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Jan Benda
2026-02-10 18:56:23 +01:00
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@@ -13,7 +13,7 @@ PT=$(wildcard *.py)
PYTHONFILES=$(filter-out plotstyle.py spectral.py examplecells.py, $(PT))
PYTHONPDFFILES=$(PYTHONFILES:.py=.pdf)
REVISION=e3814a1be539f9424c17b7bd7ef45a8826a9f1e2
REVISION=eacf6ee04c2caee1c6628d1e9d96944606ce5cbb
ifdef REBUTTALBASE
REBUTTALTEXFILE=$(REBUTTALBASE).tex

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nonlinearbaseline.tex
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@@ -34,7 +34,7 @@
\newcommand{\showkeywords}{true}
% show author contributions:
\newcommand{\showcontributions}{true}
\newcommand{\showcontributions}{false}
% figures and captions at end:
\newcommand{\figsatend}{false}
@@ -74,12 +74,12 @@
\pagestyle{myheadings}
\markright{\runningtitle}
%%%%% language %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\usepackage[english]{babel}
%%%%% fonts %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\usepackage{pslatex} % nice font for pdf file
%%%%% language %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\usepackage[english]{babel}
%%%%% section style %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\usepackage[sf,bf,it,big,clearempty]{titlesec}
\usepackage{titling}
@@ -89,11 +89,8 @@
%%%%% math %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\usepackage{amsmath}
%\usepackage{lualatex-math}
\usepackage{iftex}
% LuaTeX-spezifischen Code nur ausführen, wenn LuaTeX verwendet wird
\ifluatex
\usepackage{lualatex-math}
\fi
@@ -243,9 +240,9 @@
Number of figures & 10 \\
Number of tables & 0 \\
Number of multimedia & 0 \\
Number of words in abstract & 194 \\
Number of words in abstract & 201 \\
Number of words in introduction & 659 \\
Number of words in discussion & 1891
Number of words in discussion & 2041
\end{tabular}
\paragraph{Acknowledgements}
@@ -298,7 +295,7 @@ The transfer function that describes the linear properties of a system, is the f
Noise in nonlinear systems, however, linearizes the system's response properties \citep{Yu1989, Chialvo1997}. Also, in the limit to small stimuli, nonlinear systems can be well described by linear response theory \citep{Roddey2000, Doiron2004, Rocha2007, Sharafi2013}. With increasing stimulus amplitude, the contribution of the second-order kernel of the Volterra series becomes more relevant. For these weakly nonlinear responses analytical expressions for the second-order susceptibility have been derived for leaky-integrate-and-fire (LIF) \citep{Voronenko2017} and theta model neurons \citep{Franzen2023}. In the suprathreshold regime, in which the LIF generates a baseline firing rate in the absence of an external stimulus, the linear response function has a peak at the baseline firing rate and its harmonics (\subfigrefb{fig:lifsuscept}{A}) and the second-order susceptibility shows very distinct ridges of elevated nonlinear responses, exactly where one of two stimulus frequencies equals or both frequencies add up to the neuron's baseline firing rate (\subfigrefb{fig:lifsuscept}{B}). In experimental data, such structures in the second-order susceptibility have not been reported yet.
Here we search for such weakly nonlinear responses in electroreceptors of the two electrosensory systems of the wave-type electric fish \textit{Apteronotus leptorhynchus}, i.e. the tuberous (active) and the ampullary (passive) electrosensory system. The p-type electroreceptor afferents of the active system (P-units) are driven by the fish's high-frequency, quasi-sinusoidal electric organ discharges (EOD) and encode disturbances of it \citep{Bastian1981a}. The electroreceptors of the passive system are tuned to lower-frequency exogeneous electric fields such as caused by muscle activity of prey \citep{Kalmijn1974}. As different animals have different EOD-frequencies, being exposed to stimuli of multiple distinct frequencies is part of the animal's everyday life \citep{Benda2020,Henninger2020} and weakly nonlinear interactions may occur in the electrosensory periphery. In communication contexts \citep{Walz2014, Henninger2018} the EODs of interacting fish superimpose and lead to periodic amplitude modulations (AMs or beats) of the receiver's EOD. Nonlinear mechanisms in P-units, enable encoding of AMs in their time-dependent firing rates \citep{Bastian1981a, Walz2014, Middleton2006, Barayeu2023}. When multiple animals interact, the EOD interferences induce second-order amplitude modulations referred to as envelopes \citep{Yu2005, Fotowat2013, Stamper2012Envelope} and saturation nonlinearities allow also for the encoding of these in the electrosensory periphery \citep{Savard2011}. Field observations have shown that courting males were able to react to the extremely weak signals of distant intruding males despite the strong foreground EOD of the nearby female \citep{Henninger2018}. Weakly nonlinear interactions at particular combinations of signals can be of immediate relevance in such settings as they could boost detectability of the faint signals \citep{Schlungbaum2023}.
