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updating model_and_data

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saschuta 2024-02-29 15:45:17 +01:00
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__pycache__/utils_all_down.cpython-39.pyc
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model_and_data.pdf
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model_and_data.py
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@ -47,9 +47,9 @@ def model_and_data2(eod_metrice = False, width=0.005, nffts=['whole'], powers=[1
cells_all = [good_data[0]]
plot_style()
default_figsize(column=2, length=3.25) #4.75 0.75
grid = gridspec.GridSpec(2, 4, wspace=0.85, bottom=0.07,
hspace=0.18, left=0.09, right=0.93, top=0.94)
default_figsize(column=2, length=3.1) #.254.75 0.75
grid = gridspec.GridSpec(2, 5, wspace=0.95, bottom=0.09,
hspace=0.25, width_ratios = [1,0,1,1,1], left=0.09, right=0.93, top=0.9)
a = 0
maxs = []
@ -134,7 +134,7 @@ def model_and_data2(eod_metrice = False, width=0.005, nffts=['whole'], powers=[1
#'calc_RAM_model-2__nfft_whole_power_1_afe_0.009_RAM_RAM_additiv_cv_adapt_factor_scaled_cNoise_0.1_cSig_0.9_cutoff1_300_cutoff2_300no_sinz_length1_TrialsStim_11_a_fr_1__trans1s__TrialsNr_1_fft_o_forward_fft_i_forward_Hz_mV',
#'calc_RAM_model-2__nfft_whole_power_1_afe_0.009_RAM_RAM_additiv_cv_adapt_factor_scaled_cNoise_0.1_cSig_0.9_cutoff1_300_cutoff2_300no_sinz_length1_TrialsStim_500000_a_fr_1__trans1s__TrialsNr_1_fft_o_forward_fft_i_forward_Hz_mV',
nrs_s = [2, 3, 6, 7]#, 10, 11
nrs_s = [3, 4, 8, 9]#, 10, 11
#embed()
tr_name = trial_nr/1000000
if tr_name == 1:
@ -181,13 +181,13 @@ def model_and_data2(eod_metrice = False, width=0.005, nffts=['whole'], powers=[1
maxs.append(np.max(np.array(stack_plot)))
mins.append(np.min(np.array(stack_plot)))
col = 2
row = 3
row = 2
ax_external.set_xticks_delta(100)
ax_external.set_yticks_delta(100)
# cbar[0].set_label(nonlin_title(add_nonlin_title)) # , labelpad=100
cbar.set_label(nonlin_title(' ['+add_nonlin_title), labelpad=lp) # rotation=270,
if (s in np.arange(col - 1, 100, col)) | (s == 0):
if (s in np.arange(col - 1, 100, col)):# | (s == 0)
remove_yticks(ax_external)
else:
set_ylabel_arrow(ax_external, xpos=xpos_y_modelanddata(), ypos=0.87)
@ -214,7 +214,7 @@ def model_and_data2(eod_metrice = False, width=0.005, nffts=['whole'], powers=[1
c_sigs = [0, 0.9]#, 0.9
grid_left = [[], grid[1, 0]]#, grid[2, 0]
ax_ams = []
for g, grid_here in enumerate([grid[0, 1], grid[1, 1]]):#, grid[2, 0]
for g, grid_here in enumerate([grid[0, 2], grid[1, 2]]):#, grid[2, 0]
grid_lowpass = gridspec.GridSpecFromSubplotSpec(4, 1,
subplot_spec=grid_here, hspace=0.3,
height_ratios=[1, 1,1, 0.1])
@ -321,15 +321,15 @@ def model_and_data2(eod_metrice = False, width=0.005, nffts=['whole'], powers=[1
transform=ax_n.transAxes)
elif vers == 'second':
ax_external.text(1, 1, 'RAM', ha='right', color='red', transform=ax_external.transAxes)
ax_intrinsic.text(start_pos_modeldata(), 1, signal_component_name(), ha='right', color='purple',
ax_intrinsic.text(start_pos_modeldata(), 1.1, signal_component_name(), ha='right', color='purple',
transform=ax_intrinsic.transAxes)
ax_n.text(start_pos_modeldata(), 0.8, noise_component_name(), ha='right', color='gray',
ax_n.text(start_pos_modeldata(), 0.9, noise_component_name(), ha='right', color='gray',
transform=ax_n.transAxes)
else:
ax_n.text(start_pos_modeldata(), 0.8, noise_component_name(), ha='right', color='gray',
ax_n.text(start_pos_modeldata(), 0.9, noise_component_name(), ha='right', color='gray',
transform=ax_n.transAxes)
ax_external.text(1, 1, 'RAM', ha='right', color='red', transform=ax_external.transAxes)
ax_intrinsic.text(start_pos_modeldata(), 1, signal_component_name(), ha='right', color='purple',
ax_intrinsic.text(start_pos_modeldata(), 1.1, signal_component_name(), ha='right', color='purple',
transform=ax_intrinsic.transAxes)
#embed()
set_same_ylim(ax_ams, up='up')
@ -363,11 +363,11 @@ def start_pos_modeldata():
def signal_component_name():
return 'Signal component'#'signal noise'
return r'$\xi_{signal}$'#signal noise'
def noise_component_name():
return 'Noise component'#'intrinsic noise'
def noise_component_name():#$\xi_{noise}$noise_name =
return r'$\xi_{noise}$'#'Noise component'#'intrinsic noise'
def ypos_x_modelanddata():

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susceptibility1.pdf
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susceptibility1.tex
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@ -511,7 +511,7 @@ In a high-CV P-unit we could not find such nonlinear structures, neither in the
\begin{figure*}[hb!]
\includegraphics[width=\columnwidth]{model_and_data}
\caption{\label{model_and_data} \notejg{$\xi_{signal}$ and $\xi_{noise}$ in the figure?, reorder the figure and put the model left?} High trial numbers and a reduced internal noise reveal the non-linear structure in a LIF model with carrier. \figitem{A} $\chi_2$ surface of the electrophysiologically recorded low-CV P-unit (cell 2012-07-03-ak) driven with a weak stimulus with 1\% contrast. The plot shows that average $\chi_2$ surface (n\,=\,11). Pink edges mark the expected structure of enhanced nonlinearity. \figitem[i]{B} \textit{Standard condition} of the model simulation. The model is driven by RAM stimulus (red trace). $\xi_{signal}$ and $\xi_{noise}$ components of the total intrinsic noise ($\xi$) are shown in the center and bottom. In this condition, $\xi_{noise} = \xi$. \figitem[ii]{B} $\chi_2$ surface of the cell's LIF model counterpart (cell 2012-07-03-ak, table~\ref{modelparams}) averaged over the same number of stimulus repetitions. \figitem[iii]{B} Same as \panel[II]{B} but 1 million stimulus repetitions. \figitem[i - iii]{C} Same as in \panel[i - iii]{B} but in the \tetit{noise split} condition: there is no external RAM signal driving the model. Parts (90\%) of the total intrinsic noise ($\xi$) are treated as signal (center trace, $\xi_{signal}$) while the remaining part is treated as $\xi_{noise}$ (see methods for details).
\caption{\label{model_and_data} \notejg{$\xi_{signal}$ and $\xi_{noise}$ in the figure?, reorder the figure and put the model left?} High trial numbers and a reduced internal noise reveal the non-linear structure in a LIF model with carrier. \figitem{A} $\chi_2$ surface of the electrophysiologically recorded low-CV P-unit (cell 2012-07-03-ak) driven with a weak stimulus with 1\% contrast. The plot shows that average $\chi_2$ surface (n\,=\,11). Pink edges mark the expected structure of enhanced nonlinearity. \figitem[i]{B} \textit{Standard condition} of the model simulation. The model is driven by RAM stimulus (red trace). $\xi_{signal}$ and $\xi_{noise}$ components of the total intrinsic noise ($\xi$) are shown in the center and bottom. In this condition, $\xi_{noise} = \xi$. \figitem[ii]{B} $\chi_2$ surface of the cell's LIF model counterpart (cell 2012-07-03-ak, table~\ref{modelparams}) averaged over the same number of stimulus repetitions. \figitem[iii]{B} Same as \panel[II]{B} but 1 million stimulus repetitions. \figitem[i - iii]{C} Same as in \panel[i - iii]{B} but in the \textit{noise split} condition: there is no external RAM signal driving the model. Parts (90\%) of the total intrinsic noise ($\xi$) are treated as signal (center trace, $\xi_{signal}$) while the remaining part is treated as $\xi_{noise}$ (see methods for details).
}
\end{figure*}
@ -536,7 +536,7 @@ The smaller \fdiff{} power spectral peak observed during pure sine-wave stimulat
\end{figure*}
\subsection{Low CVs and weak stimuli are associated with strong nonlinearity}
The non-linear effects shown for single cell examples above are supported by the analysis of the pool of 222 P-units and 45 ampullary afferents recorded in 71 specimen. To compare across cells we expressed the second-order susceptibility as the nonlinearity index \nli{}, see \eqnref{eq:nli_equation}. The \nli{} characterizes the peakedness of the projections onto the diagonal of the \suscept{} matrices (e.g. \subfigref{cells_suscept}{G} at \fbase{}. It assumes high values when the \fbase{} peak in the projected diagonal is pronounced relative to the median of the diagonal and is small when there is no distinct peak. The P-unit population has higher \nli{} values up to three and depends weakly on the the CV of the ISI distribution under baseline condition. Cells with lower baseline CVs have the tendency to exhibit a stronger nonlinearity than those that have high CVs (Pearson's r\,=\,-0.16, p\,<\,0.01). The two example P-units shown before (\figrefb{cells_suscept} and \figrefb{cells_suscept_high_CV}) are highlighted with dark circles in \subfigrefb{data_overview_mod}{B, D}. The stimulus strength plays an important role. Several of the recorded neurons contribute with two dots to the data as their responses to the same RAM stimulus but with different contrasts were recorded. Higher stimulus contrasts lead to a stronger drive and thus stronger response modulations (see color code bar in \subfigref{data_overview_mod}{A}, see methods). Since P-units are heterogeneous in their susceptibility to the stimulus\citealp{Grewe2017}, their responses to the same stimulus intensity vary a lot. When replotting the \nli{} against the response modulation (\subfigrefb{data_overview_mod}{C}) a negative correlation is observed (Pearson's r=-0.35, p < 0.001) showing that cells that are strongly driven by the stimulus show less nonlinearity while those that are only weakly driven show higher nonlinearities. Whether or not a cell responds nonlinearly thus depends on both, the baseline CV (i.e. the internal noise) and the response strength.
