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@ -469,12 +464,12 @@ Weakly nonlinear responses are expected in cells with sufficiently low intrinsic
\begin{figure*}[t]
\includegraphics[width=\columnwidth]{cells_suscept.pdf}
\caption{\label{fig:cells_suscept} Linear and nonlinear stimulus encoding in a low-CV P-unit (cell identifier ``2010-06-21-ai"). \figitem{A} Interspike interval (ISI) distribution of the cell's baseline activity, i.e. the cell is driven only by the unperturbed own electric field. The low CV of the ISIs indicates quite regular firing. \figitem{B} Power spectral density of the baseline response with peaks at the cell's baseline firing rate \fbase{} and the fish's EOD frequency \feod{}. \figitem{C} Random amplitude modulation stimulus (top, with cutoff frequency of 300\,Hz) and evoked responses (spike raster, bottom) of the same P-unit. The stimulus contrast (right) measures the strength of the AM. \figitem{D} Gain of the transfer function (first-order susceptibility), \Eqnref{linearencoding_methods}, computed from the responses to 10\,\% (light purple) and 20\,\% contrast (dark purple) RAM stimulation of 10\,s duration. \figitem{E, F} Absolute value of the second-order susceptibility, \Eqnref{eq:susceptibility}, for both the low and high stimulus contrast. Pink triangles mark vertical, horizontal, and diagonal lines where \fone, \ftwo{} or \fsum{} are equal to \fbase{}. \figitem{G} Second-order susceptibilities projected onto the diagonal (means of all anti-diagonals of the matrices shown in \panel{E, F}). Dots mark \fbase{}, horizontal dashed lines mark medians of the projected susceptibilities.}
\caption{\label{fig:cells_suscept} Linear and nonlinear stimulus encoding in a low-CV P-unit (cell identifier ``2010-06-21-ai"). \figitem{A} Interspike interval (ISI) distribution of the cell's baseline activity, i.e. the cell is driven only by the unperturbed own electric field. The low CV of the ISIs indicates quite regular firing. \figitem{B} Power spectral density of the baseline response with peaks at the cell's baseline firing rate \fbase{} and the fish's EOD frequency \feod{}. \figitem{C} Random amplitude modulation stimulus (top, with cutoff frequency of 300\,Hz) and evoked responses (spike raster, bottom) of the same P-unit. The stimulus contrast (right) measures the strength of the AM. \figitem{D} Gain of the transfer function (first-order susceptibility), \eqnref{linearencoding_methods}, computed from the responses to 10\,\% (light purple) and 20\,\% contrast (dark purple) RAM stimulation of 10\,s duration. \figitem{E, F} Absolute value of the second-order susceptibility, \eqnref{eq:susceptibility}, for both the low and high stimulus contrast. Pink triangles mark vertical, horizontal, and diagonal lines where \fone, \ftwo{} or \fsum{} are equal to \fbase{}. \figitem{G} Second-order susceptibilities projected onto the diagonal (means of all anti-diagonals of the matrices shown in \panel{E, F}). Dots mark \fbase{}, horizontal dashed lines mark medians of the projected susceptibilities.}
\end{figure*}
Noise stimuli, here random amplitude modulations (RAM) of the EOD (\subfigref{fig:cells_suscept}{C}, top trace, red line), are commonly used to characterize stimulus-driven responses of sensory neurons using transfer functions (first-order susceptibility), spike-triggered averages, or stimulus-response coherences. Here, we additionally estimate the second-order susceptibility to quantify nonlinear encoding. P-unit spikes align more or less clearly with fluctuations in the RAM stimulus. A higher stimulus intensity, here a higher contrast of the RAM relative to the EOD amplitude (see methods), entrains the P-unit response more clearly (light and dark purple for low and high contrast stimuli, \subfigrefb{fig:cells_suscept}{C}). Linear encoding, quantified by the transfer function \Eqnref{linearencoding_methods}, is similar for the two RAM contrasts in this low-CV P-unit (\subfigrefb{fig:cells_suscept}{D}), as expected for a linear system. The first-order susceptibility is low for low frequencies, peaks in the range below 100\,Hz and then falls off again \notejb{Cite Moe paper and Benda2005?}.
Noise stimuli, here random amplitude modulations (RAM) of the EOD (\subfigref{fig:cells_suscept}{C}, top trace, red line), are commonly used to characterize stimulus-driven responses of sensory neurons using transfer functions (first-order susceptibility), spike-triggered averages, or stimulus-response coherences. Here, we additionally estimate the second-order susceptibility to quantify nonlinear encoding. P-unit spikes align more or less clearly with fluctuations in the RAM stimulus. A higher stimulus intensity, here a higher contrast of the RAM relative to the EOD amplitude (see methods), entrains the P-unit response more clearly (light and dark purple for low and high contrast stimuli, \subfigrefb{fig:cells_suscept}{C}). Linear encoding, quantified by the transfer function, \eqnref{linearencoding_methods}, is similar for the two RAM contrasts in this low-CV P-unit (\subfigrefb{fig:cells_suscept}{D}), as expected for a linear system. The first-order susceptibility is low for low frequencies, peaks in the range below 100\,Hz and then falls off again \notejb{Cite Moe paper and Benda2005?}.
The second-order susceptibility, \Eqnref{eq:susceptibility}, quantifies the amplitude and phase of the stimulus-evoked response at the sum \fsum{} for each combination of two stimulus frequencies \fone{} and \ftwo{}. Large values of the second-order susceptibility indicate stimulus-evoked peaks in the response spectrum at the summed frequency that cannot be explained by linear response theory. Similar to the first-order susceptibility, the second-order susceptibility can be estimated directly from the response evoked by a RAM stimulus that stimulates the neuron with a whole range of frequencies simultaneously (\subfigsref{fig:cells_suscept}{E, F}). For a LIF-model driven in the supra-threshold regime, theory predicts nonlinear interactions between the two stimulus frequencies, when the two frequencies \fone{} and \ftwo{} or their sum \fsum{} exactly match the neuron's baseline firing rate \fbase{} \citep{Voronenko2017}. Only then, additional stimulus-evoked peaks appear in the spectrum of the spiking response that would show up in the second-order susceptibility as a horizontal, a vertical, and an anti-diagonal line (\subfigrefb{fig:lifresponse}{B}, pink triangle in \subfigsref{fig:cells_suscept}{E, F}).
