diff --git a/susceptibility1.tex b/susceptibility1.tex index 799e208..1c3d5c7 100644 --- a/susceptibility1.tex +++ b/susceptibility1.tex @@ -578,31 +578,40 @@ Ampullary stimulus encoding is somewhat different in \lepto{}. The power spectru The population of ampullary cells is very homogeneous with respect to the baseline rate (131$\pm$29\,Hz) and stimulus encoding properties \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 in the full population of ampullary cells (at the baseline firing frequency) that is outside the linear coding range. Postsynaptic cells integrating ampullary input might be able to extract such nonlinear responses. How such nonlinear effects might influence prey detection should be addressed in future studies. -\subsection{Noise stimulation approximates real interactions} -Our characterization of electroreceptors is based on neuronal responses to white-noise stimuli. 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}. In the real-world situation of wave-type electric fish, the natural stimuli encoded by P-units are the sinusoidal EOD signals with distinct frequencies continuosly emitted by a small number of near by interacting fish and the resulting beating AMs \citep{Stamper2010, Henninger2020}. How informative is the second-order susceptibility estimated in the limit of vanishing noise-stimulus amplitude for the encoding of distinct frequencies with finite amplitudes? +\subsection{Nonlinear encoding in P-units} +Noise stimuli have the advantage that a range of frequencies can be measured with a single stimulus presentation and they have been successfully applied to characterize sensory coding in many systems \citep{French1973, Marmarelis1999, Borst1999, Chacron2005, Grewe2017}. In the real-world situation of wave-type electric fish, however, the natural stimuli encoded by P-units are periodic amplitude modulations of the self-generated electric field which arise from the superposition of the own and foreign EODs. Natural interactions usually occur between low numbers of close-by fish and thus the AMs are a mixture of a few distinct frequencies with substantial amplitudes \citep{Stamper2010,Fotowat2013, Henninger2020}. How informative are the second-order susceptibilities estimated using noise-stimuli for the encoding of distinct frequencies? + +We here applied broad-band noise amplitude-modulation stimuli to estimate the second-order susceptibility. The total signal power in noise stimuli is uniformly distributed over a wide frequency band while it is spectrally focused in pure sinewave stimuli. If both stimuli have the same total power, i.e. are presented with the same amplitude (contrast) the power of the sinewave stimulus, the power at these frequencies is much higher and, assuming white neuronal noise, have a much higher signal-to-noise ratio. \notejg{Ehrlich gesagt, weiss ich nicht so genau, wo du mit dem vorherigen Satz noch hinwolltest.} The absence of the linearizing effect of the noise stimuli can explain nonlinear interactions under distinct-frequency stimulation \figref{fig:motivation} while they are barely visible with strong white-noise stimulation \figref{fig:model_full}. 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 due to external white-noise stimulation. 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, in accordance with the the second-order susceptibilities found in LIF models without a carrier \citep{Voronenko2017, Schlungbaum2023}. 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 recordings we directly relate the AM waveform to the cellular responses. Responding to the AM itself requires a nonlinearity \citep{Middleton2006, Stamper2012Envelope, Savard2011, Barayeu2023}, encoding the time-course of the AM, however, is linear over a wide range of AM amplitudes and frequencies \citep{Xu1996, Benda2005, Gussin2007, Grewe2017, Savard2011}. The encoding of secondary or social envelopes that arise from relative movement or the interaction of more than two animals \citep{Stamper2012Envelope} requires an additional nonlinearity in the system that was initially attributed to downstream processing \citep{Middleton2006, Middleton2007}. Later studies showed that already the electroreceptors can encode such information when strong saturation nonlinearities occur \citep{Savard2011}. Based on our work here, we would predict that a subset of cells, with low CVs, should encode the social envelopes even under weak stimulation. The exact transitions on nonlinear encoding from a regime of weak stimuli to a regime of strong stimuli should be addressed in further studies \notejg{Here we should probably work with the new figure}. + +The observed nonlinear effects can possibly facilitate the detectability of faint signals during a three-animal interactions as found in freely behaving animals \citep{Henninger2018}, the electrosensory cocktail party. They are, however, very specific with respect to the stimulus frequencies and a given P-unit's baseline frequency. The relevant frequencies