From 086264121f1aebd4077664883dd8a75b70cec012 Mon Sep 17 00:00:00 2001 From: Jan Benda Date: Wed, 11 Feb 2026 11:00:31 +0100 Subject: [PATCH] Jan Gs great comments on manuscript and rebuttal --- nonlinearbaseline.tex | 16 ++++----- rebuttal2.tex | 81 +++++++++++++++++++++++-------------------- 2 files changed, 51 insertions(+), 46 deletions(-) diff --git a/nonlinearbaseline.tex b/nonlinearbaseline.tex index fc437c5..6163bee 100644 --- a/nonlinearbaseline.tex +++ b/nonlinearbaseline.tex @@ -275,7 +275,7 @@ Supported by SPP 2205 ``Evolutionary optimisation of neuronal processing'' by th % 250 words \section{Abstract} Spiking thresholds in neurons or rectification at synapses are essential for neuronal computations rendering neuronal processing inherently nonlinear. Nevertheless, linear response theory has been instrumental for understanding, for example, the impact of noise or neuronal synchrony on signal transmission, or the emergence of oscillatory activity, but is valid only at low stimulus amplitudes or large levels of intrinsic noise. At higher signal-to-noise ratios, however, nonlinear response components become relevant. Theoretical results for leaky integrate-and-fire neurons in the weakly nonlinear regime suggest strong responses at the sum of two input frequencies if one of these frequencies or their sum match the neuron's baseline firing rate. -We here analyze nonlinear responses in two types of primary electroreceptor afferents, the P-units of the active and the ampullary cells of the passive electrosensory system of the wave-type electric fish \textit{Apteronotus leptorhynchus}. In our combined experimental and modeling approach we identify these predicted nonlinear responses primarily in a few low-noise P-units and in more than every second ampullary cell. Our results provide experimental evidence for nonlinear responses of spike generators in the weakly nonlinear regime. We conclude that such nonlinear responses occur in any sensory neuron that operates in similar regimes particularly at near-threshold stimulus conditions. +We here analyze nonlinear responses in two types of primary electroreceptor afferents, the P-units of the active and the ampullary cells of the passive electrosensory system of the wave-type electric fish \textit{Apteronotus leptorhynchus}. In our combined experimental and modeling approach we identify these predicted nonlinear responses only in individual low-noise P-units, but in more than half of the ampullary cells. Our results provide experimental evidence for nonlinear responses of spike generators in the weakly nonlinear regime. We conclude that such nonlinear responses occur in any sensory neuron that operates in similar regimes particularly at near-threshold stimulus conditions. % max 120 words \section{Significance statement} @@ -286,7 +286,7 @@ The generation of action potentials involves a strong threshold nonlinearity. Ne \begin{figure*}[t] \includegraphics[width=\columnwidth]{lifsuscept} - \caption{\label{fig:lifsuscept} First- and second-order response functions of the leaky integrate-and-fire model. \figitem{A} Magnitude of the first-order (linear) response function $|\chi_1(f)|$, also known as the ``gain'' function, quantifies the response amplitude relative to the stimulus amplitude, both measured at the same stimulus frequency. $r$ marks the neuron's baseline firing rate. \figitem{B} Magnitude of the second-order (nonlinear) response function $|\chi_2(f_1, f_2)|$ quantifies the response at the sum of two stimulus frequencies. Because frequencies can also be negative, the sum is actually a difference in the upper-left and lower-right quadrants, since $f_2 + f_1 = f_2 - |f_1|$ for $f_1 < 0$. For linear systems, the second-order response function is zero, because linear systems do not create new frequencies and thus there is no response at the sum of the two frequencies. The plots show the analytical solutions from \citet{Lindner2001} and \citet{Voronenko2017} with $\mu = 1.1$ and $D = 0.001$. Note that the leaky integrate-and-fire model is formulated without dimensions, frequencies are given in multiples of the inverse membrane time constant.