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__pycache__/utils_all_down.cpython-39.pyc
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data_overview_mod.pdf
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data_overview_mod.py
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@ -6,7 +6,7 @@ from matplotlib import gridspec as gridspec, pyplot as plt
import numpy as np
from utils_all_down import default_settings
from utils_suseptibility import colors_overview
from utils_suseptibility import default_figsize, NLI_scorename,pearson_label, exclude_nans_for_corr, kernel_scatter, \
from utils_suseptibility import default_figsize, NLI_scorename2,pearson_label, exclude_nans_for_corr, kernel_scatter, \
plt_burst_modulation_hists, \
version_final
from utils_all import update_cell_names, load_overview_susept, make_log_ticks, p_units_to_show, save_visualization, setting_overview_score
@ -135,7 +135,7 @@ def data_overview3():
scores_here = [score,score,score]#,score]
score_name = ['max(diag5Hz)/med_diagonal_proj_fr','max(diag5Hz)/med_diagonal_proj_fr']#,'max(diag5Hz)/med_diagonal_proj_fr']#'Perc99/Med'
score_name = ['Fr/Med', 'Fr/Med']#'Fr/Med'] # 'Perc99/Med'
score_name = [NLI_scorename(), NLI_scorename(), NLI_scorename()]#NLI_scorename()] # 'Fr/Med''Perc99/Med'
score_name = [NLI_scorename2(), NLI_scorename2(), NLI_scorename2()]#NLI_scorename()] # 'Fr/Med''Perc99/Med'
ax_j = []
axls = []
axss = []
@ -172,6 +172,9 @@ def data_overview3():
max_x=max_x[v], xlim=xlimk, x_pos=1, labelpad = labelpad,
burst_fraction=burst_fraction[c], ha='right')
print(cell_type_here + ' median '+scores_here[v]+''+str(np.nanmedian(frame_file[scores_here[v]])))
print(cell_type_here + ' max ' + x_axis[v] + '' + str(np.nanmax(frame_file[x_axis[v]])))
if v == 0:
colors = colors_overview()
axx.set_title(cell_types_name[c], color = colors[cell_type_here])

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model_and_data.pdf
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model_and_data.py
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@ -138,10 +138,11 @@ def model_and_data2(eod_metrice = False, width=0.005, nffts=['whole'], powers=[1
#embed()
tr_name = trial_nr/1000000
if tr_name == 1:
tr_name = 1
titles = ['Model\n$N=11$ \n $c=1\,\%$', 'Model\n$N=%s $' % (tr_name) +'\,million \n $c=1\,\%$',
'Model\,('+noise_name().lower()+')' + '\n' + '$N=11$\n $c=0\,\%$',
'Model\,('+noise_name().lower()+')' + '\n' + '$N=%s$' % (tr_name) + '\,million \n $c=0\,\%$',
tr_name = 1#'$c=1\,\%$','$c=0\,\%$'
cs = ['$c=1\,\%$','$c=0\,\%$']
titles = ['Model\n$N=11$', 'Model\n$N=%s $' % (tr_name) +'\,million',
'Model\,('+noise_name().lower()+')' + '\n' + '$N=11$',
'Model\,('+noise_name().lower()+')' + '\n' + '$N=%s$' % (tr_name) + '\,million',
]#%
#'Model\,('+noise_name().lower()+')' + '\n' + '$N=11$\n $c=1\,\%$',
# 'Model\,('+noise_name().lower()+')' + '\n' + '$N=%s$' % (tr_name) + '\,million\n $c=1\,\%$ '
@ -316,11 +317,11 @@ def model_and_data2(eod_metrice = False, width=0.005, nffts=['whole'], powers=[1
if vers == 'first':
ax_external.text(1, 1, 'RAM', ha='right', color='red', transform=ax_external.transAxes)
ax_external.text(1, 1, 'RAM('+cs[0]+')', ha='right', color='red', transform=ax_external.transAxes)
ax_n.text(start_pos_modeldata(), 1.1, noise_component_name(), ha='right', color='gray',
transform=ax_n.transAxes)
elif vers == 'second':
ax_external.text(1, 1, 'RAM', ha='right', color='red', transform=ax_external.transAxes)
ax_external.text(1, 1, 'RAM('+cs[1]+')', ha='right', color='red', transform=ax_external.transAxes)
ax_intrinsic.text(start_pos_modeldata(), 1.1, signal_component_name(), ha='right', color='purple',
transform=ax_intrinsic.transAxes)
ax_n.text(start_pos_modeldata(), 0.9, noise_component_name(), ha='right', color='gray',

4
references.bib
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@ -878,7 +878,7 @@ pages = {811--824}
Volume = {47}
}
@article{Borst1999,
@Article{Borst1999,
author = {Borst, A. and Theunissen, F.E.},
journal = {Nature Neurosci.},
keywords = {Information, Information Theory, neural code, review, theory},
@ -1311,7 +1311,7 @@ impulse into a complex spatiotemporal electromotor pattern.},
publisher={Springer}
}
@article{Chacron2005,
@Article{Chacron2005,
title={Electroreceptor neuron dynamics shape information transmission},
author={Chacron, Maurice J and Maler, Leonard and Bastian, Joseph},
journal={Nature Neuroscience},

85
references.bib.bak
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@ -675,6 +675,24 @@ pages = {811--824}
VOLUME = {104},
PAGES = {2806-2820} }
@incollection{Benda2013,
author = {Benda, Jan and Grewe, Jan and Krahe, R{\~A}ƒ{\^A}ƒ{\~A}‚{\^A}ƒ{\~A}ƒ{\^A}‚{\~A}‚{\^A}¼diger},
booktitle = {Animal Communication and Noise},
doi = {10.1007/978-3-642-41494-7_12},
editor = {Brumm, Henrik},
isbn = {978-3-642-41493-0},
language = {English},
pages = {331-372},
publisher = {Springer Berlin Heidelberg},
refid = {876},
series = {Animal Signals and Communication},
timestamp = {2014.02.03},
title = {Neural Noise in Electrocommunication: From Burden to Benefits},
url = {http://dx.doi.org/10.1007/978-3-642-41494-7_12},
volume = {2},
year = {2013},
bdsk-url-1 = {http://dx.doi.org/10.1007/978-3-642-41494-7_12}}
@incollection{Benda2020,
title={The physics of electrosensory worlds.},
author={Benda, Jan},
@ -860,6 +878,19 @@ pages = {811--824}
Volume = {47}
}
@Article{Borst1999,
author = {Borst, A. and Theunissen, F.E.},
journal = {Nature Neurosci.},
keywords = {Information, Information Theory, neural code, review, theory},
number = {11},
owner = {grewe},
pages = {947--957},
refid = {216},
timestamp = {2008.09.26},
title = {Information theory and neural coding},
volume = {2},
year = {1999}}
@Book{Bradbury2011,
Title = {Principles of animal communication},
Author = {Bradbury, JW and Vehrencamp, SL},
@ -1280,7 +1311,7 @@ impulse into a complex spatiotemporal electromotor pattern.},
publisher={Springer}
}
@article{Chacron2005,
@Article{Chacron2005,
title={Electroreceptor neuron dynamics shape information transmission},
author={Chacron, Maurice J and Maler, Leonard and Bastian, Joseph},
journal={Nature Neuroscience},
@ -2594,6 +2625,17 @@ article{Egerland2020,
Timestamp = {2020.01.27}
}
@article{Haggard2023,
title={Coding of object location by heterogeneous neural populations with spatially dependent correlations in weakly electric fish},
author={Haggard, Myriah and Chacron, Maurice J},
journal={PLOS Computational Biology},
volume={19},
number={3},
pages={e1010938},
year={2023},
publisher={Public Library of Science San Francisco, CA USA}
}
@ARTICLE{Hagiwara1963,
AUTHOR = {S. Hagiwara and H. Morita},
TITLE = {Coding mechanisms of electroreceptor fibers in some electric fish.},
@ -4784,6 +4826,24 @@ and Keller, Clifford H.},
VOLUME = {40},
PAGES = {e106010} }
@Article{Padmanabhan2010,
Title = {Intrinsic biophysical diversity decorrelates neuronal firing while increasing information content.},
Author = {Krishnan Padmanabhan and Nathaniel N Urban},
Journal = {Nat Neurosci},
Year = {2010},
Month = {Oct},
Number = {10},
Pages = {1276--1282},
Volume = {13},
Abstract = {Although examples of variation and diversity exist throughout the nervous system, their importance remains a source of debate. Even neurons of the same molecular type have notable intrinsic differences. Largely unknown, however, is the degree to which these differences impair or assist neural coding. We examined the outputs from a single type of neuron, the mitral cells of the mouse olfactory bulb, to identical stimuli and found that each cell's spiking response was dictated by its unique biophysical fingerprint. Using this intrinsic heterogeneity, diverse populations were able to code for twofold more information than their homogeneous counterparts. In addition, biophysical variability alone reduced pair-wise output spike correlations to low levels. Our results indicate that intrinsic neuronal diversity is important for neural coding and is not simply the result of biological imprecision.},
Institution = {Department of Biological Sciences, Carnegie Mellon University, Pittsburgh, Pennsylvania, USA.},
Keywords = {Action Potentials; Animals; Animals, Newborn; Biophysical Phenomena; Biophysics; Electric Stimulation; Entropy; Kv1.2 Potassium Channel; Mice; Mice, Inbred C57BL; Models, Neurological; Nerve Net; Neurons; Olfactory Bulb; Patch-Clamp Techniques; Statistics as Topic},
Pii = {nn.2630},
Pmid = {20802489},
Refid = {792},
Timestamp = {2011.10.20},
}
@article{Palmer1982,
title={Encoding of rapid amplitude fluctuations by cochlear-nerve fibres in the guinea-pig},
author={Palmer, Alan Richard},
@ -6455,6 +6515,18 @@ groups and electrogenic mechanisms.},
PAGES = {341--354}
}
@article{Wiesenfeld1995,
author = {Wiesenfeld, K. and Moss, F.},
journal = {Nature},
keywords = {noise, statistics},
pages = {33--36},
refid = {440},
timestamp = {2008.09.26},
title = {Stochastic resonance and the benefits of noise: from ice ages to crayfish and squids},
volume = {373},
year = {1995}
}
@Article{Wilkens2002,
Title = {The electric sense of the paddlefish: a passive system for the detection and capture of zooplankton prey.},
Author = {Lon A. Wilkens and Michael H. Hofmann and Winfried Wojtenek},
@ -6660,3 +6732,14 @@ groups and electrogenic mechanisms.},
Pages = {159--173},
Volume = {192}
}
@article{Zhang2022,
title={A robust receptive field code for optic flow detection and decomposition during self-motion},
author={Zhang, Yue and Huang, Ruoyu and N{\"o}renberg, Wiebke and Arrenberg, Aristides B},
journal={Current Biology},
volume={32},
number={11},
pages={2505--2516},
year={2022},
publisher={Elsevier}
}

53
susceptibility1.tex
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@ -389,8 +389,8 @@
\newcommand{\burstcorr}{\ensuremath{{Corrected}}}
\newcommand{\cvbasecorr}{CV\ensuremath{_{BaseCorrected}}}
\newcommand{\cv}{CV\ensuremath{_{Base}}}%\cvbasecorr{}
\newcommand{\nli}{NLI\ensuremath{(\fbase{})}}%Fr$_{Burst}$
\newcommand{\nlicorr}{NLI\ensuremath{(\fbasecorr{}})}%Fr$_{Burst}$
\newcommand{\nli}{PNL\ensuremath{(\fbase{})}}%Fr$_{Burst}$
\newcommand{\nlicorr}{PNL\ensuremath{(\fbasecorr{}})}%Fr$_{Burst}$
\newcommand{\suscept}{$|\chi_{2}|$}
\newcommand{\frcolor}{pink lines}
@ -518,7 +518,9 @@ The P-unit responses can be partially explained by simple linear filters. The li
Theoretical work shows that leaky-integrate-and-fire (LIF) model neurons show a distinct pattern of nonlinear stimulus encoding when the model is driven by two cosine signals. In the context of the weakly electric fish, such a setting is part of the animal's everyday life as the sinusoidal electric-organ discharges (EODs) of neighboring animals interfere with the own field and each lead to sinusoidal amplitude modulations (AMs) that are called beats and envelopes \cite{Middleton2006, Savard2011,Stamper2012Envelope}. The p-type electroreceptor afferents of the tuberous electrosensory system, i.e. the P-units, respond to such AMs of the underlying EOD carrier and their time-dependent firing rate carries information about the stimulus' time-course. P-units are heterogeneous in their baseline firing properties \cite{Grewe2017, Hladnik2023} and differ with respect to their baseline firing rates, the sensitivity and their noisiness. We here explore the nonlinear mechanism in different cells that exhibit distinctly different levels of noise, i.e. differences in the coefficient of variation (CV) of the interspike intervals (ISI). Low-CV P-units have a less noisy, more regular, firing pattern whereas high-CV P-units show a less regular firing pattern in their baseline activity.
\subsection*{Low-CV P-units exhibit nonlinear interactions} %frequency combinations withappearing when the input frequencies are related to \fbase{} are
Second-order susceptibility is expected to be especially pronounced for low-CV cells \cite{Voronenko2017}. P-units fire action potentials probabilistically phase-locked to the self-generated EOD. Skipping of EOD cycles leads to the characteristic multimodal ISI distribution with maxima at integer multiples of the EOD period (\subfigrefb{fig:cells_suscept}{A}). In this example the ISI distribution has a CV of 0.2 which can be considered low among P-units\cite{Hladnik2023}. Spectral analysis of the baseline activity shows two major peaks, the first is located at the baseline firing rate (\fbase), the second is located at the discharge frequency of the electric organ (\feod{}) and is flanked by two smaller peaks at $\feod \pm \fbase{}$ (\subfigref{fig:cells_suscept}{B}).
