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@ -534,19 +534,17 @@ The smaller \fdiff{} power spectral peak observed during pure sine-wave stimulat
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\end{figure*}
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\subsection{Low CVs and weak stimuli are associated with strong nonlinearity}
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The non-linear effects shown for single cell examples above are supported by the analysis of the pool of 222 P-units and 45 ampullary afferents recorded in 71 specimen. To compare across cells we expressed the second-order susceptibility as the nonlinearity index \nli{}, see \eqnref{eq:nli_equation}. The \nli{} characterizes the peakedness of the projections onto the diagonal of the \suscept{} matrices (e.g. \subfigref{cells_suscept}{G} at \fbase{}. It assumes high values when the \fbase{} peak in the projected diagonal is pronounced relative to the median of the diagonal and is small when there is no distinct peak. The P-unit population has higher \nli{} values up to three and depends weakly on the the CV of the ISI distribution under baseline condition. Cells with lower baseline CVs have the tendency to exhibit a stronger nonlinearity than those that have high CVs (Pearson's r\,=\,$-0.16$, p\,<\,0.01). The two example P-units shown before (\figrefb{cells_suscept} and \figrefb{cells_suscept_high_CV}) are highlighted with dark circles in \subfigrefb{data_overview_mod}{B, D}. The stimulus strength plays an important role. Several of the recorded neurons contribute with two dots to the data as their responses to the same RAM stimulus but with different contrasts were recorded. Higher stimulus contrasts lead to a stronger drive and thus stronger response modulations (see color code bar in \subfigref{data_overview_mod}{A}, see methods). Since P-units are heterogeneous in their susceptibility to the stimulus\citealp{Grewe2017}, their responses to the same stimulus intensity vary a lot. When replotting the \nli{} against the response modulation (\subfigrefb{data_overview_mod}{C}) a negative correlation is observed (Pearson's r\,=\,$-0.35$, p\,<\,0.001 \notesr{das minus wird durch den Mathmodus länger, das will Jan immer so haben}) showing that cells that are strongly driven by the stimulus show less nonlinearity while those that are only weakly driven show higher nonlinearities. Whether or not a cell responds nonlinearly thus depends on both, the baseline CV (i.e. the internal noise) and the response strength.
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The non-linear effects shown for single cell examples above are supported by the analysis of the pool of 222 P-units and 45 ampullary afferents recorded in 71 specimen. To compare across cells we expressed the second-order susceptibility as the nonlinearity index \nli{}, see \eqnref{eq:nli_equation}. The \nli{} characterizes the peakedness of the projections onto the diagonal of the \suscept{} matrices (e.g. \subfigref{cells_suscept}{G} at \fbase{}. It assumes high values when the \fbase{} peak in the projected diagonal is pronounced relative to the median of the diagonal and is small when there is no distinct peak. The P-unit population has higher \nli{} values up to three and depends weakly on the the CV of the ISI distribution under baseline condition. Cells with lower baseline CVs have the tendency to exhibit a stronger nonlinearity than those that have high CVs (Pearson's r\,=\,$-0.16$, p\,<\,0.01). The two example P-units shown before (\figrefb{cells_suscept} and \figrefb{cells_suscept_high_CV}) are highlighted with dark circles in \subfigrefb{data_overview_mod}{B, D}. The stimulus strength plays an important role. Several of the recorded neurons contribute with two dots to the data as their responses to the same RAM stimulus but with different contrasts were recorded. Higher stimulus contrasts lead to a stronger drive and thus stronger response modulations (see color code bar in \subfigref{data_overview_mod}{A}, see methods). Since P-units are heterogeneous in their susceptibility to the stimulus\citealp{Grewe2017}, their responses to the same stimulus intensity vary a lot. When replotting the \nli{} against the response modulation (\subfigrefb{data_overview_mod}{C}) a negative correlation is observed (Pearson's r\,=\,$-0.35$, p\,<\,0.001) showing that cells that are strongly driven by the stimulus show less nonlinearity while those that are only weakly driven show higher nonlinearities. Whether or not a cell responds nonlinearly thus depends on both, the baseline CV (i.e. the internal noise) and the response strength.
