a plan for the introduction

susceptibility1.tex

@@ -523,22 +523,42 @@ We can find nonlinearities in many sensory systems such as rectification in the
 
 In the auditory or the active electric sense, for example, nonlinear processes are needed to extract envelopes, i.e. amplitude modulations of a carrier signal\cite{Joris2004, Barayeu2023} called beats. Beats are common stimuli in different sensory modalities enabling rhythm and pitch perception in human hearing \cite{Roeber1834, Plomp1967, Joris2004, Grahn2012} and providing context for electrocommunication in weakly electric fish \cite{Engler2001, Hupe2008, Henninger2018, Benda2020}.
 
+While the encoding of signals can often be well described by linear models in the sensory periphery\cite{Machens2001}, this is not true for many upstream neurons. Rather, nonlinear processes are implemented to extract special stimulus features\cite{Adelson1985,Gabbiani1996,Olshausen1996,Gollisch2009}.
 
-\notejb{What are the main findings of the manuscript? (i) 2nd order susceptibility can be measured using RAM stimuli, (ii) we see a bit of nonlinearity in low-CV P-units and strong one in ampullary cells. (ii) We find stuff, that matches the theoretical predictions (Voronekov) (iii) AMs with carrier (auditory) (iv) There is an estimation problem because of low N and RAM stimulus introducing linearizing noise. (v) noise split gives a good estimate that translates well to pure sine wave stimulation.}
+\notejb{What are the main findings of the manuscript?  (i) We find patterns that match the theoretical predictions (Voronekov) for weak stimulation. No saturation regime! (ii) we see a bit of nonlinearity in low-CV P-units and strong one in ampullary cells. Nonlinearities show up at very specific frequencies. (iii) AMs with carrier (auditory) (iv) There is an estimation problem because of low N and RAM stimulus introducing linearizing noise. (v) noise split gives a good estimate that translates well to pure sine wave stimulation. (vi) motivation by functional not mechanistic characterization in a behaviorally relevant context. (vii) 2nd order susceptibility can be measured using RAM stimuli (not new, see Victor papers)}
 
-\notejb{Strong aspects are (i) how to estimate 2nd order susceptibilities, (ii) what do they tell us about relevant stimuli, (iii) nonlinearities show up only for very specific frequencies}
+\notejb{For the estimation problem we need to cite work that also measured higher order Wiener kernels and filters (we need to find the Andrew French paper, who else? Gabbiani?). For nonlinear encoding we need to talk about linear-nonlinear models (Chinchilinsky, Gollisch, Jan Clemens) versus Wiener series (French). And we find nonlinear responses in neurons that have been considered as quite linear, but only at specific frequency combinations and low signal amplitudes.}
 
-\notejb{For the estimation problem we need to cite work that also measured higher order Wiener kernels and filters (we need to find the Andrew French paper, who else? Gabbiani? John Miller?). For nonlinear encoding we need to talk about linear-nonlinear models (Chinchilinsky, Gollisch, Jan Clemens) versus Wiener series (French). And we find nonlinear responses in neurons that have been considered as quite linear, but only at specific frequency combinations and low signal amplitudes.}
+Old system idendification: forward models Wiener VOlterra series both in time and frequency domain, mechanotransduction in spiders, visual systems.
 
-\notejb{\cite{French1973} Derivation of the Fourier transformed kernels measured with white noise.}
-\notejb{\cite{French1976} Technical issues and tests of  Fourier transformed kernels measured with white noise.}
-\notejb{\cite{Victor1977} Cat retinal ganglion cells, gratings with sum of 6 or 8 sinusoids. X - versus Y cells. Peak at f1 == f2 in Y cells. X-cells rather linear. Discussion of mechanism, where a nonlinearity comes in along the pathway}
-\notejb{\cite{Marmarelis1972} Temporal 2nd order kernels, how well do kernels predict responses}
-\notejb{\cite{Victor1988} Cat retinal ganglion cells, sum of sinusoids, very technical, one measurement similar to \citep{Victor1977}.}
-\notejb{\citep{Mitsis2007} Spider mechanorecptor. Linear filterss, multivariate nonlinearity, and threshold. Second order kernel needed for this. Gausian noise stimuli.}
-\notejb{\citep{French2001} Time kernels up to 3rd order for predicting spider mechanorecptor responses (spikes!)}
-\notejb{\citep{French1999} Review on time domoin nonlinear systems identification}
+Later reverse correlations, stimulus estimation. Of course forward prediction of the spike train is horribly nonlinear, but predicting stimulus from spikes is much more linear.
 
