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@ -3,7 +3,7 @@
@article{Abel2009,
title={{Sensitive response to low-frequency cochlear distortion products in the auditory midbrain}},
author={Abel, Cornelius and K\"ossl, Manfred},
journal={Journal of Neurophysiology},
journal={J Neurophysiol},
volume={101},
number={3},
pages={1560--1574},
@ -58,7 +58,7 @@
@ARTICLE{Aguilera2003,
AUTHOR = {Pedro A. Aguilera and Angel A. Caputi},
TITLE = {{Electroreception in \textit{G. carapo}: detection of changes in waveform of the electrosensory signals.}},
JOURNAL = {Journal of Experimental Biology},
JOURNAL = {J Exp Biol},
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VOLUME = {206},
PAGES = {989--998}
@ -165,7 +165,7 @@ journal = {Miscellaneous Publications Museum of Zoology University of Michigan},
@ARTICLE{Arnegard2006,
AUTHOR = {Matthew E. Arnegard and B. Scott Jackson and Carl D. Hopkins},
TITLE = {{Time-domain signal divergence and discrimination without receptor modification in sympatric morphs of electric fishes.}},
JOURNAL = {Journal of Experimental Biology},
JOURNAL = {J Exp Biol},
YEAR = {2006},
VOLUME = {209},
PAGES = {2182--2198}
@ -197,7 +197,7 @@ Carl D. Hopkins},
@article{Arthur1971,
title={{Properties of ``two-tone inhibition" in primary auditory neurones}},
author={Arthur, RM and Pfeiffer, RR and Suga, N},
journal={The Journal of Physiology},
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pages={593--609},
@ -217,7 +217,7 @@ Carl D. Hopkins},
@ARTICLE{Assad1999,
AUTHOR = {Christopher Assad and Brian Rasnow and Philip K. Stoddard},
TITLE = {{Electric organ discharges and electric images during electrolocation.}},
JOURNAL = {Journal of Experimental Biology},
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@ -268,7 +268,7 @@ Carl D. Hopkins},
@article{Avoli1989,
title={{Electrophysiological properties and synaptic responses in the deep layers of the human epileptogenic neocortex in vitro}},
author={Avoli, M and Olivier, A},
journal={Journal of Neurophysiology},
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volume={61},
number={3},
pages={589--606},
@ -278,7 +278,7 @@ Carl D. Hopkins},
@ARTICLE{Babineau2006,
AUTHOR = {David Babineau and Andr\'e Longtin and John E. Lewis},
TITLE = {{Modeling the electric field of weakly electric fish.}},
JOURNAL = {Journal of Experimental Biology},
JOURNAL = {J Exp Biol},
YEAR = {2006},
VOLUME = {209},
PAGES = {3636--3651}
@ -287,7 +287,7 @@ Carl D. Hopkins},
@ARTICLE{Babineau2007,
AUTHOR = {David Babineau and Andr\'e Longtin and John E. Lewis},
TITLE = {{Spatial acuity and prey detection in weakly electric fish.}},
JOURNAL = {PLoS Computational Biology},
JOURNAL = {PLoS Comput Biol},
YEAR = {2007},
VOLUME = {3},
PAGES = {e38}
@ -390,7 +390,7 @@ Beats are slow periodic amplitude modulations resulting from the superposition o
@article{Barlow1957,
title={{Change of organization in the receptive fields of the cat's retina during dark adaptation}},
author={Barlow, HB and Fitzhugh, Roo and Kuffler, SW},
journal={The Journal of Physiology},
journal={J Physiol},
volume={137},
number={3},
pages={338},
@ -443,7 +443,7 @@ Beats are slow periodic amplitude modulations resulting from the superposition o
@article{Bastian1981,
title={Electrolocation},
author={Bastian, Joseph},
journal={Journal of Comparative Physiology},
journal={J Comp Physiol},
volume={144},
number={4},
pages={465--479},
@ -673,7 +673,7 @@ pages = {811--824}
AUTHOR = {Jan Benda and Andr\'e Longtin and Leonard Maler},
TITLE = {Spike-frequency adaptation separates transient communication signals from background oscillations.},
YEAR = {2005},
JOURNAL = {Journal of Neuroscience},
JOURNAL = {J Neurosci},
VOLUME = {25},
NUMBER = {9},
PAGES = {2312--2321} }
@ -692,7 +692,7 @@ pages = {811--824}
TITLE = {Linear versus Nonlinear Signal Transmission in Neuron Models
with Adaptation-Currents or Dynamic Thresholds.},
YEAR = {2010},