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%In a P-unit population where each cell is represented not by several contrasts but by the lowest recorded contrast, \nli{} significantly correlates with the CV during baseline ($r=-0.17$, $p=0.01$), the response modulation ($r=-0.35$, $p<0.001$) and \fbase{} ($r=-0.32$, $p<0.001$).%, $\n{}=221$*, $\n{}=221$******, $\n{}=221$
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The population of ampullary cells is generally more homogeneous and have lower CVs than the P-units and show much higher \nli{} values (factor of 10). Overall, there is a negative correlation with the baseline CV (Pearson's r=-0.35, p < 0.01). The example cell shown above (\figref{ampullary}) was recorded at two different stimulus intensities and the \nli{}s are highlighted with black circles. Again, we see that cells that are strongly driven by the stimulus cluster at the bottom of the distribution and have \nli{} values close to zero. This is confirmed when the data is replotted against the response modulation, those cells that are strongly driven by the stimulus show weak nonlinearities while weakly driven neurons exhibit high values (Pearson's r=-0.59, p < 0.0001).
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The population of ampullary cells is generally more homogeneous and have lower CVs than the P-units and show much higher \nli{} values (factor of 10). Overall, there is a negative correlation with the baseline CV (Pearson's $r=-0.35$, $p < 0.01$). The example cell shown above (\figref{ampullary}) was recorded at two different stimulus intensities and the \nli{}s are highlighted with black circles. Again, we see that cells that are strongly driven by the stimulus cluster at the bottom of the distribution and have \nli{} values close to zero. This is confirmed when the data is replotted against the response modulation, those cells that are strongly driven by the stimulus show weak nonlinearities while weakly driven neurons exhibit high values (Pearson's $r=-0.59$, $p < 0.0001$).
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\section{Discussion}
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\notejg{dumped here since not strictly result...}
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These nonlinearity bands are more pronounced in ampullary cells than they were in P-units (compare \figrefb{ampullary} and \figrefb{cells_suscept}). Ampullary cells with their unimodal ISI distribution are closer than P-units to the LIF models without EOD carrier, where the predictions about the second-order susceptibility structure have mainly been elaborated on \citealp{Voronenko2017}. All here analyzed ampullary cells had CVs lower than 0.3 and exhibited strong nonlinear effects in accordance with the theoretical predictions \citealp{Voronenko2017}.
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%and here this could be confirmed experimentally.
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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 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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@ -563,6 +561,7 @@ In this chapter, the nonlinearity of P-units and ampullary cells was retrieved b
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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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%the power spectrum of the firing rate
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%The existence of the extended nonlinearity structure (\figrefb{model_full}) was
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%\bsum{} and \bdiff{}
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@ -579,6 +578,12 @@ In this chapter, the CV has been identified as an important factor influencing n
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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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%\notejg{dumped here since not strictly result...}
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%These nonlinearity bands are more pronounced in ampullary cells than they were in P-units (compare \figrefb{ampullary} and \figrefb{cells_suscept}). Ampullary cells with their unimodal ISI distribution are closer than P-units to the LIF models without EOD carrier, where the predictions about the second-order susceptibility structure have mainly been elaborated on \citealp{Voronenko2017}. All here analyzed ampullary cells had CVs lower than 0.3 and exhibited strong nonlinear effects in accordance with the theoretical predictions \citealp{Voronenko2017}.
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%and here this could be confirmed experimentally.
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%Nonlinear effects only for specific frequency combinations
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\subsection{Heterogeneity of P-units might influence nonlinearity}
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@ -586,14 +591,15 @@ In this chapter strong nonlinear interactions were found in a subpopulation of l
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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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%and \figrefb{ROC_with_nonlin}
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P-units are very heterogeneous in their baseline firing properties considering their baseline firing rates (50--450\,Hz, \citealp{Grewe2017, Hladnik2023}), their CV and also their bursting. How this heterogeneity influences nonlinear effects when an population of P-units is integrated by the next connected pyramidal cells in the electrosensory lateral line lobe (ELL) should be addressed in further studies.
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P-units are very heterogeneous in their baseline firing properties considering their baseline firing rates (50--450\,Hz, \citealp{Grewe2017, Hladnik2023}), their CV and also their bursting, the repeating firing of spikes after an EOD cycle \citealp{Chacron2004}. How this heterogeneity influences nonlinear effects when an population of P-units is integrated by the next connected pyramidal cells in the electrosensory lateral line lobe (ELL) should be addressed in further studies.
