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@ -424,7 +424,7 @@ We here analyze nonlinear responses in two types of primary electroreceptor affe
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\includegraphics[width=\columnwidth]{lifsuscept}
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\caption{\label{fig:lifresponse} First- and second-order response functions of the leaky integrate-and-fire model. \figitem{A} Magnitude of the first-order (linear) response function $|\chi_1(f)|$, also known as the ``gain'' function, quantifies the response amplitude relative to the stimulus amplitude, both measured at the same stimulus frequency. \figitem{B} Magnitude of the second-order (non-linear) response function $|\chi_2(f_1, f_2)|$ quantifies the response at the sum of two stimulus frequencies. 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 sum of the two frequencies. The plots show the analytical solutions from \citet{Lindner2001} and \citet{Voronenko2017} with $\mu = 1.1$ and $D = 0.001$. Note that the leaky integrate-and-fire model is formulated without dimensions, frequencies are given in multiples of the inverse membrane time constant.}
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\end{figure*}
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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 transfer function or first-oder susceptibility 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 neural systems \citep{Borst1999}. Nonlinear mechanisms, on the other hand, are key on different levels of neural processing. 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.
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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 transfer function or first-order susceptibility 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 neural systems \citep{Borst1999}. Nonlinear mechanisms, on the other hand, are key on different levels of neural processing. 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.
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The transfer function that describes the 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. Nonlinear interactions of two stimulus frequencies generate peaks in the response spectrum at the sum and the difference of the two. 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.
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@ -452,7 +452,7 @@ When stimulating with both foreign signals simultaneously, additional peaks appe
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\begin{figure*}[p]
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\includegraphics[width=\columnwidth]{regimes}
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\caption{\label{fig:regimes} Linear and nonlinear responses of a model P-unit in a three-fish setting in dependence on stimulus amplitudes. The model P-unit (identifier ``2018-05-08-ad'', table~\ref{modelparams}) was stimulated with two sinewaves of equal amplitude (contrast) at difference frequencies $\Delta f_1=40$\,Hz and $\Delta f_2=228$\,Hz relative the receiver's EOD frequency. $\Delta f_2$ was set to match the baseline firing rate $r$ of the P-unit. \figitem{A--D} Top: the stimulus, an amplitude modulation of the receiver's EOD resulting from the stimulation with the two sine waves. The contrasts of both beats increase from \panel{A} to \panel{D} as indicated. Middle: Spike raster of the model P-unit response. Bottom: power spectrum of the firing rate estimated from the spike raster. \figitem{A} At low stimulus contrasts the response is linear. The only peaks in the response spectrum are at the two stimulating beat frequencies (green and purple marker). \figitem{B} At moderately higher stimulus contrast, the peaks in the response spectrum at the two beat frequencies are larger. \figitem{C} At intermediate stimulus contrasts, nonlinear responses start to appear at the sum and the difference of the stimulus frequencies (orange and red marker). \figitem{D} At higher stimulus contrasts additional peaks appear in the power spectrum. \figitem{E} Amplitude of the linear (at $\Delta f_1$ and $\Delta f_2$) and nonlinear (at $\Delta f_2 - \Delta f_1$ and $\Delta f_1 + \Delta f_2$) responses of the model P-unit as a function of beat contrast (thick lines). Thin lines indicate the initial linear and quadratic dependence on stimulus amplitude for the linear and nonlinear responses, respectively. In the linear regime, below a stimulus contrast of about 1.2\,\% (left vertical line), the only peaks in the response spectrum are at the stimulus frequencies. In the weakly nonlinear regime up to a contrast of about 3.5\,\% peaks arise at the sum and the difference of the two stimulus frequencies. At stronger stimulation the amplitudes of these nonlinear repsonses deviate from the quadratic dependency on stimulus contrast.}
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\caption{\label{fig:regimes} Linear and nonlinear responses of a model P-unit in a three-fish setting in dependence on stimulus amplitudes. The model P-unit (identifier ``2018-05-08-ad'') was stimulated with two sine waves of equal amplitude (contrast) at difference frequencies $\Delta f_1=40$\,Hz and $\Delta f_2=228$\,Hz relative the receiver's EOD frequency. $\Delta f_2$ was set to match the baseline firing rate $r$ of the P-unit. \figitem{A--D} Top: the stimulus, an amplitude modulation of the receiver's EOD resulting from the stimulation with the two sine waves. The contrasts of both beats increase from \panel{A} to \panel{D} as indicated. Middle: Spike raster of the model P-unit response. Bottom: power spectrum of the firing rate estimated from the spike raster. \figitem{A} At low stimulus contrasts the response is linear. The only peaks in the response spectrum are at the two stimulating beat frequencies (green and purple marker). \figitem{B} At moderately higher stimulus contrast, the peaks in the response spectrum at the two beat frequencies are larger. \figitem{C} At intermediate stimulus contrasts, nonlinear responses start to appear at the sum and the difference of the stimulus frequencies (orange and red marker). \figitem{D} At higher stimulus contrasts additional peaks appear in the power spectrum. \figitem{E} Amplitude of the linear (at $\Delta f_1$ and $\Delta f_2$) and nonlinear (at $\Delta f_2 - \Delta f_1$ and $\Delta f_1 + \Delta f_2$) responses of the model P-unit as a function of beat contrast (thick lines). Thin lines indicate the initial linear and quadratic dependence on stimulus amplitude for the linear and nonlinear responses, respectively. In the linear regime, below a stimulus contrast of about 1.2\,\% (left vertical line), the only peaks in the response spectrum are at the stimulus frequencies. In the weakly nonlinear regime up to a contrast of about 3.5\,\% peaks arise at the sum and the difference of the two stimulus frequencies. At stronger stimulation the amplitudes of these nonlinear responses deviate from the quadratic dependency on stimulus contrast.}
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\end{figure*}
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The stimuli used in \figref{fig:motivation} had the same not-small amplitude. Whether this stimulus condition 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}.
