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references.bib
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@ -6953,3 +6953,42 @@ microelectrode recordings from visual cortex and functional implications.},
volume={86},
pages={2934--2937},
}
@Article{Sharafi2013,
title={Information filtering by synchronous spikes in a neural population.},
author={Nahal Sharafi and Jan Benda and Benjamin Lindner},
year={2013},
journal={J Comp Neurosci},
volume={34},
pages={285--301},
}
@Article{Marinazzo2007,
title={Input-driven oscillations in networks with excitatory and inhibitory neurons with dynamic synapse.},
author={Marinazzo, D. and Kappen, H. J. and Gielen, S. C. A. M.},
year={2007},
journal={Neural Computation},
volume={19},
pages={1739},
}
@Article{Rocha2007,
title={Correlation between neural spike trains increases with firing rate.},
author={de la Rocha, J. and Doiron, B. and Shea-Brown, E. and Josic, K. and Reyes, A.},
year={2007},
journal={Nature},
volume={448},
pages={802},
}
@ARTICLE{Yu1989,
author={Yu, X. and Lewis, E.R.},
journal={IEEE Transactions on Biomedical Engineering},
title={Studies with spike initiators: linearization by noise allows continuous signal modulation in neural networks},
year={1989},
volume={36},
number={1},
pages={36-43},
doi={10.1109/10.16447}}

22
susceptibility1.tex
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@ -519,7 +519,7 @@ Nonlinear processes are key to neuronal information processing. Decision making
At the heart of nonlinear system identification is the Volterra series\cite{Rieke1999}. Second-order kernels have been used to predict firing rate responses of catfish retinal ganglion cells \cite{Marmarelis1972}.
In the frequency domain second-order kernels are known as second-order response functions or susceptibilities. They quantify the amplitude of the response at the sum and difference of two stimulus frequencies. Adding also third-order kernels, spike trains of a spider mechanorecptors have beend predicted from sensory stimuli \cite{French2001}. The nonlinear nature of Y cells in contrast to the more linear responses of X cells in cat retinal ganglion cells has been demonstrated by means of second-order kernels\cite{Victor1977}. Interactions between different frequencies in the response of neurons in visual cortices of cats and monkeys have been studied using bispectra, the crucial constituent of the second-order susceptibliity \cite{Schanze1997}. Locking of chinchilla auditory nerve fibres to pure tone stimuli is captured by second-order kernels\cite{Temchin2005}. In paddlefish ampullary afferents, bursting in response to strong, natural sensory stimuli boost nonlinear responses in the bicoherence, the bispectrum normalized by stimulus and response spectra \cite{Neiman2011}.
Noise linearizes nonlinear systems \notejb{Check references, add Lindner papers}\cite{Longtin1993, Chialvo1997, Roddey2000} and therefore noisy systems can be well described by linear response theory in the limit of vanishing stimulus amplitudes \notejb{what else to cite?} \cite{Doiron2004}. When increasing stimulus amplitude, at first the contribution of the second-order kernel of the Volterra series becomes relevant in addition to the linear one. For these weakly nonlinear responses of leaky-integrate-and fire (LIF) neurons an analytical expression for the second-order susceptibility has been derived \cite{Voronenko2017} in addition to its linear response function\cite{Lindner2001}. In the superthreshold regime, where the LIF generates a baseline firing rate in the absense of an external stimulus, the linear response function has a peak at the baseline firing rate and its harmonics (\subfigrefb{fig:lifresponse}{A}) and the second-order susceptibility shows very distinct ridges of elevated nonlinear responses, exactly where two stimulus frequencies equal or add up to the neuron's baseline firing rate (\subfigrefb{fig:lifresponse}{B}). In experimental data such structures in the second-order susceptibility have not been reported yet.
