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For linear systems, the second-order response function is zero, because linear systems do not create new frequencies and thus there is no response at the summed frequencies. The plots show the analytical solutions from \citep{Lindner2001} and \citep{Voronenko2017} with $\mu = 1.1$ and $D = 0.001$.} \end{figure*} - We like to think about signal encoding in terms of linear relations with unique mapping of a given input value to a certain output of the system under consideration. Indeed, such linear methods, for example the first-oder susceptibility or transfer function shown in figure~\ref{fig:lifresponse} have been widely and successfully applied to describe and predict neuronal responses and are an invaluable tool to characterize nervous systems \citep{Borst1999}. Nonlinear mechanisms, on the other hand, are key on different levels; deciding for one action over another is a nonlinear process on the systemic level, on the cellular level, spiking neurons are inherently nonlinear. Whether an action potential is elicited depends on the membrane potential to exceed a threshold \citep{Hodgkin1952, Koch1995}. Because of such nonlinearities, understanding and predicting neuronal responses to sensory stimuli is in general a difficult task. -At the heart of nonlinear system identification is the Volterra series \citep{Rieke1999}. Second-order kernels have been used to predict firing rate responses of catfish retinal ganglion cells \citep{Marmarelis1972}. In the frequency domain, second-order kernels are known as second-order response functions or susceptibilities. They quantify the amplitude of the response at the sum and difference of two stimulus frequencies. Adding also third-order kernels, spike trains of spider mechanoreceptors have been predicted from sensory stimuli \citep{French2001}. The nonlinear nature of Y cells in contrast to the more linear responses of X cells in cat retinal ganglion cells has been demonstrated using second-order kernels \citep{Victor1977}. Interactions between different frequencies in the response of neurons in visual cortices of cats and monkeys have been studied using bispectra, the crucial constituent of the second-order susceptibility \citep{Schanze1997}. Locking of chinchilla auditory nerve fibers to pure tone stimuli is captured by second-order kernels \citep{Temchin2005}. In paddlefish ampullary afferents, bursting in response to strong, natural sensory stimuli boosts nonlinear responses in the bicoherence, the bispectrum normalized by stimulus and response spectra \citep{Neiman2011}. \notejg{much shorter!, extract the gist of it}\notejg{Question arises why is the stuff here special, if non-linear methods have been applied all over the place?} +The transfer function used to describe linear properties of a system is the first-order term of a Volterra series. Higher-order terms successively approximate nonlinear features of a system \citep{Rieke1999}. Second-order kernels have been used in the time domain to predict visual responses in catfish \citep{Marmarelis1972}. In the frequency domain, second-order kernels are known as second-order response functions or susceptibilities. They quantify the amplitude of the response at the sum and difference of two stimulus frequencies. Including higher-order terms of the Volterra series, the nonlinear nature of spider mechanoreceptors \citep{French2001}, mammalian visual systems \citep{Victor1977, Schanze1997}, locking in chinchilla auditory nerve fibers \citep{Temchin2005}, and bursting responses in paddlefish \citep{Neiman2011} have been demonstrated. -Noise linearizes nonlinear systems \citep{Yu1989, Chialvo1997} and therefore noisy neural systems can be well described by linear response theory in the limit of small stimulus amplitudes \citep{Roddey2000, Doiron2004, Rocha2007, Sharafi2013}. When increasing stimulus amplitude, at first the contribution of the second-order kernel of the Volterra series becomes relevant in addition to the linear one. For these weakly nonlinear responses of leaky-integrate-and-fire (LIF) neurons, an analytical expression for the second-order susceptibility has been derived \citep{Voronenko2017} in addition to its linear response function \citep{Lindner2001}. In the superthreshold regime, where the LIF generates a baseline firing rate in the absence of an external stimulus, the linear response function has a peak at the baseline firing rate and its harmonics (\subfigrefb{fig:lifresponse}{A}) and the second-order susceptibility shows very distinct ridges of elevated nonlinear responses, exactly where two stimulus frequencies equal or add up to the neuron's baseline firing rate (\subfigrefb{fig:lifresponse}{B}). In experimental data, such structures in the second-order susceptibility have not been reported yet. +Under certain circumstances and in the presence of noise, however, nonlinear systems can appear \citep{Yu1989, Chialvo1997} and a noisy systems can