The second-order susceptibility, \eqnref{eq:susceptibility}, quantifies the amplitude and phase of the stimulus-evoked response at the sum \fsum{} for each combination of two stimulus frequencies \fone{} and \ftwo{}. Large values of the second-order susceptibility indicate stimulus-evoked peaks in the response spectrum at the summed frequency that cannot be explained by linear response theory. Similar to the first-order susceptibility, the second-order susceptibility can be estimated directly from the response evoked by a RAM stimulus that stimulates the neuron with a whole range of frequencies simultaneously (\subfigsref{fig:cells_suscept}{E, F}). For a LIF-model driven in the supra-threshold regime, theory predicts nonlinear interactions between the two stimulus frequencies, when the two frequencies \fone{} and \ftwo{} or their sum \fsum{} exactly match the neuron's baseline firing rate \fbase{} \citep{Voronenko2017}. Only then, additional stimulus-evoked peaks appear in the spectrum of the spiking response that would show up in the second-order susceptibility as a horizontal, a vertical, and an anti-diagonal line (\subfigrefb{fig:lifresponse}{B}, pink triangle in \subfigsref{fig:cells_suscept}{E, F}).
For the low-CV P-unit, we observe a band of slightly elevated second-order susceptibility for the low RAM contrast at \fsumb{} (yellowish anti-diagonal between pink edges, \subfigrefb{fig:cells_suscept}{E}). This structure vanishes for the stronger stimulus (\subfigref{fig:cells_suscept}{F}). Further, the overall level of the second-order susceptibility is reduced with increasing stimulus strength. To quantify the structural changes in the susceptibility matrices we projected the susceptibility values onto the diagonal by averaging over the anti-diagonals (\subfigrefb{fig:cells_suscept}{G}). At low RAM contrast this projected second-order susceptibility indeed has a small peak at \fbase{} (\subfigrefb{fig:cells_suscept}{G}, dot on top line). For the higher RAM contrast, however, this peak vanishes and the overall level of the second-order susceptibility is reduced (\subfigrefb{fig:cells_suscept}{G}). The reason behind this reduction is that a RAM with a higher contrast is not only a stimulus with an increased amplitude, but also increases the total noise in the system. Increased noise is known to linearize signal transmission \citep{Longtin1993, Chialvo1997, Roddey2000, Voronenko2017} and thus the second-order susceptibility is expected to decrease.
@ -485,7 +480,7 @@ In contrast, a high-CV P-unit (CV$_{\text{base}}=0.4$) does not exhibit pronounc
\begin{figure*}[t]
\includegraphics[width=\columnwidth]{ampullary.pdf}
\caption{\label{fig:ampullary} Linear and nonlinear stimulus encoding in an ampullary afferent (cell identifier ``2012-04-26-ae"). \figitem{A} Interspike interval (ISI) distribution of the cell's baseline activity. The very low CV of the ISIs indicates almost perfect periodic spiking. \figitem{B} Power spectral density of baseline activity with peaks at the cell's baseline firing rate and its harmonics. \figitem{C} Bad-limited white noise stimulus (top, with a cutoff frequency of 150\,Hz) added to the fish's self-generated electric field and spike raster of the evoked responses (bottom) for two stimulus contrasts as indicated (right). \figitem{D} Gain of the transfer function, \Eqnref{linearencoding_methods}, of the responses to stimulation with 2\,\% (light green) and 20\,\% contrast (dark green) of 10\,s duration. \figitem{E, F} Absolute value of the second-order susceptibility, \Eqnref{eq:susceptibility}, for both stimulus contrasts as indicated. Pink triangles indicate the baseline firing rate. \figitem{G} Projections of the second-order susceptibilities in \panel{E, F} onto the diagonal. }
\caption{\label{fig:ampullary} Linear and nonlinear stimulus encoding in an ampullary afferent (cell identifier ``2012-04-26-ae"). \figitem{A} Interspike interval (ISI) distribution of the cell's baseline activity. The very low CV of the ISIs indicates almost perfect periodic spiking. \figitem{B} Power spectral density of baseline activity with peaks at the cell's baseline firing rate and its harmonics. \figitem{C} Bad-limited white noise stimulus (top, with a cutoff frequency of 150\,Hz) added to the fish's self-generated electric field and spike raster of the evoked responses (bottom) for two stimulus contrasts as indicated (right). \figitem{D} Gain of the transfer function, \eqnref{linearencoding_methods}, of the responses to stimulation with 2\,\% (light green) and 20\,\% contrast (dark green) of 10\,s duration. \figitem{E, F} Absolute value of the second-order susceptibility, \eqnref{eq:susceptibility}, for both stimulus contrasts as indicated. Pink triangles indicate the baseline firing rate. \figitem{G} Projections of the second-order susceptibilities in \panel{E, F} onto the diagonal. }
\end{figure*}
Electric fish possess an additional electrosensory system, the passive or ampullary electrosensory system, that responds to low-frequency exogenous electric stimuli. The population of ampullary afferents is much less heterogeneous, and known for the much lower CVs of their baseline ISIs (CV$_{\text{base}}=0.06$--$0.22$) \citep{Grewe2017}. Ampullary cells do not phase-lock to the EOD and the ISIs are unimodally distributed (\subfigrefb{fig:ampullary}{A}). As a consequence of the high regularity of their baseline spiking activity, the corresponding power spectrum shows distinct peaks at the baseline firing rate \fbase{} and its harmonics. Since the cells do not respond to the self-generated EOD, there is no peak at \feod{} (\subfigrefb{fig:ampullary}{B}). When driven by a low-contrast noise stimulus (note: this is no longer an AM stimulus, \subfigref{fig:ampullary}{C}), ampullary cells exhibit very pronounced bands in the second-order susceptibility, where \fsum{} is equal to \fbase{} or its harmonic (yellow diagonals in \subfigrefb{fig:ampullary}{E}), implying strong nonlinear response components at these frequency combinations (\subfigrefb{fig:ampullary}{G}, top). With higher stimulus contrasts these bands disappear (\subfigrefb{fig:ampullary}{F}), the projection onto the diagonal loses its distinct peak at \fsum{} and its overall level is reduced (\subfigrefb{fig:ampullary}{G}, bottom).