Noise linearizes nonlinear systems\cite{Yu1989,Chialvo1997} and therefore noisy neural systems can be well described by linear response theory in the limit of small stimulus amplitudes \cite{Roddey2000,Doiron2004,Rocha2007,Sharafi2013, }. When increasing stimulus amplitude, at first the contribution of the second-order kernel of the Volterra series becomes relevant in addition to the linear one. For these weakly nonlinear responses of leaky-integrate-and fire (LIF) neurons an analytical expression for the second-order susceptibility has been derived \cite{Voronenko2017} in addition to its linear response function\cite{Lindner2001}. In the superthreshold regime, where the LIF generates a baseline firing rate in the absense of an external stimulus, the linear response function has a peak at the baseline firing rate and its harmonics (\subfigrefb{fig:lifresponse}{A}) and the second-order susceptibility shows very distinct ridges of elevated nonlinear responses, exactly where two stimulus frequencies equal or add up to the neuron's baseline firing rate (\subfigrefb{fig:lifresponse}{B}). In experimental data such structures in the second-order susceptibility have not been reported yet.
Here we study weakly nonlinear repsonses in two electrosensory systems in the wave-type electric fish \textit{Apteronotus leptorhynchus}. These fish generate a quasi-sinusoidal dipolar electric field (electric organ discharge, EOD). In communication contexts\cite{Walz2014, Henninger2018} the EODs of close-by fish superimpose and lead to amplitude modulations (AMs), called beats (two-fish interaction), and modulations of beats, called envelopes (multiple-fish interaction) \cite{Yu2005,Fotowat2013}. Therefore, stimuli with multiple distinct frequencies are part of the everyday life of wave-type electric fish\cite{Benda2020} and interactions of these frequencies in the electrosensory periphery are to be expected. P-type electroreceptor afferents of the tuberous electrosensory system, the P-units, use nonlinearities to extract and encode these AMs in their time-dependent firing rates \cite{Bastian1981a,Walz2014,Middleton2006,Stamper2012Envelope,Savard2011,Barayeu2023}. P-units are heterogeneous in their baseline firing rates as well as in their intrinsic noise levels, as quantified by the coefficient of variation (CV) of the interspike intervals (ISI) \cite{Grewe2017, Hladnik2023}. Low-CV P-units have a less noisy firing pattern that is closer to pacemaker firing, whereas high-CV P-units show a more irregular firing pattern that is closer to a Poisson process. On the other hand, ampullary cells of the passive electrosensory system are homogeneous in ther response properties and have very low CVs \cite{Grewe2017}.
@ -563,7 +563,7 @@ Weakly nonlinear responses are expected in cells with sufficiently low intrinsic
\begin{figure*}[tp]
\includegraphics[width=\columnwidth]{cells_suscept}
\caption{\label{fig:cells_suscept} Estimation of linear and nonlinear stimulus encoding in a low-CV P-unit (cell identifier ``2010-06-21-ai"). \figitem{A} Interspike interval (ISI) distribution of the cell's baseline activity, i.e. the cell is driven only by the unperturbed own electric field. The low CV of the ISIs indicates quite regular firing. \figitem{B} Power spectral density of the baseline response with peaks at the cell's baseline firing rate \fbase{} and the fish's EOD frequency \feod{}. \figitem{C} Random amplitude modulation stimulus (top, with cutoff frequency of 300\,Hz) and evoked responses (spike raster, bottom) of the same P-unit. The stimulus contrast (right) reflects the strength of the AM. \figitem{D} Gain of the transfer function (first-order susceptibility), \Eqnref{linearencoding_methods}, of the responses to 10\,\% (light purple) and 20\,\% contrast (dark purple) RAM stimulation. \figitem{E, F} Absolute value of the second-order susceptibility, \Eqnref{eq:susceptibility}, for both the low and high stimulus contrast. Pink triangles mark vertical, horizontal, and diagonal lines where \fone, \ftwo{} or \fsum{} are equal to \fbase{}. \figitem{G} Second-order susceptibilities projected onto the diagonal (means of all anti-diagonals of the matrices shown in \panel{E, F}). Dots mark \fbase{}, horizontal dashed lines mark medians of the projected susceptibilities. \notejb{``Calculated based on the first frozen noise repeat.'' Really? Only one trial has been used for computing the susceptibilities?