benda/susceptibility1
1
0
Fork 0
Code Issues Pull Requests Actions Packages Projects Releases Wiki Activity

Merge branch 'main' of github.com:saschuta/susceptibility1

Browse Source
This commit is contained in:
Jan Benda 2024-11-05 09:21:32 +01:00
commit 9fce766b15
1 changed files with 17 additions and 17 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

34
susceptibility1.tex
View File

@ -566,20 +566,20 @@ Theoretical work \citep{Voronenko2017} derived analytical expressions for weakly
\subsection{Intrinsic noise limits nonlinear responses}
The pattern of elevated second-order susceptibility found in the experimental data matches the theoretical expectations only partially. Only P-units with low coefficients of variation (CV $<$ 0.25) of the interspike-interval distribution in their baseline response show the expected nonlinearities (\figref{fig:cells_suscept}, \figref{fig:model_full}, \subfigref{fig:data_overview}{A}). Such low-CV cells are rare among the 221 P-units used in this study. On the other hand, the majority of the ampullary cells have generally lower CVs (median of 0.12) and have an approximately ten-fold higher level of second-order susceptibilities where \fsumb{} (\figref{fig:ampullary}, \subfigrefb{fig:data_overview}{B}).
The CV is a proxy for the intrinsic noise in the cells \cite{}\notejb{Cite Lindner IF by rate and CV}. In both cell types, we observe a negative correlation between the second-order susceptibility at \fsumb{} and the CV, indicating that it is the level of intrinsic noise that shapes nonlinear responses. These findings are in line with previous studies showing the linearizing effects of noise in nonlinear systems \citep{Roddey2000, Chialvo1997, Voronenko2017}. Increased intrinsic noise has been demonstrated to increase the CV and reduce nonlinear phase-locking in vestibular afferents \citep{Schneider2011}. Reduced noise, on the other hand, has been associated with stronger nonlinearity in pyramidal cells of the ELL \citep{Chacron2006}. Only in cells with sufficiently low levels of intrinsic noise, weakly nonlinear responses can be observed.
The CV is a proxy for the intrinsic noise in the cells \cite{}\notejb{Cite Lindner IF by rate and CV}. In both cell types, we observe a negative correlation between the second-order susceptibility at \fsumb{} and the CV, indicating that it is the level of intrinsic noise that shapes nonlinear responses. These findings are in line with previous theoretical and experimental studies showing the linearizing effects of noise in nonlinear systems \citep{Roddey2000, Chialvo1997, Voronenko2017}. Increased intrinsic noise has been demonstrated to increase the CV and reduce nonlinear phase-locking in vestibular afferents \citep{Schneider2011}. Reduced noise, on the other hand, has been associated with stronger nonlinearity in pyramidal cells of the ELL \citep{Chacron2006}. Only in cells with sufficiently low levels of intrinsic noise, weakly nonlinear responses can be observed.
\subsection{Linearizing effects of white-noise stimulation}
Further support for the notion of noise limiting the nonlinearity comes from our P-unit LIF model \citep{Barayeu2023}. We can use this model and apply a noise-split \citep{Lindner2022} based on the Furutsu-Novikov theorem \citep{Novikov1965, Furutsu1963}, to increase the signal-to-noise ratio in the cell while keeping the overall response variability constant (see methods). Treating 90\,\% of the total noise as a signal and simulating large numbers of trials uncovers the full nonlinear structure (\figref{model_and_data}) seen in LIF neurons and the analytical derivations when driven with sine-wave stimuli \citep{Voronenko2017}.
