[intro] some modification of the fish part

susceptibility1.tex

@@ -5,15 +5,15 @@
 \author{Alexandra Barayeu\textsuperscript{1},
   Maria Schlungbaum\textsuperscript{2,3},
   Benjamin Lindner\textsuperscript{2,3},
-  Jan Benda\textsuperscript{1, 4}
-  Jan Grewe\textsuperscript{1, $\dagger$}}
+  Jan Grewe\textsuperscript{1},
+  Jan Benda\textsuperscript{1, 4, $\dagger$}}
 
 \date{\normalsize
   \textsuperscript{1} Institute for Neurobiology, Eberhard Karls Universit\"at T\"ubingen, Germany\\
   \textsuperscript{2} Bernstein Center for Computational Neuroscience Berlin, Berlin, Germany\\
   \textsuperscript{3} Department of Physics, Humboldt University Berlin, Berlin, Germany\\
   \textsuperscript{4} Bernstein Center for Computational Neuroscience Tübingen, Germany\\
-  \textsuperscript{$\dagger$} corresponding author: \url{jan.grewe@uni-tuebingen.de}}
+  \textsuperscript{$\dagger$} corresponding author: \url{jan.benda@uni-tuebingen.de}}
 
 \newcommand{\runningtitle}{}
 
@@ -395,7 +395,7 @@
 %\paragraph{Conflict of interest:}The authors declare no conflict of interest.
 %\paragraph{Author Contributions:} All authors designed the study and discussed the results. AB performed the data analyses and modeling, AB and JG drafted the paper, all authors discussed and revised the manuscript.
   
-\paragraph{Keywords:} population coding $|$ conduction delay $|$ heterogeneity $|$ electric fish $|$ mutual information
+\paragraph{Keywords:}  $|$  $|$ heterogeneity $|$ electric fish $|$ mutual information
 
 
 % Please keep the abstract below 300 words
@@ -416,16 +416,17 @@ Neuronal processing is inherently nonlinear --- spiking thresholds or rectificat
   \caption{\label{fig:lifresponse} First- (linear) and second-order response functions of the leaky integrate-and-fire model. \figitem{A} Magnitude of the first-order response function $|\chi_1(f_1)|$, also known as ``gain'' quantifies the response amplitude relative to the stimulus amplitude, both measured at the stimulus frequency. \figitem{B} Magnitude of the second-order response function $|\chi_2(f_1, f_2)|$ quantifies the response at the sum of two stimulus frequencies. 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 summed frequencies. The plots show the analytical solutions from \citep{Lindner2001} and \citep{Voronenko2017} with $\mu = 1.1$ and $D = 0.001$.}
 \end{figure*}
 
-Nonlinear processes are key to neuronal information processing. Decision-making is a fundamentally 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.
+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 have been widely and successfully applied to describe and predict neuronal responses and are an invaluable tool to characterize nervous systems \citep{Borst1999}. Nonlinear mechanisms, on the other hand, are key on different levels; 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.
 
 At the heart of nonlinear system identification is the Volterra series \citep{Rieke1999}. Second-order kernels have been used to predict firing rate responses of catfish retinal ganglion cells \citep{Marmarelis1972}.
 In the frequency domain, second-order kernels are known as second-order response functions or susceptibilities. They quantify the amplitude of the response at the sum and difference of two stimulus frequencies. Adding also third-order kernels, spike trains of spider mechanoreceptors have been predicted from sensory stimuli \citep{French2001}. The nonlinear nature of Y cells in contrast to the more linear responses of X cells in cat retinal ganglion cells has been demonstrated using second-order kernels \citep{Victor1977}. Interactions between different frequencies in the response of neurons in visual cortices of cats and monkeys have been studied using bispectra, the crucial constituent of the second-order susceptibility \citep{Schanze1997}. Locking of chinchilla auditory nerve fibers to pure tone stimuli is captured by second-order kernels \citep{Temchin2005}. In paddlefish ampullary afferents, bursting in response to strong, natural sensory stimuli boosts nonlinear responses in the bicoherence, the bispectrum normalized by stimulus and response spectra \citep{Neiman2011}.
 