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The nonlinearity in this work were found in low-CV P-units. For this nonlinear effects to sustain on a population level a selective readout from a homogeneous population of low-CV cells might be required. Since a receptive field is defined as a patch of adjacent receptors on the fish surface \citealp{Maler2009a} and P-units have heterogeneous baseline properties that do not depend on the location of the receptor on the fish body \citealp{Hladnik2023} as a result each pyramidal cell in the ELL will integrate over a heterogeneous and not homogeneous P-unit population. The center-surround receptive field organization of superficial and intermediate layer pyramidal cells in addition increases this heterogeneity of the integrated population. Heterogeneous populations with different baseline properties can be more advantageous for the encoding compared to homogeneous populations, as has been shown in P-units and models \citealp{Beiran2018, Hladnik2023}.
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The nonlinearity in this work were found in low-CV P-units. A selective readout from a homogeneous population of low-CV cells might be required for this nonlinear effects to sustain on a population level. Since a receptive field is defined as a patch of adjacent receptors on the fish surface \citealp{Maler2009a} and P-units have heterogeneous baseline properties that do not depend on the location of the receptor on the fish body \citealp{Hladnik2023} as a result each pyramidal cell in the ELL will integrate over a heterogeneous and not homogeneous P-unit population. The center-surround receptive field organization of superficial and intermediate layer pyramidal cells in addition increases this heterogeneity of the integrated population. Heterogeneous populations with different baseline properties can be more advantageous for the encoding compared to homogeneous populations, as has been shown in P-units and models \citealp{Beiran2018, Hladnik2023}.
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%\subsubsection{A heterogeneous readout is required to cover all female-intruder combinations}
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A heterogeneous readout might be not only physiologically plausible but also required since nonlinear effects depending on cell property \fbase{} and not on stimulus properties might be not behaviorally relevant. Nonlinear effects might facilitate the encoding of faint signals during a three fish setting, the electrosensory cocktail party. The EOD frequencies of the encoutered three fish would be drawn from the EOD frequency distribution of these fish and a stable faint signal detection would require a response irrespective of the individual EOD frequencies. Weather integrating from a heterogeneous population with different \fbasesolid{} (50--450\,Hz, \citealp{Grewe2017, Hladnik2023}) would cover the behaviorally relevant range in the electrosensory cocktail party should be addressed in further studies.
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Nonlinear effects might facilitate the encoding of faint signals during a three fish setting, the electrosensory cocktail party. The EOD frequencies of the encountered three fish would be drawn from the EOD frequency distribution and a stable faint signal detection would require a response irrespective of the individual EOD frequencies. Here nonlinear effects, that might influence the detection of faint signals, was found only at specific frequencies in relation to the mean baseline firing rate \fbase{}. Weather integrating from a heterogeneous population with different \fbasesolid{} (50--450\,Hz, \citealp{Grewe2017, Hladnik2023}) would cover the behaviorally relevant range in the electrosensory cocktail party could be addressed in further studies.
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%Thus, a heterogeneous readout might be not only physiologically plausible but also required since nonlinear effects depending on cell property, as the mean baseline firing rate \fbase{}, and not on stimulus properties, might be not behaviorally relevant.
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%In this work, nonlinear effects were always found only for specific frequencies in relation to \fbase{}, corresponding to findings from previous literature \citealp{Voronenko2017}.
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%Only a heterogeneous population could cover the whole stimulus space required during the electrosensory cocktail party.
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%If pyramidal cells would integrate only from P-units with the same mean baseline firing rate \fbasesolid{} not all fish encounters, relevant for the context of the electrosensory cocktail party, could be covered.
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@ -614,14 +620,14 @@ A heterogeneous readout might be not only physiologically plausible but also req
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\subsection{Encoding of secondary envelopes}%($\n{}=49$) Whether this population of envelope encoders was in addition bursty was not addressed in the corresponding study
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The RAM stimulus used in this work is an approximation of the three-fish scenario, where the two generated beats are often slowly modulated at the difference between the two beat frequencies \bdiff{}, known as secondary or social envelope \citealp{Stamper2012Envelope}.
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In previous works it was demonstrated that low-frequency secondary envelopes are extracted not in P-units but downstream of them in the ELL \citealp{Middleton2006} utilizing threshold nonlinear response curves of the involved neuron \citealp{Middleton2007}. Based on our work we would predict that only a small class of cells, with very low CVs, should encode the social envelope at the difference frequency. If the sample in that previous work \citealp{Middleton2007} did not contain low CVs cells, This could explain the conclusion that P-units were identified not as envelope encoders.
