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susceptibility1.pdf
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@ -484,8 +484,8 @@ To quantify the second-order susceptibility in a three-fish setting the noise st
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\begin{figure*}[h]%hp!
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\includegraphics{cells_suscept}%cells_suscept
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\caption{\label{cells_suscept} Response of experimentally measured P-units to RAM stimuli. Light purple -- low RAM contrast. Dark purple -- high RAM contrast. \figitem{A} Regular firing low-CV P-unit. \figitem[i]{A} Left: Interspike intervals (ISI) distribution during baseline. Right: Baseline power spectrum of the firing rate. \figitem[ii]{A} Top: EOD carrier (gray) with RAM (red). Middle: Spike trains in response to a low RAM contrast. Bottom: Spike trains in response to a high RAM contrast. \figitem[iii]{A} First-order susceptibility (see \eqnref{linearencoding_methods}). \figitem[iv]{A} Absolute value of the second-order susceptibility, \eqnref{susceptibility}, for the low RAM contrast.
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Pink lines -- edges of the structure when \fone, \ftwo{} or \fsum{} are equal to \fbase{}. Orange line -- part of the structure when \fsum{} is equal to half \feod. \figitem[v]{A} Absolute value of the second-order susceptibility for the higher RAM contrast. \figitem[vi]{A} Projected diagonals, calculated as the mean of the anti-diagonals of the matrices in \panel[iv,v]{A}. Gray dots: \fbase{}. Gray area: $\fbase{} \pm 5$\,Hz. Dashed lines: Medians of the projected diagonals. \figitem{B} Noisy high-CV P-Unit. Panels as in \panel{A}.
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\caption{\label{cells_suscept} Response of experimentally measured P-units to RAM stimuli. Light purple -- low RAM contrast. Dark purple -- high RAM contrast. Regular firing low-CV P-unit. \figitem{A} Interspike intervals (ISI) distribution during baseline. \figitem{B} Baseline power spectrum of the firing rate. \figitem{C} Top: EOD carrier (gray) with RAM (red). Middle: Spike trains in response to a low RAM contrast. Bottom: Spike trains in response to a high RAM contrast. \figitem{D} First-order susceptibility (see \eqnref{linearencoding_methods}). \figitem{E} Absolute value of the second-order susceptibility, \eqnref{susceptibility}, for the low RAM contrast.
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Pink lines -- edges of the structure when \fone, \ftwo{} or \fsum{} are equal to \fbase{}. Orange line -- part of the structure when \fsum{} is equal to half \feod. \figitem{F} Absolute value of the second-order susceptibility for the higher RAM contrast. \figitem{G} Projected diagonals, calculated as the mean of the anti-diagonals of the matrices in \panel{E--F}. Gray dots: \fbase{}. Gray area: $\fbase{} \pm 5$\,Hz. Dashed lines: Medians of the projected diagonals..
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}
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\end{figure*}
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@ -494,8 +494,8 @@ Based on the theory strong nonlinearities in spiking responses are not predicted
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\begin{figure*}[ht]%hp!
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\includegraphics{ampullary}
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\caption{\label{ampullary} Response of an experimentally measured ampullary cell. Light green -- low noise stimulus contrast. Dark green -- high noise stimulus contrast. \figitem[i]{A} Left: ISI distribution during baseline. Right: Baseline power spectrum of the firing rate. \figitem[ii]{A} Top: EOD carrier (gray) plus a band-pass limited white noise (red, see methods section \ref{rammethods}). Middle: Spike trains in response to a low noise stimulus contrast. Bottom: Spike trains in response to a high noise stimulus contrast. \figitem[iii]{A} First-order susceptibility (see \eqnref{linearencoding_methods}). \figitem[iv]{A} Absolute value of the second-order susceptibility (\eqnref{susceptibility}) for the low noise stimulus contrast.
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Pink lines -- edges of the structure when \fone, \ftwo{} or \fsum{} are equal to \fbase{}. Orange line -- part of the structure when \fsum{} is equal to half \feod. \figitem[v]{A} Second-order susceptibility matrix for the higher noise stimulus contrast. Colored lines as in \panel[iv]{A}. \figitem[vi]{A} Projected diagonals, calculated as the mean of the anti-diagonals of the matrices in \panel[iv,v]{A}. Gray dot: \fbase{}. Gray area: $\fbase{} \pm 5$\,Hz. Dashed lines: Medians of the projected diagonals.