be well described by linear response theory in the limit of small stimulus amplitudes \citep{Roddey2000, Doiron2004, Rocha2007, Sharafi2013}. With increasing stimulus amplitude, the contribution of the second-order kernel of the Volterra series becomes relevant in addition to the linear one. For these weakly nonlinear responses of leaky-integrate-and-fire (LIF) model neurons, an analytical expression for the second-order susceptibility has been derived \citep{Voronenko2017} in addition to its linear response function \citep{Lindner2001}. In the suprathreshold regime, where the LIF generates a baseline firing rate in the absence of an external stimulus, the linear response function has a peak at the baseline firing rate and its harmonics (\subfigrefb{fig:lifresponse}{A}) and the second-order susceptibility shows very distinct ridges of elevated nonlinear responses, exactly where two stimulus frequencies equal or add up to the neuron's baseline firing rate (\subfigrefb{fig:lifresponse}{B}). In experimental data, such structures in the second-order susceptibility have not been reported yet. -Here we search for such weakly nonlinear responses in electroreceptors of the two electrosensory systems of the wave-type electric fish \textit{Apteronotus leptorhynchus}, i.e. the tuberous (active) and the ampullary (passive) electrosensory system. The electroreceptors of the active system are driven by the fish's self generated electric field generated by regular, quasi-sinusoidal, discharges of their electric organ (electric organ discharge or EOD) and encode disturbances of it \citep{Bastian1981a}. The electroreceptors of the passive system are tuned to lower-frequency exogeneous electric fields such as caused by muscle activity of prey \citep{Kalmijn1974}. As different animals have different EOD-frequencies, being exposed to stimuli of multiple distinct frequencies is part of the animal's everyday life \citep{Benda2020} and weakly non-linear interactions may occur in the electrosensory periphery. In communication contexts \citep{Walz2014, Henninger2018} the EODs of interacting fish superimpose and lead to periodic amplitude modulations (AMs or beats) of the self-generated EOD when two animals interact. Non-linear mechanisms in P-type electroreceptor afferents of the tuberous electrosensory system, the P-units, enabled them to encode these AMs in their time-dependent firing rates \citep{Bastian1981a, Walz2014, Middleton2006, Barayeu2023}. When multiple animals interact, the EOD interferences induce second-order amplitude modulations referred to as envelopes \citep{Yu2005, Fotowat2013, Stamper2012Envelope} and saturation non-linearities allow also for the encoding of these in the electrosensory periphery \citep{Savard2011}. Field observations have shown that courting males were able to react to the extremely weak signals of distant intruding males despite the strong foreground EOD of the nearby female \citep{Henninger2018}. Weakly nonlinear interactions at particular combinations of signals can be of immediate relevance in such settings as they could boost detectability of the faint signals \citep{Schlungbaum2023}. +Here we search for such weakly nonlinear responses in electroreceptors of the two electrosensory systems of the wave-type electric fish \textit{Apteronotus leptorhynchus}, i.e. the tuberous (active) and the ampullary (passive) electrosensory system. The electroreceptors of the active system are driven by the fish's self generated electric field generated by regular, quasi-sinusoidal, discharges of their electric organ (electric organ discharge or EOD) and encode disturbances of it \citep{Bastian1981a}. The electroreceptors of the passive system are tuned to lower-frequency exogeneous electric fields such as caused by muscle activity of prey \citep{Kalmijn1974}. As different animals have different EOD-frequencies, being exposed to stimuli of multiple distinct frequencies is part of the animal's everyday life \citep{Benda2020,Henninger2020} and weakly non-linear interactions may occur in the electrosensory periphery. In communication contexts \citep{Walz2014, Henninger2018} the EODs of interacting fish superimpose and lead to periodic amplitude modulations (AMs or beats) of the self-generated EOD when two animals interact. Non-linear mechanisms in P-type electroreceptor afferents of the tuberous electrosensory system, the P-units, enabled them to encode these AMs in their time-dependent firing rates \citep{Bastian1981a, Walz2014, Middleton2006, Barayeu2023}. When multiple animals interact, the EOD interferences induce second-order amplitude modulations referred to as envelopes \citep{Yu2005, Fotowat2013, Stamper2012Envelope} and saturation non-linearities allow also for the encoding of these in the electrosensory periphery \citep{Savard2011}. Field observations have shown that courting males were able to react to the extremely weak signals of distant intruding males despite the strong foreground EOD of the