JOURNAL = {Journal of Neurophysiology},
JOURNAL = {J Neurophysiol},
VOLUME = {104},
PAGES = {2806-2820} }
@ -1314,7 +1314,7 @@ impulse into a complex spatiotemporal electromotor pattern.},
@article{Chacron2001,
title={Negative interspike interval correlations increase the neuronal capacity for encoding time-dependent stimuli},
author={Chacron, Maurice J and Longtin, Andre and Maler, Leonard},
journal={Journal of Neuroscience},
journal={J Neurosci},
volume={21},
number={14},
pages={5328--5343},
@ -1335,7 +1335,7 @@ impulse into a complex spatiotemporal electromotor pattern.},
@Article{Chacron2005,
title={Electroreceptor neuron dynamics shape information transmission},
author={Chacron, Maurice J and Maler, Leonard and Bastian, Joseph},
journal={Nature Neuroscience},
journal={Nature Neurosci},
volume={8},
number={5},
pages={673--678},
@ -1346,7 +1346,7 @@ impulse into a complex spatiotemporal electromotor pattern.},
@Article{Chacron2006,
title={Nonlinear information processing in a model sensory system},
author={Chacron, Maurice J},
journal={Journal of Neurophysiology},
journal={J Neurophysiol},
volume={95},
number={5},
pages={2933--2946},
@ -2075,7 +2075,7 @@ article{Egerland2020,
@Article{Engelmann2010,
Title = {Coding of stimuli by ampullary afferents in \textit{Gnathonemus petersii}.},
Author = {J. Engelmann and S. Gertz and J. Goulet and A. Schuh and G. von der Emde},
Journal = {Journal of Neurophysiology},
Journal = {J Neurophysiol},
Year = {2010},
Month = {Oct},
@ -2285,7 +2285,7 @@ article{Egerland2020,
@Article{Fotowat2013,
author = {Fotowat, H and Harrison, RR. and Krahe, R},
journal = {Journal of Neuroscience},
journal = {J Neurosci},
number = {34},
pages = {13758--13772},
title = {Statistics of the electrosensory input in the freely swimming weakly electric fish \textit{Apteronotus leptorhynchus}.},
@ -2350,7 +2350,7 @@ article{Egerland2020,
@article{Marmarelis1999,
title={Principal dynamic mode analysis of nonlinear transduction in a spider mechanoreceptor},
author={Marmarelis, Vasilis Z and Juusola, Mikko and French, Andrew S},
journal={Annals of biomedical engineering},
journal={Annals of Biomedical Engineering},
volume={27},
pages={391--402},
year={1999},
@ -2620,7 +2620,7 @@ article{Egerland2020,
@article{Gussin2007,
title={Limits of linear rate coding of dynamic stimuli by electroreceptor afferents},
author={Gussin, Daniel and Benda, Jan and Maler, Leonard},
journal={Journal of Neurophysiology},
journal={J Neurophysiol},
volume={97},
number={4},
pages={2917--2929},
@ -2905,26 +2905,10 @@ and Valenzuela, David},
Volume = {63}
}
@article{Henninger2017,
title={Court and spark in the wild: communication at the limits of sensation},
author={Henninger, J{\"o}rg and Kirschbaum, Frank and Grewe, Jan and Krahe, Ruediger and Benda, Jan},
journal={bioRxiv:114249},
year={2017},
publisher={Cold Spring Harbor Laboratory},
elocation-id = {114249},
year = {2017},
doi = {10.1101/114249},
abstract = {Sensory systems evolve in the ecological niches each species is occupying. Accordingly, the tuning of sensory neurons is expected to be adapted to the statistics of natural stimuli. For an unbiased quantification of sensory scenes we tracked natural communication behavior of the weakly electric fish Apteronotus rostratus in their Neotropical rainforest habitat with high spatio-temporal resolution over several days. In the context of courtship and aggression we observed large quantities of electrocommunication signals. Echo responses and acknowledgment signals clearly demonstrated the behavioral relevance of these signals. The known tuning properties of peripheral electrosensory neurons suggest, however, that they are barely activated by these obviously relevant signals. Frequencies of courtship signals are clearly mismatched with the frequency tuning of neuronal population activity. Our results emphasize the importance of quantifying sensory scenes derived from freely behaving animals in their natural habitats for understanding the evolution and function of neural systems.},