Second-order susceptibility is expected to be especially pronounced for low-CV cells \cite{Voronenko2017}. P-units fire action potentials probabilistically phase-locked to the self-generated EOD. Skipping of EOD cycles leads to the characteristic multimodal ISI distribution with maxima at integer multiples of the EOD period (\subfigrefb{fig:cells_suscept}{A}). In this example the ISI distribution has a CV of 0.2 which can be considered low among P-units\cite{Hladnik2023}. Spectral analysis of the baseline activity shows two major peaks, the first is located at the baseline firing rate (\fbase), the second is located at the discharge frequency of the electric organ (\feod{}) and is flanked by two smaller peaks at $\feod \pm \fbase{}$ (\subfigref{fig:cells_suscept}{B}). High-CV P-units do not exhibit pronounced nonlinearities (for more details see supplementary information: \nameref*{S1:highcvpunit} )
%
\begin{figure*}[!ht]
\includegraphics[width=\columnwidth]{cells_suscept}
@ -529,9 +531,7 @@ Noise stimuli, the random amplitude modulations (RAM, \subfigref{fig:cells_susce
Theory predicts a pattern of nonlinear susceptibility for the interaction of two cosine stimuli when \fone{}, \ftwo{} or \fsum{} are equal to \fbase{} (pink triangle in \subfigsref{fig:cells_suscept}{E, F}). To estimate the second-order susceptibility from the P-unit responses to RAM stimuli, the noise stimulus was set in relation to the corresponding neuronal response in the Fourier domain, resulting in a matrix in which the nonlinearity at the sum frequency \fsum{} in the firing rate is depicted for two noise frequencies \fone{} and \ftwo{} (\Eqnref{eq:susceptibility}, \subfigrefb{fig:cells_suscept}{E--F}). The decomposition of the RAM stimulus into the frequency components allows us to approximate the beat frequencies $\Delta f$, occurring in case of pure sine-wave stimulation (\subfigrefb{fig:motivation}{D}). Thus, the nonlinearity accessed with the RAM stimulation at \fsum{} (\subfigrefb{fig:cells_suscept}{E}) is similar to the nonlinearity appearing during pure sine-wave stimulation at \bsum{} (orange peak, \subfigrefb{fig:motivation}{D}). A band of elevated nonlinearity appears for the low RAM contrast when \fsumb{} (yellow anti-diagonal between pink edges, \subfigrefb{fig:cells_suscept}{E}) but vanishes for the stronger stimulus (20\% contrast, \subfigref{fig:cells_suscept}{F}). Further, the overall level of nonlinearity changes with the stimulus strength. To compare the structural changes in the matrices we calculated the mean of the anti-diagonals, resulting in the projected diagonal (\subfigrefb{fig:cells_suscept}{G}). For a low RAM contrast the second-order susceptibility at \fbase{} is slightly enhanced (\subfigrefb{fig:cells_suscept}{G}, gray dot on light purple line). For the higher RAM contrast, however, this is not visible and the overall second-order susceptibility is reduced (\subfigrefb{fig:cells_suscept}{G}, compare light and dark purple lines).
\subsection*{High-CV P-units do not exhibit increased nonlinear interactions at \fsum}
\notejg{it is not PLoS style to actively work with the supplementary figures... We may want to reduce this section here, move the text to the supporting information}
CVs in P-units can range up to 1.5 \cite{Grewe2017, Hladnik2023}. \nameref*{S1:highcvpunit} shows the same analysis for an example higher-CV P-unit. Similar to the low-CV cell, high-CV P-units fire at multiples of the EOD period (\subfigrefb{fig:cells_suscept_high_CV}{A}). In contrast to low-CV P-unit, however, the higher CV characterizes the noisier baseline firing pattern and the peak at \fbase{} is less pronounced in the power spectrum of the baseline activity (\subfigrefb{fig:cells_suscept_high_CV}{B}). High-CV P-units do not exhibit a clear nonlinear structure related to \fbase{} neither in the second-order susceptibility matrices (\subfigrefb{fig:cells_suscept_high_CV}{E--F}), nor in the projected diagonals (\subfigrefb{fig:cells_suscept_high_CV}{G}). The overall level of nonlinearity, however shows the same dependence on the stimulus contrast. It is much reduced for high-contrast stimuli that drive the neuron much stronger (\subfigrefb{fig:cells_suscept_high_CV}{F}).
\subsection*{Ampullary afferents exhibit strong nonlinear interactions}
@ -544,7 +544,7 @@ Irrespective of the CV, neither cell shows the complete proposed structure of no
\end{figure*}
\subsection*{Internal noise hides parts of the nonlinearity structure}
Traces of the proposed structure of second-order susceptibility are found in both ampullary and p-type electrosensory afferents. The nonlinerarity seems to depend on the CV, i.e. the level of intrinsic noise in the cells and in the next step we address whether the level of intrinsic masks the expected structures. One option to reduce the impact of this intrinsic noise is to average it out over many repetitions of the same stimulus. In the electrophysiological experiments we only have a limited recording duration and hence the number of stimulus repetitions is limited. To overcome these limitations we compared between an electrophysiologically recorded low-CV P-unit and its P-unit LIF model counterpart fitted to reproduce the P-unit behavior faithfully (see \figref{flowchart})\cite{Barayeu2023}. In the recording depicted in \subfigrefb{model_and_data}{A} the cell was stimulated with a weak RAM stimulus with the contrast of 1\,$\%$ (red, \subfigrefb{model_and_data}\,\panel[i]{B}). The across trial average ($\n{}=11$) shows the diagonal band in second-order susceptibility at \fsumb{} (yellow diagonal in pink edges) and an additional nonlinear band appeared at \fsumehalf{} (\subfigrefb{model_and_data}{A}). The matched model reproduces the same diagonal at \fsumb{} (\subfigrefb{model_and_data}\,\panel[ii]{B}) but not the diagonal at \fsumehalf. By increasing the number of trials in the simulation, the signal-to-noise ratio and thus the estimation of the nonlinearity structures can be improved. Still, even with 1 million repetitions, no changes are observable in the nonlinearity structures (\subfigrefb{model_and_data}\,\panel[iii]{B}). The increased trial count, however goes along with generally smaller second-order susceptibility values (\subfigrefb{model_and_data}\,\panel[iii]{B}).
Traces of the proposed structure of second-order susceptibility are found in both ampullary and p-type electrosensory afferents. The nonlinerarity seems to depend on the CV, i.e. the level of intrinsic noise in the cells and in the next step we address whether the level of intrinsic masks the expected structures. One option to reduce the impact of this intrinsic noise is to average it out over many repetitions of the same stimulus. In the electrophysiological experiments we only have a limited recording duration and hence the number of stimulus repetitions is limited. To overcome these limitations we compared between an electrophysiologically recorded low-CV P-unit and its P-unit LIF model counterpart fitted to reproduce the P-unit behavior faithfully (see \figref{flowchart})\cite{Barayeu2023}. In the recording depicted in \subfigrefb{model_and_data}{A} the cell was stimulated with a weak RAM stimulus with the contrast of 1\,$\%$ (red, \subfigrefb{model_and_data}\,\panel[i]{B}). The across trial average ($\n{}=11$) shows the diagonal band in second-order susceptibility at \fsumb{} (yellow diagonal in pink edges) and an additional nonlinear band appeared at \fsumehalf{} (\subfigrefb{model_and_data}{A}). The matched model reproduces the same diagonal at \fsumb{} (\subfigrefb{model_and_data}\,\panel[ii]{B}) but not the diagonal at \fsumehalf. By increasing the number of trials in the simulation, the signal-to-noise ratio and thus the estimation of the nonlinearity structures can be improved. Still, even with 1 million repetitions, no changes are observable in the nonlinearity structures (\subfigrefb{model_and_data}\,\panel[iii]{B}). The increased trial count, however goes along with generally smaller second-order susceptibility values (\subfigrefb{model_and_data}\,\panel[iii]{B}). The decrease of the second-order susceptibility follows the relation $1/ \sqrt{N} $ (\figrefb{trialnr}).
%Note that this line doesn't appear in the susceptibility matrix of the model (\subfigrefb{model_and_data}{C})Each cell has an intrinsic noise level (\subfigrefb{model_and_data}{A}, bottom).
%The signal component (purple) compensates for this total noise reduction, is not a weak signal anymore and
@ -558,7 +558,7 @@ In a high-CV P-unit we could not find such nonlinear structures, neither in the
\begin{figure*}[!hb]
\includegraphics[width=\columnwidth]{model_and_data}
\caption{\label{model_and_data} High trial numbers and a reduced internal noise reveal the nonlinear structure in a LIF model with carrier. \figitem{A} \suscept{} surface of the electrophysiologically recorded low-CV P-unit (cell 2012-07-03-ak) driven with a weak stimulus with 1\% contrast. The plot shows that average \suscept{} surface ($N=11$). Pink edges mark the expected structure of enhanced nonlinearity. \figitem[i]{B} \textit{Standard condition} of the model simulation. The model is driven by a RAM stimulus (red trace). $\xi_{signal}$ and $\xi_{noise}$ components of the total intrinsic noise ($\xi$) are shown in the center and bottom. In this condition, $\xi_{noise} = \xi$. \figitem[ii]{B} \suscept{} surface of the cell's LIF model counterpart (cell 2012-07-03-ak, table~\ref{modelparams}) averaged over the same number of stimulus repetitions. \figitem[iii]{B} Same as \panel[ii]{B} but 1 million stimulus repetitions. \figitem[i - iii]{C} Same as in \panel[i - iii]{B} but in the \textit{noise split} condition: there is no external RAM signal driving the model. Parts (90\%) of the total intrinsic noise ($\xi$) are treated as signal (center trace, $\xi_{signal}$) while the remaining part is treated as $\xi_{noise}$ (see methods for details).
\caption{\label{model_and_data} High trial numbers and a reduced internal noise reveal the nonlinear structure in a LIF model with carrier. \figitem{A} \suscept{} surface of the electrophysiologically recorded low-CV P-unit (cell 2012-07-03-ak) driven with a weak stimulus with 1\% contrast. The plot shows that average \suscept{} surface ($N=11$). Pink edges mark the expected structure of enhanced nonlinearity. \figitem[i]{B} \textit{Standard condition} of the model simulation. The model is driven by a RAM stimulus (red trace). $\xi_{signal}$ and $\xi_{noise}$ components of the total intrinsic noise ($\xi$) are shown in the center and bottom. In this condition, $\xi_{noise} = \xi$. \figitem[ii]{B} \suscept{} surface of the cell's LIF model counterpart (cell 2012-07-03-ak, table~\ref{modelparams}) averaged over the same number of stimulus repetitions. \figitem[iii]{B} Same as \panel[ii]{B} but 1 million stimulus repetitions. \figitem[i - iii]{C} Same as in \panel[i - iii]{B} but in the \textit{noise split} condition: there is no external RAM signal driving the model. Parts (90\%) of the total intrinsic noise ($\xi$) are treated as signal (center trace, $\xi_{signal}$) while the remaining part is treated as $\xi_{noise}$ (see methods for details). Note that the signal component ($\xi_{signal}$) has a lower frequency content than the noise component ($\xi_{noise}$). Adding the discarded high frequency noise to the noise component $\xi_{noise}$ does not alter the results here (not shown).
}
\end{figure*}
@ -578,16 +578,16 @@ The smaller \fdiff{} power spectral peak observed during pure sine-wave stimulat
\begin{figure*}[!ht]
\includegraphics[width=\columnwidth]{data_overview_mod}
\caption{\label{fig:data_overview_mod} Nonlinearity for a population of P-units (\panel{A, C}) and ampullary cells (\panel{B, D}). The nonlinearity index \nli{} is calculated as the maximal value in the range $\fbase{} \pm 5$\,Hz of the projected diagonal divided by its median (see \Eqnref{eq:nli_equation}). Each recorded neuron contributes at maximum with two stimulus contrasts. \figitem{A, B} There is a negative correlation between the CV during baseline and \nli. \figitem{C, D} There is a negative correlation between the CV during stimulation and \nli. \figitem{E, F} The \nli{} is plotted against the response modulation, (see methods), an indicator of the subjective stimulus strength for a cell. There is a negative correlation between response modulation and \nli. Restricting the analysis to the weakest stimulus that was presented to each unique neuron, does not change the results. The number of unique neurons is 222 for P-units and 45 for ampullary cells.
\caption{\label{fig:data_overview_mod} Nonlinearity for a population of P-units (\panel{A, C}) and ampullary cells (\panel{B, D}). The peakedness of the nonlinearity \nli{} is calculated as the maximal value in the range $\fbase{} \pm 5$\,Hz of the projected diagonal divided by its median (see \Eqnref{eq:nli_equation}). Each recorded neuron contributes at maximum with two stimulus contrasts. \figitem{A, B} There is a negative correlation between the CV during baseline and \nli. \figitem{C, D} There is a negative correlation between the CV during stimulation and \nli. \figitem{E, F} The \nli{} is plotted against the response modulation, (see methods), an indicator of the subjective stimulus strength for a cell. There is a negative correlation between response modulation and \nli. Restricting the analysis to the weakest stimulus that was presented to each unique neuron, does not change the results. The number of unique neurons is 222 for P-units and 45 for ampullary cells.