} + \caption{\label{fig:lifsuscept} First- and second-order response functions of the leaky integrate-and-fire model. \figitem{A} Magnitude of the first-order (linear) response function $|\chi_1(f)|$, also known as the ``gain'' function, quantifies the response amplitude relative to the stimulus amplitude, both measured at the same stimulus frequency. $r$ and $2r$ mark the neuron's baseline firing rate and its harmonic (dashed vertical lines). \figitem{B} Magnitude of the second-order (nonlinear) response function $|\chi_2(f_1, f_2)|$ quantifies the response at the sum of two stimulus frequencies. Because Fourier spectra have positive and negative frequencies, the sum is actually a difference in the upper-left and lower-right quadrants, since $f_2 + f_1 = f_2 - |f_1|$ for $f_1 < 0$. For linear systems, the second-order response function is zero, because linear systems do not create new frequencies and thus there is no response at the sum of the two frequencies. The plots show the analytical solutions from \citet{Lindner2001} and \citet{Voronenko2017} with $\mu = 1.1$ and $D = 0.001$. Note that the leaky integrate-and-fire model is formulated without dimensions, frequencies are given in multiples of the inverse membrane time constant.} \end{figure*} We like to think about signal encoding in terms of linear relations with unique mapping of a given input value to a certain output of the system under consideration. Indeed, such linear methods, for example the transfer function or first-order susceptibility shown in \subfigref{fig:lifsuscept}{A}, have been widely and successfully applied to describe and predict neuronal responses and are an invaluable tools to characterize neural systems \citep{Eggermont1983,Borst1999}. Nonlinear mechanisms, on the other hand, are key on different levels of neural processing. Deciding for one action over another is a nonlinear process on the systemic level. On the cellular level, spiking neurons are inherently nonlinear. Whether an action potential is elicited depends on the membrane potential to exceed a threshold \citep{Hodgkin1952, Koch1995}. Because of such nonlinearities, understanding and predicting neuronal responses to sensory stimuli is in general a difficult task. @@ -319,7 +319,7 @@ For monitoring the EOD without the stimulus, two vertical carbon rods ($11\,\cen \subsection{Stimulation}\label{rammethods} -Electric stimuli were attenuated (ATN-01M, npi-electronics, Tamm, Germany), isolated from ground (ISO-02V, npi-electronics, Tamm, Germany) and delivered via two horizontal carbon rods (30 cm length, 8 mm diameter) located $15\,\centi\meter$ parallel to each side of the fish. The fish were stimulated with band-limited white noise stimuli with a cut-off frequency of 150\,Hz (ampullary), 300 or 400\,Hz (P-units). The stimuli were generated by drawing normally distributed real and imaginary numbers for all frequencies up to the desired cut-off frequency in the Fourier domain and then applying an inverse Fourier transform \citep{Skorjanc2023}. The stimulus intensity is given as a contrast, i.e. the standard deviation of the resulting amplitude modulation relative to the fish's EOD amplitude. The contrast varied between 1 and 20\,\% (median 5\,\%) for P-units and 2.5 and 20\,\% (median 5\,\%) for ampullary cells. Only recordings with noise stimuli with a duration of at least 2\,s (maximum of 50\,s, median 10\,s) and enough repetitions to results in at least 100 FFT segments (see below, P-units: 100--1520, median 313, ampullary cells: 105 -- 3648, median 722) were included into the analysis. When ampullary cells were recorded, the noise stimuli $s(t)$ were directly applied as the stimulus and thus were simply added to the fish's own EOD: $s(t) + EOD(t)$. To create random amplitude modulations (RAM) for P-unit recordings, the noise stimulus was first multiplied with the EOD of the fish (MXS-01M; npi electronics) and then played back through the stimulation electrodes: $(1 + s(t))EOD(t)$. +Electric stimuli were attenuated (ATN-01M, npi-electronics, Tamm, Germany), isolated from ground (ISO-02V, npi-electronics, Tamm, Germany) and delivered via two horizontal carbon rods (30 cm length, 8 mm diameter) located $15\,\centi\meter$ parallel to each side of the fish. The fish were stimulated with band-limited white noise stimuli with a cut-off frequency of 150\,Hz (ampullary), 300 or 400\,Hz (P-units). The stimuli were generated by drawing normally distributed real and imaginary numbers for all frequencies up to the desired cut-off frequency in the Fourier domain and then applying an inverse Fourier transform \citep{Skorjanc2023}. The stimulus intensity is given as a contrast, i.e. the standard deviation of the resulting amplitude modulation relative to the fish's EOD amplitude. The contrast varied between 1 and 20\,\% (median 5\,\%) for P-units and 2.5 and 20\,\% (median 5\,\%) for ampullary cells. Only recordings with noise stimuli with a duration of at least 2\,s (maximum of 50\,s, median 10\,s) and enough repetitions to results in at least 100 FFT