added author summary

cover1.tex

@@ -27,12 +27,18 @@
 
 we would like to submit our manuscript ``Spike generation in
 electroreceptor afferents introduces additional spectral response
-components by weakly nonlinear interactions'' for publication in PLoS Computational
-Biology. It is the result of a collaborative effort on the theoretical side, Benjamin Linder, HU Berlin, and the experimental and numerical modeling side, Jan Grewe and myself at the University of Tuebingen, within the DFG priority program 2205 ``Evolutionary optimization of neuronal processing''.
+components by weakly nonlinear interactions'' for publication in PLoS
+Computational Biology. It is the result of a collaborative effort on
+the theoretical side, Benjamin Linder, HU Berlin, and the experimental
+and numerical modeling side, Jan Grewe and myself at the University of
+Tuebingen, within the DFG priority program 2205 ``Evolutionary
+optimization of neuronal processing''.
 
-Non-linearites are at the heart of neural computations in the brain, spike generation, for example, involves a strong non-linearity. Nevertheless, the encoding of dynamic stimuli in spike trains often can be well approximated by linear response theory, in particular when driving the neuron in the supra-threshold \textbf{(was superthreshold before...)} regime at low signal-to-noise ratios. This has been exploited in numerous theoretical studies. However, at less noise or stronger stimulus amplitudes non-linear effects become more prominent. In the weakly non-linear regime, the second term of the Volterra series becomes relevant. Benjamin Lindner developed analytical solutions of this term for the leaky integrate-and-fire neuron, which predicts non-linear interactions whenever one or the sum of two stimulus frequencies matches the neuron's baseline firing rate. In our manuscript we set out to find signatures of these interactions in electrophysiological data measured in two types of electrosensory neurons of the electric fish \textit{Apteronotus leptorhynchus}. In ampullary cells, these interactions are prominent, whereas in P-units they are harder to find. Estimating the second-order susceptibilites from real data turns out to be a hard problem as limited data leads to poor estimates. Comparison with models that have been fitted to individual P-units we are deduce the presence of non-linear interactions also in P-units. Finally, we discuss our findings and the relevance of non-linear interactions in the context of the neuroethological background.
+Computational neuroscientists have a strong interest in characterizing non-linearities and study their functional consequences. However, experimental backing of these theoretical findings are scarce. For example, encoding of dynamic stimuli in spike trains often can be well approximated by linear response theory, in particular when driving the neuron in the supra-threshold regime at low signal-to-noise ratios. This has been exploited in numerous theoretical studies. At less noise or stronger stimulus amplitudes non-linear effects become more prominent. In the weakly non-linear regime, the second term of the Volterra series becomes relevant. Benjamin Lindner developed analytical solutions of this term for the leaky integrate-and-fire neuron, which predicts non-linear interactions whenever one or the sum of two stimulus frequencies matches the neuron's baseline firing rate. Until now, however, these fundamental non-linearities arising from the core mechanism of spike generation have not been reported in any experimental data.
 
-Computational neuroscientists have a strong interest in characterizing non-linearities and study their functional consequences. However, experimental backing of these theoretical findings are scarce. With our work we set out to fill this gap and are able to confirm theoretical findings about the second-order susceptibility in real neurons. We believe that it is of strong interest to many readers of PLoS Computational Biology and can inspire future research in other sensory systems.
+With our work we set out to fill this gap. We scan a large set of electrophysiological data measured in two types of electrosensory neurons of the electric fish \textit{Apteronotus leptorhynchus} for signatures of these non-linearities. In ampullary cells, non-linear interaction between two stimulus frequencies are prominent, whereas in P-units they are harder to find. Estimating the second-order susceptibilites from real data turns out to be a hard problem as limited data leads to poor estimates. Comparison with models that have been fitted to individual P-units, we are able to deduce the presence of non-linear interactions also in some of the P-units. Finally, we discuss our findings and the relevance of non-linear interactions in the neuroethological context.
+
+We believe that this analysis of electrophysiological data close to expectations from theoretical work is of strong interest to many readers of PLoS Computational Biology and can inspire future research in other sensory systems.
 
 Best regards,\\%[-2ex]
 %\hspace*{0.17\textwidth}\includegraphics[width=0.3\textwidth]{JanBenda-Signature2020}\\

nonlinearbaseline.tex

@@ -258,14 +258,14 @@ We here analyze nonlinear responses in two types of primary electroreceptor affe
 % Please keep the Author Summary between 150 and 200 words
 % Use the first person.
 \section{Author summary}
-The generation of action potentials involves a strong threshold non-linearity. Nevertheless, the encoding of stimuli with small amplitudes by neurons with sufficient intrinsic noise can be well described as a linear system. As the stimulus amplitude is increased, non-linear effects start to appear. Initially, in the so-called weakly non-linear regime new spectral components at the sum and the difference of two stimulus frequencies start to appear. This regime has been well characterized by theoretical analysis based on simple neuron models like the leaky integrate-and-fire model. These findings predict non-linear interactions whenever one or the sum of two stimulus frequencies matches a neuron's baseline firing rate. We set out to find these signatures in a large set of electrophysiological recordings from electroreceptive neurons of a weakly electric fish. In ampullary cells these interactions are prominent, whereas in P-units they are harder to find. Estimating non-linear response kernels from limited real data turns out to be a hard problem. By comparison with models that have been fitted to individual P-units we are then able to interpret the poor estimates. The non-linear response components could boost sensory responses to weak signals emitted by distant conspecifics.
+The generation of action potentials involves a strong threshold nonlinearity. Nevertheless, the encoding of stimuli with small amplitudes by neurons with sufficient intrinsic noise can be well described as a linear system. As the stimulus amplitude is increased, new spectral components start to appear in the so called weakly nonlinear regime. This regime has been well characterized by theoretical analysis based on simple neuron models such as the leaky integrate-and-fire model. These findings predict nonlinear interactions whenever one or the sum of two stimulus frequencies matches the neuron's baseline firing rate. We set out to find these signatures in a large set of electrophysiological recordings from electroreceptive neurons of a weakly electric fish. In ampullary cells these interactions are prominent, whereas in P-units they are harder to find. Estimating nonlinear response kernels from limited real data turns out to be a hard problem. By comparison with models that have been fitted to individual P-units we are then able to interpret the poor estimates. The nonlinear response components could boost sensory responses to weak signals emitted, for example, by distant conspecifics.
 