}
\end{figure*}
%\Eqnref{response_modulation}
\subsection*{Low CVs and weak stimuli are associated with strong nonlinearity}
The nonlinear effects shown for single cell examples above are supported by the analysis of the pool of 222 P-units and 47 ampullary afferents recorded in 71 specimen. To compare across cells we expressed the second-order susceptibility as the nonlinearity index \nli{}, see \Eqnref{eq:nli_equation}. The \nli{} characterizes the peakedness of the projections onto the diagonal of the \suscept{} matrices (e.g. \subfigref{fig:cells_suscept}{G} at \fbase{}). It assumes high values when the \fbase{} peak in the projected diagonal is pronounced relative to the median of the diagonal and is small when there is no distinct peak. The P-unit population \nli{} values depend weakly on the CV of the ISI distribution under baseline condition. Cells with lower baseline CVs have the tendency to exhibit a stronger nonlinearity than those that have high CVs during baseline (\subfigrefb{fig:data_overview_mod}{A}) or during stimulation (\subfigrefb{fig:data_overview_mod}{C}) . The two example P-units shown before (\figrefb{fig:cells_suscept} and \figrefb{fig:cells_suscept_high_CV}) are highlighted with dark circles in \subfigrefb{fig:data_overview_mod}{A, C, E}. The stimulus strength plays an important role. Several of the recorded neurons contribute with two dots to the data as their responses to the same RAM stimulus but with different contrasts were recorded. Higher stimulus contrasts lead to a stronger drive and thus stronger response modulations (see color code bar in \subfigref{fig:data_overview_mod}{A}, see methods). Since P-units are heterogeneous in their susceptibility to the stimulus\cite{Grewe2017}, their responses to the same stimulus intensity vary a lot. When replotting the \nli{} against the response modulation (\subfigrefb{fig:data_overview_mod}{E}) a negative correlation is observed showing that cells that are strongly driven by the stimulus show less nonlinearity while those that are only weakly driven show higher nonlinearities. Whether or not a cell responds nonlinearity thus depends on both, the baseline CV (i.e. the internal noise) and the response strength.
The nonlinear effects shown for single cell examples above are supported by the analysis of the pool of 222 P-units and 47 ampullary afferents recorded in 71 specimen. To compare across cells we expressed the second-order susceptibility as the peakedness of the nonlinearity \nli{}, see \Eqnref{eq:nli_equation}. The \nli{} characterizes the peakedness of the projections onto the diagonal of the \suscept{} matrices (e.g. \subfigref{fig:cells_suscept}{G} at \fbase{}). It assumes high values when the \fbase{} peak in the projected diagonal is pronounced relative to the median of the diagonal and is small when there is no distinct peak. The P-unit population \nli{} values depend weakly on the CV of the ISI distribution under baseline condition. Cells with lower baseline CVs have the tendency to exhibit a stronger nonlinearity than those that have high CVs during baseline (\subfigrefb{fig:data_overview_mod}{A}) or during stimulation (\subfigrefb{fig:data_overview_mod}{C}) . The two example P-units shown before (\figrefb{fig:cells_suscept} and \figrefb{fig:cells_suscept_high_CV}) are highlighted with dark circles in \subfigrefb{fig:data_overview_mod}{A, C, E}. The stimulus strength plays an important role. Several of the recorded neurons contribute with two dots to the data as their responses to the same RAM stimulus but with different contrasts were recorded. Higher stimulus contrasts lead to a stronger drive and thus stronger response modulations (see color code bar in \subfigref{fig:data_overview_mod}{A}, see methods). Since P-units are heterogeneous in their susceptibility to the stimulus\cite{Grewe2017}, their responses to the same stimulus intensity vary a lot. When replotting the \nli{} against the response modulation (\subfigrefb{fig:data_overview_mod}{E}) a negative correlation is observed showing that cells that are strongly driven by the stimulus show less nonlinearity while those that are only weakly driven show higher nonlinearities. Whether or not a cell responds nonlinearity thus depends on both, the baseline CV (i.e. the internal noise) and the response strength.
%(Pearson's $r=-0.35$, $p<0.001$)222 P-units and 47 (Pearson's $r=-0.16$, $p<0.01$)
%In a P-unit population where each cell is represented not by several contrasts but by the lowest recorded contrast, \nli{} significantly correlates with the CV during baseline ($r=-0.17$, $p=0.01$), the response modulation ($r=-0.35$, $p<0.001$) and \fbase{} ($r=-0.32$, $p<0.001$).%, $\n{}=221$*, $\n{}=221$******, $\n{}=221$
%In a P-unit population where each cell is represented not by several contrasts but by the lowest recorded contrast, \nli{} significantly correlates with the CV during baseline ($r=-0.17$, $p=0.01$), the response modulation ($r=-0.35$, $p<0.001$) and \fbase{} ($r=-0.32$, $p<0.001$).%, $\n{}=222$*, $\n{}=222$******, $\n{}=222$
The population of ampullary cells is generally more homogeneous and have lower CVs than the P-units and show much higher \nli{} values (factor of 10). Overall, there is a negative correlation with the baseline CV. The example cell shown above (\figref{ampullary}) was recorded at two different stimulus intensities and the \nli{}s are highlighted with black circles. Again, we see that cells that are strongly driven by the stimulus cluster at the bottom of the distribution and have \nli{} values close to zero (\subfigrefb{fig:data_overview_mod}{B, D}). This is confirmed when the data is replotted against the response modulation, those cells that are strongly driven by the stimulus show weak nonlinearities while weakly driven neurons exhibit high values (\subfigrefb{fig:data_overview_mod}{F}).
@ -603,30 +603,30 @@ Nonlinearities are ubiquitous in nervous systems, they are essential to extract
Theoretical work\cite{Voronenko2017} explained analytically the occurrence of nonlinear products when a LIF model neuron is stimulated with pure sine-waves. To investigate whether the same mechanisms occur in electroreceptor afferents which are driven by AMs of a carrier and not by pure sine-waves, we followed the previous approach and quantified the second-order susceptibility from responses to white-noise stimuli \cite{Voronenko2017,Neiman2011fish,Nikias1993}. We expected to see elevated second-order susceptibility where either of the foreign signals matches the baseline firing rate ($f_1=f_{base}$ or $f_2=f_{base}$) or when the sum equals the baseline firing rate of the neuron (\fsumb{}) creating a triangular pattern of elevated \suscept{} e.g.\,\subfigref{model_and_data}\,\panel[iii]{C}. Indeed, we find traces of the same nonlinearities in the neuronal responses of p-type electroreceptor afferents. The nonlinear pattern observed in the experimental data, however, matches to the expectations only partially and only in a subset of neurons \figsref{fig:cells_suscept} and\,\ref{ampullary}. Nevertheless, the theory holds also for systems that are driven by AMs of a carrier and is thus more widely applicable.
\subsection*{Intrinsic noise limits nonlinear responses}
Only those P-units that exhibit low coefficients of variation (CV) of the interspike-interval distribution (\figref{fig:cells_suscept}{A}) in their unperturbed baseline response show the expected nonlinerities (\subfigref{fig:data_overview_mod}{A}). Such low-CV cells are rare among the 221 P-units that we used in this study. The afferents of the passive electrosensory system, the ampullary cells, however have generally lower CVs and show a much clearer nonlinearity pattern than the low-CV P-unit exemplified here (compare \figsref{fig:cells_suscept} and \ref{ampullary}). The single ampullary cell featured in \figref{ampullary} is a representative of the majority of cells analyzed here. All ampullary cells have CVs below 0.4 with a median around 0.1 and the observed \nli{}s are 10-fold higher than in P-units.
Only those P-units that exhibit low coefficients of variation (CV) of the interspike-interval distribution (\figref{fig:cells_suscept}{A}) in their unperturbed baseline response show the expected nonlinerities (\subfigref{fig:data_overview_mod}{A}). Such low-CV cells are rare among the 222 P-units that we used in this study. The afferents of the passive electrosensory system, the ampullary cells, however have generally lower CVs and show a much clearer nonlinearity pattern than the low-CV P-unit exemplified here (compare \figsref{fig:cells_suscept} and \ref{ampullary}). The single ampullary cell featured in \figref{ampullary} is a representative of the majority of ampullary cells analyzed here. All ampullary cells have CVs below 0.4 with a median around 0.12 and the observed \nli{}s are 10-fold higher than in P-units.
The CV serves as a proxy for the intrinsic noise in the cells. In both cell types, we observe a negative correlation between \nli{} and the CV, indicating that it is the level of intrinsic noise that plays a role here. These findings are in line with previous studies that propose that noise linearizes the system\cite{Roddey2000, Chialvo1997, Voronenko2017}. More intrinsic noise has been demonstrated to increase the CV and reduce nonlinear phase-locking in vestibular afferents\cite{Schneider2011}. Reduced noise, on the other hand, has been associated with stronger nonlinearity in pyramidal cells of the ELL\cite{Chacron2006}. Further support for the notion of noise limiting the nonlinearity comes from our P-unit LIF model that faithfully reproduces P-unit activity\cite{Barayeu2023}. We can use this model and the noise-split according to the Novikov-Furutsu theorem\cite{Novikov1965,Furutsu1963} to increase the signal-to-noise ratio in the cell while keeping the overall response variability constant (see methods). Treating 90\% of the total noise as signal and simulating large numbers of trial uncovers the full nonlinearity structure (\figref{model_and_data}) seen in LIF neurons and the analytical derivations when driven with sine-wave stimuli\cite{Voronenko2017}.
\subsection*{Noise stimulation approximates the real three-fish interaction}
Our analysis is based on the neuronal responses to white noise stimulus sequences. For the P-units, the stimulus was a random amplitude modulation (RAM) while it was a direct noise stimulus for the ampullary cells. In true white noise, each frequency component contributes with equal amplitude. This has the methodological advantage that all behaviorally relevant frequencies can be measured with a single stimulus presentation and is a widely used approach to characterize sensory coding\cite{Borst1999}\notejg{better references?}. In contrast to the situation with individual frequencies (direct sine-waves or sinusoidal AMs) the total power of the stimulus is equally distributed on all frequencies leading to a weaker signal-to-noise ratio. This explains that the nonlinearity pattern in the electroreceptor recordings only partially matches the expectation \figsref{fig:cells_suscept},\,\ref{ampullary} while the single-frequency stimulation shows nonlinear interference when the individual stimulus frequencies ($f_1, f_2, \Delta f_1, \Delta f_2$) match the baseline firing rate (\figref{fig:motivation}). With the noise-splitting trick, we could show that the full nonlinearity pattern is present but covered by the intrinsic noise. We thus conclude that the presence of the anti-diagonal pattern in the \suscept{} matrix is sufficient to conclude that the same nonlinear interactions happen here. This also validates the application of the white noise approach to characterize the full \suscept{} matrix instead of using all combinations of individual frequencies.
Our analysis is based on the neuronal responses to white noise stimulus sequences. For the P-units, the stimulus was a random amplitude modulation (RAM) while it was a direct noise stimulus for the ampullary cells. In true white noise, each frequency component contributes with equal amplitude. This has the methodological advantage that all behaviorally relevant frequencies can be measured with a single stimulus presentation and is a widely used approach to characterize sensory coding\cite{Borst1999,Chacron2005, Grewe2017}. In contrast to the situation with individual frequencies (direct sine-waves or sinusoidal AMs) the total power of the stimulus is equally distributed on all frequencies leading to a weaker signal-to-noise ratio. This explains that the nonlinearity pattern in the electroreceptor recordings only partially matches the expectation (\figsref{fig:cells_suscept},\,\ref{ampullary}) while the single-frequency stimulation shows nonlinear interference when the individual stimulus frequencies ($f_1, f_2, \Delta f_1, \Delta f_2$) match the baseline firing rate (\figref{fig:motivation}). With the noise-splitting trick, we could show that the full nonlinearity pattern is present but covered by the intrinsic noise. We thus conclude that the presence of the anti-diagonal pattern in the \suscept{} matrix is sufficient to conclude that the same nonlinear interactions happen here. This also validates the application of the white noise approach to characterize the full \suscept{} matrix instead of using all combinations of individual frequencies.
% The nonlinearity of ampullary cells in paddlefish \cite{Neiman2011fish} has been previously accessed with bandpass limited white noise.
% Here it was demonstrated that the second-order susceptibility for the two RAM noise input frequencies \fone{} and \ftwo{} can approximate a three-fish setting, where the driving force for the P-unit are two beats with frequencies \bone{} and \btwo{}. This was confirmed by a low-CV P-unit, where nonlinearities in the P-unit response occurred at the sum and difference of the beat frequencies for pure sine-wave stimulation (\figrefb{fig:motivation}). In this P-unit the nonlinearity appeared in a three-wave setting with \bone{} being close to \fbase{}, corresponding to a frequency combination on the vertical line at \foneb{} in the second-order susceptibility (\subfigrefb{fig:model_full}{B}). This implies that even if only the diagonal structure can be accessed with noise stimulation in the second-order susceptibility in an experiment (\subfigrefb{fig:model_full}{A}), it can be taken as an indicator that the whole nonlinear structure should be present during pure sine-wave stimulation in the same cell. With this RAM stimulation was demonstrated to be an effective method to scan the three-fish or two-beat plane and estimate the whole theoretically predicted nonlinear structure in experimentally recorded cells.
\subsection*{Heterogeneity of P-units might influence nonlinearity}\notejg{Better title...}
The observed nonlinear effects might facilitate the detectability of faint signals during a three fish setting, the electrosensory cocktail party. They are, however, very specific with respect to the relation of stimulus frequencies and the P-unit baseline frequency. The EOD frequencies of the interacting fish would be drawn from the distributions of EOD frequencies found in male and female fish\cite{Henninger2018, Henninger2020}\notejg{more refs on eod freq distributions?}. To be behaviorally relevant the faint signal detection would require reliable neuronal signalling irrespective of the individual EOD frequencies.