URL = {https://www.biorxiv.org/content/early/2017/08/12/114249},
eprint = {https://www.biorxiv.org/content/early/2017/08/12/114249.full.pdf},
journal = {bioRxiv}
}
@Article{Henninger2018,
Title = {Statistics of natural communication signals observed in the wild identify important yet neglected stimulus regimes in weakly electric fish.},
Author = {J\"org Henninger and R\"udiger Krahe and Frank Kirschbaum and Jan Grewe and Jan Benda},
Journal = {Journal of Neuroscience},
Journal = {J Neurosci},
Year = {2018},
Pages = {5456--5465},
Volume = {38}
@ -2933,7 +2917,7 @@ and Valenzuela, David},
@Article{Henninger2020,
Title = {Tracking activity patterns of a multispecies community of gymnotiform weakly electric fish in their neotropical habitat without tagging},
Author = {Henninger, J{\"o}rg and Krahe, R{\"u}diger and Sinz, Fabian and Benda, Jan},
Journal = {Journal of Experimental Biology},
Journal = {J Exp Biol},
Year = {2020},
Volume = {223},
@ -2979,7 +2963,7 @@ and Valenzuela, David},
@Article{Hladnik2023,
AUTHOR = {T. C. Hladnik and J. Grewe},
TITLE = {Receptive field sizes and neuronal encoding bandwidth are constrained by axonal conduction delays},
JOURNAL = {PLoS Computational Biology},
JOURNAL = {PLoS Comput Biol},
YEAR = {2023},
NUMBER = {19},
VOLUME = {8},
@ -3241,9 +3225,9 @@ in active electroreception.},
}
@article{Hunter2007,
title={Matplotlib: A 2D graphics environment},
title={Matplotlib: A {2D} graphics environment},
author={Hunter, John D},
journal={Computing in science \& engineering},
journal={Computing in Science \& Engineering},
volume={9},
number={3},
pages={90--95},
@ -3706,7 +3690,7 @@ and Sch{\"a}ffler, Livia},
@article{Koch1995,
title={Do neurons have a voltage or a current threshold for action potential initiation?},
author={Koch, Christof and Bernander, {\"O}jvind and Douglas, Rodney J},
journal={Journal of computational neuroscience},
journal={J Comput Neurosci},
volume={2},
pages={63--82},
year={1995},
@ -4089,7 +4073,7 @@ We collected weakly electric gymnotoid fish in the vicinity of Manaus, Amazonas,
@article{Longtin1993,
title={Stochastic resonance in neuron models},
author={Longtin, Andr{\'e}},
journal={Journal of statistical physics},
journal={J Statistical Physics},
volume={70},
pages={309--327},
year={1993},
@ -4193,7 +4177,7 @@ We collected weakly electric gymnotoid fish in the vicinity of Manaus, Amazonas,
AUTHOR = {Leonard Maler},
TITLE = {Receptive field organization across multiple electrosensory maps. {I}. Columnar organization and estimation of receptive field size.},
YEAR = {2009},
JOURNAL = {Journal of Comparative Neurology},
JOURNAL = {J Comp Neurol},
VOLUME = {516},
PAGES = {376--393} }
@ -5278,7 +5262,7 @@ and Keller, Clifford H.},
@Article{Roddey2000,
title={Assessing the performance of neural encoding models in the presence of noise},
author={Roddey, J Cooper and Girish, B and Miller, John P},
journal={Journal of Computational Neuroscience},
journal={J Comput Neurosci},
volume={8},
pages={95--112},
year={2000},
@ -5627,7 +5611,7 @@ groups and electrogenic mechanisms.},
@article{Schneider2011,
title={In vivo conditions induce faithful encoding of stimuli by reducing nonlinear synchronization in vestibular sensory neurons},
author={Schneider, Adam D and Cullen, Kathleen E and Chacron, Maurice J},
journal={PLoS Computational Biology},
journal={PLoS Comput Biol},
volume={7},
number={7},
pages={e1002120},
@ -5658,7 +5642,7 @@ groups and electrogenic mechanisms.},
@Article{Chialvo1997,
title={Stochastic resonance in models of neuronal ensembles},
author={Chialvo, Dante R and Longtin, Andr{\'e} and M{\"u}ller-Gerking, Johannes},
journal={Physical Review E},
journal={Phys Rev E},
volume={55},
number={2},
pages={1798},
@ -5891,7 +5875,7 @@ groups and electrogenic mechanisms.},
@Article{Sinz2020,