} \noteab{Yes. In the population statistics of the last 20 years I was trying to maximize the number cells and maximize the duration. In the end the susceptibility is calculated based on 10 seconds. The susceptibility has to be calculated with newly initiated noise, with 10 seconds and 0.5 nfft durations this results in 20 trials. For the overview figure the mean of all NLI values for each frozen noise was calculated. It is not possible to take the mean over several frozen noise matrices since the baseline properties sometimes change and this disrupts the shape of the second order susceptibility.}}
\caption{\label{fig:cells_suscept} Estimation of linear and nonlinear stimulus encoding in a low-CV P-unit (cell identifier ``2010-06-21-ai"). \figitem{A} Interspike interval (ISI) distribution of the cell's baseline activity, i.e. the cell is driven only by the unperturbed own electric field. The low CV of the ISIs indicates quite regular firing. \figitem{B} Power spectral density of the baseline response with peaks at the cell's baseline firing rate \fbase{} and the fish's EOD frequency \feod{}. \figitem{C} Random amplitude modulation stimulus (top, with cutoff frequency of 300\,Hz) and evoked responses (spike raster, bottom) of the same P-unit. The stimulus contrast (right) measures the strength of the AM. \figitem{D} Gain of the transfer function (first-order susceptibility), \Eqnref{linearencoding_methods}, computed from the responses to 10\,\% (light purple) and 20\,\% contrast (dark purple) RAM stimulation of 10\,s duration. \figitem{E, F} Absolute value of the second-order susceptibility, \Eqnref{eq:susceptibility}, for both the low and high stimulus contrast. Pink triangles mark vertical, horizontal, and diagonal lines where \fone, \ftwo{} or \fsum{} are equal to \fbase{}. \figitem{G} Second-order susceptibilities projected onto the diagonal (means of all anti-diagonals of the matrices shown in \panel{E, F}). Dots mark \fbase{}, horizontal dashed lines mark medians of the projected susceptibilities.}
\end{figure*}
Noise stimuli, here random amplitude modulations (RAM) of the EOD (\subfigref{fig:cells_suscept}{C}, top trace, red line), are commonly used to characterize stimulus driven responses of sensory neurons by means of transfer functions (first-order susceptibility), spike-triggered averages, or stimulus-response coherences. Here, we additionally estimate the second-order susceptibility to quantify nonlinear encoding. P-unit spikes align more or less clearly to fluctuations in the RAM stimulus. A higher stimulus intensity, here a higher contrast of the RAM relative to the EOD amplitude (see methods), entrains the P-unit response more clearly (light and dark purple for low and high contrast stimuli, \subfigrefb{fig:cells_suscept}{C}). Linear encoding, quantified by the transfer function \Eqnref{linearencoding_methods}, is similar for the two RAM contrasts in this low-CV P-unit (\subfigrefb{fig:cells_suscept}{D}), as expected for a linear system. The first-order susceptibility is low for low frequencies, peaks in the range below 100\,Hz and then falls off again \notejb{Cite Moe paper?}.
@ -577,7 +577,7 @@ In contrast, a high-CV P-unit (CV$_{\text{base}}=0.4$) does not exhibit pronounc
\begin{figure*}[t]
\includegraphics[width=\columnwidth]{ampullary}
\caption{\label{fig:ampullary} Estimation of linear and nonlinear stimulus encoding in an ampullary afferent (cell identifier ``2012-04-26-ae"). \figitem{A} Interspike interval (ISI) distribution of the cell's baseline activity. The very low CV of the ISIs indicates almost perfect periodic spiking. \figitem{B} Power spectral density of baseline activity with peaks at the cell's baseline firing rate and its harmonics. \figitem{C} Bad-limited white noise stimulus (top, with cutoff frequency of 150\,Hz) added to the fish's self-generated electric field and spike raster of the evoked responses (bottom) for two stimulus contrasts as indicated (right). \figitem{D} Gain of the transfer function, \Eqnref{linearencoding_methods}, of the responses to stimulation with 2\,\% (light green) and 20\,\% contrast (dark green). \figitem{E, F} Absolute value of the second-order susceptibility, \Eqnref{eq:susceptibility}, for both stimulus contrasts as indicated. Pink triangles indicate baseline firing rate. \figitem{G} Projections of the second-order susceptibilities in \panel{E, F} onto the diagonal. \notejb{`` Calculated based on the first frozen noise repeat.''}