@ -501,11 +496,11 @@ In the example recordings shown above (\figsrefb{fig:cells_suscept} and \fref{fi
%\notejb{Since the model overestimated the sensitivity of the real P-unit, we adjusted the RAM contrast to 0.9\,\%, such that the resulting spike trains had the same CV as the electrophysiological recorded P-unit during the 2.5\,\% contrast stimulation (see table~\ref{modelparams} for model parameters).} \notejb{chi2 scale is higher than in real cell}
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, dendritic preprocessing, and parameters fitted to the experimentally recorded P-unit (\figrefb{flowchart}) \citep{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}).
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, dendritic preprocessing, and parameters fitted to the experimentally recorded P-unit (\figrefb{flowchart}) \citep{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}).
In model simulations we can increase the number of trials beyond what would be experimentally possible, here to one million (\subfigrefb{model_and_data}\,\panel[iii]{B}). The estimate of the second-order susceptibility indeed improves. It gets less noisy and the diagonal at \fsum{} is emphasized. However, the expected vertical and horizontal lines at \foneb{} and \ftwob{} are still mainly missing.
Using a broadband stimulus increases the effective input-noise level. This may linearize signal transmission and suppress potential nonlinear responses \citep{Longtin1993, Chialvo1997, Roddey2000, Voronenko2017}. Assuming that the intrinsic noise level in this P-unit is small enough, the full expected structure of the second-order susceptibility 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 \citep{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}) \citep{Egerland2020}. In line with an alternative derivation of the Furutsu-Novikov theorem \citep{Lindner2022}, we can split the total noise and consider a fraction of it as a stimulus. This allows us 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 $\sqrt{2D \, c_{\rm{noise}}} \cdot \xi(t)$ 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 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 \signalnoise, 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]{C}), but not for a number of trials comparable to the experiment (\subfigrefb{model_and_data}\,\panel[ii]{C}). 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}).
Using a broadband stimulus increases the effective input-noise level. This may linearize signal transmission and suppress potential nonlinear responses \citep{Longtin1993, Chialvo1997, Roddey2000, Voronenko2017}. Assuming that the intrinsic noise level in this P-unit is small enough, the full expected structure of the second-order susceptibility 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 \citep{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}) \citep{Egerland2020}. In line with an alternative derivation of the Furutsu-Novikov theorem \citep{Lindner2022}, we can split the total noise and consider a fraction of it as a stimulus. This allows us 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 $\sqrt{2D \, c_{\rm{noise}}}\;\xi(t)$ 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 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 \signalnoise, 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]{C}), but not for a number of trials comparable to the experiment (\subfigrefb{model_and_data}\,\panel[ii]{C}). 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 trials goes along with a substantial reduction of second-order susceptibility values (\subfigrefb{model_and_data}\,\panel[iii]{C}), that saturate in its peak values for $N>10^6$ (\figrefb{fig:trialnr}). This demonstrates the limited reliability of an estimate of the second-order susceptibility that is based on 11 trials only. However, we would like to point out that already the limited number of trials used in the experiments reveals key features of the nonlinear response.
@ -517,25 +512,24 @@ Using the RAM stimulation we found pronounced nonlinear responses in the limit t
\begin{figure*}[t]
\includegraphics[width=\columnwidth]{model_full.pdf}
\caption{\label{fig:model_full} Using second-order susceptibility to predict responses to sine-wave stimuli. \figitem[]{A} Absolute value of the second-order susceptibility, \Eqnref{eq:susceptibility}, for both positive and negative frequencies. \susceptf{} was estimated from $N=10^6$ trials of model simulations in the noise-split condition (cell 2013-01-08-aa, see table~\ref{modelparams} for model parameters). White lines indicate zero frequencies. Nonlinear responses at \fsum{} are quantified in the upper right and lower left quadrants. Nonlinear responses at \fdiff{} are quantified in the upper left and lower right quadrants. The baseline firing rate of this cell was at $\fbase=120$\,Hz. The position of the orange/red letters corresponds to the beat frequencies used for the stimulation with pure sine waves in the subsequent panels and indicates the sum/difference of those beat frequencies. \figitem{B--E} Black line -- power spectral density of model simulations in response to stimulation with two pure sine waves, \fone{} and \ftwo, in addition to the receiving fish's own EOD (three-fish scenario). The contrast of beat beats is 0.02. Colored circles highlight the height of selected peaks in the power spectrum. Grey line -- power spectral density of model in the baseline condition. \figitem{B} The sum of the two beat frequencies match \fbase{}. \figitem{C} The difference of \fone{} and \ftwo{} match \fbase{}. \figitem{D} Only the first beat frequency matches \fbase{}. \figitem{E} None of the two beat frequencies matches \fbase{}.}
\caption{\label{fig:model_full} Using second-order susceptibility to predict responses to sine-wave stimuli. \figitem[]{A} Absolute value of the second-order susceptibility, \eqnref{eq:susceptibility}, for both positive and negative frequencies. \susceptf{} was estimated from $N=10^6$ trials of model simulations in the noise-split condition (cell 2013-01-08-aa, see table~\ref{modelparams} for model parameters). White lines indicate zero frequencies. Nonlinear responses at \fsum{} are quantified in the upper right and lower left quadrants. Nonlinear responses at \fdiff{} are quantified in the upper left and lower right quadrants. The baseline firing rate of this cell was at $\fbase=120$\,Hz. The position of the orange/red letters corresponds to the beat frequencies used for the stimulation with pure sine waves in the subsequent panels and indicates the sum/difference of those beat frequencies. \figitem{B--E} Black line -- power spectral density of model simulations in response to stimulation with two pure sine waves, \fone{} and \ftwo, in addition to the receiving fish's own EOD (three-fish scenario). The contrast of beat beats is 0.02. Colored circles highlight the height of selected peaks in the power spectrum. Grey line -- power spectral density of model in the baseline condition. \figitem{B} The sum of the two beat frequencies match \fbase{}. \figitem{C} The difference of \fone{} and \ftwo{} match \fbase{}. \figitem{D} Only the first beat frequency matches \fbase{}. \figitem{E} None of the two beat frequencies matches \fbase{}.}
\end{figure*}
However, the second-order susceptibility \Eqnref{eq:susceptibility} is a spectral measure that is based on Fourier transforms and thus is also defined for negative stimulus frequencies. The full \susceptf{} matrix is symmetric with respect to the origin. In the upper-right and lower-left quadrants of \susceptf{}, stimulus-evoked responses at \fsum{} are shown, whereas in the lower-right and upper-left quadrants nonlinear responses at the difference \fdiff{} are shown (\figref{fig:model_full}). The vertical and horizontal lines at \foneb{} and \ftwob{} are very pronounced in the upper-right quadrant of \subfigrefb{fig:model_full}{A} for nonlinear responses at \fsum{} and extend into the upper-left quadrant (representing \fdiff), where they fade out towards more negative $f_1$ frequencies. The peak in the response power-spectrum at \fdiff{} evoked by pure sine-wave stimulation (\subfigrefb{fig:motivation}{D}) is predicted by the horizontal line in the upper-left quadrant (\subfigrefb{fig:model_full}{A}, \citealp{Schlungbaum2023}).