Title = {Simultaneous spike-time locking to multiple frequencies},
Author = {Sinz, Fabian H. and Sachgau, Carolin and Henninger, J\"org and Benda, Jan and Grewe, Jan},
Journal = {Journal of Neurophysiology},
Journal = {J Neurophysiol},
Year = {2020},
Number = {6},
Pages = {2355-2372},
@ -5983,7 +5967,7 @@ groups and electrogenic mechanisms.},
@ARTICLE{Stamper2012Envelope,
AUTHOR = {Sarah A. Stamper and Manu S. Madhav and Noah J. Cowan and Eric S. Fortune},
TITLE = {Beyond the {Jamming Avoidance Response}: weakly electric fish respond to the envelope of social electrosensory signals.},
JOURNAL = {Journal of Experimental Biology},
JOURNAL = {J Exp Biol},
YEAR = {2012},
VOLUME = {215},
PAGES = {4196--4207}
@ -6106,7 +6090,7 @@ groups and electrogenic mechanisms.},
@article{Stoewer2014,
title={File format and library for neuroscience data and metadata},
author={Stoewer, Adrian and Kellner, Christian Johannes and Benda, Jan and Wachtler, Thomas and Grewe, Jan},
journal={Frontiers of Neuroinformatics},
journal={Front Neuroinf},
volume={8},
number={27},
pages={10--3389},
@ -6513,7 +6497,7 @@ groups and electrogenic mechanisms.},
AUTHOR = {Henriette Walz and Jan Grewe and Jan Benda},
TITLE = {Static frequency tuning accounts for changes in neural synchrony evoked by transient communication signals.},
YEAR = {2014},
JOURNAL = {Journal of Neurophysiology},
JOURNAL = {J Neurophysiol},
VOLUME = {112},
PAGES = {752--765} }
@ -6731,7 +6715,7 @@ groups and electrogenic mechanisms.},
@article{Xu1996,
title={Logarithmic time course of sensory adaptation in electrosensory afferent nerve fibers in a weakly electric fish},
author={Xu, Zhian and Payne, Jeremy R and Nelson, Mark E},
journal={Journal of Neurophysiology},
journal={J Neurophysiol},
volume={76},
number={3},
pages={2020--2032},
@ -6772,7 +6756,7 @@ groups and electrogenic mechanisms.},
@ARTICLE{Yu2005,
AUTHOR = {Na Yu and Ginette Hup\'e and Charles Garfinkle and John E. Lewis and Andr\'e Longtin},
TITLE = {Coding conspecific identity and motion in the electric sense.},
JOURNAL = {PLoS Computational Biology},
JOURNAL = {PLoS Comput Biol},
YEAR = {2005},
VOLUME = {8},
PAGES = {e1002564}
@ -7032,10 +7016,19 @@ microelectrode recordings from visual cortex and functional implications.},
}
@Article{Vilela2009,
title={Are the input parameters of white noise driven integrate and fire neurons uniquely determined by rate and CV?},
title={Are the input parameters of white noise driven integrate and fire neurons uniquely determined by rate and {CV}?},
author={Rafael D. Vilela and Benjamin Lindner},
year={2009},
journal={J Theor Biol},
volume={257},
pages={90–-99},
}
@Article{Franzen2023,
title={The steady state and response to a periodic stimulation of the firing rate for a theta neuron with correlated noise.},
author={Jannik Franzen and Lukas Ramelow and Benjamin Lindner},
year={2023},
journal={J Comput Neurosci},
volume={51},
pages={107–-128},
}

16
susceptibility1.tex
View File

@ -421,7 +421,7 @@ We like to think about signal encoding in terms of linear relations with unique
The transfer function used to describe linear properties of a system is the first-order term of a Volterra series. Higher-order terms successively approximate nonlinear features of a system \citep{Rieke1999}. Second-order kernels have been used in the time domain to predict visual responses in catfish \citep{Marmarelis1972}. In the frequency domain, second-order kernels are known as second-order response functions or susceptibilities. Nonlinear interactions of two stimulus frequencies generate peaks in the response spectrum at the sum and the difference of the two. Including higher-order terms of the Volterra series, the nonlinear nature of spider mechanoreceptors \citep{French2001}, mammalian visual systems \citep{Victor1977, Schanze1997}, locking in chinchilla auditory nerve fibers \citep{Temchin2005}, and bursting responses in paddlefish \citep{Neiman2011} have been demonstrated.