\caption{\label{fig:ampullary} Estimation of linear and nonlinear stimulus encoding in an ampullary afferent (cell identifier ``2012-04-26-ae"). \figitem{A} Interspike interval (ISI) distribution of the cell's baseline activity. The very low CV of the ISIs indicates almost perfect periodic spiking. \figitem{B} Power spectral density of baseline activity with peaks at the cell's baseline firing rate and its harmonics. \figitem{C} Bad-limited white noise stimulus (top, with cutoff frequency of 150\,Hz) added to the fish's self-generated electric field and spike raster of the evoked responses (bottom) for two stimulus contrasts as indicated (right). \figitem{D} Gain of the transfer function, \Eqnref{linearencoding_methods}, of the responses to stimulation with 2\,\% (light green) and 20\,\% contrast (dark green). \figitem{E, F} Absolute value of the second-order susceptibility, \Eqnref{eq:susceptibility}, for both stimulus contrasts as indicated. Pink triangles indicate baseline firing rate. \figitem{G} Projections of the second-order susceptibilities in \panel{E, F} onto the diagonal. \notejb{`` Wurden die ampullaeren auch auf 10s ausgewertet?.''}
}
\end{figure*}
@ -592,8 +592,7 @@ In the example recordings shown above (\figsrefb{fig:cells_suscept} and \fref{fi
\begin{figure*}[tp]
\includegraphics[width=\columnwidth]{model_and_data}
\caption{\label{model_and_data} Estimating second-order susceptibilities in the limit of weak stimuli. \figitem{A} \suscept{} estimated from $N=11$ trials of an electrophysiological recording of the\notejb{same as in Fig 2?} \noteab{No it is a different cell. In this cell we do not have several contrasts recorded, but a model.} low-CV P-unit (cell 2012-07-03-ak, see table~\ref{modelparams} for model parameters) driven with a weak RAM stimulus. The stimulus strength was with 2.5\,\% contrast in the electrophysiologically recorded P-unit. The contrast for the P-unit model was with 0.009\,\%, thus reproducing the CV of the electrophysiolgoical recorded P-unit during the 2.5\,\% contrast stimulation. Pink edges mark baseline firing rate where enhanced nonlinear responses are expected. \figitem[i]{B} \textit{Standard condition} of model simulations with intrinsic noise (bottom) and a RAM stimulus (top). \figitem[ii]{B} \suscept{} estimated from simulations of the cell's LIF model counterpart (cell 2012-07-03-ak, table~\ref{modelparams}) based on the same number of trials as in the electrophysiological recording. \figitem[iii]{B} Same as \panel[ii]{B} but using $10^6$ 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. Instead, a large part (90\,\%) of the total intrinsic noise is treated as signal and is presented as an equivalent amplitude modulation (\signalnoise, center), while the intrinsic noise is reduced to 10\,\% of its original strength (see methods for details). In addition to one million trials, this reveals the full expected structure of the second-order susceptibility. \notejb{to methods: ``Note that the signal component \signalnoise{} is added as an amplitude modulation and is thus limited with respect to its spectral content by the Nyquist frequency of the carrier, half the EOD frequency. It thus has a reduced high frequency content as compared to the intrinsic noise. Adding this discarded high frequency components to the intrinsic noise does not affect the results here (not shown).''}
}
\caption{\label{model_and_data} Estimating second-order susceptibilities in the limit of weak stimuli. \figitem{A} \suscept{} estimated from $N=11$ trials of an electrophysiological recording of another low-CV P-unit (cell 2012-07-03-ak) driven with a weak RAM stimulus. \notejb{The stimulus strength was with 2.5\,\% contrast in the electrophysiologically recorded P-unit. The contrast for the P-unit model was with 0.009\,\%, thus reproducing the CV of the electrophysiolgoical recorded P-unit during the 2.5\,\% contrast stimulation (see table~\ref{modelparams} for model parameters).