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nonlinearbaseline.tex
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\sffamily\bfseries\LARGE Spike generation in electroreceptor afferents
introduces\\ additional spectral response components by\\ weakly
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\section{Abstract}
Spiking thresholds in neurons or rectification at synapses are essential for neuronal computations rendering neuronal processing inherently nonlinear. Nevertheless, linear response theory has been instrumental for understanding, for example, the impact of noise or neuronal synchrony on signal transmission, or the emergence of oscillatory activity, but is valid only at low stimulus amplitudes or large levels of intrinsic noise. At higher signal-to-noise ratios, however, nonlinear response components become relevant. Theoretical results for leaky integrate-and-fire neurons in the weakly nonlinear regime suggest strong responses at the sum of two input frequencies if one of these frequencies or their sum match the neuron's baseline firing rate.
We here analyze nonlinear responses in two types of primary electroreceptor afferents, the P-units of the active and the ampullary cells of the passive electrosensory system of the wave-type electric fish \textit{Apteronotus leptorhynchus}. In our combined experimental and modeling approach we identify these predicted nonlinear responses only in individual low-noise P-units, but in more than half of the ampullary cells. Our results provide experimental evidence for nonlinear responses of spike generators in the weakly nonlinear regime. We conclude that such nonlinear responses occur in any sensory neuron that operates in similar regimes particularly at near-threshold stimulus conditions.
We here analyze nonlinear responses in two types of primary electroreceptor afferents, the P-units of the active and the ampullary cells of the passive electrosensory system of the wave-type electric fish \textit{Apteronotus leptorhynchus} of either sex. In our combined experimental and modeling approach we identify these predicted nonlinear responses only in individual low-noise P-units, but in more than half of the ampullary cells. Our results provide experimental evidence for nonlinear responses of spike generators in the weakly nonlinear regime. We conclude that such nonlinear responses occur in any sensory neuron that operates in similar regimes particularly at near-threshold stimulus conditions.
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\section{Significance statement}
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\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 172 P-units and 30 ampullary electroreceptor afferents recorded in 80 weakly electric fish of both sexes of the species \textit{Apteronotus leptorhynchus}. Fish were obtained from a commercial supplier for tropical fish (Aquarium Glaser GmbH, Rodgau, Germany) and 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.
Within this project, we re-evaluated datasets that were recorded between 2010 and 2023 at the \secret{Ludwig Maximilian University (LMU) M\"unchen} and the \secret{Eberhard-Karls University T\"ubingen}. All experimental protocols complied with national and \secret{European} law and were approved by the respective Ethics Committees of the \secret{Ludwig-Maximilians Universität München} (permit no. \secret{55.2-1-54-2531-135-09}) and the \secret{Eberhard-Karls Unversität Tübingen} (permit no. \secret{ZP 1/13} and \secret{ZP 1/16}).
The final sample consisted of 172 P-units and 30 ampullary electroreceptor afferents recorded in 80 weakly electric fish of both sexes of the species \textit{Apteronotus leptorhynchus}. Fish were obtained from a commercial supplier for tropical fish (\secret{Aquarium Glaser GmbH, Rodgau, Germany}) and 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 of 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{Electrophysiological recordings}
For the recordings fish were positioned centrally in the experimental tank, with the major parts of their body submerged into the water. Those body parts that were above the water surface were covered with paper tissue to avoid drying of the skin. Local analgesia was refreshed in intervals of two hours by cutaneous application of Lidocaine (2\,\%; bela-pharm, Vechta, Germany) around the surgical wounds. Electrodes (borosilicate; 1.5\,mm outer diameter; GB150F-8P; Science Products, Hofheim, Germany) were pulled to a resistance of 50--100\,\mega\ohm{} (model P-97; Sutter Instrument, Novato, CA) and filled with 1\,M KCl solution. Electrodes were fixed in a microdrive (Luigs-Neumann, Ratingen, Germany) and lowered into the nerve. Recordings of electroreceptor afferents were amplified and lowpass filtered at 10\,kHz (SEC-05, npi-electronics, Tamm, Germany, operated in bridge mode). All signals, neuronal recordings, recorded EOD, and the generated stimulus, were digitized with sampling rates of 20 or 40\,kHz (PCI-6229, National Instruments, Austin, TX). RELACS (\url{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.