segments (see below, P-units: 100--1520, median 313, ampullary cells: 105 -- 3648, median 722) were included into the analysis. When ampullary cells were recorded, the noise stimuli $s(t)$ were directly applied as the stimulus and thus were simply added to the fish's own EOD: $s(t) + EOD(t)$. To create random amplitude modulations (RAM) for P-unit recordings, the noise stimulus was first multiplied with the EOD of the fish (MXS-01M; npi electronics) and then played back through the stimulation electrodes: $EOD(t) + s(t)EOD(t) = (1 + s(t))EOD(t)$. \subsection{Data analysis} Data analysis was done in Python 3 using the packages matplotlib \citep{Hunter2007}, numpy \citep{Walt2011}, scipy \citep{scipy2020}, nixio \citep{Stoewer2014}, and thunderlab (\url{https://github.com/bendalab/thunderlab}). @@ -338,7 +338,7 @@ To characterize the relation between the spiking response evoked by white-noise \label{fourier} X(\omega) = \sum_{k=0}^{N-1} \, x_k e^{- i \omega k} \end{equation} -for $N=512$ long segments of $T=N \Delta t = 256$\,ms duration with no overlap, resulting in a spectral resolution of about 4\,Hz. In \figref{fig:regimes} we used $N=2^{18}$ and $\Delta t=0.1$\,ms. Note, that for a real Fourier integral a factor $\Delta t$ is missing. For simplicity we use angular frequencies $\omega=2\pi f$ instead of frequencies $f$. Also note, that the Fourier transform of a signal results in both positive as well as negative frequencies. This is because the signal is decomposed into complex exponentials, not sine or cosine waves. To get back a real-valued sine or cosine wave, the components at positive and negative frequencies need to be combined. For example, the sum $e^{-\omega t} + e^{+\omega t}$ of two complex exponentials at frequencies $\pm\omega$ is $2\cos(\omega t)$. +for $N=512$ long segments of $T=N \Delta t = 256$\,ms duration with no overlap, resulting in a spectral resolution of about 4\,Hz. In \figref{fig:regimes} we used $N=2^{18}$ and $\Delta t=0.1$\,ms (26.2\,s). Note, that for a real Fourier integral a factor $\Delta t$ is missing. For simplicity we use angular frequencies $\omega=2\pi f$ instead of frequencies $f$. Also note, that the Fourier transform of a signal results in both positive as well as negative frequencies. This is because the signal is decomposed into complex exponentials, not sine or cosine waves. To get back a real-valued sine or cosine wave, the components at positive and negative frequencies need to be combined. For example, the sum $e^{-\omega t} + e^{+\omega t}$ of two complex exponentials at frequencies $\pm\omega$ is $2\cos(\omega t)$. In the experimental data the duration of the noise stimuli varied and they were presented once or repeatedly (frozen noise). For the analysis we discarded the responses within the initial 200\,ms of stimulation in each trial. To make the recordings comparable we always used the first 100 segments from as many trials as needed for the following analysis. @@ -436,7 +436,7 @@ First, the input $y(t)$ is thresholded by setting negative values to zero: \label{eq:dendrite} \tau_{d} \frac{d V_{d}}{d t} = -V_{d}+ \lfloor y(t) \rfloor_{0} \end{equation} -the threshold operation is required for extracting the amplitude modulation from the input \citep{Barayeu2024}. As detailed in the discussion, the details of the threshold nonlinearity \eqnref{eq:threshold2} and the low-pass filter \eqnref{eq:dendrite} do not matter here, since we compute all cross-spectra between the spiking response and the amplitude-modulation $s(t)$ --- and not the full input signal $y(t)$, \eqnref{eq:ram_equation}. +the threshold operation is required for extracting the amplitude modulation from the input \citep{Barayeu2024}. As detailed in the discussion, the intricacies of the threshold nonlinearity \eqnref{eq:threshold2} and the low-pass filter \eqnref{eq:dendrite} do not matter here, as all cross-spectra were computed between the spiking response and the amplitude-modulation $s(t)$ --- and not the full input signal $y(t)$, \eqnref{eq:ram_equation}. The low-pass filter models passive signal conduction in the afferent's dendrite (\subfigrefb{flowchart}{B}) from the synapse to the spike initiation zone. $\tau_{d}$ is the effective membrane time constant of this part of the dendrite. Dendritic low-pass filtering was also necessary to reproduce the loose coupling of P-unit spikes to the EOD while maintaining high sensitivity at small amplitude modulations. @@ -498,10 +498,10 @@ When stimulating with both foreign signals simultaneously, additional peaks appe \begin{figure*}[p] \includegraphics[width=\columnwidth]{regimes} - \caption{\label{fig:regimes} Linear