nearby female \citep{Henninger2018}. Weakly nonlinear interactions at particular combinations of signals can be of immediate relevance in such settings as they could boost detectability of the faint signals \citep{Schlungbaum2023}. %The population of P-units is heterogeneous with respect to their baseline firing rates as well as in their intrinsic noise levels, as quantified by the coefficient of variation (CV) of the interspike intervals (ISI) \citep{Grewe2017, Hladnik2023}. Low-CV P-units have a less noisy firing pattern that is closer to pacemaker firing, whereas high-CV P-units show a more irregular firing pattern that is closer to a Poisson process. The population of ampullary cells of the passive electrosensory system, on the other hand, is homogeneous in their response properties and CVs are low \citep{Grewe2017}. @@ -440,7 +440,7 @@ Here we search for such weakly nonlinear responses in electroreceptors of the tw Theoretical work on leaky integrate-and-fire and conductance-based models suggests a distinct structure of the second-order response function for neurons with low levels of intrinsic noise driven in the supra-threshold regime with low stimulus amplitudes (\figrefb{fig:lifresponse}, \citealp{Voronenko2017}). Here, we explored a large set of recordings of P-units and ampullary cells of the active and passive electrosensory systems of the brown ghost knifefish \textit{Apteronotus leptorhynchus} to search for such weakly nonlinear responses in real neurons. Our work is supported by simulations of LIF-based models of P-unit spiking. We start with demonstrating the basic concepts using example P-units and models. \subsection{Nonlinear responses in P-units stimulated with two beat frequencies}\notejg{stimulated with two beats?} -Without external stimulation, a P-unit is driven by the fish's own EOD alone (with EOD frequency \feod{}) and spontaneously fires action potentials at the baseline rate \fbase{}. Accordingly, the power spectrum of the baseline activity has a peak at \fbase{} (\subfigrefb{fig:motivation}{A}). Superposition of the receiver's EOD with an EOD of another fish with frequency $f_1$ results in a beat, a periodic amplitude modulation of the receiver's EOD. The frequency of the beat is given by the difference frequency $\Delta f_1 = f_1 - \feod$ between the two fish \notejg{(only valid for $f_1 < f_{EOD}/2)$}. P-units encode this beat in their firing rate \citep{Bastian1981a, Barayeu2023} and consequently, the power spectrum of the response has a peak at the beat frequency (\subfigrefb{fig:motivation}{B}). A second peak at the first harmonic of the beat frequency is indicative of a nonlinear process that here is easily identified by the clipping of the P-unit's firing rate at zero. Pairing the fish with another fish with a higher beat frequency $\Delta f_2 = f_2 - \feod$ results in a weaker response with a single peak in the response power spectrum (\subfigrefb{fig:motivation}{C}). The weaker response to this beat can be explained by the beat tuning of the cell \citep{Walz2014}. Note that $\Delta f_2$ has been deliberately chosen to match the recorded P-unit's baseline firing rate. +Without external stimulation, a P-unit is driven by the fish's own EOD alone (with EOD frequency \feod{}) and spontaneously fires action potentials at the baseline rate \fbase{}. Accordingly, the power spectrum of the baseline activity has a peak at \fbase{} (\subfigrefb{fig:motivation}{A}). Superposition of the receiver's EOD with an EOD of another fish with frequency $f_1$ results in a beat, a periodic amplitude modulation of the receiver's EOD. The frequency of the beat is given by the difference frequency $\Delta f_1 = f_1 - \feod$ between the two fish \notejg{(only valid for $f_1 < f_{EOD}/2)$}. P-units encode this beat in their firing rate \citep{Bastian1981a, Barayeu2023} and consequently, the power spectrum of the response has a peak at the beat frequency (\subfigrefb{fig:motivation}{B}). A second peak at the first harmonic of the beat frequency is indicative of a nonlinear process that here is easily identified by the clipping of the P-unit's firing rate at zero. Pairing the fish with another fish with a higher beat frequency $\Delta f_2 = f_2 - \feod > \Delta f_1$ results in a weaker response with a single peak in the response power spectrum, suggesting a linear response (\subfigrefb{fig:motivation}{C}). The weaker response to this beat can be explained by the beat tuning of the cell \citep{Walz2014}. Note that $\Delta f_2$ has been deliberately chosen to match the recorded P-unit's baseline firing rate. When stimulating with both foreign signals simultaneously, additional peaks appear in the response power spectrum at the sum \bsum{} and the difference frequency \bdiff{} (\subfigrefb{fig:motivation}{D}). Thus, the cellular response is not equal to the sum of the responses to the two beats presented separately. These peaks at the sum and the difference of the two stimulus frequencies are a hallmark