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\caption{\label{ampullary} Response of an experimentally measured ampullary cell. Light green -- low noise stimulus contrast. Dark green -- high noise stimulus contrast. \figitem{A} ISI distribution during baseline. \figitem{B} Baseline power spectrum of the firing rate. \figitem{C} Top: EOD carrier (gray) plus a band-pass limited white noise (red, see methods section \ref{rammethods}). Middle: Spike trains in response to a low noise stimulus contrast. Bottom: Spike trains in response to a high noise stimulus contrast. \figitem{D} First-order susceptibility (see \eqnref{linearencoding_methods}). \figitem{E} Absolute value of the second-order susceptibility (\eqnref{susceptibility}) for the low noise stimulus contrast.
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Pink lines -- edges of the structure when \fone, \ftwo{} or \fsum{} are equal to \fbase{}. Orange line -- part of the structure when \fsum{} is equal to half \feod. \figitem{F} Second-order susceptibility matrix for the higher noise stimulus contrast. Colored lines as in \panel{E}. \figitem{G} Projected diagonals, calculated as the mean of the anti-diagonals of the matrices in \panel{E--F}. Gray dot: \fbase{}. Gray area: $\fbase{} \pm 5$\,Hz. Dashed lines: Medians of the projected diagonals.
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}
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\end{figure*}
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@ -926,8 +926,8 @@ A big portion of the total noise was assigned to the signal component ($c_{signa
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\begin{figure*}[hp]%hp!
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\includegraphics{cells_suscept_high_CV}
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\caption{\label{cells_suscept_high_CV} Response of experimentally measured P-units to RAM stimuli. Light purple -- low RAM contrast. Dark purple -- high RAM contrast. \figitem{A} Regular firing low-CV P-unit. \figitem[i]{A} Left: Interspike intervals (ISI) distribution during baseline. Right: Baseline power spectrum of the firing rate. \figitem[ii]{A} Top: EOD carrier (gray) with RAM (red). Middle: Spike trains in response to a low RAM contrast. Bottom: Spike trains in response to a high RAM contrast. \figitem[iii]{A} First-order susceptibility (see \eqnref{linearencoding_methods}). \figitem[iv]{A} Absolute value of the second-order susceptibility, \eqnref{susceptibility}, for the low RAM contrast.
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Pink lines -- edges of the structure when \fone, \ftwo{} or \fsum{} are equal to \fbase{}. Orange line -- part of the structure when \fsum{} is equal to half \feod. \figitem[v]{A} Absolute value of the second-order susceptibility for the higher RAM contrast. \figitem[vi]{A} Projected diagonals, calculated as the mean of the anti-diagonals of the matrices in \panel[iv,v]{A}. Gray dots: \fbase{}. Gray area: $\fbase{} \pm 5$\,Hz. Dashed lines: Medians of the projected diagonals. \figitem{B} Noisy high-CV P-Unit. Panels as in \panel{A}.
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\caption{\label{cells_suscept_high_CV} Response of experimentally measured P-units to RAM stimuli. Light purple -- low RAM contrast. Dark purple -- high RAM contrast. Noisy high-CV P-Unit. \figitem{A} Interspike intervals (ISI) distribution during baseline. \figitem{B} Baseline power spectrum of the firing rate. \figitem{C} Top: EOD carrier (gray) with RAM (red). Middle: Spike trains in response to a low RAM contrast. Bottom: Spike trains in response to a high RAM contrast. \figitem{D} First-order susceptibility (see \eqnref{linearencoding_methods}). \figitem{E} Absolute value of the second-order susceptibility, \eqnref{susceptibility}, for the low RAM contrast.
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Pink lines -- edges of the structure when \fone, \ftwo{} or \fsum{} are equal to \fbase{}. Orange line -- part of the structure when \fsum{} is equal to half \feod. \figitem{F} Absolute value of the second-order susceptibility for the higher RAM contrast. \figitem{G} Projected diagonals, calculated as the mean of the anti-diagonals of the matrices in \panel{E--F}. Gray dots: \fbase{}. Gray area: $\fbase{} \pm 5$\,Hz. Dashed lines: Medians of the projected diagonals.
