Benjamins last comments on manuscript and rebuttal

nonlinearbaseline.tex

@@ -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: $EOD(t) + s(t)EOD(t) = (1 + s(t))EOD(t)$.
+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 Gaussian white noise stimuli, i.e. signals with equal power at all frequencies up to a cut-off frequency and with a stationary Gaussian probability density. For the ampullary cells we chose a cut-off frequency of 150\,Hz, whereas for the P-units we used either 300 or 400\,Hz. 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{Billah1990,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}).
 
@@ -677,7 +677,7 @@ Making assumptions about the nonlinearities in a system also reduces the amount
 
 The afferents of the passive electrosensory system, the ampullary cells, exhibit strong second-order susceptibilities (\figref{fig:dataoverview}). Ampullary cells more or less directly translate external low-frequency electric fields into afferent spikes, much like in the standard LIF and theta models used by Lindner and colleagues \citep{Voronenko2017,Franzen2023}. Indeed, we observe in the ampullary cells similar second-order nonlinearities as in LIF models.
 
-Ampullary stimuli originate from the muscle potentials induced by prey movement \citep{Kalmijn1974, Engelmann2010, Neiman2011fish}. For a single prey items such as \textit{Daphnia}, these potentials are often periodic but the simultaneous activity of a swarm of prey resembles Gaussian white noise \citep{Neiman2011fish}. Linear and nonlinear encoding in ampullary cells has been studied in great detail in the paddlefish \citep{Neiman2011fish}. The power spectrum of the baseline response shows two main peaks: One peak at the baseline firing frequency, a second one at the oscillation frequency of primary receptor cells in the epithelium, plus interactions of both. Linear encoding in the paddlefish shows a gap at the epithelial oscillation frequency, instead, nonlinear responses are very pronounced there.
+Ampullary stimuli originate from the muscle potentials induced by prey movement \citep{Kalmijn1974, Engelmann2010, Neiman2011fish}. For a single prey item such as \textit{Daphnia}, these potentials are often periodic but the simultaneous activity of a swarm of prey resembles Gaussian white noise \citep{Neiman2011fish}. Linear and nonlinear encoding in ampullary cells has been studied in great detail in the paddlefish \citep{Neiman2011fish}. The power spectrum of the baseline response shows two main peaks: One peak at the baseline firing frequency, a second one at the oscillation frequency of primary receptor cells in the epithelium, plus interactions of both. Linear encoding in the paddlefish shows a gap at the epithelial oscillation frequency, instead, nonlinear responses are very pronounced there.
 
 Ampullary stimulus encoding is somewhat different in \lepto{}. The power spectrum of the spontaneous response is dominated by only the baseline firing rate and its harmonics, a second oscillator is not visible. The baseline firing frequency, however, is outside the linear coding range \citep{Grewe2017} while it is within the linear coding range in paddlefish \citep{Neiman2011fish}. Interestingly, the nonlinear response in the paddlefish ampullary cells increases with stimulus intensity while it disappears in our case (\figrefb{fig:dataoverview}~\panel[ii]{C}) indicating that paddlefish data have been recorded above the weakly-nonlinear regime. 
 
@@ -687,13 +687,13 @@ The population of ampullary cells is very homogeneous with respect to the baseli
 
 In contrast to the ampullary cells, P-units respond to the amplitude modulation of the self-generated EOD. Extracting the AM requires a (threshold) nonlinearity \citep{Middleton2006, Stamper2012Envelope, Savard2011, Barayeu2023}. This nonlinearity, however, does not show up in our estimates of the susceptibilities, because in our analysis we directly relate the AM waveform to the recorded cellular responses. Encoding the time-course of the AM, however, has been shown to be linear over a wide range of AM amplitudes and frequencies \citep{Xu1996, Benda2005, Gussin2007, Grewe2017, Savard2011}. In contrast, we here have demonstrated nonlinear interactions originating from the spike generator for broad-band noise stimuli with small amplitudes and for stimulation with two distinct frequencies. Both settings have not been studied yet.
 
