updating discussion
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@ -6,3 +6,4 @@ __pychache__
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*.out
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*.log
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*synctex*
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susceptibility1.pdf
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susceptibility1.pdf
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susceptibility1.pdf
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@ -576,18 +576,18 @@ The P-unit population has higher baseline CVs and lower \nli{} values (\subfigre
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\section{Discussion}
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In this work, the second-order susceptibility in spiking responses of P-units was characterized for the scenario where at least three fish are present. A low-CV subpopulation of P-units in \lepto{} was demonstrated to have increased nonlinearity values when \fone{}, \ftwo{}, \fsum{} or \fdiff{} were equal to the mean baseline firing rate \fbase{}. Ampullary cells, with even lower CVs, exhibited even stronger nonlinearity. These nonlinear structures in low-CV cells confirmed the predictions from the nonlinear theory described in \citet{Voronenko2017}. High-CV P-units did not exhibit any nonlinear interactions. The nonlinearities appearing for the noise frequencies \fone{}, \ftwo{} were confirmed as a proxy for nonlinearities that might arise in response to the beat frequencies \bone{}, \btwo{}, when at least three fish are present.
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In this work, the second-order susceptibility in spiking responses of P-units was characterized for the scenario where at least three fish are present. A low-CV subpopulation of P-units in \lepto{} was demonstrated to have increased nonlinearity values when \fone{}, \ftwo{}, \fsum{} or \fdiff{} were equal to the mean baseline firing rate \fbase{}. Ampullary cells, with even lower CVs, exhibited even stronger nonlinearity. These nonlinear structures in low-CV cells confirmed the predictions from the weakly nonlinear theory \citealp{Voronenko2017}. High-CV P-units did not exhibit any nonlinear interactions. The nonlinearities appearing for the noise frequencies \fone{}, \ftwo{} were confirmed as a proxy for nonlinearities that might arise in response to the beat frequencies \bone{}, \btwo{}, when at least three fish are present.
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%\fsum{} or \fdiff{} were equal to \fbase{} \bsum{} or \bdiff{},
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\subsection{Methodological implications}%implying that the
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\subsubsection{Full nonlinear structure present in LIF models with and without carrier}%implying that the
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%\subsection{Methodological implications}%implying that the
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\subsection{Full nonlinear structure present in LIF models with and without carrier}%implying that the
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%occurring nonlinearities could only be explained based on a (\foneb{}, \ftwob{}, \fsumb{}, \fdiffb{},
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In this chapter, it was demonstrated that in small homogeneous populations of electrophysiological recorded cells, a diagonal structure appeared when the sum of the input frequencies was equal to \fbase{} in the second-order susceptibility (\figrefb{cells_suscept}, \figrefb{ampullary}). The size of a population is limited by the possible recording duration in an experiment, but there are no such limitations in P-unit models, where an extended nonlinear structure with diagonals, vertical and horizontal lines at \fsumb{}, \fdiffb{}, \foneb{} or \ftwob{} appeared with increased stimulus realizations (\figrefb{model_full}). This nonlinear structure is in line with the one predicted in the framework of the nonlinear response theory, which was developed based on LIF models without carrier (\figrefb{plt_RAM_didactic2}, \citealp{Voronenko2017}) and was here confirmed to be valid in models with carrier (\figrefb{model_and_data}).
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In this chapter, it was demonstrated that in small homogeneous populations of electrophysiological recorded cells, a diagonal structure appeared when the sum of the input frequencies was equal to \fbase{} in the second-order susceptibility (\figrefb{cells_suscept}, \figrefb{ampullary}). The size of a population is limited by the possible recording duration in an experiment, but there are no such limitations in P-unit models, where an extended nonlinear structure with diagonals, vertical and horizontal lines at \fsumb{}, \fdiffb{}, \foneb{} or \ftwob{} appeared with increased stimulus realizations (\figrefb{model_full}). This nonlinear structure is in line with the one predicted in the framework of the nonlinear response theory, which was developed based on LIF models without carrier \citealp{Voronenko2017} and was here confirmed to be valid in models with carrier (\figrefb{model_and_data}).
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\subsubsection{Noise stimulation approximates the nonlinearity in a three-fish setting}%implying that the
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\subsection{Noise stimulation approximates the nonlinearity in a three-fish setting}%implying that the
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In this chapter, the nonlinearity of P-units and ampullary cells was retrieved based on white noise stimulation, where all behaviorally relevant frequencies were simultaneously present in the stimulus. The nonlinearity of LIF models (\citealp{Egerland2020}) and of ampullary cells in paddlefish \citealp{Neiman2011fish} has been previously accessed with bandpass limited white noise.