\subsection{Linearization by white-noise stimulation}
Not only the intrinsic noise but also the stimulaion with external white-noise linearizes the cells. This applies to both, the stimulation with AMs in P-units (\subfigrefb{fig:data_overview}{E}) and direct stimulation in ampullary cells (\subfigrefb{fig:data_overview}{F}). The stronger the effective stimulus, the less pronounced are the peaks in second-order susceptibility. In order to characterize the non-linearity of the cells in the limit to vanishing stimulus amplitudes we utilized the Furutsu-Novikov theorem \citep{Novikov1965, Furutsu1963}. Following \citet{Lindner2022}, a substantial part of the intrinsic noise of a P-unit model \citep{Barayeu2023} is treated as signal. Performing this noise-split trick we can estimate the weakly nonlinear response without the linearizing effect of an additional external white noise stimulus. The model of a low-CV P-unit then shows the full nonlinear structure (\figref{model_and_data}) known from analytical derivations and simulations of basic LIF models driven with pairs of sine-wave stimuli \citep{Voronenko2017}.
%
\subsection{Noise stimulation approximates the real three-fish interaction}
\subsection{Noise stimulation approximates real interactions}
Our analysis is based on the neuronal responses to white-noise stimulus sequences. %For the P-units, the stimulus was a random amplitude modulation (RAM) while it was a direct noise stimulus for the ampullary cells.
These broad-band stimuli have the advantage that all behaviorally relevant frequencies can be measured with a single stimulus presentation and it is a widely used approach to characterize sensory coding \citep{French1973, Marmarelis1999, Borst1999, Chacron2005, Grewe2017}. However, these stimuli also increase the total level of noise in the system and may have a linearizing effect on signal transmission.
These broad-band stimuli have the advantage that all behaviorally relevant frequencies can be measured with a single stimulus presentation and it is a widely used approach to characterize sensory coding \citep{French1973, Marmarelis1999, Borst1999, Chacron2005, Grewe2017}. However, asdiscussed above, these stimuli also increase the total level of noise in the system and linearize the system. In the real-world situation, the natural stimuli are periodic signals defined by the EODs of small groups of interacting fish and the resulting AMs \citep{Stamper2010}. How well can we extrapolate from the white-noise analysis to the pure sinewave situation?
In contrast to the situation with individual frequencies (direct sine-waves or sinusoidal AMs) the total signal power of the noise stimulus is equally distributed on all frequencies leading to a weaker signal-to-noise ratio. This explains that the nonlinearity pattern in the electroreceptor recordings only partially matches the expectation (\figsref{fig:cells_suscept},\,\ref{fig:ampullary}) while the single-frequency stimulation shows nonlinear interference when the individual stimulus frequencies ($f_1, f_2, \Delta f_1, \Delta f_2$) match the baseline firing rate (\figref{fig:motivation}, \subfigref{fig:model_full}{B--E}). Applying the Furutsu-Novikov theorem in the model simulations strongly suggests that low-CV cells do have the full nonlinearity pattern that is, however, covered by the intrinsic noise. We thus conclude that the presence of the anti-diagonal pattern in the \suscept{} matrix is sufficient to infer that the same nonlinear interactions happen here. This also validates the application of the white noise approach to characterize the full \suscept{} matrix instead of using all combinations of frequencies.
%In our P-unit models, we were able to make use of the Furutsu-Novikov theorem to estimate nonlinear signal transmission for zero-amplitude stimulation. Only with this procedure and many trials, we find for low-CV P-units all the pronounced ridges in the second-order susceptibility that we would expect according to theory \citep{Voronenko2017}.
In the real-world situation, the natural stimuli are periodic signals defined by the EODs of the interacting fish and the resulting AMs. How well can we extrapolate from the white-noise analysis to the pure sinewave situation?
In contrast to the situation with individual frequencies (direct sine-waves or sinusoidal AMs) the total signal power of the noise stimulus is equally distributed on all frequencies leading to a weaker signal-to-noise ratio. This explains that the nonlinearity pattern in the electroreceptor recordings only partially matches the expectation (\figsref{fig:cells_suscept},\,\ref{fig:ampullary}) while the single-frequency stimulation shows nonlinear interference when the individual stimulus frequencies ($f_1, f_2, \Delta f_1, \Delta f_2$) match the baseline firing rate (\figref{fig:motivation}, \subfigref{fig:model_full}{B--E}). With the noise-splitting trick, we could show \notejg{too hard?} that low-CV cells do have the full nonlinearity pattern that is, however, covered by the intrinsic noise. We thus conclude that the presence of the anti-diagonal pattern in the \suscept{} matrix is sufficient to infer that the same nonlinear interactions happen here. This also validates the application of the white noise approach to characterize the full \suscept{} matrix instead of using all combinations of frequencies.