and nonlinear responses of a model P-unit in a three-fish setting in dependence on stimulus amplitudes. The model P-unit (identifier ``2018-05-08-ad'') was stimulated with two sine waves of equal amplitude (contrast) at difference frequencies $\Delta f_1=40$\,Hz and $\Delta f_2=228$\,Hz relative the receiver's EOD frequency $f_{EOD}=656$\,Hz. $\Delta f_2$ was set to match the baseline firing rate $r$ of the P-unit. \figitem{A--D} Top row: the stimulus, an amplitude modulation of the receiver's EOD resulting from the stimulation with the two sine waves. The contrasts of both beats increase from \panel{A} to \panel{D} as indicated. Second row: Spike raster of the model P-unit response. Third row: average firing rate and, bottom row: power spectrum estimated from 1000 trials of spike trains in response to a 26.2\,s long stimulus. \figitem{A} At low stimulus contrasts the response to the two beat frequencies is linear. These frequencies are present in the response spectrum (green and purple circles), the peak at $\Delta f_2$ (purple) enhances the peak at baseline firing rate (blue). The largest peak, however, is always the one at the EOD frequency of the receiver (black), reflecting the locking of P-unit spikes to the receiver's own EOD. The peak at $f_{EOD}$ is also flanked by harmonics of $\Delta f_1$ (gray triangles), where $f_{EOD} + \Delta f_1 = f_1$ is the EOD frequency of the first fish. \figitem{B} At moderately higher stimulus contrast, the peaks in the response spectrum at the two beat frequencies become larger. In addition, two peaks at $f_{EOD} - \Delta f_2$ and $f_{EOD} + \Delta f_2$ appear (blue diamonds), the latter is the EOD frequency of the second fish, $f_2$. These peaks indicate nonlinear processes acting on the three EOD frequencies from which the beat frequencies are generated. \figitem{C} At intermediate stimulus contrasts, nonlinear responses start to appear at the sum and the difference of the beat frequencies (orange and red circles). Similarly, side peaks appear at $\pm \Delta f_1$ around $f_{EOD} \pm \Delta f_2$ (purple squares) as well as harmonics of $\Delta f_1$ (small green circles). \figitem{D} At higher stimulus contrasts many more peaks appear in the power spectrum. Most of them are interactions of $f_{EOD}$ and $\Delta f_2$ with harmonics of $\Delta f_1$ ($k=1, 2, ...$), since the cell responds strongest to $\Delta f_1$. \figitem{E} Amplitude of the linear (at $\Delta f_1$ and $\Delta f_2$) and nonlinear (at $\Delta f_2 - \Delta f_1$ and $\Delta f_1 + \Delta f_2$) responses of the model P-unit as a function of beat contrast (thick lines). Thin lines indicate the initial linear and quadratic dependence on stimulus amplitude for the linear and nonlinear responses, respectively. In the linear regime, below a stimulus contrast of about 1.2\,\% (left vertical line), the only peaks in the response spectrum are at the stimulus frequencies. In the weakly nonlinear regime up to a contrast of about 3.5\,\% peaks arise at the sum and the difference of the two stimulus frequencies. At stronger stimulation the amplitudes of these nonlinear responses deviate from the quadratic dependency on stimulus contrast.} + \caption{\label{fig:regimes} Linear and nonlinear responses of a model P-unit in a three-fish setting in dependence on stimulus amplitudes. The model P-unit (identifier ``2018-05-08-ad'') was stimulated with two sine waves of equal amplitude (contrast) at difference frequencies $\Delta f_1=40$\,Hz and $\Delta f_2=228$\,Hz relative the receiver's EOD frequency $f_{EOD}=656$\,Hz. $\Delta f_2$ was set to match the baseline firing rate $r$ of the P-unit. \figitem{A--D} Top row: the stimulus, an amplitude modulation of the receiver's EOD resulting from the stimulation with the two sine waves. The contrasts of both beats increase from \panel{A} to \panel{D} as indicated. Second row: Spike raster of the model P-unit response. Third row: average firing rate and, bottom row: power spectrum estimated from 1000 trials of spike train responses. \figitem{A} At low stimulus contrasts the response to the two beat frequencies is linear. These frequencies are present in the response spectrum (green and purple circles), the peak at $\Delta f_2$ (purple) enhances the peak at baseline firing rate (blue). The largest peak, however, is always the one at the EOD frequency of the receiver (black), reflecting the locking of P-unit spikes to the receiver's own EOD. The peak at $f_{EOD}$ is also flanked by harmonics of $\Delta f_1$ (gray triangles), where $f_{EOD} + \Delta f_1 = f_1$ is the EOD