However, the second-order susceptibility \eqnref{eq:susceptibility} is a spectral measure that is based on Fourier transforms and thus is also defined for negative stimulus frequencies. The full \susceptf{} matrix is symmetric with respect to the origin. In the upper-right and lower-left quadrants of \susceptf{}, stimulus-evoked responses at \fsum{} are shown, whereas in the lower-right and upper-left quadrants nonlinear responses at the difference \fdiff{} are shown (\figref{fig:model_full}). The vertical and horizontal lines at \foneb{} and \ftwob{} are very pronounced in the upper-right quadrant of \subfigrefb{fig:model_full}{A} for nonlinear responses at \fsum{} and extend into the upper-left quadrant (representing \fdiff), where they fade out towards more negative $f_1$ frequencies. The peak in the response power-spectrum at \fdiff{} evoked by pure sine-wave stimulation (\subfigrefb{fig:motivation}{D}) is predicted by the horizontal line in the upper-left quadrant (\subfigrefb{fig:model_full}{A}, \citealp{Schlungbaum2023}).
Is it possible to predict nonlinear responses in a three-fish setting based on second-order susceptibility matrices estimated from RAM stimulation (\subfigrefb{fig:model_full}{A})? We test this by stimulating the same model with two weak beats (\subfigrefb{fig:model_full}{B--E}). If we choose a frequency combination in which the sum of the two beat frequencies is equal to the model's baseline firing rate \fbase{}, a peak at the sum of the two beat frequencies appears in the power spectrum of the response (\subfigrefb{fig:model_full}{B}), as expected from \suscept. If instead we choose two beat frequencies that differ from \fbase{}, a peak is present at the difference frequency (\subfigrefb{fig:model_full}{C}). If only one of the beat frequency matches \fbase{}, both, a peak at the sum and at the difference frequency are present in the P-unit response (\subfigrefb{fig:model_full}{D}). And if none of these conditions are met, neither a peak at the sum nor at the difference of the two beat frequencies appears (\subfigrefb{fig:model_full}{E}).
\begin{figure*}[tp]
\includegraphics[width=\columnwidth]{data_overview_mod.pdf}
\caption{\label{fig:data_overview} Nonlinear responses in P-units and ampullary cells. The second-order susceptibility is condensed into the peakedness of the nonlinearity, \nli{} \Eqnref{eq:nli_equation}, that relates the amplitude of the projected susceptibility at a cell's baseline firing rate to its median (see \subfigrefb{fig:cells_suscept}{G}). Each of the recorded neurons contributes at maximum with two stimulus contrasts. Black squares and circles highlight recordings conducted in a single cell. Squares in \panel{A, C, E} correspond to the cell in \figrefb{fig:cells_suscept} and circles to the cell in \figrefb{fig:cells_suscept_high_CV}. Squares in \panel{B, D, F} correspond to the cell in \figrefb{fig:ampullary}. \figitem{A, B} There is a negative correlation between the CV during baseline and \nli. \figitem{C, D} There is a negative correlation between the CV during stimulation and \nli. \figitem{E, F} \nli{} is plotted against the response modulation, (see methods), an indicator of the subjective stimulus strength for a cell. There is a negative correlation between response modulation and \nli. Restricting the analysis to the weakest stimulus that was presented to each unique neuron, does not change the results. The number of unique neurons is 221 for P-units and 45 for ampullary cells.
\caption{\label{fig:data_overview} Nonlinear responses in P-units and ampullary cells. The second-order susceptibility is condensed into the peakedness of the nonlinearity, \nli{} \eqnref{eq:nli_equation}, that relates the amplitude of the projected susceptibility at a cell's baseline firing rate to its median (see \subfigrefb{fig:cells_suscept}{G}). Each of the recorded neurons contributes at maximum with two stimulus contrasts. Black squares and circles highlight recordings conducted in a single cell. Squares in \panel{A, C, E} correspond to the cell in \figrefb{fig:cells_suscept} and circles to the cell in \figrefb{fig:cells_suscept_high_CV}. Squares in \panel{B, D, F} correspond to the cell in \figrefb{fig:ampullary}. \figitem{A, B} There is a negative correlation between the CV during baseline and \nli. \figitem{C, D} There is a negative correlation between the CV during stimulation and \nli. \figitem{E, F} \nli{} is plotted against the response modulation, (see methods), an indicator of the subjective stimulus strength for a cell. There is a negative correlation between response modulation and \nli. Restricting the analysis to the weakest stimulus that was presented to each unique neuron, does not change the results. The number of unique neurons is 221 for P-units and 45 for ampullary cells.