of nonlinear interactions that by definition are absent in linear systems. @@ -453,11 +453,12 @@ When stimulating with both foreign signals simultaneously, additional peaks appe The stimuli used in \figref{fig:motivation} had the same not-small amplitude. Whether this stimulus conditions falls into the weakly nonlinear regime as in \citet{Voronenko2017} is not clear. In order to illustrate how the responses to two beat frequencies develop over a range of amplitudes we use a stochastic leaky-integrate-and-fire (LIF) based P-unit model fitted to a specific electrophysiologically measured cell \citep{Barayeu2023}. -At very low stimulus contrasts (less than approximately 0.5\,\%) the spectrum has peaks only at the stimulating beat frequencies (note, $|\Delat f_2| = f_{base}$, \subfigref{fig:nonlin_regime}{A}). The amplitudes of these peaks initially increase linearly with stimulus amplitude (\subfigref{fig:nonlin_regime}{E}), an indication of the linear response at low stimulus amplitudes. +At very low stimulus contrasts (less than approximately 0.5\,\% relative to the receiver's EOD amplitude) the spectrum has small peaks only at the beat frequencies (note, $|\Delat f_2| = f_{base}$, \subfigref{fig:nonlin_regime}{A}, green and purple). The amplitudes of these peaks initially increase linearly with stimulus amplitude (\subfigref{fig:nonlin_regime}{E}), an indication of the linear response at lowest stimulus amplitudes. + +The linear regime is followed by the weakly non-linear regime. In addition to peaks at the stimulus frequencies, peaks at the sum and the difference of the stimulus frequencies appear in the response spectrum (\subfigref{fig:nonlin_regime}{B}, orange and red). The amplitudes of these two peaks initially increase quadratically with stimulus amplitude (\subfigref{fig:nonlin_regime}{E}). -For higher stimulus contrasts, the linear regime is followed by the weakly non-linear regime. In addition to peaks at the beat frequencies (green and purple), peaks at the sum and the difference of the stimulus frequencies appear in the response spectrum (red and yellow, \subfigref{fig:nonlin_regime}{B}). The amplitudes of these two peaks initially increase quadratically with stimulus amplitude (\subfigref{fig:nonlin_regime}{E}). +At higher stimulus amplitudes (\subfigref{fig:nonlin_regime}{C \& D}) additional peaks appear in the response spectrum. The linear response and the weakly-nonlinear response start to deviate from their linear and quadratic dependence on amplitude (\subfigref{fig:nonlin_regime}{E}). The responses may even decrease for intermediate stimulus contrasts (\subfigref{fig:nonlin_regime}{D}). At high stimulus contrasts, additional non-linearities, in particular clipping of the firing rate, in the system shape the respones. -At higher stimulus amplitudes (\subfigref{fig:nonlin_regime}{C \& D}) additional peaks appear in the response spectrum. The linear response and the weakly-nonlinear response start to deviate from their linear and quadratic dependence on amplitude (\subfigref{fig:nonlin_regime}{E}). The responses may even decrease for intermediate stimulus contrasts (\subfigref{fig:nonlin_regime}{D}). For this example, we have chosen two specific stimulus (beat) frequencies. One of these matching the P-unit's baseline firing rate. In the following, however, we are interested in how the non-linear responses depend on different combinations of stimulus frequencies in the weakly nonlinear regime. @@ -471,7 +472,7 @@ Weakly nonlinear responses are expected in cells with sufficiently low intrinsic \caption{\label{fig:cells_suscept} Linear and nonlinear stimulus encoding in a low-CV P-unit (cell identifier ``2010-06-21-ai"). \figitem{A} Interspike interval (ISI) distribution of the cell's baseline activity, i.e. the cell is driven only by the unperturbed own electric field. The low CV of the ISIs indicates quite regular firing. \figitem{B} Power spectral density of the baseline response with peaks at the cell's baseline firing rate \fbase{} and the fish's EOD frequency \feod{}. \figitem{C} Random amplitude modulation stimulus (top, with cutoff frequency of 300\,Hz) and evoked responses (spike raster, bottom) of the same P-unit. The stimulus contrast (right) measures the strength of the AM. \figitem{D} Gain of the transfer function (first-order susceptibility), \Eqnref{linearencoding_methods}, computed from the responses to 10\,\% (light purple) and 20\,\% contrast (dark purple) RAM stimulation of 10\,s duration. \figitem{E, F} Absolute value of the second-order susceptibility, \Eqnref{eq:susceptibility}, for both the low and high stimulus contrast. Pink triangles mark vertical, horizontal, and diagonal lines where \fone, \ftwo{} or \fsum{} are equal to \fbase{}. \figitem{G} Second-order susceptibilities projected onto the diagonal (means of all anti-diagonals of the matrices shown in \panel{E, F}). Dots mark \fbase{}, horizontal dashed lines mark medians of the projected susceptibilities.