Noise in nonlinear systems, however, linearizes the system's response properties \citep{Yu1989, Chialvo1997}. Vice versa, in the limit to small stimuli, nonlinear systems can be well described by linear response theory \citep{Roddey2000, Doiron2004, Rocha2007, Sharafi2013}. With increasing stimulus amplitude, the contribution of the second-order kernel of the Volterra series becomes more relevant. For these weakly nonlinear responses of leaky-integrate-and-fire (LIF) model neurons, an analytical expression for the second-order susceptibility has been derived \citep{Lindner2001, Voronenko2017}. In the suprathreshold regime, where the LIF generates a baseline firing rate in the absence of an external stimulus, the linear response function has a peak at the baseline firing rate and its harmonics (\subfigrefb{fig:lifresponse}{A}) and the second-order susceptibility shows very distinct ridges of elevated nonlinear responses, exactly where two stimulus frequencies equal or add up to the neuron's baseline firing rate (\subfigrefb{fig:lifresponse}{B}). In experimental data, such structures in the second-order susceptibility have not been reported yet.
Noise in nonlinear systems, however, linearizes the system's response properties \citep{Yu1989, Chialvo1997}. Vice versa, in the limit to small stimuli, nonlinear systems can be well described by linear response theory \citep{Roddey2000, Doiron2004, Rocha2007, Sharafi2013}. With increasing stimulus amplitude, the contribution of the second-order kernel of the Volterra series becomes more relevant. For these weakly nonlinear responses analytical expressions for the second-order susceptibility have been derived for leaky-integrate-and-fire (LIF) \citep{Lindner2001, Voronenko2017} and theta model neurons \citep{Franzen2023}. In the suprathreshold regime, where the LIF generates a baseline firing rate in the absence of an external stimulus, the linear response function has a peak at the baseline firing rate and its harmonics (\subfigrefb{fig:lifresponse}{A}) and the second-order susceptibility shows very distinct ridges of elevated nonlinear responses, exactly where two stimulus frequencies equal or add up to the neuron's baseline firing rate (\subfigrefb{fig:lifresponse}{B}). In experimental data, such structures in the second-order susceptibility have not been reported yet.
Here we search for such weakly nonlinear responses in electroreceptors of the two electrosensory systems of the wave-type electric fish \textit{Apteronotus leptorhynchus}, i.e. the tuberous (active) and the ampullary (passive) electrosensory system. The p-type electroreceptors of the active system (P-units) are driven by the fish's high-frequency, quasi-sinusoidal electric organ discharges (EOD) and encode disturbances of it \citep{Bastian1981a}. The electroreceptors of the passive system are tuned to lower-frequency exogeneous electric fields such as caused by muscle activity of prey \citep{Kalmijn1974}. As different animals have different EOD-frequencies, being exposed to stimuli of multiple distinct frequencies is part of the animal's everyday life \citep{Benda2020,Henninger2020} and weakly non-linear interactions may occur in the electrosensory periphery. In communication contexts \citep{Walz2014, Henninger2018} the EODs of interacting fish superimpose and lead to periodic amplitude modulations (AMs or beats) of the receiver's EOD. Non-linear mechanisms in P-units, enable encoding of AMs in their time-dependent firing rates \citep{Bastian1981a, Walz2014, Middleton2006, Barayeu2023}. When multiple animals interact, the EOD interferences induce second-order amplitude modulations referred to as envelopes \citep{Yu2005, Fotowat2013, Stamper2012Envelope} and saturation non-linearities allow also for the encoding of these in the electrosensory periphery \citep{Savard2011}. Field observations have shown that courting males were able to react to the extremely weak signals of distant intruding males despite the strong foreground EOD of the nearby female \citep{Henninger2018}. Weakly nonlinear interactions at particular combinations of signals can be of immediate relevance in such settings as they could boost detectability of the faint signals \citep{Schlungbaum2023}.
@ -475,7 +475,7 @@ Weakly nonlinear responses are expected in cells with sufficiently low intrinsic
Noise stimuli, here random amplitude modulations (RAM) of the EOD (\subfigref{fig:cells_suscept}{C}, top trace, red line), are commonly used to characterize stimulus-driven responses of sensory neurons using transfer functions (first-order susceptibility), spike-triggered averages, or stimulus-response coherences. Here, we additionally estimate the second-order susceptibility to quantify nonlinear encoding. P-unit spikes align more or less clearly with fluctuations in the RAM stimulus. A higher stimulus intensity, here a higher contrast of the RAM relative to the EOD amplitude (see methods), entrains the P-unit response more clearly (light and dark purple for low and high contrast stimuli, \subfigrefb{fig:cells_suscept}{C}). Linear encoding, quantified by the transfer function, \eqnref{linearencoding_methods}, is similar for the two RAM contrasts in this low-CV P-unit (\subfigrefb{fig:cells_suscept}{D}), as expected for a linear system. The first-order susceptibility is low for low frequencies, peaks in the range below 100\,Hz and then falls off again \citep{Benda2005}.
