no more notes

journalsabbrv.bib

@@ -106,7 +106,7 @@
 @STRING( PLoSBiol = "PLoS Biol" )
 @STRING( PLoSComputBiol = "PLoS Comput Biol" )
 @STRING( PLoSONE = "PLoS ONE" )
-@STRING( PNAS = "P Natl Acad Sci" )
+@STRING( PNAS = "Proc Natl Acad Sci" )
 @STRING( PhilTransRSocLondBBiolSci = "Phil Trans R Soc Lond B Biol Sci" )
 @STRING( PhysicaD = "Physica D" )
 @STRING( PhysiolRep = "Physiol Rep" )

references.bib

@@ -33,14 +33,11 @@
   publisher={Springer}
 }
 
-@Inbook{Baker2019,
+@incollection{Baker2019,
 author="Baker, Clare V. H.",
-editor="Carlson, Bruce A.
-and Sisneros, Joseph A.
-and Popper, Arthur N.
-and Fay, Richard R.",
+editor="Carlson, Bruce A. and Sisneros, Joseph A. and Popper, Arthur N. and Fay, Richard R.",
 title="The Development and Evolution of Lateral Line Electroreceptors: Insights from Comparative Molecular Approaches",
-bookTitle="Electroreception: Fundamental Insights from Comparative Approaches",
+booktitle="Electroreception: Fundamental Insights from Comparative Approaches",
 year="2019",
 publisher="Springer International Publishing",
 address="Cham",
@@ -472,7 +469,7 @@ Beats are slow periodic amplitude modulations resulting from the superposition o
 @Article{Bastian1981a,
   Title                    = {Electrolocation {I}. {H}ow Electroreceptors of \textit{{A}pteronotus albifrons} Code for Moving Objects and Other Electrical Stimuli.},
   Author                   = {Joseph Bastian},
-  Journal                  = {Journal of Comparative Physiology},
+  Journal                  = {J Comp Physiol},
   Year                     = {1981},
   Pages                    = {465--479},
   Volume                   = {144},
@@ -740,7 +737,7 @@ pages = {811--824}
   pages={228--254},
   year={2020},
   publisher={Elsevier, Academic Press},
-  booktitle={The senses: a comprehensive reference}
+  booktitle={The Senses: a Comprehensive Reference}
 }
 
 @article{Benda2021,
@@ -2617,7 +2614,7 @@ article{Egerland2020,
 @Article{Grewe2017,
   Title                    = {Synchronous spikes are necessary but not sufficient for a synchrony code in populations of spiking neurons},
   Author                   = {Grewe, Jan and Kruscha, Alexandra and Lindner, Benjamin and Benda, Jan},
-  Journal                  = {Proceedings of the National Academy of Sciences},
+  Journal                  = {Proc Natl Acad Sci},
   Year                     = {2017},
   Number                   = {10},
   Pages                    = {E1977-E1985},

susceptibility1.tex

@@ -416,8 +416,7 @@ We here analyze nonlinear responses in two types of primary electroreceptor affe
 
 \begin{figure*}[t]
   \includegraphics[width=\columnwidth]{plot_chi2.pdf}
-  \notejg{units on the figure?}
-  \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 $D = 0.001$.}
+  \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 $D = 0.001$. Note that here the leaky integrate-and-fire model is formulated dimensionless, e.g. 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-oder susceptibility shown in figure~\ref{fig:lifresponse}, have been widely and successfully applied to describe and predict neuronal responses and are an invaluable tool to characterize neural systems \citep{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.
 
@@ -427,11 +426,6 @@ Noise in nonlinear systems, however, linearizes the system's response properties
 
