updated citation ins intro

susceptibility1.tex

@@ -498,9 +498,9 @@
 \section*{Introduction}
 %with nonlinearities being observed in all sensory modalities
 
-Nonlinear processes are fundamental in neuronal information processing. On the systemic level: deciding to take one or another action is a nonlinear process. On a finer scale, neurons are inherently nonlinear: whether an action potential is elicited depends on the membrane potential to exceed a certain threshold\cite{Adelson1985, Brincat2004, Chacron2000, Chacron2001, Nelson1997, Gussin2007, Middleton2007, Longtin2008}. In conjunction with neuronal noise, nonlinear mechanism facilitate the encoding of weak stimuli via stochastic resonance\cite{Wiesenfeld1995, Stocks2000,Neiman2011fish}. We can find nonlinearities in many sensory systems such as rectification in the transduction machinery of inner hair cells \cite{Peterson2019}, signal rectification in electroreceptor cells \cite{Chacron2000, Chacron2001}, or in complex cells of the visual system \cite{Adelson1985}. In the auditory or the active electric sense, for example, nonlinear processes are needed to extract envelopes, i.e. amplitude modulations of a carrier signal\cite{Joris2004, Barayeu2023} called beats. Beats are common stimuli in different sensory modalities enabling rhythm and pitch perception in human hearing \cite{Roeber1834, Plomp1967, Joris2004, Grahn2012} and providing context for electrocommunication in weakly electric fish \cite{Engler2001, Hupe2008, Henninger2018, Benda2020}.
+Nonlinear processes are fundamental in neuronal information processing. On the systemic level: deciding to take one or another action is a nonlinear process. On a finer scale, neurons are inherently nonlinear: whether an action potential is elicited depends on the membrane potential to exceed a certain threshold\cite{Brincat2004, Chacron2000, Chacron2001, Nelson1997, Gussin2007, Middleton2007, Longtin2008}. In conjunction with neuronal noise, nonlinear mechanism facilitate the encoding of weak stimuli via stochastic resonance\cite{Wiesenfeld1995, Stocks2000,Neiman2011fish}. We can find nonlinearities in many sensory systems such as rectification in the transduction machinery of inner hair cells \cite{Peterson2019}, signal rectification in electroreceptor cells \cite{Chacron2000, Chacron2001}, or in complex cells of the visual system \cite{Adelson1985}. In the auditory or the active electric sense, for example, nonlinear processes are needed to extract envelopes, i.e. amplitude modulations of a carrier signal\cite{Joris2004, Barayeu2023} called beats. Beats are common stimuli in different sensory modalities enabling rhythm and pitch perception in human hearing \cite{Roeber1834, Plomp1967, Joris2004, Grahn2012} and providing context for electrocommunication in weakly electric fish \cite{Engler2001, Hupe2008, Henninger2018, Benda2020}.%Adelson1985, 
 
-While the sensory periphery can often be well described by linear models, this is not true for many upstream neurons. Rather, nonlinear processes are implemented to extract special stimulus features\cite{Gabbiani1996}. In active electrosensation, the self-generated electric field (electric organ discharge, EOD)\cite{Salazar2013} that is quasi sinusoidal in wavetype electric fish and acts as the carrier signal that is amplitude modulated in the context of communication\cite{Benda2013, Fotowat2013, Walz2014, Henninger2018} as well as 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}. P-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 ot 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} encoding the time-course of the AM is linear over a wide range\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}. 
+While the sensory periphery can often be well described by linear models, this is not true for many upstream neurons. Rather, nonlinear processes are implemented to extract special stimulus features\cite{Gabbiani1996}. In active electrosensation, the self-generated electric field (electric organ discharge, EOD)\cite{Salazar2013} that is quasi sinusoidal in wavetype electric fish and acts as the carrier signal that is amplitude modulated in the context of communication\cite{Benda2013, Fotowat2013, Walz2014, Henninger2018} as well as 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}. P-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 ot 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} encoding the time-course of the AM is linear over a wide range\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}. 
 
 \begin{figure*}[!ht]
   \includegraphics[width=\columnwidth]{motivation}
@@ -597,8 +597,10 @@ The population of ampullary cells is generally more homogeneous and have lower C
 
