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further work on P-unit discussion

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susceptibility1.tex
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@ -566,7 +566,7 @@ In order to characterize weakly nonlinear responses of the cells in the limit to
\notejb{Say that previous work on second-order nonlinearities did not see the weakly nonlinear regime, because of the linearizing effects} \notejb{Say that previous work on second-order nonlinearities did not see the weakly nonlinear regime, because of the linearizing effects}
\subsection{Characterizing nonlinear processes from limited experimental data} \subsection{Characterizing nonlinear coding from limited experimental data}
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}. 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} \subsection{Nonlinear encoding in ampullary cells}
@ -581,9 +581,14 @@ The population of ampullary cells is very homogeneous with respect to the baseli
\subsection{Nonlinear encoding in P-units} \subsection{Nonlinear encoding in P-units}
Noise stimuli have the advantage that a range of frequencies can be measured with a single stimulus presentation and they have been successfully applied to characterize sensory coding in many systems \citep{French1973, Marmarelis1999, Borst1999, Chacron2005, Grewe2017}. In the real-world situation of wave-type electric fish, however, the natural stimuli encoded by P-units are periodic amplitude modulations of the self-generated electric field which arise from the superposition of the own and foreign EODs. Natural interactions usually occur between low numbers of close-by fish and thus the AMs are a mixture of a few distinct frequencies with substantial amplitudes \citep{Stamper2010,Fotowat2013, Henninger2020}. How informative are the second-order susceptibilities estimated using noise-stimuli for the encoding of distinct frequencies? Noise stimuli have the advantage that a range of frequencies can be measured with a single stimulus presentation and they have been successfully applied to characterize sensory coding in many systems \citep{French1973, Marmarelis1999, Borst1999, Chacron2005, Grewe2017}. In the real-world situation of wave-type electric fish, however, the natural stimuli encoded by P-units are periodic amplitude modulations of the self-generated electric field which arise from the superposition of the own and foreign EODs. Natural interactions usually occur between low numbers of close-by fish and thus the AMs are a mixture of a few distinct frequencies with substantial amplitudes \citep{Stamper2010,Fotowat2013, Henninger2020}. How informative are the second-order susceptibilities estimated using noise-stimuli for the encoding of distinct frequencies?
We here applied broad-band noise amplitude-modulation stimuli to estimate the second-order susceptibility. The total signal power in noise stimuli is uniformly distributed over a wide frequency band while it is spectrally focused in pure sinewave stimuli. If both stimuli have the same total power, i.e. are presented with the same amplitude (contrast) the power of the sinewave stimulus, the power at these frequencies is much higher and, assuming white neuronal noise, have a much higher signal-to-noise ratio. \notejg{Ehrlich gesagt, weiss ich nicht so genau, wo du mit dem vorherigen Satz noch hinwolltest.} The absence of the linearizing effect of the noise stimuli can explain nonlinear interactions under distinct-frequency stimulation \figref{fig:motivation} while they are barely visible with strong white-noise stimulation \figref{fig:model_full}. Applying the Furutsu-Novikov theorem in the model simulations strongly suggests that low-CV cells do have the full nonlinearity pattern that is, however, covered due to external white-noise stimulation. We thus conclude that the presence of the anti-diagonal pattern in the \suscept{} matrix is sufficient to infer that the same nonlinear interactions happen here, in accordance with the the second-order susceptibilities found in LIF models without a carrier \citep{Voronenko2017, Schlungbaum2023}. This also validates the application of the white noise approach to characterize the full \suscept{} matrix instead of using all combinations of frequencies. We here applied broad-band noise amplitude-modulation stimuli to estimate the second-order susceptibility. The total signal power in noise stimuli is uniformly distributed over a wide frequency band while it is spectrally focused in pure sinewave stimuli.
% If both stimuli have the same total power, i.e. are presented with the same amplitude (contrast), then at the frequency of the sinewave stimulus the power of the noise stimulus is much lower than the one of the sinewave stimulus.
The linearizing effect of such narrow-band signals is much weaker in comparison to a broad-band noise stimulus of the same total power \notejb{Benjamin: can we say it like that? Is there a reference for it?}. This explains why we can observe nonlinear interactions between sinewave stimuli with distinct frequencies \figref{fig:motivation} although they are barely visible when stimulating with strong broad-band noise \figref{model_and_data}. As discussed above in the context of broad-band noise stimulation, we expect interactions with a cell's baseline frequency only for the very few P-units with low enough CVs of the baseline interspike intervals.
%We thus conclude that the presence of the anti-diagonal pattern in the \suscept{} matrix is sufficient to infer that the same nonlinear interactions happen here, in accordance with the the second-order susceptibilities found in LIF models without a carrier \citep{Voronenko2017, Schlungbaum2023}. This also validates the application of the white noise approach to characterize the full \suscept{} matrix instead of using all combinations of frequencies.
