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@ -315,7 +315,7 @@ Respiration was then switched to normal tank water and the fish was transferred
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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.
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\subsection{Identification of P-units and ampullary cells}
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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.
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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. P-units are clearly distinguised from T-type electrorecptors, that we did not analyze here, by having firing rates much lower than the EOD frequency of the fish (no 1:1 locking to the EOD) \notejb{CITE TUNIT PAPER}. 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.
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\subsection{Electric field recordings}
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For monitoring the EOD without the stimulus, two vertical carbon rods ($11\,\centi\meter$ long, 8\,mm diameter) in a head-tail configuration were placed isopotential to the stimulus. Their signal was differentially amplified with a gain factor between 100 and 500 (depending on the recorded animal) and band-pass filtered (3 to 1500\,Hz pass-band, DPA2-FX; npi electronics, Tamm, Germany). For an estimate of the transdermal potential that drives the electroreceptors, two silver wires spaced by 1\,cm were located next to the left gill of the fish and orthogonal to the fish's longitudinal body axis (amplification 100 to 500 times, band-pass filtered with 3 to 1\,500\,Hz pass-band, DPA2-FX; npi-electronics, Tamm, Germany). This local EOD measurement recorded the combination of the fish's own EOD and the applied stimulus.
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@ -434,12 +434,14 @@ First, the input $y(t)$ is thresholded by setting negative values to zero:
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\label{eq:threshold2}
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\lfloor y(t) \rfloor_0 = \left\{ \begin{array}{rcl} y(t) & ; & y(t) \ge 0 \\ 0 & ; & y(t) < 0 \end{array} \right.
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\end{equation}
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(\subfigrefb{flowchart}{A}). This thresholds models the transfer function of the synapses between the primary receptor cells and the afferent. Together with a low-pass filter
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(\subfigrefb{flowchart}{A}). This threshold models the transfer function of the synapses between the primary receptor cells and the afferent \notejb{CITE A MOSER PAPER} and may also include nonlinear properties introduced by low-threshold Kv1 channels known to be present in the afference \citep{TroySmith2006,Nogueira2013}. Together with a low-pass filter
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\begin{equation}
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\label{eq:dendrite}
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\tau_{d} \frac{d V_{d}}{d t} = -V_{d}+ \lfloor y(t) \rfloor_{0}
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\end{equation}
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the threshold operation is required for extracting the amplitude modulation from the input \citep{Barayeu2024}. The low-pass filter models passive signal conduction in the afferent's dendrite (\subfigrefb{flowchart}{B}) and $\tau_{d}$ is the membrane time constant of the dendrite. Dendritic low-pass filtering was also necessary to reproduce the loose coupling of P-unit spikes to the EOD while maintaining high sensitivity at small amplitude modulations.
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the threshold operation is required for extracting the amplitude modulation from the input \citep{Barayeu2024}. As detailed in the discussion, the details of the threshold nonlinearity \eqnref{eq:threshold2} and the low-pass filter \eqnref{eq:dendrite} do not matter here, since we compute all cross-spectra between the spiking response and the amplitude-modulation $s(t)$ --- and not the full input signal $y(t)$, \eqnref{eq:ram_equation}.
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The low-pass filter models passive signal conduction in the afferent's dendrite (\subfigrefb{flowchart}{B}) and $\tau_{d}$ is the membrane time constant of the dendrite. Dendritic low-pass filtering was also necessary to reproduce the loose coupling of P-unit spikes to the EOD while maintaining high sensitivity at small amplitude modulations.
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The dendritic voltage $V_d(t)$ is then fed into a stochastic leaky integrate-and-fire (LIF) model with adaptation,
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\begin{equation}
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@ -544,7 +546,7 @@ Electric fish possess an additional electrosensory system, the passive or ampull
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\subsection{Model-based estimation of the second-order susceptibility}
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In the example recordings shown above (\figsrefb{fig:punit} and \fref{fig:ampullary}), we only observe nonlinear responses where the sum of the two stimulus frequencies matches the neuron's baseline firing rate, but not where either og the frequencies alone matches the baseline rate. In the following, we investigate this discrepancy to the theoretical expectations \citep{Voronenko2017,Franzen2023}.
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In the example recordings shown above (\figsrefb{fig:punit} and \fref{fig:ampullary}), we only observe nonlinear responses where the sum of the two stimulus frequencies matches the neuron's baseline firing rate, but not where either of the frequencies alone matches the baseline rate. In the following, we investigate this discrepancy to the theoretical expectations \citep{Voronenko2017,Franzen2023}.
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\begin{figure*}[p]
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\includegraphics[width=\columnwidth]{noisesplit}
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@ -577,7 +579,9 @@ The SI($r$) correlates with the CVs of the cell's baseline interspike intervals
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\subsection{Weakly nonlinear interactions vanish for higher stimulus contrasts}
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As pointed out above, the weakly nonlinear regime can only be observed for sufficiently weak stimuli. In the model cells we estimated second-order susceptibilities for RAM stimuli with a contrast of 1, 3, and 10\,\%. The estimates for 1\,\% contrast (\figrefb{fig:modelsusceptcontrasts}\,\panel[ii]{E}) were quite similar to the estimates from the noise-split method, corresponding to a stimulus contrast of 0\,\% ($r=0.97$, $p\ll 0.001$). Thus, RAM stimuli with 1\,\% contrast are sufficiently small to not destroy weakly nonlinear interactions by their linearizing effect. At this low contrast, 51\,\% of the model cells have an SI($r$) value greater than 1.2.
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\notebl{As pointed out above, the weakly nonlinear regime can only be observed for sufficiently weak \notejb{noise} stimuli: strong \notejb{noise} simuli act as a linearizing background noise and thus shift the neural signal transmission into a regime free of nonlinearities. What exactly constitutes a 'strong' stimulus causing an effective increase of the background noise, also depends on all the other cellular properties of the neuron, or on the parameters in our leaky integrate-and-fire neuron. This is what we explore now by varying the contrast of RAM stimuli driving our model cells.}
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In the model cells we estimated second-order susceptibilities for RAM stimuli with a contrast of 1, 3, and 10\,\%. The estimates for 1\,\% contrast (\figrefb{fig:modelsusceptcontrasts}\,\panel[ii]{E}) were quite similar to the estimates from the noise-split method, corresponding to a stimulus contrast of 0\,\% ($r=0.97$, $p\ll 0.001$). Thus, RAM stimuli with 1\,\% contrast are sufficiently small to not destroy weakly nonlinear interactions by their linearizing effect. At this low contrast, 51\,\% of the model cells have an SI($r$) value greater than 1.2.
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At a RAM contrast of 3\,\% the SI($r$) values become smaller (\figrefb{fig:modelsusceptcontrasts}\,\panel[iii]{E}). Only 7 cells (18\,\%) have SI($r$) values exceeding 1.2. Finally, at 10\,\% the SI($r$) values of all cells drop below 1.2, except for three cells (8\,\%, \figrefb{fig:modelsusceptcontrasts}\,\panel[iv]{E}). The cell shown in \subfigrefb{fig:modelsusceptcontrasts}{A} is one of them. At 10\,\% contrast the SI($r$) values are no longer correlated with the ones in the noise-split configuration ($r=0.32$, $p=0.05$). To summarize, the regime of distinct nonlinear interactions at frequencies matching the baseline firing rate extends in this set of P-unit model cells to stimulus contrasts ranging from a few percents to about 10\,\%.
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