 Nonlinearities are ubiquitous in nervous systems, they are essential to extract certain features such as amplitude modulations of a carrier. Here, we analyzed the nonlinearity in primary electroreceptor afferents of the weakly electric fish \lepto{}. Under natural stimulus conditions, when the electric fields of two or more animals produce interference patterns that contain information of the context of the encounter, such nonlinear encoding is required to enable responding to AMs or also so-called envelopes. The work presented here is inspired by observations of electric fish interactions in the natural habitat. In these observed electrosensory cocktail-parties, a second male is intruding the conversations of a courting dyad. The courting male detects the intruder at a large distance where the foreign electric signal is strongly attenuated due to spatial distance and tiny compared to the signal emitted by the close-by female. Such a three-animal situation is exemplified in \figref{fig:motivation} for a rather specific combination of interfering EODs. There, we observe that the electroreceptor response contains components that result from nonlinear interference (\subfigref{fig:motivation}{D}) that might help to solve this detection task. Based on theoretical work we work out the circumstances under which electroreceptor afferents show such nonlinearities.
 
+
+%\,\panel[iii]{C}
 \subsection*{Theory applies to systems with and without carrier}
-Theoretical work\cite{Voronenko2017} explained analytically the occurrence of nonlinear products when a LIF model neuron is stimulated with pure sine-waves. To investigate whether the same mechanisms occur in electroreceptor afferents which are driven by AMs of a carrier and not by pure sine-waves, we followed the previous approach and quantified the second-order susceptibility from responses to white-noise stimuli\cite{Voronenko2017,Neiman2011fish,Nikias1993}. We expected to see elevated second-order susceptibility where either of the foreign signals matches the baseline firing rate ($f_1=f_{base}$ or $f_2=f_{base}$) or when the sum equals the baseline firing rate of the neuron (\fsumb{}) creating a triangular pattern of elevated \suscept{} e.g.\,\subfigref{model_and_data}{C}{iii}. Indeed, we find traces of the same nonlinearities in the neuronal responses of p-type electroreceptor afferents. The nonlinear pattern observed in the experimental data, however, matches to the expectations only partially and only in a subset of neurons \figsref{fig:cells_suscept} and\,\ref{ampullary}. Nevertheless, the theory holds also for systems that are driven by AMs of a carrier and is thus more widely applicable.
+Theoretical work\cite{Voronenko2017} explained analytically the occurrence of nonlinear products when a LIF model neuron is stimulated with pure sine-waves. To investigate whether the same mechanisms occur in electroreceptor afferents which are driven by AMs of a carrier and not by pure sine-waves, we followed the previous approach and quantified the second-order susceptibility from responses to white-noise stimuli \cite{Voronenko2017,Neiman2011fish,Nikias1993}. We expected to see elevated second-order susceptibility where either of the foreign signals matches the baseline firing rate ($f_1=f_{base}$ or $f_2=f_{base}$) or when the sum equals the baseline firing rate of the neuron (\fsumb{}) creating a triangular pattern of elevated \suscept{} e.g.\,\subfigref{model_and_data}\,\panel[iii]{C}. Indeed, we find traces of the same nonlinearities in the neuronal responses of p-type electroreceptor afferents. The nonlinear pattern observed in the experimental data, however, matches to the expectations only partially and only in a subset of neurons \figsref{fig:cells_suscept} and\,\ref{ampullary}. Nevertheless, the theory holds also for systems that are driven by AMs of a carrier and is thus more widely applicable.
 
 \subsection*{Intrinsic noise limits nonlinear responses}
 Only those P-units that exhibit low coefficients of variation (CV) of the interspike-interval distribution (\figref{fig:cells_suscept}{A}) in their unperturbed baseline response show the expected nonlinerities (\subfigref{fig:data_overview_mod}{A}). Such low-CV cells are rare among the 221 P-units that we used in this study. The afferents of the passive electrosensory system, the ampullary cells, however have generally lower CVs and show a much clearer nonlinearity pattern than the low-CV P-unit exemplified here (compare \figsref{fig:cells_suscept} and \ref{ampullary}). The single ampullary cell featured in \figref{ampullary} is a representative of the majority of cells analyzed here. All ampullary cells have CVs below 0.4 with a median around 0.1 and the observed \nli{}s are 10-fold higher than in P-units.

