discussion

susceptibility1.tex

@@ -566,11 +566,20 @@ 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}
 
-\subsection{Alternatives to the Volterra series}
-Estimating the infinite Volterra series from limited experimental data is usually limited to the first two or three kernels, which might not be sufficient for a proper prediction of the neuronal response  \citep{French2001}. As we have demonstrated, even a proper estimation of just the second-order susceptibility in the weakly nonlinear regime is not feasable 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}. On the other hand, 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, Chacron?}, because backward models do not need to generate action potentials \citep{Rieke1999}.
+\subsection{Characterizing nonlinear processes 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}.
+
+\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.
+
+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.
+
+Ampullary stimulus encoding is somewhat different in \lepto{}. The power spectrum of the spontaneous response is dominated by only the baseline firing rate and its harmonics, a second oscillator is not visible. The baseline firing frequency, however, is outside the linear coding range \citep{Grewe2017} while it is within the linear coding range in paddlefish \citep{Neiman2011fish}. Interestingly, the nonlinear response in the paddlefish ampullaries increases with stimulus intensity while it disappears in our case (\subfigrefb{fig:dataoverview}{F}) indicating that paddlefish data have been recorded above the weakly-nonlinear regime. 
+
+The population of ampullary cells is very homogeneous with respect to the baseline rate (131$\pm$29\,Hz) and stimulus encoding properties \citep{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 in the full population of ampullary cells (at the baseline firing frequency) that is outside the linear coding range. Postsynaptic cells integrating ampullary input might be able to extract such nonlinear responses. How such nonlinear effects might influence prey detection should be addressed in future studies.
 
 \subsection{Noise stimulation approximates real interactions}
-Our characterization of electrorecptors is based on neuronal responses to white-noise stimuli. These broad-band stimuli have the advantage that all behaviorally relevant frequencies can be measured with a single stimulus presentation and it is a widely used approach to characterize sensory coding \citep{French1973, Marmarelis1999, Borst1999, Chacron2005, Grewe2017}. In the real-world situation of wave-type electric fish, the natural stimuli encoded by P-units are the sinusoidal EOD signals with distinct frequencies continuosly emitted by a small number of near by interacting fish and the resulting beating AMs \citep{Stamper2010, Henninger2020}. How informative is the second-order susceptibility estimated in the limit of vanishing noise-stimulus amplitude for the encoding of distinct frequencies with finite amplitudes?
+Our characterization of electroreceptors is based on neuronal responses to white-noise stimuli. These broad-band stimuli have the advantage that all behaviorally relevant frequencies can be measured with a single stimulus presentation and it is a widely used approach to characterize sensory coding \citep{French1973, Marmarelis1999, Borst1999, Chacron2005, Grewe2017}. In the real-world situation of wave-type electric fish, the natural stimuli encoded by P-units are the sinusoidal EOD signals with distinct frequencies continuosly emitted by a small number of near by interacting fish and the resulting beating AMs \citep{Stamper2010, Henninger2020}. How informative is the second-order susceptibility estimated in the limit of vanishing noise-stimulus amplitude for the encoding of distinct frequencies with finite amplitudes?
 
 % 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}).
 
@@ -598,8 +607,7 @@ Outside courtship behavior, the encoding of secondary or social envelopes is a c
 % Envelope extraction requires a nonlinearity:
 % In the context of social signaling among three fish, we observe an AM of the AM, also referred to as second-order envelope or just social envelope \citep{Middleton2006, Savard2011, Stamper2012Envelope}. Encoding this again requires nonlinearities \citep{Middleton2006} and it was shown that a subpopulation of P-units is sensitive to envelopes \citep{Savard2011} and exhibit nonlinearities e.g. when driven by strong stimuli \citep{Nelson1997, Chacron2004}.
 
-\subsection{Nonlinear encoding in ampullary cells}
-The afferents of the passive electrosensory system, the ampullary cells, were found to exhibit much stronger nonlinearities than P-units (\figref{fig:dataoverview}). The adequate stimulus for this system is a direct stimulation not an amplitude modulation. In this respect, the ampullary cells are closer to the LIF models used by Voroneko and colleagues \citep{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 \citep{Kalmijn1974, Engelmann2010, Neiman2011fish}. The simultaneous muscle activity of a swarm of prey (such as \textit{Daphnia}) resembles Gaussian white noise \citep{Neiman2011fish}. Our results show some similarities with the analyses by Neiman and Russel \citep{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 can be well described using linear models for low frequencies but there are also nonlinearities in both systems. Linear encoding in the paddlefish shows a gap at the epithelial oscillation frequency, 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 \citep{Grewe2017} while it is within the linear coding range in paddlefish \citep{Neiman2011fish}. Interestingly, the nonlinear response in the paddlefish ampullaries increases with stimulus intensity while it disappears in our case (\subfigrefb{fig:dataoverview}{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 \citep{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 to 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. How such nonlinear effects in ampullary cells might influence prey detection should be addressed in future studies.
+
 
 \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