frequency of the first fish. \figitem{B} At moderately higher stimulus contrast, the peaks in the response spectrum at the two beat frequencies become larger. In addition, two peaks at $f_{EOD} - \Delta f_2$ and $f_{EOD} + \Delta f_2$ appear (blue diamonds), the latter is the EOD frequency of the second fish, $f_2$. These peaks indicate nonlinear processes acting on the three EOD frequencies from which the beat frequencies are generated. \figitem{C} At intermediate stimulus contrasts, nonlinear responses start to appear at the sum and the difference of the beat frequencies (orange and red circles). Similarly, side peaks appear at $\pm \Delta f_1$ around $f_{EOD} \pm \Delta f_2$ (purple squares) as well as harmonics of $\Delta f_1$ (small green circles). \figitem{D} At higher stimulus contrasts many more peaks appear in the power spectrum. Most of them are interactions of $f_{EOD}$ and $\Delta f_2$ with harmonics of $\Delta f_1$ ($k=1, 2, ...$), since the cell responds strongest to $\Delta f_1$. \figitem{E} Amplitude of the linear (at $\Delta f_1$ and $\Delta f_2$) and nonlinear (at $\Delta f_2 - \Delta f_1$ and $\Delta f_1 + \Delta f_2$) responses of the model P-unit as a function of beat contrast (thick lines). Thin lines indicate the initial linear and quadratic dependence on stimulus amplitude for the linear and nonlinear responses, respectively. In the linear regime, below a stimulus contrast of about 1.2\,\% (left vertical line), the only peaks in the response spectrum are at the stimulus frequencies. In the weakly nonlinear regime up to a contrast of about 3.5\,\% peaks arise at the sum and the difference of the two stimulus frequencies. At stronger stimulation the amplitudes of these nonlinear responses deviate from the quadratic dependency on stimulus contrast.} \end{figure*} -The stimuli used in \figref{fig:twobeats} had the same not-small amplitude resulting in AM contrasts of 10\,\% that evoke strong modulations in a P-unit's firing rate response. Whether this stimulus condition falls into the weakly nonlinear regime as in \citet{Voronenko2017} is not clear. In order to illustrate how the responses to two beat frequencies develop over a range of amplitudes we use a stochastic leaky-integrate-and-fire (LIF) based P-unit model fitted to a specific electrophysiologically measured cell \citep{Barayeu2023}. +The stimuli used in \figref{fig:twobeats} had the same AM contrasts of 10\,\% that evoke rather strong modulations in a P-unit's firing rate response. Whether these definitely not-small stimulus amplitudes fall into the weakly nonlinear regime as in \citet{Voronenko2017} is not clear. In order to illustrate how the responses to two beat frequencies develop over a range of amplitudes we use a stochastic leaky-integrate-and-fire (LIF) based P-unit model fitted to a specific electrophysiologically measured cell \citep{Barayeu2023}. At very low stimulus contrasts (in the example cell less than approximately 1.2\,\% relative to the receiver's EOD amplitude) the spectrum has small peaks at the beat frequencies (\subfigref{fig:regimes}{A,B}, green and purple). The amplitudes of these peaks initially increase linearly with stimulus amplitude (\subfigref{fig:regimes}{E}, thin lines), an indication of the linear response at lowest stimulus amplitudes. The largest peak at the receiver's EOD frequency indicates the stochastic locking of the P-unit spikes to the EOD, and its side-peaks are remnants of the two stimulating EOD frequencies \citep{Sinz2020}. @@ -574,7 +574,7 @@ The SI($r$) correlates with the CVs of the cell's baseline interspike intervals \subsection{Weakly nonlinear interactions vanish for higher stimulus contrasts} -As pointed out above, strong noise simuli act as a linearizing background noise and thus shift the neural signal transmission into a regime free of nonlinearities. Sine-wave stimuli, as for example used in the introductory figure~\ref{fig:regimes}, do not have such a linearizing effect. What exactly constitutes a 'strong' stimulus causing an effective increase of the background noise, also depends on all the other cellular properties of the neuron, or on the parameters in our leaky integrate-and-fire neuron. This is what we explore now by varying the contrast of RAM stimuli driving our model cells. +As pointed out above, noise stimuli act as a linearizing background noise and thus may shift the neural signal transmission into a regime free of nonlinearities (see discussion). Sine-wave stimuli, as for example used in the introductory figure~\ref{fig:regimes}, do not have such a linearizing effect. At which