   
 \section{Introduction}
 
 \begin{figure*}[t]
   \includegraphics[width=\columnwidth]{lifsuscept}
-  \caption{\label{fig:lifsuscept} First- and second-order response functions of the leaky integrate-and-fire model. \figitem{A} Magnitude of the first-order (linear) response function $|\chi_1(f)|$, also known as the ``gain'' function, quantifies the response amplitude relative to the stimulus amplitude, both measured at the same stimulus frequency. \figitem{B} Magnitude of the second-order (non-linear) response function $|\chi_2(f_1, f_2)|$ quantifies the response at the sum of two stimulus frequencies. Because frequencies can also be negative, the sum is actually a difference in the upper-left and lower-right quadrants, since $f_2 + f_1 = f_2 - |f_1|$ for $f_1 < 0$. For linear systems, the second-order response function is zero, because linear systems do not create new frequencies and thus there is no response at the sum of the two frequencies. The plots show the analytical solutions from \citet{Lindner2001} and \citet{Voronenko2017} with $\mu = 1.1$ and $D = 0.001$. Note that the leaky integrate-and-fire model is formulated without dimensions, frequencies are given in multiples of the inverse membrane time constant.}
+  \caption{\label{fig:lifsuscept} First- and second-order response functions of the leaky integrate-and-fire model. \figitem{A} Magnitude of the first-order (linear) response function $|\chi_1(f)|$, also known as the ``gain'' function, quantifies the response amplitude relative to the stimulus amplitude, both measured at the same stimulus frequency. \figitem{B} Magnitude of the second-order (nonlinear) response function $|\chi_2(f_1, f_2)|$ quantifies the response at the sum of two stimulus frequencies. Because frequencies can also be negative, the sum is actually a difference in the upper-left and lower-right quadrants, since $f_2 + f_1 = f_2 - |f_1|$ for $f_1 < 0$. For linear systems, the second-order response function is zero, because linear systems do not create new frequencies and thus there is no response at the sum of the two frequencies. The plots show the analytical solutions from \citet{Lindner2001} and \citet{Voronenko2017} with $\mu = 1.1$ and $D = 0.001$. Note that the leaky integrate-and-fire model is formulated without dimensions, frequencies are given in multiples of the inverse membrane time constant.}
 \end{figure*}
 
 We like to think about signal encoding in terms of linear relations with unique mapping of a given input value to a certain output of the system under consideration. Indeed, such linear methods, for example the transfer function or first-order susceptibility shown in \subfigref{fig:lifsuscept}{A}, have been widely and successfully applied to describe and predict neuronal responses and are an invaluable tools to characterize neural systems \citep{Eggermont1983,Borst1999}. Nonlinear mechanisms, on the other hand, are key on different levels of neural processing. Deciding for one action over another is a nonlinear process on the systemic level. On the cellular level, spiking neurons are inherently nonlinear. Whether an action potential is elicited depends on the membrane potential to exceed a threshold \citep{Hodgkin1952, Koch1995}. Because of such nonlinearities, understanding and predicting neuronal responses to sensory stimuli is in general a difficult task.
@@ -600,7 +600,7 @@ Values larger than one indicate a sharp ridge in the susceptibility matrix close
 
 \subsection{Leaky integrate-and-fire models for P-units}\label{lifmethods}
 
-Modified leaky integrate-and-fire (LIF) models were constructed to reproduce the specific firing properties of P-units \citep{Chacron2001, Sinz2020, Barayeu2023}. Its basic components (static non-linearity, low-pass filtering and spike generation) are equivalent to models of hair cells in auditory systems \citep{Eggermont1983}.  The sole driving input into the P-unit model during baseline, i.e. when no external stimulus was given, is the fish's own EOD, modeled as a cosine wave
+Modified leaky integrate-and-fire (LIF) models were constructed to reproduce the specific firing properties of P-units \citep{Chacron2001, Sinz2020, Barayeu2023}. Its basic components (static nonlinearity, low-pass filtering and spike generation) are equivalent to models of hair cells in auditory systems \citep{Eggermont1983}.  The sole driving input into the P-unit model during baseline, i.e. when no external stimulus was given, is the fish's own EOD, modeled as a cosine wave
 \begin{equation}
     \label{eq:eod}
     y(t) = y_{EOD}(t) = \cos(2\pi f_{EOD} t)