-Noise stimuli have the advantage that a range of frequencies can be measured with a single stimulus presentation and they have been successfully applied to characterize sensory coding in many systems \citep{French1973, Marmarelis1999, Borst1999, Chacron2005, Grewe2017}. The natural stimuli encoded by P-units are, however, periodic amplitude modulations of the self-generated electric field which arise from the superposition of the own and foreign EODs. Such interactions usually occur between low numbers of close-by fish and thus the AMs are a mixture of a few distinct frequencies with substantial amplitudes \citep{Stamper2010,Fotowat2013, Henninger2020}. How informative are the second-order susceptibilities observed under noise stimulation for the encoding of distinct frequencies? Broadband noise stimuli introduce additional noise that linearizes the dynamics of the system. In contrast, pure sine wave stimulation is spectrally focused and drives the system on the background of the intrinsic noise. This explains why we can observe nonlinear interactions between sine wave stimuli with distinct frequencies and substantial power (\figrefb{fig:twobeats}) although these interactions vanish when stimulating with noise stimuli of similar contrast (\figrefb{fig:modelsusceptcontrasts}).
+Noise stimuli have the advantage that a range of frequencies can be measured with a single stimulus presentation and they have been successfully applied to characterize sensory coding in many systems \citep{French1973, Marmarelis1999, Borst1999, Chacron2005, Grewe2017}. The natural stimuli encoded by P-units are, however, periodic amplitude modulations of the self-generated electric field which arise from the superposition of the own and foreign EODs. Such interactions usually occur between low numbers of close-by fish and thus the AMs are a mixture of a few distinct frequencies with substantial amplitudes \citep{Stamper2010,Fotowat2013, Henninger2020}. How informative are the second-order susceptibilities observed under noise stimulation for the encoding of distinct frequencies? Broadband noise stimuli introduce additional noise that linearizes the dynamics of the system. In contrast, a pure sine wave stimulation is spectrally focused and drives the system on the background of the intrinsic noise. This explains why we can observe nonlinear interactions between sine wave stimuli with distinct frequencies and substantial power (\figrefb{fig:twobeats}) although these interactions vanish when stimulating with noise stimuli of similar contrast (\figrefb{fig:modelsusceptcontrasts}).
 
 The encoding of secondary AMs or social envelopes that arise from relative movement or the interaction of more than two animals \citep{Stamper2012Envelope} requires another nonlinearity in addition to the one needed for extracting the AM. Initially, this nonlinearity was attributed to downstream processing \citep{Middleton2006, Middleton2007}. Later studies showed that already the electroreceptors can encode such information whenever the firing rate saturates at zero or the maximum rate at the EOD frequency \citep{Savard2011}. Based on our work, we predict that P-units with low CVs encode the social envelopes even under weak stimulation, whenever the resulting beat frequencies match or add up to the baseline firing rate. Then difference frequencies show up in the response spectrum that characterize the slow envelope.
 
 The weakly nonlinear interactions in low-CV P-units could facilitate the detectability of faint signals during three-animal interactions as found in courtship contexts in the wild \citep{Henninger2018}. The detection of a faint, distant intruder male could be improved by the presence of a nearby strong female stimulus, because of the nonlinear interaction terms \citep{Schlungbaum2023}. This boosting effect is, however, very specific with respect to the stimulus frequencies and a given P-unit's baseline frequency. The population of P-units is very heterogeneous in their baseline firing rates and CVs (50--450\,Hz and 0.1--1.4, respectively, \subfigref{fig:dataoverview}{B}, \citealp{Grewe2017, Hladnik2023}). The range of baseline firing rates thus covers substantial parts of the beat frequencies that may occur during animal interactions \citep{Henninger2018, Henninger2020}, while the number of P-units showing weakly nonlinear responses is small. Whether and how this information is specifically maintained and read out by pyramidal cells in the electrosensory lateral line lobe (ELL) in the hindbrain onto which P-units converge \citep{Krahe2014, Maler2009a} is an open question.
 
-Electric fish are able to slowly modulate their EOD frequency, as for example during the so-called jamming-avoidance-response \citep{Fortune2020}. Such behaviors modify the resulting beat frequency by a few Hertz. This could in principle increase the chance that the now slowly changing beat frequency matches at some point the baseline firing rate of a P-unit, where the weakly nonlinear responses then enhance the detectability of another conspecific \citep{Schlungbaum2023}. Furthermore, transient changes in EOD frequency on timescales of tens of milliseconds up to a few seconds are known as chirps and rises, respectively, and are involved in courtship and aggression behaviors \citep{Henninger2018, Raab2021}. How the encoding of such transient frequency modulations is affected by the nonlinearities described here is an open question, since the presented analysis focuses on stationary signals.
+Electric fish are able to slowly modulate their EOD frequency, as for example during the so-called jamming-avoidance-response \citep{Fortune2020}. Such behaviors modify the resulting beat frequency by a few Hertz. This could in principle increase the chance that the now slowly changing beat frequency matches at some point the baseline firing rate of a P-unit, where the weakly nonlinear responses then enhance the detectability of another conspecific \citep{Schlungbaum2023}. Furthermore, transient changes in EOD frequency on timescales of tens of milliseconds up to a few seconds are known as chirps and rises, respectively, and are involved in courtship and aggression behaviors \citep{Henninger2018, Raab2021}. How the encoding of such transient frequency modulations is affected by the nonlinearities described here is another open question, since the presented analysis focuses on stationary signals.
 