% The nonlinearity of ampullary cells in paddlefish \citep{Neiman2011fish} has been previously accessed with bandpass limited white noise.
@ -587,21 +587,21 @@ In contrast to the situation with individual frequencies (direct sine-waves or s
\subsection{Selective readout versus integration of heterogeneous populations}% Nonlinearity might be influenced once integrating from a P-unit population with heterogeneous baseline properties}%Heterogeneity of P-units might influence
The observed nonlinear effects might facilitate the detectability of faint signals during a three-fish setting, the electrosensory cocktail party. These nonlinear effects are, however, very specific with respect to the relation of stimulus frequencies and the P-unit baseline frequency. The EOD frequencies of the interacting fish would be drawn from the distributions of EOD frequencies found in male and female fish \citep{Hopkins1974Eigen, Meyer1987, Henninger2018, Henninger2020}. To be behaviorally relevant the faint signal detection would require reliable neuronal signaling irrespective of the individual EOD frequencies.
P-units, however, are very heterogeneous in their baseline firing properties \citep{Grewe2017, Hladnik2023}. The baseline firing rates vary in wide ranges (50--450\,Hz). This range covers substantial parts of the beat frequencies that may occur during animal interactions which is limited to frequencies below the Nyquist frequency (0 -- \feod/2) \citep{Barayeu2023}. It is thus likely that there are P-units that match approximately the specificities of the different encounters.
The observed nonlinear effects can possibly facilitate the detectability of faint signals during a three-fish setting, the electrosensory cocktail party \citep{Henninger2018}. These nonlinear effects are, however, very specific with respect to the relation of stimulus frequencies and a given P-unit's baseline frequency. The EOD frequencies of the interacting fish would be drawn from the distributions of EOD frequencies found in male and female fish \citep{Hopkins1974Eigen, Meyer1987, Henninger2018, Henninger2020}. To be behaviorally relevant the faint signal detection would require reliable neuronal signaling irrespective of the individual EOD frequencies.
P-units, however, are very heterogeneous in their baseline firing properties \citep{Grewe2017, Hladnik2023}. The baseline firing rates vary in wide ranges (50--450\,Hz). This range covers substantial parts of the beat frequencies that may occur during animal interactions which is limited to frequencies below the Nyquist frequency (0 -- \feod/2) \citep{Barayeu2023}. It is thus likely that there are P-units that approximately match the specificities of the different encounters.
On the other hand, the nonlinearity was found only in low-CV P-units (with white noise stimulation). The CVs are also very heterogeneous (0.1--1.4, \figref{fig:data_overview}\panel{A}) in our sample. Only a small fraction of the P-units have a sufficiently low level of intrinsic noise and will exhibit nonlinear responses. The P-units project to the ELL \citep{Krahe2014} and the integrating pyramidal cells in the different segments receive inputs in the range of 10s to 1000s of neurons \citep{Maler2009a}. Since the receptive fields of the pyramidal neurons are patches of adjacent receptors on the fish's body surface \citep{Bastian2002, Maler2009a, Haggard2023} and the input heterogeneity does not depend on the location of the receptor on the fish body \citep{Hladnik2023} the pyramidal cell input population will be heterogeneous. Heterogeneity was shown to be generally advantageous for the encoding in this \citep{Hladnik2023} and other systems \citep{Padmanabhan2010, Beiran2018}. At the same time, it contradicts the apparent need for a selective readout of low-CV cells to maintain information arising through nonlinear interactions.