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@ -489,10 +489,10 @@ Overall we observed in 17\,\% of the 159 P-units ridges where the stimulus frequ
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\begin{figure*}[p]
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\includegraphics[width=\columnwidth]{ampullaryexamplecell}
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\caption{\label{fig:ampullary} Linear and nonlinear stimulus encoding in example ampullary afferents. \figitem{A} Interspike interval (ISI) distribution of the cell's baseline activity (cell identifier ``2012-05-15-ac''). The very low CV of the ISIs indicates almost perfect periodic spiking. \figitem{B} Power spectral density of baseline activity with peaks at the cell's baseline firing rate and its harmonics. Ampullary afferents do not respond to the fish's EOD frequency, $f_{\text{EOD}}$ --- a sharp peak at $f_{\text{EOD}}$ is missing. \figitem{C} Band-limited white noise stimulus (top, red, with a cutoff frequency of 150\,Hz) added to the fish's self-generated electric field (no amplitude modulation!) and spike raster of the evoked responses (bottom) for two stimulus contrasts as indicated (right). \figitem{D} Gain of the transfer function, \eqnref{linearencoding_methods}, of the responses to stimulation with 5\,\% (light green) and 10\,\% contrast (dark green) of 10\,s duration. \figitem{E, F} Absolute value of the second-order susceptibility, \eqnref{chi2}, for both stimulus contrasts as indicated. Both show a clear anti-diagonal where the two stimulus frequencies add up to the afferent's baseline firing rate. \figitem{G} Projections of the second-order susceptibilities in \panel{E, F} onto the diagonal. \figitem{H} ISI distributions (top) and second-order susceptibilites (bottom) of three more example afferents with clear anti-diagonals (``2010-11-26-an'', ``2010-11-08-aa'', ``2011-02-18-ab''). \figitem{I} Some ampullary afferents do not show any structure in their second-order susceptibility (``2014-01-16-aj'').}
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\caption{\label{fig:ampullary} Linear and nonlinear stimulus encoding in example ampullary afferents. \figitem{A} Interspike interval (ISI) distribution of the cell's baseline activity (cell identifier ``2012-05-15-ac''). The very low CV of the ISIs indicates almost perfect periodic spiking. \figitem{B} Power spectral density of baseline activity with peaks at the cell's baseline firing rate and its harmonics. Ampullary afferents do not respond to the fish's EOD frequency, $f_{\text{EOD}}$ --- a sharp peak at $f_{\text{EOD}}$ is missing. \figitem{C} Band-limited white noise stimulus (top, red, with a cutoff frequency of 150\,Hz) added to the fish's self-generated electric field (no amplitude modulation!) and spike raster of the evoked responses (bottom) for two stimulus contrasts as indicated (right). \figitem{D} Gain of the transfer function, \eqnref{linearencoding_methods}, of the responses to stimulation with 5\,\% (light green) and 10\,\% contrast (dark green) of 10\,s duration. \figitem{E, F} Absolute value of the second-order susceptibility, \eqnref{chi2}, for both stimulus contrasts as indicated. Both show a clear anti-diagonal where the two stimulus frequencies add up to the afferent's baseline firing rate. \figitem{G} Projections of the second-order susceptibilities in \panel{E, F} onto the diagonal. \figitem{H} ISI distributions (top) and second-order susceptibilities (bottom) of three more example afferents with clear anti-diagonals (``2010-11-26-an'', ``2010-11-08-aa'', ``2011-02-18-ab''). \figitem{I} Some ampullary afferents do not show any structure in their second-order susceptibility (``2014-01-16-aj'').}
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\end{figure*}
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Electric fish possess an additional electrosensory system, the passive or ampullary electrosensory system, that responds to low-frequency exogenous electric stimuli. The population of ampullary afferents is much less heterogeneous, and known for the much lower CVs of their baseline ISIs ($0.06 < \text{CV}_{\text{base}} < 0.22$, \citealp{Grewe2017}). Ampullary cells do not phase-lock to the high-frequency EOD and the ISIs are unimodally distributed (\subfigrefb{fig:ampullary}{A}). As a consequence of the high regularity of their baseline spiking activity, the corresponding power spectrum shows distinct peaks at the baseline firing rate $r$ and its harmonics. Since the cells do not respond to the self-generated EOD, there is no sharp peak at \feod{} (\subfigrefb{fig:ampullary}{B}). When driven by a band-limited white noise stimulus (note: for ampullary afferents this is not an AM stimulus, \subfigref{fig:ampullary}{C}), ampullary afferents exhibit very pronounced ridges in the second-order susceptibility, where $f_1 + f_2$ is equal to $r$ or its harmonics (yellow anti-diagonals in \subfigrefb{fig:ampullary}{E--H}), implying strong nonlinear response components at these frequency combinations (\subfigrefb{fig:ampullary}{G}, top). With higher stimulus contrasts these bands get weaker (\subfigrefb{fig:ampullary}{F}), the projection onto the diagonal loses its distinct peak at $r$, and its overall level is reduced (\subfigrefb{fig:ampullary}{G}, bottom). Some ampullary afferents (27\,\% of 30 afferents), however, do not show any such structure in their second-order susceptibility (\subfigrefb{fig:ampullary}{I}).
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Electric fish possess an additional electrosensory system, the passive or ampullary electrosensory system, that responds to low-frequency exogenous electric stimuli. The population of ampullary afferents is much less heterogeneous, and known for the much lower CVs of their baseline ISIs ($0.06 < \text{CV}_{\text{base}} < 0.22$, \citealp{Grewe2017}). Ampullary cells do not phase-lock to the high-frequency EOD and the ISIs have an unimodal distribution (\subfigrefb{fig:ampullary}{A}). As a consequence of the high regularity of their baseline spiking activity, the corresponding power spectrum shows distinct peaks at the baseline firing rate $r$ and its harmonics. Since the cells do not respond to the self-generated EOD, there is no sharp peak at \feod{} (\subfigrefb{fig:ampullary}{B}). When driven by a band-limited white noise stimulus (note: for ampullary afferents this is not an AM stimulus, \subfigref{fig:ampullary}{C}), ampullary afferents exhibit very pronounced ridges in the second-order susceptibility, where $f_1 + f_2$ is equal to $r$ or its harmonics (yellow anti-diagonals in \subfigrefb{fig:ampullary}{E--H}), implying strong nonlinear response components at these frequency combinations (\subfigrefb{fig:ampullary}{G}, top). With higher stimulus contrasts these bands get weaker (\subfigrefb{fig:ampullary}{F}), the projection onto the diagonal loses its distinct peak at $r$, and its overall level is reduced (\subfigrefb{fig:ampullary}{G}, bottom). Some ampullary afferents (27\,\% of 30 afferents), however, do not show any such structure in their second-order susceptibility (\subfigrefb{fig:ampullary}{I}).
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\subsection{Model-based estimation of the second-order susceptibility}
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@ -500,7 +500,7 @@ In the example recordings shown above (\figsrefb{fig:punit} and \fref{fig:ampull
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\begin{figure*}[p]
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\includegraphics[width=\columnwidth]{noisesplit}
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\caption{\label{fig:noisesplit} Estimation of second-order susceptibilities. \figitem{A} \suscept{} (right) estimated from $N=198$ 256\,ms long FFT segments of an electrophysiological recording of another P-unit (cell ``2017-07-18-ai'', $r=78$\,Hz, CV$_{\text{base}}=0.22$) driven with a RAM stimulus with contrast 5\,\% (left). \figitem[i]{B} \textit{Standard condition} of model simulations with intrinsic noise (bottom) and a RAM stimulus (top). \figitem[ii]{B} \suscept{} estimated from simulations of the cell's LIF model counterpart (cell ``2017-07-18-ai'', table~\ref{modelparams}) based on the same RAM contrast and number of $N=100$ FFT segments. As in the electrophysiological recording only a weak anti-diagonal is visible. \figitem[iii]{B} Same as \panel[ii]{B} but using $10^6$ FFT segments. Now, the expected triangular structure is revealed. \figitem[iv]{B} Convergence of the \suscept{} estimate as a function of FFT segments. \figitem{C} At a lower stimulus contrast of 1\,\% the estimate did not converge yet even for $10^6$ FFT segments. The triangular structure is not revealed yet. \figitem[i]{D} Same as in \panel[i]{B} but in the \textit{noise split} condition: there is no external RAM signal (red) driving the model. Instead, a large part (90\,\%) of the total intrinsic noise is treated as a signal and is presented as an equivalent amplitude modulation ($s_{\xi}(t)$, orange, 10.6\,\% contrast), while the intrinsic noise is reduced to 10\,\% of its original strength (bottom, see methods for details). \figitem[i]{D} 100 FFT segments are still not sufficient for estimating \suscept{}. \figitem[iii]{D} Simulating one million segments reveals the full expected trangular structure of the second-order susceptibility. \figitem[iv]{D} In the noise-split condition, the \suscept{} estimate converges already at about $10^{4}$ FFT segments.}
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\caption{\label{fig:noisesplit} Estimation of second-order susceptibilities. \figitem{A} \suscept{} (right) estimated from $N=198$ 256\,ms long FFT segments of an electrophysiological recording of another P-unit (cell ``2017-07-18-ai'', $r=78$\,Hz, CV$_{\text{base}}=0.22$) driven with a RAM stimulus with contrast 5\,\% (left). \figitem[i]{B} \textit{Standard condition} of model simulations with intrinsic noise (bottom) and a RAM stimulus (top). \figitem[ii]{B} \suscept{} estimated from simulations of the cell's LIF model counterpart (cell ``2017-07-18-ai'') based on the same RAM contrast and number of $N=100$ FFT segments. As in the electrophysiological recording only a weak anti-diagonal is visible. \figitem[iii]{B} Same as \panel[ii]{B} but using $10^6$ FFT segments. Now, the expected triangular structure is revealed. \figitem[iv]{B} Convergence of the \suscept{} estimate as a function of FFT segments. \figitem{C} At a lower stimulus contrast of 1\,\% the estimate did not converge yet even for $10^6$ FFT segments. The triangular structure is not revealed yet. \figitem[i]{D} Same as in \panel[i]{B} but in the \textit{noise split} condition: there is no external RAM signal (red) driving the model. Instead, a large part (90\,\%) of the total intrinsic noise is treated as a signal and is presented as an equivalent amplitude modulation ($s_{\xi}(t)$, orange, 10.6\,\% contrast), while the intrinsic noise is reduced to 10\,\% of its original strength (bottom, see methods for details). \figitem[i]{D} 100 FFT segments are still not sufficient for estimating \suscept{}. \figitem[iii]{D} Simulating one million segments reveals the full expected triangular structure of the second-order susceptibility. \figitem[iv]{D} In the noise-split condition, the \suscept{} estimate converges already at about $10^{4}$ FFT segments.}
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\end{figure*}
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One simple reason could be the lack of data, i.e. the estimation of the second-order susceptibility is not good enough. Electrophysiological recordings are limited in time, and therefore only a limited number of trials, here repeated presentations of the same frozen RAM stimulus, are available. In our data set we have 1 to 199 trials (median: 10) of RAM stimuli with a duration ranging from 2 to 5\,s (median: 8\,s), resulting in a total duration of 30 to 400\,s. Using a resolution of 0.5\,ms and FFT segments of 512 samples this yields 105 to 1520 available FFT segments for a specific RAM stimulus. As a consequence, the cross-spectra, \eqnref{crossxss}, 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 the P-unit estimated from the same low number of FFT (fast fourier transform) segments as in the experiment ($N=100$, compare faint anti-diagonal in the bottom left corner of the second-order susceptibility in \panel[ii]{A} and \panel[ii]{B} in \figrefb{fig:noisesplit}).
