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@ -544,23 +544,23 @@ High-CV P-units do not exhibit pronounced nonlinearities even at low stimulus co
\subsection*{Ampullary afferents exhibit strong nonlinear interactions}
Irrespective of the CV, neither cell shows the complete proposed structure of nonlinear interactions. \lepto{} posses an additional electrosensory system, the passive or ampullary electrosensory system that responds to low-frequency exogeneous electric stimuli. The population of ampullary afferents is much less heterogeneous, and known for the much lower CVs of the ISI distributions (0.08--0.22)\cite{Grewe2017}. Ampullary cells do not phase-lock to the EOD and the ISI are unimodally distributed (\subfigrefb{ampullary}{A}). According to the low irregularity of the baseline response, the power spectrum shows a very distinct peak at \fbase{} and harmonics of it. Since the cells do not fire phase locked to the EOD, there is no peak at \feod{} (\subfigrefb{ampullary}{B}). When driven by a noise stimulus with a low contrast (note: this is not an AM but an added stimulus to the self-generated EOD, \subfigref{ampullary}{C}), ampullary cells exhibit very pronounced bands when \fsum{} is equal to \fbase{} or its harmonic in the second-order susceptibility matrix implying that this cell is especially nonlinear at these frequency combinations (yellow diagonals, \subfigrefb{ampullary}{E, G}, light green). With higher stimulus contrasts these bands disappear (\subfigrefb{ampullary}{F}) and the projected diagonal is decreased and lacks the distinct peak at \fsum{} (\subfigrefb{ampullary}{G}, dark green).
Irrespective of the CV, neither cell shows the complete proposed structure of nonlinear interactions. \lepto{} posses an additional electrosensory system, the passive or ampullary electrosensory system that responds to low-frequency exogeneous electric stimuli. The population of ampullary afferents is much less heterogeneous, and known for the much lower CVs of the ISI distributions (0.08--0.22)\cite{Grewe2017}. Ampullary cells do not phase-lock to the EOD and the ISI are unimodally distributed (\subfigrefb{fig:ampullary}{A}). According to the low irregularity of the baseline response, the power spectrum shows a very distinct peak at \fbase{} and harmonics of it. Since the cells do not fire phase locked to the EOD, there is no peak at \feod{} (\subfigrefb{fig:ampullary}{B}). When driven by a noise stimulus with a low contrast (note: this is not an AM but an added stimulus to the self-generated EOD, \subfigref{fig:ampullary}{C}), ampullary cells exhibit very pronounced bands when \fsum{} is equal to \fbase{} or its harmonic in the second-order susceptibility matrix implying that this cell is especially nonlinear at these frequency combinations (yellow diagonals, \subfigrefb{fig:ampullary}{E, G}, light green). With higher stimulus contrasts these bands disappear (\subfigrefb{fig:ampullary}{F}) and the projected diagonal is decreased and lacks the distinct peak at \fsum{} (\subfigrefb{fig:ampullary}{G}, dark green).
\begin{figure*}[!ht]
\includegraphics[width=\columnwidth]{ampullary}
\caption{\label{ampullary} Estimation of linear and nonlinear stimulus encoding in an ampullary afferent. Calculated based on the first frozen noise repeat. \figitem{A} Interspike interval (ISI) distribution of the cell's baseline activity. The CV of the ISIs is a dimensionless measure quantifying the response regularity. Zero CV would indicate perfect regularity. \figitem{B} Power-spectrum of the baseline response. \figitem{C} White noise stimulus (top) added to the fish's self-generated electric field and responses (spike raster in the lower traces). The stimulus contrast reflects the strength of the stimulus in relation to the own electric field amplitude. \figitem{D} Transfer function (first-order susceptibility, \Eqnref{linearencoding_methods}) of the responses to 2\% (light green) and 20\% contrast (dark green) stimulation. \figitem{E, F} Absolute value of the second-order susceptibility, \Eqnref{eq:susceptibility}, for the low, and high stimulus contrasts. Pink angles -- edges of the structure when \fone, \ftwo{} or \fsum{} are equal to \fbase{}. \figitem{G} Projected diagonals, calculated as the mean of the anti-diagonals of the matrices in \panel{E, F}. Gray dots mark \fbase{}. Horizontal dashed lines: medians of the projected diagonals.