\subsection*{Nonlinearity might be influenced once integrating from a P-unit population with heterogeneous baseline properties}%Heterogeneity of P-units might influence
The observed nonlinear effects might facilitate the detectability of faint signals during a three fish setting, the electrosensory cocktail party. They are, however, very specific with respect to the relation of stimulus frequencies and the P-unit baseline frequency. The EOD frequencies of the interacting fish would be drawn from the distributions of EOD frequencies found in male and female fish\cite{Hopkins1974Eigen, Meyer1987, Henninger2018, Henninger2020}. To be behaviorally relevant the faint signal detection would require reliable neuronal signaling irrespective of the individual EOD frequencies.
P-units, however are very heterogeneous in their baseline firing properties\cite{Grewe2017, Hladnik2023}. The baseline firing rates vary in wide ranges (50--450\,Hz). This range covers substantial parts of the beat frequencies that may occur during animal interactions which is limited to frequencies below the Nyquist frequency (0 -- \feod/2)\cite{Barayeu2023}. It is thus likely that there are P-units which match approximately to the specificities of the different encounters.
On the other hand, the nonlinearity was found only in low-CV P-units (with white noise stimulation). The CVs are also very heterogeneous (0.1--1.5\notejg{CVs aus Abbildung einfuegen}, \figref{fig:data_overview_mod}\panel{A}) in our sample. Only a small fraction of the P-units has a sufficiently low level of intrinsic noise and will exhibit nonlinear responses. The P-units project to the ELL\cite{Krahe2014} and the integrating pyramidal cells in the different segments receive inputs in the range of 10s to 1000s of neurons\cite{Maler2009a}. Since the receptive fields of the pyramidal neurons are patches of adjacent receptors on the fish surface \cite{Bastian2002, Maler2009a, Haggard2023} and the input heterogeneity does not depend on the location of the receptor on the fish body \cite{Hladnik2023} the pyramidal cell input population will be heterogeneous. Heterogeneity was shown to be generally advantageous for the encoding in this\cite{Hladnik2023} and other systems\cite{Padmanabhan2010, Beiran2018}. At the same time it contradicts the apparent need for a selective readout of low-CV cells to maintain information arising through nonlinear interactions.
On the other hand, the nonlinearity was found only in low-CV P-units (with white noise stimulation). The CVs are also very heterogeneous (0.1--1.4, \figref{fig:data_overview_mod}\panel{A}) in our sample. Only a small fraction of the P-units has a sufficiently low level of intrinsic noise and will exhibit nonlinear responses. The P-units project to the ELL\cite{Krahe2014} and the integrating pyramidal cells in the different segments receive inputs in the range of 10s to 1000s of neurons\cite{Maler2009a}. Since the receptive fields of the pyramidal neurons are patches of adjacent receptors on the fish surface \cite{Bastian2002, Maler2009a, Haggard2023} and the input heterogeneity does not depend on the location of the receptor on the fish body \cite{Hladnik2023} the pyramidal cell input population will be heterogeneous. Heterogeneity was shown to be generally advantageous for the encoding in this\cite{Hladnik2023} and other systems\cite{Padmanabhan2010, Beiran2018}. At the same time it contradicts the apparent need for a selective readout of low-CV cells to maintain information arising through nonlinear interactions.
A possible readout mechanism should be the topic of future studies that need to take into account that the nonlinearities are stronger in pure sine-wave stimulation and the fraction of cells that show it under naturalistic stimulation might be larger than expected from the distribution of CVs.
\subsection*{Behavioral relevance of nonlinear interactions}
The behavioral relevance of the weak signal detection in P-units is evident from the courtship context observed in freely interacting animals\cite{Henninger2018}. Outside courtship behavior, the encoding of secondary or social envelopes is a common need\cite{Stamper2012Envelope}. In a previous study it was demonstrated that information about low-frequency secondary envelopes would not be present in P-units responses but would arise thorough nonlinear processing downstream in the ELL \cite{Middleton2006,Middleton2007}. Based on our work we would predict that only a small subset of cells, with low CVs, should encode the social envelopes under white noise stimulation. An absence of low-CVs cells in the population analyzed in the previous studies could explain their conclusions. On the other hand, another study showed that P-units with strong nonlinearities, low firing rates and high CVs could indeed encode social envelopes\cite{Savard2011}. These findings are in contrast to the previously mentioned work\cite{Middleton2007} and, at first glance, also to our results. The missing link, that has not been considered in this work, might be the bursting of P-units, the repeated firing of spikes after one EOD period interleaved with longer intervals of quietness\cite{Chacron2004}. Bursting was not explicitly addressed in the previous work, still the reported high CVs of the envelope encoding P-units indicate a higher rate of bursting\cite{Savard2011}. How bursts influence the second-order susceptibility of will be addressed in following works (in preparation).
The behavioral relevance of the weak signal detection in P-units is evident from the courtship context observed in freely interacting animals\cite{Henninger2018}. Outside courtship behavior, the encoding of secondary or social envelopes is a common need\cite{Stamper2012Envelope}. In a previous study it was demonstrated that information about low-frequency secondary envelopes would not be present in P-units' responses but would arise thorough nonlinear processing downstream in the ELL \cite{Middleton2006,Middleton2007}. Based on our work we would predict that only a small subset of cells, with low CVs, should encode the social envelopes under white noise stimulation. An absence of low-CVs cells in the population analyzed in the previous studies could explain their conclusions. On the other hand, another study showed that P-units with strong nonlinearities, low firing rates and high CVs could indeed encode social envelopes\cite{Savard2011}. These findings are in contrast to the previously mentioned work\cite{Middleton2007} and, at first glance, also to our results. The missing link, that has not been considered in this work, might be the bursting of P-units, the repeated firing of spikes after one EOD period interleaved with longer intervals of quietness\cite{Chacron2004}. Bursting was not explicitly addressed in the previous work, still the reported high CVs of the envelope encoding P-units indicate a higher rate of bursting\cite{Savard2011}. How bursts influence the second-order susceptibility of P-units will be addressed in following works (in preparation).
Sinusoidal AMs are relevant in interactions with a few fish. We can understand the noise as the presence of many animals with individual EOD frequencies at the same time. Under noise stimulation, strong nonlinearities were demonstrated for weak stimuli but to decrease for stronger noise stimuli (\figrefb{fig:cells_suscept}). As long as the noise signal is weak, those fish are distant and the nonlinearity is maintained. An increasing stimulus amplitude would indicate that many fish are close to the receiver and a decrease of nonlinear effects can be observed. These findings imply that the nonlinear effects arising in the presence of three fish decline the more fish join. \lepto{} usually prefers small groups of fish\cite{Stamper2010}. Thus, the described second-order susceptibility might still be behaviorally relevant under natural conditions. The decline of nonlinear effects when several fish are present might be an adaptive process reducing the number of frequencies represented in its primary sensory afferents to a minimum. Such representation would still leave room to create nonlinear effects at later processing steps in higher-order neurons.
Sinusoidal AMs are relevant in interactions with a few fish. We can understand the noise as the presence of many animals with individual EOD frequencies at the same time. Under noise stimulation, nonlinearities were demonstrated to be strong for weak stimuli but were shown to decrease for stronger noise stimuli (\figrefb{fig:cells_suscept}). As long as the noise signal is weak, those fish are distant and the nonlinearity is maintained. An increasing stimulus amplitude would indicate that many fish are close to the receiver and a decrease of nonlinear effects can be observed. These findings imply that the nonlinear effects arising in the presence of three fish decline the more fish join. \lepto{} usually prefers small groups of fish\cite{Stamper2010}. Thus, the described second-order susceptibility might still be behaviorally relevant under natural conditions. The decline of nonlinear effects when several fish are present might be an adaptive process reducing the number of frequencies represented in its primary sensory afferents to a minimum. Such representation would still leave room to create nonlinear effects at later processing steps in higher-order neurons.
The afferents of the passive electrosensory system, the ampullary cells, were found to exhibit much stronger nonlinearities than P-units (\figref{fig:data_overview_mod}). The adequate stimulus for this system is a direct stimulation not an amplitude modulation. In this sense the ampullary cells are closer to the LIF models used by Voroneko and colleagues\cite{Voronenko2017} and we can thus expect that the same nonlinear mechanisms are at work here. For the ampullary system, the sinewave stimulation might is not as relevant as for the P-unit system. Ampullary cells encode low-frequency exogenous electric signals such as muscle potentials induced by prey movement\cite{Kalmijn1974, Engelmann2010, Neiman2011fish}. The simultaneous muscle activity of a swarm of prey (such as \textit{Daphnia}) resembles Gaussian white noise\cite{Neiman2011fish}, similar to the stimuli used here. Our results show some similarities with the analyses by Neiman and Russel\cite{Neiman2011fish} who study the encoding in ampullary afferents in the paddlefish. There, the power spectrum of the spontaneous also shows a peak at the baseline frequency (internal oscillator) but also at the oscillation frequency measured at the epithelium and interactions of both. Most of the latter disappear in the case of external stimulation, though. Here we find only peaks at the baseline frequency of the neuron and its harmonics. There are interesting similarities and dissimilarities; stimulus encoding in the paddlefish as well as in the brown ghost is very linear for low frequencies and there are nonlinearities in both systems. Linear encoding in the paddlefish shows a gap in the spectrum at the frequency of the epitheliar oscillation, instead the nonlinear response is very pronounced there. In \lepto{}, the dominating frequency under baseline conditions is the baseline firing rate, and we do see increased nonlinearity in this frequency range. The baseline frequency, however, is outside the linear coding range\cite{Grewe2017} while it is within the linear coding range in paddlefish\cite{Neiman2011fish}. Interestingly, the nonlinear response in the paddlefish ampullaries increases with stimulus intensity while it disappears in our case (\figref{fig:data_overview_mod}\panel{D}). The population of ampullary cells is much more homogeneous with respect to the baseline rate (131$\pm$29\,Hz) and stimulus encoding properties than the P-units\cite{Grewe2017}. This implies that, if the stimulus contains the appropriate frequency components that sum up to the baseline rate, there should be a nonlinear response at a frequency that is similar for the full population of ampullary cells (the baseline frequency) that is outside the linear coding range. Postsynaptic cells integrating ampullary input might be able to extract this nonlinear response from the input population. How such nonlinear effects in ampullary cells might influence prey detection should be addressed in further studies.
The afferents of the passive electrosensory system, the ampullary cells, were found to exhibit much stronger nonlinearities than P-units (\figref{fig:data_overview_mod}). The adequate stimulus for this system is a direct stimulation not an amplitude modulation. In this sense the ampullary cells are closer to the LIF models used by Voroneko and colleagues\cite{Voronenko2017} and we can thus expect that the same nonlinear mechanisms are at work here. For the ampullary system, the sinewave stimulation is not as relevant as for the P-unit system. Ampullary cells encode low-frequency exogenous electric signals such as muscle potentials induced by prey movement\cite{Kalmijn1974, Engelmann2010, Neiman2011fish}. The simultaneous muscle activity of a swarm of prey (such as \textit{Daphnia}) resembles Gaussian white noise\cite{Neiman2011fish}, similar to the stimuli used here. Our results show some similarities with the analyses by Neiman and Russel\cite{Neiman2011fish} who study the encoding in ampullary afferents in the paddlefish. There, the power spectrum of the spontaneous activity also shows a peak at the baseline frequency (internal oscillator) but also at the oscillation frequency measured at the epithelium and interactions of both. Most of the latter disappear in the case of external stimulation, though. Here we find only peaks at the baseline frequency of the neuron and its harmonics. There are interesting similarities and dissimilarities; stimulus encoding in the paddlefish as well as in the brown ghost is very linear for low frequencies and there are nonlinearities in both systems. Linear encoding in the paddlefish shows a gap in the spectrum at the frequency of the epitheliar oscillation, instead the nonlinear response is very pronounced there. In \lepto{}, the dominating frequency under baseline conditions is the baseline firing rate, and we do see increased nonlinearity in this frequency range. The baseline frequency, however, is outside the linear coding range\cite{Grewe2017} while it is within the linear coding range in paddlefish\cite{Neiman2011fish}. Interestingly, the nonlinear response in the paddlefish ampullaries increases with stimulus intensity while it disappears in our case (\subfigrefb{fig:data_overview_mod}{F}). The population of ampullary cells is much more homogeneous with respect to the baseline rate (131$\pm$29\,Hz) and stimulus encoding properties than the P-units\cite{Grewe2017}. This implies that, if the stimulus contains the appropriate frequency components that sum up to the baseline rate, there should be a nonlinear response at a frequency that is similar for the full population of ampullary cells (the baseline frequency) that is outside the linear coding range. Postsynaptic cells integrating ampullary input might be able to extract this nonlinear response from the input population. How such nonlinear effects in ampullary cells might influence prey detection should be addressed in further studies.
\subsection*{Conclusion}
We have demonstrated that there are pronounced nonlinear responses in the primary electrosensory afferences of the weakly electric fish \lepto{}, systems that are very often characterized using linear methods. The observed nonlinearities match the expectations from previous theoretical studies\cite{Voronenko2017}. We can confirm that the theory applies also to systems that are encoding amplitude modulations of a carrier signal. Comparisons of P-units and ampullary cells showed that it is the level of intrinsic noise that determines how strongly nonlinear the system acts. Using the second-oder susceptibility estimated from the responses to white noise stimuli provides an easy way to determine the nonlinearity of the system under study. P-units share several features with mammalian
@ -734,14 +734,13 @@ The second-order susceptibility was calculated by dividing the higher-order cros
% \chi_{2}(\omega_{1}, \omega_{2}) = \frac{TN \sum_{n=1}^N \int_{0}^{T} dt\,r_{n}(t) e^{-i(\omega_{1}+\omega_{2})t} \int_{0}^{T}dt'\,s_{n}(t')e^{i \omega_{1}t'} \int_{0}^{T} dt''\,s_{n}(t'')e^{i \omega_{2}t''}}{2 \sum_{n=1}^N \int_{0}^{T} dt\, s_{n}(t)e^{-i \omega_{1}t} \int_{0}^{T} dt'\,s_{n}(t')e^{i \omega_{1}t'} \sum_{n=1}^N \int_{0}^{T} dt\,s_{n}(t)e^{-i \omega_{2}t} \int_{0}^{T} dt'\,s_{n}(t')e^{i \omega_{2}t'}}
% \end{split}
% \end{equation}
% \notejg{Wofuer genau brauchen wir equation 9?}
The absolute value of a second-order susceptibility matrix is visualized in \figrefb{fig:model_full}. There the upper right and the lower left quadrants characterize the nonlinearity in the response $r(t)$ at the sum frequency of the two input frequencies. The lower right and upper left quadrants characterize the nonlinearity in the response $r(t)$ at the difference of the input frequencies.