The non-linear effects shown for single cell examples above are supported by the analysis of the pool of 222 P-units and 45 ampullary afferents recorded in 71 specimen. To compare across cells we expressed the second-order susceptibility as the nonlinearity index \nli{}, see \eqnref{eq:nli_equation}. The \nli{} characterizes the peakedness of the projections onto the diagonal of the \suscept{} matrices (e.g. \subfigref{cells_suscept}{G} at \fbase{}. It assumes high values when the \fbase{} peak in the projected diagonal is pronounced relative to the median of the diagonal and is small when there is no distinct peak. The P-unit population has higher \nli{} values up to three and depends weakly on the the CV of the ISI distribution under baseline condition. Cells with lower baseline CVs have the tendency to exhibit a stronger nonlinearity than those that have high CVs (Pearson's r\,=\,$-0.16$, p\,<\,0.01). The two example P-units shown before (\figrefb{cells_suscept} and \figrefb{cells_suscept_high_CV}) are highlighted with dark circles in \subfigrefb{data_overview_mod}{B, D}. The stimulus strength plays an important role. Several of the recorded neurons contribute with two dots to the data as their responses to the same RAM stimulus but with different contrasts were recorded. Higher stimulus contrasts lead to a stronger drive and thus stronger response modulations (see color code bar in \subfigref{data_overview_mod}{A}, see methods). Since P-units are heterogeneous in their susceptibility to the stimulus\citealp{Grewe2017}, their responses to the same stimulus intensity vary a lot. When replotting the \nli{} against the response modulation (\subfigrefb{data_overview_mod}{C}) a negative correlation is observed (Pearson's r\,=\,$-0.35$, p\,<\,0.001 \notesr{das minus wird durch den Mathmodus länger, das will Jan immer so haben}) showing that cells that are strongly driven by the stimulus show less nonlinearity while those that are only weakly driven show higher nonlinearities. Whether or not a cell responds nonlinearly thus depends on both, the baseline CV (i.e. the internal noise) and the response strength.
%In a P-unit population where each cell is represented not by several contrasts but by the lowest recorded contrast, \nli{} significantly correlates with the CV during baseline ($r=-0.17$, $p=0.01$), the response modulation ($r=-0.35$, $p<0.001$) and \fbase{} ($r=-0.32$, $p<0.001$).%, $\n{}=221$*, $\n{}=221$******, $\n{}=221$
The population of ampullary cells is generally more homogeneous and have lower CVs than the P-units and show much higher \nli{} values (factor of 10). Overall, there is a negative correlation with the baseline CV (Pearson's r=-0.35, p < 0.01). The example cell shown above (\figref{ampullary}) was recorded at two different stimulus intensities and the \nli{}s are highlighted with black circles. Again, we see that cells that are strongly driven by the stimulus cluster at the bottom of the distribution and have \nli{} values close to zero. This is confirmed when the data is replotted against the response modulation, those cells that are strongly driven by the stimulus show weak nonlinearities while weakly driven neurons exhibit high values (Pearson's r=-0.59, p < 0.0001).

157
susceptibility1.tex.bak
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@ -107,7 +107,7 @@
%%%%% figures %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% references to panels of a figure within the caption:
\newcommand{\figitem}[2][]{\newline\ifthenelse{\equal{#1}{}}{\textsf{\bfseries #2}}{\textsf{\bfseries #2}}$_{\sf #1}$}
\newcommand{\figitem}[2][]{\ifthenelse{\equal{#1}{}}{\textsf{\bfseries #2}}{\textsf{\bfseries #2}}$_{\sf #1}$}
% references to panels of a figure within the text:
%\newcommand{\panel}[2][]{\textsf{#2}}
\newcommand{\panel}[2][]{\ifthenelse{\equal{#1}{}}{\textsf{#2}}{\textsf{#2}$_{\sf #1}$}}%\ifthenelse{\equal{#1}{}}{\textsf{#2}}{\textsf{#2}$_{\sf #1}$}
@ -469,115 +469,86 @@ In the communication context of weakly electric fish white noise stimuli are imp
\end{figure*}
\section{Results}
Theoretical work shows that leaky-integrate-and-fire (LIF) model neurons show a distinct pattern of non-linear stimulus encoding when the model is driven by two cosine signals. In the context of the weakly electric fish, such a setting is part of the animal's everyday life as the sinusoidal electric-organ discharges (EODs) of neighboring animals interfere with the own field and each lead to sinusoidal amplitude modulations (AMs) that are called beats and envelopes\cite{Middleton2006, Savard2011,Stamper2012Envelope}. The p-type electroreceptor afferents of the tuberous electrosensory system, i.e. the P-units, respond to such AMs of the underlying EOD carrier and their time-dependent firing rate carries information about the stimulus' time-course. P-units are heterogeneous in their baseline firing properties \citealp{Grewe2017, Hladnik2023} and differ with respect to their baseline firing rates, the sensitivity and their noisiness. We here explore the non-linear mechanism in different in cells that exhibit distinctly different levels of noise, i.e. differences in the coefficient of variation (CV) of the interspike intervals (ISI). Low-CV P-units have a less noisy, more regular, firing pattern whereas high-CV P-units show a less regular firing pattern in their baseline activity.
\subsection{Low-CV P-units exhibit nonlinear interactions} %frequency combinations withappearing when the input frequencies are related to \fbase{} are
Second-order susceptibility is expected to be especially pronounced for low-CV cells \citealp{Voronenko2017}. P-units fire action potentials probabilistically phase-locked to the self-generated EOD. Skipping of EOD cycles leads to the characteristic multimodal ISI distribution with maxima at integer multiples of the EOD period (\subfigrefb{cells_suscept}{A}). In this example the ISI distribution has a CV of 0.2 which can be considered low among P-units\citealp{Hladnik2023}. Spectral analysis of the baseline activity shows two major peaks, the first is located at the baseline firing rate (\fbase), the second is located at the discharge frequency of the electric organ (\feod{}) and is flanked by two smaller peaks at $\feod \pm \fbase{}$ \subfigref{cells_suscept}{B}.
P-units are heterogeneous in their baseline firing properties \citealp{Grewe2017, Hladnik2023} and differ in their noisiness, which is represented by the coefficient of variation (CV) of the interspike intervals (ISI). Low-CV P-units have a regular firing pattern and are less noisy, whereas high-CV P-units have a less regular firing pattern.
Second-order susceptibility is expected to be especially pronounced for low-CV cells \citealp{Voronenko2017}. In the following first low-CV P-units will be addressed in \subfigrefb{cells_suscept}{A}.
P-units probabilistically phase-lock to the EOD of the fish, firing at the same phase but not in every EOD cycle, resulting in a multimodal ISI histogram with maxima at integer multiples of the EOD period (\subfigrefb{cells_suscept}{A}). The strongest peak in the baseline power spectrum of the firing rate of a P-unit is the \feod{} peak, and the second strongest peak is the mean baseline firing rate \fbase{} peak (\subfigrefb{cells_suscept}{B}). The power spectrum of P-units is symmetric around half \feod, with baseline peaks appearing at $\feod \pm \fbase{}$.
Noise stimuli, as random amplitude modulations (RAM) of the EOD, are common stimuli during P-unit recordings. In the following, the amplitude of the noise stimulus will be quantified as the standard deviation and will be expressed as a contrast (unit \%) in relation to the receiver EOD. The spikes of P-units slightly align with the RAM stimulus with a low contrast (light purple) and are stronger driven in response to a higher RAM contrast (dark purple, \subfigrefb{cells_suscept}{C}). The linear encoding (see \eqnref{linearencoding_methods}) is comparable between the two RAM contrasts in this low-CV P-unit (\subfigrefb{cells_suscept}{D}).%visualized by the gain of the transfer function,\suscept{}
\begin{figure*}[!h]
\includegraphics{cells_suscept}
\caption{\label{cells_suscept} \notejg{dashed lines still a little faint} Estimation of linear and non-linear stimulus encoding in a low-CV P-unit. \textbf{A} Interspike-interval (ISI) distribution of the cell's baseline activity, i.e. the cell is driven only by the unperturbed own field. The CV of the ISIs is a dimensionless measure quantifying the response regularity. Zero CV would indicate perfect regularity. \textbf{B} Power-spectrum of the baseline response. \textbf{C} Random amplitude modulation stimulus (top) and responses (spike raster in the lower traces) to the same P-unit. The stimulus contrast reflects the strength of the AM. \textbf{D} Transfer function (first-order susceptibility, \eqnref{linearencoding_methods}) of the responses to 10\% (light purple) and 20\% contrast (dark purple) RAM stimulation. \textbf{E, F} Absolute value of the second-order susceptibility, \eqnref{eq:susceptibility}, for the low, and high stimulus contrasts. Pink angles -- edges of the structure when \fone, \ftwo{} or \fsum{} are equal to \fbase{}. \textbf{G} Projected diagonals, calculated as the mean of the anti-diagonals of the matrices in \panel{E, F}. Gray dots mark \fbase{}. Horizontal dashed lines: medians of the projected diagonals.}
\end{figure*}
Noise stimuli, the random amplitude modulations (RAM, \subfigref{cells_suscept}{C, top trace, red line}) of the EOD, are commonly used to characterize the stimulus driven responses of P-units by means of the transfer function (first-order susceptibility), the spike-triggered average, or the stimulus-response coherence. Here, we additionally estimate the second-order susceptibility to capture non-linear encoding. Depending on the stimulus intensity which is given as the contrast, i.e. the amplitude of the noise stimulus in relation to the EOD amplitude (see methods), the spikes align more or less clearly to AMs of the stimulus (light and dark purple for low and high contrast stimuli, \subfigrefb{cells_suscept}{C}). The linear encoding (see \eqnref{linearencoding_methods}) is similar for the two RAM contrasts in this low-CV P-unit (\subfigrefb{cells_suscept}{D}). First-order susceptibility is low for low frequencies, peaks in the range below 100\% and then falls off again.
To quantify the second-order susceptibility in a three-fish setting the noise stimulus was set in relation to the corresponding P-unit response in the Fourier domain, resulting in a matrix where the nonlinearity at the sum frequency \fsum{} in the firing rate is depicted for two noise frequencies \fone{} and \ftwo{} (\eqnref{susceptibility}, \subfigrefb{cells_suscept}{E--F}). Note that the RAM stimulus can be decomposed in frequencies $f$, that approximate the beat frequencies $\Delta f$, occurring in case of pure sine-wave stimulation (\subfigrefb{motivation}{D}). Thus the nonlinearity accessed with the RAM stimulation at \fsum{} (\subfigrefb{cells_suscept}{E}) is comparable to the nonlinearity appearing during pure sine-wave stimulation at \bsum{} (orange peak, \subfigrefb{motivation}{D}). Based on the theory \citealp{Voronenko2017} nonlinearities should arise when \fone{}, \ftwo{} or \fsum{} are equal to \fbase{} (upper right quadrant in \figrefb{plt_RAM_didactic2}), which would imply a triangular nonlinearity shape highlighted by the pink triangle corners in \subfigrefb{cells_suscept}{E--F}. A slight diagonal nonlinearity band appears for the low RAM contrast when \fsumb{} is satisfied (yellow diagonal between pink edges, \subfigrefb{cells_suscept}{E}). Since the matrix contains only anti-diagonal elements, the structural changes were quantified by the mean of the anti-diagonals, resulting in the projected diagonal (\subfigrefb{cells_suscept}{G}). For a low RAM contrast the \fbase{} peak in the projected diagonal is slightly enhanced (\subfigrefb{cells_suscept}{G}, gray dot on light purple line). For the higher RAM contrast, the overall second-order susceptibility is reduced (\subfigrefb{cells_suscept}{F}), with no pronounced \fbase{} peak in the projected diagonal (\subfigrefb{cells_suscept}{G}, dark purple line). In addition, there is an offset between the projected diagonals, demonstrating that the second-order susceptibility is reduced for RAM stimuli with a higher contrast (\subfigrefb{cells_suscept}{G}).