} Pink edges mark baseline firing rate where enhanced nonlinear responses are expected. \figitem[i]{B} \textit{Standard condition} of model simulations with intrinsic noise (bottom) and a RAM stimulus (top). \figitem[ii]{B} \suscept{} estimated from simulations of the cell's LIF model counterpart (cell 2012-07-03-ak, table~\ref{modelparams}) based on the same number of trials as in the electrophysiological recording. \figitem[iii]{B} Same as \panel[ii]{B} but using $10^6$ 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. Instead, a large part (90\,\%) of the total intrinsic noise is treated as signal and is presented as an equivalent amplitude modulation (\signalnoise, center), while the intrinsic noise is reduced to 10\,\% of its original strength (see methods for details). In addition to one million trials, this reveals the full expected structure of the second-order susceptibility.}
\end{figure*}
One reason could be simply too little data for a good estimate of the second-order susceptibility. Electrophysiological recordings are limited in time, and therefore responses to only a limited number of trials, i.e. repetitions of the same RAM stimulus, are available. As a consequence, the cross-spectra, \Eqnref{eq:crosshigh}, are insufficiently averaged and the full structure of the second-order susceptibility might be hidden in finite-data noise. This experimental limitation can be overcome by using a computational model for the P-unit, a stochastic leaky integrate-and-fire model with adaptation current and dendritic preprocessing, and parameters fitted to the experimentally recorded P-unit (\figrefb{flowchart}) \cite{Barayeu2023}. The model faithfully reproduces the second-order susceptibility of a low-CV cell estimated from the same low number of trials as in the experiment ($\n{}=11$, compare \panel{A} and \panel[ii]{B} in \figrefb{model_and_data}).
@ -618,11 +617,6 @@ We estimated the second-order susceptibility of P-unit responses using RAM stimu
However, the second-order susceptibility \Eqnref{eq:susceptibility} is a spectral measure that is based on Fourier transforms and thus is also defined for negative stimulus frequencies. The full \susceptf{} matrix is symmetric with respect to the origin. In the upper-right and lower-left quadrants of \susceptf{}, stimulus-evoked responses at \fsum{} are shown, whereas in the lower-right and upper-left quadrants nonlinear responses at the difference \fdiff{} are shown (\figref{fig:model_full}). The vertical and horizontal lines at \foneb{} and \ftwob{} are very pronounced in the upper-right quadrant of \subfigrefb{fig:model_full}{A} for nonlinear responses at \fsum{} and extend into the upper-left quadrant (representing \fdiff) where they fade out towards more negative $f_1$ frequencies. The peak in the response power-spectrum at \fdiff{} evoked by pure sine-wave stimulation (\subfigrefb{fig:motivation}{D}) is predicted by the horizontal line in the upper-left quadrant (\subfigrefb{fig:model_full}{A}, \cite{Schlungbaum2023}).
Is it possible based on the second-order susceptibility estimated by means of RAM stimuli (\subfigrefb{fig:model_full}{A}) to predict nonlinear responses in a three-fish setting? We can test this by stimulating the same model with two beats with weak amplitudes (\subfigrefb{fig:model_full}{C--F}). If we choose a frequency combination where the sum of the two beat frequencies is equal to the model's baseline firing rate \fbase{}, a peak at the sum of the two beat frequencies appears in the power spectrum of the response (\subfigrefb{fig:model_full}{C}), as expected from \suscept. If instead we choose two beat frequencies that differ by \fbase{}, a peak is present at the difference frequency (\subfigrefb{fig:model_full}{D}). If only one beat frequency is equal to \fbase{}, both a peak at the sum and at the difference frequency is present in the P-unit response (\subfigrefb{fig:model_full}{E}). And if none of these conditions are met, neither a peak at the sum nor at the difference of the two beat frequencies appears (\subfigrefb{fig:model_full}{F}).
\notejb{Note in the discussion that \suscept{} predicts stimulus evoked responses whereas in the power spectrum we see peaks even if they are unrelated to the stimulus. Like the baseline firing rate.}
% TO DISCUSSION:
%Even though the second-order susceptibilities here were estimated from data and models with an modulated (EOD) carrier (\figrefb{fig:model_full}) they are in good accordance with the second-order susceptibilities found in LIF models without a carrier\cite{Voronenko2017, Schlungbaum2023}.