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}
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\end{figure*}
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@ -472,35 +472,35 @@ In this work, the second-order susceptibility in the spiking responses of P-unit
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P-units are heterogeneous in their baseline firing properties \citealp{Grewe2017, Hladnik2023} and differ in their noisiness, which is represented by the coefficient of variation (CV) of the interspike intervals (ISI). Low-CV P-units have a regular firing pattern and are less noisy, whereas high-CV P-units have a less regular firing pattern.
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Second-order susceptibility is expected to be especially pronounced for low-CV cells \citealp{Voronenko2017}. In the following first low-CV P-units will be addressed in \subfigrefb{cells_suscept}{A}.
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P-units probabilistically phase-lock to the EOD of the fish, firing at the same phase but not in every EOD cycle, resulting in a multimodal ISI histogram with maxima at integer multiples of the EOD period (\subfigrefb{cells_suscept}\,\panel[i]{A}, left). The strongest peak in the baseline power spectrum of the firing rate of a P-unit is the \feod{} peak, and the second strongest peak is the mean baseline firing rate \fbase{} peak (\subfigrefb{cells_suscept}\,\panel[i]{A}, right). The power spectrum of P-units is symmetric around half \feod, with baseline peaks appearing at $\feod \pm \fbase{}$.
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P-units probabilistically phase-lock to the EOD of the fish, firing at the same phase but not in every EOD cycle, resulting in a multimodal ISI histogram with maxima at integer multiples of the EOD period (\subfigrefb{cells_suscept}{A}). The strongest peak in the baseline power spectrum of the firing rate of a P-unit is the \feod{} peak, and the second strongest peak is the mean baseline firing rate \fbase{} peak (\subfigrefb{cells_suscept}{B}). The power spectrum of P-units is symmetric around half \feod, with baseline peaks appearing at $\feod \pm \fbase{}$.
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Noise stimuli, as random amplitude modulations (RAM) of the EOD, are common stimuli during P-unit recordings. In the following, the amplitude of the noise stimulus will be quantified as the standard deviation and will be expressed as a contrast (unit \%) in relation to the receiver EOD. The spikes of P-units slightly align with the RAM stimulus with a low contrast (light purple) and are stronger driven in response to a higher RAM contrast (dark purple, \subfigrefb{cells_suscept}\,\panel[ii]{A}). The linear encoding (see \eqnref{linearencoding_methods}) is comparable between the two RAM contrasts in this low-CV P-unit (\subfigrefb{cells_suscept}\,\panel[iii]{A}).%visualized by the gain of the transfer function,\suscept{}
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Noise stimuli, as random amplitude modulations (RAM) of the EOD, are common stimuli during P-unit recordings. In the following, the amplitude of the noise stimulus will be quantified as the standard deviation and will be expressed as a contrast (unit \%) in relation to the receiver EOD. The spikes of P-units slightly align with the RAM stimulus with a low contrast (light purple) and are stronger driven in response to a higher RAM contrast (dark purple, \subfigrefb{cells_suscept}{C}). The linear encoding (see \eqnref{linearencoding_methods}) is comparable between the two RAM contrasts in this low-CV P-unit (\subfigrefb{cells_suscept}{D}).%visualized by the gain of the transfer function,\suscept{}