The second-order susceptibility, \eqnref{eq:susceptibility}, quantifies the amplitude and phase of the stimulus-evoked response at the sum \fsum{} for each combination of two stimulus frequencies \fone{} and \ftwo{}. Large values of the second-order susceptibility indicate stimulus-evoked peaks in the response spectrum at the summed frequency that cannot be explained by linear response theory. Similar to the first-order susceptibility, the second-order susceptibility can be estimated directly from the response evoked by a RAM stimulus that stimulates the neuron with a whole range of frequencies simultaneously (\subfigsref{fig:cells_suscept}{E, F}). For a LIF-model driven in the supra-threshold regime, theory predicts nonlinear interactions between the two stimulus frequencies, when the two frequencies \fone{} and \ftwo{} or their sum \fsum{} exactly match the neuron's baseline firing rate \fbase{} \citep{Voronenko2017}. Only then, additional stimulus-evoked peaks appear in the spectrum of the spiking response that would show up in the second-order susceptibility as a horizontal, a vertical, and an anti-diagonal line (\subfigrefb{fig:lifresponse}{B}, pink triangle in \subfigsref{fig:cells_suscept}{E, F}).
The second-order susceptibility, \eqnref{eq:susceptibility}, quantifies the amplitude and phase of the stimulus-evoked response at the sum \fsum{} for each combination of two stimulus frequencies \fone{} and \ftwo{}. Large values of the second-order susceptibility indicate stimulus-evoked peaks in the response spectrum at the summed frequency that cannot be explained by linear response theory. Similar to the first-order susceptibility, the second-order susceptibility can be estimated directly from the response evoked by a RAM stimulus that stimulates the neuron with a whole range of frequencies simultaneously (\subfigsref{fig:cells_suscept}{E, F}). For LIF and theta neuron models driven in the supra-threshold regime, theory predicts nonlinear interactions between the two stimulus frequencies, when the two frequencies \fone{} and \ftwo{} or their sum \fsum{} exactly match the neuron's baseline firing rate \fbase{} \citep{Voronenko2017,Franzen2023}. Only then, additional stimulus-evoked peaks appear in the spectrum of the spiking response that would show up in the second-order susceptibility as a horizontal, a vertical, and an anti-diagonal line (\subfigrefb{fig:lifresponse}{B}, pink triangle in \subfigsref{fig:cells_suscept}{E, F}).
For the low-CV P-unit, we observe a band of slightly elevated second-order susceptibility for the low RAM contrast at \fsumb{} (yellowish anti-diagonal between pink edges, \subfigrefb{fig:cells_suscept}{E}). This structure vanishes for the stronger stimulus (\subfigref{fig:cells_suscept}{F}). Further, the overall level of the second-order susceptibility is reduced with increasing stimulus strength. To quantify the structural changes in the susceptibility matrices we projected the susceptibility values onto the diagonal by averaging over the anti-diagonals (\subfigrefb{fig:cells_suscept}{G}). At low RAM contrast this projected second-order susceptibility indeed has a small peak at \fbase{} (\subfigrefb{fig:cells_suscept}{G}, dot on top line). For the higher RAM contrast, however, this peak vanishes and the overall level of the second-order susceptibility is reduced (\subfigrefb{fig:cells_suscept}{G}). The reason behind this reduction is that a RAM with a higher contrast is not only a stimulus with an increased amplitude, but also increases the total noise in the system. Increased noise is known to linearize signal transmission \citep{Longtin1993, Chialvo1997, Roddey2000, Voronenko2017} and thus the second-order susceptibility is expected to decrease.
@ -493,7 +493,7 @@ Electric fish possess an additional electrosensory system, the passive or ampull
\subsection{Model-based estimation of the nonlinear structure}
In the example recordings shown above (\figsrefb{fig:cells_suscept} and \fref{fig:ampullary}), we only observe nonlinear responses on the anti-diagonal of the second-order susceptibility, where the sum of the two stimulus frequencies matches the neuron's baseline firing rate, which is in line with theoretical expectations \citep{Voronenko2017}. However, a pronounced nonlinear response at frequencies \foneb{} or \ftwob{}, although predicted by theory, cannot be observed. In the following, we investigate how these discrepancies can be understood.