+Shift away from series to linear-nonlinear models, very successfull in forward modelling of sensory neurons.
+
+Mostly linear characterization by transfer functions and coherences. Rarely bispectra/bicoherences. But then mostly at higher stimulus amplitudes where saturation of fI curve gets relevant.
+
+Recent theoretical work makes predictions for bispectra in the limit to small stimulus amplitudes. Very specific combinations of stimulus frequencies at f1 + f2 = fb evoke nonlinear responses.
+
+We set out to find these nonlinear responses in electroreceptor afferents. Advantage high baseline rate, good for estimation problem.
+
+\begin{itemize}
+\item \notejb{\cite{French1973} Derivation of the Fourier transformed kernels measured with white noise.}
+\item \notejb{\cite{French1976} Technical issues and tests of  Fourier transformed kernels measured with white noise.}
+\item \notejb{\cite{Victor1977} Cat retinal ganglion cells, gratings with sum of 6 or 8 sinusoids. X - versus Y cells. Peak at f1 == f2 in Y cells. X-cells rather linear. Discussion of mechanism, where a nonlinearity comes in along the pathway}
+\item \notejb{\cite{Marmarelis1972} Temporal 2nd order kernels, how well do kernels predict responses}
+\item \notejb{\cite{Victor1988} Cat retinal ganglion cells, sum of sinusoids, very technical, one measurement similar to \cite{Victor1977}.}
+\item \notejb{\citep{Nikias1993} Third order spectra or bispectra. Very technical overview to higher order spectra}
+\item \notejb{\cite{Mitsis2007} Spider mechanorecptor. Linear filterss, multivariate nonlinearity, and threshold. Second order kernel needed for this. Gausian noise stimuli.}
+\item \notejb{\cite{French2001} Time kernels up to 3rd order for predicting spider mechanorecptor responses (spikes!)}
+\item \notejb{\cite{French1999} Review on time domoin nonlinear systems identification}
+\item \notejb{\cite{Theunissen1996} Linear backward stimulus reconstruction in the context of information theory/signal-to-noise ratios}
+\item \notejb{\cite{Wessel1996} Same as Theunissen1996 but for P-units}
+\item \notejb{\cite{Schanze1997} lots of bispectra, visual cortex MUA recordings}
+\item \notejb{\cite{Chichilnisky2001} Linear Nonlinear Poisson model}
+\item \notejb{\cite{Gollisch2009} Linear Nonlinear models in retina}
+\item \notejb{\cite{Clemens2013} Grasshoppper model for female preferences}
+\item \notejb{\cite{Neimann2011} cross bispectrum, bicoherence, mutual information, saturating nonlinearities, `` ampullary electroreceptors of paddlefish are perfectly suited to linearly encode weak low-frequency stimuli.''}
+\end{itemize}
 
 \noteab{The nonlinearity of a system has been accessed with the use of wiener kernels \cite{French1973,French1976}, measuring the system response to white noise stimulation. Besides that the nonlinearity of a system has been addressed by pure sinewave simulation, considering the Fourier transform of the Volterra series \cite{Victor1977,Victor1980,Shapley1979}. The estimates of the nonlinearity with both methods, white noise and sinewave stimulation, was shown to yield similar results \cite{Vitor1979}. Nonlinearity was investigated, not addressing the system properties, but focusing on the quadratic phase coupling of the two input frequencies \cite{Nikias1993, Neiman2011fish}. With these approaches nonlinearity at the sum of two input frequencies was quantified in retinal cells \cite{Shapley1979} for stimuli with small amplitudes, in ampullary cells \cite{Neiman2011fish}, in the EEG of sleep \cite{Barnett1971,Bullock1997} and in mechanorecetors \cite{French1976}. Second-order responses have been quantified in not amplitude modulated \cite{Neiman2011fish} and amplitude modulated systems \cite{Victor1977,Victor1980,Shapley1979}.}
 
@@ -548,9 +568,9 @@ In the auditory or the active electric sense, for example, nonlinear processes a
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-\noteab{Ich weiß nicht ob du in die Literatur reingeschaut hast, aber so eine Dreiecksstruktur finden wir schon in früheren Arbeiten! Vor allen in denen von Viktor.}
+While the encoding of signals can often be well described by linear models in the sensory periphery\cite{Machens2001}, this is not true for many upstream neurons. Rather, nonlinear processes are implemented to extract special stimulus features\cite{Adelson1985,Gabbiani1996,Olshausen1996,Gollisch2009}.
 