In our P-unit recordings we directly relate the AM waveform to the cellular responses. Responding to the AM itself requires a nonlinearity \citep{Middleton2006, Stamper2012Envelope, Savard2011, Barayeu2023}, encoding the time-course of the AM, however, is linear over a wide range of AM amplitudes and frequencies \citep{Xu1996, Benda2005, Gussin2007, Grewe2017, Savard2011}. The encoding of secondary or social envelopes that arise from relative movement or the interaction of more than two animals \citep{Stamper2012Envelope} requires an additional nonlinearity in the system that was initially attributed to downstream processing \citep{Middleton2006, Middleton2007}. Later studies showed that already the electroreceptors can encode such information when strong saturation nonlinearities occur \citep{Savard2011}. Based on our work here, we would predict that a subset of cells, with low CVs, should encode the social envelopes even under weak stimulation. The exact transitions on nonlinear encoding from a regime of weak stimuli to a regime of strong stimuli should be addressed in further studies \notejg{Here we should probably work with the new figure}. Extracting the AM from the stimulus already requires a (threshold) nonlinearity \citep{Middleton2006, Stamper2012Envelope, Savard2011, Barayeu2023}. This nonlinearity, however, does not show up in our estimates of the susceptibilities, because in our analysis we directly relate the AM waveform to the recorded cellular responses. Encoding the time-course of the AM, however, has been shown to be linear over a wide range of AM amplitudes and frequencies \citep{Xu1996, Benda2005, Gussin2007, Grewe2017, Savard2011}. In contrast, we here have demonstrated nonlinear interactions originating from the spike generator for broad-band noise stimuli in the limit of vanishing amplitudes and for stimulation with two distinct frequencies. Both settings have not been studied yet.
The encoding of secondary or social envelopes that arise from relative movement or the interaction of more than two animals \citep{Stamper2012Envelope} requires an additional nonlinearity in the system that was initially attributed to downstream processing \citep{Middleton2006, Middleton2007}. Later studies showed that already the electroreceptors can encode such information when strong saturation nonlinearities occur \citep{Savard2011}. Based on our work here, we would predict that a subset of cells, with low CVs, should encode the social envelopes even under weak stimulation. The exact transitions on nonlinear encoding from a regime of weak stimuli to a regime of strong stimuli should be addressed in further studies (\figrefb{fig:nonlin_regime}).
The observed nonlinear effects can possibly facilitate the detectability of faint signals during a three-animal interactions as found in freely behaving animals \citep{Henninger2018}, the electrosensory cocktail party. They are, however, very specific with respect to the stimulus frequencies and a given P-unit's baseline frequency. The relevant frequencies are determined by combination of EOD frequencies observed in male and female fish \citep{Hopkins1974Eigen, Meyer1987, Henninger2018, Henninger2020} and the AM frequencies are limited to a range below half of each fish's EOD frequency (0 -- \feod/2, a restriction arising from the sampling theorem) \citep{Barayeu2023}. The population of P-units is very heterogeneous in their baseline firing properties \citep{Grewe2017, Hladnik2023}. The baseline firing rates and the observed CVs vary in wide ranges (50--450\,Hz and 0.1--1.4, respectively, \figref{fig:dataoverview}\panel{A}). The range of baseline firing rates thus covers substantial parts of the beat frequencies that may occur during animal interactions, while the number of low-CV P-units is small. For the response components arising due to the nonlinearities described here to be behaviorally relevant requires this information to survive the convergence onto the pyramidal cells in the electrosensory lateral line lobe (ELL) in the hindbrain with different tuning properties and levels of convergence \citep{Krahe2014, Maler2009a}. Since the receptive fields of the pyramidal neurons are patches of adjacent receptors on the fish's body surface \citep{Bastian2002, Maler2009a, Haggard2023} and the input heterogeneity does not depend on the location of the receptor on the fish body \citep{Hladnik2023} the pyramidal cell input population will be heterogeneous which 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. The observed nonlinear effects can possibly facilitate the detectability of faint signals during a three-animal interactions as found in freely behaving animals \citep{Henninger2018}, the electrosensory cocktail party. They are, however, very specific with respect to the stimulus frequencies and a given P-unit's baseline frequency. The relevant frequencies are determined by combination of EOD frequencies observed in male and female fish \citep{Hopkins1974Eigen, Meyer1987, Henninger2018, Henninger2020} and the AM frequencies are limited to a range below half of each fish's EOD frequency (0 -- \feod/2, a restriction arising from the sampling theorem) \citep{Barayeu2023}. The population of P-units is very heterogeneous in their baseline firing properties \citep{Grewe2017, Hladnik2023}. The baseline firing rates and the observed CVs vary in wide ranges (50--450\,Hz and 0.1--1.4, respectively, \figref{fig:dataoverview}\panel{A}). The range of baseline firing rates thus covers substantial parts of the beat frequencies that may occur during animal interactions, while the number of low-CV P-units is small. For the response components arising due to the nonlinearities described here to be behaviorally relevant requires this information to survive the convergence onto the pyramidal cells in the electrosensory lateral line lobe (ELL) in the hindbrain with different tuning properties and levels of convergence \citep{Krahe2014, Maler2009a}. Since the receptive fields of the pyramidal neurons are patches of adjacent receptors on the fish's body surface \citep{Bastian2002, Maler2009a, Haggard2023} and the input heterogeneity does not depend on the location of the receptor on the fish body \citep{Hladnik2023} the pyramidal cell input population will be heterogeneous which 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.