Here we search for such weakly nonlinear responses in electroreceptors of the two electrosensory systems of the wave-type electric fish \textit{Apteronotus leptorhynchus}, i.e. the tuberous (active) and the ampullary (passive) electrosensory system. The p-type electroreceptor afferents of the active system (P-units) are driven by the fish's high-frequency, quasi-sinusoidal electric organ discharges (EOD) and encode disturbances of it \citep{Bastian1981a}. The electroreceptors of the passive system are tuned to lower-frequency exogeneous electric fields such as caused by muscle activity of prey \citep{Kalmijn1974}. As different animals have different EOD-frequencies, being exposed to stimuli of multiple distinct frequencies is part of the animal's everyday life \citep{Benda2020,Henninger2020} and weakly nonlinear interactions may occur in the electrosensory periphery. In communication contexts \citep{Walz2014, Henninger2018} the EODs of interacting fish superimpose and lead to periodic amplitude modulations (AMs or beats) of the receiver's EOD. Nonlinear mechanisms in P-units, enable encoding of AMs in their time-dependent firing rates \citep{Bastian1981a, Walz2014, Middleton2006, Barayeu2023}. When multiple animals interact, the superpositions of the EODs induce second-order amplitude modulations referred to as envelopes \citep{Yu2005, Fotowat2013, Stamper2012Envelope} and saturation nonlinearities allow also for the encoding of these in the electrosensory periphery \citep{Savard2011}. Field observations have shown that courting males were able to react to the extremely weak signals of distant intruding males despite the strong foreground EOD of the nearby female \citep{Henninger2018}. Weakly nonlinear interactions at particular combinations of signals can be of immediate relevance in such settings as they could boost detectability of the faint signals \citep{Schlungbaum2023}.
\section{Materials and Methods}
@@ -315,16 +312,16 @@ Respiration was then switched to normal tank water and the fish was transferred
For the recordings fish were positioned centrally in the experimental tank, with the major parts of their body submerged into the water. Those body parts that were above the water surface were covered with paper tissue to avoid drying of the skin. Local analgesia was refreshed in intervals of two hours by cutaneous application of Lidocaine (2\,\%; bela-pharm, Vechta, Germany) around the surgical wounds. Electrodes (borosilicate; 1.5\,mm outer diameter; GB150F-8P; Science Products, Hofheim, Germany) were pulled to a resistance of 50--100\,\mega\ohm{} (model P-97; Sutter Instrument, Novato, CA) and filled with 1\,M KCl solution. Electrodes were fixed in a microdrive (Luigs-Neumann, Ratingen, Germany) and lowered into the nerve. Recordings of electroreceptor afferents were amplified and lowpass filtered at 10\,kHz (SEC-05, npi-electronics, Tamm, Germany, operated in bridge mode). All signals, neuronal recordings, recorded EOD, and the generated stimulus, were digitized with sampling rates of 20 or 40\,kHz (PCI-6229, National Instruments, Austin, TX). RELACS (\url{https://github.com/relacs/relacs}) running on a Linux computer was used for online spike and EOD detection, stimulus generation, and calibration. Recorded data was then stored on the hard drive for offline analysis.
\subsection{Identification of P-units and ampullary cells}
Recordings were classified as P-units if baseline action potentials phase locked to the EOD with vectors strengths between 0.7 and 0.95, a baseline firing rate larger than 30\,Hz, a serial correlation of subsequent interspike intervals below zero, a coefficient of variation of baseline interspike intervals below 1.5 und during stimulation below 2. P-units are clearly distinguised from T-type electrorecptors, that we did not analyze here, by having firing rates much lower than the EOD frequency of the fish (no 1:1 locking to the EOD) \notejb{CITE TUNIT PAPER}. As ampullary cells we classified recordings with vector strengths below 0.15, baseline firing rate above 10\,Hz, baseline CV below 0.18, CV during stimulation below 1.0, and a response modulation during stimulation below 80\,Hz \citep{Grewe2017}. We here selected only those cells of which the neuron's baseline activity as well as the responses to band-limited white noise stimuli were recorded.