} \end{figure*} -Noise stimuli, here random amplitude modulations (RAM) of the EOD (\subfigref{fig:cells_suscept}{C}, top trace, red line), are commonly used to characterize stimulus-driven responses of sensory neurons using transfer functions (first-order susceptibility), spike-triggered averages, or stimulus-response coherences. Here, we additionally estimate the second-order susceptibility to quantify nonlinear encoding. P-unit spikes align more or less clearly with fluctuations in the RAM stimulus. A higher stimulus intensity, here a higher contrast of the RAM relative to the EOD amplitude (see methods), entrains the P-unit response more clearly (light and dark purple for low and high contrast stimuli, \subfigrefb{fig:cells_suscept}{C}). Linear encoding, quantified by the transfer function \Eqnref{linearencoding_methods}, is similar for the two RAM contrasts in this low-CV P-unit (\subfigrefb{fig:cells_suscept}{D}), as expected for a linear system. The first-order susceptibility is low for low frequencies, peaks in the range below 100\,Hz and then falls off again% \notejb{Cite Moe paper?}. +Noise stimuli, here random amplitude modulations (RAM) of the EOD (\subfigref{fig:cells_suscept}{C}, top trace, red line), are commonly used to characterize stimulus-driven responses of sensory neurons using transfer functions (first-order susceptibility), spike-triggered averages, or stimulus-response coherences. Here, we additionally estimate the second-order susceptibility to quantify nonlinear encoding. P-unit spikes align more or less clearly with fluctuations in the RAM stimulus. A higher stimulus intensity, here a higher contrast of the RAM relative to the EOD amplitude (see methods), entrains the P-unit response more clearly (light and dark purple for low and high contrast stimuli, \subfigrefb{fig:cells_suscept}{C}). Linear encoding, quantified by the transfer function \Eqnref{linearencoding_methods}, is similar for the two RAM contrasts in this low-CV P-unit (\subfigrefb{fig:cells_suscept}{D}), as expected for a linear system. The first-order susceptibility is low for low frequencies, peaks in the range below 100\,Hz and then falls off again \notejb{Cite Moe paper and Benda2005?}. The second-order susceptibility, \Eqnref{eq:susceptibility}, quantifies the amplitude and phase of the stimulus-evoked response at the sum \fsum{} for each combination of two stimulus frequencies \fone{} and \ftwo{}. Large values of the second-order susceptibility indicate stimulus-evoked peaks in the response spectrum at the summed frequency that cannot be explained by linear response theory. Similar to the first-order susceptibility, the second-order susceptibility can be estimated directly from the response evoked by a RAM stimulus that stimulates the neuron with a whole range of frequencies simultaneously (\subfigsref{fig:cells_suscept}{E, F}). For a LIF-model driven in the supra-threshold regime, theory predicts nonlinear interactions between the two stimulus frequencies, when the two frequencies \fone{} and \ftwo{} or their sum \fsum{} exactly match the neuron's baseline firing rate \fbase{} \citep{Voronenko2017}. Only then, additional stimulus-evoked peaks appear in the spectrum of the spiking response that would show up in the second-order susceptibility as a horizontal, a vertical, and an anti-diagonal line (\subfigrefb{fig:lifresponse}{B}, pink triangle in \subfigsref{fig:cells_suscept}{E, F}). @@ -500,11 +501,11 @@ In the example recordings shown above (\figsrefb{fig:cells_suscept} and \fref{fi %\notejb{Since the model overestimated the sensitivity of the real P-unit, we adjusted the RAM contrast to 0.9\,\%, such that the resulting spike trains had the same CV as the electrophysiological recorded P-unit during the 2.5\,\% contrast stimulation (see table~\ref{modelparams} for model parameters).} \notejb{chi2 scale is higher than in real cell} -One reason could be simply too little data for a good estimate of the second-order susceptibility. Electrophysiological recordings are limited in time, and therefore only a limited number of trials, i.e. repetitions of the same RAM stimulus, are available. As a consequence, the cross-spectra, \Eqnref{eq:crosshigh}, are insufficiently averaged and the full structure of the second-order susceptibility might be hidden in finite-data noise. This experimental limitation can be overcome by using a computational model for the P-unit, a stochastic leaky integrate-and-fire model with adaptation current and dendritic preprocessing, and parameters fitted to the experimentally recorded P-unit (\figrefb{flowchart}) \citep{Barayeu2023}. The model faithfully reproduces the second-order susceptibility of a low-CV cell estimated from the same low number of trials as in the experiment ($\n{}=11$, compare \panel{A} and \panel[ii]{B} in \figrefb{model_and_data}). +One reason could be simply too little data for a good estimate of the second-order susceptibility. Electrophysiological recordings are limited