diff --git a/susceptibility1.tex b/susceptibility1.tex index c85b055..7693ae3 100644 --- a/susceptibility1.tex +++ b/susceptibility1.tex @@ -661,7 +661,8 @@ The neurons were classified into cell types during the recording by the experime The electric field of the fish was recorded in two ways: 1. we measured the so-called \textit{global EOD} with two vertical carbon rods ($11\,\centi\meter$ long, 8\,mm diameter) in a head-tail configuration (\figrefb{Setup}, green bars). The electrodes were placed isopotential to the stimulus. This signal was differentially amplified with a 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). 2. The so-called \textit{local EOD} was measured with 1\,cm-spaced silver wires 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, \figrefb{Setup}, red markers). This local measurement recorded the combination of the fish's own field and the applied stimulus and thus serves as a proxy of the transdermal potential that drives the electroreceptors. \subsection{Stimulation} -The stimulus was isolated from the ground (ISO-02V, npi-electronics, Tamm, Germany) and delivered via two horizontal carbon rods (30 cm length, 8 mm diameter) located $15\,\centi\meter$ laterally to the fish (\figrefb{Setup}, gray bars). The stimulus was calibrated with respect to the local EOD. +The stimulus was isolated from the ground (ISO-02V, npi-electronics, Tamm, Germany) and delivered via two horizontal carbon rods (30 cm length, 8 mm diameter) located $15\,\centi\meter$ laterally to the fish (\figrefb{Setup}, gray bars). The stimulus was calibrated with respect t>>>>>>> Stashed changes +o the local EOD. \begin{figure*}[h!]%(\subfigrefb{beat_amplitudes}{B}). \includegraphics[width=\columnwidth]{Settup} @@ -727,7 +728,8 @@ The second-order cross-spectrum that depends on the two frequencies $\omega_1$ a \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{suscept>>>>>>> Stashed changes + ibility0} %\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} @@ -809,44 +811,44 @@ 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. Model cell identifier 2012-07-03-ak (see table~\ref{modelparams} for model parameters). \figitem[i]{A}--\,\panel[i]{\textbf{D}} Rectification of the input. Positive values are maintained and negative discarded (see box on the left). \figitem[ii]{A}--\,\panel[ii]{\textbf{D}} Dendritic low-pass filtering. \figitem[iii]{A}--\,\panel[iii]{\textbf{D}} The noise component in $\sqrt{2D}\,\xi(t)$ \eqnref{LIF} or $\sqrt{2D \, c_{noise}}\,\xi(t)$ in \eqnref{Noise_split_intrinsic}. \figitem[iv]{A}--\,\panel[iv]{\textbf{D}} Spikes generation in the LIF model. Spikes are generated when the voltage of 1 is crossed (markers). Then the voltage is again reset to 0. \figitem[v]{A}--\,\panel[v]{\textbf{D}} Power spectrum of the spikes above. The first peak in panel \panel[v]{A} is the \fbase{} peak. The peak at 1 is the \feod{} peak. The other two peaks are at $\feod{} \pm \fbase{}$. \figitem{A} Baseline condition: The input to the model is a sinus with frequency \feod{}. \figitem{B} The EOD carrier is multiplied with a band-pass limited random amplitude modulation (RAM) with a contrast of 2\,$\%$, as in \eqnref{ram_equation}. \figitem{C} The EOD carrier is multiplied with a band-pass limited RAM signal with a contrast of 20\,$\%$. \figitem{D} The total noise of the model is split into a signal component regulated by $c_{signal}$ in \eqnref{ram_split}, and a noise competent regulated by $c_{noise}$ in \eqnref{Noise_split_intrinsic}. The intrinsic noise in \panel[iii]{D} is reduced compared to \panel[iii]{A}--\panel[iii]{C}. To maintain the CV during the noise split in \panel{D} comparable to the CV during the baseline in \panel{A} the RAM contrast is increased in \panel[i]{D}.} + \caption{\label{flowchart} \notejg{Change numbering of panels?, Remove column C?} + 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.} \end{figure*} -%\figitem[i]{C}$RAM(t)$ \subsection{Numerical implementation} -The model's ODEs were integrated by the Euler forward method with a time-step of $\Delta t = 0.00005$\,s. 