 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 p-type electroreceptors of the active system (P-units) are driven by the fish's high-frequency, quasi-sinusoidal electric organ discharges (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,Henninger2020} and weakly nonlinear 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 receiver's EOD. Nonlinear mechanisms in P-units, enable encoding of 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 nonlinearities 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}.
-
-%\notejb{in Voronenko they talk about second-order response functions and not of susceptibilities}\notesr{Ja das ist wahr, aber so wie ich das verstehe sind das synonyme, oder etwa nicht?}
-
-
 \section{Results}
 
 \begin{figure*}[t]
@@ -451,7 +445,9 @@ When stimulating with both foreign signals simultaneously, additional peaks appe
 \subsection{Linear and weakly nonlinear regimes}
 \begin{figure*}[tp]
   \includegraphics[width=\columnwidth]{nonlin_regime.pdf}
-  \caption{\label{fig:nonlin_regime} \notejg{legend: switch sum and difference, fbase to the front} Linear and nonlinear responses of a model P-unit in a three-fish setting in dependence on increasing stimulus amplitudes. The model P-unit was stimulated with two sinewaves of equal amplitude (contrast) at difference frequencies $\bone=30$\,Hz and $\btwo=130$\,Hz relative the receiver's EOD frequency. \btwo{} was set to match the baseline firing rate \fbase{} of the P-unit. The model used has the cell identifier 2013-01-08-aa (table~\ref{modelparams}). \figitem{A--D} Top: the stimulus, an amplitude modulation of the receiver's EOD resulting from the stimulation with the two sine waves. The contrasts of both beats increase from \panel{A} to \panel{D} as indicated. Middle: Spike raster of the model P-unit response. Bottom: power spectrum of the firing rate estimated from the spike raster with a Gaussian kernel ($\sigma=1$\,ms). \figitem{A} At very low stimulus contrasts the response is linear. The only peaks in the response spectrum are at the two stimulating beat frequencies (green and purple marker). \figitem{B} At higher stimulus contrasts, nonlinear responses appear at the sum and the difference of the stimulus frequencies (orange and red marker). \figitem{C} At even stronger stimulation additional peaks appear in the power spectrum. \figitem{D} At a contrast of 10\,\% the response at the sum of the stimulus frequencies almost disappears. \figitem{E} Amplitude of the linear (\bone{}, \btwo{}) and nonlinear (\bdiff{}, \bsum{}) responses of the model P-unit as a function of beat contrast (thick lines). Thin lines indicate the initial linear and quadratic dependence on stimulus amplitude for the linear and nonlinear responses, respectively.}
+  \notejb{I will recalculate the whole figure to show more details in the low contrast regime.}\\
+  \notejg{legend: switch sum and difference, fbase to the front}
+  \caption{\label{fig:nonlin_regime} Linear and nonlinear responses of a model P-unit in a three-fish setting in dependence on increasing stimulus amplitudes. The model P-unit was stimulated with two sinewaves of equal amplitude (contrast) at difference frequencies $\bone=30$\,Hz and $\btwo=130$\,Hz relative the receiver's EOD frequency. \btwo{} was set to match the baseline firing rate \fbase{} of the P-unit. The model used has the cell identifier 2013-01-08-aa (table~\ref{modelparams}). \figitem{A--D} Top: the stimulus, an amplitude modulation of the receiver's EOD resulting from the stimulation with the two sine waves. The contrasts of both beats increase from \panel{A} to \panel{D} as indicated. Middle: Spike raster of the model P-unit response. Bottom: power spectrum of the firing rate estimated from the spike raster with a Gaussian kernel ($\sigma=1$\,ms). \figitem{A} At very low stimulus contrasts the response is linear. The only peaks in the response spectrum are at the two stimulating beat frequencies (green and purple marker). \figitem{B} At higher stimulus contrasts, nonlinear responses appear at the sum and the difference of the stimulus frequencies (orange and red marker). \figitem{C} At even stronger stimulation additional peaks appear in the power spectrum. \figitem{D} At a contrast of 10\,\% the response at the sum of the stimulus frequencies almost disappears. \figitem{E} Amplitude of the linear (\bone{}, \btwo{}) and nonlinear (\bdiff{}, \bsum{}) responses of the model P-unit as a function of beat contrast (thick lines). Thin lines indicate the initial linear and quadratic dependence on stimulus amplitude for the linear and nonlinear responses, respectively.}
 \end{figure*}
 