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\,\%; \secret{bela-pharm, Vechta, Germany}) around the surgical wounds. Electrodes (borosilicate; 1.5\,mm outer diameter; GB150F-8P; \secret{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 (\secret{Luigs-Neumann, Ratingen, Germany}) and lowered into the nerve. Recordings of electroreceptor afferents were amplified and lowpass filtered at 10\,kHz (\secret{SEC-05, npi-electronics, Tamm, Germany}, operated in bridge mode). All signals, neuronal recordings, recorded EOD, and the generated stimulus, were digitized with sampling rates of 20 or 40\,kHz (PCI-6229, National Instruments, Austin, TX). RELACS (\url{https://github.com/relacs/relacs}) running on a Linux computer was used for online spike and EOD detection, stimulus generation, and calibration. Recorded data was then stored on the hard drive for offline analysis.
\subsection{Identification of P-units and ampullary cells}
Recordings were classified as P-units if baseline action potentials phase locked to the EOD with vectors strengths between 0.7 and 0.95, a baseline firing rate larger than 30\,Hz, a serial correlation of subsequent interspike intervals below zero, a coefficient of variation of baseline interspike intervals below 1.5 and during stimulation below 2. P-units are clearly distinguished from T-type electroreceptors, 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). 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.
\subsection{Electric field recordings}
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.
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; \secret{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; \secret{npi-electronics, Tamm, Germany}). This local EOD measurement recorded the combination of the fish's own EOD and the applied stimulus.
\subsection{Stimulation}\label{rammethods}
Electric stimuli were attenuated (ATN-01M, npi-electronics, Tamm, Germany), isolated from ground (ISO-02V, npi-electronics, Tamm, Germany) and delivered via two horizontal carbon rods (30 cm length, 8 mm diameter) located $15\,\centi\meter$ parallel to each side of the fish. The fish were stimulated with band-limited Gaussian white noise stimuli, i.e. signals with equal power at all frequencies up to a cut-off frequency and with a stationary Gaussian probability density. For the ampullary cells we chose a cut-off frequency of 150\,Hz, whereas for the P-units we used either 300 or 400\,Hz. The stimuli were generated by drawing normally distributed real and imaginary numbers for all frequencies up to the desired cut-off frequency in the Fourier domain and then applying an inverse Fourier transform \citep{Billah1990,Skorjanc2023}. The stimulus intensity is given as a contrast, i.e. the standard deviation of the resulting amplitude modulation relative to the fish's EOD amplitude. The contrast varied between 1 and 20\,\% (median 5\,\%) for P-units and 2.5 and 20\,\% (median 5\,\%) for ampullary cells. Only recordings with noise stimuli with a duration of at least 2\,s (maximum of 50\,s, median 10\,s) and enough repetitions to results in at least 100 FFT segments (see below, P-units: 100--1520, median 313, ampullary cells: 105 -- 3648, median 722) were included into the analysis. When ampullary cells were recorded, the noise stimuli $s(t)$ were directly applied as the stimulus and thus were simply added to the fish's own EOD: $s(t) + EOD(t)$. To create random amplitude modulations (RAM) for P-unit recordings, the noise stimulus was first multiplied with the EOD of the fish (MXS-01M; npi electronics) and then played back through the stimulation electrodes: $EOD(t) + s(t)EOD(t) = (1 + s(t))EOD(t)$.