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In previous works it was demonstrated that low-frequency secondary envelopes are extracted not in P-units but downstream of them in the ELL \citealp{Middleton2006} utilizing threshold nonlinear response curves of the involved neuron \citealp{Middleton2007}. Based on our work we would predict that only a small class of cells, with very low CVs, should encode the social envelope at the difference frequency. If the sample in that previous work \citealp{Middleton2007} did not contain low CVs cells, this could explain the conclusion that P-units were not identified as envelope encoders.
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On the other hand in previous literature the encoding of social envelopes was attributed to a subpopulation of P-units with strong nonlinearities, low firing rates and high CVs \citealp{Savard2011}. These findings are in contrast to the findings in the previously mentioned work \citealp{Middleton2007} and on first glance also to our findings. The missing link, that has not been considered in this work, might be bursting of P-units, the repeated firing of spikes after one EOD period interleaved with quiescence (unpublished work). Bursting was not explicitly addressed in the work \citealp{Savard2011}, still the high CVs of the envelope encoding P-units indicate a higher rate of bursting. How bursts influence the second-order susceptibility of P-units will be addressed in following works (in preparation).
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On the other hand in previous literature the encoding of social envelopes was attributed to a subpopulation of P-units with strong nonlinearities, low firing rates and high CVs \citealp{Savard2011}. These findings are in contrast to the findings in the previously mentioned work \citealp{Middleton2007} and on first glance also to our findings. The missing link, that has not been considered in this work, might be bursting of P-units, the repeated firing of spikes after one EOD period interleaved with quiescence (unpublished work). Bursting was not explicitly addressed in the previous work, still the high CVs of the envelope encoding P-units indicate a higher rate of bursting \citealp{Savard2011}. How bursts influence the second-order susceptibility of P-units will be addressed in following works (in preparation).
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%This small percentage of the low-CV cells would be in line with no P-units found in the work.
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\subsection{More fish would decrease second-order susceptibility}%
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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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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 (\citealp{Henninger2018}) will be addressed in the next chapter.
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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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@ -534,19 +534,17 @@ The smaller \fdiff{} power spectral peak observed during pure sine-wave stimulat
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\end{figure*}
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\subsection{Low CVs and weak stimuli are associated with strong nonlinearity}
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The non-linear effects shown for single cell examples above are supported by the analysis of the pool of 222 P-units and 45 ampullary afferents recorded in 71 specimen. To compare across cells we expressed the second-order susceptibility as the nonlinearity index \nli{}, see \eqnref{eq:nli_equation}. The \nli{} characterizes the peakedness of the projections onto the diagonal of the \suscept{} matrices (e.g. \subfigref{cells_suscept}{G} at \fbase{}. It assumes high values when the \fbase{} peak in the projected diagonal is pronounced relative to the median of the diagonal and is small when there is no distinct peak. The P-unit population has higher \nli{} values up to three and depends weakly on the the CV of the ISI distribution under baseline condition. Cells with lower baseline CVs have the tendency to exhibit a stronger nonlinearity than those that have high CVs (Pearson's r\,=\,$-0.16$, p\,<\,0.01). The two example P-units shown before (\figrefb{cells_suscept} and \figrefb{cells_suscept_high_CV}) are highlighted with dark circles in \subfigrefb{data_overview_mod}{B, D}. The stimulus strength plays an important role. Several of the recorded neurons contribute with two dots to the data as their responses to the same RAM stimulus but with different contrasts were recorded. Higher stimulus contrasts lead to a stronger drive and thus stronger response modulations (see color code bar in \subfigref{data_overview_mod}{A}, see methods). Since P-units are heterogeneous in their susceptibility to the stimulus\citealp{Grewe2017}, their responses to the same stimulus intensity vary a lot. When replotting the \nli{} against the response modulation (\subfigrefb{data_overview_mod}{C}) a negative correlation is observed (Pearson's r\,=\,$-0.35$, p\,<\,0.001 \notesr{das minus wird durch den Mathmodus länger, das will Jan immer so haben}) showing that cells that are strongly driven by the stimulus show less nonlinearity while those that are only weakly driven show higher nonlinearities. Whether or not a cell responds nonlinearly thus depends on both, the baseline CV (i.e. the internal noise) and the response strength.