Recordings were classified as P-units if baseline action potentials phase locked to the EOD with vectors strengths between 0.7 and 0.95, a baseline firing rate larger than 30\,Hz, a serial correlation of subsequent interspike intervals below zero, a coefficient of variation of baseline interspike intervals below 1.5 and during stimulation below 2. P-units are clearly distinguished from T-type electroreceptors, that we did not analyze here, by having firing rates much lower than the EOD frequency of the fish (no 1:1 locking to the EOD). As ampullary cells we classified recordings with vector strengths below 0.15, baseline firing rate above 10\,Hz, baseline CV below 0.18, CV during stimulation below 1.0, and a response modulation during stimulation below 80\,Hz \citep{Grewe2017}. We here selected only those cells of which the neuron's baseline activity as well as the responses to band-limited white noise stimuli were recorded.
\subsection{Electric field recordings}
For monitoring the EOD without the stimulus, two vertical carbon rods ($11\,\centi\meter$ long, 8\,mm diameter) in a head-tail configuration were placed isopotential to the stimulus. Their signal was differentially amplified with a gain factor between 100 and 500 (depending on the recorded animal) and band-pass filtered (3 to 1500\,Hz pass-band, DPA2-FX; npi electronics, Tamm, Germany). For an estimate of the transdermal potential that drives the electroreceptors, two silver wires spaced by 1\,cm were located next to the left gill of the fish and orthogonal to the fish's longitudinal body axis (amplification 100 to 500 times, band-pass filtered with 3 to 1\,500\,Hz pass-band, DPA2-FX; npi-electronics, Tamm, Germany). This local EOD measurement recorded the combination of the fish's own EOD and the applied stimulus.
\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 stimulus intensity is given as a contrast, i.e. the standard deviation of the noise stimulus 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 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 white noise was directly applied as the stimulus. 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).
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: $(1 + s(t))EOD(t)$.
\subsection{Data analysis} Data analysis was done in Python 3 using the packages matplotlib \citep{Hunter2007}, numpy \citep{Walt2011}, scipy \citep{scipy2020}, pandas \citep{Mckinney2010}, nixio \citep{Stoewer2014}, and thunderlab (\url{https://github.com/bendalab/thunderlab}).
\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}).
\paragraph{Code accessibility}
The P-unit model parameters and spectral analysis algorithms are available at \url{https://github.com/bendalab/punitmodel/tree/v1}.
@@ -341,7 +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 freqiencies $\pm\omega$ 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. 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.
@@ -422,7 +419,7 @@ To mimic the interaction with other fish, the EODs of a second or third fish wit
\end{equation}
For two fish, $c_2 = 0$.
Random amplitude modulations (RAMs) were simulated by first generating the AM as a band-limited white noise stimulus $s(t)$. For this, random real and imaginary numbers were drawn from Gaussian distributions for each frequency component in the range from 0 to 300\,Hz in the Fourier domain \citep{Billah1990,Skorjanc2023}. By means of the inverse Fourier transform, the time course of the RAM stimulus, $s(t)$, was generated. The input to the model was then
Random amplitude modulations (RAMs) were simulated by first generating the AM as a band-limited white noise stimulus $s(t)$. For this, random real and imaginary numbers were drawn from normal distributions for each frequency component in the range from 0 to 300\,Hz in the Fourier domain \citep{Billah1990,Skorjanc2023}. By means of the inverse Fourier transform, the time course of the RAM stimulus, $s(t)$, was generated. The input to the model was then
\begin{equation}
\label{eq:ram_equation}
y(t) = (1+ s(t)) \cos(2\pi f_{EOD} t)
@@ -434,14 +431,14 @@ First, the input $y(t)$ is thresholded by setting negative values to zero:
\label{eq:threshold2}
\lfloor y(t) \rfloor_0 = \left\{ \begin{array}{rcl} y(t) & ; & y(t) \ge 0 \\ 0 & ; & y(t) < 0 \end{array} \right.