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@ -514,7 +514,7 @@ Using a broadband stimulus increases the effective input-noise level. This may l
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\begin{figure}[p]
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\includegraphics[width=\columnwidth]{modelsusceptcontrasts}
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\caption{\label{fig:modelsusceptcontrasts}Dependence of second order susceptibility on stimulus contrast. \figitem{A} Second-order susceptibilities estimated for increasing stimulus contrasts of $c=0, 1, 3$ and $10$\,\% as indicated ($N=10^7$ FFT segments for $c=1$\,\%, $N=10^6$ segments for all other contrasts). $c=0$\,\% refers to the noise-split configuration (limit to vanishing external RAM signal, see \subfigrefb{fig:noisesplit}{D}). Shown are simulations of the P-unit model cell ``2017-07-18-ai'' (table~\ref{modelparams}) with a baseline firing rate of $82$\,Hz and CV$_{\text{base}}=0.23$. The cell shows a clear triangular pattern in its second-order susceptibility even up to a contrast of $10$\,\%. Note, that for $c=1$\,\% (\panel[ii]{D}), the estimate did not converge yet. \figitem{B} Cell ``2012-12-13-ao'' (baseline firing rate of $146$\,Hz, CV$=0.23$) also has strong interactions at its baseline firing rate that survive up to a stimulus contrast of $3$\,\%. \figitem{C} Model cell ``2012-12-20-ac'' (baseline firing rate of $212$\,Hz, CV$=0.26$) shows a weak triangular structure in the second-order susceptibility that vanishes for stimulus contrasts larger than $1$\,\%. \figitem{D} Cell ``2013-01-08-ab'' (baseline firing rate of $218$\,Hz, CV$=0.55$) does not show any triangular pattern in its second-order susceptibility. Nevertheless, interactions between low stimulus frequencies become substantial at higher contrasts. \figitem{E} The presence of an elevated second-order susceptibility where the stimulus frequency add up to the neuron's baseline frequency, can be identified by the susceptibility index (SI($r$), \eqnref{siindex2}) greater than one (horizontal black line). The SI($r$) (density to the right) is plotted as a function of the model neuron's baseline CV for all $39$ model cells (table~\ref{modelparams}). Model cells have been visually categorized based on the strong (11 cells) or weak (5 cells) presence of a triangular pattern in their second-order susceptibility estimated in the noise-split configuration (legend). The cells from \panel{A--D} are marked by black circles. Pearson's correlation coefficients $r$, the corresponding significance level $p$ and regression line (dashed gray line) are indicated. The higher the stimulus contrast, the less cells show weakly nonlinear interactions as expressed by the triangular structure in the second-order susceptibility.}
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\caption{\label{fig:modelsusceptcontrasts}Dependence of second order susceptibility on stimulus contrast. \figitem{A} Second-order susceptibilities estimated for increasing stimulus contrasts of $c=0, 1, 3$ and $10$\,\% as indicated ($N=10^7$ FFT segments for $c=1$\,\%, $N=10^6$ segments for all other contrasts). $c=0$\,\% refers to the noise-split configuration (limit to vanishing external RAM signal, see \subfigrefb{fig:noisesplit}{D}). Shown are simulations of the P-unit model cell ``2017-07-18-ai'') with a baseline firing rate of $82$\,Hz and CV$_{\text{base}}=0.23$. The cell shows a clear triangular pattern in its second-order susceptibility even up to a contrast of $10$\,\%. Note, that for $c=1$\,\% (\panel[ii]{D}), the estimate did not converge yet. \figitem{B} Cell ``2012-12-13-ao'' (baseline firing rate of $146$\,Hz, CV$=0.23$) also has strong interactions at its baseline firing rate that survive up to a stimulus contrast of $3$\,\%. \figitem{C} Model cell ``2012-12-20-ac'' (baseline firing rate of $212$\,Hz, CV$=0.26$) shows a weak triangular structure in the second-order susceptibility that vanishes for stimulus contrasts larger than $1$\,\%. \figitem{D} Cell ``2013-01-08-ab'' (baseline firing rate of $218$\,Hz, CV$=0.55$) does not show any triangular pattern in its second-order susceptibility. Nevertheless, interactions between low stimulus frequencies become substantial at higher contrasts. \figitem{E} The presence of an elevated second-order susceptibility where the stimulus frequency add up to the neuron's baseline frequency, can be identified by the susceptibility index (SI($r$), \eqnref{siindex}) greater than one (horizontal black line). The SI($r$) (density to the right) is plotted as a function of the model neuron's baseline CV for all $39$ model cells. Model cells have been visually categorized based on the strong (11 cells) or weak (5 cells) presence of a triangular pattern in their second-order susceptibility estimated in the noise-split configuration (legend). The cells from \panel{A--D} are marked by black circles. Pearson's correlation coefficients $r$, the corresponding significance level $p$ and regression line (dashed gray line) are indicated. The higher the stimulus contrast, the less cells show weakly nonlinear interactions as expressed by the triangular structure in the second-order susceptibility.}
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\end{figure}
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\subsection{Weakly nonlinear interactions in many model cells}
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By just looking at the second-order susceptibilities estimated using the noise-split method (first column of \figrefb{fig:modelsusceptcontrasts}) we can readily identify strong triangular patterns in 11 of the 39 model cells (28\,\%, see \figrefb{fig:modelsusceptcontrasts}\,\panel[i]{A}\&\panel[i]{B} for two examples). In another 5 cells (13\,\%) the triangle is much weaker and sits on top of a smooth bump of elevated second-order susceptibility (\figrefb{fig:modelsusceptcontrasts}\,\panel[i]{C} shows an example). The remaining 23 model cells (59\,\%) show no triangle (see \figrefb{fig:modelsusceptcontrasts}\,\panel[i]{D} for an example).
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This categorization is supported by the susceptibility index, SI($r$), \eqnref{siindex2}, which quantifies the height of the ridge where the stimulus frequencies add up to the neuron's baseline firing rate relative to the background. Values above one indicate an elevated ridge. The absence of such a ridge results in values close to one. Indeed, the cells showing only a weak triangle (orange) arise out of values around one and the cells showing strong triangles (red) have consistently SI($r$) values exceeding 1.8 (\figrefb{fig:modelsusceptcontrasts}\,\panel[i]{E}).
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This categorization is supported by the susceptibility index, SI($r$), \eqnref{siindex}, which quantifies the height of the ridge where the stimulus frequencies add up to the neuron's baseline firing rate relative to the background. Values above one indicate an elevated ridge. The absence of such a ridge results in values close to one. Indeed, the cells showing only a weak triangle (orange) arise out of values around one and the cells showing strong triangles (red) have consistently SI($r$) values exceeding 1.8 (\figrefb{fig:modelsusceptcontrasts}\,\panel[i]{E}).