\paragraph{Nonlinearity index}\label{projected_method}
We expect to see nonlinear susceptibility when $\omega_1 + \omega_2 = \fbase{}$. To characterize this we calculated the nonlinearity index (NLI) as
We expect to see nonlinear susceptibility when $\omega_1 + \omega_2 = \fbase{}$. To characterize this we calculated the peakedness of the nonlinearity (PNL) as
\begin{equation}
\label{eq:nli_equation}
NLI(\fbase{}) = \frac{\max_{\fbase{}-5\,\rm{Hz} \leq f \leq \fbase{}+5\,\rm{Hz}} D(f)}{\mathrm{med}(D(f))}
\nli{} = \frac{\max_{\fbase{}-5\,\rm{Hz} \leq f \leq \fbase{}+5\,\rm{Hz}} D(f)}{\mathrm{med}(D(f))}
\end{equation}
For this index, the second-order susceptibility matrix was projected onto the diagonal $D(f)$, by taking the mean of the anti-diagonals. The peakedness at the frequency $\fbase{}$ in $D(f)$ was quantified by finding the maximum of $D(f)$ in the range $\fbase{} \pm 5$\,Hz (\subfigrefb{fig:cells_suscept}{G}) and dividing it by the median of $D(f)$.
@ -894,7 +893,15 @@ A big portion of the total noise was assigned to the signal component ($c_{signa
\newpage
\section*{Supporting information}
\%subsection*{High-CV P-units do not exhibit increased nonlinear interactions at \fsum}
\paragraph*{S1 Second-order susceptibility of high-CV P-unit}
CVs in P-units can range up to 1.5 \cite{Grewe2017, Hladnik2023}. We show the same analysis for an example higher-CV P-unit. Similar to the low-CV cell, high-CV P-units fire at multiples of the EOD period (\subfigrefb{fig:cells_suscept_high_CV}{A}). In contrast to low-CV P-unit, however, the higher CV characterizes the noisier baseline firing pattern and the peak at \fbase{} is less pronounced in the power spectrum of the baseline activity (\subfigrefb{fig:cells_suscept_high_CV}{B}). High-CV P-units do not exhibit a clear nonlinear structure related to \fbase{} neither in the second-order susceptibility matrices (\subfigrefb{fig:cells_suscept_high_CV}{E--F}), nor in the projected diagonals (\subfigrefb{fig:cells_suscept_high_CV}{G}). The overall level of nonlinearity, however shows the same dependence on the stimulus contrast. It is much reduced for high-contrast stimuli that drive the neuron much stronger (\subfigrefb{fig:cells_suscept_high_CV}{F}).
\label{S1:highcvpunit}
\begin{figure*}[!ht]
\includegraphics[width=\columnwidth]{cells_suscept_high_CV}
@ -906,7 +913,7 @@ A big portion of the total noise was assigned to the signal component ($c_{signa
\begin{figure*}[hp]%hp!
\includegraphics{trialnr}
\caption{\label{fig:trialnr} Change of the signal-to-noise ratio depending on the stimulus repetition number.
\caption{\label{fig:trialnr} Change of the median second-order susceptibility depending on the stimulus repetition number $\n{}$.
}
\end{figure*}

53
susceptibility1.tex.bak
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@ -389,8 +389,8 @@
\newcommand{\burstcorr}{\ensuremath{{Corrected}}}
\newcommand{\cvbasecorr}{CV\ensuremath{_{BaseCorrected}}}
\newcommand{\cv}{CV\ensuremath{_{Base}}}%\cvbasecorr{}
\newcommand{\nli}{NLI\ensuremath{(\fbase{})}}%Fr$_{Burst}$
\newcommand{\nlicorr}{NLI\ensuremath{(\fbasecorr{}})}%Fr$_{Burst}$
\newcommand{\nli}{PNL\ensuremath{(\fbase{})}}%Fr$_{Burst}$
\newcommand{\nlicorr}{PNL\ensuremath{(\fbasecorr{}})}%Fr$_{Burst}$
\newcommand{\suscept}{$|\chi_{2}|$}
\newcommand{\frcolor}{pink lines}
@ -518,7 +518,9 @@ The P-unit responses can be partially explained by simple linear filters. The li
Theoretical work shows that leaky-integrate-and-fire (LIF) model neurons show a distinct pattern of nonlinear stimulus encoding when the model is driven by two cosine signals. In the context of the weakly electric fish, such a setting is part of the animal's everyday life as the sinusoidal electric-organ discharges (EODs) of neighboring animals interfere with the own field and each lead to sinusoidal amplitude modulations (AMs) that are called beats and envelopes \cite{Middleton2006, Savard2011,Stamper2012Envelope}. The p-type electroreceptor afferents of the tuberous electrosensory system, i.e. the P-units, respond to such AMs of the underlying EOD carrier and their time-dependent firing rate carries information about the stimulus' time-course. P-units are heterogeneous in their baseline firing properties \cite{Grewe2017, Hladnik2023} and differ with respect to their baseline firing rates, the sensitivity and their noisiness. We here explore the nonlinear mechanism in different cells that exhibit distinctly different levels of noise, i.e. differences in the coefficient of variation (CV) of the interspike intervals (ISI). Low-CV P-units have a less noisy, more regular, firing pattern whereas high-CV P-units show a less regular firing pattern in their baseline activity.
\subsection*{Low-CV P-units exhibit nonlinear interactions} %frequency combinations withappearing when the input frequencies are related to \fbase{} are
Second-order susceptibility is expected to be especially pronounced for low-CV cells \cite{Voronenko2017}. P-units fire action potentials probabilistically phase-locked to the self-generated EOD. Skipping of EOD cycles leads to the characteristic multimodal ISI distribution with maxima at integer multiples of the EOD period (\subfigrefb{fig:cells_suscept}{A}). In this example the ISI distribution has a CV of 0.2 which can be considered low among P-units\cite{Hladnik2023}. Spectral analysis of the baseline activity shows two major peaks, the first is located at the baseline firing rate (\fbase), the second is located at the discharge frequency of the electric organ (\feod{}) and is flanked by two smaller peaks at $\feod \pm \fbase{}$ (\subfigref{fig:cells_suscept}{B}).
Second-order susceptibility is expected to be especially pronounced for low-CV cells \cite{Voronenko2017}. P-units fire action potentials probabilistically phase-locked to the self-generated EOD. Skipping of EOD cycles leads to the characteristic multimodal ISI distribution with maxima at integer multiples of the EOD period (\subfigrefb{fig:cells_suscept}{A}). In this example the ISI distribution has a CV of 0.2 which can be considered low among P-units\cite{Hladnik2023}. Spectral analysis of the baseline activity shows two major peaks, the first is located at the baseline firing rate (\fbase), the second is located at the discharge frequency of the electric organ (\feod{}) and is flanked by two smaller peaks at $\feod \pm \fbase{}$ (\subfigref{fig:cells_suscept}{B}). High-CV P-units do not exhibit pronounced nonlinearities (for more details see supplementary information: \nameref*{S1:highcvpunit} )
%
\begin{figure*}[!ht]
\includegraphics[width=\columnwidth]{cells_suscept}
@ -529,9 +531,7 @@ Noise stimuli, the random amplitude modulations (RAM, \subfigref{fig:cells_susce
Theory predicts a pattern of nonlinear susceptibility for the interaction of two cosine stimuli when \fone{}, \ftwo{} or \fsum{} are equal to \fbase{} (pink triangle in \subfigsref{fig:cells_suscept}{E, F}). To estimate the second-order susceptibility from the P-unit responses to RAM stimuli, the noise stimulus was set in relation to the corresponding neuronal response in the Fourier domain, resulting in a matrix in which the nonlinearity at the sum frequency \fsum{} in the firing rate is depicted for two noise frequencies \fone{} and \ftwo{} (\Eqnref{eq:susceptibility}, \subfigrefb{fig:cells_suscept}{E--F}). The decomposition of the RAM stimulus into the frequency components allows us to approximate the beat frequencies $\Delta f$, occurring in case of pure sine-wave stimulation (\subfigrefb{fig:motivation}{D}). Thus, the nonlinearity accessed with the RAM stimulation at \fsum{} (\subfigrefb{fig:cells_suscept}{E}) is similar to the nonlinearity appearing during pure sine-wave stimulation at \bsum{} (orange peak, \subfigrefb{fig:motivation}{D}). A band of elevated nonlinearity appears for the low RAM contrast when \fsumb{} (yellow anti-diagonal between pink edges, \subfigrefb{fig:cells_suscept}{E}) but vanishes for the stronger stimulus (20\% contrast, \subfigref{fig:cells_suscept}{F}). Further, the overall level of nonlinearity changes with the stimulus strength. To compare the structural changes in the matrices we calculated the mean of the anti-diagonals, resulting in the projected diagonal (\subfigrefb{fig:cells_suscept}{G}). For a low RAM contrast the second-order susceptibility at \fbase{} is slightly enhanced (\subfigrefb{fig:cells_suscept}{G}, gray dot on light purple line). For the higher RAM contrast, however, this is not visible and the overall second-order susceptibility is reduced (\subfigrefb{fig:cells_suscept}{G}, compare light and dark purple lines).
\subsection*{High-CV P-units do not exhibit increased nonlinear interactions at \fsum}
\notejg{it is not PLoS style to actively work with the supplementary figures... We may want to reduce this section here, move the text to the supporting information}
CVs in P-units can range up to 1.5 \cite{Grewe2017, Hladnik2023}. \nameref*{S1:highcvpunit} shows the same analysis for an example higher-CV P-unit. Similar to the low-CV cell, high-CV P-units fire at multiples of the EOD period (\subfigrefb{fig:cells_suscept_high_CV}{A}). In contrast to low-CV P-unit, however, the higher CV characterizes the noisier baseline firing pattern and the peak at \fbase{} is less pronounced in the power spectrum of the baseline activity (\subfigrefb{fig:cells_suscept_high_CV}{B}). High-CV P-units do not exhibit a clear nonlinear structure related to \fbase{} neither in the second-order susceptibility matrices (\subfigrefb{fig:cells_suscept_high_CV}{E--F}), nor in the projected diagonals (\subfigrefb{fig:cells_suscept_high_CV}{G}). The overall level of nonlinearity, however shows the same dependence on the stimulus contrast. It is much reduced for high-contrast stimuli that drive the neuron much stronger (\subfigrefb{fig:cells_suscept_high_CV}{F}).
\subsection*{Ampullary afferents exhibit strong nonlinear interactions}
@ -544,7 +544,7 @@ Irrespective of the CV, neither cell shows the complete proposed structure of no
\end{figure*}
\subsection*{Internal noise hides parts of the nonlinearity structure}
Traces of the proposed structure of second-order susceptibility are found in both ampullary and p-type electrosensory afferents. The nonlinerarity seems to depend on the CV, i.e. the level of intrinsic noise in the cells and in the next step we address whether the level of intrinsic masks the expected structures. One option to reduce the impact of this intrinsic noise is to average it out over many repetitions of the same stimulus. In the electrophysiological experiments we only have a limited recording duration and hence the number of stimulus repetitions is limited. To overcome these limitations we compared between an electrophysiologically recorded low-CV P-unit and its P-unit LIF model counterpart fitted to reproduce the P-unit behavior faithfully (see \figref{flowchart})\cite{Barayeu2023}. In the recording depicted in \subfigrefb{model_and_data}{A} the cell was stimulated with a weak RAM stimulus with the contrast of 1\,$\%$ (red, \subfigrefb{model_and_data}\,\panel[i]{B}). The across trial average ($\n{}=11$) shows the diagonal band in second-order susceptibility at \fsumb{} (yellow diagonal in pink edges) and an additional nonlinear band appeared at \fsumehalf{} (\subfigrefb{model_and_data}{A}). The matched model reproduces the same diagonal at \fsumb{} (\subfigrefb{model_and_data}\,\panel[ii]{B}) but not the diagonal at \fsumehalf. By increasing the number of trials in the simulation, the signal-to-noise ratio and thus the estimation of the nonlinearity structures can be improved. Still, even with 1 million repetitions, no changes are observable in the nonlinearity structures (\subfigrefb{model_and_data}\,\panel[iii]{B}). The increased trial count, however goes along with generally smaller second-order susceptibility values (\subfigrefb{model_and_data}\,\panel[iii]{B}).