Theory predicts a pattern of non-linear susceptibility for the interaction of two cosine stimuli when \fone{}, \ftwo{} or \fsum{} are equal to \fbase{} (pink triangle in \subfigsref{cells_suscept}{E, F}). To estimate the second-order susceptibility from the P-unit responses to RAM stimuli, the noise stimulus was set in relation to the corresponding neuronal response in the Fourier domain, resulting in a matrix in which the nonlinearity at the sum frequency \fsum{} in the firing rate is depicted for two noise frequencies \fone{} and \ftwo{} (\eqnref{eq:susceptibility}, \subfigrefb{cells_suscept}{E--F}). The decomposition of the RAM stimulus into the frequency components allows us to approximate the beat frequencies $\Delta f$, occurring in case of pure sine-wave stimulation (\subfigrefb{motivation}{D}). Thus, the nonlinearity accessed with the RAM stimulation at \fsum{} (\subfigrefb{cells_suscept}{E}) is similar to the nonlinearity appearing during pure sine-wave stimulation at \bsum{} (orange peak, \subfigrefb{motivation}{D}). A band of elevated nonlinearity appears for the low RAM contrast when \fsumb{} (yellow anti-diagonal between pink edges, \subfigrefb{cells_suscept}{E}) but vanishes for the stronger stimulus (20\% contrast, \subfigref{cells_suscept}{F}). Further, the overall level of nonlinearity changes with the stimulus strength. To compare the structural changes in the matrices we calculated the mean of the anti-diagonals, resulting in the projected diagonal (\subfigrefb{cells_suscept}{G}). For a low RAM contrast the second-order susceptibility at \fbase{} is slightly enhanced (\subfigrefb{cells_suscept}{G}, gray dot on light purple line). For the higher RAM contrast, however, this is not visible and the overall second-order susceptibility is reduced (\subfigrefb{cells_suscept}{G}, compare light and dark purple lines).
%There a triangle is plotted not only if the frequency combinations are equal to the \fbase{} fundamental but also to the \fbase{} harmonics (two triangles further away from the origin).
%In this figure a part of \fsumehalf{} is marked with the orange diagonal line.
\subsection{High-CV P-units do not exhibit increased nonlinear interactions at \fsum}
CVs in P-units can range up to 1.5 \citealp{Grewe2017, Hladnik2023}. \Figref{cells_suscept_high_CV} shows the same analysis for an example higher-CV P-unit. Similar to the low-CV cell, high-CV P-units fire at multiples of the EOD period (\subfigrefb{cells_suscept_high_CV}{A}). In contrast to low-CV P-unit, however, the higher CV characterizes the noisier baseline firing pattern and the peak at \fbase{} is less pronounced in the power spectrum of the baseline activity (\subfigrefb{cells_suscept_high_CV}{B}). High-CV P-units do not exhibit a clear nonlinear structure related to \fbase{} neither in the second-order susceptibility matrices (\subfigrefb{cells_suscept_high_CV}{E--F}), nor in the projected diagonals (\subfigrefb{cells_suscept_high_CV}{G}). The overall level of nonlinearity, however shows the same dependence on the stimulus contrast. It is much reduced for high-contrast stimuli that drive the neuron much stronger (\subfigrefb{cells_suscept}{F}).
\begin{figure*}[h]%hp!
\includegraphics{cells_suscept}%cells_suscept
\caption{\label{cells_suscept} Response of experimentally measured P-units to RAM stimuli. Light purple -- low RAM contrast. Dark purple -- high RAM contrast. Regular firing low-CV P-unit. \figitem{A} Interspike intervals (ISI) distribution during baseline. \figitem{B} Baseline power spectrum of the firing rate. \figitem{C} Top: EOD carrier (gray) with RAM (red). Middle: Spike trains in response to a low RAM contrast. Bottom: Spike trains in response to a high RAM contrast. \figitem{D} First-order susceptibility (see \eqnref{linearencoding_methods}). \figitem{E} Absolute value of the second-order susceptibility, \eqnref{susceptibility}, for the low RAM contrast.
Pink lines -- edges of the structure when \fone, \ftwo{} or \fsum{} are equal to \fbase{}. Orange line -- part of the structure when \fsum{} is equal to half \feod. \figitem{F} Absolute value of the second-order susceptibility for the higher RAM contrast. \figitem{G} Projected diagonals, calculated as the mean of the anti-diagonals of the matrices in \panel{E--F}. Gray dots: \fbase{}. Gray area: $\fbase{} \pm 5$\,Hz. Dashed lines: Medians of the projected diagonals..
}
\end{figure*}
\subsection{High-CV P-units do not exhibit any nonlinear interactions}%frequency combinations
Based on the theory strong nonlinearities in spiking responses are not predicted for cells with irregular firing properties and high CVs \citealp{Voronenko2017}. CVs in P-units can range up to 1.5 \citealp{Grewe2017, Hladnik2023} and as a next step the second-order susceptibility of high-CV P-units will be presented. As low-CV P-units, high-CV P-units fire at multiples of the EOD period (\subfigrefb{cells_suscept_high_CV}{A}). In contrast to low-CV P-units high-CV P-units are noisier in their firing pattern and have a less pronounced mean baseline firing rate peak \fbase{} in the power spectrum of their firing rate during baseline (\subfigrefb{cells_suscept_high_CV}{B}). High-CV P-units do not exhibit any nonlinear structures related to \fbase{} neither in the second-order susceptibility matrices (\subfigrefb{cells_suscept_high_CV}{E--F}), nor in the projected diagonals (\subfigrefb{cells_suscept_high_CV}{G}). As in low-CV P-units (\subfigrefb{cells_suscept}{F}), the mean second-order susceptibility decreases with higher RAM contrasts in high-CV P-units (\subfigrefb{cells_suscept_high_CV}{F}).
\subsection{Ampullary afferents exhibit strong nonlinear interactions}
Irrespective of the CV, neither cell shows the complete proposed structure of non-linear interactions. \lepto{} posses an additional electrosensory system, the passive or ampullary electrosensory system that responds to low-frequency exogeneous electric stimuli. The population of ampullary afferents is much less heterogeneous, and known for the much lower CVs of the ISI distributions (0.08--0.22)\citealp{Grewe2017}. Ampullary cells do not phase-lock to the EOD and the ISI are unimodally distributed (\subfigrefb{ampullary}{A}). According to the low irregularity of the baseline response, the power spectrum shows a very distinct peak at \fbase{} and harmonics of it. Since the cells do not fire phase locked to the EOD, there is no peak at \feod{} (\subfigrefb{ampullary}{B}). When driven by a noise stimulus with a low contrast (note: this is not an AM but an added stimulus to the self-generated EOD, \subfigref{ampullary}{C}), ampullary cells exhibit very pronounced bands when \fsum{} is equal to \fbase{} or its harmonic in the second-order susceptibility matrix implying that this cell is especially nonlinear at these frequency combinations (yellow diagonals, \subfigrefb{ampullary}{E, G light green}). With higher stimulus contrasts these bands disappear (\subfigrefb{ampullary}{F}) and the projected diagonal is decreased and lacks the distinct peak at \fsum{}(\subfigrefb{ampullary}{G}, dark green).
\begin{figure*}[ht]%hp!
\includegraphics{ampullary}
\caption{\label{ampullary} Response of an experimentally measured ampullary cell. Light green -- low noise stimulus contrast. Dark green -- high noise stimulus contrast. \figitem{A} ISI distribution during baseline. \figitem{B} Baseline power spectrum of the firing rate. \figitem{C} Top: EOD carrier (gray) plus a band-pass limited white noise (red, see methods section \ref{rammethods}). Middle: Spike trains in response to a low noise stimulus contrast. Bottom: Spike trains in response to a high noise stimulus contrast. \figitem{D} First-order susceptibility (see \eqnref{linearencoding_methods}). \figitem{E} Absolute value of the second-order susceptibility (\eqnref{susceptibility}) for the low noise stimulus contrast.
Pink lines -- edges of the structure when \fone, \ftwo{} or \fsum{} are equal to \fbase{}. Orange line -- part of the structure when \fsum{} is equal to half \feod. \figitem{F} Second-order susceptibility matrix for the higher noise stimulus contrast. Colored lines as in \panel{E}. \figitem{G} Projected diagonals, calculated as the mean of the anti-diagonals of the matrices in \panel{E--F}. Gray dot: \fbase{}. Gray area: $\fbase{} \pm 5$\,Hz. Dashed lines: Medians of the projected diagonals.
\includegraphics{ampullary}
\caption{\label{ampullary} Estimation of linear and non-linear stimulus encoding in an ampullary afferent. \textbf{A} Interspike-interval (ISI) distribution of the cell's baseline activity. The CV of the ISIs is a dimensionless measure quantifying the response regularity. Zero CV would indicate perfect regularity. \textbf{B} Power-spectrum of the baseline response. \textbf{C} White noise stimulus (top) added to the fish's self-generated electric field and responses (spike raster in the lower traces). The stimulus contrast reflects the strength of the stimulus in relation to the own electric field amplitude. \textbf{D} Transfer function (first-order susceptibility, \eqnref{linearencoding_methods}) of the responses to 2\% (light green) and 20\% contrast (dark green) stimulation. \textbf{E, F} Absolute value of the second-order susceptibility, \eqnref{eq:susceptibility}, for the low, and high stimulus contrasts. Pink angles -- edges of the structure when \fone, \ftwo{} or \fsum{} are equal to \fbase{}. \textbf{G} Projected diagonals, calculated as the mean of the anti-diagonals of the matrices in \panel{E, F}. Gray dots mark \fbase{}. Horizontal dashed lines: medians of the projected diagonals.
