update figure

susceptibility1.tex.bak

@@ -531,6 +531,12 @@ Nonlinear processes are key to neuronal information processing. Decision making
 
 \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}.}
 
+\begin{figure*}[t]
+  \includegraphics[width=\columnwidth]{plot_chi2}
+  \caption{\label{fig:plot_chi2} Nonlinearity predicted based on the analytic results in \cite{Voronenko2017}. \figitem{A} Second-order susceptibility. \figitem{B} First-order susceptibility.
+  }
+\end{figure*}
+
 
 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 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}.