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To quantify the second-order susceptibility in a three-fish setting the noise stimulus was set in relation to the corresponding P-unit response in the Fourier domain, resulting in a matrix where the nonlinearity at the sum frequency \fsum{} in the firing rate is depicted for two noise frequencies \fone{} and \ftwo{} (\eqnref{susceptibility}, \subfigrefb{cells_suscept}\,\panel[iv]{A}--\panel[v]{A}). Note that the RAM stimulus can be decomposed in frequencies $f$, that approximate the beat frequencies $\Delta f$, occurring in case of pure sine-wave stimulation (\subfigrefb{motivation}{D}). Thus the nonlinearity accessed with the RAM stimulation at \fsum{} (\subfigrefb{cells_suscept}\,\panel[iv]{A}) is comparable to the nonlinearity appearing during pure sine-wave stimulation at \bsum{} (orange peak, \subfigrefb{motivation}{D}). Based on the theory \citealp{Voronenko2017} nonlinearities should arise when \fone{}, \ftwo{} or \fsum{} are equal to \fbase{} (upper right quadrant in \figrefb{plt_RAM_didactic2}), which would imply a triangular nonlinearity shape highlighted by the pink triangle corners in \subfigrefb{cells_suscept}\,\panel[iv]{A}--\panel[v]{A}. A slight diagonal nonlinearity band appears for the low RAM contrast when \fsumb{} is satisfied (yellow diagonal between pink edges, \subfigrefb{cells_suscept}\,\panel[iv]{A}). Since the matrix contains only anti-diagonal elements, the structural changes were quantified by the mean of the anti-diagonals, resulting in the projected diagonal (\subfigrefb{cells_suscept}\,\panel[vi]{A}). For a low RAM contrast the \fbase{} peak in the projected diagonal is slightly enhanced (\subfigrefb{cells_suscept}\,\panel[vi]{A}, gray dot on light purple line). For the higher RAM contrast, the overall second-order susceptibility is reduced (\subfigrefb{cells_suscept}\,\panel[v]{A}), with no pronounced \fbase{} peak in the projected diagonal (\subfigrefb{cells_suscept}\,\panel[vi]{A}, dark purple line). In addition, there is an offset between the projected diagonals, demonstrating that the second-order susceptibility is reduced for RAM stimuli with a higher contrast (\subfigrefb{cells_suscept}\,\panel[vi]{A}).
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To quantify the second-order susceptibility in a three-fish setting the noise stimulus was set in relation to the corresponding P-unit response in the Fourier domain, resulting in a matrix where the nonlinearity at the sum frequency \fsum{} in the firing rate is depicted for two noise frequencies \fone{} and \ftwo{} (\eqnref{susceptibility}, \subfigrefb{cells_suscept}{E--F}). Note that the RAM stimulus can be decomposed in frequencies $f$, that approximate the beat frequencies $\Delta f$, occurring in case of pure sine-wave stimulation (\subfigrefb{motivation}{D}). Thus the nonlinearity accessed with the RAM stimulation at \fsum{} (\subfigrefb{cells_suscept}{E}) is comparable to the nonlinearity appearing during pure sine-wave stimulation at \bsum{} (orange peak, \subfigrefb{motivation}{D}). Based on the theory \citealp{Voronenko2017} nonlinearities should arise when \fone{}, \ftwo{} or \fsum{} are equal to \fbase{} (upper right quadrant in \figrefb{plt_RAM_didactic2}), which would imply a triangular nonlinearity shape highlighted by the pink triangle corners in \subfigrefb{cells_suscept}{E--F}. A slight diagonal nonlinearity band appears for the low RAM contrast when \fsumb{} is satisfied (yellow diagonal between pink edges, \subfigrefb{cells_suscept}{E}). Since the matrix contains only anti-diagonal elements, the structural changes were quantified by the mean of the anti-diagonals, resulting in the projected diagonal (\subfigrefb{cells_suscept}{G}). For a low RAM contrast the \fbase{} peak in the projected diagonal is slightly enhanced (\subfigrefb{cells_suscept}{G}, gray dot on light purple line). For the higher RAM contrast, the overall second-order susceptibility is reduced (\subfigrefb{cells_suscept}{F}), with no pronounced \fbase{} peak in the projected diagonal (\subfigrefb{cells_suscept}{G}, dark purple line). In addition, there is an offset between the projected diagonals, demonstrating that the second-order susceptibility is reduced for RAM stimuli with a higher contrast (\subfigrefb{cells_suscept}{G}).
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%There a triangle is plotted not only if the frequency combinations are equal to the \fbase{} fundamental but also to the \fbase{} harmonics (two triangles further away from the origin).
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%In this figure a part of \fsumehalf{} is marked with the orange diagonal line.
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\begin{figure*}[h]%hp!