susceptibility1.tex.bak

@@ -498,9 +498,9 @@
 \section*{Introduction}
 %with nonlinearities being observed in all sensory modalities
 
-Nonlinear processes are fundamental in neuronal information processing. On the systemic level: deciding to take one or another action is a nonlinear process. On a finer scale, neurons are inherently nonlinear: whether an action potential is elicited depends on the membrane potential to exceed a certain threshold\cite{Adelson1985, Brincat2004, Chacron2000, Chacron2001, Nelson1997, Gussin2007, Middleton2007, Longtin2008}. In conjunction with neuronal noise, nonlinear mechanism facilitate the encoding of weak stimuli via stochastic resonance\cite{Wiesenfeld1995, Stocks2000,Neiman2011fish}. We can find nonlinearities in many sensory systems such as rectification in the transduction machinery of inner hair cells \cite{Peterson2019}, signal rectification in electroreceptor cells \cite{Chacron2000, Chacron2001}, or in complex cells of the visual system \cite{Adelson1985}. In the auditory or the active electric sense, for example, nonlinear processes are needed to extract envelopes, i.e. amplitude modulations of a carrier signal\cite{Joris2004, Barayeu2023} called beats. Beats are common stimuli in different sensory modalities enabling rhythm and pitch perception in human hearing \cite{Roeber1834, Plomp1967, Joris2004, Grahn2012} and providing context for electrocommunication in weakly electric fish \cite{Engler2001, Hupe2008, Henninger2018, Benda2020}.
+Nonlinear processes are fundamental in neuronal information processing. On the systemic level: deciding to take one or another action is a nonlinear process. On a finer scale, neurons are inherently nonlinear: whether an action potential is elicited depends on the membrane potential to exceed a certain threshold\cite{Brincat2004, Chacron2000, Chacron2001, Nelson1997, Gussin2007, Middleton2007, Longtin2008}. In conjunction with neuronal noise, nonlinear mechanism facilitate the encoding of weak stimuli via stochastic resonance\cite{Wiesenfeld1995, Stocks2000,Neiman2011fish}. We can find nonlinearities in many sensory systems such as rectification in the transduction machinery of inner hair cells \cite{Peterson2019}, signal rectification in electroreceptor cells \cite{Chacron2000, Chacron2001}, or in complex cells of the visual system \cite{Adelson1985}. In the auditory or the active electric sense, for example, nonlinear processes are needed to extract envelopes, i.e. amplitude modulations of a carrier signal\cite{Joris2004, Barayeu2023} called beats. Beats are common stimuli in different sensory modalities enabling rhythm and pitch perception in human hearing \cite{Roeber1834, Plomp1967, Joris2004, Grahn2012} and providing context for electrocommunication in weakly electric fish \cite{Engler2001, Hupe2008, Henninger2018, Benda2020}.%Adelson1985, 
 
-While the sensory periphery can often be well described by linear models, this is not true for many upstream neurons. Rather, nonlinear processes are implemented to extract special stimulus features\cite{Gabbiani1996}. In active electrosensation, the self-generated electric field (electric organ discharge, EOD)\cite{Salazar2013} that is quasi sinusoidal in wavetype electric fish and acts as the carrier signal that is amplitude modulated in the context of communication\cite{Benda2013, Fotowat2013, Walz2014, Henninger2018} as well as 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}. P-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 ot 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} encoding the time-course of the AM is linear over a wide range\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}. 
+While the sensory periphery can often be well described by linear models, this is not true for many upstream neurons. Rather, nonlinear processes are implemented to extract special stimulus features\cite{Gabbiani1996}. In active electrosensation, the self-generated electric field (electric organ discharge, EOD)\cite{Salazar2013} that is quasi sinusoidal in wavetype electric fish and acts as the carrier signal that is amplitude modulated in the context of communication\cite{Benda2013, Fotowat2013, Walz2014, Henninger2018} as well as 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}. P-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 ot 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} encoding the time-course of the AM is linear over a wide range\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}. 
 
 \begin{figure*}[!ht]
   \includegraphics[width=\columnwidth]{motivation}
@@ -597,8 +597,10 @@ The population of ampullary cells is generally more homogeneous and have lower C
 