On the other hand, this nonlinearity was found only in low-CV P-units (with white noise stimulation). The CVs are also very heterogeneous (0.1--1.4, \figref{fig:data_overview}\panel{A}) in our sample. Only a small fraction of the P-units have a sufficiently low level of intrinsic noise and will exhibit nonlinear responses. The P-units project to the ELL \citep{Krahe2014} and the integrating pyramidal cells in the different segments receive inputs in the range of 10s to 1000s of neurons \citep{Maler2009a}. Since the receptive fields of the pyramidal neurons are patches of adjacent receptors on the fish's body surface \citep{Bastian2002, Maler2009a, Haggard2023} and the input heterogeneity does not depend on the location of the receptor on the fish body \citep{Hladnik2023} the pyramidal cell input population will be heterogeneous. Even though heterogeneity was shown to be generally advantageous for the encoding in this \citep{Hladnik2023} and other systems \citep{Padmanabhan2010, Beiran2018} it contradicts the apparent need for a selective readout of low-CV cells to maintain information arising through nonlinear interactions.
A possible readout mechanism should be the topic of future studies that need to take into account that the nonlinearities are stronger in pure sine-wave stimulation and the fraction of cells that show it under naturalistic stimulation might be larger than expected from the distribution of CVs.
\subsection{Behavioral relevance of nonlinear interactions}
The behavioral relevance of the weak signal detection in P-units is evident from the courtship context observed in freely interacting animals \citep{Henninger2018}. Outside courtship behavior, the encoding of secondary or social envelopes is a common need \citep{Stamper2012Envelope}. In a previous study, it was demonstrated that information about low-frequency secondary envelopes would not be present in P-units' responses but would arise through nonlinear processing downstream in the ELL \citep{Middleton2006, Middleton2007}. Based on our work we would predict that only a small subset of cells, with low CVs, should encode the social envelopes under white noise stimulation. An absence of low-CV cells in the population analyzed in the previous studies could explain their conclusions. On the other hand, another study showed that P-units with strong nonlinearities, low firing rates and, high CVs could indeed encode social envelopes \citep{Savard2011}. These findings are in contrast to the previously mentioned work \citep{Middleton2007} and, at first glance, also to our results. The missing link, that has not been considered in this work, might be the bursting of P-units, the repeated firing of spikes after one EOD period interleaved with longer intervals of quietness \citep{Chacron2004}. Bursting was not explicitly addressed in the previous work, still, the reported high CVs of the envelope encoding P-units indicate a higher rate of bursting \citep{Savard2011}. How bursts influence the second-order susceptibility of P-units will be addressed in the following works \citep{Barayeu2024}. Note that in this work we operated in a regime of weak stimuli and that the envelope encoding addressed in \citep{Savard2011,Middleton2007} operates in a regime of strong stimuli, where the firing rate is saturated. The exact transition from the nonlinearities in a regime of weak stimuli to a regime of strong stimuli could be addressed in further P-unit studies.
\subsection{Nonlinear encoding in electroreceptors}
Outside courtship behavior, the encoding of secondary or social envelopes is a common need \citep{Stamper2012Envelope}. In a previous study, it was demonstrated that information about low-frequency secondary envelopes would not be present in P-units' responses but would arise through nonlinear processing downstream in the ELL \citep{Middleton2006, Middleton2007}. Based on our work we would predict that only a small subset of cells, with low CVs, should encode the social envelopes under white noise stimulation. An absence of low-CV cells in the population analyzed in the previous studies could explain their conclusions. On the other hand, another study showed that P-units with strong nonlinearities, low firing rates and, high CVs could indeed encode social envelopes \citep{Savard2011}. These findings are in contrast to the previously mentioned work \citep{Middleton2007} and, at first glance, also to our results. The missing link, that has not been considered in this work, might be the bursting of P-units, the repeated firing of spikes after one EOD period interleaved with longer intervals of quietness \citep{Chacron2004}. Bursting was not explicitly addressed in the previous work, still, the reported high CVs of the envelope encoding P-units indicate a higher rate of bursting \citep{Savard2011}. How bursts influence the second-order susceptibility of P-units will be addressed in the following works \citep{Barayeu2024, Schlungbaum2024}. Note that in this work we operated in a regime of weak stimuli and that the envelope-encoding addressed in \citep{Savard2011, Middleton2007} operates in a regime of strong stimuli, where firing rates are saturated. The exact transition from the nonlinearities in a regime of weak stimuli to a regime of strong stimuli should be addressed in further P-unit studies.