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The SI($r$) correlates with the cell's CV of its baseline interspike intervals ($r=-0.60$, $p<0.001$). The lower the cell's CV$_{\text{base}}$, the higher the SI($r$) value and thus the stronger the triangular structure of its second-order susceptibility. The model cells with the most distinct triangular pattern in their second-order susceptibility are the ones with the lowest CVs, hinting at low intrinsic noise levels.
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@ -540,7 +540,7 @@ At a RAM contrast of 3\,\% the SI($r$) values become smaller (\figrefb{fig:model
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\subsection{Weakly nonlinear interactions can be deduced from limited data}
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Estimating second-order susceptibilities reliably requires large numbers (millions) of FFT segments (\figrefb{fig:noisesplit}). Electrophysiological measurements, however, suffer from limited recording durations and hence limited numbers of available FFT segments and estimating weakly nonlinear interactions from just a few hundred segments appears futile. The question arises, to what extend such limited-data estimates are still informative?
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The second-order susceptibility matrices that are based on only 100 segments look flat and noisy, lacking the triangular structure (\subfigref{fig:modelsusceptlown}{B}). The anti-diagonal ridge, however, where the sum of the stimulus frequencies matches the neuron's baseline firing rate, seems to be present whenever the converged estimate shows a clear triangular structure (compare \subfigref{fig:modelsusceptlown}{B} and \subfigref{fig:modelsusceptlown}{A}). The SI($r$) characterizes the height of the ridge in the second-oder susceptibility plane at the neuron's baseline firing rate $r$. Comparing SI($r$) values based on 100 FFT segements to the ones based on one or ten million segments for all 39 model cells (\subfigrefb{fig:modelsusceptlown}{C}) supports this impression. They correlate quite well at contrasts of 1\,\% and 3\,\% ($r=0.9$, $p\ll 0.001$). At a contrast of 10\,\% this correlation is weaker ($r=0.38$, $p<0.05$), because there are only three cells left with SI($r$) values greater than 1.2. Despite the good correlations, care has to be taken to set a threshold on the SI($r$) values for deciding whether a triangular structure would emerge for a much higher number of segments. Because at low number of segments the estimates are noisier, there could be false positives for a too low threshold. Setting the threshold to 1.8 avoids false positives for the price of a few false negatives.
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The second-order susceptibility matrices that are based on only 100 segments look flat and noisy, lacking the triangular structure (\subfigref{fig:modelsusceptlown}{B}). The anti-diagonal ridge, however, where the sum of the stimulus frequencies matches the neuron's baseline firing rate, seems to be present whenever the converged estimate shows a clear triangular structure (compare \subfigref{fig:modelsusceptlown}{B} and \subfigref{fig:modelsusceptlown}{A}). The SI($r$) characterizes the height of the ridge in the second-oder susceptibility plane at the neuron's baseline firing rate $r$. Comparing SI($r$) values based on 100 FFT segments to the ones based on one or ten million segments for all 39 model cells (\subfigrefb{fig:modelsusceptlown}{C}) supports this impression. They correlate quite well at contrasts of 1\,\% and 3\,\% ($r=0.9$, $p\ll 0.001$). At a contrast of 10\,\% this correlation is weaker ($r=0.38$, $p<0.05$), because there are only three cells left with SI($r$) values greater than 1.2. Despite the good correlations, care has to be taken to set a threshold on the SI($r$) values for deciding whether a triangular structure would emerge for a much higher number of segments. Because at low number of segments the estimates are noisier, there could be false positives for a too low threshold. Setting the threshold to 1.8 avoids false positives for the price of a few false negatives.
|
||||
|
||||
Overall, observing SI($r$) values greater than about 1.8, even for a number of FFT segments as low as one hundred, seems to be a reliable indication for a triangular structure in the second-order susceptibility at the corresponding stimulus contrast. Small stimulus contrasts of 1\,\% are less informative, because of their bad signal-to-noise ratio. Intermediate stimulus contrasts around 3\,\% seem to be optimal, because there, most cells still have a triangular structure in their susceptibility and the signal-to-noise ratio is better. At RAM stimulus contrasts of 10\,\% or higher the signal-to-noise ratio is even better, but only few cells remain with weak triangularly shaped susceptibilities that might be missed as a false positives.
|
||||
|
||||
@ -548,8 +548,8 @@ Overall, observing SI($r$) values greater than about 1.8, even for a number of F
|
||||
\includegraphics[width=\columnwidth]{dataoverview}
|
||||
\caption{\label{fig:dataoverview} Nonlinear responses in P-units and
|
||||
ampullary afferents. The second-order susceptibility is condensed
|
||||
into the susceptibility index, SI($r$) \eqnref{siindex},
|
||||
that quantifies the relative amplitude of the ridge where the two
|
||||
into the susceptibility index, SI($r$) \eqnref{siindex}, that
|
||||
quantifies the relative amplitude of the ridge where the two
|
||||
stimulus frequencies add up to the cell's baseline firing rate $r$
|
||||
(see \subfigrefb{fig:punit}{G}). In both the models and the
|
||||
experimental data, the SI($r$) was estimated based on 100 FFT
|
||||
@ -567,17 +567,16 @@ Overall, observing SI($r$) values greater than about 1.8, even for a number of F
|
||||
right. The horizontal dashed line marks a threshold for SI($r$)
|
||||
values at 1.8 and the percentages to the right denote the
|
||||
fractions of samples above and below this threshold. \figitem{A}
|
||||
The SI($r$) of all 39 model P-units (table~\ref{modelparams})
|
||||
measured with RAM stimuli with a cutoff frequency of 300\,Hz. The
|
||||
black square marks the cell from \subfigrefb{fig:noisesplit}{C},
|
||||
the circles the four cells shown in
|
||||
\subfigref{fig:modelsusceptcontrasts}{A--D}, and the triangle the
|
||||
cell from \subfigref{fig:modelsusceptlown}{A--B}. \figitem{B}
|
||||
Electrophysiological data from 172 P-units. Each cell contributes
|
||||
on average with 2 (min. 1, max. 10) RAM stimulus presentations to
|
||||
the $n=376$ data points. The RAMs had cutoff frequencies of
|
||||
300\,Hz (352 samples) and 400\,Hz (24 samples). The two black
|
||||
triangles mark the responses of the example P-unit from
|
||||
The SI($r$) of all 39 model P-units measured with RAM stimuli with
|
||||
a cutoff frequency of 300\,Hz. The black square marks the cell
|
||||
from \subfigrefb{fig:noisesplit}{C}, the circles the four cells
|
||||
shown in \subfigref{fig:modelsusceptcontrasts}{A--D}, and the
|
||||
triangle the cell from \subfigref{fig:modelsusceptlown}{A--B}.
|
||||
\figitem{B} Electrophysiological data from 172 P-units. Each cell
|
||||
contributes on average with 2 (min. 1, max. 10) RAM stimulus
|
||||
presentations to the $n=376$ data points. The RAMs had cutoff
|
||||
frequencies of 300\,Hz (352 samples) and 400\,Hz (24 samples). The
|
||||
two black triangles mark the responses of the example P-unit from
|
||||
\subfigrefb{fig:punit}{E,F}, the circles the other four examples
|
||||
from \subfigrefb{fig:punit}{H}, and the triangle the unit from
|
||||
\subfigrefb{fig:noisesplit}{A}. \figitem{C} Recordings from 30
|
||||
@ -592,13 +591,13 @@ Overall, observing SI($r$) values greater than about 1.8, even for a number of F
|
||||
\subsection{Low CVs and weak responses predict weakly nonlinear responses}
|
||||
Now we are prepared to evaluate our pool of 39 P-unit model cells, 172 P-units, and 30 ampullary afferents recorded in 80 specimen of \textit{Apteronotus leptorhynchus}. For comparison across cells we condensed the structure of the second-order susceptibilities into SI($r$) values, \eqnref{siindex}. In order to make the data comparable, both model and experimental SI($r$) estimates, \eqnref{siindex}, are based on 100 FFT segments.