Electric stimuli were attenuated (ATN-01M, \secret{npi-electronics, Tamm, Germany}), isolated from ground (ISO-02V, \secret{npi-electronics, Tamm, Germany}) and delivered via two horizontal carbon rods (30 cm length, 8 mm diameter) located $15\,\centi\meter$ parallel to each side of the fish. The fish were stimulated with band-limited Gaussian white noise stimuli, i.e. signals with equal power at all frequencies up to a cut-off frequency and with a stationary Gaussian probability density. For the ampullary cells we chose a cut-off frequency of 150\,Hz, whereas for the P-units we used either 300 or 400\,Hz. The stimuli were generated by drawing normally distributed real and imaginary numbers for all frequencies up to the desired cut-off frequency in the Fourier domain and then applying an inverse Fourier transform \citep{Billah1990,Skorjanc2023}. The stimulus intensity is given as a contrast, i.e. the standard deviation of the resulting amplitude modulation relative to the fish's EOD amplitude. The contrast varied between 1 and 20\,\% (median 5\,\%) for P-units and 2.5 and 20\,\% (median 5\,\%) for ampullary cells. Only recordings with noise stimuli with a duration of at least 2\,s (maximum of 50\,s, median 10\,s) and enough repetitions to results in at least 100 FFT segments (see below, P-units: 100--1520, median 313, ampullary cells: 105 -- 3648, median 722) were included into the analysis. When ampullary cells were recorded, the noise stimuli $s(t)$ were directly applied as the stimulus and thus were simply added to the fish's own EOD: $s(t) + EOD(t)$. To create random amplitude modulations (RAM) for P-unit recordings, the noise stimulus was first multiplied with the EOD of the fish (MXS-01M; npi electronics) and then played back through the stimulation electrodes: $EOD(t) + s(t)EOD(t) = (1 + s(t))EOD(t)$.
\subsection{Data analysis} Data analysis was done in Python 3 using the packages matplotlib \citep{Hunter2007}, numpy \citep{Walt2011}, scipy \citep{scipy2020}, nixio \citep{Stoewer2014}, and thunderlab (\url{https://github.com/bendalab/thunderlab}).
\subsection{Data analysis} Data analysis was done in Python 3 using the packages matplotlib \citep{Hunter2007}, numpy \citep{Walt2011}, scipy \citep{scipy2020}, nixio \citep{Stoewer2014}, and thunderlab (\secret{\url{https://github.com/bendalab/thunderlab}}).
\paragraph{Code accessibility}
The P-unit model parameters and spectral analysis algorithms are available at \url{https://github.com/bendalab/punitmodel/tree/v1}.
The P-unit model parameters and spectral analysis algorithms are available at \secret{\url{https://github.com/bendalab/punitmodel/tree/v1}}.
\paragraph{Baseline analysis}\label{baselinemethods}
The baseline firing rate $r$ was calculated as the number of spikes divided by the duration of the baseline recording (median 32\,s). The coefficient of variation (CV) of the interspike intervals (ISI) is their standard deviation relative to their mean: $\rm{CV}_{\rm base} = \sqrt{\langle (ISI- \langle ISI \rangle) ^2 \rangle} / \langle ISI \rangle$. If the baseline was recorded several times in a recording, the measures from the longest recording were taken.
@@ -456,7 +470,7 @@ Whenever the membrane voltage $V_m(t)$ crosses the spiking threshold $\theta=1$,
The P-unit models were integrated by the Euler forward method with a time-step of $\Delta t = 0.05$\,ms. For each trial of a simulation, $V_{m}$ was drawn from a uniform distribution between 0 and 1 and the initial value of $A$ was jittered by adding a random number drawn from a normal distribution with standard deviation of 2\,\% of its initial value. Then the first 500\,ms of any simulation were discarded to remove remaining transients.
The eight free parameters of the P-unit model $\beta$, $\tau_m$, $\mu$, $D$, $\tau_A$, $\Delta_A$, $\tau_d$, and $t_{ref}$, were fitted to both the baseline activity (baseline firing rate, CV of ISIs, serial correlation of ISIs at lag one, and vector strength of spike coupling to EOD) and the responses to step increases and decreases in EOD amplitude (onset and steady-state responses, effective adaptation time constant, \citealp{Benda2005}) of recorded P-units. Model parameters of all 39 cells are summarized in file \texttt{models.csv} of our \texttt{punitmodel} repository at \url{https://github.com/bendalab/punitmodel/tree/v1}.
The eight free parameters of the P-unit model $\beta$, $\tau_m$, $\mu$, $D$, $\tau_A$, $\Delta_A$, $\tau_d$, and $t_{ref}$, were fitted to both the baseline activity (baseline firing rate, CV of ISIs, serial correlation of ISIs at lag one, and vector strength of spike coupling to EOD) and the responses to step increases and decreases in EOD amplitude (onset and steady-state responses, effective adaptation time constant, \citealp{Benda2005}) of recorded P-units. Model parameters of all 39 cells are summarized in file \texttt{models.csv} of our \texttt{punitmodel} repository at \secret{\url{https://github.com/bendalab/punitmodel/tree/v1}}.
\subsection{Noise split}