}
\end{figure*}
\subsection{Ampullary cells exhibit strong nonlinear interactions}%with lower CVs as P-units
\lepto{} posses another primary sensory afferent population, the ampullary cells, with overall low \fbase{} (80--200\,Hz) and low CV values (0.08--0.22, \citealp{Grewe2017}). Ampullary cells do not phase-lock to the EOD, with no maxima at multiples of the EOD period and smoothly unimodal distributed ISIs (\subfigrefb{ampullary}{A}). Ampullary cells do not have a peak at \feod{} in the baseline power spectrum of the firing rate with no symmetry around it (\subfigrefb{ampullary}{B}). Instead, the \fbase{} peak is very pronounced with clear harmonics. When being exposed to a noise stimulus with a low contrast, ampullary cells exhibit very pronounced bands when \fsum{} is equal to \fbase{} or its harmonic in the second-order susceptibility matrix, implying that this cell is especially nonlinear at these frequency combinations (yellow diagonals, \subfigrefb{ampullary}{E}). With higher noise stimuli contrasts these bands disappear (\subfigrefb{ampullary}{F}) and the projected diagonal is lowered (\subfigrefb{ampullary}{G}, dark green).
These nonlinearity bands are more pronounced in ampullary cells than they were in P-units (compare \figrefb{ampullary} and \figrefb{cells_suscept}). Ampullary cells with their unimodal ISI distribution are closer than P-units to the LIF models without EOD carrier, where the predictions about the second-order susceptibility structure have mainly been elaborated on \citealp{Voronenko2017}. All here analyzed ampullary cells had CVs lower than 0.3 and exhibited strong nonlinear effects in accordance with the theoretical predictions \citealp{Voronenko2017}.
%and here this could be confirmed experimentally.
\subsection{Full nonlinear structure visible only in P-unit models}
In the following nonlinear interactions were systematically compared between an electrophysiologically recorded low-CV P-unit and the according P-unit LIF models with a RAM contrast of 1\,$\%$ (red, \subfigrefb{model_and_data}\,\panel[i]{A}). For a homogeneous population with size $\n{}=11$ one could observe a diagonal band in the absolute value of the second-order susceptibility at \fsumb{} of the recorded P-unit (yellow diagonal in pink edges, \subfigrefb{model_and_data}\,\panel[ii]{A}) and in the according model (\subfigrefb{model_and_data}\,\panel[iii]{A}). A nonlinear band appeared at \fsumehalf{}, but only in the recorded P-unit (orange line, \subfigrefb{model_and_data}\,\panel[ii]{A}). The signal-to-noise ratio and estimation of the nonlinearity structures can be improved if the number of RAM stimulus realizations is increased. Models have the advantage that they allow for data amounts that cannot be acquired experimentally. Still, even if a RAM stimulus is generated 1 million times, no changes are observable in the nonlinearity structures in the model second-order susceptibility (\subfigrefb{model_and_data}\,\panel[iv]{A}). An improved signal-to-noise ratio with 1 million stimuli is associated with smaller second-order susceptibility values (\subfigrefb{model_and_data}\,\panel[iv]{A}).
% \subsection{Full nonlinear structure visible only in P-unit models}
\subsection{Internal noise hides parts of the nonlinearity structure}
Traces of the proposed structure of second-order susceptibility are found in both ampullary and p-type electrosensory afferents. The nonlinerarity seems to depend on the CV, i.e. the level of intrinsic noise in the cells and in the next step we address whether the level of intrinsic masks the expected structures. One option to reduce the impact of this intrinsic noise is to average it out over many repetitions of the same stimulus. In the electrophysiological experiments we only have a limited recording duration and hence the number of stimulus repetitions is limited. To overcome these limitations we compared between an electrophysiologically recorded low-CV P-unit and its P-unit LIF model counterpart fitted to reproduce the P-unit behavior faithfully (see \figref{flowchart})\citealp{Barayeu2023}. In the recording depicted in \subfigrefb{model_and_data}\,\panel[II]{A} the cell was stimulated with a weak RAM stimulus with the contrast of 1\,$\%$ (red, \subfigrefb{model_and_data}\,\panel[i]{A}). The across trial average ($\n{}=11$) shows the diagonal band in second-order susceptibility at \fsumb{} (yellow diagonal in pink edges) and an additional nonlinear band appeared at \fsumehalf{} (\subfigrefb{model_and_data}\,\panel[ii]{A}). The matched model reproduces the same diagonal at \fsumb{} (\subfigrefb{model_and_data}\,\panel[iii]{A}) but not the diagonal at \fsumehalf. By increasing the number of trials in the simulation, the signal-to-noise ratio and thus the estimation of the nonlinearity structures can be improved. Still, even with 1 million repetitions, no changes are observable in the nonlinearity structures (\subfigrefb{model_and_data}\,\panel[iv]{A}). The increased trial count, however goes along with generally smaller second-order susceptibility values (\subfigrefb{model_and_data}\,\panel[iv]{A}).\notejg{a problem, that we use a new noise for each trial?}
%Note that this line doesn't appear in the susceptibility matrix of the model (\subfigrefb{model_and_data}{C})Each cell has an intrinsic noise level (\subfigrefb{model_and_data}{A}, bottom).
%The signal component (purple) compensates for this total noise reduction, is not a weak signal anymore and
Based on the Novikov-Furutsu Theorem \citealp{Novikov1965, Furutsu1963} the intrinsic noise of a LIF model can be split up into several independent noise processes with the same correlation function. Based on this a weak RAM signal as the input to the P-unit model (red in \subfigrefb{model_and_data}\,\panel[i]{A}) can be approximated by a model where no RAM stimulus is present (red) but instead the total noise of the model is split up into a reduced intrinsic noise component (gray) and a signal component (purple), maintaining the CV and \fbase{} as during baseline (\subfigrefb{model_and_data}\,\panel[i]{B}, see methods section \ref{intrinsicsplit_methods} for more details). This signal component (purple) can be used for the calculation of the second-order susceptibility. With the reduced noise component the signal-to-noise ratio increases and the number of stimulus realizations can be reduced. This noise split cannot be applied in experimentally measured cells. If this noise split is applied in the model with $\n{}=11$ stimulus realizations the nonlinearity at \fsumb{} is still present (\subfigrefb{model_and_data}\,\panel[iii]{B}). If instead, the RAM stimulus is drawn 1 million times the diagonal nonlinearity band is complemented by vertical and horizontal lines appearing at \foneb{} and \ftwob{} (\subfigrefb{model_and_data}\,\panel[iv]{B}). These nonlinear structures correspond to the ones observed in previous works \citealp{Voronenko2017}. If now a weak RAM stimulus is added (\subfigrefb{model_and_data}\,\panel[i]{C}, red), simultaneously the noise is split up into a noise and signal component (gray and purple) and the calculation is performed on the sum of the signal component and the RAM (red plus purple) only the diagonal band is present with $\n{}=11$ (\subfigrefb{model_and_data}\,\panel[iii]{C}) but also with 1\,million stimulus realizations (\subfigrefb{model_and_data}\,\panel[iv]{C}).
Based on the Novikov-Furutsu Theorem \citealp{Novikov1965, Furutsu1963} the intrinsic noise of a LIF model can be split up into several independent noise processes with the same correlation function. We make use of this and split the intrinsic noise $\xi$ into two parts: 90\% are then treated as signal ($\xi_{signal}$) while the remaining 10\% are treated as noise ($\xi_{noise}$, see methods for details). With this, the signal-to-noise ratio in the simulation can be arbitrarily varied and the combination of many repetitions and noise-split indeed reveals the triangular shape shown theoretically and for LIF models without carrier\citealp{Voronenko2017}(\subfigrefb{model_and_data}\,\panel[iv]{B}). Adding an additional, independent, RAM stimulus to the simulation does not heavily influence the qualitative observations but the nonlinearity becomes weaker (compare \subfigrefb{model_and_data}\,\panel[iii]{C} and \panel[iv]{C})
If a high-CV P-unit is investigated (not shown), there would be no nonlinear structures, neither in the electrophysiologically recorded data nor in the according model, corresponding to the theoretical predictions \citealp{Voronenko2017}.
% Based on this a weak RAM signal as the input to the P-unit model (red in \subfigrefb{model_and_data}\,\panel[i]{A}) can be approximated by a model where no RAM stimulus is present (red) but instead the total noise of the model is split up into a reduced intrinsic noise component (gray) and a signal component (purple), maintaining the CV and \fbase{} as during baseline (\subfigrefb{model_and_data}\,\panel[i]{B}, see methods section \ref{intrinsicsplit_methods} for more details). This signal component (purple) can be used for the calculation of the second-order susceptibility. With the reduced noise component the signal-to-noise ratio increases and the number of stimulus realizations can be reduced. This noise split cannot be applied in experimentally measured cells. If this noise split is applied in the model with $\n{}=11$ stimulus realizations the nonlinearity at \fsumb{} is still present (\subfigrefb{model_and_data}\,\panel[iii]{B}). If instead, the RAM stimulus is drawn 1 million times the diagonal nonlinearity band is complemented by vertical and horizontal lines appearing at \foneb{} and \ftwob{} (\subfigrefb{model_and_data}\,\panel[iv]{B}). These nonlinear structures correspond to the ones observed in previous works \citealp{Voronenko2017}. If now a weak RAM stimulus is added (\subfigrefb{model_and_data}\,\panel[i]{C}, red), simultaneously the noise is split up into a noise and signal component (gray and purple) and the calculation is performed on the sum of the signal component and the RAM (red plus purple) only the diagonal band is present with $\n{}=11$ (\subfigrefb{model_and_data}\,\panel[iii]{C}) but also with 1\,million stimulus realizations (\subfigrefb{model_and_data}\,\panel[iv]{C}).
In a high-CV P-unit we could not find such nonlinear structures, neither in the electrophysiologically recorded data nor in the respective model (not shown), corresponding to the theoretical predictions \citealp{Voronenko2017}.\notejg{How can we understand this? If we take a larger chunk of the intrinsic noise should it not help? Is there more than just noise that changes the structure?}
% (see methods, \eqnref{Noise_split_intrinsic}, \citealp{Novikov1965, Furutsu1963}) or its harmonics
\begin{figure*}[hb!]
\includegraphics[width=\columnwidth]{model_and_data}% is equal to \fbase is equal to half the \feod
\caption{\label{model_and_data} The influence of the RAM stimulus realization number $\n$, the RAM contrast $c$, and the split of the total intrinsic noise in a signal and noise component on the nonlinearity structures of the second-order susceptibility of an electrophysiologically recorded low-CV P-unit and its LIF model (see table~\ref{modelparams} for model parameters of 2012-07-03-ak). Pink lines in the matrices mark the edges of the structure when \fone{}, \ftwo{} or \fsum{} are equal to \fbase{}. The orange line in the matrices marks a part of the line at \fsumehalf.