|
||||
|
||||
In the P-unit models, each model cell contributed with three RAM stimulus presentations with contrasts of 1, 3, and 10\,\%, resulting in $n=117$ data points. 19 (16\,\%) had SI($r$) values larger than 1.8, indicating the expected ridges at the baseline firing rate in their second-order susceptiility. The lower the cell's baseline CV, i.e. the less intrinsic noise, the higher the SI($r$) (\figrefb{fig:dataoverview}\,\panel[i]{A}).
|
||||
In the P-unit models, each model cell contributed with three RAM stimulus presentations with contrasts of 1, 3, and 10\,\%, resulting in $n=117$ data points. 19 (16\,\%) had SI($r$) values larger than 1.8, indicating the expected ridges at the baseline firing rate in their second-order susceptibility. The lower the cell's baseline CV, i.e. the less intrinsic noise, the higher the SI($r$) (\figrefb{fig:dataoverview}\,\panel[i]{A}).
|
||||
|
||||
The effective stimulus strength also plays a role in predicting the SI($r$) values. We quantify the effect of stimulus strength on a cell's response by the response modulation --- the standard deviation of a cell's firing rate in response to a RAM stimulus. The lower the response modulation, i.e. the weaker the effective stimulus, the higher the S($r$) (\figrefb{fig:dataoverview}\,\panel[ii]{A}). Although there is a tendency of low stimulus contrasts to evoke lower response modulations, response modulations evoked by each of the three contrasts overlap substantially, emphasizing the strong heterogeneity of the P-units' sensitivity \citep{Grewe2017}. Cells with high SI($r$) values are the ones with baseline firing rate below 200\,Hz (\figrefb{fig:dataoverview}\,\panel[iii]{A}).
|
||||
|
||||
In comparison to the experimentally measured P-unit recordings, the model cells are skewed to lower baseline CVs (Mann-Whitney $U=13986$, $p=3\times 10^{-9}$), because the models are not able to reproduce bursting, which we observe in many P-units and which leads to high CVs. Also the response modulation of the models is skewed to lower values (Mann-Whitney $U=15312$, $p=7\times 10^{-7}$), because in the measured cells, response modulation is positively correlated with baseline CV (Pearson $R=0.45$, $p=1\times 10^{-19}$), i.e. bursting cells are more sensitive. Median baseline firing rate in the models is by 53\,Hz smaller than in the experimental data (Mann-Whitney $U=17034$, $p=0.0002$).
|
||||
|
||||
In the experimentally measured P-units, each of the $172$ units contributes on average with two RAM stimulus presentations, presented at contrasts ranging from 1 to 20\,\% to the 376 samples. Despite the mentioned differences between the P-unit models and the measured data, the SI($r$) values do not differ between models and data (median of 1.3, Mann-Whitney $U=19702$, $p=0.09$) and also 16\,\% of the samples from all presented stimulus contrasts exceed the threshold of 1.8. The SI($r$) values of the P-unit population correlate weakly with the CV of the baseline ISIs that range from 0.18 to 1.35 (median 0.49). Cells with lower baseline CVs tend to have more pronounced ridges in their second-order susceptibilites than those with higher baseline CVs (\figrefb{fig:dataoverview}\,\panel[i]{B}).
|
||||
In the experimentally measured P-units, each of the $172$ units contributes on average with two RAM stimulus presentations, presented at contrasts ranging from 1 to 20\,\% to the 376 samples. Despite the mentioned differences between the P-unit models and the measured data, the SI($r$) values do not differ between models and data (median of 1.3, Mann-Whitney $U=19702$, $p=0.09$) and also 16\,\% of the samples from all presented stimulus contrasts exceed the threshold of 1.8. The SI($r$) values of the P-unit population correlate weakly with the CV of the baseline ISIs that range from 0.18 to 1.35 (median 0.49). Cells with lower baseline CVs tend to have more pronounced ridges in their second-order susceptibilities than those with higher baseline CVs (\figrefb{fig:dataoverview}\,\panel[i]{B}).
|
||||
|
||||
Samples with weak responses to a stimulus, be it an insensitive P-unit or a weak stimulus, have higher SI($r$) values and thus a more pronounced ridge in the second-order susceptibility in comparison to strongly responding cells, most of them having flat second-order susceptibilities (\figrefb{fig:dataoverview}\,\panel[ii]{B}). P-units with low or high baseline firing rates can have large SI($r$) (\figrefb{fig:dataoverview}\,\panel[iii]{B}). How pronounced nonlinear response components are in P-units thus depends on the baseline CV (a proxy for the internal noise level), and the response strength during stimulation (effective output noise).
|
||||
|
||||
@ -606,7 +605,7 @@ The population of ampullary cells is generally more homogeneous, with lower base
|
||||
|
||||
\begin{figure*}[t]
|
||||
\includegraphics[width=\columnwidth]{model_full.pdf}
|
||||
\caption{\label{fig:model_full} Second-order susceptibility qualitatively predicts responses to sine-wave stimuli. \figitem[]{A} Absolute value of the second-order susceptibility, \eqnref{chi2}, for both positive and negative frequencies. \susceptf{} was estimated from $N=10^6$ FFT segments of model simulations in the noise-split condition (cell ``2013-01-08-aa'', table~\ref{modelparams}). Dashed white lines mark 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 2\,\%. 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{}.}
|
||||
\caption{\label{fig:model_full} Second-order susceptibility qualitatively predicts responses to sine-wave stimuli. \figitem[]{A} Absolute value of the second-order susceptibility, \eqnref{chi2}, for both positive and negative frequencies. \susceptf{} was estimated from $N=10^6$ FFT segments of model simulations in the noise-split condition (cell ``2013-01-08-aa''). Dashed white lines mark 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 2\,\%. 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*}
|
||||
|
||||
\subsection{Second-order susceptibility explains nonlinear responses to pure sinewave stimuli}
|
||||
@ -629,10 +628,10 @@ The CV is a proxy for the intrinsic noise in the cells (\citealp{Vilela2009}, no
|
||||
\subsection{Linearization by white-noise stimulation}
|
||||
Not only the intrinsic noise but also the stimulation with external white-noise linearizes the cells. This applies to both, the stimulation with AMs in P-units (\figrefb{fig:dataoverview}\,\panel[ii]{B}) and direct stimulation in ampullary cells (\figrefb{fig:dataoverview}\,\panel[ii]{C}). The stronger the effective stimulus, the less pronounced are the ridges in the second-order susceptibility (see \subfigref{fig:punit}{E\&F} for a P-unit example and \subfigref{fig:ampullary}{E\&F} for an ampullary cell). This linearizing effect of noise stimuli limits the weakly nonlinear regime to small stimulus amplitudes. At higher stimulus amplitudes, however, other nonlinearities of the system eventually show up in the second-order susceptibility.
|
||||
|
||||
In order to characterize weakly nonlinear responses 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. 41\,\% of the model cells then show the full nonlinear structure (\figref{fig:modelsusceptcontrast}\,\panel[i]{E}) known from analytical derivations and simulations of basic LIF and theta models driven with pairs of sine-wave stimuli \citep{Voronenko2017,Franzen2023}. Previous studies on second-order nonlinearities have not observed the weakly nonlinear regime, probably because of the linearizing effects of strong noise stimuli \citep{Victor1977, Schanze1997, Neiman2011}.
|
||||
In order to characterize weakly nonlinear responses 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. 41\,\% of the model cells then show the full nonlinear structure (\figref{fig:modelsusceptcontrasts}\,\panel[i]{E}) known from analytical derivations and simulations of basic LIF and theta models driven with pairs of sine-wave stimuli \citep{Voronenko2017,Franzen2023}. Previous studies on second-order nonlinearities have not observed the weakly nonlinear regime, probably because of the linearizing effects of strong noise stimuli \citep{Victor1977, Schanze1997, Neiman2011}.
|
||||
|
||||
\subsection{Characterizing nonlinear coding from limited experimental data}
|
||||
Estimating the Volterra series from limited experimental data is usually restricted to the first two or three orders, which might not be sufficient for a proper prediction of the neuronal response \citep{French2001}. As we have demonstrated, a proper estimation of just the second-order susceptibility in the weakly nonlinear regime is challenging in electrophysiological experiments (\figrefb{modelsusceptlown}). Our minimum of 100 FFT segements corresponds to just 26\,s of stimulation and thus is not a challenge. However, the estimates of the second-order susceptibilities start to converge only beyond 10\,000 FFT segments, corresponding to 43\,min of stimulation. Often, however, more than one million segments were needed which requires 71 hours of recording, which is clearly out of reach. We have demonstrated that even in non-converged estimates, the presence of a anti-diagonal ridge is sufficient to predict a triangular pattern in a converged estimate.