In the example recordings shown above (\figsrefb{fig:cells_suscept} and \fref{fig:ampullary}), we only observe nonlinear responses on the anti-diagonal of the second-order susceptibility, where the sum of the two stimulus frequencies matches the neuron's baseline firing rate, which is in line with theoretical expectations \citep{Voronenko2017,Franzen2023}. However, a pronounced nonlinear response at frequencies \foneb{} or \ftwob{}, although predicted by theory, cannot be observed. In the following, we investigate how these discrepancies can be understood.
\begin{figure*}[t]
\includegraphics[width=\columnwidth]{model_and_data.pdf}
@ -547,7 +547,7 @@ The population of ampullary cells is generally more homogeneous, with lower base
\section{Discussion}
Theoretical work \citep{Voronenko2017} derived analytical expressions for weakly-nonlinear responses in a LIF model neuron driven by two sine waves with distinct frequencies. We here looked for such nonlinear responses in two types of electroreceptor afferents that differ in their intrinsic noise levels \citep{Grewe2017} using white-noise stimuli to estimate second-order susceptibilities. Following \citet{Voronenko2017} we expected to observe increased levels of second-order susceptibility where either of the stimulus frequencies alone or the sum of the stimulus frequencies matches the baseline firing rate ($f_1=\fbase{}$, $f_2=\fbase{}$ or \fsumb{}). We find traces of these nonlinear responses in most of the low-noise ampullary afferents and only those P-units with very low intrinsic noise levels. Complementary model simulations demonstrate, in the limit to vanishing stimulus amplitudes and extremely high number of repetitions, that the second order susceptibilities estimated from the electrophysiological data are indeed indicative of the theoretically expected weakly nonlinear responses.
Theoretical work \citep{Voronenko2017,Franzen2023} derived analytical expressions for weakly-nonlinear responses in LIF and theta model neurons driven by two sine waves with distinct frequencies. We here looked for such nonlinear responses in two types of electroreceptor afferents that differ in their intrinsic noise levels \citep{Grewe2017} using white-noise stimuli to estimate second-order susceptibilities. Following \citet{Voronenko2017} we expected to observe increased levels of second-order susceptibility where either of the stimulus frequencies alone or the sum of the stimulus frequencies matches the baseline firing rate ($f_1=\fbase{}$, $f_2=\fbase{}$ or \fsumb{}). We find traces of these nonlinear responses in most of the low-noise ampullary afferents and only those P-units with very low intrinsic noise levels. Complementary model simulations demonstrate, in the limit to vanishing stimulus amplitudes and extremely high number of repetitions, that the second order susceptibilities estimated from the electrophysiological data are indeed indicative of the theoretically expected weakly nonlinear responses.
% EOD locking:
% Phase-locking to the own field also leads to a representation of \feod{} in the P-unit firing rate (see \figref{fig:cells_suscept}\panel{B}) \citep{Sinz2020}.
@ -562,7 +562,7 @@ The CV is a proxy for the intrinsic noise in the cells \citep{Vilela2009}. In bo
\subsection{Linearization by white-noise stimulation}
Not only the intrinsic noise but also the stimulaion with external white-noise linearizes the cells. This applies to both, the stimulation with AMs in P-units (\subfigrefb{fig:dataoverview}{E}) and direct stimulation in ampullary cells (\subfigrefb{fig:dataoverview}{F}). The stronger the effective stimulus, the less pronounced are the peaks in second-order susceptibility. This linearizing effect of noise stimuli limits the weakly nonlinear regime to small stimulus amplitudes. At higher stimulus amplitudes, however, other non-linearities of the system eventually show up in the second-order susceptibility.
In order to characterize weakly nonlinear responses of the cells in the limit to vanishing stimulus amplitudes we utilized the Furutsu-Novikov theorem \citep{Novikov1965, Furutsu1963}. Following \citet{Lindner2022}, a substantial part of the intrinsic noise of a P-unit model \citep{Barayeu2023} is treated as signal. Performing this noise-split trick we can estimate the weakly nonlinear response without the linearizing effect of an additional external white noise stimulus. The model of a low-CV P-unit then shows the full nonlinear structure (\figref{model_and_data}) known from analytical derivations and simulations of basic LIF models driven with pairs of sine-wave stimuli \citep{Voronenko2017}.
In order to characterize weakly nonlinear responses of the cells in the limit to vanishing stimulus amplitudes we utilized the Furutsu-Novikov theorem \citep{Novikov1965, Furutsu1963}. Following \citet{Lindner2022}, a substantial part of the intrinsic noise of a P-unit model \citep{Barayeu2023} is treated as signal. Performing this noise-split trick we can estimate the weakly nonlinear response without the linearizing effect of an additional external white noise stimulus. The model of a low-CV P-unit then shows the full nonlinear structure (\figref{model_and_data}) known from analytical derivations and simulations of basic LIF and theta models driven with pairs of sine-wave stimuli \citep{Voronenko2017,Franzen2023}.