\figitem[i]{A},\,\panel[i]{\textbf{B}},\,\panel[i]{\textbf{C}} Red -- RAM stimulus. The total intrinsic noise can be split into a noise component (gray) and a signal component (purple), maintaining the same CV and \fbase{} as before the split (see methods section \ref{intrinsicsplit_methods}). The calculation is performed on the sum of the signal component (purple) and the RAM (red) in \eqnref{susceptibility}.
\figitem[ii]{A} Absolute value of the second-order susceptibility of an electrophysiologically recorded P-unit, with $\n{}=11$ RAM stimulus realizations.
\figitem[iii]{A},\,\panel[iii]{\textbf{B}},\,\panel[iii]{\textbf{C}} Absolute value of the model second-order susceptibility with $\n{}=11$ RAM stimulus realizations.
\figitem[iv]{A},\,\panel[iv]{\textbf{B}},\,\panel[iv]{\textbf{C}} Absolute value of the model second-order susceptibility with 1 million RAM stimulus realizations.
\figitem{A} RAM contrast of 1\,$\%$. The band at \fsumb{} is visible in the matrices.
\figitem{B} No RAM stimulus, but a total noise split into a signal component (purple) and a noise component (gray). The band at \fsumb{} is visible in \panel[iii]{B} and \panel[iv]{B}. Besides that horizontal and vertical nonlinearities appear at \foneb{} and \ftwob{} in \panel[iv]{B}.
\figitem{C} A RAM stimulus (red) and a total noise split into a signal component (purple) and a noise component (gray). Only the band at \fsumb{} is visible in the matrices.
\includegraphics[width=\columnwidth]{model_and_data}
\caption{\label{model_and_data} \notejg{$\xi_{signal}$ and $\xi_{noise}$ in the figure?, reorder the figure and put the model left?} High trial numbers and a reduced internal noise reveal the non-linear structure in a LIF model with carrier. \figitem{A} $\chi_2$ surface of the electrophysiologically recorded low-CV P-unit (cell 2012-07-03-ak) driven with a weak stimulus with 1\% contrast. The plot shows that average $\chi_2$ surface (n\,=\,11). Pink edges mark the expected structure of enhanced nonlinearity. \figitem[i]{B} \textit{Standard condition} of the model simulation. The model is driven by RAM stimulus (red trace). $\xi_{signal}$ and $\xi_{noise}$ components of the total intrinsic noise ($\xi$) are shown in the center and bottom. In this condition, $\xi_{noise} = \xi$. \figitem[ii]{B} $\chi_2$ surface of the cell's LIF model counterpart (cell 2012-07-03-ak, table~\ref{modelparams}) averaged over the same number of stimulus repetitions. \figitem[iii]{B} Same as \panel[II]{B} but 1 million stimulus repetitions. \figitem[i - iii]{C} Same as in \panel[i - iii]{B} but in the \textit{noise split} condition: there is no external RAM signal driving the model. Parts (90\%) of the total intrinsic noise ($\xi$) are treated as signal (center trace, $\xi_{signal}$) while the remaining part is treated as $\xi_{noise}$ (see methods for details).
}
\end{figure*}
\subsection{Second-order susceptibility can explain nonlinear peaks in pure sinewave stimulation}
We calculated the second-order susceptibility surfaces at \fsumb{} by extracting the respective spectral component of the P-unit responses to RAM stimuli. How does this relate to the pure sinewave situation that approximates the interference of real animals? The relevant signals are the the beat frequencies \fone{} and \ftwo{} that arise from the interference of the receiving fish's own and either of the foreign EODs \figref{motivation}. The response power spectrum showed peaks from nonlinear interaction at the sum of the two beat frequencies (orange marker, \subfigrefb{motivation}{D}) and at the difference between the two beat frequencies (purple marker, \subfigrefb{motivation}{D}). In the example $\Delta f_{1}$ was similar to \fbase{}, corresponding to the vertical line in the RAM-based second-order susceptibility matrix in \subfigrefb{model_and_data}\,\panel[iii]{B}. In the three-fish example, there was a second, less prominent, nonlinearity at the difference between the two beat frequencies (purple circle, \subfigrefb{motivation}{D}), that cannot be explained with the second-order susceptibility matrix in \subfigrefb{model_and_data}\,\panel[iv]{B}, in which only the nonlinearity at \fsum{} in the response is addressed (\eqnref{eq:crosshigh}).
\begin{figure*}[h]
\includegraphics[width=\columnwidth]{model_full}
\caption{\label{model_full} Second-order susceptibility of an electrophysiologically recorded P-unit and of the corresponding model (see table~\ref{modelparams} for model parameters of 2012-07-03-ak). White lines -- coordinate axis. The nonlinearity at \fsum{} in the firing rate is quantified in the upper right and lower left quadrants (\eqnref{susceptibility}). The nonlinearity at \fdiff{} in the firing rate is quantified in the upper left and lower right quadrants. Mean baseline firing rate $\fbase{}=120$\,Hz. \figitem{A} Absolute value of the second-order susceptibility of an electrophysiologically recorded P-unit. RAM stimulus realizations $N=11$. Diagonal bands appear when the sum of the frequencies \fsum{} or the difference \fdiff{} is equal to \fbase{}. \figitem{B} Absolute value of the model second-order susceptibility, with $N=1$\,million stimulus realizations and the intrinsic noise split (see methods section \ref{intrinsicsplit_methods}). The diagonals, that were present in \panel{A}, are complemented by vertical and horizontal lines when \fone{} or \ftwo{} are equal to \fbase{}.}
\caption{\label{model_full} Second-order susceptibility of an electrophysiologically recorded P-unit and of the corresponding model (see table~\ref{modelparams} for model parameters of 2012-07-03-ak). White lines -- coordinate axis. The nonlinearity at \fsum{} in the firing rate is quantified in the upper right and lower left quadrants (\eqnref{eq:susceptibility}). The nonlinearity at \fdiff{} in the firing rate is quantified in the upper left and lower right quadrants. Mean baseline firing rate $\fbase{}=120$\,Hz. \figitem{A} Absolute value of the second-order susceptibility of an electrophysiologically recorded P-unit. RAM stimulus realizations $N=11$. Diagonal bands appear when the sum of the frequencies \fsum{} or the difference \fdiff{} is equal to \fbase{}. \figitem{B} Absolute value of the model second-order susceptibility, with $N=1$\,million stimulus realizations and the intrinsic noise split (see methods section \ref{intrinsicsplit_methods}). The diagonals, that were present in \panel{A}, are complemented by vertical and horizontal lines when \fone{} or \ftwo{} are equal to \fbase{}. Note that the different scale of the second-order susceptibility is associated with the higher signal-to-ratio in case of 1 million repeats in \panel{B}.}
\end{figure*}
\subsection{Similar nonlinear effects with RAM and sine-wave stimulation}
In the previous paragraphs, the nonlinearity at \fsum{} in the P-unit response was identified for the RAM frequencies \fone{} and \ftwo{}. This RAM-based second-order susceptibility can be used to approximate the nonlinearity in the three-fish setting, where two beats with frequencies \bone{} and \btwo{} are the driving forces for the P-unit response. In the previously shown three-fish setting a nonlinear peak occurred at the sum of the two beat frequencies (orange circle, \subfigrefb{motivation}{D}). In that example $\Delta f_{1}$ was similar to \fbase{}, corresponding to the vertical line in the RAM-based second-order susceptibility matrix in \subfigrefb{model_and_data}\,\panel[iv]{B}. In the three-fish example, there was a second less prominent nonlinearity at the difference of the two beat frequencies (purple circle, \subfigrefb{motivation}{D}), that cannot be explained with the second-order susceptibility matrix in \subfigrefb{model_and_data}\,\panel[iv]{B}, where only the nonlinearity at \fsum{} in the response is addressed.
Instead, the full second-order susceptibility matrix in \figrefb{model_full}, which depicts nonlinearities in the P-unit response at \fsum{} in the upper right and lower left quadrants and nonlinearities at \fdiff{} in the lower right and upper left quadrants (\eqnref{susceptibility}, \citealp{Voronenko2017}), has to be considered. Once calculating this full second-order susceptibility matrix based on the experimentally recorded data (\subfigrefb{model_full}{A}) and the corresponding model (\subfigrefb{model_full}{B}), one can observe that the diagonal structures are present in the upper right quadrant and for the lower right quadrants. The vertical and horizontal lines at \foneb{} and \ftwob{} are very pronounced in the upper right quadrant for the nonlinearity at \fsum{} and are prolonged to the lower right quadrant with lower nonlinearity values at \fdiff{} in the P-unit response. Note that the different scale of the second-order susceptibility is associated with the higher signal-to-ratio in case of 1 million repeats in \subfigrefb{model_full}{B}.
%, that quantifies the nonlinearity at \fdiff{} in the response , that quantifies the nonlinearity at \fsum{} in the response,
The second-order susceptibility can also be calculated for the full matrix including negative frequency components of the respective spectra in \eqnsref{eq:crosshigh} and (\ref{eq:susceptibility}). The resulting \suscept{} matrix is symmetric with respect to the origin and shows the \suscept{} at \fsum{} in the upper-right and lower-left quadrants and \suscept{} for the differences \fdiff{} in the lower-right and upper-left quadrants\citealp{Voronenko2017} (\figref{model_full}). The vertical and horizontal lines at \foneb{} and \ftwob{} are very pronounced in the upper-right quadrant of \subfigref{model_full}{B} for the nonlinearity at \fsum{} and extend into the lower-right quadrant (representing \fdiff)fading out towards more negative $f_2$ frequencies. \suscept{} values at \fsum{} match the \fsum{} peak seen in \figref{motivation}.
The small \fdiff{} peak in the power spectrum of the firing rate appearing during pure sine-wave stimulation (\subfigrefb{motivation}{D}) can be explained by the vertical line in the lower right quadrant (\subfigrefb{model_full}{B}, \citealp{Schlungbaum2023}). The here presented full second-order susceptibilities matrix was retrieved based on data and models with EOD carrier (\figrefb{model_full}) and is in accordance with the second-order susceptibilities calculated based on models without a carrier (\citealp{Voronenko2017, Schlungbaum2023}).
% When pure sine wave stimulation is happening it is expected that both nonlinear effects observed at \fsum{} and \fdiff{} (upper right and lower right quadrant, \subfigrefb{model_full}{B}) for a stimulation with positive frequencies \citealp{Schlungbaum2023}.
The smaller \fdiff{} power spectral peak observed during pure sine-wave stimulation (\subfigrefb{motivation}{D}) can be explained by the fading out vertical line in the lower-right quadrant (\subfigrefb{model_full}{B}, \citealp{Schlungbaum2023}). Even though the second-order susceptibilities were here estimated form data and models with an modulated (EOD) carrier (\figrefb{model_full}) they are in good accordance with the second-order susceptibilities found in LIF models without a carrier\citealp{Voronenko2017, Schlungbaum2023}.
