nonlinearbaseline2025/rebuttal2.tex

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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}

\issue{The manuscript "Spike generation in electroreceptor afferents
  introduces additional spectral response components by weakly
  nonlinear interactions" submitted to eNeuro represents a noteworthy
  advancement in the field, as it elucidates that, under an often
  naturally occurring scenario the non linear responses of the pair
  electroreceptor-primary afferent ensemble may intervene in signal
  encoding. The manuscript shows that during the reception of signals
  originating distantly from multiple individual conspecifics,
  electroreceptor primary afferents may exhibit nonlinear responses
  allowing the fish to guess the presence of more than one
  individual. This is articulated with clarity through a
  straightforward leaky-integrate-and-fire model of electroreceptor
  responsiveness. The illustrations are both lucid and enhance the
  comprehension of the results section. The discussion is
  sound. However, the intrinsic value of the manuscript would likely
  be obscure without a more "biologist-friendly" approach. I would
  like to offer several suggestions that may serve to either enhance
  the manuscript or inspire future research endeavors.}

\response{Thank you for trying to make our manuscript more
  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
  axon-like dendritic innervation receptor cell terminals within the
  electroreceptor organ. This analogical nonlinear response may have
  its origin in the branched anatomy of the dendrite-axon terminals,
  easily verifiable by anatomical studies, and the presence, hardly
  demonstrable but plausible, of ion channel diversity; see, for
  example, Trigo, F. F. (2019) Antidromic analog signaling. Frontiers
  in Cellular Neuroscience, 13, 354. for a discussion of the general
  case and the study by Troy Smith, Unguez and Weber (2006, Fig. 3) in
  which receptor cells of tuberous electroreceptor organs and their
  afferents from Apteronotus leptorhynchus were labeled to varying
  degrees by six anti-Kv1 antibodies. Kv1.1 and Kv1.4 immunoreactivity
  was intense in the afferent axons of electroreceptor organs. It is
  noteworthy that Kv1 are low-threshold channels and, in some cases,
  exhibit a prolonged refractory period (Nogueira and Caputi,
  2013). These sources of nonlinearity could be mentioned to
  strengthen the links between well-written theoretical analysis and
  the practical field of experimental physiology.}

\response{You are right, there are more nonlinear mechanisms
  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
  articles.}

\issue{Second, and along the same lines, the discussion could be
  improved by mentioning the effects and significance of these
  nonlinearities when the recipient fish makes changes in its EOD
  frequency in at least two cases: a) sustained changes, as in
  interference avoidance responses, and b) transient changes, as in
  chirps.}

\response{We added a paragraph addressing JARs, chirps, and rises to
  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
  purpose of some procedures should be explained in some way. While
  the meaning seems clear as the reader scrolls through the results, a
  first reading of the methods, although accurate, does not offer the
  biology reader a quick and intuitive approach to the study.}

\issue{Next, I list some minor more detailed comments that may clarify
  the design and methods and facilitate their understanding by a
  broader audience.}

\issue{In general, you referenced P receptors and ampullary
  structures; however, what about T receptors? How can one distinguish
  between T and P in the recordings? Might it be possible that the
  negative results observed in certain receptors are attributable to
  the type of receptor (P or T)? Did you postulate, as suggested by
  Viancour (1979), that there exists a continuum of responsiveness
  between the extreme profiles of P (signal amplitude) and T (signal
  slope)?}

\response{T-units are characterized by 1:1 locking to the EOD, i.e. by
  having a baseline firing rate matching the EOD frequency. We
  definitely have no T-units in our data set, since our P-unit firing
  rates are well below the EOD frequencies. This we explain now in the
  ``Identification of P-units and ampullary cells'' section in the
  methods.}

\issue{In line 147, rather than using the term
  "laterally," I believe it would enhance clarity to state "parallel
  to each side of the fish," as the orientation of the electrodes may
  otherwise remain ambiguous.}

\response{Done.}

\issue{Furthermore, no commentary or discussion is provided regarding
  the fact that the stimulation procedure, which is transverse to the
  main axis of the body, neglects to account for the effects on the
  field foveal perioral region where the majority of receptors are
  located.}

\response{As stated in ``Experimental subjects and procedures'', all
  recordings were done in the posterior lateral line nerve. So we did
  not record from the foveal perioral region, and hence this problem
  is not relevant.}

\issue{Line 148, the phrase "band limited white noise" lacks
  clarity. Upon my initial reading, I assumed that the cutoff limit
  you referenced pertained to a low pass filter applicable to both
  ampullary and P-type tuberous receptors; however, it could indeed be
  interpreted as the opposite. In a strict sense, all "white noise
  stimuli" are band-passed. The duration of the stimulus establishes a
  lower cutoff for the band pass in one instance, while the
  responsiveness of the stimulation apparatus delineates the upper
  cutoff limits in another. Nevertheless, once one comprehends the
  objective of the experiment, the implicit significance of the white
  noise filtering becomes exceedingly apparent. Thus, this description
  could benefit from greater clarity to avoid the need to explore the
  results first in order to understand well.}

\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
  approaches for the ampullary and P receptors: a) in the case of
  ampullary receptors, you presented white noise and incrementally
  elevated its amplitude (variance) until the mean amplitude of the
  averaged sine wave recorded via local electrodes adjacent to the
  gills exhibited an increase of 1 to 5\%, is this correct?}