amplitudes a noise stimulus effectively linearizes the system, also depends on cellular properties of the neuron, or on the parameters in our leaky integrate-and-fire neuron. This is what we explore now by varying the contrast of RAM stimuli driving our model cells. In the model cells we estimated second-order susceptibilities for RAM stimuli with a contrast of 1, 3, and 10\,\%. The estimates for 1\,\% contrast (\figrefb{fig:modelsusceptcontrasts}\,\panel[ii]{E}) were quite similar to the estimates from the noise-split method, corresponding to a stimulus contrast of 0\,\% ($r=0.97$, $p\ll 0.001$). Thus, RAM stimuli with 1\,\% contrast are sufficiently small to not destroy weakly nonlinear interactions by their linearizing effect. At this low contrast, 51\,\% of the model cells have an SI($r$) value greater than 1.2. diff --git a/rebuttal2.tex b/rebuttal2.tex index 2bdb5ff..cabb6a5 100644 --- a/rebuttal2.tex +++ b/rebuttal2.tex @@ -33,6 +33,10 @@ \begin{document} +Thank you for your valuable feedback. Line numbers mentioned in our +responses refer to the new version of the manuscript, not the redlined +one. + \issue{\large Reviewer \#1} \issue{The manuscript "Spike generation in electroreceptor afferents @@ -55,7 +59,8 @@ the manuscript or inspire future research endeavors.} \response{Thank you for trying to make our manuscript more - biologist-friendly!} + biologist-friendly! And yes, some of your comments indeed inspired + our thinking for future projects.} \issue{First, I should point out that beyond the presence of a threshold-induced nonlinearity, the complex structure of the @@ -81,11 +86,7 @@ potentially contributing to the threshold nonlinearity. We now mention this in the methods when introducing the threshold nonlinearity (after eq. 13) and cite the corresponding - manuscripts. We also added a sentence there reiterating what we have - in the discussion, namely that the details of this nonlinearity are - not important in the context of the present manuscript, since we - compute all cross-spectra between the resulting amplitude modulation - and the spikes responses.} + manuscripts.} \issue{Second, and along the same lines, the discussion could be improved by mentioning the effects and significance of these @@ -167,7 +168,7 @@ averaged sine wave recorded via local electrodes adjacent to the gills exhibited an increase of 1 to 5\%, is this correct?} -\response{No! We increased the amplitude of the white noise until the +\response{We increased the amplitude of the white noise until the standard deviation (not the mean) of the resulting modulation of the EOD reached 1 to 5\,\%. We rephrased the description of the stimulation and hope that this is clearer now (lines 164--168).} @@ -194,14 +195,15 @@ \response{You are right about the phase shifts and that this does not ``significantly impact individual receptors response''. This is a standard stimulation procedure for characterizing receptor responses - that are located mainly on the sides of a fish's flat body. See, for - example, Hladnik and Grewe, 2023. And yes, this will probably impact - relative spike timing in distinct receptors and thus may also impact - the JAR mechanisms. However, this manuscript is about single - receptor responses and not about T-units, and we feel it is already - complicated enough. Therefore we would rather prefer to not open up - all these issues, since they are not relevant for the results we - present.} + that are located mainly on the sides of a fish's flat body as the + recording site is on the posterior branch of the lateral line + nerve. See, for example, Hladnik and Grewe, 2023. And yes, this will + probably impact relative spike timing in distinct receptors and thus + may also impact the JAR mechanisms. However, this manuscript is + about single receptor responses and not about T-units, and we feel + it is already complicated enough. Therefore we would rather prefer + to not open up all these issues, since they are not relevant for the + results we present.} \issue{Line 238. Are you referring to the terminal non-myelinated branches that connect receptor cells to the initial Ranvier node? @@ -246,9 +248,9 @@ P-units, but "much stronger" does not clearly convey this, especially in the abstract.} -\response{We changed the sentence to ``... we identify these - predicted nonlinear responses primarily in a few low-noise P-units - and in more than every second ampullary cell.''} +\response{We changed the sentence to ``... identify these predicted + nonlinear responses only in individual low-noise P-units, but in + more than half of the ampullary cells.''