% The two example P-units shown before are highlighted with dark markers in \subfigrefb{fig:data_overview}{A, C, E} (squares -- \figrefb{fig:cells_suscept}, circles -- \figrefb{fig:cells_suscept_high_CV}).
% Several of the recorded neurons contribute with two samples to the population analysis as their responses have been recorded to two different contrasts of the same RAM stimulus. Higher stimulus contrasts lead to a stronger drive and thus stronger response modulations (see color code bar in \subfigref{fig:data_overview}{A}, see methods).
% The example cell shown above (\figref{fig:ampullary}) was recorded at two different stimulus intensities and the \nli{} values are highlighted with black squares.
}
\end{figure*}
%\Eqnref{response_modulation}
\subsection{Low CVs and weak stimuli are associated with strong nonlinearity}
All the statements about nonlinear encoding in p-type and ampullary electroreceptor afferents based on single-cell examples shown above are supported by the analysis of our pool of 221 P-units and 47 ampullary afferents recorded in 71 specimen. For comparison across cells we summarize the structure of the second-order susceptibilities in a single number, the peakedness of the projected nonlinearity at \fbase{} (\nli{}) \Eqnref{eq:nli_equation}, that characterizes the size of the expected peak of the projections of a \suscept{} matrix onto its diagonal at the baseline firing rate (e.g. \subfigref{fig:cells_suscept}{G}). \nli{} assumes high values when the peak at \fbase{} is pronounced relative to the median of projections onto the diagonal and is small when there is no distinct peak. The \nli{} values of the P-unit population depend weakly on the CV of the baseline ISI distribution. Cells with lower baseline CVs tend to have more pronounced peaks in their projections than those that have high baseline CVs (\subfigrefb{fig:data_overview}{A}). This negative correlation is more pronounced against the CV measured during stimulation (\subfigrefb{fig:data_overview}{C}).
All the statements about nonlinear encoding in p-type and ampullary electroreceptor afferents based on single-cell examples shown above are supported by the analysis of our pool of 221 P-units and 47 ampullary afferents recorded in 71 specimen. For comparison across cells we summarize the structure of the second-order susceptibilities in a single number, the peakedness of the projected nonlinearity at \fbase{} (\nli{}) \eqnref{eq:nli_equation}, that characterizes the size of the expected peak of the projections of a \suscept{} matrix onto its diagonal at the baseline firing rate (e.g. \subfigref{fig:cells_suscept}{G}). \nli{} assumes high values when the peak at \fbase{} is pronounced relative to the median of projections onto the diagonal and is small when there is no distinct peak. The \nli{} values of the P-unit population depend weakly on the CV of the baseline ISI distribution. Cells with lower baseline CVs tend to have more pronounced peaks in their projections than those that have high baseline CVs (\subfigrefb{fig:data_overview}{A}). This negative correlation is more pronounced against the CV measured during stimulation (\subfigrefb{fig:data_overview}{C}).
The effective stimulus strength also plays an important role. We quantify the effect of stimulus strength on a cell's response by the response modulation --- the standard deviation of a cell's firing rate in response to a RAM stimulus. P-units are heterogeneous in their sensitivity, their response modulations to the same stimulus contrast vary a lot \citep{Grewe2017}. Cells with weak responses to a stimulus, be it an insensitive cell or a weak stimulus, have higher \nli{} values and thus a more pronounced ridge in the second-order susceptibility at \fsumb{} in comparison to strongly responding cells that basically have flat second-order susceptibilities (\subfigrefb{fig:data_overview}{E}). How pronounced nonlinear response components are in P-units thus depends on the baseline CV (a proxy for the internal noise level), and both the CV and response strength during stimulation (effective output noise).
%(Pearson's $r=-0.35$, $p<0.001$)221 P-units and 47 (Pearson's $r=-0.16$, $p<0.01$)
@ -672,7 +666,7 @@ The average firing rate during stimulation, $r_s = \langle r(t) \rangle_t$, is g
\paragraph{Spectral analysis}\label{susceptibility_methods}
The neuron is driven by the stimulus and thus its spiking response depends on the time course of the stimulus. To characterize the relation between stimulus $s(t)$ and response $x(t)$, we calculated the first- and second-order susceptibilities in the frequency domain.
Fourier transforms $\tilde s_T(\omega)$ and $\tilde x_T(\omega)$ of $s(t)$ and $x(t)$, respectively, were computed according to $\tilde x_T(\omega) = \int_{0}^{T} \, x(t) \cdot e^{- i \omega t}\,dt$ for $T=0.5$\,s long segments with no overlap, resulting in a spectral resolution of 2\,Hz. For simplicity we use angular frequencies $\omega=2\pi f$ instead of frequencies $f$. Most stimuli had a duration of 10\,s and were chopped into 20 segments. Spectral measures were computed for single trials of neural responses, series of spike times, \Eqnref{eq:spikes}.
Fourier transforms $\tilde s_T(\omega)$ and $\tilde x_T(\omega)$ of $s(t)$ and $x(t)$, respectively, were computed according to $\tilde x_T(\omega) = \int_{0}^{T} \, x(t) e^{- i \omega t}\,dt$ for $T=0.5$\,s long segments with no overlap, resulting in a spectral resolution of 2\,Hz. For simplicity we use angular frequencies $\omega=2\pi f$ instead of frequencies $f$. Most stimuli had a duration of 10\,s and were chopped into 20 segments. Spectral measures were computed for single trials of neural responses, series of spike times, \eqnref{eq:spikes}.