 The stimuli used in \figref{fig:motivation} had the same not-small amplitude. Whether this stimulus conditions falls into the weakly nonlinear regime as in \citet{Voronenko2017} is not clear. In order to illustrate how the responses to two beat frequencies develop over a range of amplitudes we use a stochastic leaky-integrate-and-fire (LIF) based P-unit model fitted to a specific electrophysiologically measured cell \citep{Barayeu2023}.
@@ -519,7 +515,7 @@ Using the RAM stimulation we found pronounced nonlinear responses in the limit t
 
 \begin{figure*}[t]
   \includegraphics[width=\columnwidth]{model_full.pdf}
-  \caption{\label{fig:model_full} \notejg{Legend without Delta, order of legend, dots higher} Using second-order susceptibility to predict responses to sine-wave stimuli. \figitem[]{A} Absolute value of the second-order susceptibility,  \eqnref{eq:susceptibility}, for both positive and negative frequencies. \susceptf{} was estimated from $N=10^6$ trials of model simulations in the noise-split condition (cell 2013-01-08-aa, see table~\ref{modelparams} for model parameters). White lines indicate zero frequencies. Nonlinear responses at \fsum{} are quantified in the upper right and lower left quadrants.  Nonlinear responses at \fdiff{} are quantified in the upper left and lower right quadrants. The baseline firing rate of this cell was at $\fbase=120$\,Hz. The position of the orange/red letters corresponds to the beat frequencies used for the stimulation with pure sine waves in the subsequent panels and indicates the sum/difference of those beat frequencies.  \figitem{B--E} Black line -- power spectral density of model simulations in response to stimulation with two pure sine waves, \fone{} and \ftwo, in addition to the receiving fish's own EOD (three-fish scenario). The contrast of beat beats is 0.02.  Colored circles highlight the height of selected peaks in the power spectrum. Grey line -- power spectral density of model in the baseline condition. \figitem{B} The sum of the two beat frequencies match \fbase{}. \figitem{C} The difference of \fone{} and \ftwo{} match \fbase{}. \figitem{D} Only the first beat frequency matches \fbase{}. \figitem{E} None of the two beat frequencies matches \fbase{}.}
+  \caption{\label{fig:model_full} Using second-order susceptibility to predict responses to sine-wave stimuli. \figitem[]{A} Absolute value of the second-order susceptibility,  \eqnref{eq:susceptibility}, for both positive and negative frequencies. \susceptf{} was estimated from $N=10^6$ trials of model simulations in the noise-split condition (cell 2013-01-08-aa, see table~\ref{modelparams} for model parameters). White lines indicate zero frequencies. Nonlinear responses at \fsum{} are quantified in the upper right and lower left quadrants.  Nonlinear responses at \fdiff{} are quantified in the upper left and lower right quadrants. The baseline firing rate of this cell was at $\fbase=120$\,Hz. The position of the orange/red letters corresponds to the beat frequencies used for the stimulation with pure sine waves in the subsequent panels and indicates the sum/difference of those beat frequencies.  \figitem{B--E} Black line -- power spectral density of model simulations in response to stimulation with two pure sine waves, \fone{} and \ftwo, in addition to the receiving fish's own EOD (three-fish scenario). The contrast of beat beats is 0.02.  Colored circles highlight the height of selected peaks in the power spectrum. Grey line -- power spectral density of model in the baseline condition. \figitem{B} The sum of the two beat frequencies match \fbase{}. \figitem{C} The difference of \fone{} and \ftwo{} match \fbase{}. \figitem{D} Only the first beat frequency matches \fbase{}. \figitem{E} None of the two beat frequencies matches \fbase{}.}
 \end{figure*}
 