|
||||
Estimating the Volterra series from limited experimental data is usually restricted to the first two or three orders, which might not be sufficient for a proper prediction of the neuronal response \citep{French2001}. As we have demonstrated, a proper estimation of just the second-order susceptibility in the weakly nonlinear regime is challenging in electrophysiological experiments (\figrefb{fig:modelsusceptlown}). Our minimum of 100 FFT segments corresponds to just 26\,s of stimulation and thus is not a challenge. However, the estimates of the second-order susceptibilities start to converge only beyond 10\,000 FFT segments, corresponding to 43\,min of stimulation. Often, however, more than one million segments were needed which requires 71 hours of recording, which is clearly out of reach. We have demonstrated that even in non-converged estimates, the presence of a anti-diagonal ridge is sufficient to predict a triangular pattern in a converged estimate.
|
||||
|
||||
Making assumptions about the nonlinearities in a system also reduces the amount of data needed for parameter estimation. In particular, models combining linear filtering with static nonlinearities \citep{Chichilnisky2001} have been successful in capturing functionally relevant neuronal computations in visual \citep{Gollisch2009} as well as auditory systems \citep{Clemens2013}. Linear methods based on backward models for estimating the stimulus from neuronal responses, have been extensively used to quantify information transmission in neural systems \citep{Theunissen1996, Borst1999, Wessel1996, Machens2001}, because backward models do not need to generate action potentials that involve strong nonlinearities \citep{Rieke1999}.
|
||||
|
||||
@ -641,17 +640,17 @@ The afferents of the passive electrosensory system, the ampullary cells, exhibit
|
||||
|
||||
Ampullary stimuli originate from the muscle potentials induced by prey movement \citep{Kalmijn1974, Engelmann2010, Neiman2011fish}. For a single prey items such as \textit{Daphnia}, these potentials are often periodic but the simultaneous activity of a swarm of prey resembles Gaussian white noise \citep{Neiman2011fish}. Linear and nonlinear encoding in ampullary cells has been studied in great detail in the paddlefish \citep{Neiman2011fish}. The power spectrum of the baseline response shows two main peaks: One peak at the baseline firing frequency, a second one at the oscillation frequency of primary receptor cells in the epithelium, and interactions of both. Linear encoding in the paddlefish shows a gap at the epithelial oscillation frequency, instead, nonlinear responses are very pronounced there.
|
||||
|
||||
Ampullary stimulus encoding is somewhat different in \lepto{}. The power spectrum of the spontaneous response is dominated by only the baseline firing rate and its harmonics, a second oscillator is not visible. The baseline firing 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:dataoverview}{F}) indicating that paddlefish data have been recorded above the weakly-nonlinear regime.
|
||||
Ampullary stimulus encoding is somewhat different in \lepto{}. The power spectrum of the spontaneous response is dominated by only the baseline firing rate and its harmonics, a second oscillator is not visible. The baseline firing 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 ampullary cells increases with stimulus intensity while it disappears in our case (\subfigrefb{fig:dataoverview}{F}) indicating that paddlefish data have been recorded above the weakly-nonlinear regime.
|
||||
|
||||
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, the resulting nonlinear response appears at the baseline rate that is similar in the full population of ampullary cells and 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{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 observed under noise stimulation for the encoding of distinct frequencies?
|
||||
|
||||
We here applied broad-band noise amplitude-modulation stimuli to estimate the second-order susceptibility. Broadband noise stimuli introduce additional noise that linearizes the dynamics of the system. In contrast, pure sinewave stimulation is spectrally focussed and drives the system on the background of the intrinsic noise. This explains why we can observe nonlinear interactions between sinewave stimuli with distinct frequencies (\figrefb{fig:motivation}) although they are barely visible when stimulating with strong broad-band noise (\figrefb{fig:noisesplit}).
|
||||
We here applied broad-band noise amplitude-modulation stimuli to estimate the second-order susceptibility. Broadband noise stimuli introduce additional noise that linearizes the dynamics of the system. In contrast, pure sine wave stimulation is spectrally focused and drives the system on the background of the intrinsic noise. This explains why we can observe nonlinear interactions between sine wave stimuli with distinct frequencies (\figrefb{fig:motivation}) although they are barely visible when stimulating with strong broad-band noise (\figrefb{fig:noisesplit}).
|
||||
|
||||
%As discussed above in the context of broad-band noise stimulation, we expect interactions with a cell's baseline frequency only for the very few P-units with low enough CVs of the baseline interspike intervals.
|
||||
%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.
|
||||
%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 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.
|
||||
|
||||
Extracting the AM from the stimulus already requires a (threshold) nonlinearity \citep{Middleton2006, Stamper2012Envelope, Savard2011, Barayeu2023}. This nonlinearity, however, does not show up in our estimates of the susceptibilities, because in our analysis we directly relate the AM waveform to the recorded cellular responses. Encoding the time-course of the AM, however, has been shown to be linear over a wide range of AM amplitudes and frequencies \citep{Xu1996, Benda2005, Gussin2007, Grewe2017, Savard2011}. In contrast, we here have demonstrated nonlinear interactions originating from the spike generator for broad-band noise stimuli with small amplitudes and for stimulation with two distinct frequencies. Both settings have not been studied yet.
|
||||
|
||||
@ -680,7 +679,7 @@ Before surgery, the animals were deeply anesthetized via bath application of a s
|
||||
Respiration was then switched to normal tank water and the fish was transferred to the experimental tank.
|
||||
|
||||
\subsection{Electrophysiological recordings}
|
||||
For the recordings fish were positioned centrally in the experimental tank, with the major parts of their body submerged into the water. Those body parts that were above the water surface were covered with paper tissue to avoid drying of the skin. Local analgesia was refreshed in intervals of two hours by cutaneous application of Lidocaine (2\,\%; bela-pharm, Vechta, Germany) around the surgical wounds. Electrodes (borosilicate; 1.5\,mm outer diameter; GB150F-8P; Science Products, Hofheim, Germany) were pulled to a resistance of 50--100\,\mega\ohm{} (model P-97; Sutter Instrument, Novato, CA) and filled with 1\,M KCl solution. Electrodes were fixed in a microdrive (Luigs-Neumann, Ratingen, Germany) and lowered into the nerve. Recordings of electroreceptor afferents were amplified and lowpass filtered at 10\,kHz (SEC-05, npi-electronics, Tamm, Germany, operated in bridge mode). All signals, neuronal recordings, recorded EOD, and the generated stimulus, were digitized with sampling rates of 20 or 40\,kHz (PCI-6229, National Instruments, Austin, TX). RELACS (\url{www.relacs.net}) running on a Linux computer was used for online spike and EOD detection, stimulus generation, and calibration. Recorded data was then stored on the hard drive for offline analysis.
|
||||
For the recordings fish were positioned centrally in the experimental tank, with the major parts of their body submerged into the water. Those body parts that were above the water surface were covered with paper tissue to avoid drying of the skin. Local analgesia was refreshed in intervals of two hours by cutaneous application of Lidocaine (2\,\%; bela-pharm, Vechta, Germany) around the surgical wounds. Electrodes (borosilicate; 1.5\,mm outer diameter; GB150F-8P; Science Products, Hofheim, Germany) were pulled to a resistance of 50--100\,\mega\ohm{} (model P-97; Sutter Instrument, Novato, CA) and filled with 1\,M KCl solution. Electrodes were fixed in a microdrive (Luigs-Neumann, Ratingen, Germany) and lowered into the nerve. Recordings of electroreceptor afferents were amplified and lowpass filtered at 10\,kHz (SEC-05, npi-electronics, Tamm, Germany, operated in bridge mode). All signals, neuronal recordings, recorded EOD, and the generated stimulus, were digitized with sampling rates of 20 or 40\,kHz (PCI-6229, National Instruments, Austin, TX). RELACS (\url{https://github.com/relacs/relacs}) running on a Linux computer was used for online spike and EOD detection, stimulus generation, and calibration. Recorded data was then stored on the hard drive for offline analysis.