\notejb{Say that previous work on second-order nonlinearities did not see the weakly nonlinear regime, because of the linearizing effects}
@ -570,7 +570,7 @@ In order to characterize weakly nonlinear responses of the cells in the limit to
Estimating the Volterra series from limited experimental data is usually restricted to the first two or three orders, which might not be sufficient for a proper prediction of the neuronal response \citep{French2001}. A proper estimation of just the second-order susceptibility in the weakly nonlinear regime is challenging in electrophysiological experiments. Making assumptions about the nonlinearities in a system reduces the amount of data needed for parameter estimation. In particular, models combining linear filtering with static nonlinearities \citep{Chichilnisky2001} have been successful in capturing functionally relevant neuronal computations in visual \citep{Gollisch2009} as well as auditory systems \citep{Clemens2013}. Linear methods based on backward models for estimating the stimulus from neuronal responses, have been extensively used to quantify information transmission in neural systems \citep{Theunissen1996, Borst1999, Wessel1996, Machens2001}, because backward models do not need to generate action potentials that involve strong nonlinearities \citep{Rieke1999}.
\subsection{Nonlinear encoding in ampullary cells}
The afferents of the passive electrosensory system, the ampullary cells, exhibit strong second-order susceptibilities (\figref{fig:dataoverview}). Ampullary cells more or less directly translate external low-frequency electric fields into afferent spikes, much like in the standard LIF models used by Voroneko and colleagues \citep{Voronenko2017}. Indeed, we observe in the ampullary cells similar second-order nonlinearities as in LIF models.
The afferents of the passive electrosensory system, the ampullary cells, exhibit strong second-order susceptibilities (\figref{fig:dataoverview}). Ampullary cells more or less directly translate external low-frequency electric fields into afferent spikes, much like in the standard LIF and theta models used by Lindner and colleagues \citep{Voronenko2017,Franzen2023}. Indeed, we observe in the ampullary cells similar second-order nonlinearities as in LIF models.
Ampullary stimuli originate from the muscle potentials induced by prey movement \citep{Kalmijn1974, Engelmann2010, Neiman2011fish}. For a single prey items such as \textit{Daphnia}, these potentials are often periodic but the simultaneous activity of a swarm of prey resembles Gaussian white noise \citep{Neiman2011fish}. Linear and nonlinear encoding in ampullary cells has been studied in great detail in the paddlefish \citep{Neiman2011fish}. The power spectrum of the baseline response shows two main peaks: One peak at the baseline firing frequency, a second one at the oscillation frequency of primary receptor cells in the epithelium, and interactions of both. Linear encoding in the paddlefish shows a gap at the epithelial oscillation frequency, instead, nonlinear responses are very pronounced there.
@ -608,7 +608,7 @@ The weakly nonlinear interactions in low-CV P-units could facilitate the detecta
\subsection{Conclusions and outlook}
We have demonstrated that there are pronounced nonlinear responses in the primary electrosensory afferents of the weakly electric fish \lepto{}. The observed nonlinearities match the expectations from previous theoretical studies \citep{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-order 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
We have demonstrated that there are pronounced nonlinear responses in the primary electrosensory afferents of the weakly electric fish \lepto{}. The observed nonlinearities match the expectations from previous theoretical studies \citep{Voronenko2017,Franzen2023}. 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-order 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
auditory nerve fibers and such nonlinear effects might also be expected in the auditory system during the encoding of amplitude modulations \citep{Joris2004}. Nevertheless, the theory holds for systems that are directly driven by a sinusoidal input (ampullary cells) and systems that are driven by amplitude modulations of a carrier (P-units) and is thus widely applicable.
@ -713,7 +713,7 @@ normalizes the second-order cross-spectrum by the spectral power at the two stim
Throughout the manuscript we only show the absolute values of the complex-valued second-order susceptibility matrix and ignore the corresponding phases.
\paragraph{Nonlinearity index}\label{projected_method}
We expected to see elevated values in the second-order susceptibility at $\omega_1 + \omega_2 = \fbase$ \citep{Voronenko2017}. To characterize this in a single number we computed the peakedness of the nonlinearity (PNL) defined as
We expected to see elevated values in the second-order susceptibility at $\omega_1 + \omega_2 = \fbase$ \citep{Voronenko2017,Franzen2023}. To characterize this in a single number we computed the peakedness of the nonlinearity (PNL) defined as
\begin{equation}
\label{eq:nli_equation}
\nli{} = \frac{ \max D(\fbase{}-5\,\rm{Hz} \leq f \leq \fbase{}+5\,\rm{Hz})}{\mathrm{median}(D(f))}