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Here it was demonstrated that the second-order susceptibility for the two RAM noise input frequencies \fone{} and \ftwo{} can approximate a three-fish setting, where the driving force for the P-unit are two beats with frequencies \bone{} and \btwo{}. This was confirmed by a low-CV P-unit, where nonlinearities in the P-unit response occurred at the sum and difference of the beat frequencies for pure sine-wave stimulation (\figrefb{motivation}). In this P-unit the nonlinearity appeared in a three-wave setting with \bone{} being close to \fbase{}, corresponding to a frequency combination on the vertical line at \foneb{} in the second-order susceptibility (\subfigrefb{model_full}{B}). This implies that even if only the diagonal structure can be accessed with noise stimulation in the second-order susceptibility in an experiment (\subfigrefb{model_full}{A}), it can be taken as an indicator that the whole nonlinear structure should be present during pure sine-wave stimulation in the same cell. With this RAM stimulation was demonstrated to be an effective method to scan the three-fish or two-beat plane and estimate the whole theoretically predicted nonlinear structure in experimentally recorded cells.
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@ -600,14 +600,16 @@ Here it was demonstrated that the second-order susceptibility for the two RAM no
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%. It could be shown that if the nonlinearity at \fsumb{} could be found with a low stimulus repeat number in electrophysiologically recorded P-units,
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%\subsection{Nonlinearity and CV}%
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\subsection{Nonlinearity and CV}%
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In this chapter, the CV has been identified as an important factor influencing nonlinearity in the spiking response, with strong nonlinearity in low-CV cells (\subfigrefb{data_overview_mod}{A--B}). These findings of strong nonlinearity in low-CV cells are in line with previous literature, where it has been proposed that noise linearizes the system \citealp{Roddey2000, Chialvo1997, Voronenko2017}. More noise has been demonstrated to increase the CV and reduce nonlinear phase-locking in vestibular afferents \citealp{Schneider2011}. Reduced noise of low-CV cells has been associated with stronger nonlinearity in pyramidal cells of the ELL \citealp{Chacron2006}.
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\subsection{Ampullary cells}%
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In this chapter strong nonlinear interactions were found in a subpopulation of low-CV P-units and low-CV ampullary cells (\subfigrefb{data_overview_mod}{A, B}). Ampullary cells have lower CVs than P-units, do not phase-lock to the EOD, and have unimodal ISI distributions. With this, ampullary cells have very similar properties as the LIF models, where the theory of weakly nonlinear interactions was developed on \citealp{Voronenko2017}. Almost all here investigated ampullary cells had pronounced nonlinearity bands in the second-order susceptibility for small stimulus amplitudes (\figrefb{ampullary}). With this ampullary cells are an experimentally recorded cell population close to the LIF models that fully meets the theoretical predictions of low-CV cells having very strong nonlinearities \citealp{Voronenko2017}. Ampullary cells encode low-frequency changes in the environment e.g. prey movement \citealp{Engelmann2010, Neiman2011fish}. The activity of a prey swarm of \textit{Daphnia} strongly resembles Gaussian white noise \citealp{Neiman2011fish}, as it has been used in this chapter. How such nonlinear effects in ampullary cells might influence prey detection could be addressed in further studies.
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%Nonlinear effects only for specific frequency combinations
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\subsubsection{The readout from P-units in pyramidal cells is heterogeneous}
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\subsection{The readout from P-units in pyramidal cells is heterogeneous}
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%is required to cover all female-intruder combinations} %gain_via_mean das ist der Frequenz plane faktor
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%associated with nonlinear effects as \fdiffb{} and \fsumb{} were with the same firing rate low-CV P-units
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@ -639,7 +641,9 @@ The RAM stimulus used in this chapter is an approximation of the three-fish scen
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When using noise stimulation strong nonlinearity was demonstrated to appear for small noise stimuli but to decrease for stronger noise stimuli (\figrefb{cells_suscept}). A white noise stimulus is a proxy of many fish being present simultaneously. When the noise amplitude is small, those fish are distant and the nonlinearity is maintained. When the stimulus amplitude increases, many fish close to the receiver are mimicked and a decrease of nonlinear effects can be observed. These findings imply that the nonlinear effects arising in the presence of three fish decline the more fish join. \eigen{} can usually be found in groups of three to four fish \citealp{Tan2005} and \lepto{} in groups of two \citealp{Stamper2010}. Thus the here described second-order susceptibility might still be behaviorally relevant for both species. The decline of nonlinear effects when several fish are present might be adaptive for the receiver, reducing the number of frequencies represented in its primary sensory afferents to a minimum. Such representation would still leave room to create nonlinear effects at later processing steps in higher-order neurons. How nonlinear effects might influence the three-fish setting known as the electrosensory cocktail party (see section \ref{cocktail party}, \citealp{Henninger2018}) will be addressed in the next chapter.