Sinusoidal AMs are relevant in interactions with a few fish. We can understand the noise as the presence of many animals with individual EOD frequencies at the same time. Under noise stimulation, nonlinearities were demonstrated to be strong for weak stimuli but were shown to decrease for stronger noise stimuli (\figrefb{fig:cells_suscept}). As long as the noise signal is weak, those fish are distant and the nonlinearity is maintained. An increasing stimulus amplitude would indicate that many fish are close to the receiver 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. \lepto{} usually prefers small groups of fish \citep{Stamper2010}. Thus, the described second-order susceptibility might still be behaviorally relevant under natural conditions. The decline of nonlinear effects when several fish are present might be an adaptive process 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.
%Sinusoidal AMs are relevant in interactions with a few fish. We can understand the noise as the presence of many animals with individual EOD frequencies at the same time. Under noise stimulation, nonlinearities were demonstrated to be strong for weak stimuli but were shown to decrease for stronger noise stimuli (\figrefb{fig:cells_suscept}). As long as the noise signal is weak, those fish are distant and the nonlinearity is maintained. An increasing stimulus amplitude would indicate that many fish are close to the receiver 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. \lepto{} usually prefers small groups of fish \citep{Stamper2010}. Thus, the described second-order susceptibility might still be behaviorally relevant under natural conditions. The decline of nonlinear effects when several fish are present might be an adaptive process 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.
The afferents of the passive electrosensory system, the ampullary cells, were found to exhibit much stronger nonlinearities than P-units (\figref{fig:data_overview}). The adequate stimulus for this system is a direct stimulation not an amplitude modulation. In this sense, the ampullary cells are closer to the LIF models used by Voroneko and colleagues \citep{Voronenko2017} and we can thus expect that the same nonlinear mechanisms are at work here. For the ampullary system, the sinewave stimulation is not as relevant as for the P-unit system. Ampullary cells encode low-frequency exogenous electric signals such as muscle potentials induced by prey movement \citep{Kalmijn1974, Engelmann2010, Neiman2011fish}. The simultaneous muscle activity of a swarm of prey (such as \textit{Daphnia}) resembles Gaussian white noise \citep{Neiman2011fish}, similar to the stimuli used here. Our results show some similarities with the analyses by Neiman and Russel \citep{Neiman2011fish} who study the encoding in ampullary afferents in the paddlefish. There, the power spectrum of the spontaneous activity also shows a peak at the baseline frequency (internal oscillator) but also at the oscillation frequency measured at the epithelium and interactions of both. Most of the latter disappear in the case of external stimulation, though. Here we find only peaks at the baseline frequency of the neuron and its harmonics. There are interesting similarities and dissimilarities; stimulus encoding in the paddlefish as well as in the brown ghost is very linear for low frequencies and there are nonlinearities in both systems. Linear encoding in the paddlefish shows a gap in the spectrum at the frequency of the epithelial oscillation, instead, the nonlinear response is very pronounced there. In \lepto{}, the dominating frequency under baseline conditions is the baseline firing rate, and we do see increased nonlinearity in this frequency range. The baseline frequency, however, is outside the linear coding range \citep{Grewe2017} while it is within the linear coding range in paddlefish \citep{Neiman2011fish}. Interestingly, the nonlinear response in the paddlefish ampullaries increases with stimulus intensity while it disappears in our case (\subfigrefb{fig:data_overview}{F}). The population of ampullary cells is much more homogeneous with respect to the baseline rate (131$\pm$29\,Hz) and stimulus encoding properties than the P-units \citep{Grewe2017}. This implies that, if the stimulus contains the appropriate frequency components that sum up to the baseline rate, there should be a nonlinear response at a frequency that is similar to the full population of ampullary cells (the baseline frequency) that is outside the linear coding range. Postsynaptic cells integrating ampullary input might be able to extract this nonlinear response from the input population. How such nonlinear effects in ampullary cells might influence prey detection should be addressed in further studies.