-While the encoding of signals can often be well described by linear models in the sensory periphery\cite{Machens2001}, this is not true for many upstream neurons. Rather, nonlinear processes are implemented to extract special stimulus features\cite{Adelson1985,Gabbiani1996,Olshausen1996,Gollisch2009}. In active electrosensation, the self-generated electric field (electric organ discharge, EOD) that is quasi sinusoidal in wavetype electric fish acts as the carrier signal that is amplitude modulated in the context of communication\cite{Walz2014, Henninger2018, Benda2020}, object detection and navigation\cite{Fotowat2013, Nelson1999}. In social contexts, the interference of the EODs of two interacting animals result in a characteristic periodic amplitude modulation, the so-called beat. The beat amplitude is defined by the smaller EOD amplitude, its frequency is defined as the difference between the two EOD frequencies ($\Delta f = f-\feod{}$, valid for $f < \feod{}/2$)\cite{Barayeu2023}. Cutaneous electroreceptor organs that are distributed over the bodies of these fish \cite{Carr1982} are tuned to the own field\cite{Hopkins1976,Viancour1979}. Probability-type electroreceptor afferents (P-units) innervate these organs via ribbon synapses\cite{Szabo1965, Wachtel1966} and project to the hindbrain where they trifurcate and synapse onto pyramidal cells in the electrosensory lateral line lobe (ELL)\cite{Krahe2014}. The P-units of the gymnotiform electric fish \lepto{} encode such amplitude modulations (AMs) by modulation of their firing rate\cite{Gabbiani1996}. They fire probabilistically but phase-locked to the own EOD and the skipping of cycles leads to their characteristic multimodal interspike-interval distribution. Even though the extraction of the AM itself requires a \notejb{nonlinearity}\cite{Middleton2006,Stamper2012Envelope,Savard2011,Barayeu2023} encoding the time-course of the AM is linear over a wide range of AM amplitudes and frequencies\cite{Xu1996,Benda2005,Gussin2007,Grewe2017,Savard2011}. In the context of social signalling among three fish we observe an AM of the AM, also referred to as second-order envelope or just social envelope\cite{Middleton2006, Savard2011, Stamper2012Envelope}. Encoding this again requires nonlinearities\cite{Middleton2006} and it was shown that a subpopulation of P-units are sensitive to envelopes\cite{Savard2011} and exhibit nonlinearities e.g. when driven by strong stimuli\cite{Nelson1997,Chacron2004}.
+In active electrosensation, the self-generated electric field (electric organ discharge, EOD) that is quasi sinusoidal in wavetype electric fish acts as the carrier signal that is amplitude modulated in the context of communication\cite{Walz2014, Henninger2018, Benda2020}, object detection and navigation\cite{Fotowat2013, Nelson1999}. In social contexts, the interference of the EODs of two interacting animals result in a characteristic periodic amplitude modulation, the so-called beat. The beat amplitude is defined by the smaller EOD amplitude, its frequency is defined as the difference between the two EOD frequencies ($\Delta f = f-\feod{}$, valid for $f < \feod{}/2$)\cite{Barayeu2023}. Cutaneous electroreceptor organs that are distributed over the bodies of these fish \cite{Carr1982} are tuned to the own field\cite{Hopkins1976,Viancour1979}. Probability-type electroreceptor afferents (P-units) innervate these organs via ribbon synapses\cite{Szabo1965, Wachtel1966} and project to the hindbrain where they trifurcate and synapse onto pyramidal cells in the electrosensory lateral line lobe (ELL)\cite{Krahe2014}. The P-units of the gymnotiform electric fish \lepto{} encode such amplitude modulations (AMs) by modulation of their firing rate\cite{Gabbiani1996}. They fire probabilistically but phase-locked to the own EOD and the skipping of cycles leads to their characteristic multimodal interspike-interval distribution. Even though the extraction of the AM itself requires a \notejb{nonlinearity}\cite{Middleton2006,Stamper2012Envelope,Savard2011,Barayeu2023} encoding the time-course of the AM is linear over a wide range of AM amplitudes and frequencies\cite{Xu1996,Benda2005,Gussin2007,Grewe2017,Savard2011}. In the context of social signalling among three fish we observe an AM of the AM, also referred to as second-order envelope or just social envelope\cite{Middleton2006, Savard2011, Stamper2012Envelope}. Encoding this again requires nonlinearities\cite{Middleton2006} and it was shown that a subpopulation of P-units are sensitive to envelopes\cite{Savard2011} and exhibit nonlinearities e.g. when driven by strong stimuli\cite{Nelson1997,Chacron2004}.
 
 \begin{figure*}[t]
   \includegraphics[width=\columnwidth]{motivation}