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The non-linear effects shown for single cell examples above are supported by the analysis of the pool of 222 P-units and 45 ampullary afferents recorded in 71 specimen. To compare across cells we expressed the second-order susceptibility as the nonlinearity index \nli{}, see \eqnref{eq:nli_equation}. The \nli{} characterizes the peakedness of the projections onto the diagonal of the \suscept{} matrices (e.g. \subfigref{cells_suscept}{G} at \fbase{}. It assumes high values when the \fbase{} peak in the projected diagonal is pronounced relative to the median of the diagonal and is small when there is no distinct peak. The P-unit population has higher \nli{} values up to three and depends weakly on the the CV of the ISI distribution under baseline condition. Cells with lower baseline CVs have the tendency to exhibit a stronger nonlinearity than those that have high CVs (Pearson's r\,=\,$-0.16$, p\,<\,0.01). The two example P-units shown before (\figrefb{cells_suscept} and \figrefb{cells_suscept_high_CV}) are highlighted with dark circles in \subfigrefb{data_overview_mod}{B, D}. The stimulus strength plays an important role. Several of the recorded neurons contribute with two dots to the data as their responses to the same RAM stimulus but with different contrasts were recorded. Higher stimulus contrasts lead to a stronger drive and thus stronger response modulations (see color code bar in \subfigref{data_overview_mod}{A}, see methods). Since P-units are heterogeneous in their susceptibility to the stimulus\citealp{Grewe2017}, their responses to the same stimulus intensity vary a lot. When replotting the \nli{} against the response modulation (\subfigrefb{data_overview_mod}{C}) a negative correlation is observed (Pearson's r\,=\,$-0.35$, p\,<\,0.001) showing that cells that are strongly driven by the stimulus show less nonlinearity while those that are only weakly driven show higher nonlinearities. Whether or not a cell responds nonlinearly thus depends on both, the baseline CV (i.e. the internal noise) and the response strength.
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%In a P-unit population where each cell is represented not by several contrasts but by the lowest recorded contrast, \nli{} significantly correlates with the CV during baseline ($r=-0.17$, $p=0.01$), the response modulation ($r=-0.35$, $p<0.001$) and \fbase{} ($r=-0.32$, $p<0.001$).%, $\n{}=221$*, $\n{}=221$******, $\n{}=221$
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The population of ampullary cells is generally more homogeneous and have lower CVs than the P-units and show much higher \nli{} values (factor of 10). Overall, there is a negative correlation with the baseline CV (Pearson's r=-0.35, p < 0.01). The example cell shown above (\figref{ampullary}) was recorded at two different stimulus intensities and the \nli{}s are highlighted with black circles. Again, we see that cells that are strongly driven by the stimulus cluster at the bottom of the distribution and have \nli{} values close to zero. This is confirmed when the data is replotted against the response modulation, those cells that are strongly driven by the stimulus show weak nonlinearities while weakly driven neurons exhibit high values (Pearson's r=-0.59, p < 0.0001).
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The population of ampullary cells is generally more homogeneous and have lower CVs than the P-units and show much higher \nli{} values (factor of 10). Overall, there is a negative correlation with the baseline CV (Pearson's $r=-0.35$, $p < 0.01$). The example cell shown above (\figref{ampullary}) was recorded at two different stimulus intensities and the \nli{}s are highlighted with black circles. Again, we see that cells that are strongly driven by the stimulus cluster at the bottom of the distribution and have \nli{} values close to zero. This is confirmed when the data is replotted against the response modulation, those cells that are strongly driven by the stimulus show weak nonlinearities while weakly driven neurons exhibit high values (Pearson's $r=-0.59$, $p < 0.0001$).
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\section{Discussion}
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\notejg{dumped here since not strictly result...}
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These nonlinearity bands are more pronounced in ampullary cells than they were in P-units (compare \figrefb{ampullary} and \figrefb{cells_suscept}). Ampullary cells with their unimodal ISI distribution are closer than P-units to the LIF models without EOD carrier, where the predictions about the second-order susceptibility structure have mainly been elaborated on \citealp{Voronenko2017}. All here analyzed ampullary cells had CVs lower than 0.3 and exhibited strong nonlinear effects in accordance with the theoretical predictions \citealp{Voronenko2017}.
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%and here this could be confirmed experimentally.