are determined by combination of EOD frequencies observed in male and female fish \citep{Hopkins1974Eigen, Meyer1987, Henninger2018, Henninger2020} and the AM frequencies are limited to a range below half of each fish's EOD frequency (0 -- \feod/2, a restriction arising from the sampling theorem) \citep{Barayeu2023}. The population of P-units is very heterogeneous in their baseline firing properties \citep{Grewe2017, Hladnik2023}. The baseline firing rates and the observed CVs vary in wide ranges (50--450\,Hz and 0.1--1.4, respectively, \figref{fig:dataoverview}\panel{A}). The range of baseline firing rates thus covers substantial parts of the beat frequencies that may occur during animal interactions, while the number of low-CV P-units is small. For the response components arising due to the nonlinearities described here to be behaviorally relevant requires this information to survive the convergence onto the pyramidal cells in the electrosensory lateral line lobe (ELL) in the hindbrain with different tuning properties and levels of convergence \citep{Krahe2014, 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 which 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. + +%that the nonlinearity pattern in the electroreceptor recordings. only partially matches the expectation (\figsref{fig:cells_suscept},\,\ref{fig:ampullary}); while 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. + +%\subsection{Noise stimulation approximates real interactions} +%Our characterization of electroreceptors is based on neuronal responses to white-noise stimuli. 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}. In the real-world situation of wave-type electric fish, the natural stimuli encoded by P-units are the sinusoidal EOD signals with distinct frequencies continuosly emitted by a small number of near by interacting fish and the resulting beating AMs \citep{Stamper2010, Henninger2020}. How informative is the second-order susceptibility estimated in the limit of vanishing noise-stimulus amplitude for the encoding of distinct frequencies with finite amplitudes? % 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}). -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. % 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{fig: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{fig: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{fig: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. -\subsection{Beats are essentiallly sine wave stimuli} -Even though the second-order susceptibilities here were estimated from data and models with a modulated (EOD) carrier (\figrefb{fig:model_full}) they are in good accordance with the second-order susceptibilities found in LIF models without a carrier \citep{Voronenko2017, Schlungbaum2023}. +% \subsection{Beats are essentiallly sine wave stimuli} +% Even though the second-order susceptibilities here were estimated from data and models with a modulated (EOD) carrier (\figrefb{fig:model_full}) they are in good accordance with the second-order susceptibilities found in LIF models without a carrier \citep{Voronenko2017, Schlungbaum2023}. -% AM extraction is linear: -Even though the extraction of the AM itself requires a nonlinearity \citep{Middleton2006, Stamper2012Envelope, Savard2011, Barayeu2023} encoding the time-course of the AM is linear over a wide range of AM amplitudes and frequencies \citep{Xu1996, Benda2005, Gussin2007, Grewe2017, Savard2011}. +% % AM extraction is linear: +% Even though the extraction of the AM itself requires a nonlinearity \citep{Middleton2006, Stamper2012Envelope, Savard2011, Barayeu2023} encoding the time-course of the AM is linear over a wide range of AM amplitudes and frequencies \citep{Xu1996, Benda2005, Gussin2007, Grewe2017, Savard2011}. -\subsection{Selective readout versus integration of heterogeneous populations} +%\subsection{Selective readout versus integration of heterogeneous populations} -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. +%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, 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:dataoverview}\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. +%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:dataoverview}\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. -\subsection{Nonlinear encoding in P-units} -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. +%\subsection{Nonlinear encoding in P-units} +%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. % Envelope extraction requires a nonlinearity: % In the context of social signaling among three fish, we observe an AM of the AM, also referred to as second-order envelope or just social envelope \citep{Middleton2006, Savard2011, Stamper2012Envelope}. Encoding this again requires nonlinearities \citep{Middleton2006} and it was shown that a subpopulation of P-units is sensitive to envelopes \citep{Savard2011} and exhibit nonlinearities e.g. when driven by strong stimuli \citep{Nelson1997, Chacron2004}.