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{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} 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} \subsection{Model parameters}\label{paramtext} -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 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 responses to step increases and decreases in EOD amplitude (onset-state and steady-state responses, -effective adaptation time constant) of 42 specific P-units for a fixed power of $p=1$ (table~\ref{modelparams}, \citealp{Ott2020}). When modifying the model (e.g. varying the threshold nonlinearity or the power $p$ in \eqnref{dendrite}) the bias current $\mu$ was adapted to restore the original mean baseline firing rate. For each stimulus repetition the start parameters $A$, $V_{d}$ and $V_{m}$ were drawn randomly from a starting value distribution, retrieved from a 100\,s baseline simulation after an initial 100\,s transient that was dicarded. +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). This resulted in a set of 42 specific P-units (table~\ref{modelparams}, \citealp{Ott2020}). 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 random amplitude modulation (RAM) input to the model created using the same random sequences as were used in the electrophysiological experiments. The input to the model was then +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. A 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} x(t) = (1+ RAM(t)) \cdot \cos(2\pi f_{EOD} t) \end{equation} +From each simulation run, the first second was discarded and the analysis was based on the last second of the data. The resulting spectra thus have a spectral resolution of 1\,Hz. % \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. \subsection{Model noise split into a noise and a stimulus component}\label{intrinsicsplit_methods} -\notejg{This has large overlaps with the results text... where to keep it?} -Based on the Novikov-Furutsu Theorem \citealp{Novikov1965, Furutsu1963} the total noise of a LIF model can be split up into several independent noise processes with the same correlation function. Based on this theorem the internal noise can be split into two parts. One part can then be treated as (additional) input signal and used to calculate the cross-spectra \eqnref{susceptibility}. While maintaining the variability in the system, the effective signal-to-noise ratio can thus be increased. The relation of the total noise treated as signal and the noise is regulated by $c_{signal}$ in \eqnref{ram_split} and the noise component is regulated by $c_{noise}$ in \eqnref{Noise_split_intrinsic}. + +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. \begin{equation} \label{ram_split} - x(t) = (1+ \sqrt{\rho \, 2D \,c_{signal}} \cdot RAM(t)) \cdot \cos(2\pi f_{EOD} t) + 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} \tau_{d} \frac{d V_{d}}{d t} = -V_{d}+ \lfloor x(t) \rfloor_{0} @@ -864,19 +866,9 @@ Based on the Novikov-Furutsu Theorem \citealp{Novikov1965, Furutsu1963} the tota \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}). Both components have to add up to the initial 100\,$\%$ of the total noise, otherwise the Novikov-Furutsu Theorem \citealp{Novikov1965, Furutsu1963} would not be applicable. This is easily achieved in a model without a carrier if the condition $c_{signal}+c_{noise}=1$ is satisfied \citealp{Egerland2020}. In the here used LIF model with EOD carrier, this is more complicated since the noise stimulus $RAM(t)$ is first multiplied with the carrier (\eqnref{ram_split}), the signal is then subjected to rectification and subsequent dendritic low-pass filtering and becomes colored (\eqnref{Noise_split_intrinsic_dendrite}). This is the component that is added to the noise component in \eqnref{Noise_split_intrinsic} and should in sum lead to a total noise of 100\,\%. - - To compensate for these transformations, the generated noise $RAM(t)$ was scaled up by a factor $\rho$ (\eqnref{ram_split}, red in \subfigrefb{flowchart}\,\panel[i]{D}). The $\rho$ scaling factor was found by bisecting the space of possible $\rho$ scaling factors by minimizing the error between the baseline CV (only carrier present) and the CV during stimulation (total noise split with $c_{signal}$ and $c_{noise}$). The assumption behind this approach was that as long the CV stays the same between baseline and stimulation both components have added up to 100\,$\%$ of the total noise and the noise split is valid. - - - - - - - - - +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 \citep{Egerland2020}. 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. +\notejg{Etwas kurz, die Tabelle. Oder benutzen wir hier tatsaechlich nur die zwei Modelle?} \begin{table*}[hp!] \caption{\label{modelparams} Model parameters of LIF models, fitted to 2 electrophysiologically recorded P-units (\citealp{Ott2020}). See section \ref{lifmethods} for model and parameter description.} \begin{center}