in time, and therefore responses to only a limited number of trials, i.e. repetitions of the same RAM stimulus, are available. As a consequence, the cross-spectra, \Eqnref{eq:crosshigh}, are insufficiently averaged and the full structure of the second-order susceptibility might be hidden in finite-data noise. This experimental limitation can be overcome by using a computational model for the P-unit, a stochastic leaky integrate-and-fire model with adaptation current, dendritic preprocessing, and parameters fitted to the experimentally recorded P-unit (\figrefb{flowchart}) \citep{Barayeu2023}. The model faithfully reproduces the second-order susceptibility of a low-CV cell estimated from the same low number of trials as in the experiment ($\n{}=11$, compare \panel{A} and \panel[ii]{B} in \figrefb{model_and_data}). In model simulations we can increase the number of trials beyond what would be experimentally possible, here to one million (\subfigrefb{model_and_data}\,\panel[iii]{B}). The estimate of the second-order susceptibility indeed improves. It gets less noisy and the diagonal at \fsum{} is emphasized. However, the expected vertical and horizontal lines at \foneb{} and \ftwob{} are still mainly missing. -Using a broadband stimulus increases the effective input-noise level and this may linearize signal transmission and suppress potential nonlinear responses \citep{Longtin1993, Chialvo1997, Roddey2000, Voronenko2017}. Assuming that the intrinsic noise level in this P-unit is small enough, the full expected structure of the second-order susceptibility should appear in the limit of weak AMs. Again, this cannot be done experimentally, because the problem of insufficient averaging becomes even more severe for weak AMs (low contrast). In the model, however, we know the time course of the intrinsic noise and can use this knowledge to determine the susceptibilities by input-output correlations via the Furutsu-Novikov theorem \citep{Furutsu1963, Novikov1965}. This theorem, in its simplest form, states that the cross-spectrum $S_{x\eta}(\omega)$ of a Gaussian noise $\eta(t)$ driving a nonlinear system and the system's output $x(t)$ is proportional to the linear susceptibility according to $S_{x\eta}(\omega)=\chi(\omega)S_{\eta\eta}(\omega)$. Here $\chi(\omega)$ characterizes the linear response to an infinitely weak signal $s(t)$ in the presence of the background noise $\eta(t)$. Likewise, the nonlinear susceptibility can be determined in an analogous fashion from higher-order input-output cross-spectra (see methods, equations \eqref{eq:crosshigh} and \eqref{eq:susceptibility}) \citep{Egerland2020}. In line with an alternative derivation of the Furutsu-Novikov theorem \citep{Lindner2022}, we can split the total noise and consider a fraction of it as a stimulus. This allows us to calculate the susceptibility from the cross-spectrum between the output and this stimulus fraction of the noise. Adapting this approach to our P-unit model (see methods), we replace the intrinsic noise by an approximately equivalent RAM stimulus $s_\xi(t)$ and a weak remaining intrinsic noise $\sqrt{2D \, c_{\rm{noise}}} \cdot \xi(t)$ with $c_\text{noise} = 0.1$ (see methods, equations \eqref{eq:ram_split}, \eqref{eq:Noise_split_intrinsic}, \eqref{eq:Noise_split_intrinsic_dendrite}, \subfigrefb{model_and_data}\,\panel[i]{C}). We tune the amplitude of the RAM stimulus $s_\xi(t)$ such that the output firing rate and variability (CV) are the same as in the baseline activity (i.e. full intrinsic noise $\sqrt{2D}\xi(t)$ in the voltage equation but no RAM) and compute the cross-spectra between the RAM part of the noise $s_\xi(t)$ and the output spike train. This procedure has two consequences: (i) by means of the cross-spectrum between the output and \signalnoise, which is a large fraction of the noise, the signal-to-noise ratio of the measured susceptibilities is drastically improved; (ii) the total noise in the system has been reduced (by what was before the external RAM stimulus $s(t)$), which makes the system more nonlinear. For both reasons we now see the expected nonlinear features in the second-order susceptibility for a sufficient number of trials (\subfigrefb{model_and_data}\,\panel[iii]{C}), but not for a number of trials comparable to the experiment (\subfigrefb{model_and_data}\,\panel[ii]{C}). In addition to the strong response for \fsumb{}, we now also observe pronounced nonlinear responses at \foneb{} and \ftwob{} (vertical and horizontal lines, \subfigrefb{model_and_data}\,\panel[iii]{C}). +Using a broadband stimulus increases the effective input-noise level. This may linearize signal transmission and suppress potential nonlinear responses \citep{Longtin1993, Chialvo1997, Roddey2000, Voronenko2017}. Assuming that the intrinsic noise level in this P-unit is small enough, the full expected structure of the second-order susceptibility should appear in the limit of weak AMs. Again, this cannot be done experimentally, because the problem