Traces of the proposed structure of second-order susceptibility are found in both ampullary and p-type electrosensory afferents. The nonlinerarity seems to depend on the CV, i.e. the level of intrinsic noise in the cells and in the next step we address whether the level of intrinsic masks the expected structures. One option to reduce the impact of this intrinsic noise is to average it out over many repetitions of the same stimulus. In the electrophysiological experiments we only have a limited recording duration and hence the number of stimulus repetitions is limited. To overcome these limitations we compared between an electrophysiologically recorded low-CV P-unit and its P-unit LIF model counterpart fitted to reproduce the P-unit behavior faithfully (see \figref{flowchart})\cite{Barayeu2023}. In the recording depicted in \subfigrefb{model_and_data}{A} the cell was stimulated with a weak RAM stimulus with the contrast of 1\,$\%$ (red, \subfigrefb{model_and_data}\,\panel[i]{B}). The across trial average ($\n{}=11$) shows the diagonal band in second-order susceptibility at \fsumb{} (yellow diagonal in pink edges) and an additional nonlinear band appeared at \fsumehalf{} (\subfigrefb{model_and_data}{A}). The matched model reproduces the same diagonal at \fsumb{} (\subfigrefb{model_and_data}\,\panel[ii]{B}) but not the diagonal at \fsumehalf. By increasing the number of trials in the simulation, the signal-to-noise ratio and thus the estimation of the nonlinearity structures can be improved. Still, even with 1 million repetitions, no changes are observable in the nonlinearity structures (\subfigrefb{model_and_data}\,\panel[iii]{B}). The increased trial count, however goes along with generally smaller second-order susceptibility values (\subfigrefb{model_and_data}\,\panel[iii]{B}). The decrease of the second-order susceptibility follows the relation $1/ \sqrt{N} $ (\figrefb{trialnr}).
%Note that this line doesn't appear in the susceptibility matrix of the model (\subfigrefb{model_and_data}{C})Each cell has an intrinsic noise level (\subfigrefb{model_and_data}{A}, bottom).
%The signal component (purple) compensates for this total noise reduction, is not a weak signal anymore and
@ -558,7 +558,7 @@ In a high-CV P-unit we could not find such nonlinear structures, neither in the
\begin{figure*}[!hb]
\includegraphics[width=\columnwidth]{model_and_data}
\caption{\label{model_and_data} High trial numbers and a reduced internal noise reveal the nonlinear structure in a LIF model with carrier. \figitem{A} \suscept{} surface of the electrophysiologically recorded low-CV P-unit (cell 2012-07-03-ak) driven with a weak stimulus with 1\% contrast. The plot shows that average \suscept{} surface ($N=11$). Pink edges mark the expected structure of enhanced nonlinearity. \figitem[i]{B} \textit{Standard condition} of the model simulation. The model is driven by a RAM stimulus (red trace). $\xi_{signal}$ and $\xi_{noise}$ components of the total intrinsic noise ($\xi$) are shown in the center and bottom. In this condition, $\xi_{noise} = \xi$. \figitem[ii]{B} \suscept{} surface of the cell's LIF model counterpart (cell 2012-07-03-ak, table~\ref{modelparams}) averaged over the same number of stimulus repetitions. \figitem[iii]{B} Same as \panel[ii]{B} but 1 million stimulus repetitions. \figitem[i - iii]{C} Same as in \panel[i - iii]{B} but in the \textit{noise split} condition: there is no external RAM signal driving the model. Parts (90\%) of the total intrinsic noise ($\xi$) are treated as signal (center trace, $\xi_{signal}$) while the remaining part is treated as $\xi_{noise}$ (see methods for details).
\caption{\label{model_and_data} High trial numbers and a reduced internal noise reveal the nonlinear structure in a LIF model with carrier. \figitem{A} \suscept{} surface of the electrophysiologically recorded low-CV P-unit (cell 2012-07-03-ak) driven with a weak stimulus with 1\% contrast. The plot shows that average \suscept{} surface ($N=11$). Pink edges mark the expected structure of enhanced nonlinearity. \figitem[i]{B} \textit{Standard condition} of the model simulation. The model is driven by a RAM stimulus (red trace). $\xi_{signal}$ and $\xi_{noise}$ components of the total intrinsic noise ($\xi$) are shown in the center and bottom. In this condition, $\xi_{noise} = \xi$. \figitem[ii]{B} \suscept{} surface of the cell's LIF model counterpart (cell 2012-07-03-ak, table~\ref{modelparams}) averaged over the same number of stimulus repetitions. \figitem[iii]{B} Same as \panel[ii]{B} but 1 million stimulus repetitions. \figitem[i - iii]{C} Same as in \panel[i - iii]{B} but in the \textit{noise split} condition: there is no external RAM signal driving the model. Parts (90\%) of the total intrinsic noise ($\xi$) are treated as signal (center trace, $\xi_{signal}$) while the remaining part is treated as $\xi_{noise}$ (see methods for details). Note that the signal component ($\xi_{signal}$) has a lower frequency content than the noise component ($\xi_{noise}$). Adding the discarded high frequency noise to the noise component $\xi_{noise}$ does not alter the results here (not shown).
}
\end{figure*}
@ -578,16 +578,16 @@ The smaller \fdiff{} power spectral peak observed during pure sine-wave stimulat
\begin{figure*}[!ht]
\includegraphics[width=\columnwidth]{data_overview_mod}
\caption{\label{fig:data_overview_mod} Nonlinearity for a population of P-units (\panel{A, C}) and ampullary cells (\panel{B, D}). The nonlinearity index \nli{} is calculated as the maximal value in the range $\fbase{} \pm 5$\,Hz of the projected diagonal divided by its median (see \Eqnref{eq:nli_equation}). Each recorded neuron contributes at maximum with two stimulus contrasts. \figitem{A, B} There is a negative correlation between the CV during baseline and \nli. \figitem{C, D} There is a negative correlation between the CV during stimulation and \nli. \figitem{E, F} The \nli{} is plotted against the response modulation, (see methods), an indicator of the subjective stimulus strength for a cell. There is a negative correlation between response modulation and \nli. Restricting the analysis to the weakest stimulus that was presented to each unique neuron, does not change the results. The number of unique neurons is 222 for P-units and 45 for ampullary cells.
\caption{\label{fig:data_overview_mod} Nonlinearity for a population of P-units (\panel{A, C}) and ampullary cells (\panel{B, D}). The peakedness of the nonlinearity \nli{} is calculated as the maximal value in the range $\fbase{} \pm 5$\,Hz of the projected diagonal divided by its median (see \Eqnref{eq:nli_equation}). Each recorded neuron contributes at maximum with two stimulus contrasts. \figitem{A, B} There is a negative correlation between the CV during baseline and \nli. \figitem{C, D} There is a negative correlation between the CV during stimulation and \nli. \figitem{E, F} The \nli{} is plotted against the response modulation, (see methods), an indicator of the subjective stimulus strength for a cell. There is a negative correlation between response modulation and \nli. Restricting the analysis to the weakest stimulus that was presented to each unique neuron, does not change the results. The number of unique neurons is 222 for P-units and 45 for ampullary cells.
}
\end{figure*}
%\Eqnref{response_modulation}
\subsection*{Low CVs and weak stimuli are associated with strong nonlinearity}
The nonlinear effects shown for single cell examples above are supported by the analysis of the pool of 222 P-units and 47 ampullary afferents recorded in 71 specimen. To compare across cells we expressed the second-order susceptibility as the nonlinearity index \nli{}, see \Eqnref{eq:nli_equation}. The \nli{} characterizes the peakedness of the projections onto the diagonal of the \suscept{} matrices (e.g. \subfigref{fig:cells_suscept}{G} at \fbase{}). It assumes high values when the \fbase{} peak in the projected diagonal is pronounced relative to the median of the diagonal and is small when there is no distinct peak. The P-unit population \nli{} values depend weakly on the CV of the ISI distribution under baseline condition. Cells with lower baseline CVs have the tendency to exhibit a stronger nonlinearity than those that have high CVs during baseline (\subfigrefb{fig:data_overview_mod}{A}) or during stimulation (\subfigrefb{fig:data_overview_mod}{C}) . The two example P-units shown before (\figrefb{fig:cells_suscept} and \figrefb{fig:cells_suscept_high_CV}) are highlighted with dark circles in \subfigrefb{fig:data_overview_mod}{A, C, E}. The stimulus strength plays an important role. Several of the recorded neurons contribute with two dots to the data as their responses to the same RAM stimulus but with different contrasts were recorded. Higher stimulus contrasts lead to a stronger drive and thus stronger response modulations (see color code bar in \subfigref{fig:data_overview_mod}{A}, see methods). Since P-units are heterogeneous in their susceptibility to the stimulus\cite{Grewe2017}, their responses to the same stimulus intensity vary a lot. When replotting the \nli{} against the response modulation (\subfigrefb{fig:data_overview_mod}{E}) a negative correlation is observed showing that cells that are strongly driven by the stimulus show less nonlinearity while those that are only weakly driven show higher nonlinearities. Whether or not a cell responds nonlinearity thus depends on both, the baseline CV (i.e. the internal noise) and the response strength.
The nonlinear effects shown for single cell examples above are supported by the analysis of the pool of 222 P-units and 47 ampullary afferents recorded in 71 specimen. To compare across cells we expressed the second-order susceptibility as the peakedness of the nonlinearity \nli{}, see \Eqnref{eq:nli_equation}. The \nli{} characterizes the peakedness of the projections onto the diagonal of the \suscept{} matrices (e.g. \subfigref{fig:cells_suscept}{G} at \fbase{}). It assumes high values when the \fbase{} peak in the projected diagonal is pronounced relative to the median of the diagonal and is small when there is no distinct peak. The P-unit population \nli{} values depend weakly on the CV of the ISI distribution under baseline condition. Cells with lower baseline CVs have the tendency to exhibit a stronger nonlinearity than those that have high CVs during baseline (\subfigrefb{fig:data_overview_mod}{A}) or during stimulation (\subfigrefb{fig:data_overview_mod}{C}) . The two example P-units shown before (\figrefb{fig:cells_suscept} and \figrefb{fig:cells_suscept_high_CV}) are highlighted with dark circles in \subfigrefb{fig:data_overview_mod}{A, C, E}. The stimulus strength plays an important role. Several of the recorded neurons contribute with two dots to the data as their responses to the same RAM stimulus but with different contrasts were recorded. Higher stimulus contrasts lead to a stronger drive and thus stronger response modulations (see color code bar in \subfigref{fig:data_overview_mod}{A}, see methods). Since P-units are heterogeneous in their susceptibility to the stimulus\cite{Grewe2017}, their responses to the same stimulus intensity vary a lot. When replotting the \nli{} against the response modulation (\subfigrefb{fig:data_overview_mod}{E}) a negative correlation is observed showing that cells that are strongly driven by the stimulus show less nonlinearity while those that are only weakly driven show higher nonlinearities. Whether or not a cell responds nonlinearity thus depends on both, the baseline CV (i.e. the internal noise) and the response strength.
%(Pearson's $r=-0.35$, $p<0.001$)222 P-units and 47 (Pearson's $r=-0.16$, $p<0.01$)
%In a P-unit population where each cell is represented not by several contrasts but by the lowest recorded contrast, \nli{} significantly correlates with the CV during baseline ($r=-0.17$, $p=0.01$), the response modulation ($r=-0.35$, $p<0.001$) and \fbase{} ($r=-0.32$, $p<0.001$).%, $\n{}=221$*, $\n{}=221$******, $\n{}=221$
%In a P-unit population where each cell is represented not by several contrasts but by the lowest recorded contrast, \nli{} significantly correlates with the CV during baseline ($r=-0.17$, $p=0.01$), the response modulation ($r=-0.35$, $p<0.001$) and \fbase{} ($r=-0.32$, $p<0.001$).%, $\n{}=222$*, $\n{}=222$******, $\n{}=222$
The population of ampullary cells is generally more homogeneous and have lower CVs than the P-units and show much higher \nli{} values (factor of 10). Overall, there is a negative correlation with the baseline CV. The example cell shown above (\figref{ampullary}) was recorded at two different stimulus intensities and the \nli{}s are highlighted with black circles. Again, we see that cells that are strongly driven by the stimulus cluster at the bottom of the distribution and have \nli{} values close to zero (\subfigrefb{fig:data_overview_mod}{B, D}). This is confirmed when the data is replotted against the response modulation, those cells that are strongly driven by the stimulus show weak nonlinearities while weakly driven neurons exhibit high values (\subfigrefb{fig:data_overview_mod}{F}).
@ -603,30 +603,30 @@ Nonlinearities are ubiquitous in nervous systems, they are essential to extract
Theoretical work\cite{Voronenko2017} explained analytically the occurrence of nonlinear products when a LIF model neuron is stimulated with pure sine-waves. To investigate whether the same mechanisms occur in electroreceptor afferents which are driven by AMs of a carrier and not by pure sine-waves, we followed the previous approach and quantified the second-order susceptibility from responses to white-noise stimuli \cite{Voronenko2017,Neiman2011fish,Nikias1993}. We expected to see elevated second-order susceptibility where either of the foreign signals matches the baseline firing rate ($f_1=f_{base}$ or $f_2=f_{base}$) or when the sum equals the baseline firing rate of the neuron (\fsumb{}) creating a triangular pattern of elevated \suscept{} e.g.\,\subfigref{model_and_data}\,\panel[iii]{C}. Indeed, we find traces of the same nonlinearities in the neuronal responses of p-type electroreceptor afferents. The nonlinear pattern observed in the experimental data, however, matches to the expectations only partially and only in a subset of neurons \figsref{fig:cells_suscept} and\,\ref{ampullary}. Nevertheless, the theory holds also for systems that are driven by AMs of a carrier and is thus more widely applicable.
\subsection*{Intrinsic noise limits nonlinear responses}
Only those P-units that exhibit low coefficients of variation (CV) of the interspike-interval distribution (\figref{fig:cells_suscept}{A}) in their unperturbed baseline response show the expected nonlinerities (\subfigref{fig:data_overview_mod}{A}). Such low-CV cells are rare among the 221 P-units that we used in this study. The afferents of the passive electrosensory system, the ampullary cells, however have generally lower CVs and show a much clearer nonlinearity pattern than the low-CV P-unit exemplified here (compare \figsref{fig:cells_suscept} and \ref{ampullary}). The single ampullary cell featured in \figref{ampullary} is a representative of the majority of cells analyzed here. All ampullary cells have CVs below 0.4 with a median around 0.1 and the observed \nli{}s are 10-fold higher than in P-units.