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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 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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@ -563,6 +561,7 @@ In this chapter, the nonlinearity of P-units and ampullary cells was retrieved b
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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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%the power spectrum of the firing rate
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%The existence of the extended nonlinearity structure (\figrefb{model_full}) was
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%\bsum{} and \bdiff{}
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@ -579,6 +578,12 @@ In this chapter, the CV has been identified as an important factor influencing n
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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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%\notejg{dumped here since not strictly result...}
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%These nonlinearity bands are more pronounced in ampullary cells than they were in P-units (compare \figrefb{ampullary} and \figrefb{cells_suscept}). Ampullary cells with their unimodal ISI distribution are closer than P-units to the LIF models without EOD carrier, where the predictions about the second-order susceptibility structure have mainly been elaborated on \citealp{Voronenko2017}. All here analyzed ampullary cells had CVs lower than 0.3 and exhibited strong nonlinear effects in accordance with the theoretical predictions \citealp{Voronenko2017}.
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%and here this could be confirmed experimentally.
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%Nonlinear effects only for specific frequency combinations
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\subsection{Heterogeneity of P-units might influence nonlinearity}
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@ -586,14 +591,15 @@ In this chapter strong nonlinear interactions were found in a subpopulation of l
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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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%and \figrefb{ROC_with_nonlin}
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P-units are very heterogeneous in their baseline firing properties considering their baseline firing rates (50--450\,Hz, \citealp{Grewe2017, Hladnik2023}), their CV and also their bursting. How this heterogeneity influences nonlinear effects when an population of P-units is integrated by the next connected pyramidal cells in the electrosensory lateral line lobe (ELL) should be addressed in further studies.
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P-units are very heterogeneous in their baseline firing properties considering their baseline firing rates (50--450\,Hz, \citealp{Grewe2017, Hladnik2023}), their CV and also their bursting, the repeating firing of spikes after an EOD cycle \citealp{Chacron2004}. How this heterogeneity influences nonlinear effects when an population of P-units is integrated by the next connected pyramidal cells in the electrosensory lateral line lobe (ELL) should be addressed in further studies.
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The nonlinearity in this work were found in low-CV P-units. For this nonlinear effects to sustain on a population level a selective readout from a homogeneous population of low-CV cells might be required. Since a receptive field is defined as a patch of adjacent receptors on the fish surface \citealp{Maler2009a} and P-units have heterogeneous baseline properties that do not depend on the location of the receptor on the fish body \citealp{Hladnik2023} as a result each pyramidal cell in the ELL will integrate over a heterogeneous and not homogeneous P-unit population. The center-surround receptive field organization of superficial and intermediate layer pyramidal cells in addition increases this heterogeneity of the integrated population. Heterogeneous populations with different baseline properties can be more advantageous for the encoding compared to homogeneous populations, as has been shown in P-units and models \citealp{Beiran2018, Hladnik2023}.
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The nonlinearity in this work were found in low-CV P-units. A selective readout from a homogeneous population of low-CV cells might be required for this nonlinear effects to sustain on a population level. Since a receptive field is defined as a patch of adjacent receptors on the fish surface \citealp{Maler2009a} and P-units have heterogeneous baseline properties that do not depend on the location of the receptor on the fish body \citealp{Hladnik2023} as a result each pyramidal cell in the ELL will integrate over a heterogeneous and not homogeneous P-unit population. The center-surround receptive field organization of superficial and intermediate layer pyramidal cells in addition increases this heterogeneity of the integrated population. Heterogeneous populations with different baseline properties can be more advantageous for the encoding compared to homogeneous populations, as has been shown in P-units and models \citealp{Beiran2018, Hladnik2023}.
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%\subsubsection{A heterogeneous readout is required to cover all female-intruder combinations}
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A heterogeneous readout might be not only physiologically plausible but also required since nonlinear effects depending on cell property \fbase{} and not on stimulus properties might be not behaviorally relevant. Nonlinear effects might facilitate the encoding of faint signals during a three fish setting, the electrosensory cocktail party. The EOD frequencies of the encoutered three fish would be drawn from the EOD frequency distribution of these fish and a stable faint signal detection would require a response irrespective of the individual EOD frequencies. Weather integrating from a heterogeneous population with different \fbasesolid{} (50--450\,Hz, \citealp{Grewe2017, Hladnik2023}) would cover the behaviorally relevant range in the electrosensory cocktail party should be addressed in further studies.