\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 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
  output, adjusting the amplitude until the local signal experienced
  an enhancement of 1 to 5\% in average, is this interpretation
  accurate?  Since the head to tail EOD and the local signals over the
  body are out of phase this process induces both amplitude and phase
  modulation of the stimulus signal, which will be contingent upon the
  phase lag of the local EOD at the receptor site in relation to the
  head-to-tail EOD. This phase lag, as reported in the literature,
  exhibits a shift ranging from pi to 2pi between a receptor situated
  at the head and another at the tail. (I posit that this may not
  significantly impact individual receptors response; however, how
  does this influence the relative timing among distinct receptors,
  and what is its correlation with jamming avoidance mechanisms?)
  Furthermore, does this form of noise modulation exert a comparable
  effect on the flanks (i.e., the apex of the derivative) as it does
  on the peaks of the signal themselves? How does this affect the
  recruitment of P and T receptors?}

\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 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?
  The peripheral afferent constitutes a myelinated and active dendrite
  whose distal branches receive synapses from receptor
  cells. Consequently, there exists a summation occurring at some
  juncture, likely at the first node, that facilitates the generation
  of an action potential. Otherwise, the signal would not be
  effectively propagated from the receptors to the ganglia where the
  somata reside. Receptors across various species exhibit notable
  differences; some are myelinated within the electroreceptor organ,
  while others display the first node external to the electroreceptor
  organ. Could you discuss this aspect, considering the anatomical
  structure of the receptor in your species?}

\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 260--261).}


\issue{\large Reviewer \#2}

\issue{This work is a nice contribution to our general understanding
  of nonlinearities in sensory coding, and to our detailed
  understanding of behaviorally relevant information processing in the
  electrosensory systems of weakly electric fish. I have several
  suggestions for the authors.}

\issue{(1) Abstract, line 29. "...if these frequencies or their sum
  match the neuron's baseline firing rate" is not quite accurate
  because "these frequencies" implies BOTH input frequencies must
  match the baseline firing rate. I think you mean to state, "...if
  one of these frequencies or their sum match the neuron's baseline
  firing rate."}

\response{Your are right! We changed the sentence as suggested.}

\issue{(2) Abstract, line 33. The wording here is unclear,
  specifically what you mean by "much stronger." Much stronger what
  exactly? I think you mean to refer to the fact that these nonlinear
  responses were more common and stronger in ampullary units than
  P-units, but "much stronger" does not clearly convey this,
  especially in the abstract.}

\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
  this needs to be made clear in the figure legend.}

\response{We added a brief explanatory sentence to the caption.}

\issue{(4) Figure 1B. "Because frequencies can also be negative..."
  This is unclear and needs more explanation, especially because there
  are no negative frequencies in your actual data. How can frequencies
  be negative?}

\response{We added a few sentences following equation (1) to motivate
  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
  potential df2 harmonics and fEOD. Figure 5 extends to higher
  frequencies to illustrate these and it is not clear why these are
  clipped in these two figures.}

\response{You are right. In figure 4 we show now the spectrum up to
  950\,Hz, such that $f_{EOD}$ and its interactions with $f_1$ and
  $f_2$ are included. We labeled the additional peaks and expanded the
  figure caption accordingly. In figure 3 we stay with the small
  range, because we have so little data for this special setting where
  one of the beat frequencies approximately matches the P-units
  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, 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
  on convolution with a 2 ms Gaussian kernel (line 169)? It seems that
  example data should effectively illustrate how the data were
  actually analyzed. More fundamentally, why would a 2-fold difference
  in kernel width be appropriate for presentation vs. analysis?}

\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. 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
  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
  believe the envelope is represented by both peaks, correct?}

\response{No, what is shown is the power spectrum of the spike
  response, not the one of the amplitude modulation or envelope of the
  stimulus. We added a sentence to the end of the figure caption to
  make this clear.\\
  If it were the power spectrum of the signal after it passed
  a non-linearity (rectification or thresholding at zero), then there
  could be also peaks at the sum and difference of the beat
  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 (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 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
  by these traces, as responses to two different contrasts are shown.}

\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
  shown in E, but E shows the low contrast and F shows the high
  contrast.}

\response{Good catch! Fixed.}

\issue{(11) Lines 453-465, Figure 8. This section was confusing to
  me. Why does second-order susceptibility decrease as stimulus
  contrast increases, when theory predicts that higher signal-to-noise
  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
  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. 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 (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
  of this study to be expanded upon with respect to (1) behavioral
  relevance for signal detection in weakly electric fish, and (2)
  comparative relevance for other modalities and species. Speculation
  is fine so long as it is clearly indicated as such. It might work
  best to expand upon and distribute the information in lines 655-666
  throughout the discussion at relevant points, rather than as an
  afterthought. The conclusion section in lines 667-675 could then
  reiterate these points briefly and delve into more detail on
  comparative considerations.}

\response{We also like to see more on this, but we feel that we
  already speculated enough. Without further studies on the readout of
  the receptor responses, we cannot make any convincing claim about
  whether and how weakly nonlinear interactions are actually utilized
  in a neural system. The problem is that a match of one of the
  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.
  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 (lines 715--716).}

\end{document}