\end{equation}
(\subfigrefb{flowchart}{A}). This threshold models the transfer function of the synapses between the primary receptor cells and the afferent \notejb{CITE A MOSER PAPER} and may also include nonlinear properties introduced by low-threshold Kv1 channels known to be present in the afference \citep{TroySmith2006,Nogueira2013}. Together with a low-pass filter
(\subfigrefb{flowchart}{A}). This threshold models the transfer function of the synapses between the primary receptor cells and the afferent \citep{Moser2016} and may also include nonlinear properties introduced by low-threshold Kv1 channels known to be present in the afference \citep{TroySmith2006,Nogueira2013}. Together with a low-pass filter
\begin{equation}
\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 low-pass filter models passive signal conduction in the afferent's dendrite (\subfigrefb{flowchart}{B}) and $\tau_{d}$ is the membrane time constant 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.
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.
The dendritic voltage $V_d(t)$ is then fed into a stochastic leaky integrate-and-fire (LIF) model with adaptation,
\begin{equation}
@@ -486,7 +483,7 @@ Both, the reduced intrinsic noise and the RAM stimulus, need to replace the orig
\begin{figure*}[t]
\includegraphics[width=\columnwidth]{twobeats}
\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 circle). \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 a periodic amplitude modulation, referred to as beat, with beat frequeny $\Delta f_1=33$\,Hz. The P-unit strongly responds to this beat (purple). \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 is faster as the difference between the EOD frequencies is larger. The P-unit repsonse to this faster beat is weaker (green). \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. Note, the spectrum of the raw signal (top row, gray) has power only at the three EOD frequencies $f_{EOD}$, $f_1$, and $f_2$.}
\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 circle). \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 a periodic amplitude modulation, referred to as beat, with beat frequency $\Delta f_1=33$\,Hz. The P-unit strongly responds to this beat (purple). \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 is faster as the difference between the EOD frequencies is larger. The P-unit response to this faster beat is weaker (green). \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 nonlinear interactions between the two frequencies in the P-unit. Note, the spectrum of the raw signal (top row, gray) 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.
@@ -501,19 +498,18 @@ 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 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 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 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.}
\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}.
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.
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}.
This linear regime is followed by the weakly nonlinear regime (in the example cell between approximately 1.2\,\% and 3.5\,\% stimulus contrast). In addition to the peaks at the stimulus frequencies, peaks at the sum and the difference of the stimulus frequencies appear in the response spectrum (\subfigref{fig:regimes}{C}, orange and red). The amplitudes of these two peaks initially increase quadratically with stimulus amplitude (\subfigref{fig:regimes}{E}, thin lines). Note, that we have chosen $\Delta f_2$ to match the baseline firing rate $f_{base}$ of the neuron.
This linear regime is followed by the weakly nonlinear regime (in the example cell between approximately 1.2\,\% and 2.6\,\% stimulus contrast). In addition to the peaks at the stimulating beat frequencies, peaks at the sum and the difference of the beat frequencies appear in the response spectrum (\subfigref{fig:regimes}{C}, orange and red). The amplitudes of these two peaks initially increase quadratically with stimulus amplitude (\subfigref{fig:regimes}{E}, thin lines). Note, that we have chosen $\Delta f_2$ to match the baseline firing rate $r$ of the neuron.
At higher stimulus amplitudes, the linear response and the weakly-nonlinear response begin to deviate from their linear and quadratic dependency on amplitude (\subfigrefb{fig:regimes}{E}) and additional peaks appear in the response spectrum (\subfigrefb{fig:regimes}{D}). At high stimulus contrasts, additional nonlinearities in the system, in particular clipping of the firing rate, shape the responses.
For this example, we chose very specific stimulus (beat) frequencies. %One of these matching the P-unit's baseline firing rate.
In the following, however, we are interested in how the nonlinear responses depend on different combinations of stimulus frequencies in the weakly nonlinear regime. For the sake of simplicity we will drop the $\Delta$ notation even though P-unit stimuli are beats.
For this example, we chose very specific stimulus (beat) frequencies. In the following, however, we are interested in how the nonlinear responses depend on different combinations of stimulus frequencies in the weakly nonlinear regime. For the sake of simplicity we will drop the $\Delta$ notation even though P-unit stimuli are beats.