of insufficient averaging becomes even more severe for weak AMs (low contrast). In the model, however, we know the time course of the intrinsic noise and can use this knowledge to determine the susceptibilities by input-output correlations via the Furutsu-Novikov theorem \citep{Furutsu1963, Novikov1965}. This theorem, in its simplest form, states that the cross-spectrum $S_{x\eta}(\omega)$ of a Gaussian noise $\eta(t)$ driving a nonlinear system and the system's output $x(t)$ is proportional to the linear susceptibility according to $S_{x\eta}(\omega)=\chi(\omega)S_{\eta\eta}(\omega)$. Here $\chi(\omega)$ characterizes the linear response to an infinitely weak signal $s(t)$ in the presence of the background noise $\eta(t)$. Likewise, the nonlinear susceptibility can be determined in an analogous fashion from higher-order input-output cross-spectra (see methods, equations \eqref{eq:crosshigh} and \eqref{eq:susceptibility}) \citep{Egerland2020}. In line with an alternative derivation of the Furutsu-Novikov theorem \citep{Lindner2022}, we can split the total noise and consider a fraction of it as a stimulus. This allows us to calculate the susceptibility from the cross-spectrum between the output and this stimulus fraction of the noise. Adapting this approach to our P-unit model (see methods), we replace the intrinsic noise by an approximately equivalent RAM stimulus $s_\xi(t)$ and a weak remaining intrinsic noise $\sqrt{2D \, c_{\rm{noise}}} \cdot \xi(t)$ with $c_\text{noise} = 0.1$ (see methods, equations \eqref{eq:ram_split}, \eqref{eq:Noise_split_intrinsic}, \eqref{eq:Noise_split_intrinsic_dendrite}, \subfigrefb{model_and_data}\,\panel[i]{C}). We tune the amplitude of the RAM stimulus $s_\xi(t)$ such that the output firing rate and variability (CV) are the same as in the baseline activity (i.e. full intrinsic noise $\sqrt{2D}\xi(t)$ in the voltage equation but no RAM) and compute the cross-spectra between the RAM part of the noise $s_\xi(t)$ and the output spike train. This procedure has two consequences: (i) by means of the cross-spectrum between the output and \signalnoise, which is a large fraction of the noise, the signal-to-noise ratio of the measured susceptibilities is drastically improved; (ii) the total noise in the system has been reduced (by what was before the external RAM stimulus $s(t)$), which makes the system more nonlinear. For both reasons we now see the expected nonlinear features in the second-order susceptibility for a sufficient number of trials (\subfigrefb{model_and_data}\,\panel[iii]{C}), but not for a number of trials comparable to the experiment (\subfigrefb{model_and_data}\,\panel[ii]{C}). In addition to the strong response for \fsumb{}, we now also observe pronounced nonlinear responses at \foneb{} and \ftwob{} (vertical and horizontal lines, \subfigrefb{model_and_data}\,\panel[iii]{C}). Note, that the increased number of trials goes along with a substantial reduction of second-order susceptibility values (\subfigrefb{model_and_data}\,\panel[iii]{C}), that saturate in its peak values for $N>10^6$ (\figrefb{fig:trialnr}). This demonstrates the limited reliability of an estimate of the second-order susceptibility that is based on 11 trials only. However, we would like to point out that already the limited number of trials used in the experiments reveals key features of the nonlinear response. @@ -519,7 +520,8 @@ Using the RAM stimulation we found pronounced nonlinear responses in the limit t \caption{\label{fig:model_full} Using second-order susceptibility to predict responses to sine-wave stimuli. \figitem[]{A} Absolute value of the second-order susceptibility, \Eqnref{eq:susceptibility}, for both positive and negative frequencies. \susceptf{} was estimated from $N=10^6$ trials of model simulations in the noise-split condition (cell 2013-01-08-aa, see table~\ref{modelparams} for model parameters). White lines indicate zero frequencies. Nonlinear responses at \fsum{} are quantified in the upper right and lower left quadrants. Nonlinear responses at \fdiff{} are quantified in the upper left and lower right quadrants. The baseline firing rate of this cell was at $\fbase=120$\,Hz. The position of the orange/red letters corresponds to the beat frequencies used for the stimulation with pure sine waves in the subsequent panels and indicates the sum/difference of those beat frequencies. \figitem{B--E} Black line -- power spectral density of model simulations in response to stimulation with two pure sine waves, \fone{} and \ftwo, in addition to the receiving fish's own EOD (three-fish scenario). The contrast of beat beats is 0.02. Colored circles highlight the height of selected peaks in the power spectrum. Grey line -- power spectral density of model in the baseline condition. \figitem{B} The sum of the two beat frequencies match \fbase{}. \figitem{C} The difference of \fone{} and \ftwo{} match \fbase{}. \figitem{D} Only the first beat frequency matches \fbase{}. \figitem{E} None of the two beat frequencies matches \fbase{}.