\notejg{Only show the model?}
\notejg{Why do we see peaks at the vertical lines in the three fish setting but not in the RAM situation? SNR? Discussion?}
\begin{figure*}[ht!]
\includegraphics[width=\columnwidth]{data_overview_mod}
\caption{\label{data_overview_mod} Nonlinearity for a population of ampullary cells (\panel{A, C}) and P-units (\panel{B, D}). \nli{} is calculated as the maximal value in the range $\fbase{} \pm 5$\,Hz of the projected diagonal divided by its median (see \eqnref{nli_equation}). There are maximally two noise contrasts per cell in a population. \figitem{A, B} There is a negative correlation between the CV during baseline and \nli. \figitem{C, D} Response modulation, \eqnref{response_modulation}, is an indicator of the subjective stimulus strength for a cell. There is a negative correlation between response modulation and \nli.
\caption{\label{data_overview_mod} Nonlinearity for a population of P-units (\panel{A, C}) and ampullary cells (\panel{B, D}). The nonlinearity indec (\nli{}) is calculated as the maximal value in the range $\fbase{} \pm 5$\,Hz of the projected diagonal divided by its median (see \eqnref{eq:nli_equation}). Each recorded neuron contributes at maximum with two stimulus contrasts. \figitem{A, B} There is a negative correlation between the CV during baseline and \nli. \figitem{C, D} The \nli{} is plotted against the response modulation, \eqnref{response_modulation}, an indicator of the subjective stimulus strength for a cell. There is a negative correlation between response modulation and \nli. Restricting the analysis to the weakest stimulus that was presented to each unique neuron, does not change the results. The number of unique neurons is 222 for P-units and 45 for ampullary cells.
\notejg{switch order of P-units and ampullaries.}
}
\end{figure*}
%\label{response_modulation}
%Second-order susceptibility for all frequencies
\subsection{Low CVs are associated with strong nonlinearity on a population level}%when considering
So far second-order susceptibility was illustrated only with single-cell examples (\figrefb{cells_suscept}, \figrefb{ampullary}). For a P-unit comparison on a population level, the second-order susceptibility of P-units was expressed in a nonlinearity index \nli{}, see \eqnref{nli_equation}, that characterized the peakedness of the \fbase{} peak in the projected diagonal (\subfigrefb{cells_suscept}{G}). \nli{} has high values when the \fbase{} peak in the projected diagonal is especially pronounced, as in the low-CV ampullary cell (\subfigrefb{ampullary}{G}, light green). The two noise stimulus contrasts of this ampullary cell are highlighted in the population statics of ampullary cells with dark circles (\subfigrefb{data_overview_mod}{A}). The higher noise stimulus contrast is associated with a less pronounced peak in the projected diagonal (\subfigrefb{ampullary}{G}, dark green) and is represented with a lower \nli{} value (\subfigrefb{data_overview_mod}{A}, dark circle close to the origin). In an ampullary cell population, there is a negative correlation between the CV during baseline and \nli{}, meaning that the diagonals are pronounced for low-CV cells and disappear towards high-CV cells (\subfigrefb{data_overview_mod}{A}). Since the same stimulus can be strong for some cells and faint for others, the noise stimulus contrast is not directly comparable between cells. A better estimation of the subjective stimulus strength is the response modulation of the cell (see methods section \ref{response_modulation}). Ampullary cells with stronger response modulations have lower \nli{} scores (red in \subfigrefb{data_overview_mod}{A}, \subfigrefb{data_overview_mod}{C}). The so far shown population statistics comprised several RAM contrasts per cell and if instead each ampullary cell is represented with the lowest recorded contrast, then \nli{} significantly correlates with the CV during baseline ($r=-0.46$, $p<0.001$), the response modulation ($r=-0.6$, $p<0.001$) but not with \fbase{} ($r=0.2$, $p=0.16$).%, $\n{}=51$, $\n{}=51$, $\n{}=51${*}{*}{*}^*^*^*each cell can contribute several RAM contrasts in
The P-unit population has higher baseline CVs and lower \nli{} values (\subfigrefb{data_overview_mod}{B}) that are weaker correlated than in the population of ampullary cells. The negative correlation (\subfigrefb{data_overview_mod}{B}) is increased when \nli{} is plotted against the response modulation of P-units (\subfigrefb{data_overview_mod}{D}). The two example P-units shown before (\figrefb{cells_suscept}) are highlighted with dark circles in \subfigrefb{data_overview_mod}{B, D}. High-CV P-units and strongly driven P-units have lower \nli{} values (\subfigrefb{data_overview_mod}{B, D}). In a P-unit population where each cell is represented not by several contrasts but by the lowest recorded contrast, \nli{} significantly correlates with the CV during baseline ($r=-0.17$, $p=0.01$), the response modulation ($r=-0.35$, $p<0.001$) and \fbase{} ($r=-0.32$, $p<0.001$).%, $\n{}=221$*, $\n{}=221$******, $\n{}=221$
\subsection{Low CVs and weak stimuli are associated with strong nonlinearity}
The non-linear effects shown for single cell examples above are supported by the analysis of the pool of 222 P-units and 45 ampullary afferents recorded in 71 specimen. To compare across cells we expressed the second-order susceptibility as the nonlinearity index \nli{}, see \eqnref{eq:nli_equation}. The \nli{} characterizes the peakedness of the projections onto the diagonal of the \suscept{} matrices (e.g. \subfigref{cells_suscept}{G} at \fbase{}. It assumes high values when the \fbase{} peak in the projected diagonal is pronounced relative to the median of the diagonal and is small when there is no distinct peak. The P-unit population has higher \nli{} values up to three and depends weakly on the the CV of the ISI distribution under baseline condition. Cells with lower baseline CVs have the tendency to exhibit a stronger nonlinearity than those that have high CVs (Pearson's r\,=\,$-0.16$, p\,<\,0.01). The two example P-units shown before (\figrefb{cells_suscept} and \figrefb{cells_suscept_high_CV}) are highlighted with dark circles in \subfigrefb{data_overview_mod}{B, D}. The stimulus strength plays an important role. Several of the recorded neurons contribute with two dots to the data as their responses to the same RAM stimulus but with different contrasts were recorded. Higher stimulus contrasts lead to a stronger drive and thus stronger response modulations (see color code bar in \subfigref{data_overview_mod}{A}, see methods). Since P-units are heterogeneous in their susceptibility to the stimulus\citealp{Grewe2017}, their responses to the same stimulus intensity vary a lot. When replotting the \nli{} against the response modulation (\subfigrefb{data_overview_mod}{C}) a negative correlation is observed (Pearson's r\,=\,$-0.35$, p\,<\,0.001 \notesr{das minus wird durch den Mathmodus länger, das will Jan immer so haben}) showing that cells that are strongly driven by the stimulus show less nonlinearity while those that are only weakly driven show higher nonlinearities. Whether or not a cell responds nonlinearly thus depends on both, the baseline CV (i.e. the internal noise) and the response strength.
%In a P-unit population where each cell is represented not by several contrasts but by the lowest recorded contrast, \nli{} significantly correlates with the CV during baseline ($r=-0.17$, $p=0.01$), the response modulation ($r=-0.35$, $p<0.001$) and \fbase{} ($r=-0.32$, $p<0.001$).%, $\n{}=221$*, $\n{}=221$******, $\n{}=221$
The population of ampullary cells is generally more homogeneous and have lower CVs than the P-units and show much higher \nli{} values (factor of 10). Overall, there is a negative correlation with the baseline CV (Pearson's r=-0.35, p < 0.01). The example cell shown above (\figref{ampullary}) was recorded at two different stimulus intensities and the \nli{}s are highlighted with black circles. Again, we see that cells that are strongly driven by the stimulus cluster at the bottom of the distribution and have \nli{} values close to zero. This is confirmed when the data is replotted against the response modulation, those cells that are strongly driven by the stimulus show weak nonlinearities while weakly driven neurons exhibit high values (Pearson's r=-0.59, p < 0.0001).
\section{Discussion}
In this work, the second-order susceptibility in spiking responses of P-units was characterized for the scenario where at least three fish are present. A low-CV subpopulation of P-units in \lepto{} was demonstrated to have increased nonlinearity values when \fone{}, \ftwo{}, \fsum{} or \fdiff{} were equal to the mean baseline firing rate \fbase{}. Ampullary cells, with even lower CVs, exhibited even stronger nonlinearity. These nonlinear structures in low-CV cells confirmed the predictions from the weakly nonlinear theory \citealp{Voronenko2017}. High-CV P-units did not exhibit any nonlinear interactions. The nonlinearities appearing for the noise frequencies \fone{}, \ftwo{} were confirmed as a proxy for nonlinearities that might arise in response to the beat frequencies \bone{}, \btwo{}, when at least three fish are present.
\notejg{dumped here since not strictly result...}
These nonlinearity bands are more pronounced in ampullary cells than they were in P-units (compare \figrefb{ampullary} and \figrefb{cells_suscept}). Ampullary cells with their unimodal ISI distribution are closer than P-units to the LIF models without EOD carrier, where the predictions about the second-order susceptibility structure have mainly been elaborated on \citealp{Voronenko2017}. All here analyzed ampullary cells had CVs lower than 0.3 and exhibited strong nonlinear effects in accordance with the theoretical predictions \citealp{Voronenko2017}.
%and here this could be confirmed experimentally.
In this work, the second-order susceptibility in spiking responses of P-units was characterized for the scenario where at least three fish are present. A low-CV subpopulation of P-units in \lepto{} was demonstrated to have increased nonlinearity values when \fone{}, \ftwo{}, \fsum{} or \fdiff{} were equal to the mean baseline firing rate \fbase{}. Ampullary cells, with even lower CVs, exhibited even stronger nonlinearity. These nonlinear structures in low-CV cells confirmed the predictions from the nonlinear theory described in \citet{Voronenko2017}. High-CV P-units did not exhibit any nonlinear interactions. The nonlinearities appearing for the noise frequencies \fone{}, \ftwo{} were confirmed as a proxy for nonlinearities that might arise in response to the beat frequencies \bone{}, \btwo{}, when at least three fish are present.
%\fsum{} or \fdiff{} were equal to \fbase{} \bsum{} or \bdiff{},
@ -665,14 +636,16 @@ auditory nerve fibers and such nonlinear effects might also be expected in the a
\subsection{Experimental subjects and procedures}
Within this project we re-evaluated datasets that were recorded between 2010 and 2023 at the Ludwig Maximilian University (LMU) M\"unchen and the Eberhard-Karls University T\"ubingen. All experimental protocols complied with national and European law and were approved by the respective Ethics Committees of the Ludwig-Maximilians Universität München (permit no. 55.2-1-54-2531-135-09) and the Eberhard-Karls Unversität Tübingen (permit no. ZP 1/13 and ZP 1/16).