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\includegraphics{cells_suscept}%cells_suscept
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\caption{\label{cells_suscept} Response of experimentally measured P-units to RAM stimuli. Light purple -- low RAM contrast. Dark purple -- high RAM contrast. \figitem{A} Regular firing low-CV P-unit. \figitem[i]{A} Left: Interspike intervals (ISI) distribution during baseline. Right: Baseline power spectrum of the firing rate. \figitem[ii]{A} Top: EOD carrier (gray) with RAM (red). Middle: Spike trains in response to a low RAM contrast. Bottom: Spike trains in response to a high RAM contrast. \figitem[iii]{A} First-order susceptibility (see \eqnref{linearencoding_methods}). \figitem[iv]{A} Absolute value of the second-order susceptibility, \eqnref{susceptibility}, for the low RAM contrast.
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Pink lines -- edges of the structure when \fone, \ftwo{} or \fsum{} are equal to \fbase{}. Orange line -- part of the structure when \fsum{} is equal to half \feod. \figitem[v]{A} Absolute value of the second-order susceptibility for the higher RAM contrast. \figitem[vi]{A} Projected diagonals, calculated as the mean of the anti-diagonals of the matrices in \panel[iv,v]{A}. Gray dots: \fbase{}. Gray area: $\fbase{} \pm 5$\,Hz. Dashed lines: Medians of the projected diagonals. \figitem{B} Noisy high-CV P-Unit. Panels as in \panel{A}.
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\caption{\label{cells_suscept} Response of experimentally measured P-units to RAM stimuli. Light purple -- low RAM contrast. Dark purple -- high RAM contrast. Regular firing low-CV P-unit. \figitem{A} Interspike intervals (ISI) distribution during baseline. \figitem{B} Baseline power spectrum of the firing rate. \figitem{C} Top: EOD carrier (gray) with RAM (red). Middle: Spike trains in response to a low RAM contrast. Bottom: Spike trains in response to a high RAM contrast. \figitem{D} First-order susceptibility (see \eqnref{linearencoding_methods}). \figitem{E} Absolute value of the second-order susceptibility, \eqnref{susceptibility}, for the low RAM contrast.
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Pink lines -- edges of the structure when \fone, \ftwo{} or \fsum{} are equal to \fbase{}. Orange line -- part of the structure when \fsum{} is equal to half \feod. \figitem{F} Absolute value of the second-order susceptibility for the higher RAM contrast. \figitem{G} Projected diagonals, calculated as the mean of the anti-diagonals of the matrices in \panel{E--F}. Gray dots: \fbase{}. Gray area: $\fbase{} \pm 5$\,Hz. Dashed lines: Medians of the projected diagonals..
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}
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\end{figure*}
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\subsection{High-CV P-units do not exhibit any nonlinear interactions}%frequency combinations
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Based on the theory strong nonlinearities in spiking responses are not predicted for cells with irregular firing properties and high CVs \citealp{Voronenko2017}. CVs in P-units can range up to 1.5 \citealp{Grewe2017, Hladnik2023} and as a next step the second-order susceptibility of high-CV P-units will be presented. As low-CV P-units, high-CV P-units fire at multiples of the EOD period (\subfigrefb{cells_suscept_high_CV}\,\panel[i]{A}, left). In contrast to low-CV P-units high-CV P-units are noisier in their firing pattern and have a less pronounced mean baseline firing rate peak \fbase{} in the power spectrum of their firing rate during baseline (\subfigrefb{cells_suscept_high_CV}\,\panel[i]{A}, right). High-CV P-units do not exhibit any nonlinear structures related to \fbase{} neither in the second-order susceptibility matrices (\subfigrefb{cells_suscept_high_CV}\,\panel[iv]{A}--\panel[v]{A}), nor in the projected diagonals (\subfigrefb{cells_suscept_high_CV}\,\panel[vi]{A}). As in low-CV P-units (\subfigrefb{cells_suscept}\,\panel[v]{A}), the mean second-order susceptibility decreases with higher RAM contrasts in high-CV P-units (\subfigrefb{cells_suscept_high_CV}\,\panel[v]{A}).
