worked on rebuttal

nonlinearbaseline.tex

@@ -341,7 +341,7 @@ 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. 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 $f$.
+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. 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 freqiencies $\pm\omega$ 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.
 
@@ -486,15 +486,15 @@ Both, the reduced intrinsic noise and the RAM stimulus, need to replace the orig
 
 \begin{figure*}[t]
   \includegraphics[width=\columnwidth]{twobeats}
-  \caption{\label{fig:motivation} Nonlinearity in an electrophysiologically recorded P-unit of \lepto{} in a three-fish setting (cell identifier ``2021-08-03-ac"). Receiver with EOD frequency $f_{\rm EOD} =664$\,Hz encounters fish with EOD frequencies $f_{1}=631$\,Hz and $f_{2}=797$\,Hz. Both foreign signals have the same strength relative to the own field amplitude (10\,\% contrast). Top row: Sketch of signal processing in the nonlinear system (black box). Second row: Superposition of the receiver EOD with the EODs of other fish, colored line highlights the amplitude modulation. Third row: Three trials of spike trains of the recorded P-unit. Fourth row: Firing rate, estimated by convolution of the spike trains with a Gaussian kernel. Bottom row: Power spectrum of the spike trains. \figitem{A} Baseline condition: The cell is driven by the self-generated field alone. The baseline firing rate $r = 139$\,Hz dominates the power spectrum (blue). \figitem{B} The receiver's EOD and a single conspecific with an EOD frequency $f_{1}=631$\,Hz are present. Superposition of the two EODs induces an amplitude modulation, referred to as beat, with beat frequeny $\Delta f_1=33$\,Hz (purple). The P-unit strongly responds to this beat. \figitem{C} The receiver and a fish with an EOD frequency $f_{2}=797$\,Hz are present. The resulting beat $\Delta f_2=133$\,Hz (green) is faster as the difference between the EOD frequencies is larger. The P-unit repsonse to this faster beat is weaker. \figitem{D} All three fish with EOD frequencies $f_{\rm EOD}$, $f_1$, and $f_2$ are present. Additional peaks occur in the power spectrum of the spike response at the sum (orange) and difference (red) of the two beat frequencies, indicating non-linear interactions between the two frequencies in the P-unit.}
+  \caption{\label{fig:twobeats} Nonlinearity in an electrophysiologically recorded P-unit of \lepto{} in a three-fish setting (cell identifier ``2021-08-03-ac"). Receiver with EOD frequency $f_{\rm EOD} =664$\,Hz encounters fish with EOD frequencies $f_{1}=631$\,Hz and $f_{2}=797$\,Hz. Both foreign signals have the same strength relative to the own field amplitude (10\,\% contrast). Top row: Sketch of signal processing in the nonlinear system (black box). Second row: Superposition of the receiver EOD with the EODs of other fish, colored line highlights the amplitude modulation. Third row: Three trials of spike trains of the recorded P-unit. Fourth row: Firing rate, estimated by convolution of the spike trains with a Gaussian kernel. Bottom row: Power spectrum of the spike trains. \figitem{A} Baseline condition: The cell is driven by the self-generated field alone. The baseline firing rate $r = 139$\,Hz dominates the power spectrum (blue). \figitem{B} The receiver's EOD and a single conspecific with an EOD frequency $f_{1}=631$\,Hz are present. Superposition of the two EODs induces an amplitude modulation, referred to as beat, with beat frequeny $\Delta f_1=33$\,Hz (purple). The P-unit strongly responds to this beat. \figitem{C} The receiver and a fish with an EOD frequency $f_{2}=797$\,Hz are present. The resulting beat $\Delta f_2=133$\,Hz (green) is faster as the difference between the EOD frequencies is larger. The P-unit repsonse to this faster beat is weaker. \figitem{D} All three fish with EOD frequencies $f_{\rm EOD}$, $f_1$, and $f_2$ are present. Additional peaks occur in the power spectrum of the spike response at the sum (orange) and difference (red) of the two beat frequencies, indicating non-linear interactions between the two frequencies in the P-unit. Remember, the spectrum of the raw signal (top row, gray line) has power only at the three EOD frequencies $f_{EOD}$, $f_1$, and $f_2$.}
 \end{figure*}
 
 We explored a large set of electrophysiological data from primary afferents of the active and passive electrosensory system, P-units and ampullary cells \citep{Grewe2017, Hladnik2023}, that were recorded in the brown ghost knifefish \textit{Apteronotus leptorhynchus}. We re-analyzed this dataset to search for weakly nonlinear responses that have been predicted in previous theoretical work \citep{Voronenko2017}. Additional simulations of LIF-based models of P-unit spiking help to interpret the experimental findings in this theoretical framework. We start with demonstrating the basic concepts using example P-units and respective models and then compare the population of recordings in both cell types.
 