The afferents of the passive electrosensory system, the ampullary cells, were found to exhibit much stronger nonlinearities than P-units (\figref{fig:data_overview}). The adequate stimulus for this system is a direct stimulation not an amplitude modulation. In this respect, the ampullary cells are closer to the LIF models used by Voroneko and colleagues \citep{Voronenko2017} and we can thus expect that the same nonlinear mechanisms are at work here. For the ampullary system, the sinewave stimulation is not as relevant as for the P-unit system. Ampullary cells encode low-frequency exogenous electric signals such as muscle potentials induced by prey movement \citep{Kalmijn1974, Engelmann2010, Neiman2011fish}. The simultaneous muscle activity of a swarm of prey (such as \textit{Daphnia}) resembles Gaussian white noise \citep{Neiman2011fish}. Our results show some similarities with the analyses by Neiman and Russel \citep{Neiman2011fish} who study the encoding in ampullary afferents in the paddlefish. There, the power spectrum of the spontaneous activity also shows a peak at the baseline frequency (internal oscillator) but also at the oscillation frequency measured at the epithelium and interactions of both. Most of the latter disappear in the case of external stimulation, though. Here we find only peaks at the baseline frequency of the neuron and its harmonics. There are interesting similarities and dissimilarities; stimulus encoding in the paddlefish as well as in the brown ghost can be well described using linear models for low frequencies but there are also nonlinearities in both systems. Linear encoding in the paddlefish shows a gap at the epithelial oscillation frequency, instead, the nonlinear response is very pronounced there. In \lepto{}, the dominating frequency under baseline conditions is the baseline firing rate, and we do see increased nonlinearity in this frequency range. The baseline frequency, however, is outside the linear coding range \citep{Grewe2017} while it is within the linear coding range in paddlefish \citep{Neiman2011fish}. Interestingly, the nonlinear response in the paddlefish ampullaries increases with stimulus intensity while it disappears in our case (\subfigrefb{fig:data_overview}{F}). The population of ampullary cells is much more homogeneous with respect to the baseline rate (131$\pm$29\,Hz) and stimulus encoding properties than the P-units \citep{Grewe2017}. This implies that, if the stimulus contains the appropriate frequency components that sum up to the baseline rate, there should be a nonlinear response at a frequency that is similar to the full population of ampullary cells (the baseline frequency) that is outside the linear coding range. Postsynaptic cells integrating ampullary input might be able to extract this nonlinear response. How such nonlinear effects in ampullary cells might influence prey detection should be addressed in future studies.
\subsection{Conclusion}
We have demonstrated that there are pronounced nonlinear responses in the primary electrosensory afferents of the weakly electric fish \lepto{}, systems that are very often characterized using linear methods. The observed nonlinearities match the expectations from previous theoretical studies \citep{Voronenko2017}. We can confirm that the theory applies also to systems that are encoding amplitude modulations of a carrier signal. Comparisons of P-units and ampullary cells showed that it is the level of intrinsic noise that determines how strongly nonlinear the system acts. Using the second-order susceptibility estimated from the responses to white noise stimuli provides an easy way to determine the nonlinearity of the system under study. P-units share several features with mammalian
\subsection{Conclusions and outlook}
We have demonstrated that there are pronounced nonlinear responses in the primary electrosensory afferents of the weakly electric fish \lepto{}. The observed nonlinearities match the expectations from previous theoretical studies \citep{Voronenko2017}. We can confirm that the theory applies also to systems that are encoding amplitude modulations of a carrier signal. Comparisons of P-units and ampullary cells showed that it is the level of intrinsic noise that determines how strongly nonlinear the system acts. Using the second-order susceptibility estimated from the responses to white noise stimuli provides an easy way to determine the nonlinearity of the system under study. P-units share several features with mammalian
auditory nerve fibers and such nonlinear effects might also be expected in the auditory system during the encoding of amplitude modulations \citep{Joris2004}. Nevertheless, the theory holds for systems that are directly driven by a sinusoidal input (ampullary cells) and systems that are driven by amplitude modulations of a carrier (P-units) and is thus widely applicable.
\section{Methods}