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Based on the theory strong nonlinearities in spiking responses are not predicted for cells with irregular firing properties and high CVs \citealp{Voronenko2017}. CVs in P-units can range up to 1.5 \citealp{Grewe2017, Hladnik2023} and as a next step the second-order susceptibility of high-CV P-units will be presented. As low-CV P-units, high-CV P-units fire at multiples of the EOD period (\subfigrefb{cells_suscept_high_CV}{A}). In contrast to low-CV P-units high-CV P-units are noisier in their firing pattern and have a less pronounced mean baseline firing rate peak \fbase{} in the power spectrum of their firing rate during baseline (\subfigrefb{cells_suscept_high_CV}{B}). High-CV P-units do not exhibit any nonlinear structures related to \fbase{} neither in the second-order susceptibility matrices (\subfigrefb{cells_suscept_high_CV}{E--F}), nor in the projected diagonals (\subfigrefb{cells_suscept_high_CV}{G}). As in low-CV P-units (\subfigrefb{cells_suscept}{F}), the mean second-order susceptibility decreases with higher RAM contrasts in high-CV P-units (\subfigrefb{cells_suscept_high_CV}{F}).
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\begin{figure*}[ht]%hp!
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\includegraphics{ampullary}
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\caption{\label{ampullary} Response of an experimentally measured ampullary cell. Light green -- low noise stimulus contrast. Dark green -- high noise stimulus contrast. \figitem[i]{A} Left: ISI distribution during baseline. Right: Baseline power spectrum of the firing rate. \figitem[ii]{A} Top: EOD carrier (gray) plus a band-pass limited white noise (red, see methods section \ref{rammethods}). Middle: Spike trains in response to a low noise stimulus contrast. Bottom: Spike trains in response to a high noise stimulus contrast. \figitem[iii]{A} First-order susceptibility (see \eqnref{linearencoding_methods}). \figitem[iv]{A} Absolute value of the second-order susceptibility (\eqnref{susceptibility}) for the low noise stimulus contrast.
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\caption{\label{ampullary} Response of an experimentally measured ampullary cell. Light green -- low noise stimulus contrast. Dark green -- high noise stimulus contrast. \figitem{A} Left: ISI distribution during baseline. Right: Baseline power spectrum of the firing rate. \figitem[ii]{A} Top: EOD carrier (gray) plus a band-pass limited white noise (red, see methods section \ref{rammethods}). Middle: Spike trains in response to a low noise stimulus contrast. Bottom: Spike trains in response to a high noise stimulus contrast. \figitem[iii]{A} First-order susceptibility (see \eqnref{linearencoding_methods}). \figitem[iv]{A} Absolute value of the second-order susceptibility (\eqnref{susceptibility}) for the low noise stimulus contrast.
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Pink lines -- edges of the structure when \fone, \ftwo{} or \fsum{} are equal to \fbase{}. Orange line -- part of the structure when \fsum{} is equal to half \feod. \figitem[v]{A} Second-order susceptibility matrix for the higher noise stimulus contrast. Colored lines as in \panel[iv]{A}. \figitem[vi]{A} Projected diagonals, calculated as the mean of the anti-diagonals of the matrices in \panel[iv,v]{A}. Gray dot: \fbase{}. Gray area: $\fbase{} \pm 5$\,Hz. Dashed lines: Medians of the projected diagonals.
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}
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\end{figure*}
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\subsection{Ampullary cells exhibit strong nonlinear interactions}%with lower CVs as P-units
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\lepto{} posses another primary sensory afferent population, the ampullary cells, with overall low \fbase{} (80--200\,Hz) and low CV values (0.08--0.22, \citealp{Grewe2017}). Ampullary cells do not phase-lock to the EOD, with no maxima at multiples of the EOD period and smoothly unimodal distributed ISIs (\subfigrefb{ampullary}\,\panel[i]{A}, left). Ampullary cells do not have a peak at \feod{} in the baseline power spectrum of the firing rate with no symmetry around it (\subfigrefb{ampullary}\,\panel[i]{A}, right). Instead, the \fbase{} peak is very pronounced with clear harmonics. When being exposed to a noise stimulus with a low contrast, 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}\,\panel[iv]{A}). With higher noise stimuli contrasts these bands disappear (\subfigrefb{ampullary}\,\panel[v]{A}) and the projected diagonal is lowered (\subfigrefb{ampullary}\,\panel[vi]{A}, dark green).