The power spectrum of the stimulus $s(t)$ was estimated as
\begin{equation}
@ -725,7 +719,7 @@ If the same RAM was recorded several times in a cell, each trial resulted in a s
\begin{figure*}[t]
\includegraphics[width=\columnwidth]{flowchart.pdf}
\caption{\label{flowchart}
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) band-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.}
Architecture of the P-unit model. Each row illustrates subsequent processing steps for three different stimulation regimes: (i) baseline activity without external stimulus, only the fish's self-generated EOD (the carrier, \eqnref{eq:EOD}) is present. (ii) RAM stimulation, \eqnref{eq:ram_equation}. The amplitude of the EOD carrier is modulated with a weak (2\,\% contrast) band-limited white-noise stimulus. (iii) Noise split, \eqnsref{eq:ram_split}--\eqref{eq:Noise_split_intrinsic}, where 90\,\% of the intrinsic noise is replaced by a RAM stimulus, whose amplitude is scaled to maintain the mean firing rate and the CV of the ISIs of the model's baseline activity. As an example, simulations of the model for cell ``2012-07-03-ak'' are shown (table~\ref{modelparams}). \figitem{A} The stimuli are thresholded, \eqnref{eq:threshold2}, by setting all negative values to zero. \figitem{B} Subsequent low-pass filtering, \eqnref{eq:dendrite}, attenuates the carrier and carves out the AM signal. \figitem{C} Intrinsic Gaussian white-noise is added to the signals shown in \panel{B}. Note the reduced internal noise amplitude in the noise split (iii) condition. \figitem{D} Spiking output of the LIF model, \eqnsref{eq:LIF}--\eqref{spikethresh}, in response to the sum of \panel{B} and \panel{C}. \figitem{E} Power spectra of the LIF neuron's spiking activity. Both, baseline activity (\panel[i]{E}) and noise split (\panel[iii]{E}), have the same peaks in the response spectrum at $\fbase$, $f_{EOD} - \fbase$, $f_{EOD}$, and $f_{EOD} + \fbase$. With RAM stimulation (\panel[ii]{E}), the peak at the baseline firing rate, $\fbase$, is washed out.}
\end{figure*}
\subsection{Leaky integrate-and-fire models}\label{lifmethods}
@ -737,7 +731,7 @@ Modified leaky integrate-and-fire (LIF) models were constructed to reproduce the
\end{equation}
with EOD frequency $f_{EOD}$ and an amplitude normalized to one.
To mimic the interaction with other fish, the EODs of a second or third fish with EOD frequencies $f_1$ and $f_2$, respectively, were added to the normalized EOD, \Eqnref{eq:eod}, of the receiving fish according to their contrasts, $c_1$ and $c_2$ at the position of the receiving fish:
To mimic the interaction with other fish, the EODs of a second or third fish with EOD frequencies $f_1$ and $f_2$, respectively, were added to the normalized EOD, \eqnref{eq:eod}, of the receiving fish according to their contrasts, $c_1$ and $c_2$ at the position of the receiving fish:
\begin{equation}
\label{eq:modelbeats}
\carrierinput = \cos(2\pi f_{EOD} t) + c_1 \cos(2\pi f_1 t) + c_2\cos(2\pi f_2 t)
@ -749,7 +743,7 @@ Random amplitude modulations (RAMs) were simulated by first generating the AM as
\label{eq:ram_equation}
y(t) = (1+ s(t)) \cos(2\pi f_{EOD} t)
\end{equation}
For each of the stimulus and response segments needed for the spectral analysis, \Eqnsref{powereq}--\eqref{eq:susceptibility}, a simulation was run. The first second was discarded and the analysis was based on the last second of the data. The resulting spectra thus have a spectral resolution of 1\,Hz.
For each of the stimulus and response segments needed for the spectral analysis, \eqnsref{powereq}--\eqref{eq:susceptibility}, a simulation was run. The first second was discarded and the analysis was based on the last second of the data. The resulting spectra thus have a spectral resolution of 1\,Hz.
First, the input \carrierinput{} is thresholded by setting negative values to zero:
\begin{equation}
@ -766,9 +760,9 @@ the threshold operation is required for extracting the amplitude modulation from
The dendritic voltage $V_d(t)$ is then fed into a stochastic leaky integrate-and-fire (LIF) model with adaptation,
\begin{equation}
\label{eq:LIF}
\tau_{m} \frac{d V_{m}}{d t} = - V_{m} + \mu + \alpha V_{d} - A + \sqrt{2D}\xi(t)
\tau_{m} \frac{d V_{m}}{d t} = - V_{m} + \mu + \alpha V_{d} - A + \sqrt{2D}\;\xi(t)
\end{equation}
where $\tau_{m}$ is the membrane time-constant, $\mu$ is a fixed bias current, $\alpha$ is a scaling factor for $V_{d}$, $A$ is an inhibiting adaptation current, and $\sqrt{2D}\xi(t)$ is a white noise with strength $D$. Note, that all state variables, membrane voltages $V_d$ and $V_m$ as well as the adaptation current, are dimensionless.
where $\tau_{m}$ is the membrane time-constant, $\mu$ is a fixed bias current, $\alpha$ is a scaling factor for $V_{d}$, $A$ is an inhibiting adaptation current, and $\sqrt{2D}\;\xi(t)$ is a white noise with strength $D$. Note, that all state variables, membrane voltages $V_d$ and $V_m$ as well as the adaptation current, are dimensionless.
The adaptation current $A$ follows
\begin{equation}
@ -783,13 +777,14 @@ Whenever the membrane voltage $V_m(t)$ crosses the spiking threshold $\theta=1$,
V_m(t) \ge \theta \; : \left\{ \begin{array}{rcl} V_m & \mapsto & 0 \\ A & \mapsto & A + \Delta A/\tau_A \end{array} \right.
\end{equation}
The P-unit models were integrated by the Euler forward method with a time-step of $\Delta t = 0.05$\,ms. The intrinsic noise $\xi(t)$ (\Eqnref{eq:LIF}, \subfigrefb{flowchart}{C}) was added by drawing a random number from a normal distribution $\mathcal{N}(0,\,1)$ with zero mean and standard deviation of one in each time step $i$. This number was multiplied with $\sqrt{2D}$ and divided by $\sqrt{\Delta t}$. For each simulation, the variables $A$, $V_{d}$ and $V_{m}$ were drawn from a distribution of initial values, estimated from a 100\,s long simulation of baseline activity after an initial 100\,s long integration that was discarded as a transient.