\caption{\label{fig:ampullary} Estimation of linear and nonlinear stimulus encoding in an ampullary afferent. Calculated based on the first frozen noise repeat. \figitem{A} Interspike interval (ISI) distribution of the cell's baseline activity. The CV of the ISIs is a dimensionless measure quantifying the response regularity. Zero CV would indicate perfect regularity. \figitem{B} Power-spectrum of the baseline response. \figitem{C} White noise stimulus (top) added to the fish's self-generated electric field and responses (spike raster in the lower traces). The stimulus contrast reflects the strength of the stimulus in relation to the own electric field amplitude. \figitem{D} Transfer function (first-order susceptibility, \Eqnref{linearencoding_methods}) of the responses to 2\% (light green) and 20\% contrast (dark green) stimulation. \figitem{E, F} Absolute value of the second-order susceptibility, \Eqnref{eq:susceptibility}, for the low, and high stimulus contrasts. Pink angles -- edges of the structure when \fone, \ftwo{} or \fsum{} are equal to \fbase{}. \figitem{G} Projected diagonals, calculated as the mean of the anti-diagonals of the matrices in \panel{E, F}. Gray dots mark \fbase{}. Horizontal dashed lines: medians of the projected diagonals.
}
\end{figure*}
\subsection*{Model-based estimation of the nonlinear structure}
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 expec\-tations\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 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 according to $S_{x\eta}(\omega)=\chi(\omega)S_{\eta\eta}(\omega)$. Here $\chi(\omega)$ characterizes the linear response to an infinitely weak signal $s(t)$ 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 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}).
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 according to $S_{x\eta}(\omega)=\chi(\omega)S_{\eta\eta}(\omega)$. Here $\chi(\omega)$ characterizes the linear response to an infinitely weak signal $s(t)$ 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, equations \eqref{eq:crosshigh} and \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, equations \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 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 a statistical estimate that is based on 11 trials only. 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 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{fig:trialnr}). This demonstrates the limited reliability of a statistical estimate that is based on 11 trials only. 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.
With high levels of intrinsic noise, we would not expect the nonlinear response features to survive. Indeed, we do not find these features in a high-CV P-unit and its corresponding model (not shown).
@ -597,7 +597,7 @@ The nonlinear effects shown for single cell examples above are supported by the
%(Pearson's $r=-0.35$, $p<0.001$)222 P-units and 47 (Pearson's $r=-0.16$, $p<0.01$)
%In a P-unit population where each cell is represented not by several contrasts but by the lowest recorded contrast, \nli{} significantly correlates with the CV during baseline ($r=-0.17$, $p=0.01$), the response modulation ($r=-0.35$, $p<0.001$) and \fbase{} ($r=-0.32$, $p<0.001$).%, $\n{}=222$*, $\n{}=222$******, $\n{}=222$
The population of ampullary cells is generally more homogeneous and have lower CVs than the P-units and show much higher \nli{} values (factor of 10). Overall, there is a negative correlation with the baseline CV. The example cell shown above (\figref{ampullary}) was recorded at two different stimulus intensities and the \nli{}s are highlighted with black circles. Again, we see that cells that are strongly driven by the stimulus cluster at the bottom of the distribution and have \nli{} values close to zero (\subfigrefb{fig:data_overview_mod}{B, D}). This is confirmed when the data is replotted against the response modulation, those cells that are strongly driven by the stimulus show weak nonlinearities while weakly driven neurons exhibit high values (\subfigrefb{fig:data_overview_mod}{F}).