\subsection{Nonlinear signal transmission in P-units}
@@ -586,7 +582,7 @@ At a RAM contrast of 3\,\% the SI($r$) values become smaller (\figrefb{fig:model
\begin{figure}[tp]
\includegraphics[width=\columnwidth]{modelsusceptlown}
\caption{\label{fig:modelsusceptlown}Inferring the triangular structure of the second-order susceptibility from limited data. \figitem{A} Reliably estimating the structure of the second-order susceptibility requires a high number of FFT segments $N$ in the order of one or even ten millions. As an example, susceptibilities of the model cell ``2012-12-21-ak-invivo-1'' (baseline firing rate of 157\,Hz, CV=0.15) are shown for the noise-split configuration ($c=0$\,\%) and RAM stimulus contrasts of $c=1$, $3$, and $10$\,\% as indicated. For contrasts below $10$\,\% this cell shows a nice triangular pattern in its susceptibilities, quite similar to the introductory example of a LIF in \figrefb{fig:lifsuscept}. \figitem{B} However, with limited data of $N=100$ trials the susceptibility estimates are noisy and show much less structure, except for the anti-diagonal at the cell's baseline firing rate. The SI($r$) quantifies the height of this ridge where the two stimulus frequencies add up to the neuron's baseline firing rate. \figitem{C} Correlations between the estimates of SI($r$) based on 100 FFT segments (density to the right) with the converged ones based on one or ten million segments at a given stimulus contrast for all $n=39$ model cells. The black circle marks the model cell shown in \panel{A} and \panel{B}. The black diagonal line is the identity line and the dashed line is a linear regression. The correlation coefficient and corresponding significance level are indicated in the top left corner. The thin vertical line is a threshold at 1.2, the thin horizontal line a threshold at 1.8. The number of cells within each of the resulting four quadrants denote the false positives (top left), true positives (top right), true negatives (bottom left), and false negatives (bottom right) for predicting a triangular structure in the converged susceptibility estimate from the estimates based on only 100 segments.}
\caption{\label{fig:modelsusceptlown}Inferring the triangular structure of the second-order susceptibility from limited data. \figitem{A} Reliably estimating the structure of the second-order susceptibility requires a high number of FFT segments $N$ in the order of one or even ten millions. As an example, susceptibilities of the model cell ``2012-12-21-ak'' (baseline firing rate of 157\,Hz, CV=0.15) are shown for the noise-split configuration ($c=0$\,\%) and RAM stimulus contrasts of $c=1$, $3$, and $10$\,\% as indicated. For contrasts below $10$\,\% this cell shows a nice triangular pattern in its susceptibilities, quite similar to the introductory example of a LIF in \figrefb{fig:lifsuscept}. \figitem{B} However, with limited data of $N=100$ trials the susceptibility estimates are noisy and show much less structure, except for the anti-diagonal at the cell's baseline firing rate. The SI($r$) quantifies the height of this ridge where the two stimulus frequencies add up to the neuron's baseline firing rate. \figitem{C} Correlations between the estimates of SI($r$) based on 100 FFT segments (density to the right) with the converged ones based on one or ten million segments at a given stimulus contrast for all $n=39$ model cells. The black circle marks the model cell shown in \panel{A} and \panel{B}. The black diagonal line is the identity line and the dashed line is a linear regression. The correlation coefficient and corresponding significance level are indicated in the top left corner. The thin vertical line is a threshold at 1.2, the thin horizontal line a threshold at 1.8. The number of cells within each of the resulting four quadrants denote the false positives (top left), true positives (top right), true negatives (bottom left), and false negatives (bottom right) for predicting a triangular structure in the converged susceptibility estimate from the estimates based on only 100 segments.}
\end{figure}
\subsection{Weakly nonlinear interactions can be deduced from limited data}
@@ -693,13 +689,15 @@ In contrast to the ampullary cells, P-units respond to the amplitude modulation
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.
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.
\subsection{Conclusions}
We have demonstrated pronounced nonlinear responses in primary electrosensory afferents at weak stimulus amplitudes and sufficiently low intrinsic noise levels. The observed nonlinearities match the expectations from previous theoretical studies \citep{Voronenko2017,Franzen2023}. The resulting nonlinear components introduce spectral components not present in the original stimulus, that may provide an edge in the context of signal detection problems at stimulus amplitudes close to threshold \citep{Schlungbaum2023}. Electrosensory afferents share an evolutionary history with hair cells \citep{Baker2019} and share many response properties with mammalian auditory nerve fibers \citep{Barayeu2023, Joris2004}. Thus, we expect weakly nonlinear responses for near-threshold stimulation in auditory nerve fibers as well.