 \subsection{Nonlinear responses in P-units stimulated with two frequencies}
-Without external stimulation, a P-unit is driven by the fish's own EOD alone (with a specific EOD frequency $f_{\rm EOD}$) and spontaneously fires action potentials at the baseline rate $r$. Accordingly, the power spectrum of the baseline activity has a peak at $r$ (\subfigrefb{fig:motivation}{A}). In the communication context, this animal (the receiver) is exposed to the EODs of one or many foreign fish. Superposition of the receiver's EOD with an EOD of another fish with frequency $f_1$ results in a beat, a periodic amplitude modulation of the receiver's EOD. The frequency of the beat is given by the difference frequency $\Delta f_1 = f_1 - f_{\rm EOD}$ between the two fish. P-units encode this beat in their firing rate \citep{Bastian1981a} and consequently the power spectrum of the response has a peak at the beat frequency (\subfigrefb{fig:motivation}{B}). A second peak at the first harmonic of the beat frequency is indicative of a nonlinear process that here is associated with the clipping of the P-unit's firing rate at zero \citep{Barayeu2023}. Pairing the fish with another fish at a higher beat frequency $\Delta f_2 = f_2 - f_{\rm EOD} > \Delta f_1$ results in a weaker response with a single peak in the response power spectrum, suggesting a linear response (\subfigrefb{fig:motivation}{C}). The weaker response to this beat can be explained by the beat tuning of the cell \citep{Walz2014}. Note, $\Delta f_2$ has been deliberately chosen to match the recorded P-unit's baseline firing rate.
+Without external stimulation, a P-unit is driven by the fish's own EOD alone (with a specific EOD frequency $f_{\rm EOD}$) and spontaneously fires action potentials at the baseline rate $r$. Accordingly, the power spectrum of the baseline activity has a peak at $r$ (\subfigrefb{fig:twobeats}{A}). In the communication context, this animal (the receiver) is exposed to the EODs of one or many foreign fish. Superposition of the receiver's EOD with an EOD of another fish with frequency $f_1$ results in a beat, a periodic amplitude modulation of the receiver's EOD. The frequency of the beat is given by the difference frequency $\Delta f_1 = f_1 - f_{\rm EOD}$ between the two fish. P-units encode this beat in their firing rate \citep{Bastian1981a} and consequently the power spectrum of the response has a peak at the beat frequency (\subfigrefb{fig:twobeats}{B}). A second peak at the first harmonic of the beat frequency is indicative of a nonlinear process that here is associated with the clipping of the P-unit's firing rate at zero \citep{Barayeu2023}. Pairing the fish with another fish at a higher beat frequency $\Delta f_2 = f_2 - f_{\rm EOD} > \Delta f_1$ results in a weaker response with a single peak in the response power spectrum, suggesting a linear response (\subfigrefb{fig:twobeats}{C}). The weaker response to this beat can be explained by the beat tuning of the cell \citep{Walz2014}. Note, $\Delta f_2$ has been deliberately chosen to match the recorded P-unit's baseline firing rate.
 
-When stimulating with both foreign signals simultaneously, additional peaks appear in the response power spectrum at the sum $\Delta f_1 + \Delta f_2$ and the difference frequency $\Delta f_2 - \Delta f_1$ (\subfigrefb{fig:motivation}{D}). Thus, the cellular response is not equal to the sum of the responses to the two beats presented separately. These additional peaks at the sum and the difference of the two stimulus frequencies are a hallmark of nonlinear interactions that, by definition, are absent in linear systems.
+When stimulating with both foreign signals simultaneously, additional peaks appear in the response power spectrum at the sum $\Delta f_1 + \Delta f_2$ and the difference frequency $\Delta f_2 - \Delta f_1$ (\subfigrefb{fig:twobeats}{D}). Thus, the cellular response is not equal to the sum of the responses to the two beats presented separately. These additional peaks at the sum and the difference of the two stimulus frequencies are a hallmark of nonlinear interactions that, by definition, are absent in linear systems.
 
 
 \subsection{Linear and weakly nonlinear regimes}
@@ -504,7 +504,7 @@ When stimulating with both foreign signals simultaneously, additional peaks appe
   \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. $\Delta f_2$ was set to match the baseline firing rate $r$ of the P-unit. \figitem{A--D} Top: 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. Middle: Spike raster of the model P-unit response. Bottom: power spectrum of the firing rate estimated from the spike raster. \figitem{A} At low stimulus contrasts the response is linear. The only peaks in the response spectrum are at the two stimulating beat frequencies (green and purple circles), the latter enhancing the peak at baseline firig 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 own EOD. The peak at $f_{EOD}$ is also flanked by harmonics of $\Delta f_1$ (gray triangles).  \figitem{B} At moderately higher stimulus contrast, the peaks in the response spectrum at the two beat frequencies become larger. In addition, a peak at $f_{EOD} - \Delta f_2 $ appears (blue diamond). \figitem{C} At intermediate stimulus contrasts, nonlinear responses start to appear at the sum and the difference of the stimulus frequencies (orange and red). Additional peaks appear at harmonics of $\Delta f_1$ (small green circles) and at $f_{EOD} - \Delta f_2 \pm \Delta f_1$ (purple squares). \figitem{D} At higher stimulus contrasts further peaks appear in the power spectrum. Most of them are interactions of $f_{EOD}$ and $\Delta f_2$ with harmonics of $\Delta f_1$ ($k$ is an integer). \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:motivation} had the same not-small amplitude. 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 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}.
 