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\subsection{Conclusion} In this chapter, noise stimulation was confirmed as a method to access the second-order susceptibility in P-unit responses when at least three fish or two beats were present. Nonlinear effects were identified in experimentally recorded non-bursty low-CV cells and bursty high-CV P-units. It was demonstrated that the theory of weakly nonlinear interactions \citealp{Voronenko2017} is valid in P-units where the stimulus is an amplitude modulation of the carrier and not the whole signal. Nonlinearity was found to be decreased the more fish were present, thus keeping the signal representation in the firing rate simple.
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\subsection{Conclusion} In this work, noise stimulation was confirmed as a method to access the second-order susceptibility in P-unit responses when at least three fish or two beats were present. It was demonstrated that the theory of weakly nonlinear interactions \citealp{Voronenko2017} is valid in P-units where the stimulus is an amplitude modulation of the carrier and not the whole signal. Nonlinear effects were identified in experimentally recorded low-CV cells primary sensory afferents, the P-units and the ampullary cells. P-units share several features with mammalian
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auditory nerve fibers and such nonlinear effects might also be expected in the auditory system during the encoding of amplitude modulations \citet{Joris2004}.
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\section{Methods}
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@ -452,13 +452,13 @@ When the receiver fish with EOD frequency \feod{} is alone, a peak at \fbase{} i
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If the receiver fish \feod{} in a three fish setting is always the same, then the varied parameters are the EOD frequencies of the encountered fish, resulting in a two-dimensional stimulus space spanned by the two beat frequencies. It is challenging to access the whole stimulus space in an electrophysiological experiment. Instead, white noise stimulation, where all behaviorally relevant frequencies are present at the same time, has been used to access second-order susceptibility \citealp{Neiman2011fish, Nikias1993}.
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In the communication context of weakly electric fish white noise stimuli are implemented as random amplitude modulations (RAM) of the EOD and the white noise and whether it is possible to access the second-order susceptibility of P-units will be addressed in this work. %the presented method is still and are not a direct input to the neuron
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In the communication context of weakly electric fish white noise stimuli are implemented as random amplitude modulations (RAM) of the EOD and the white noise and whether it is possible to access the second-order susceptibility of P-units will be addressed in this work. It will be demonstrated that some low-CV P-units exhibit nonlinearities in relation to \fbase{}, as predicted by the theory of weakly nonlinear interactions \citealp{Voronenko2017}. %the presented method is still and are not a direct input to the neuron
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%is not the direct input to the neurons. with RAM stimuli
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It will be demonstrated that some low-CV P-units exhibit nonlinearities in relation to \fbase{}, as predicted by the theory of weakly nonlinear interactions \citealp{Voronenko2017}.
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%
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%In this work, the second-order susceptibility in the spiking responses of P-units will be accessed with white noise stimulation.
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%White noise stimulation will be confirmed as a method to access the second-order susceptibility in P-units.
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%The influence of the baseline firing properties, such as the CV, on nonlinear interactions will be investigated.
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@ -576,7 +576,7 @@ The P-unit population has higher baseline CVs and lower \nli{} values (\subfigre
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\section{Discussion}
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In this work, the second-order susceptibility in spiking responses of P-units was characterized for the scenario where at least three fish are present. A low-CV subpopulation of P-units in \lepto{} was demonstrated to have increased nonlinearity values when \fone{}, \ftwo{}, \fsum{} or \fdiff{} were equal to the mean baseline firing rate \fbase{}. Ampullary cells, with even lower CVs, exhibited even stronger nonlinearity. These nonlinear structures in low-CV cells confirmed the predictions from the nonlinear theory described in \citet{Voronenko2017}. High-CV P-units did not exhibit any nonlinear interactions. The nonlinearities appearing for the noise frequencies \fone{}, \ftwo{} were confirmed as a proxy for nonlinearities that might arise in response to the beat frequencies \bone{}, \btwo{}, when at least three fish are present.
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In this work, the second-order susceptibility in spiking responses of P-units was characterized for the scenario where at least three fish are present. A low-CV subpopulation of P-units in \lepto{} was demonstrated to have increased nonlinearity values when \fone{}, \ftwo{}, \fsum{} or \fdiff{} were equal to the mean baseline firing rate \fbase{}. Ampullary cells, with even lower CVs, exhibited even stronger nonlinearity. These nonlinear structures in low-CV cells confirmed the predictions from the weakly nonlinear theory \citet{Voronenko2017}. High-CV P-units did not exhibit any nonlinear interactions. The nonlinearities appearing for the noise frequencies \fone{}, \ftwo{} were confirmed as a proxy for nonlinearities that might arise in response to the beat frequencies \bone{}, \btwo{}, when at least three fish are present.