We have demonstrated pronounced nonlinear responses in primary electrosensory afferents at weak stimulus amplitudes and sufficiently low intrinsic noise levels. The observed nonlinearities match the expectations from previous theoretical studies \citep{Voronenko2017,Franzen2023}. The resulting nonlinear components introduce spectral components not present in the original stimulus, that may provide an edge in the context of signal detection problems at stimulus amplitudes close to threshold \citep{Schlungbaum2023}. Electrosensory afferents share an evolutionary history with hair cells \citep{Baker2019} and share many response properties with mammalian auditory nerve fibers \citep{Barayeu2023, Joris2004}. Thus, we expect weakly nonlinear responses for near-threshold stimulation in auditory nerve fibers as well. These could boost or distort responses to two simultaneously presented tones and thus might play a role in forming perception of music.
%\bibliographystyle{apalike}%alpha}%}%alpha}%apalike}
@@ -720,4 +718,4 @@ We have demonstrated pronounced nonlinear responses in primary electrosensory af
% LocalWords: Furutsu Novikov interspike durations Apteronotus
% LocalWords: leptorhynchus nonlinearity linearizing spectrally
% LocalWords: electroreceptors lowpass differentially isopotential
% LocalWords: transdermal convolved
% LocalWords: transdermal convolved electrophysiologically

112
rebuttal2.tex
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@@ -54,7 +54,8 @@
like to offer several suggestions that may serve to either enhance
the manuscript or inspire future research endeavors.}
\response{}
\response{Thank you for trying to make our manuscript more
biologist-friendly!}
\issue{First, I should point out that beyond the presence of a
threshold-induced nonlinearity, the complex structure of the
@@ -67,7 +68,7 @@
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 leptorhinchus were labeled to varying
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,
@@ -76,16 +77,15 @@
strengthen the links between well-written theoretical analysis and
the practical field of experimental physiology.}
\response{READ PAPERS AND CITE THEM You are right, there are more
nonlinear mechanisms potentially contributiong 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 paragraph to the methods reiterating
what we have in the discusssion, that the details of this
nonlinearity are not so important in the context of the present
manuscript, since we compute all cross-spectra between the resulting
amplitude modulation and the spikes responses. MAYBE ALSO IN THE
CONCLUSION.}
\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
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.}
\issue{Second, and along the same lines, the discussion could be
improved by mentioning the effects and significance of these
@@ -94,7 +94,8 @@
interference avoidance responses, and b) transient changes, as in
chirps.}
\response{HERE: STATIONARY ANALYSIS, CHIRPS ARE TRANSIENT, NEEDS RESEARCH. JAR SIGNALS are slow but change continously so they might hit the baseline frequency Add something somewhere around line 660.}
\response{We added a paragraph addressing JARs, chirps, and rises to
the discussion (lines 695 -- 703).}
\issue{Finally, the precise description of the methods could be
expanded for reaching a broader biology audience; in particular, the
@@ -103,8 +104,6 @@
first reading of the methods, although accurate, does not offer the
biology reader a quick and intuitive approach to the study.}
\response{IMPROVE METHOD DESCRIPTION AS DESCRIBED BELOW}
\issue{Next, I list some minor more detailed comments that may clarify
the design and methods and facilitate their understanding by a
broader audience.}
@@ -118,11 +117,12 @@
between the extreme profiles of P (signal amplitude) and T (signal
slope)?}
\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 explain now in the ``Identification of P-units and
ampullary cells'' section in the methods.}
\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
@@ -137,7 +137,10 @@
field foveal perioral region where the majority of receptors are
located.}
\response{ADD SOMETHING TO STIMULATION SECTION LINE 110}
\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
@@ -153,7 +156,8 @@
could benefit from greater clarity to avoid the need to explore the
results first in order to understand well.}
\response{STATE TYPE OF FILTERING IN STIMULATION SECTION, CITE ALES SKORJANC}
\response{We added a sentence that describes how we generate those
stimuli in the Fourier domain (lines 156--160).}
\issue{Line 154. This procedure elicits a modulation of the envelope
of the reafferent signal. To achieve this, you adopted distinct
@@ -163,7 +167,10 @@
averaged sine wave recorded via local electrodes adjacent to the
gills exhibited an increase of 1 to 5\%, is this correct?}
\response{NO! EXPLAIN AND ENHANCE STIMULATION SECTION}
\response{No! 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).}
\issue{b) with regard to P receptors, you multiplied the head-to-tail
ongoing signal by a white noise signal and played the resultant
@@ -184,7 +191,17 @@
on the peaks of the signal themselves? How does this affect the
recruitment of P and T receptors?}
\response{ALL THESE DETAILS DO NOT MATTER AT THE LEVL OF INDIVIDUAL P-UNITS. SEE HLADNIK.}
\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.}
\issue{Line 238. Are you referring to the terminal non-myelinated
branches that connect receptor cells to the initial Ranvier node?