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Nonlinear effects might facilitate the encoding of faint signals during a three fish setting, the electrosensory cocktail party. The EOD frequencies of the encountered three fish would be drawn from the EOD frequency distribution and a stable faint signal detection would require a response irrespective of the individual EOD frequencies. Here nonlinear effects, that might influence the detection of faint signals, was found only at specific frequencies in relation to the mean baseline firing rate \fbase{}. Weather integrating from a heterogeneous population with different \fbasesolid{} (50--450\,Hz, \citealp{Grewe2017, Hladnik2023}) would cover the behaviorally relevant range in the electrosensory cocktail party could be addressed in further studies.
|
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|
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%Thus, a heterogeneous readout might be not only physiologically plausible but also required since nonlinear effects depending on cell property, as the mean baseline firing rate \fbase{}, and not on stimulus properties, might be not behaviorally relevant.
|
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%In this work, nonlinear effects were always found only for specific frequencies in relation to \fbase{}, corresponding to findings from previous literature \citealp{Voronenko2017}.
|
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%Only a heterogeneous population could cover the whole stimulus space required during the electrosensory cocktail party.
|
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%If pyramidal cells would integrate only from P-units with the same mean baseline firing rate \fbasesolid{} not all fish encounters, relevant for the context of the electrosensory cocktail party, could be covered.
|
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@ -614,14 +620,14 @@ A heterogeneous readout might be not only physiologically plausible but also req
|
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\subsection{Encoding of secondary envelopes}%($\n{}=49$) Whether this population of envelope encoders was in addition bursty was not addressed in the corresponding study
|
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The RAM stimulus used in this work is an approximation of the three-fish scenario, where the two generated beats are often slowly modulated at the difference between the two beat frequencies \bdiff{}, known as secondary or social envelope \citealp{Stamper2012Envelope}.
|
||||
|
||||
In previous works it was demonstrated that low-frequency secondary envelopes are extracted not in P-units but downstream of them in the ELL \citealp{Middleton2006} utilizing threshold nonlinear response curves of the involved neuron \citealp{Middleton2007}. Based on our work we would predict that only a small class of cells, with very low CVs, should encode the social envelope at the difference frequency. If the sample in that previous work \citealp{Middleton2007} did not contain low CVs cells, This could explain the conclusion that P-units were identified not as envelope encoders.
|
||||
In previous works it was demonstrated that low-frequency secondary envelopes are extracted not in P-units but downstream of them in the ELL \citealp{Middleton2006} utilizing threshold nonlinear response curves of the involved neuron \citealp{Middleton2007}. Based on our work we would predict that only a small class of cells, with very low CVs, should encode the social envelope at the difference frequency. If the sample in that previous work \citealp{Middleton2007} did not contain low CVs cells, this could explain the conclusion that P-units were not identified as envelope encoders.
|
||||
|
||||
On the other hand in previous literature the encoding of social envelopes was attributed to a subpopulation of P-units with strong nonlinearities, low firing rates and high CVs \citealp{Savard2011}. These findings are in contrast to the findings in the previously mentioned work \citealp{Middleton2007} and on first glance also to our findings. The missing link, that has not been considered in this work, might be bursting of P-units, the repeated firing of spikes after one EOD period interleaved with quiescence (unpublished work). Bursting was not explicitly addressed in the work \citealp{Savard2011}, still the high CVs of the envelope encoding P-units indicate a higher rate of bursting. How bursts influence the second-order susceptibility of P-units will be addressed in following works (in preparation).
|
||||
On the other hand in previous literature the encoding of social envelopes was attributed to a subpopulation of P-units with strong nonlinearities, low firing rates and high CVs \citealp{Savard2011}. These findings are in contrast to the findings in the previously mentioned work \citealp{Middleton2007} and on first glance also to our findings. The missing link, that has not been considered in this work, might be bursting of P-units, the repeated firing of spikes after one EOD period interleaved with quiescence (unpublished work). Bursting was not explicitly addressed in the previous work, still the high CVs of the envelope encoding P-units indicate a higher rate of bursting \citealp{Savard2011}. How bursts influence the second-order susceptibility of P-units will be addressed in following works (in preparation).
|
||||
|
||||
%This small percentage of the low-CV cells would be in line with no P-units found in the work.
|
||||
|
||||
\subsection{More fish would decrease second-order susceptibility}%
|
||||
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.
|
||||
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 (\citealp{Henninger2018}) will be addressed in the next chapter.
|
||||
|
||||
|
||||
\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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|
||||
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