 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 only 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.
 
@@ -545,7 +545,7 @@ Electric fish possess an additional electrosensory system, the passive or ampull
 
 
 \subsection{Model-based estimation of the second-order susceptibility}
-In the example recordings shown above (\figsrefb{fig:punit} and \fref{fig:ampullary}), we only observe nonlinear responses where the sum of the two stimulus frequencies matches the neuron's baseline firing rate, but not where either of the frequencies alone matches the baseline rate. In the following, we investigate this discrepancy to the theoretical expectations \citep{Voronenko2017,Franzen2023}.
+In the example recordings shown above (\figsrefb{fig:punit} and \fref{fig:ampullary}), we only observe nonlinear responses where the sum of the two stimulus frequencies matches the neuron's baseline firing rate, but not where either of the frequencies alone matches the baseline rate. In the following, we investigate this discrepancy to the theoretical expectations (\figrefb{fig:lifsuscept}, \citealp{Voronenko2017,Franzen2023}).
 
 \begin{figure*}[p]
   \includegraphics[width=\columnwidth]{noisesplit}
@@ -578,7 +578,7 @@ The SI($r$) correlates with the CVs of the cell's baseline interspike intervals
 
 
 \subsection{Weakly nonlinear interactions vanish for higher stimulus contrasts}
-\notebl{As pointed out above, the weakly nonlinear regime can only be observed for sufficiently weak \notejb{noise} stimuli: strong \notejb{noise} simuli act as a linearizing background noise and thus shift the neural signal transmission into a regime free of nonlinearities. 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, 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.
 
 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.
 
@@ -691,7 +691,7 @@ 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:motivation}) 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, 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 an 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.
 

rebuttal2.tex

@@ -121,7 +121,7 @@
 \response{In \textit{Apteronotus} 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. This we eplain now in the ``Identification of P-units and
+  set. 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
@@ -211,8 +211,6 @@
   electrosensory systems of weakly electric fish. I have several
   suggestions for the authors.}
 
-\response{}
-
 \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
@@ -244,7 +242,8 @@
   are no negative frequencies in your actual data. How can frequencies
   be negative?}
 
-\response{UH... LETS WRITE SOMETHING}
+\response{We added a few sentences following equation (1) to motivate
+  the existence of negative frequencies in Fourier transforms.}
 
 \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
@@ -253,7 +252,7 @@
   clipped in these two figures.}
 
 \response{You are right. In figure 4 we show now the spectrum up to
-  750\,Hz, such that $f_{EOD}§ and its interactions with $\Delta f_2$
+  750\,Hz, such that $f_{EOD}$ and its interactions with $\Delta f_2$
   and harmonics are included. We labeled the additonal peaks
   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
@@ -288,9 +287,8 @@
 \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.}
-
-\response{If it were the power spectrum of the signal after it passed
+  make this clear.\\
+  If it were the power spectrum of the signal after it passed
   a non-linearity (rectification or threhsolding 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
@@ -302,7 +300,8 @@
 \issue{(8) Line 302. "not-small amplitude" is arbitrary and
   vague. Please be clearer and more precise.}
 
-\response{}
+\response{We added ``resulting in AM contrasts of 10\,\% that evoke
+  strong modulations in a P-unit's firing rate response''.}
 
 \issue{(9) Figures 5C and 6C. For the stimuli with the red RAM
   waveforms, please make it clear which contrast is being represented
@@ -322,7 +321,15 @@
   contrast increases, when theory predicts that higher signal-to-noise
   ratios should result in larger nonlinearities?}
 
-\response{NOISE STIMULUS LINEARIZES. BENJI. BUt also highlight the difference between noise and sinewae stimulation in other parts of the manuscript.}
+\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 itselfs
+  adds background noise to the system that linearizes the
+  response. That is why 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.}
 
 \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

regimes.py

@@ -31,7 +31,7 @@ spec_trials = 1000   # set to zero to only recompute firng rates
 sigma = 0.001
 nfft = 2**18
 
-recompute = True
+recompute = False
 
 
 def load_data(file_path):