|
||||
|
||||
\subsection{Identification of P-units and ampullary cells}
|
||||
Recordings were classified as P-units if baseline action potentials phase locked to the EOD with vectors strengths between 0.7 and 0.95, a baseline firing rate larger than 30\,Hz, a serial correlation of subsequent interspike intervals below zero, a coefficient of variation of baseline interspike intervals below 1.5 und during stimulation below 2. As ampullary cells we classified recordings with vector strengths below 0.15, baseline firing rate above 10\,Hz, baseline CV below 0.18, CV during stimulation below 1.0, and a response modulation during stimulation below 80\,Hz \citep{Grewe2017}. We here selected only those cells of which the neuron's baseline activity as well as the responses to band-limited white noise stimuli were recorded.
|
||||
@ -693,9 +692,7 @@ For monitoring the EOD without the stimulus, two vertical carbon rods ($11\,\cen
|
||||
Electric stimuli were attenuated (ATN-01M, npi-electronics, Tamm, Germany), isolated from 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.
|
||||
The fish were stimulated with band-limited white noise stimuli with a cut-off frequency of 150\,Hz (ampullary), 300 or 400\,Hz (P-units). The stimulus intensity is given as a contrast, i.e. the standard deviation of the noise stimulus relative to the fish's EOD amplitude. The contrast varied between 1 and 20\,\% (median 5\,\%) for P-units and 2.5 and 20\,\% (median 5\,\%) for ampullary cells. Only noise stimuli with a duration of at least 2\,s (maximum of 50\,s, median 10\,s) and enough repetitions to results in at least 100 FFT segments (see below, P-units: 100--1520, median 313, ampullary cells: 105 -- 3648, median 722) were included into the analysis. When ampullary cells were recorded, the white noise was directly applied as the stimulus. To create random amplitude modulations (RAM) for P-unit recordings, the noise stimulus was first multiplied with the EOD of the fish (MXS-01M; npi electronics).
|
||||
|
||||
% and between 2.5 and 40\,\% for \eigen
|
||||
|
||||
\subsection{Data analysis} Data analysis was done in Python 3 using the packages matplotlib \citep{Hunter2007}, numpy \citep{Walt2011}, scipy \citep{scipy2020}, pandas \citep{Mckinney2010}, nixio \citep{Stoewer2014}, and thunderlab (\url{https://github.com/bendalab/thunderlab}).
|
||||
\subsection{Data analysis} Data analysis was done in Python 3 using the packages matplotlib \citep{Hunter2007}, numpy \citep{Walt2011}, scipy \citep{scipy2020}, pandas \citep{Mckinney2010}, nixio \citep{Stoewer2014}, and thunderlab (\url{https://github.com/bendalab/thunderlab}). The P-unit model parameters and spectral analysis algorithms are available at \url{https://github.com/bendalab/punitmodel/tree/v1}.
|
||||
|
||||
\paragraph{Baseline analysis}\label{baselinemethods}
|
||||
The baseline firing rate $r$ was calculated as the number of spikes divided by the duration of the baseline recording (median 32\,s). The coefficient of variation (CV) of the interspike intervals (ISI) is their standard deviation relative to their mean: $\rm{CV}_{\rm base} = \sqrt{\langle (ISI- \langle ISI \rangle) ^2 \rangle} / \langle ISI \rangle$. If the baseline was recorded several times in a recording, the measures from the longest recording were taken.
|
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@ -703,35 +700,11 @@ The baseline firing rate $r$ was calculated as the number of spikes divided by t
|
||||
\paragraph{White noise analysis} \label{response_modulation}
|
||||
When stimulated with band-limited white noise stimuli, neuronal activity is modulated around the average firing rate that is similar to the baseline firing rate and in that way encodes the time-course of the stimulus. For an estimate of the time-dependent firing rate $r(t)$ we convolved each spike train with normalized Gaussian kernels with standard deviation of 2\,ms and averaged the resulting single-trail firing rates over trials. The response modulation quantifies the variation of $r(t)$ computed as the standard deviation in time $\sigma_{s} = \sqrt{\langle (r(t)-\langle r(t) \rangle_t )^2\rangle_t}$, where $\langle \cdot \rangle_t$ denotes averaging over time.
|
||||
|
||||
% Spiking activity
|
||||
% \begin{equation}
|
||||
% \label{eq:spikes}
|
||||
% x_k(t) = \sum_i\delta(t-t_{k,i})
|
||||
% \end{equation}
|
||||
% is recorded for each stimulus presentation $k$, as a train of times $t_{k,i}$ where action potentials occured.
|
||||
% %If only a single trial was recorded or is used for the analysis, we drop the trial index $k$.
|
||||
|
||||
% The single-trial firing rate
|
||||
% \begin{equation}
|
||||
% r_k(t) = x_k(t) * K(t) = \int_{-\infty}^{+\infty} x_k(t') K(t-t') \, \mathrm{d}t'
|
||||
% \end{equation}
|
||||
% was estimated by convolving the spike train with a kernel. We used a Gaussian kernel
|
||||
% \begin{equation}
|
||||
% K(t) = {\scriptstyle \frac{1}{\sigma\sqrt{2\pi}}} e^{-\frac{t^2}{2\sigma^2}}
|
||||
% \end{equation}
|
||||
% with standard deviation $\sigma$ set to 2\,ms. Averaging over $n$ repeated stimulus presentations results in the trial averaged firing rate
|
||||
% \begin{equation}
|
||||
% \label{eq:rate}
|
||||
% r(t) = \left\langle r_k(t) \right\rangle _k = \frac{1}{n} \sum_{k=1}^n r_k(t)
|
||||
% \end{equation}
|
||||
|
||||
% The average firing rate during stimulation, $r_s = \langle r(t) \rangle_t$, is given by the temporal average $\langle \cdot \rangle_t$ over the duration of the stimulus of the trial-averaged firing rate. To quantify how strongly a neuron was driven by the stimulus, we computed the response modulation as the standard deviation $\sigma_{s} = \sqrt{\langle (r(t)-r_s)^2\rangle_t}$ of the trial-averaged firing rate.
|
||||
|
||||
\paragraph{Spectral analysis}\label{susceptibility_methods}
|
||||
To characterize the relation between the spiking response evoked by white-noise stimuli, we estimated the first- and second-order susceptibilities in the frequency domain. For this we converted spike times into binary vectors $x_k$ with $\Delta t = 0.5$\,ms wide bins that are set to 2\,kHz where a spike occurred and zero otherwise. Fast Fourier transforms (FFT) $S(\omega)$ and $X(\omega)$ of the stimulus $s_k$ (also down-sampled to a sampling rate of 2\,kHz) and $x_k$, respectively, were computed numerically according to
|
||||
\begin{equation}
|
||||
\label{fourier}
|
||||
X(\omega) = \sum_{k=0}^{n} \, x_k e^{- i \omega k}
|
||||
X(\omega) = \sum_{k=0}^{n-1} \, x_k e^{- i \omega k}
|
||||
\end{equation}
|
||||
for $n=512$ long segments of $T=n \Delta t = 256$\,ms duration with no overlap, resulting in a spectral resolution of about 4\,Hz. Note, that for a real Fourier integral a factor $\Delta t$ is missing. For simplicity we use angular frequencies $\omega=2\pi f$ instead of frequencies $f$.