} \issue{(3) Figure 1A. "r" needs to be clearly defined here. Based on the text, it seems to be the baseline firing rate of the neuron, but @@ -262,7 +264,8 @@ be negative?} \response{We added a few sentences following equation (1) to motivate - the existence of negative frequencies in Fourier transforms.} + the existence of negative frequencies in Fourier transforms. And we + added a hint in the caption of figure 1B.} \issue{(5) Figure 3 and 4. Why are the power spectra clipped at such low frequencies? This makes it impossible to see peaks due to @@ -279,8 +282,8 @@ baseline firing rate (only three trials of 500ms duration). This is why the power spectra are very noisy. Also, for an introductory figure we prefer to only show the few peaks that are relevant for - the rest of the manuscript, such that the reader does not get - overwhelmed.} + the rest of the manuscript, to not overwhelm the reader right at the + start.} \issue{(6) Figure 3. Why are these example firing rates based on convolution with a 1 ms Gaussian kernel if the analyses were based @@ -289,16 +292,16 @@ actually analyzed. More fundamentally, why would a 2-fold difference in kernel width be appropriate for presentation vs. analysis?} -\response{This was for ``historical'' reasons. We now decided to use the - 1\,ms kernel for all figures and analysis. In doing so we also added - panels showing firing rates in addition to the response spectra in - figure 4. Using the more narrow kernel better reveals the details of - the time course of the firing rate and this way improves the - connection between the firing rate and the response spectra. In - figure 10, middle column, the range of possible values of the - response modulations is a bit enlarged by using the 1\,ms kernel, - but the correlations and their significance did not change a lot - either.} +\response{Thank you for addressing this inconsistency. This was for + ``historical'' reasons. We now decided to use the 1\,ms kernel for + all figures and analysis. In doing so we also added panels showing + firing rates in addition to the response spectra in figure 4. Using + the more narrow kernel better reveals the details of the time course + of the firing rate and this way improves the connection between the + firing rate and the response spectra. In figure 10, middle column, + the range of possible values of the response modulations is a bit + enlarged by using the 1\,ms kernel, but the correlations and their + significance did not change a lot either.} \issue{(7) Figure 3D legend. The relationship between 2nd order AM (envelope) and the two nonlinear peaks should be made clear. I @@ -320,14 +323,13 @@ \issue{(8) Line 302. "not-small amplitude" is arbitrary and vague. Please be clearer and more precise.} -\response{We added ``resulting in AM contrasts of 10\,\% that evoke - strong modulations in a P-unit's firing rate response''.} +\response{We rephrased to two sentences in lines 323 -- 325.} \issue{(9) Figures 5C and 6C. For the stimuli with the red RAM waveforms, please make it clear which contrast is being represented by these traces, as responses to two different contrasts are shown.} -\response{We added the shown stimulus contrasts to both figures.} +\response{We added the contrast values to both figures.} \issue{(10) Figure 5E, F. The legend states that second-order susceptibility for both the low and high stimulus contrasts are @@ -342,7 +344,7 @@ ratios should result in larger nonlinearities?} \response{Yes, in figure 4 increasing stimulus contrast results in - stronger nonlinearities. There the stimuli are narrow-band sine + stronger nonlinearities. There, the stimuli are narrow-band sine waves. However, as pointed out in the context of figure 7, when using a broad-band noise stimulus instead, this stimulus by itself adds background noise to the system that linearizes the @@ -371,8 +373,11 @@ stimulating frequencies or their sum with the neuron's baseline firing rate is required. This is all addressed in the (now second final) paragraph of the ``Nonlinear encoding in P-units'' section. - However, we agree that the comparative aspect of the conclusion - could be expanded. We therefore added one more final speculative - sentence to the conclusion.} + Nevertheless, in response to reviewer \#1, we added another + paragraph discussing various behaviors that modulate the EOD + frequency and how these may exploit the weakly nonlinear + interactions. However, we agree that the comparative aspect of the + conclusion could be expanded. We therefore added one more final + speculative sentence to the conclusion.} \end{document}