The final sample consisted of 222 P-units and 45 ampullary electroreceptor afferents recorded in 71 weakly electric fish of the species \lepto{}. The original electrophysiological recordings were performed on male and female weakly electric fish of the species \lepto{} that were obtained from a commercial supplier for tropical fish (Aquarium Glaser GmbH, Rodgau,
The final sample consisted of 221 P-units and 45 ampullary electroreceptor afferents recorded in 71 weakly electric fish of the species \lepto{}. The original electrophysiological recordings were performed on male and female weakly electric fish of the species \lepto{} that were obtained from a commercial supplier for tropical fish (Aquarium Glaser GmbH, Rodgau,
Germany). The fish were kept in tanks with a water temperature of $25\pm1\,^\circ$C and a conductivity of around $270\,\micro\siemens\per\centi\meter$ under a 12\,h:12\,h light-dark cycle.
Before surgery, the animals were deeply anesthetized via bath application with a solution of MS222 (120\,mg/l, PharmaQ, Fordingbridge, UK) buffered with Sodium Bicarbonate (120\,mg/l). The posterior anterior lateral line nerve (pALLN) was exposed by making a small cut into the skin covering the nerve. The cut was placed dorsal of the operculum just before the nerve descends towards the anterior lateral line ganglion (ALLNG). Those parts of the skin that were to be cut were locally anesthetized by cutaneous application of liquid lidocaine hydrochloride (20\,mg/ml, bela-pharm GmbH). During the surgery water supply was ensured by a mouthpiece to maintain anesthesia with a solution of MS222 (100\,mg/l) buffered with Sodium Bicarbonate (100\,mg/l). After surgery fish were immobilized by intramuscular injection of from 25\,$\micro$l to 50\,$\micro$l of tubocurarine (5\,mg/ml dissolved in fish saline; Sigma-Aldrich).
Respiration was then switched to normal tank water and the fish was transferred to the experimental tank.
\subsection{Experimental setup}
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 reapplication 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 (\figrefb{Setup}, blue triangle). 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 reapplication of Lidocaine (2\,\%; bela-pharm, Vechta, Germany) around the surgical wounds. Electrodes (borosilicate; 1.5\,mm outer diameter; GB1These nonlinearity bands are more pronounced in ampullary cells than they were in P-units (compare \figrefb{ampullary} and \figrefb{cells_suscept}). Ampullary cells with their unimodal ISI distribution are closer than P-units to the LIF models without EOD carrier, where the predictions about the second-order susceptibility structure have mainly been elaborated on \citealp{Voronenko2017}. All here analyzed ampullary cells had CVs lower than 0.3 and exhibited strong nonlinear effects in accordance with the theoretical predictions \citealp{Voronenko2017}.
%and here this could be confirmed experimentally.
50F-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 (\figrefb{Setup}, blue triangle). 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.
\subsection{Identification of P-units and ampullary cells}
The neurons were classified into cell types during the recording by the experimenter. P-units were classified based on mean baseline firing rates of 50--450\,Hz and a clear phase-locking to the EOD and their responses to amplitude modulations of their own EOD\citealp{Grewe2017, Hladnik2023}. Ampullary cells were classified based on mean firing rates of 80--200\,Hz absent phase-locking to the EOD and responses to low-frequency sinusoidal stimuli\citealp{Grewe2017}. We here selected only those cells of which the neuron's baseline activity as well as the responses to frozen noise stimuli were recorded.
@ -743,12 +716,12 @@ From $S_{rs}(\omega)$ and $ S_{ss}(\omega)$ we calculated the linear susceptibil
\end{equation}
The second-order cross-spectrum that depends on the two frequencies $\omega_1$ and $\omega_2$ was calculated according to
\begin{equation}
\label{crosshigh}
\label{eq:crosshigh}
S_{rss} (\omega_{1},\omega_{2}) = \frac{\langle \tilde r (\omega_{1}+\omega_{2}) \tilde s^* (\omega_{1})\tilde s^* (\omega_{2}) \rangle}{T}
\end{equation}
The second-order susceptibility was calculated by dividing the higher-order cross-spectrum by the spectral power at the respective frequencies.
\begin{equation}
\label{susceptibility0}
\label{eq:susceptibility}
%\begin{split}
\chi_{2}(\omega_{1}, \omega_{2}) = \frac{S_{rss} (\omega_{1},\omega_{2})}{2S_{ss} (\omega_{1}) S_{ss} (\omega_{2})}
%\end{split}
@ -767,7 +740,7 @@ The absolute value of a second-order susceptibility matrix is visualized in \fig
\notejg{use of $f_{Base}$ or $f_{base}$ or $f_0$ should be consistent throughout the manuscript.}
We expect to see non-linear susceptibility when $\omega_1 + \omega_2 = f_{Base}$. To characterize this we calculated the nonlinearity index (NLI) as
\begin{equation}
\label{nli_equation}
\label{eq:nli_equation}
NLI(f_{Base}) = \frac{\max_{f_{Base}-5\,\rm{Hz} \leq f \leq f_{Base}+5\,\rm{Hz}} D(f)}{\mathrm{med}(D(f))}
\end{equation}
\notejg{sollte es $D(\omega)$ sein?}
@ -780,35 +753,35 @@ If the same frozen noise was recorded several times in a cell, each noise repeti
Leaky integrate-and-fire (LIF) models with a carrier were constructed to reproduce the specific firing properties of P-units \citealp{Chacron2001,Sinz2020}. The sole driving input into the P-unit model during baseline, i.e. when no external stimulus was given, is the fish's own EOD modeled as a cosine wave
\begin{equation}
\label{eod}
\label{eq:eod}
x(t) = x_{EOD}(t) = \cos(2\pi f_{EOD} t)
\end{equation}
with the EOD frequency $f_{EOD}$ and an amplitude normalized to one.
In the model, the input $x(t)$ was then first thresholded to model the synapse between the primary receptor cells and the afferent.
\begin{equation}
\label{threshold2}
\label{eq:threshold2}
\lfloor x(t) \rfloor_0 = \left\{ \begin{array}{rcl} x(t) & ; & x(t) \ge 0 \\ 0 & ; & x(t) < 0 \end{array} \right.
\end{equation}
$\lfloor x(t) \rfloor_{0}$ denotes the threshold operation that sets negative values to zero (box left to \subfigrefb{flowchart}\,\panel[i]{A}).
The resulting receptor signal was then low-pass filtered to approximate passive signal conduction in the afferent's dendrite (box left to \subfigrefb{flowchart}\,\panel[ii]{A}).
\begin{equation}
\label{dendrite}
\label{eq:dendrite}
\tau_{d} \frac{d V_{d}}{d t} = -V_{d}+ \lfloor x(t) \rfloor_{0}
\end{equation}
with $\tau_{d}$ as the dendritic time constant. Dendritic low-pass filtering was necessary to reproduce the loose coupling of P-unit spikes to the EOD while maintaining high sensitivity at small amplitude modulations. Because the input was dimensionless, the dendritic voltage was dimensionless, too. The combination of threshold and low-pass filtering extracts the amplitude modulation of the input $x(t)$.
The dendritic voltage $V_d(t)$ was the input to a leaky integrate-and-fire (LIF) model
\begin{equation}
\label{LIF}
\label{eq:LIF}
\tau_{m} \frac{d V_{m}}{d t} = - V_{m} + \mu + \alpha V_{d} - A + \sqrt{2D}\xi(t)
\end{equation}
where $\tau_{m}$ is the membrane time-constant, $\mu$ is a fixed bias current, $\alpha$ is a scaling factor for $V_{d}$, $A$ is an inhibiting adaptation current, and $\sqrt{2D}\xi(t)$ is a white noise with strength $D$. All state variables except $\tau_m$ are dimensionless.
The adaptation current $A$ followed
\begin{equation}
\label{adaptation}
\label{eq:adaptation}
\tau_{A} \frac{d A}{d t} = - A
\end{equation}
with adaptation time constant $\tau_A$.
@ -830,13 +803,13 @@ Whenever the membrane voltage $V_m(t)$ crossed the spiking threshold $\theta=1$
\begin{figure*}[hb!]
\includegraphics[width=\columnwidth]{flowchart}
\caption{\label{flowchart}
Flowchart of a LIF P-unit model with EOD carrier. The main steps of the model are depicted in the left model column. The three columns show the relevant signals in three different settings. (i) the baseline situation, no external stimulus, only the animal's self-generated EOD (i.e. the carrier) is present (ii) RAM stimulation, the carrier is amplitude modulated with a weak (2\% contrast) stimulus, (iii) Noise split condition in which 90\% of the internal noise is used as a driving stimulus scaled with the correction factor $\rho$ (see text). Note: that the firing rate and the CV of the ISI distribution is the same in this and the baseline condition. (Model cell identifier 2012-07-03-ak, see table~\ref{modelparams} for model parameters. \textbf{A} Thresholding: a simple linear threshold was applied to the EOD carrier (eq,\,\ref{eod}) The red line on top depicts the amplitude modulation (AM). \textbf{B} Dendritic low-pass filtering attenuates the carrier. \textbf{C} An Gaussian noise is added to the signal in B. Note the reduced internal noise amplitude in the split (iii) condition. \textbf{D} Spiking output of the LIF model in response to the addition of B and C. \textbf{E} Power spectra of the LIF neuron's spiking activity. Under the baseline condition (D$_i$) there are several peaks at, from left to right, the baseline firing rate $f_{base}$, $f_{EOD} - f_{base}$ $f_{EOD}$, and $f_{EOD} + f_{base}$, in the stimulus driven regime, there is only a peak at $f_{eod}$, while under the noise split condition (D$_iii$) again all peaks are present.}
Flowchart of a LIF P-unit model with EOD carrier. The main steps of the model are depicted in the left model column. The three columns show the relevant signals in three different settings. (i) the baseline situation, no external stimulus, only the animal's self-generated EOD (i.e. the carrier) is present (ii) RAM stimulation, the carrier is amplitude modulated with a weak (2\% contrast) stimulus, (iii) Noise split condition in which 90\% of the internal noise is used as a driving stimulus scaled with the correction factor $\rho$ (see text). Note: that the firing rate and the CV of the ISI distribution is the same in this and the baseline condition. (Model cell identifier 2012-07-03-ak, see table~\ref{modelparams} for model parameters. \textbf{A} Thresholding: a simple linear threshold was applied to the EOD carrier (eq,\,\ref{eq:eod}) The red line on top depicts the amplitude modulation (AM). \textbf{B} Dendritic low-pass filtering attenuates the carrier. \textbf{C} An Gaussian noise is added to the signal in B. Note the reduced internal noise amplitude in the split (iii) condition. \textbf{D} Spiking output of the LIF model in response to the addition of B and C. \textbf{E} Power spectra of the LIF neuron's spiking activity. Under the baseline condition (D$_i$) there are several peaks at, from left to right, the baseline firing rate $f_{base}$, $f_{EOD} - f_{base}$ $f_{EOD}$, and $f_{EOD} + f_{base}$, in the stimulus driven regime, there is only a peak at $f_{eod}$, while under the noise split condition (D$_iii$) again all peaks are present.}
\end{figure*}
\subsection{Numerical implementation}
The model's ODEs were integrated by the Euler forward method with a time-step of $\Delta t = 0.05$\,ms. The intrinsic noise $\xi(t)$ (\eqnref{LIF}, \subfigrefb{flowchart}\,\panel[iii]{A}) was added by drawing a random number from a normal distribution $\mathcal{N}(0,\,1)$ with zero mean and standard deviation of one in each time step $i$. This number was multiplied with $\sqrt{2D}$ and divided by $\sqrt{\Delta t}$:
The model's ODEs were integrated by the Euler forward method with a time-step of $\Delta t = 0.05$\,ms. The intrinsic noise $\xi(t)$ (\eqnref{eq:LIF}, \subfigrefb{flowchart}\,\panel[iii]{A}) was added by drawing a random number from a normal distribution $\mathcal{N}(0,\,1)$ with zero mean and standard deviation of one in each time step $i$. This number was multiplied with $\sqrt{2D}$ and divided by $\sqrt{\Delta t}$:
\begin{equation}
\label{LIFintegration}
\label{eq:LIFintegration}
V_{m_{i+1}} = V_{m_i} + \left(-V_{m_i} + \mu + \alpha V_{d_i} - A_i + \sqrt{\frac{2D}{\Delta t}}\mathcal{N}(0,\,1)_i\right) \frac{\Delta t}{\tau_m}
\end{equation}
@ -844,11 +817,11 @@ The model's ODEs were integrated by the Euler forward method with a time-step of
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 (mean 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-like increases and decreases in EOD amplitude (onset-state and steady-state responses, effective adaptation time constant). For each simulation, the start parameters $A$, $V_{d}$ and $V_{m}$ were drawn from a random starting value distribution, estimated from a 100\,s baseline simulation after an initial 100\,s of simulation that was discarded as a transient.