 However, the second-order susceptibility \eqnref{eq:susceptibility} is a spectral measure that is based on Fourier transforms and thus is also defined for negative stimulus frequencies. The full \susceptf{} matrix is symmetric with respect to the origin. In the upper-right and lower-left quadrants of \susceptf{}, stimulus-evoked responses at \fsum{} are shown, whereas in the lower-right and upper-left quadrants nonlinear responses at the difference \fdiff{} are shown (\figref{fig:model_full}). The vertical and horizontal lines at \foneb{} and \ftwob{} are very pronounced in the upper-right quadrant of \subfigrefb{fig:model_full}{A} for nonlinear responses at \fsum{} and extend into the upper-left quadrant (representing \fdiff), where they fade out towards more negative $f_1$ frequencies. The peak in the response power-spectrum at \fdiff{} evoked by pure sine-wave stimulation (\subfigrefb{fig:motivation}{D}) is predicted by the horizontal line in the upper-left quadrant (\subfigrefb{fig:model_full}{A}, \citealp{Schlungbaum2023}). 
@@ -563,9 +559,7 @@ The CV is a proxy for the intrinsic noise in the cells (\citealp{Vilela2009}, no
 \subsection{Linearization by white-noise stimulation}
 Not only the intrinsic noise but also the stimulation with external white-noise linearizes the cells. This applies to both, the stimulation with AMs in P-units (\subfigrefb{fig:dataoverview}{E}) and direct stimulation in ampullary cells (\subfigrefb{fig:dataoverview}{F}). The stronger the effective stimulus, the less pronounced are the peaks in second-order susceptibility (see \subfigref{fig:punit}{E\&F} for a P-unit example and \subfigref{fig:ampullary}{E\&F} for an ampullary cell). This linearizing effect of noise stimuli limits the weakly nonlinear regime to small stimulus amplitudes. At higher stimulus amplitudes, however, other nonlinearities of the system eventually show up in the second-order susceptibility.
 
-In order to characterize weakly nonlinear responses of the cells in the limit to vanishing stimulus amplitudes we utilized the Furutsu-Novikov theorem \citep{Novikov1965, Furutsu1963}. Following \citet{Lindner2022}, a substantial part of the intrinsic noise of a P-unit model \citep{Barayeu2023} is treated as signal. Performing this noise-split trick we can estimate the weakly nonlinear response without the linearizing effect of an additional external white noise stimulus. The model of a low-CV P-unit then shows the full nonlinear structure (\figref{model_and_data}) known from analytical derivations and simulations of basic LIF and theta models driven with pairs of sine-wave stimuli \citep{Voronenko2017,Franzen2023}. 
-
-\notejb{Say that previous work on second-order nonlinearities did not see the weakly nonlinear regime, because of the linearizing effects}
+In order to characterize weakly nonlinear responses of the cells in the limit to vanishing stimulus amplitudes we utilized the Furutsu-Novikov theorem \citep{Novikov1965, Furutsu1963}. Following \citet{Lindner2022}, a substantial part of the intrinsic noise of a P-unit model \citep{Barayeu2023} is treated as signal. Performing this noise-split trick we can estimate the weakly nonlinear response without the linearizing effect of an additional external white noise stimulus. The model of a low-CV P-unit then shows the full nonlinear structure (\figref{model_and_data}) known from analytical derivations and simulations of basic LIF and theta models driven with pairs of sine-wave stimuli \citep{Voronenko2017,Franzen2023}. Previous studies on second-order nonlinearities have not observed the weakly nonlinear regime, probably because of the linearizing effects of strong noise stimuli \citep{Victor1977, Schanze1997, Neiman2011}.
 
 \subsection{Characterizing nonlinear coding 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}.
@@ -619,7 +613,7 @@ For monitoring the EOD without the stimulus, two vertical carbon rods ($11\,\cen
 
 
 \subsection{Stimulation}\label{rammethods}
-ELectric stimuli were isolated from the ground (ISO-02V, npi-electronics, Tamm, Germany) and delivered via two horizontal carbon rods (30 cm length, 8 mm diameter) located $15\,\centi\meter$ laterally to the fish. \notejb{attenuator?}
+Electric stimuli were attenuated (ATN-01M, npi-electronics, Tamm, Germany), isolated from ground (ISO-02V, npi-electronics, Tamm, Germany) and delivered via two horizontal carbon rods (30 cm length, 8 mm diameter) located $15\,\centi\meter$ laterally to the fish.
 
 The fish were stimulated with band-limited white noise stimuli with a cut-off frequency of 150, 300 or 400\,Hz. The stimulus intensity is given as the contrast, i.e. the standard deviation of the white noise stimulus in relation to the fish's EOD amplitude. The contrast varied between 1 and 20\,\%. Only cells with at least 10\,s of white noise stimulation were included for the analysis. When ampullary cells were recorded, the white noise was directly applied as the stimulus. To create random amplitude modulations (RAM) for P-unit recordings, the EOD of the fish was multiplied with the desired random amplitude modulation profile (MXS-01M; npi electronics).