 Nonlinearities are ubiquitous in nervous systems, they are essential to extract certain features such as amplitude modulations of a carrier. Here, we analyzed the nonlinearity in primary electroreceptor afferents of the weakly electric fish \lepto{}. Under natural stimulus conditions, when the electric fields of two or more animals produce interference patterns that contain information of the context of the encounter, such nonlinear encoding is required to enable responding to AMs or also so-called envelopes. The work presented here is inspired by observations of electric fish interactions in the natural habitat. In these observed electrosensory cocktail-parties, a second male is intruding the conversations of a courting dyad. The courting male detects the intruder at a large distance where the foreign electric signal is strongly attenuated due to spatial distance and tiny compared to the signal emitted by the close-by female. Such a three-animal situation is exemplified in \figref{fig:motivation} for a rather specific combination of interfering EODs. There, we observe that the electroreceptor response contains components that result from nonlinear interference (\subfigref{fig:motivation}{D}) that might help to solve this detection task. Based on theoretical work we work out the circumstances under which electroreceptor afferents show such nonlinearities.
 
+
+%\,\panel[iii]{C}
 \subsection*{Theory applies to systems with and without carrier}
-Theoretical work\cite{Voronenko2017} explained analytically the occurrence of nonlinear products when a LIF model neuron is stimulated with pure sine-waves. To investigate whether the same mechanisms occur in electroreceptor afferents which are driven by AMs of a carrier and not by pure sine-waves, we followed the previous approach and quantified the second-order susceptibility from responses to white-noise stimuli\cite{Voronenko2017,Neiman2011fish,Nikias1993}. We expected to see elevated second-order susceptibility where either of the foreign signals matches the baseline firing rate ($f_1=f_{base}$ or $f_2=f_{base}$) or when the sum equals the baseline firing rate of the neuron (\fsumb{}) creating a triangular pattern of elevated \suscept{} e.g.\,\subfigref{model_and_data}{C}{iii}. Indeed, we find traces of the same nonlinearities in the neuronal responses of p-type electroreceptor afferents. The nonlinear pattern observed in the experimental data, however, matches to the expectations only partially and only in a subset of neurons \figsref{fig:cells_suscept} and\,\ref{ampullary}. Nevertheless, the theory holds also for systems that are driven by AMs of a carrier and is thus more widely applicable.
+Theoretical work\cite{Voronenko2017} explained analytically the occurrence of nonlinear products when a LIF model neuron is stimulated with pure sine-waves. To investigate whether the same mechanisms occur in electroreceptor afferents which are driven by AMs of a carrier and not by pure sine-waves, we followed the previous approach and quantified the second-order susceptibility from responses to white-noise stimuli \cite{Voronenko2017,Neiman2011fish,Nikias1993}. We expected to see elevated second-order susceptibility where either of the foreign signals matches the baseline firing rate ($f_1=f_{base}$ or $f_2=f_{base}$) or when the sum equals the baseline firing rate of the neuron (\fsumb{}) creating a triangular pattern of elevated \suscept{} e.g.\,\subfigref{model_and_data}\,\panel[iii]{C}. Indeed, we find traces of the same nonlinearities in the neuronal responses of p-type electroreceptor afferents. The nonlinear pattern observed in the experimental data, however, matches to the expectations only partially and only in a subset of neurons \figsref{fig:cells_suscept} and\,\ref{ampullary}. Nevertheless, the theory holds also for systems that are driven by AMs of a carrier and is thus more widely applicable.
 
 \subsection*{Intrinsic noise limits nonlinear responses}
 Only those P-units that exhibit low coefficients of variation (CV) of the interspike-interval distribution (\figref{fig:cells_suscept}{A}) in their unperturbed baseline response show the expected nonlinerities (\subfigref{fig:data_overview_mod}{A}). Such low-CV cells are rare among the 221 P-units that we used in this study. The afferents of the passive electrosensory system, the ampullary cells, however have generally lower CVs and show a much clearer nonlinearity pattern than the low-CV P-unit exemplified here (compare \figsref{fig:cells_suscept} and \ref{ampullary}). The single ampullary cell featured in \figref{ampullary} is a representative of the majority of cells analyzed here. All ampullary cells have CVs below 0.4 with a median around 0.1 and the observed \nli{}s are 10-fold higher than in P-units.