 Noise linearizes nonlinear systems \citep{Yu1989, Chialvo1997} and therefore noisy neural systems can be well described by linear response theory in the limit of small stimulus amplitudes \citep{Roddey2000, Doiron2004, Rocha2007, Sharafi2013}. When increasing stimulus amplitude, at first the contribution of the second-order kernel of the Volterra series becomes relevant in addition to the linear one. For these weakly nonlinear responses of leaky-integrate-and-fire (LIF) neurons, an analytical expression for the second-order susceptibility has been derived \citep{Voronenko2017} in addition to its linear response function \citep{Lindner2001}. In the superthreshold regime, where the LIF generates a baseline firing rate in the absence of an external stimulus, the linear response function has a peak at the baseline firing rate and its harmonics (\subfigrefb{fig:lifresponse}{A}) and the second-order susceptibility shows very distinct ridges of elevated nonlinear responses, exactly where two stimulus frequencies equal or add up to the neuron's baseline firing rate (\subfigrefb{fig:lifresponse}{B}). In experimental data, such structures in the second-order susceptibility have not been reported yet.
 
-Here we study weakly nonlinear responses in two electrosensory systems in the wave-type electric fish \textit{Apteronotus leptorhynchus}. These fish generate a quasi-sinusoidal dipolar electric field (electric organ discharge, EOD). In communication contexts \citep{Walz2014, Henninger2018} the EODs of close-by fish superimpose and lead to amplitude modulations (AMs), called beats (two-fish interaction), and modulations of beats, called envelopes (multiple-fish interaction) \citep{Yu2005, Fotowat2013}. Therefore, stimuli with multiple distinct frequencies are part of the everyday life of wave-type electric fish \citep{Benda2020} and interactions of these frequencies in the electrosensory periphery are to be expected. P-type electroreceptor afferents of the tuberous electrosensory system, the P-units, use nonlinearities to extract and encode these AMs in their time-dependent firing rates \citep{Bastian1981a, Walz2014, Middleton2006, Stamper2012Envelope, Savard2011, Barayeu2023}. P-units are heterogeneous in their baseline firing rates as well as in their intrinsic noise levels, as quantified by the coefficient of variation (CV) of the interspike intervals (ISI) \citep{Grewe2017, Hladnik2023}. Low-CV P-units have a less noisy firing pattern that is closer to pacemaker firing, whereas high-CV P-units show a more irregular firing pattern that is closer to a Poisson process. On the other hand, ampullary cells of the passive electrosensory system are homogeneous in their response properties and have very low CVs \citep{Grewe2017}.
+Here we search for such weakly nonlinear responses in electroreceptors of the two electrosensory systems of the wave-type electric fish \textit{Apteronotus leptorhynchus}, i.e. the tuberous (active) and the ampullary (passive) electrosensory system. The electroreceptors of the active system are driven by the fish's self generated electric field generated by regular, quasi-sinusoidal, discharges of their electric organ (electric organ discharge or EOD) and encode disturbances of it \citep{Bastian1981a}. The electroreceptors of the passive system are tuned to lower-frequency exogeneous electric fields such as caused by muscle activity of prey \citep{Kalmijn1974}. As different animals have different EOD-frequencies, being exposed to stimuli of multiple distinct frequencies is part of the animal's everyday life \citep{Benda2020} and weakly non-linear interactions may occur in the electrosensory periphery. In communication contexts \citep{Walz2014, Henninger2018} the EODs of interacting fish superimpose and lead to periodic amplitude modulations (AMs or beats) of the self-generated EOD when two animals interact. Non-linear mechanisms in P-type electroreceptor afferents of the tuberous electrosensory system, the P-units, enabled them to encode these AMs in their time-dependent firing rates \citep{Bastian1981a, Walz2014, Middleton2006, Barayeu2023}. When multiple animals interact, the EOD interferences induce second-order amplitude modulations referred to as envelopes \citep{Yu2005, Fotowat2013, Stamper2012Envelope} and saturation non-linearities allow also for the encoding of these in the electrosensory periphery \citep{Savard2011}. Field observations have shown that courting males were able to react to the extremely weak signals of distant intruding males despite the strong foreground EOD of the nearby female \citep{Henninger2018}. Weakly nonlinear interactions at particular combinations of signals can be of immediate relevance in such settings as they could boost detectability of the faint signals \citep{Schlungbaum2023}.
+
+%The population of P-units is heterogeneous with respect to their baseline firing rates as well as in their intrinsic noise levels, as quantified by the coefficient of variation (CV) of the interspike intervals (ISI) \citep{Grewe2017, Hladnik2023}. Low-CV P-units have a less noisy firing pattern that is closer to pacemaker firing, whereas high-CV P-units show a more irregular firing pattern that is closer to a Poisson process. The population of ampullary cells of the passive electrosensory system, on the other hand, is homogeneous in their response properties and CVs are low \citep{Grewe2017}.
 
-Field observations have shown that courting males were able to react to distant intruder males despite the strong EOD of the nearby female \citep{Henninger2018}. Weakly nonlinear responses are of direct relevance in this setting since they could boost responses to the faint signal of the distant intruder and thus improve detection \citep{Schlungbaum2023}.