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\lepto{} posses another primary sensory afferent population, the ampullary cells, with overall low \fbase{} (80--200\,Hz) and low CV values (0.08--0.22, \citealp{Grewe2017}). Ampullary cells do not phase-lock to the EOD, with no maxima at multiples of the EOD period and smoothly unimodal distributed ISIs (\subfigrefb{ampullary}{A}). Ampullary cells do not have a peak at \feod{} in the baseline power spectrum of the firing rate with no symmetry around it (\subfigrefb{ampullary}{B}). Instead, the \fbase{} peak is very pronounced with clear harmonics. When being exposed to a noise stimulus with a low contrast, 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}). With higher noise stimuli contrasts these bands disappear (\subfigrefb{ampullary}{F}) and the projected diagonal is lowered (\subfigrefb{ampullary}{G}, dark green).
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These nonlinearity bands are more pronounced in ampullary cells than they were in P-units (compare \figrefb{ampullary} and \figrefb{cells_suscept}). Ampullary cells with their unimodal ISI distribution are closer than P-units to the LIF models without EOD carrier, where the predictions about the second-order susceptibility structure have mainly been elaborated on \citealp{Voronenko2017}. All here analyzed ampullary cells had CVs lower than 0.3 and exhibited strong nonlinear effects in accordance with the theoretical predictions \citealp{Voronenko2017}.
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%and here this could be confirmed experimentally.
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@ -558,7 +558,7 @@ The small \fdiff{} peak in the power spectrum of the firing rate appearing durin
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%Second-order susceptibility for all frequencies
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\subsection{Low CVs are associated with strong nonlinearity on a population level}%when considering
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So far second-order susceptibility was illustrated only with single-cell examples (\figrefb{cells_suscept}, \figrefb{ampullary}). For a P-unit comparison on a population level, the second-order susceptibility of P-units was expressed in a nonlinearity index \nli{}, see \eqnref{nli_equation}, that characterized the peakedness of the \fbase{} peak in the projected diagonal (\subfigrefb{cells_suscept}\,\panel[vi]{A}). \nli{} has high values when the \fbase{} peak in the projected diagonal is especially pronounced, as in the low-CV ampullary cell (\subfigrefb{ampullary}\,\panel[vi]{A}, light green). The two noise stimulus contrasts of this ampullary cell are highlighted in the population statics of ampullary cells with dark circles (\subfigrefb{data_overview_mod}{A}). The higher noise stimulus contrast is associated with a less pronounced peak in the projected diagonal (\subfigrefb{ampullary}\,\panel[vi]{A}, dark green) and is represented with a lower \nli{} value (\subfigrefb{data_overview_mod}{A}, dark circle close to the origin). In an ampullary cell population, there is a negative correlation between the CV during baseline and \nli{}, meaning that the diagonals are pronounced for low-CV cells and disappear towards high-CV cells (\subfigrefb{data_overview_mod}{A}). Since the same stimulus can be strong for some cells and faint for others, the noise stimulus contrast is not directly comparable between cells. A better estimation of the subjective stimulus strength is the response modulation of the cell (see methods section \ref{response_modulation}). Ampullary cells with stronger response modulations have lower \nli{} scores (red in \subfigrefb{data_overview_mod}{A}, \subfigrefb{data_overview_mod}{C}). The so far shown population statistics comprised several RAM contrasts per cell and if instead each ampullary cell is represented with the lowest recorded contrast, then \nli{} significantly correlates with the CV during baseline ($r=-0.46$, $p<0.001$), the response modulation ($r=-0.6$, $p<0.001$) but not with \fbase{} ($r=0.2$, $p=0.16$).%, $\n{}=51$, $\n{}=51$, $\n{}=51${*}{*}{*}^*^*^*each cell can contribute several RAM contrasts in