Only those P-units that exhibit low coefficients of variation (CV) of the interspike-interval distribution (\figref{fig:cells_suscept}{A}) in their unperturbed baseline response show the expected nonlinerities (\subfigref{fig:data_overview_mod}{A}). Such low-CV cells are rare among the 222 P-units that we used in this study. The afferents of the passive electrosensory system, the ampullary cells, however have generally lower CVs and show a much clearer nonlinearity pattern than the low-CV P-unit exemplified here (compare \figsref{fig:cells_suscept} and \ref{ampullary}). The single ampullary cell featured in \figref{ampullary} is a representative of the majority of ampullary cells analyzed here. All ampullary cells have CVs below 0.4 with a median around 0.12 and the observed \nli{}s are 10-fold higher than in P-units.
The CV serves as a proxy for the intrinsic noise in the cells. In both cell types, we observe a negative correlation between \nli{} and the CV, indicating that it is the level of intrinsic noise that plays a role here. These findings are in line with previous studies that propose that noise linearizes the system\cite{Roddey2000, Chialvo1997, Voronenko2017}. More intrinsic noise has been demonstrated to increase the CV and reduce nonlinear phase-locking in vestibular afferents\cite{Schneider2011}. Reduced noise, on the other hand, has been associated with stronger nonlinearity in pyramidal cells of the ELL\cite{Chacron2006}. Further support for the notion of noise limiting the nonlinearity comes from our P-unit LIF model that faithfully reproduces P-unit activity\cite{Barayeu2023}. We can use this model and the noise-split according to the Novikov-Furutsu theorem\cite{Novikov1965,Furutsu1963} to increase the signal-to-noise ratio in the cell while keeping the overall response variability constant (see methods). Treating 90\% of the total noise as signal and simulating large numbers of trial uncovers the full nonlinearity structure (\figref{model_and_data}) seen in LIF neurons and the analytical derivations when driven with sine-wave stimuli\cite{Voronenko2017}.
\subsection*{Noise stimulation approximates the real three-fish interaction}
Our analysis is based on the neuronal responses to white noise stimulus sequences. For the P-units, the stimulus was a random amplitude modulation (RAM) while it was a direct noise stimulus for the ampullary cells. In true white noise, each frequency component contributes with equal amplitude. This has the methodological advantage that all behaviorally relevant frequencies can be measured with a single stimulus presentation and is a widely used approach to characterize sensory coding\cite{Borst1999}\notejg{better references?}. In contrast to the situation with individual frequencies (direct sine-waves or sinusoidal AMs) the total power of the stimulus is equally distributed on all frequencies leading to a weaker signal-to-noise ratio. This explains that the nonlinearity pattern in the electroreceptor recordings only partially matches the expectation \figsref{fig:cells_suscept},\,\ref{ampullary} while the single-frequency stimulation shows nonlinear interference when the individual stimulus frequencies ($f_1, f_2, \Delta f_1, \Delta f_2$) match the baseline firing rate (\figref{fig:motivation}). With the noise-splitting trick, we could show that the full nonlinearity pattern is present but covered by the intrinsic noise. We thus conclude that the presence of the anti-diagonal pattern in the \suscept{} matrix is sufficient to conclude that the same nonlinear interactions happen here. This also validates the application of the white noise approach to characterize the full \suscept{} matrix instead of using all combinations of individual frequencies.
Our analysis is based on the neuronal responses to white noise stimulus sequences. For the P-units, the stimulus was a random amplitude modulation (RAM) while it was a direct noise stimulus for the ampullary cells. In true white noise, each frequency component contributes with equal amplitude. This has the methodological advantage that all behaviorally relevant frequencies can be measured with a single stimulus presentation and is a widely used approach to characterize sensory coding\cite{Borst1999,Chacron2005, Grewe2017}. In contrast to the situation with individual frequencies (direct sine-waves or sinusoidal AMs) the total power of the stimulus is equally distributed on all frequencies leading to a weaker signal-to-noise ratio. This explains that the nonlinearity pattern in the electroreceptor recordings only partially matches the expectation (\figsref{fig:cells_suscept},\,\ref{ampullary}) while the single-frequency stimulation shows nonlinear interference when the individual stimulus frequencies ($f_1, f_2, \Delta f_1, \Delta f_2$) match the baseline firing rate (\figref{fig:motivation}). With the noise-splitting trick, we could show that the full nonlinearity pattern is present but covered by the intrinsic noise. We thus conclude that the presence of the anti-diagonal pattern in the \suscept{} matrix is sufficient to conclude that the same nonlinear interactions happen here. This also validates the application of the white noise approach to characterize the full \suscept{} matrix instead of using all combinations of individual frequencies.
% The nonlinearity of ampullary cells in paddlefish \cite{Neiman2011fish} has been previously accessed with bandpass limited white noise.
% Here it was demonstrated that the second-order susceptibility for the two RAM noise input frequencies \fone{} and \ftwo{} can approximate a three-fish setting, where the driving force for the P-unit are two beats with frequencies \bone{} and \btwo{}. This was confirmed by a low-CV P-unit, where nonlinearities in the P-unit response occurred at the sum and difference of the beat frequencies for pure sine-wave stimulation (\figrefb{fig:motivation}). In this P-unit the nonlinearity appeared in a three-wave setting with \bone{} being close to \fbase{}, corresponding to a frequency combination on the vertical line at \foneb{} in the second-order susceptibility (\subfigrefb{fig:model_full}{B}). This implies that even if only the diagonal structure can be accessed with noise stimulation in the second-order susceptibility in an experiment (\subfigrefb{fig:model_full}{A}), it can be taken as an indicator that the whole nonlinear structure should be present during pure sine-wave stimulation in the same cell. With this RAM stimulation was demonstrated to be an effective method to scan the three-fish or two-beat plane and estimate the whole theoretically predicted nonlinear structure in experimentally recorded cells.
\subsection*{Heterogeneity of P-units might influence nonlinearity}\notejg{Better title...}
The observed nonlinear effects might facilitate the detectability of faint signals during a three fish setting, the electrosensory cocktail party. They are, however, very specific with respect to the relation of stimulus frequencies and the P-unit baseline frequency. The EOD frequencies of the interacting fish would be drawn from the distributions of EOD frequencies found in male and female fish\cite{Henninger2018, Henninger2020}\notejg{more refs on eod freq distributions?}. To be behaviorally relevant the faint signal detection would require reliable neuronal signalling irrespective of the individual EOD frequencies.
\subsection*{Nonlinearity might be influenced once integrating from a P-unit population with heterogeneous baseline properties}%Heterogeneity of P-units might influence
The observed nonlinear effects might facilitate the detectability of faint signals during a three fish setting, the electrosensory cocktail party. They are, however, very specific with respect to the relation of stimulus frequencies and the P-unit baseline frequency. The EOD frequencies of the interacting fish would be drawn from the distributions of EOD frequencies found in male and female fish\cite{Hopkins1974Eigen, Meyer1987, Henninger2018, Henninger2020}. To be behaviorally relevant the faint signal detection would require reliable neuronal signaling irrespective of the individual EOD frequencies.
P-units, however are very heterogeneous in their baseline firing properties\cite{Grewe2017, Hladnik2023}. The baseline firing rates vary in wide ranges (50--450\,Hz). This range covers substantial parts of the beat frequencies that may occur during animal interactions which is limited to frequencies below the Nyquist frequency (0 -- \feod/2)\cite{Barayeu2023}. It is thus likely that there are P-units which match approximately to the specificities of the different encounters.
On the other hand, the nonlinearity was found only in low-CV P-units (with white noise stimulation). The CVs are also very heterogeneous (0.1--1.5\notejg{CVs aus Abbildung einfuegen}, \figref{fig:data_overview_mod}\panel{A}) in our sample. Only a small fraction of the P-units has a sufficiently low level of intrinsic noise and will exhibit nonlinear responses. The P-units project to the ELL\cite{Krahe2014} and the integrating pyramidal cells in the different segments receive inputs in the range of 10s to 1000s of neurons\cite{Maler2009a}. Since the receptive fields of the pyramidal neurons are patches of adjacent receptors on the fish surface \cite{Bastian2002, Maler2009a, Haggard2023} and the input heterogeneity does not depend on the location of the receptor on the fish body \cite{Hladnik2023} the pyramidal cell input population will be heterogeneous. Heterogeneity was shown to be generally advantageous for the encoding in this\cite{Hladnik2023} and other systems\cite{Padmanabhan2010, Beiran2018}. At the same time it contradicts the apparent need for a selective readout of low-CV cells to maintain information arising through nonlinear interactions.
On the other hand, the nonlinearity was found only in low-CV P-units (with white noise stimulation). The CVs are also very heterogeneous (0.1--1.4, \figref{fig:data_overview_mod}\panel{A}) in our sample. Only a small fraction of the P-units has a sufficiently low level of intrinsic noise and will exhibit nonlinear responses. The P-units project to the ELL\cite{Krahe2014} and the integrating pyramidal cells in the different segments receive inputs in the range of 10s to 1000s of neurons\cite{Maler2009a}. Since the receptive fields of the pyramidal neurons are patches of adjacent receptors on the fish surface \cite{Bastian2002, Maler2009a, Haggard2023} and the input heterogeneity does not depend on the location of the receptor on the fish body \cite{Hladnik2023} the pyramidal cell input population will be heterogeneous. Heterogeneity was shown to be generally advantageous for the encoding in this\cite{Hladnik2023} and other systems\cite{Padmanabhan2010, Beiran2018}. At the same time it contradicts the apparent need for a selective readout of low-CV cells to maintain information arising through nonlinear interactions.
A possible readout mechanism should be the topic of future studies that need to take into account that the nonlinearities are stronger in pure sine-wave stimulation and the fraction of cells that show it under naturalistic stimulation might be larger than expected from the distribution of CVs.
\subsection*{Behavioral relevance of nonlinear interactions}
The behavioral relevance of the weak signal detection in P-units is evident from the courtship context observed in freely interacting animals\cite{Henninger2018}. Outside courtship behavior, the encoding of secondary or social envelopes is a common need\cite{Stamper2012Envelope}. In a previous study it was demonstrated that information about low-frequency secondary envelopes would not be present in P-units responses but would arise thorough nonlinear processing downstream in the ELL \cite{Middleton2006,Middleton2007}. Based on our work we would predict that only a small subset of cells, with low CVs, should encode the social envelopes under white noise stimulation. An absence of low-CVs cells in the population analyzed in the previous studies could explain their conclusions. On the other hand, another study showed that P-units with strong nonlinearities, low firing rates and high CVs could indeed encode social envelopes\cite{Savard2011}. These findings are in contrast to the previously mentioned work\cite{Middleton2007} and, at first glance, also to our results. The missing link, that has not been considered in this work, might be the bursting of P-units, the repeated firing of spikes after one EOD period interleaved with longer intervals of quietness\cite{Chacron2004}. Bursting was not explicitly addressed in the previous work, still the reported high CVs of the envelope encoding P-units indicate a higher rate of bursting\cite{Savard2011}. How bursts influence the second-order susceptibility of will be addressed in following works (in preparation).
The behavioral relevance of the weak signal detection in P-units is evident from the courtship context observed in freely interacting animals\cite{Henninger2018}. Outside courtship behavior, the encoding of secondary or social envelopes is a common need\cite{Stamper2012Envelope}. In a previous study it was demonstrated that information about low-frequency secondary envelopes would not be present in P-units' responses but would arise thorough nonlinear processing downstream in the ELL \cite{Middleton2006,Middleton2007}. Based on our work we would predict that only a small subset of cells, with low CVs, should encode the social envelopes under white noise stimulation. An absence of low-CVs cells in the population analyzed in the previous studies could explain their conclusions. On the other hand, another study showed that P-units with strong nonlinearities, low firing rates and high CVs could indeed encode social envelopes\cite{Savard2011}. These findings are in contrast to the previously mentioned work\cite{Middleton2007} and, at first glance, also to our results. The missing link, that has not been considered in this work, might be the bursting of P-units, the repeated firing of spikes after one EOD period interleaved with longer intervals of quietness\cite{Chacron2004}. Bursting was not explicitly addressed in the previous work, still the reported high CVs of the envelope encoding P-units indicate a higher rate of bursting\cite{Savard2011}. How bursts influence the second-order susceptibility of P-units will be addressed in following works (in preparation).
Sinusoidal AMs are relevant in interactions with a few fish. We can understand the noise as the presence of many animals with individual EOD frequencies at the same time. Under noise stimulation, strong nonlinearities were demonstrated for weak stimuli but to decrease for stronger noise stimuli (\figrefb{fig:cells_suscept}). As long as the noise signal is weak, those fish are distant and the nonlinearity is maintained. An increasing stimulus amplitude would indicate that many fish are close to the receiver and a decrease of nonlinear effects can be observed. These findings imply that the nonlinear effects arising in the presence of three fish decline the more fish join. \lepto{} usually prefers small groups of fish\cite{Stamper2010}. Thus, the described second-order susceptibility might still be behaviorally relevant under natural conditions. The decline of nonlinear effects when several fish are present might be an adaptive process reducing the number of frequencies represented in its primary sensory afferents to a minimum. Such representation would still leave room to create nonlinear effects at later processing steps in higher-order neurons.