The P-unit models were integrated by the Euler forward method with a time-step of $\Delta t = 0.05$\,ms. The intrinsic noise $\xi(t)$ (\eqnref{eq:LIF}, \subfigrefb{flowchart}{C}) was added by drawing a random number from a normal distribution $\mathcal{N}(0,\,1)$ with zero mean and standard deviation of one in each time step $i$. This number was multiplied with $\sqrt{2D}$ and divided by $\sqrt{\Delta t}$. For each simulation, the variables $A$, $V_{d}$ and $V_{m}$ were drawn from a distribution of initial values, estimated from a 100\,s long simulation of baseline activity after an initial 100\,s long integration that was discarded as a transient.
%\begin{equation}
% \label{eq:LIFintegration}
% V_{m_{i+1}} = V_{m_i} + \left(-V_{m_i} + \mu + \alpha V_{d_i} - A_i + \sqrt{\frac{2D}{\Delta t}}\mathcal{N}(0,\,1)_i\right) \frac{\Delta t}{\tau_m}
%\end{equation}
\paragraph{Fitting the model to recorded P-units} The eight free parameters of the P-unit model $\beta$, $\tau_m$, $\mu$, $D$, $\tau_A$, $\Delta_A$, $\tau_d$, and $t_{ref}$, were fitted to both the baseline activity (baseline firing rate, CV of ISIs, serial correlation of ISIs at lag one, and vector strength of spike coupling to EOD) and the responses to step increases and decreases in EOD amplitude (onset and steady-state responses, effective adaptation time constant, \citealp{Benda2005}) of recorded P-units (table~\ref{modelparams}).
%\paragraph{Fitting the model to recorded P-units}
The eight free parameters of the P-unit model $\beta$, $\tau_m$, $\mu$, $D$, $\tau_A$, $\Delta_A$, $\tau_d$, and $t_{ref}$, were fitted to both the baseline activity (baseline firing rate, CV of ISIs, serial correlation of ISIs at lag one, and vector strength of spike coupling to EOD) and the responses to step increases and decreases in EOD amplitude (onset and steady-state responses, effective adaptation time constant, \citealp{Benda2005}) of recorded P-units (table~\ref{modelparams}).
\begin{table*}[tp]
\caption{\label{modelparams} Model parameters of LIF models, fitted to 3 electrophysiologically recorded P-units \citep{Ott2020}.}
@ -803,60 +798,19 @@ The P-unit models were integrated by the Euler forward method with a time-step o
\end{tabular}
\end{table*}
\subsection{Model noise split into a noise and a stimulus component}\label{intrinsicsplit_methods}
Based on the Furutsu-Novikov theorem \citep{Furutsu1963,Novikov1965,Lindner2022,Egerland2020}, we split the total noise, $\sqrt{2D}\xi$, of a LIF model, \Eqnref{eq:LIF}, into two parts: (i) One part is treated as a driving input signal $s_\xi(t)$, a RAM stimulus where frequencies above 300\,Hz are discarded (\Eqnref{eq:ram_split}), and used to calculate the cross-spectra in \Eqnref{eq:crosshigh} and (ii) the remaining noise $\sqrt{2D \, c_{\rm{noise}}} \cdot \xi(t)$ that is treated as pure noise (\Eqnref{eq:Noise_split_intrinsic}). In this way, the effective signal-to-noise ratio can be increased while maintaining the total noise in the system.
%\sqrt{\rho \, 2D \,c_{\rm{signal}}} \cdot \xi(t)
%(1-c_{\rm{signal}})\cdot\xi$c_{\rm{noise}} = 1-c_{\rm{signal}}$
%c_{\rm{signal}} \cdot \xi
\begin{equation}
\label{eq:ram_split}
y(t) = (1+ s_\xi(t)) \cdot \cos(2\pi f_{EOD} t)
\end{equation}
\begin{equation}
\subsection{Noise split}
\label{intrinsicsplit_methods}
Based on the Furutsu-Novikov theorem \citep{Furutsu1963,Novikov1965,Lindner2022,Egerland2020}, we split the total noise, $\sqrt{2D}\;\xi(t)$, of a LIF model, \eqnref{eq:LIF}, into two parts. The first part is the intrinsic noise term, $\sqrt{2D \, c_{\rm{noise}}}\;\xi(t)$, whose strength is reduced by a factor $c_{\rm{noise}}=0.1$ (\subfigrefb{flowchart}\,\panel[iii]{C}). The second part replaces the now missing intrinsic noise by a driving input signal $s_{\xi}(t)$, a RAM stimulus with frequencies up to 300\,Hz (\subfigrefb{flowchart}\,\panel[iii]{A}). The LIF model with splitted noise then reads
\begin{eqnarray}
\label{eq:ram_split}
y(t) & = & (1+ s_\xi(t)) \cos(2\pi f_{EOD} t) \\
\label{eq:Noise_split_intrinsic_dendrite}
\tau_{d} \frac{d V_{d}}{d t} = -V_{d}+ \lfloor y(t) \rfloor_{0}
\end{equation}
\begin{equation}
\tau_{d} \frac{d V_{d}}{d t} & = & -V_{d}+ \lfloor y(t) \rfloor_{0} \\
\label{eq:Noise_split_intrinsic}
\tau_{m} \frac{d V_{m}}{d t} = - V_{m} + \mu + \alpha V_{d} - A + \sqrt{2D \, c_{\rm{noise}}} \cdot \xi(t)
\end{equation}
% das stimmt so, das c kommt unter die Wurzel!