The population of ampullary cells is generally more homogeneous and have lower CVs than the P-units and show much higher \nli{} values (factor of 10). Overall, there is a negative correlation with the baseline CV. The example cell shown above (\figref{fig:ampullary}) was recorded at two different stimulus intensities and the \nli{}s are highlighted with black circles. Again, we see that cells that are strongly driven by the stimulus cluster at the bottom of the distribution and have \nli{} values close to zero (\subfigrefb{fig:data_overview_mod}{B, D}). This is confirmed when the data is replotted against the response modulation, those cells that are strongly driven by the stimulus show weak nonlinearities while weakly driven neurons exhibit high values (\subfigrefb{fig:data_overview_mod}{F}).
%(Pearson's $r=-0.35$, $p < 0.01$) (Pearson's $r=-0.59$, $p < 0.0001$)
@ -608,10 +608,10 @@ Nonlinearities are ubiquitous in nervous systems, they are essential to extract
%\,\panel[iii]{C}
\subsection*{Theory applies to systems with and without carrier}
Theoretical work\cite{Voronenko2017} explained analytically the occurrence of nonlinear products when a LIF model neuron is stimulated with pure sine-waves. To investigate whether the same mechanisms occur in electroreceptor afferents which are driven by AMs of a carrier and not by pure sine-waves, we followed the previous approach and quantified the second-order susceptibility from responses to white-noise stimuli \cite{Voronenko2017,Neiman2011fish,Nikias1993}. We expected to see elevated second-order susceptibility where either of the foreign signals matches the baseline firing rate ($f_1=f_{base}$ or $f_2=f_{base}$) or when the sum equals the baseline firing rate of the neuron (\fsumb{}) creating a triangular pattern of elevated \suscept{} e.g.\,\subfigref{model_and_data}\,\panel[iii]{C}. Indeed, we find traces of the same nonlinearities in the neuronal responses of p-type electroreceptor afferents. The nonlinear pattern observed in the experimental data, however, matches to the expectations only partially and only in a subset of neurons \figsref{fig:cells_suscept} and\,\ref{ampullary}. Nevertheless, the theory holds also for systems that are driven by AMs of a carrier and is thus more widely applicable.
Theoretical work\cite{Voronenko2017} explained analytically the occurrence of nonlinear products when a LIF model neuron is stimulated with pure sine-waves. To investigate whether the same mechanisms occur in electroreceptor afferents which are driven by AMs of a carrier and not by pure sine-waves, we followed the previous approach and quantified the second-order susceptibility from responses to white-noise stimuli \cite{Voronenko2017,Neiman2011fish,Nikias1993}. We expected to see elevated second-order susceptibility where either of the foreign signals matches the baseline firing rate ($f_1=f_{base}$ or $f_2=f_{base}$) or when the sum equals the baseline firing rate of the neuron (\fsumb{}) creating a triangular pattern of elevated \suscept{} e.g.\,\subfigref{model_and_data}\,\panel[iii]{C}. Indeed, we find traces of the same nonlinearities in the neuronal responses of p-type electroreceptor afferents. The nonlinear pattern observed in the experimental data, however, matches to the expectations only partially and only in a subset of neurons \figsref{fig:cells_suscept} and\,\ref{fig:ampullary}. Nevertheless, the theory holds also for systems that are driven by AMs of a carrier and is thus more widely applicable.
\subsection*{Intrinsic noise limits nonlinear responses}
Only those P-units that exhibit low coefficients of variation (CV) of the interspike-interval distribution (\figref{fig:cells_suscept}{A}) in their unperturbed baseline response show the expected nonlinerities (\subfigref{fig:data_overview_mod}{A}). Such low-CV cells are rare among the 222 P-units that we used in this study. The afferents of the passive electrosensory system, the ampullary cells, however have generally lower CVs and show a much clearer nonlinearity pattern than the low-CV P-unit exemplified here (compare \figsref{fig:cells_suscept} and \ref{ampullary}). The single ampullary cell featured in \figref{ampullary} is a representative of the majority of ampullary cells analyzed here. All ampullary cells have CVs below 0.4 with a median around 0.12 and the observed \nli{}s are 10-fold higher than in P-units.