Sinusoidal AMs are relevant in interactions with a few fish. We can understand the noise as the presence of many animals with individual EOD frequencies at the same time. Under noise stimulation, nonlinearities were demonstrated to be strong for weak stimuli but were shown to decrease for stronger noise stimuli (\figrefb{fig:cells_suscept}). As long as the noise signal is weak, those fish are distant and the nonlinearity is maintained. An increasing stimulus amplitude would indicate that many fish are close to the receiver and a decrease of nonlinear effects can be observed. These findings imply that the nonlinear effects arising in the presence of three fish decline the more fish join. \lepto{} usually prefers small groups of fish\cite{Stamper2010}. Thus, the described second-order susceptibility might still be behaviorally relevant under natural conditions. The decline of nonlinear effects when several fish are present might be an adaptive process reducing the number of frequencies represented in its primary sensory afferents to a minimum. Such representation would still leave room to create nonlinear effects at later processing steps in higher-order neurons.
The afferents of the passive electrosensory system, the ampullary cells, were found to exhibit much stronger nonlinearities than P-units (\figref{fig:data_overview_mod}). The adequate stimulus for this system is a direct stimulation not an amplitude modulation. In this sense the ampullary cells are closer to the LIF models used by Voroneko and colleagues\cite{Voronenko2017} and we can thus expect that the same nonlinear mechanisms are at work here. For the ampullary system, the sinewave stimulation might is not as relevant as for the P-unit system. Ampullary cells encode low-frequency exogenous electric signals such as muscle potentials induced by prey movement\cite{Kalmijn1974, Engelmann2010, Neiman2011fish}. The simultaneous muscle activity of a swarm of prey (such as \textit{Daphnia}) resembles Gaussian white noise\cite{Neiman2011fish}, similar to the stimuli used here. Our results show some similarities with the analyses by Neiman and Russel\cite{Neiman2011fish} who study the encoding in ampullary afferents in the paddlefish. There, the power spectrum of the spontaneous also shows a peak at the baseline frequency (internal oscillator) but also at the oscillation frequency measured at the epithelium and interactions of both. Most of the latter disappear in the case of external stimulation, though. Here we find only peaks at the baseline frequency of the neuron and its harmonics. There are interesting similarities and dissimilarities; stimulus encoding in the paddlefish as well as in the brown ghost is very linear for low frequencies and there are nonlinearities in both systems. Linear encoding in the paddlefish shows a gap in the spectrum at the frequency of the epitheliar oscillation, instead the nonlinear response is very pronounced there. In \lepto{}, the dominating frequency under baseline conditions is the baseline firing rate, and we do see increased nonlinearity in this frequency range. The baseline frequency, however, is outside the linear coding range\cite{Grewe2017} while it is within the linear coding range in paddlefish\cite{Neiman2011fish}. Interestingly, the nonlinear response in the paddlefish ampullaries increases with stimulus intensity while it disappears in our case (\figref{fig:data_overview_mod}\panel{D}). The population of ampullary cells is much more homogeneous with respect to the baseline rate (131$\pm$29\,Hz) and stimulus encoding properties than the P-units\cite{Grewe2017}. This implies that, if the stimulus contains the appropriate frequency components that sum up to the baseline rate, there should be a nonlinear response at a frequency that is similar for the full population of ampullary cells (the baseline frequency) that is outside the linear coding range. Postsynaptic cells integrating ampullary input might be able to extract this nonlinear response from the input population. How such nonlinear effects in ampullary cells might influence prey detection should be addressed in further studies.
The afferents of the passive electrosensory system, the ampullary cells, were found to exhibit much stronger nonlinearities than P-units (\figref{fig:data_overview_mod}). The adequate stimulus for this system is a direct stimulation not an amplitude modulation. In this sense the ampullary cells are closer to the LIF models used by Voroneko and colleagues\cite{Voronenko2017} and we can thus expect that the same nonlinear mechanisms are at work here. For the ampullary system, the sinewave stimulation is not as relevant as for the P-unit system. Ampullary cells encode low-frequency exogenous electric signals such as muscle potentials induced by prey movement\cite{Kalmijn1974, Engelmann2010, Neiman2011fish}. The simultaneous muscle activity of a swarm of prey (such as \textit{Daphnia}) resembles Gaussian white noise\cite{Neiman2011fish}, similar to the stimuli used here. Our results show some similarities with the analyses by Neiman and Russel\cite{Neiman2011fish} who study the encoding in ampullary afferents in the paddlefish. There, the power spectrum of the spontaneous activity also shows a peak at the baseline frequency (internal oscillator) but also at the oscillation frequency measured at the epithelium and interactions of both. Most of the latter disappear in the case of external stimulation, though. Here we find only peaks at the baseline frequency of the neuron and its harmonics. There are interesting similarities and dissimilarities; stimulus encoding in the paddlefish as well as in the brown ghost is very linear for low frequencies and there are nonlinearities in both systems. Linear encoding in the paddlefish shows a gap in the spectrum at the frequency of the epitheliar oscillation, instead the nonlinear response is very pronounced there. In \lepto{}, the dominating frequency under baseline conditions is the baseline firing rate, and we do see increased nonlinearity in this frequency range. The baseline frequency, however, is outside the linear coding range\cite{Grewe2017} while it is within the linear coding range in paddlefish\cite{Neiman2011fish}. Interestingly, the nonlinear response in the paddlefish ampullaries increases with stimulus intensity while it disappears in our case (\subfigrefb{fig:data_overview_mod}{F}). The population of ampullary cells is much more homogeneous with respect to the baseline rate (131$\pm$29\,Hz) and stimulus encoding properties than the P-units\cite{Grewe2017}. This implies that, if the stimulus contains the appropriate frequency components that sum up to the baseline rate, there should be a nonlinear response at a frequency that is similar for the full population of ampullary cells (the baseline frequency) that is outside the linear coding range. Postsynaptic cells integrating ampullary input might be able to extract this nonlinear response from the input population. How such nonlinear effects in ampullary cells might influence prey detection should be addressed in further studies.
\subsection*{Conclusion}
We have demonstrated that there are pronounced nonlinear responses in the primary electrosensory afferences of the weakly electric fish \lepto{}, systems that are very often characterized using linear methods. The observed nonlinearities match the expectations from previous theoretical studies\cite{Voronenko2017}. We can confirm that the theory applies also to systems that are encoding amplitude modulations of a carrier signal. Comparisons of P-units and ampullary cells showed that it is the level of intrinsic noise that determines how strongly nonlinear the system acts. Using the second-oder susceptibility estimated from the responses to white noise stimuli provides an easy way to determine the nonlinearity of the system under study. P-units share several features with mammalian
@ -734,14 +734,13 @@ The second-order susceptibility was calculated by dividing the higher-order cros
% \chi_{2}(\omega_{1}, \omega_{2}) = \frac{TN \sum_{n=1}^N \int_{0}^{T} dt\,r_{n}(t) e^{-i(\omega_{1}+\omega_{2})t} \int_{0}^{T}dt'\,s_{n}(t')e^{i \omega_{1}t'} \int_{0}^{T} dt''\,s_{n}(t'')e^{i \omega_{2}t''}}{2 \sum_{n=1}^N \int_{0}^{T} dt\, s_{n}(t)e^{-i \omega_{1}t} \int_{0}^{T} dt'\,s_{n}(t')e^{i \omega_{1}t'} \sum_{n=1}^N \int_{0}^{T} dt\,s_{n}(t)e^{-i \omega_{2}t} \int_{0}^{T} dt'\,s_{n}(t')e^{i \omega_{2}t'}}
% \end{split}
% \end{equation}
% \notejg{Wofuer genau brauchen wir equation 9?}
The absolute value of a second-order susceptibility matrix is visualized in \figrefb{fig:model_full}. There the upper right and the lower left quadrants characterize the nonlinearity in the response $r(t)$ at the sum frequency of the two input frequencies. The lower right and upper left quadrants characterize the nonlinearity in the response $r(t)$ at the difference of the input frequencies.
\paragraph{Nonlinearity index}\label{projected_method}
We expect to see nonlinear susceptibility when $\omega_1 + \omega_2 = \fbase{}$. To characterize this we calculated the nonlinearity index (NLI) as
We expect to see nonlinear susceptibility when $\omega_1 + \omega_2 = \fbase{}$. To characterize this we calculated the peakedness of the nonlinearity (PNL) as
\begin{equation}
\label{eq:nli_equation}
NLI(\fbase{}) = \frac{\max_{\fbase{}-5\,\rm{Hz} \leq f \leq \fbase{}+5\,\rm{Hz}} D(f)}{\mathrm{med}(D(f))}
\nli{} = \frac{\max_{\fbase{}-5\,\rm{Hz} \leq f \leq \fbase{}+5\,\rm{Hz}} D(f)}{\mathrm{med}(D(f))}
\end{equation}
For this index, the second-order susceptibility matrix was projected onto the diagonal $D(f)$, by taking the mean of the anti-diagonals. The peakedness at the frequency $\fbase{}$ in $D(f)$ was quantified by finding the maximum of $D(f)$ in the range $\fbase{} \pm 5$\,Hz (\subfigrefb{fig:cells_suscept}{G}) and dividing it by the median of $D(f)$.
@ -894,7 +893,15 @@ A big portion of the total noise was assigned to the signal component ($c_{signa
\newpage
\section*{Supporting information}
\%subsection*{High-CV P-units do not exhibit increased nonlinear interactions at \fsum}
\paragraph*{S1 Second-order susceptibility of high-CV P-unit}
CVs in P-units can range up to 1.5 \cite{Grewe2017, Hladnik2023}. We show the same analysis for an example higher-CV P-unit. Similar to the low-CV cell, high-CV P-units fire at multiples of the EOD period (\subfigrefb{fig:cells_suscept_high_CV}{A}). In contrast to low-CV P-unit, however, the higher CV characterizes the noisier baseline firing pattern and the peak at \fbase{} is less pronounced in the power spectrum of the baseline activity (\subfigrefb{fig:cells_suscept_high_CV}{B}). High-CV P-units do not exhibit a clear nonlinear structure related to \fbase{} neither in the second-order susceptibility matrices (\subfigrefb{fig:cells_suscept_high_CV}{E--F}), nor in the projected diagonals (\subfigrefb{fig:cells_suscept_high_CV}{G}). The overall level of nonlinearity, however shows the same dependence on the stimulus contrast. It is much reduced for high-contrast stimuli that drive the neuron much stronger (\subfigrefb{fig:cells_suscept_high_CV}{F}).
\label{S1:highcvpunit}
\begin{figure*}[!ht]
\includegraphics[width=\columnwidth]{cells_suscept_high_CV}
@ -906,7 +913,7 @@ A big portion of the total noise was assigned to the signal component ($c_{signa
\begin{figure*}[hp]%hp!
\includegraphics{trialnr}
\caption{\label{fig:trialnr} Change of the signal-to-noise ratio depending on the stimulus repetition number.
\caption{\label{fig:trialnr} Change of the median second-order susceptibility depending on the stimulus repetition number $\n{}$.
}
\end{figure*}

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trialnr.py
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@ -167,7 +167,7 @@ def trialnr(eod_metrice = False, width=0.005, nffts=['whole'], powers=[1], cells
ax.set_xscale('log')
ax.set_yscale('log')
ax.set_xlabel('Trials')
ax.set_xlabel('Trials [$N$]')
ax.set_ylabel('$\chi_{2}$\,[Hz]')
''' ax = plt.subplot(1,3,2)

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utils_paper.py
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@ -4113,9 +4113,8 @@ def plt_data_suscept_single(ax, cbar_label, cell, cells, f, fig, file_names_excl
# , fontsize=7) # + cell_type
if amp_given:
amp = amp_given
ax.text(1.1,1.05,'Recorded P-unit \n$N=%s$'%(snippets)+'\n $c=%s$' %(
amp) + '$\,\%$ ',ha = 'right', transform=ax.transAxes) #, fontsize7= + cell_type# cell[0:13] + stack_final.celltype.unique()[0] + 'S.Nr ' + str(
#ä snippets)' EODf ' + str(np.round(eod_fr)) +
ax.text(1.1,1.05,'Recorded P-unit \n$N=%s$'%(snippets),ha = 'right', transform=ax.transAxes) #, fontsize7= + cell_type# cell[0:13] + stack_final.celltype.unique()[0] + 'S.Nr ' + str(
#ä snippets)' EODf ' + str(np.round(eod_fr)) ++'\n $c=%s$' %(amp) + '$\,\%$ '
#+ 'fr$_{S}$=' + str('\n'
# int(np.round(fr_stim))) + 'Hz'
#+ ' cv$_{S}$=' + str(np.round(cv_stim, 2))
@ -18798,6 +18797,9 @@ def diagonal_xlabel():
def diagonal_xlabel_nothz():
return '$(f_{1}+f_{2})/f_{EOD}$'
def NLI_scorename2():
return 'PNL$(f_{Base})$'
def NLI_scorename():
return 'NLI$(f_{Base})$'
@ -18809,12 +18811,18 @@ def join_x(axts_all):
def join_y(axts_all):
axts_all[0].get_shared_y_axes().join(*axts_all)
if axts_all[0].get_ylim()[-1] != axts_all[1].get_ylim()[-1]:
first = axts_all[0].get_ylim()[-1]
second = axts_all[1].get_ylim()[-1]
starting_val = np.min([first[0], second[0]])
end_val = np.max([first[1], second[1]])
axts_all[0].set_ylim(starting_val,end_val)
axts_all[1].set_ylim(starting_val,end_val)
try:
first = axts_all[0].get_ylim()[-1]
second = axts_all[1].get_ylim()[-1]
starting_val = np.min([first[0], second[0]])
end_val = np.max([first[1], second[1]])
axts_all[0].set_ylim(starting_val, end_val)
axts_all[1].set_ylim(starting_val, end_val)
except:
print('joiny something')
#embed()
def find_peaks_simple(eodf, freq1,freq2, name, color1, color2):