} \end{figure*} -However, the second-order susceptibility \Eqnref{eq:susceptibility} is a spectral measure that is based on Fourier transforms and thus is also defined for negative stimulus frequencies. The full \susceptf{} matrix is symmetric with respect to the origin. In the upper-right and lower-left quadrants of \susceptf{}, stimulus-evoked responses at \fsum{} are shown, whereas in the lower-right and upper-left quadrants nonlinear responses at the difference \fdiff{} are shown (\figref{fig:model_full}). The vertical and horizontal lines at \foneb{} and \ftwob{} are very pronounced in the upper-right quadrant of \subfigrefb{fig:model_full}{A} for nonlinear responses at \fsum{} and extend into the upper-left quadrant (representing \fdiff) where they fade out towards more negative $f_1$ frequencies. The peak in the response power-spectrum at \fdiff{} evoked by pure sine-wave stimulation (\subfigrefb{fig:motivation}{D}) is predicted by the horizontal line in the upper-left quadrant (\subfigrefb{fig:model_full}{A}, \citealp{Schlungbaum2023}). +However, the second-order susceptibility \Eqnref{eq:susceptibility} is a spectral measure that is based on Fourier transforms and thus is also defined for negative stimulus frequencies. The full \susceptf{} matrix is symmetric with respect to the origin. In the upper-right and lower-left quadrants of \susceptf{}, stimulus-evoked responses at \fsum{} are shown, whereas in the lower-right and upper-left quadrants nonlinear responses at the difference \fdiff{} are shown (\figref{fig:model_full}). The vertical and horizontal lines at \foneb{} and \ftwob{} are very pronounced in the upper-right quadrant of \subfigrefb{fig:model_full}{A} for nonlinear responses at \fsum{} and extend into the upper-left quadrant (representing \fdiff), where they fade out towards more negative $f_1$ frequencies. The peak in the response power-spectrum at \fdiff{} evoked by pure sine-wave stimulation (\subfigrefb{fig:motivation}{D}) is predicted by the horizontal line in the upper-left quadrant (\subfigrefb{fig:model_full}{A}, \citealp{Schlungbaum2023}). + Is it possible to predict nonlinear responses in a three-fish setting based on second-order susceptibility matrices estimated from RAM stimulation (\subfigrefb{fig:model_full}{A})? We test this by stimulating the same model with two weak beats (\subfigrefb{fig:model_full}{B--E}). If we choose a frequency combination in which the sum of the two beat frequencies is equal to the model's baseline firing rate \fbase{}, a peak at the sum of the two beat frequencies appears in the power spectrum of the response (\subfigrefb{fig:model_full}{B}), as expected from \suscept. If instead we choose two beat frequencies that differ from \fbase{}, a peak is present at the difference frequency (\subfigrefb{fig:model_full}{C}). If only one of the beat frequency matches \fbase{}, both, a peak at the sum and at the difference frequency are present in the P-unit response (\subfigrefb{fig:model_full}{D}). And if none of these conditions are met, neither a peak at the sum nor at the difference of the two beat frequencies appears (\subfigrefb{fig:model_full}{E}). \begin{figure*}[tp] @@ -632,7 +634,7 @@ The neurons were classified into cell types during the recording by the experime The stimulus was isolated from the ground (ISO-02V, npi-electronics, Tamm, Germany) and delivered via two horizontal carbon rods (30 cm length, 8 mm diameter) located $15\,\centi\meter$ laterally to the fish (\figrefb{fig:setup}, gray bars). The stimulus was calibrated with respect to the local EOD. \begin{figure*}[t] - \includegraphics[width=\columnwidth]{Settup} + %\includegraphics[width=\columnwidth]{Settup} \caption{\label{fig:setup} Electrophysiolocical recording setup. The fish, depicted as a black scheme and surrounded by isopotential lines, was positioned in the center of the tank. Blue triangle -- electrophysiological recordings were conducted in the posterior anterior lateral line nerve (pALLN). Gray horizontal bars -- electrodes for the stimulation. Green vertical bars -- electrodes to measure the \textit{global EOD} placed isopotential to the stimulus, i.e. recording fish's unperturbed EOD. Red dots -- electrodes to measure the \textit{local EOD} picking up the combination of fish's EOD and the stimulus. The local EOD was measured with a distance of 1 \,cm between the electrodes. All measured signals were amplified, filtered, and stored for offline analysis.} \end{figure*} @@ -881,6 +883,11 @@ In the here used model a small portion of the original noise was assigned to the % \bibliographystyle{apalike} %or any other style you like %\bibliography{references} %\bibliography{journalsabbrv,references} + + +\setcounter{figure}{0} +\renewcommand{\thefigure}{S\arabic{figure}} + \newpage \section{Supporting information} diff --git a/trialnr.pdf b/trialnr.pdf index a7d105a..e728aa0 100644 Binary files a/trialnr.pdf and b/trialnr.pdf differ