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@ -601,13 +601,15 @@ Here it was demonstrated that the second-order susceptibility for the two RAM no
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\subsection{Nonlinearity and CV}%
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\subsubsection{Nonlinearity and CV}%
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In this chapter, the CV has been identified as an important factor influencing nonlinearity in the spiking response, with strong nonlinearity in low-CV cells (\subfigrefb{data_overview_mod}{A--B}). These findings of strong nonlinearity in low-CV cells are in line with previous literature, where it has been proposed that noise linearizes the system \citealp{Roddey2000, Chialvo1997, Voronenko2017}. More noise has been demonstrated to increase the CV and reduce nonlinear phase-locking in vestibular afferents \citealp{Schneider2011}. Reduced noise of low-CV cells has been associated with stronger nonlinearity in pyramidal cells of the ELL \citealp{Chacron2006}.
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\subsubsection{Ampullary cells}%
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In this chapter strong nonlinear interactions were found in a subpopulation of low-CV P-units and low-CV ampullary cells (\subfigrefb{data_overview_mod}{A, B}). Ampullary cells have lower CVs than P-units, do not phase-lock to the EOD, and have unimodal ISI distributions. With this, ampullary cells have very similar properties as the LIF models, where the theory of weakly nonlinear interactions was developed on \citealp{Voronenko2017}. Almost all here investigated ampullary cells had pronounced nonlinearity bands in the second-order susceptibility for small stimulus amplitudes (\figrefb{ampullary}). With this ampullary cells are an experimentally recorded cell population close to the LIF models that fully meets the theoretical predictions of low-CV cells having very strong nonlinearities \citealp{Voronenko2017}. Ampullary cells encode low-frequency changes in the environment e.g. prey movement \citealp{Engelmann2010, Neiman2011fish}. The activity of a prey swarm of \textit{Daphnia} strongly resembles Gaussian white noise \citealp{Neiman2011fish}, as it has been used in this chapter. How such nonlinear effects in ampullary cells might influence prey detection could be addressed in further studies.
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%Nonlinear effects only for specific frequency combinations
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\subsubsection{The readout from P-units in pyramidal cells is heterogeneous}
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\subsection{The readout from P-units in pyramidal cells is heterogeneous}
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%is required to cover all female-intruder combinations} %gain_via_mean das ist der Frequenz plane faktor
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%associated with nonlinear effects as \fdiffb{} and \fsumb{} were with the same firing rate low-CV P-units
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@ -639,7 +641,9 @@ The RAM stimulus used in this chapter is an approximation of the three-fish scen
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When using noise stimulation strong nonlinearity was demonstrated to appear for small noise stimuli but to decrease for stronger noise stimuli (\figrefb{cells_suscept}). A white noise stimulus is a proxy of many fish being present simultaneously. When the noise amplitude is small, those fish are distant and the nonlinearity is maintained. When the stimulus amplitude increases, many fish close to the receiver are mimicked and a decrease of nonlinear effects can be observed. These findings imply that the nonlinear effects arising in the presence of three fish decline the more fish join. \eigen{} can usually be found in groups of three to four fish \citealp{Tan2005} and \lepto{} in groups of two \citealp{Stamper2010}. Thus the here described second-order susceptibility might still be behaviorally relevant for both species. The decline of nonlinear effects when several fish are present might be adaptive for the receiver, reducing the number of frequencies represented in its primary sensory afferents to a minimum. Such representation would still leave room to create nonlinear effects at later processing steps in higher-order neurons. How nonlinear effects might influence the three-fish setting known as the electrosensory cocktail party (see section \ref{cocktail party}, \citealp{Henninger2018}) will be addressed in the next chapter.
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\subsection{Conclusion} In this chapter, noise stimulation was confirmed as a method to access the second-order susceptibility in P-unit responses when at least three fish or two beats were present. Nonlinear effects were identified in experimentally recorded non-bursty low-CV cells and bursty high-CV P-units. It was demonstrated that the theory of weakly nonlinear interactions \citealp{Voronenko2017} is valid in P-units where the stimulus is an amplitude modulation of the carrier and not the whole signal. Nonlinearity was found to be decreased the more fish were present, thus keeping the signal representation in the firing rate simple.
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\subsection{Conclusion} In this work, noise stimulation was confirmed as a method to access the second-order susceptibility in P-unit responses when at least three fish or two beats were present. It was demonstrated that the theory of weakly nonlinear interactions \citealp{Voronenko2017} is valid in P-units where the stimulus is an amplitude modulation of the carrier and not the whole signal. Nonlinear effects were identified in experimentally recorded low-CV cells primary sensory afferents, the P-units and the ampullary cells. P-units share several features with mammalian
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auditory nerve fibers and such nonlinear effects might also be expected in the auditory system during the encoding of amplitude modulations \citet{Joris2004}.
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\section{Methods}
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