@@ -200,7 +217,9 @@
organ. Could you discuss this aspect, considering the anatomical
structure of the receptor in your species?}
\response{READ LITERATURE AND SAY A FEW WORDS}
\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).}
\issue{\large Reviewer \#2}
@@ -252,15 +271,16 @@
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 $\Delta f_2$
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
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, such that the reader does not get overwhelmed.}
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, such that the reader does not get
overwhelmed.}
\issue{(6) Figure 3. Why are these example firing rates based on
convolution with a 1 ms Gaussian kernel if the analyses were based
@@ -269,13 +289,13 @@
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
\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, middel column, the range of possible values of the
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.}
@@ -289,12 +309,12 @@
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 threhsolding at zero), then there
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 (I guess this is what you have in
and Andre Longtin and colleges (I guess this is what you had in
mind).}
\issue{(8) Line 302. "not-small amplitude" is arbitrary and
@@ -324,12 +344,12 @@
\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
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.\\
We added a whole paragraph at the beginning of this section to make
this clear.}
this clear (line 477 -- 482).}
\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
@@ -343,6 +363,16 @@
reiterate these points briefly and delve into more detail on
comparative considerations.}
\response{UH. LETS THINK ABOUT IT.}
\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.
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}

22
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@@ -4178,6 +4178,17 @@ We collected weakly electric gymnotoid fish in the vicinity of Manaus, Amazonas,
Timestamp = {2020.02.11}
}
@article{Fortune2020,
title={Spooky interaction at a distance in cave and surface dwelling electric fishes.},
author={Eric S. Fortune and Nicole Andanar and Manu Madhav and Ravikrishnan P. Jayakumar and Noah J. Cowan and Maria Elina Bichuette and Daphne Soares},
journal={Front Integr Neurosci},
volume={14},
pages={561524},
year={2020}
}
@article{Maier2008,
title={Integration of bimodal looming signals through neuronal coherence in the temporal lobe},
author={Maier, Joost X and Chandrasekaran, Chandramouli and Ghazanfar, Asif A},
@@ -5158,7 +5169,7 @@ and Keller, Clifford H.},
publisher={American Psychological Association}
}
@Article{Raab2019,
@Article{Raa2019,
Title = {Dominance in Habitat Preference and Diurnal Explorative Behavior of the Weakly Electric Fish \textit{Apteronotus leptorhynchus}},
Author = {Raab, Till and Linhart, Laura and Wurm, Anna and Benda, Jan},
Journal = {Frontiers in Integrative Neuroscience},
@@ -5173,6 +5184,15 @@ and Keller, Clifford H.},
Url = {https://www.frontiersin.org/article/10.3389/fnint.2019.00021}
}
@article{Raab2021,
title={Electrocommunication signals indicate motivation to compete during dyadic interactions of an electric fish.},
author={Till Raab and Sercan Bayezit and Saskia Erdle and Jan Benda},
journal={J Exp Biol},
volume={224},
pages={jeb242905},
year={2021}
}
@Article{Rao1999,
Title = {Predictive coding in the visual cortex: a functional interpretation of some extra-classical receptive-field effects.},
Author = {Rajesh P.N. Rao and Dana H. Ballard},

1
regimes.py
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@@ -296,7 +296,6 @@ def plot_psd(ax, s, path, contrast, spikes, nfft, dt, beatf1, beatf2, eodf):
# zorder=40, **s.psF02m)
#ax.plot(eodf - beatf2 + 3*beatf1, decibel(peak_ampl(freqs, psd, eodf - beatf2 + 2*beatf1)) + offsm,
# zorder=40, **s.psF02m)
ax.set_xlim(0, 750)
ax.set_xlim(0, 950)
ax.set_ylim(-60, 0)
ax.set_xticks_delta(200)