|
||||
|
||||
@ -795,7 +768,7 @@ Values larger than one indicate a sharp ridge in the susceptibility matrix close
|
||||
\begin{figure*}[t]
|
||||
\includegraphics[width=\columnwidth]{flowchart.pdf}
|
||||
\caption{\label{flowchart}
|
||||
Architecture of the P-unit model. Each row illustrates subsequent processing steps for three different stimulation regimes: (i) baseline activity without external stimulus, only the fish's self-generated EOD (the carrier, \eqnref{eq:eod}) is present. (ii) RAM stimulation, \eqnref{eq:ram_equation}. The amplitude of the EOD carrier is modulated with a weak (2\,\% contrast) band-limited white-noise stimulus. (iii) Noise split, \eqnsref{eq:ram_split}--\eqref{eq:Noise_split_intrinsic}, where 90\,\% of the intrinsic noise is replaced by a RAM stimulus, whose amplitude is scaled to maintain the mean firing rate and the CV of the ISIs of the model's baseline activity. As an example, simulations of the model for cell ``2012-07-03-ak'' are shown (table~\ref{modelparams}). \figitem{A} The stimuli are thresholded, \eqnref{eq:threshold2}, by setting all negative values to zero. \figitem{B} Subsequent low-pass filtering, \eqnref{eq:dendrite}, attenuates the carrier and carves out the AM signal. \figitem{C} Intrinsic Gaussian white-noise is added to the signals shown in \panel{B}. Note the reduced internal noise amplitude in the noise split (iii) condition. \figitem{D} Spiking output of the LIF model, \eqnsref{eq:LIF}--\eqref{spikethresh}, in response to the sum of \panel{B} and \panel{C}. \figitem{E} Power spectra of the LIF neuron's spiking activity. Both, baseline activity (\panel[i]{E}) and noise split (\panel[iii]{E}), have the same peaks in the response spectrum at $\fbase$, $f_{EOD} - \fbase$, $f_{EOD}$, and $f_{EOD} + \fbase$. With RAM stimulation (\panel[ii]{E}), the peak at the baseline firing rate, $\fbase$, is washed out.}
|
||||
Architecture of the P-unit model. Each row illustrates subsequent processing steps for three different stimulation regimes: (i) baseline activity without external stimulus, only the fish's self-generated EOD (the carrier, \eqnref{eq:eod}) is present. (ii) RAM stimulation, \eqnref{eq:ram_equation}. The amplitude of the EOD carrier is modulated with a weak (2\,\% contrast) band-limited white-noise stimulus. (iii) Noise split, \eqnsref{eq:ram_split}--\eqref{eq:Noise_split_intrinsic}, where 90\,\% of the intrinsic noise is replaced by a RAM stimulus, whose amplitude is scaled to maintain the mean firing rate and the CV of the ISIs of the model's baseline activity. As an example, simulations of the model for cell ``2012-07-03-ak'' are shown. \figitem{A} The stimuli are thresholded, \eqnref{eq:threshold2}, by setting all negative values to zero. \figitem{B} Subsequent low-pass filtering, \eqnref{eq:dendrite}, attenuates the carrier and carves out the AM signal. \figitem{C} Intrinsic Gaussian white-noise is added to the signals shown in \panel{B}. Note the reduced internal noise amplitude in the noise split (iii) condition. \figitem{D} Spiking output of the LIF model, \eqnsref{eq:LIF}--\eqref{spikethresh}, in response to the sum of \panel{B} and \panel{C}. \figitem{E} Power spectra of the LIF neuron's spiking activity. Both, baseline activity (\panel[i]{E}) and noise split (\panel[iii]{E}), have the same peaks in the response spectrum at $\fbase$, $f_{EOD} - \fbase$, $f_{EOD}$, and $f_{EOD} + \fbase$. With RAM stimulation (\panel[ii]{E}), the peak at the baseline firing rate, $\fbase$, is washed out.}
|
||||
\end{figure*}
|
||||
|
||||
\subsection{Leaky integrate-and-fire models for P-units}\label{lifmethods}
|
||||
@ -853,22 +826,9 @@ Whenever the membrane voltage $V_m(t)$ crosses the spiking threshold $\theta=1$,
|
||||
V_m(t) \ge \theta \; : \left\{ \begin{array}{rcl} V_m & \mapsto & 0 \\ A & \mapsto & A + \Delta A/\tau_A \end{array} \right.
|
||||
\end{equation}
|
||||
|
||||
The P-unit models were integrated by the Euler forward method with a time-step of $\Delta t = 0.05$\,ms. For each trial of a simulation, $V_{m}$ was drawn from a uniform distribution between 0 and 1 and the initial value of $A$ was jittered by adding a random number drawn from a normal distribution with standard deviation of 2\,\% of its initial value. Then the first 500\,ms of any simulation were discarded to remove remainig transients.
|
||||
The P-unit models were integrated by the Euler forward method with a time-step of $\Delta t = 0.05$\,ms. For each trial of a simulation, $V_{m}$ was drawn from a uniform distribution between 0 and 1 and the initial value of $A$ was jittered by adding a random number drawn from a normal distribution with standard deviation of 2\,\% of its initial value. Then the first 500\,ms of any simulation were discarded to remove remaining transients.
|
||||
|
||||
The eight free parameters of the P-unit model $\beta$, $\tau_m$, $\mu$, $D$, $\tau_A$, $\Delta_A$, $\tau_d$, and $t_{ref}$, were fitted to both the baseline activity (baseline firing rate, CV of ISIs, serial correlation of ISIs at lag one, and vector strength of spike coupling to EOD) and the responses to step increases and decreases in EOD amplitude (onset and steady-state responses, effective adaptation time constant, \citealp{Benda2005}) of recorded P-units (table~\ref{modelparams}).
|
||||
|
||||
\notejb{tag git repo and insert reference to it}
|
||||
% \begin{table*}[tp]
|
||||
% \caption{\label{modelparams} Model parameters of LIF models, fitted to 3 electrophysiologically recorded P-units \citep{Ott2020}.}
|
||||
% \begin{tabular}{lrrrrrrrr}
|
||||
% \hline
|
||||
% \bfseries cell & \bfseries $\mu$ & \bfseries $\beta$ & \bfseries $\tau_{m}$/ms & \bfseries $D$/$\mathbf{ms}$ & \bfseries $\tau_{A}$/ms & \bfseries $\Delta_A$ & \bfseries $\tau_{d}$/ms & \bfseries $t_{ref}$/ms \\\hline
|
||||
% 2012-07-03-ak& $-1.32$& $10.6$& $1.38$& $0.001$& $96.05$& $0.01$& $1.18$& $0.12$ \\
|
||||
% 2013-01-08-aa& $0.59$& $4.5$& $1.20$& $0.001$& $37.52$& $0.01$& $1.18$& $0.38$ \\
|
||||
% 2018-05-08-ae& $-21.09$& $139.6$& $1.49$& $0.214$& $123.69$& $0.16$& $3.93$& $1.31$ \\
|
||||
% \hline
|
||||
% \end{tabular}
|
||||
% \end{table*}
|
||||
The eight free parameters of the P-unit model $\beta$, $\tau_m$, $\mu$, $D$, $\tau_A$, $\Delta_A$, $\tau_d$, and $t_{ref}$, were fitted to both the baseline activity (baseline firing rate, CV of ISIs, serial correlation of ISIs at lag one, and vector strength of spike coupling to EOD) and the responses to step increases and decreases in EOD amplitude (onset and steady-state responses, effective adaptation time constant, \citealp{Benda2005}) of recorded P-units. Model parameters of all 39 cells are summarized in file \texttt{models.csv} of our \texttt{punitmodel} repository at \url{https://github.com/bendalab/punitmodel/tree/v1}.
|
||||
|
||||
|
||||
\subsection{Noise split}
|
||||
@ -892,3 +852,14 @@ Both, the reduced intrinsic noise and the RAM stimulus, need to replace the orig
|
||||
|
||||
|
||||
\end{document}
|
||||
|
||||
% LocalWords: electrosensory electroreceptor afferents ampullary EOD
|
||||
% LocalWords: Volterra mechanoreceptors paddlefish linearizes EODs
|
||||
% LocalWords: suprathreshold detectability knifefish quadratically
|
||||
% LocalWords: nonlinearities probabilistically multimodal coherences
|
||||
% LocalWords: entrains LIF ISIs unimodal ISI linearize Interspike
|
||||
% LocalWords: electrophysiological FFT dendritic preprocessing
|
||||
% LocalWords: Furutsu Novikov interspike durations Apteronotus
|
||||
% LocalWords: leptorhynchus nonlinearity linearizing spectrally
|
||||
% LocalWords: electroreceptors lowpass differentially isopotential
|
||||
% LocalWords: transdermal convolved
|
||||
|
||||
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Reference in New Issue
Block a user