added author summary

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)