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So far second-order susceptibility was illustrated only with single-cell examples (\figrefb{cells_suscept}, \figrefb{ampullary}). For a P-unit comparison on a population level, the second-order susceptibility of P-units was expressed in a nonlinearity index \nli{}, see \eqnref{nli_equation}, that characterized the peakedness of the \fbase{} peak in the projected diagonal (\subfigrefb{cells_suscept}{G}). \nli{} has high values when the \fbase{} peak in the projected diagonal is especially pronounced, as in the low-CV ampullary cell (\subfigrefb{ampullary}{G}, light green). The two noise stimulus contrasts of this ampullary cell are highlighted in the population statics of ampullary cells with dark circles (\subfigrefb{data_overview_mod}{A}). The higher noise stimulus contrast is associated with a less pronounced peak in the projected diagonal (\subfigrefb{ampullary}{G}, dark green) and is represented with a lower \nli{} value (\subfigrefb{data_overview_mod}{A}, dark circle close to the origin). In an ampullary cell population, there is a negative correlation between the CV during baseline and \nli{}, meaning that the diagonals are pronounced for low-CV cells and disappear towards high-CV cells (\subfigrefb{data_overview_mod}{A}). Since the same stimulus can be strong for some cells and faint for others, the noise stimulus contrast is not directly comparable between cells. A better estimation of the subjective stimulus strength is the response modulation of the cell (see methods section \ref{response_modulation}). Ampullary cells with stronger response modulations have lower \nli{} scores (red in \subfigrefb{data_overview_mod}{A}, \subfigrefb{data_overview_mod}{C}). The so far shown population statistics comprised several RAM contrasts per cell and if instead each ampullary cell is represented with the lowest recorded contrast, then \nli{} significantly correlates with the CV during baseline ($r=-0.46$, $p<0.001$), the response modulation ($r=-0.6$, $p<0.001$) but not with \fbase{} ($r=0.2$, $p=0.16$).%, $\n{}=51$, $\n{}=51$, $\n{}=51${*}{*}{*}^*^*^*each cell can contribute several RAM contrasts in
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The P-unit population has higher baseline CVs and lower \nli{} values (\subfigrefb{data_overview_mod}{B}) that are weaker correlated than in the population of ampullary cells. The negative correlation (\subfigrefb{data_overview_mod}{B}) is increased when \nli{} is plotted against the response modulation of P-units (\subfigrefb{data_overview_mod}{D}). The two example P-units shown before (\figrefb{cells_suscept}) are highlighted with dark circles in \subfigrefb{data_overview_mod}{B, D}. High-CV P-units and strongly driven P-units have lower \nli{} values (\subfigrefb{data_overview_mod}{B, D}). 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{} ($r=-0.32$, $p<0.001$).%, $\n{}=221$*, $\n{}=221$******, $\n{}=221$
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@ -750,7 +750,7 @@ We expect to see non-linear susceptibility when $\omega_1 + \omega_2 = f_{Base}$
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NLI(f_{Base}) = \frac{\max_{f_{Base}-5\,\rm{Hz} \leq f \leq f_{Base}+5\,\rm{Hz}} D(f)}{\mathrm{med}(D(f))}
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\end{equation}
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\notejg{sollte es $D(\omega)$ sein?}
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For this index, the second-order susceptibility matrix was projected onto the diagonal $D(f)$, by taking the mean of the anti-diagonals. The peakedness at the frequency $f_{Base}$ in $D(f)$ was quantified by finding the maximum of $D(f)$ in the range $f_{Base} \pm 5$\,Hz (\subfigrefb{cells_suscept}\,\panel[vi]{A}, gray area) and dividing it by the median of $D(f)$.
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For this index, the second-order susceptibility matrix was projected onto the diagonal $D(f)$, by taking the mean of the anti-diagonals. The peakedness at the frequency $f_{Base}$ in $D(f)$ was quantified by finding the maximum of $D(f)$ in the range $f_{Base} \pm 5$\,Hz (\subfigrefb{cells_suscept}{G}, gray area) and dividing it by the median of $D(f)$.
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If the same frozen noise was recorded several times in a cell, each noise repetition resulted in a separate second-order susceptibility matrix. The mean of the corresponding \nli{} values was used for the population statistics in \figref{data_overview_mod}.
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\notejg{should go to the legend: calculated based on the first frozen noise repeat.}
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