output $x(t)$ is proportional to the linear susceptibility $S_{x\eta}(\omega)=\chi(\omega)S_{\eta\eta}(\omega)$. Here $\chi(\omega)$ characterizes the linear response to an infinitely weak signal $\varepsilon s(t)$ in the presence of the background noise $\eta(t)$. Likewise, the nonlinear susceptibility can be determined in an analogous fashion from higher-order input-output cross-spectra (see methods \eqref{eq:crosshigh}, \eqref{eq:susceptibility}) \cite{Egerland2020}. In line with an alternative derivation of the Furutsu-Novikov theorem \cite{Lindner2022}, we can split the total noise and consider a fraction of it as stimulus. This allows to calculate the susceptibility from the cross-spectrum between the output and this stimulus fraction of the noise. Adapting this approach to our P-unit model (see methods), we replace the intrinsic noise by an approximately equivalent RAM stimulus $s_\xi(t)$ and a weak remaining intrinsic noise ($\sqrt{2D \, c_{noise}} \cdot \xi(t)$, with $c_\text{noise} = 0.1$, see methods \eqref{eq:ram_split}, \eqref{eq:Noise_split_intrinsic}, \eqref{eq:Noise_split_intrinsic_dendrite}, \subfigrefb{model_and_data}\,\panel[i]{C}). We tune the amplitude / standard deviation of the RAM stimulus $s_\xi(t)$ such that the output firing rate and variability (CV) are the same as in the baseline activity (i.e. full intrinsic noise $\sqrt{2D}\xi(t)$ in the voltage equation but no RAM) and compute the second- and third-order cross-spectra between the RAM part of the noise $s_\xi(t)$ and the output spike train. This procedure has two consequences: (i) by means of the cross-spectrum between the output and $s_\xi(t)$, which is a large fraction of the noise, the signal-to-noise ratio of the measured susceptibilities is drastically improved; (ii) the total noise in the system has been reduced (by what was before the external RAM stimulus $s(t)$), which makes the system more nonlinear. For both reasons we now see the expected nonlinear features in the second-order susceptibility for a sufficient number of trials (\subfigrefb{model_and_data}\,\panel[iii]{B}), but not for a number of trials comparable to the experiment (\subfigrefb{model_and_data}\,\panel[ii]{B}). In addition to the strong response for \fsumb{}, we now also observe pronounced nonlinear responses at \foneb{} and \ftwob{} (vertical and horizontal lines, \subfigrefb{model_and_data}\,\panel[iii]{C}). Note, that the increased number of repetitions goes along with a substantial reduction of second-order susceptibility values (\subfigrefb{model_and_data}\,\panel[iii]{B}), that saturate in its peak values for $N>10^5$ (\figrefb{trialnr}). This demonstrates the limited reliability of the statistics that is based on 11 trials. However, we would like to point out that already the limited number of trials as used in the experiments reveals key features of the nonlinear response. -The nonlinerarity seems to depend on the CV, which is presumably related to the level of intrinsic noise in the cells. 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. - -%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 -Based on the Novikov-Furutsu Theorem \cite{Novikov1965, Furutsu1963} the intrinsic noise of a LIF model can be split up into several independent noise processes with the same correlation function. We make use of this and split the intrinsic noise $\xi$ into two parts: 90\% are then treated as signal ($\xi_{signal}$) while the remaining 10\% are treated as noise ($\xi_{noise}$, see methods for details). With this, the signal-to-noise ratio in the simulation can be arbitrarily varied and the combination of many repetitions and noise-split indeed reveals the triangular shape shown theoretically and for LIF models without carrier\cite{Voronenko2017}(\subfigrefb{model_and_data}\,\panel[iii]{C}). - -%Adding an additional, independent, RAM stimulus to the simulation does not heavily influence the qualitative observations but the nonlinearity becomes weaker (compare \subfigrefb{model_and_data}\,\panel[iii]{C} and \panel[iv]{C}) - -% 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 \cite{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 \cite{Voronenko2017}. +In the 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 \cite{Voronenko2017}. \begin{figure*}[!hb] \includegraphics[width=\columnwidth]{model_and_data}