 \subsection{Conclusions}
 

rebuttal2.tex

@@ -33,9 +33,9 @@
 
 \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.
+We would like to thank both reviewers for their valuable
+feedback. Note that line numbers mentioned in our following responses refer to
+the new version of the manuscript, not the redlined one.
 
 \issue{\large Reviewer \#1}
 
@@ -86,7 +86,7 @@ one.
   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.}
+  articles.}
 
 \issue{Second, and along the same lines, the discussion could be
   improved by mentioning the effects and significance of these
@@ -96,7 +96,7 @@ one.
   chirps.}
 
 \response{We added a paragraph addressing JARs, chirps, and rises to
-  the discussion (lines 695 -- 703).}
+  the discussion (lines 697--705).}
 
 \issue{Finally, the precise description of the methods could be
   expanded for reaching a broader biology audience; in particular, the
@@ -157,8 +157,24 @@ one.
   could benefit from greater clarity to avoid the need to explore the
   results first in order to understand well.}
 
-\response{We added a sentence that describes how we generate those
-  stimuli in the Fourier domain (lines 156--160).}
+\response{We are sorry for the confusion. The cutoff frequencies
+  stated are pure stimulus parameters and not related to the filtering
+  performed by the respective neurons. ``White noise'' refers to a
+  time series that has equal power at all frequencies (like white
+  light) --- this choice of signal is agnostic with respect to the
+  preferred time scales of the system because all frequencies (or,
+  timescales) appear equally on the stimulus side. Bandpass-limited
+  white noise has equal power at all frequencies up to a cutoff
+  frequency that the experimenters choose in order to distribute the
+  total power over a reasonable frequency range in which they expect a
+  measurable response of the system under investigation. The choice
+  was different for ampullary receptors and P-units as stated in the
+  manuscript, but the stated values are not related with the actual
+  bandpass filtering that the neurons perform on the input
+  stimulus. The latter are quantified in the paper when we look at the
+  linear and nonlinear response functions of the cells. We completely
+  rewrote the description of the white-noise stimuli in the methods
+  sections (lines 155--160).}
 
 \issue{Line 154. This procedure elicits a modulation of the envelope
   of the reafferent signal. To achieve this, you adopted distinct
@@ -171,7 +187,7 @@ one.
 \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).}
+  stimulation and hope that this is clearer now (lines 166--169).}
 
 \issue{b) with regard to P receptors, you multiplied the head-to-tail
   ongoing signal by a white noise signal and played the resultant
@@ -221,7 +237,7 @@ one.
 
 \response{Exactly. We slightly expanded our description to make clear
   that we talk about the signal transduction until it reaches the
-  spike initiation zone (lines 258 -- 259).}
+  spike initiation zone (lines 260--261).}
 
 
 \issue{\large Reviewer \#2}
@@ -294,7 +310,8 @@ one.
 
 \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
+  all figures and analysis. We changed the sentence in the methods
+  accordingly (line 183). 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
@@ -317,13 +334,13 @@ one.
   frequencies. However, since they are close to the higher one of the
   two beat frequencies they do not show up in the AM as obviously as
   for the settings used in the social envelope papers by Eric Fortune
-  and Andre Longtin and colleges (I guess this is what you had in
+  and Andre Longtin and colleges (we guess this is what you had in
   mind).}
 
 \issue{(8) Line 302. "not-small amplitude" is arbitrary and
   vague. Please be clearer and more precise.}
 
-\response{We rephrased to two sentences in lines 323 -- 325.}
+\response{We rephrased to two sentences in lines 325--327.}
 
 \issue{(9) Figures 5C and 6C. For the stimuli with the red RAM
   waveforms, please make it clear which contrast is being represented
@@ -347,11 +364,17 @@ one.
   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
-  response. That is why the susceptibilities estimated from noise
-  stimuli decrease for higher stimulus contrasts.\\
+  adds background noise to the system that linearizes the response. In
+  this context, it is crucial to realize that the (linear and
+  nonlinear) transfer of a nonlinear system like a neuron depends on
+  the background noise. A Gaussian noise stimulus acts here both (i)
+  as a signal that evokes a response (linear and nonlinear) and (ii)
+  as an additional background noise linearizing the (linear and
+  nonlinear) response. In the context of our study it implies the
+  susceptibilities estimated from noise stimuli decrease for higher
+  stimulus contrasts.\\
   We added a whole paragraph at the beginning of this section to make
-  this clear (line 477 -- 482).}
+  this clear (lines 479--484).}
 
 \issue{(12) Lines 655-675. This was a very nice end to the discussion,
   but I would like to see more. I would like the broader significance
@@ -378,6 +401,6 @@ one.
   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.}
+  speculative sentence to the conclusion (lines 715--716).}
 
 \end{document}