Only those P-units that exhibit low coefficients of variation (CV) of the interspike-interval distribution (\figref{fig:cells_suscept}{A}) in their unperturbed baseline response show the expected nonlinerities (\subfigref{fig:data_overview_mod}{A}). Such low-CV cells are rare among the 222 P-units that we used in this study. The afferents of the passive electrosensory system, the ampullary cells, however have generally lower CVs and show a much clearer nonlinearity pattern than the low-CV P-unit exemplified here (compare \figsref{fig:cells_suscept} and \ref{fig:ampullary}). The single ampullary cell featured in \figref{fig:ampullary} is a representative of the majority of ampullary cells analyzed here. All ampullary cells have CVs below 0.4 with a median around 0.12 and the observed \nli{}s are 10-fold higher than in P-units.
The CV serves as a proxy for the intrinsic noise in the cells. In both cell types, we observe a negative correlation between \nli{} and the CV, indicating that it is the level of intrinsic noise that plays a role here. These findings are in line with previous studies that propose that noise linearizes the system\cite{Roddey2000, Chialvo1997, Voronenko2017}. More intrinsic noise has been demonstrated to increase the CV and reduce nonlinear phase-locking in vestibular afferents\cite{Schneider2011}. Reduced noise, on the other hand, has been associated with stronger nonlinearity in pyramidal cells of the ELL\cite{Chacron2006}. Further support for the notion of noise limiting the nonlinearity comes from our P-unit LIF model that faithfully reproduces P-unit activity\cite{Barayeu2023}. We can use this model and the noise-split according to the Novikov-Furutsu theorem\cite{Novikov1965,Furutsu1963} to increase the signal-to-noise ratio in the cell while keeping the overall response variability constant (see methods). Treating 90\% of the total noise as signal and simulating large numbers of trial uncovers the full nonlinearity structure (\figref{model_and_data}) seen in LIF neurons and the analytical derivations when driven with sine-wave stimuli\cite{Voronenko2017}.
@ -620,7 +620,7 @@ Our analysis is based on the neuronal responses to white noise stimulus sequence
In the natural situation, the stimuli are periodic signals defined by the difference frequencies. Ho well can we extrapolate from the white noise analysis to the pure sinewave situation? \notejg{Predictions from the X2 matrix and the equations in Voronekov}
In contrast to the situation with individual frequencies (direct sine-waves or sinusoidal AMs) the total power of the stimulus is equally distributed on all frequencies leading to a weaker signal-to-noise ratio. This explains that the nonlinearity pattern in the electroreceptor recordings only partially matches the expectation (\figsref{fig:cells_suscept},\,\ref{ampullary}) while the single-frequency stimulation shows nonlinear interference when the individual stimulus frequencies ($f_1, f_2, \Delta f_1, \Delta f_2$) match the baseline firing rate (\figref{fig:motivation}). With the noise-splitting trick, we could show that the full nonlinearity pattern is present but covered by the intrinsic noise. We thus conclude that the presence of the anti-diagonal pattern in the \suscept{} matrix is sufficient to conclude that the same nonlinear interactions happen here. This also validates the application of the white noise approach to characterize the full \suscept{} matrix instead of using all combinations of individual frequencies.
In contrast to the situation with individual frequencies (direct sine-waves or sinusoidal AMs) the total power of the stimulus is equally distributed on all frequencies leading to a weaker signal-to-noise ratio. This explains that the nonlinearity pattern in the electroreceptor recordings only partially matches the expectation (\figsref{fig:cells_suscept},\,\ref{fig:ampullary}) while the single-frequency stimulation shows nonlinear interference when the individual stimulus frequencies ($f_1, f_2, \Delta f_1, \Delta f_2$) match the baseline firing rate (\figref{fig:motivation}). With the noise-splitting trick, we could show that the full nonlinearity pattern is present but covered by the intrinsic noise. We thus conclude that the presence of the anti-diagonal pattern in the \suscept{} matrix is sufficient to conclude that the same nonlinear interactions happen here. This also validates the application of the white noise approach to characterize the full \suscept{} matrix instead of using all combinations of individual frequencies.
% The nonlinearity of ampullary cells in paddlefish \cite{Neiman2011fish} has been previously accessed with bandpass limited white noise.