%\begin{equation}
% \label{Noise_split_intrinsic}
% V_{m_{i+1}} = V_{m_i} + \left(-V_{m_i} + \mu + \alpha V_{d_i} - A_i + \sqrt{\frac{2D c_{\rm{noise}}}{\Delta t}}\mathcal{N}(0,\,1)_i\right) \frac{\Delta t}{\tau_m}
%\end{equation}
\tau_{m} \frac{d V_{m}}{d t} & = & - V_{m} + \mu + \alpha V_{d} - A + \sqrt{2D \, c_{\rm{noise}}}\;\xi(t)
\end{eqnarray}
Both, the reduced intrinsic noise and the RAM stimulus, need to replace the original intrinsic noise. Because the RAM stimulus is band-limited and undergoes some transformations before it is added to the reduced intrinsic noise, it is not \textit{a priori} clear, what the amplitude of the RAM should be. We bisected the amplitude of $s_\xi(t)$, until the CV of the resulting interspike intervals matched the one of the original model's baseline activity. The second-order cross-spectra, \eqnref{eq:crosshigh}, were computed between the RAM stimulus $s_{\xi}(t)$ and the spike train $x(t)$ it evoked. In this way, the effective signal-to-noise ratio can be increased while maintaining the total noise in the system.
In the here used model a small portion of the original noise was assigned to the noise component ($c_{\rm{noise}} = 0.1$, \subfigrefb{flowchart}\,\panel[iii]{C}) and a big portion used as the signal component ($c_{\rm{signal}} = 0.9$, \subfigrefb{flowchart}\,\panel[iii]{A}). For the noise split to be valid \citep{Lindner2022} both components must add up to the initial 100\,\% of the total noise and the baseline properties as the firing rate and the CV of the model are maintained. This is easily achieved in a model without a carrier if the condition $c_{\rm{signal}}+c_{\rm{noise}}=1$ is satisfied. The situation here is more complicated. After the original noise was split into a signal component with $c_{\rm{signal}}$, the frequencies above 300\,Hz were discarded and the signal strength was reduced after the dendritic low pass filtering. To compensate for these transformations the initial signal component was multiplied with the factor $\rho$, keeping the baseline CV (only carrier) and the CV during the noise split comparable, and resulting in $s_\xi(t)$. $\rho$ was found by bisecting the plane of possible factors and minimizing the error between the CV during baseline and stimulation.
%that was found by minimizing the error between the
%Furutsu-Novikov Theorem \citep{Novikov1965, Furutsu1963}\Eqnref{eq:ram_split}, (red in \subfigrefb{flowchart}\,\panel[iii]{A}) bisecting the space of possible $\rho$ scaling factors
%$\rho$ a scaling factor that compensates (see below) for the signal transformations the amplitude modulation stimulus undergoes in the model, i.e. the threshold and the dendritic lowpass.
%In our case the model has a carrier (the fish's self-generated EOD) and we thus want to drive the model with an amplitude modulation stimulus
%\notejb{to methods: ``Note that the signal component \signalnoise{} is added as an amplitude modulation and is thus limited with respect to its spectral content by the Nyquist frequency of the carrier, half the EOD frequency. It thus has a reduced high-frequency content as compared to the intrinsic noise. Adding these discarded high-frequency components to the intrinsic noise does not affect the results here (not shown).''}
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@ -877,7 +831,7 @@ CVs in P-units can range up to 1.5 \citep{Grewe2017, Hladnik2023}. We show the s
\label{S1:highcvpunit}
\begin{figure*}[!ht]
\includegraphics[width=\columnwidth]{cells_suscept_high_CV.pdf}
\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.}
\caption{\label{fig:cells_suscept_high_CV} Response of a noisy P-units (cell ``2018-08-24-af") with a relatively high baseline 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*}
@ -888,29 +842,3 @@ CVs in P-units can range up to 1.5 \citep{Grewe2017, Hladnik2023}. We show the s
\end{figure*}
\end{document}
%\begin{itemize}
%\item \notejb{ \citep{French1973} Derivation of the Fourier transformed kernels measured with white noise.}
%\item \notejb{ \citep{French1976} Technical issues and tests of Fourier transformed kernels measured with white noise.}
%\item \notejb{ \citep{Victor1977} Cat retinal ganglion cells, gratings with sum of 6 or 8 sinusoids. X - versus Y cells. Peak at f1 == f2 in Y cells. X-cells rather linear. Discussion of mechanism, where a nonlinearity comes in along the pathway}
%\item \notejb{ \citep{Marmarelis1972} Temporal 2nd order kernels, how well do kernels predict responses, catfish retinal ganglion cells}
%\item \notejb{ \citep{Marmarelis1973} Temporal 2nd order kernels, how well do kernels predict responses}
%\item \notejb{ \citep{Victor1988} Cat retinal ganglion cells, the sum of sinusoids, very technical, one measurement similar to \citep{Victor1977}.}
%\item \notejb{\citep{Nikias1993} Third order spectra or bispectra. Very technical overview to higher order spectra}
%\item \notejb{ \citep{Mitsis2007} Spider mechanoreceptor. Linear filters, multivariate nonlinearity, and threshold. The second-order kernel is needed for this. Gaussian noise stimuli.}
%\item \notejb{ \citep{French2001} Time kernels up to 3rd order for predicting spider mechanoreceptor responses (spikes!)}
%\item \notejb{ \citep{French1999} Review on time domain nonlinear systems identification}
%\item \notejb{ \citep{Temchin2005, RecioSpinosa2005} 2nd order Wiener kernel for predicting chinchilla auditory nerve fiber firing rate responses. Strong 2nd order blob at characteristic frequency}
%\item \notejb{ \citep{Schanze1997} lots of bispectra, visual cortex MUA recordings}
%\item \notejb{ \citep{Theunissen1996} Linear backward stimulus reconstruction in the context of information theory/signal-to-noise ratios}
%\item \notejb{ \citep{Wessel1996} Same as Theunissen1996 but for P-units}
%\item \notejb{ \citep{Neiman2011} cross bispectrum, bicoherence, mutual information, saturating nonlinearities, `` ampullary electroreceptors of paddlefish are perfectly suited to linearly encode weak low-frequency stimuli.''}
%\item \notejb{ \citep{Chichilnisky2001} Linear Nonlinear Poisson model}
%\item \notejb{ \citep{Gollisch2009} Linear Nonlinear models in retina}
%\item \notejb{ \citep{Clemens2013} Grasshoppper model for female preferences}
%\end{itemize}