\begin{figure*}[tp]
\includegraphics[width=\columnwidth]{data_overview_mod}
\caption{\label{fig:data_overview} Nonlinear responses in P-units and ampullary cells. The second-order susceptibility is condensed into the peakedness of the nonlinearity, \nli{} \Eqnref{eq:nli_equation}, that relates the amplitude of the projected susceptibility at a cell's baseline firing rate to its median (see \subfigrefb{fig:cells_suscept}{G}). Each of the recorded neurons contributes at maximum with two stimulus contrasts. Black squares and circles highlight recordings conducted in a single cell. Squares in \panel{A, C, E} correspond to the cell in \figrefb{fig:cells_suscept} and circles to the cell in \figrefb{fig:cells_suscept_high_CV}. Squares in \panel{B, D, F} correspond to the cell in \figrefb{fig:ampullary}. \figitem{A, B} There is a negative correlation between the CV during baseline and \nli. \figitem{C, D} There is a negative correlation between the CV during stimulation and \nli. \figitem{E, F} \nli{} is plotted against the response modulation, (see methods), 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 221 for P-units and 45 for ampullary cells.
@ -664,6 +658,12 @@ The appearing difference peak is known as the social envelope\cite{Stamper2012En
\notejb{Estimating the infinite Volterra series from limited experimental data is usually limited to the first two or three kernels, that then might not be sufficient for a proper prediction of the neuronal response \cite{French2001}. Making assumptions about the nonlinearities in a system reduces the amount of data needed for parameter estimation. In particular, models combining linear filtering with static nonlinearities\cite{Chichilnisky2001}, have been successful in capturing functionally relevant neuronal computations in visual \cite{Gollisch2009} as well as auditory systems\cite{Clemens2013}. On the other hand, linear methods based on backward models for estimating the stimulus from neuronal responses, have been extensively used to quantify information transmission in neural systems \cite{Theunissen1996,Borst1999,Wessel1996,Chacron?}, because backward models do not need to generate action potentials \cite{Rieke1999}.}
\notejb{Note in the discussion that \suscept{} predicts stimulus evoked responses whereas in the power spectrum we see peaks even if they are unrelated to the stimulus. Like the baseline firing rate.}
% TO DISCUSSION:
%Even though the second-order susceptibilities here were estimated from data and models with an modulated (EOD) carrier (\figrefb{fig:model_full}) they are in good accordance with the second-order susceptibilities found in LIF models without a carrier\cite{Voronenko2017, Schlungbaum2023}.
%\,\panel[iii]{C}
\subsection*{Theory applies to systems with and without carrier}
Theoretical work\cite{Voronenko2017} explained analytically the occurrence of nonlinear products when a LIF model neuron is stimulated with pure sine-waves. To investigate whether the same mechanisms occur in electroreceptor afferents which are driven by AMs of a carrier and not by pure sine-waves, we followed the previous approach and quantified the second-order susceptibility from responses to white-noise stimuli \cite{Voronenko2017, Egerland2020, Neiman2011fish,Nikias1993}. We expected to see elevated second-order susceptibility where either of the foreign signals matches the baseline firing rate ($f_1=f_{base}$ or $f_2=f_{base}$) or when the sum equals the baseline firing rate of the neuron (\fsumb{}) creating a triangular pattern of elevated \suscept{} e.g.\,\subfigref{model_and_data}\,\panel[iii]{C}. Indeed, we find traces of the same nonlinearities in the neuronal responses of p-type electroreceptor afferents. The nonlinear pattern observed in the experimental data, however, matches to the expectations only partially and only in a subset of neurons (\figsref{fig:cells_suscept} and\,\ref{fig:ampullary}). Nevertheless, the theory holds also for systems that are driven by AMs of a carrier and is thus more widely applicable.
@ -957,6 +957,8 @@ In the here used model a small portion of the original noise was assigned to the
\end{center}
\end{table*}% 2013-01-08-aa % 2012-07-03-ak
\notejb{to methods: ``Note that the signal component \signalnoise{} is added as an amplitude modulation and is thus limited with respect to its spectral content by the Nyquist frequency of the carrier, half the EOD frequency. It thus has a reduced high frequency content as compared to the intrinsic noise. Adding this discarded high frequency components to the intrinsic noise does not affect the results here (not shown).''}
% Either type in your references using
% \begin{thebibliography}{}