\subsection{Stimuli for the model}
The model neurons were driven with similar stimuli as the real neurons in the original experiments. To mimic the interaction with one or two foreign animals the receiving fish's EOD (eq.\,\ref{eod}) was normalized to an amplitude of one and the respective EODs of a second or third fish were added.
The model neurons were driven with similar stimuli as the real neurons in the original experiments. To mimic the interaction with one or two foreign animals the receiving fish's EOD (eq.\,\ref{eq:eod}) was normalized to an amplitude of one and the respective EODs of a second or third fish were added.
The random amplitude modulation (RAM) input to the model was created by drawing random amplitude and phases from Gaussian distributions for each frequency component in the range $0-300$ Hz. An inverse Fourier transform was applied to get the final amplitude RAM time-course. The input to the model was then
\begin{equation}
\label{ram_equation}
\label{eq:ram_equation}
x(t) = (1+ RAM(t)) \cdot \cos(2\pi f_{EOD} t)
\end{equation}
@ -856,18 +829,18 @@ From each simulation run, the first second was discarded and the analysis was ba
% \subsection{Second-order susceptibility analysis of the model}
% %\subsubsection{Model second-order nonlinearity}
% The second-order susceptibility in the model was calculated with \eqnref{susceptibility}, resulting in matrices as in \figrefb{model_and_data} and \figrefb{model_full}. For this, the model neuron was presented the input $x(t)$ for 2\,s, with the first second being dismissed as the transient. The second-order susceptibility calculation was performed on the last second, resulting in a frequency resolution of 1\,Hz.
% The second-order susceptibility in the model was calculated with \eqnref{eq:susceptibility}, resulting in matrices as in \figrefb{model_and_data} and \figrefb{model_full}. For this, the model neuron was presented the input $x(t)$ for 2\,s, with the first second being dismissed as the transient. The second-order susceptibility calculation was performed on the last second, resulting in a frequency resolution of 1\,Hz.
\subsection{Model noise split into a noise and a stimulus component}\label{intrinsicsplit_methods}
According to the Novikov-Furutsu Theorem \citealp{Novikov1965, Furutsu1963} the total noise of a LIF ($\xi$) model can be split up into several independent noise processes with the same correlation function. Here we split the internal noise into two parts: (i) One part is treated as a driving input signal ($\xi_{signal} = c_{signal} \cdot \xi$) and used to calculate the cross-spectra \eqnref{crosshigh} and (ii) the remaining noise ($\xi_{noise} = (1-c_{signal})\cdot\xi$) that is treated as pure noise. In this way the effective signal-to-noise ratio can be increased while maintaining the total noise in the system. In our case the model has a carrier (the fish's self-generated EOD) and we thus want to drive the model with an amplitude modulation stimulus.
According to the Novikov-Furutsu Theorem \citealp{Novikov1965, Furutsu1963} the total noise of a LIF ($\xi$) model can be split up into several independent noise processes with the same correlation function. Here we split the internal noise into two parts: (i) One part is treated as a driving input signal ($\xi_{signal} = c_{signal} \cdot \xi$) and used to calculate the cross-spectra \eqnref{eq:crosshigh} and (ii) the remaining noise ($\xi_{noise} = (1-c_{signal})\cdot\xi$) that is treated as pure noise. In this way the effective signal-to-noise ratio can be increased while maintaining the total noise in the system. In our case the model has a carrier (the fish's self-generated EOD) and we thus want to drive the model with an amplitude modulation stimulus.
\begin{equation}
\label{ram_split}
\label{eq:ram_split}
x(t) = (1+ \sqrt{\rho \, 2D \,c_{signal}} \cdot \xi(t)) \cdot \cos(2\pi f_{EOD} t)
\end{equation}
with $\rho$ a scaling factor that compensates (see below) for the signal transformations the amplitude modulation stimulus undergoes in the model, i.e. the threshold and the dendritic lowpass.
\begin{equation}
\label{Noise_split_intrinsic_dendrite}
\label{eq:Noise_split_intrinsic_dendrite}
\tau_{d} \frac{d V_{d}}{d t} = -V_{d}+ \lfloor x(t) \rfloor_{0}
\end{equation}
@ -878,12 +851,12 @@ with $\rho$ a scaling factor that compensates (see below) for the signal transfo
%\end{equation}
\begin{equation}
\label{Noise_split_intrinsic}
\label{eq:Noise_split_intrinsic}
\tau_{m} \frac{d V_{m}}{d t} = - V_{m} + \mu + \alpha V_{d} - A + \sqrt{2D \, c_{noise}}\,\xi(t)
\end{equation}
% das stimmt so, das c kommt unter die Wurzel!
A big portion of the total noise was assigned to the signal component ($c_{signal} = 0.9$) and the remaining part to the noise component ($c_{noise} = 0.1$, \subfigrefb{flowchart}\,\panel[iii]{D}). For the application of the Novikov-Furutsu Theorem \citealp{Novikov1965, Furutsu1963} it is critical that both components add up to the initial 100\,$\%$ of the total noise. This is easily achieved in a model without a carrier if the condition $c_{signal}+c_{noise}=1$ is satisfied. The situation here is more complicated. To compensate for the transformations the signal undergoes before it enters the LIF core, $\xi_{signal}(t)$ was scaled up by the factor $\rho$ (\eqnref{ram_split}, red in \subfigrefb{flowchart}\,\panel[i]{D}). $\rho$ was found by bisecting the space of possible $\rho$ scaling factors by minimizing the error between the baseline CV (only carrier) and the CV during stimulation with noise split.
A big portion of the total noise was assigned to the signal component ($c_{signal} = 0.9$) and the remaining part to the noise component ($c_{noise} = 0.1$, \subfigrefb{flowchart}\,\panel[iii]{D}). For the application of the Novikov-Furutsu Theorem \citealp{Novikov1965, Furutsu1963} it is critical that both components add up to the initial 100\,$\%$ of the total noise. This is easily achieved in a model without a carrier if the condition $c_{signal}+c_{noise}=1$ is satisfied. The situation here is more complicated. To compensate for the transformations the signal undergoes before it enters the LIF core, $\xi_{signal}(t)$ was scaled up by the factor $\rho$ (\eqnref{eq:ram_split}, red in \subfigrefb{flowchart}\,\panel[i]{D}). $\rho$ was found by bisecting the space of possible $\rho$ scaling factors by minimizing the error between the baseline CV (only carrier) and the CV during stimulation with noise split.
\notejg{Etwas kurz, die Tabelle. Oder benutzen wir hier tatsaechlich nur die zwei Modelle?}
@ -936,7 +909,7 @@ A big portion of the total noise was assigned to the signal component ($c_{signa
\begin{figure*}[hp]%hp!
\includegraphics{cells_suscept_high_CV}
\caption{\label{cells_suscept_high_CV} Response of experimentally measured P-units to RAM stimuli. Light purple -- low RAM contrast. Dark purple -- high RAM contrast. Noisy high-CV P-Unit. \figitem{A} Interspike intervals (ISI) distribution during baseline. \figitem{B} Baseline power spectrum of the firing rate. \figitem{C} Top: EOD carrier (gray) with RAM (red). Middle: Spike trains in response to a low RAM contrast. Bottom: Spike trains in response to a high RAM contrast. \figitem{D} First-order susceptibility (see \eqnref{linearencoding_methods}). \figitem{E} Absolute value of the second-order susceptibility, \eqnref{susceptibility}, for the low RAM contrast.
\caption{\label{cells_suscept_high_CV} Response of experimentally measured P-units to RAM stimuli. Light purple -- low RAM contrast. Dark purple -- high RAM contrast. Noisy high-CV P-Unit. \figitem{A} Interspike intervals (ISI) distribution during baseline. \figitem{B} Baseline power spectrum of the firing rate. \figitem{C} Top: EOD carrier (gray) with RAM (red). Middle: Spike trains in response to a low RAM contrast. Bottom: Spike trains in response to a high RAM contrast. \figitem{D} First-order susceptibility (see \eqnref{linearencoding_methods}). \figitem{E} Absolute value of the second-order susceptibility, \eqnref{eq:susceptibility}, for the low RAM contrast.
Pink lines -- edges of the structure when \fone, \ftwo{} or \fsum{} are equal to \fbase{}. Orange line -- part of the structure when \fsum{} is equal to half \feod. \figitem{F} Absolute value of the second-order susceptibility for the higher RAM contrast. \figitem{G} Projected diagonals, calculated as the mean of the anti-diagonals of the matrices in \panel{E--F}. Gray dots: \fbase{}. Gray area: $\fbase{} \pm 5$\,Hz. Dashed lines: Medians of the projected diagonals.
}
\end{figure*}