From 68099e1b7e47231f9d5f77bb6aaa51363edb21b2 Mon Sep 17 00:00:00 2001 From: Jan Benda Date: Tue, 1 Jul 2025 18:34:17 +0200 Subject: [PATCH] added Aertsen papers --- nonlinearbaseline.tex | 12 +++++------- references.bib | 30 +++++++++++++++++++++++++++++- 2 files changed, 34 insertions(+), 8 deletions(-) diff --git a/nonlinearbaseline.tex b/nonlinearbaseline.tex index 767ad9e..81efb63 100644 --- a/nonlinearbaseline.tex +++ b/nonlinearbaseline.tex @@ -244,6 +244,7 @@ %\paragraph{Author Contributions:} All authors designed the study and discussed the results. AB performed the data analyses and modeling, AB and JG drafted the paper, all authors discussed and revised the manuscript. \begin{keywords} +\item Volterra series \item second-order susceptibility \item electric fish \item nonlinear coding @@ -261,9 +262,6 @@ We here analyze nonlinear responses in two types of primary electroreceptor affe %\section{Author summary} %Weakly electric fish use their self-generated electric field to detect a wide range of behaviorally relevant stimuli. Intriguingly, they show detection performances of stimuli that are (i) extremely weak and (ii) occur in the background of strong foreground signals, reminiscent of what is often described as the cocktail party problem. Such performances are achieved by boosting the signal detection through nonlinear mechanisms. We here analyze nonlinear encoding in two different populations of primary electrosensory afferents of the weakly electric fish. We derive the rules under which nonlinear effects can be observed in both electrosensory subsystems. In a combined experimental and modeling approach we generalize the approach of nonlinear susceptibility to systems that respond to amplitude modulations of a carrier signal. -\note{Ad Aertsen paper suggestions:} -\url{https://brainworks.biologie.uni-freiburg.de/1981/journal%20papers/Aertsen-BiolCyb-1981b.pdf} \url{https://brainworks.biologie.uni-freiburg.de/1983/journal%20papers/eggermont-quartrevbiophys-1983.pdf} \url{https://brainworks.biologie.uni-freiburg.de/1981/journal%20papers/The%20Phonocrome%20(1981).pdf} - \section{Introduction} @@ -271,9 +269,9 @@ We here analyze nonlinear responses in two types of primary electroreceptor affe \includegraphics[width=\columnwidth]{lifsuscept} \caption{\label{fig:lifresponse} First- and second-order response functions of the leaky integrate-and-fire model. \figitem{A} Magnitude of the first-order (linear) response function $|\chi_1(f)|$, also known as the ``gain'' function, quantifies the response amplitude relative to the stimulus amplitude, both measured at the same stimulus frequency. \figitem{B} Magnitude of the second-order (non-linear) response function $|\chi_2(f_1, f_2)|$ quantifies the response at the sum of two stimulus frequencies. For linear systems, the second-order response function is zero, because linear systems do not create new frequencies and thus there is no response at the sum of the two frequencies. The plots show the analytical solutions from \citet{Lindner2001} and \citet{Voronenko2017} with $\mu = 1.1$ and $D = 0.001$. Note that the leaky integrate-and-fire model is formulated without dimensions, frequencies are given in multiples of the inverse membrane time constant.} \end{figure*} -We like to think about signal encoding in terms of linear relations with unique mapping of a given input value to a certain output of the system under consideration. Indeed, such linear methods, for example the transfer function or first-order susceptibility shown in figure~\ref{fig:lifresponse}, have been widely and successfully applied to describe and predict neuronal responses and are an invaluable tool to characterize neural systems \citep{Borst1999}. Nonlinear mechanisms, on the other hand, are key on different levels of neural processing. Deciding for one action over another is a nonlinear process on the systemic level. On the cellular level, spiking neurons are inherently nonlinear. Whether an action potential is elicited depends on the membrane potential to exceed a threshold \citep{Hodgkin1952, Koch1995}. Because of such nonlinearities, understanding and predicting neuronal responses to sensory stimuli is in general a difficult task. +We like to think about signal encoding in terms of linear relations with unique mapping of a given input value to a certain output of the system under consideration. Indeed, such linear methods, for example the transfer function or first-order susceptibility shown in figure~\ref{fig:lifresponse}, have been widely and successfully applied to describe and predict neuronal responses and are an invaluable tool to characterize neural systems \citep{Eggermont1983,Borst1999}. Nonlinear mechanisms, on the other hand, are key on different levels of neural processing. Deciding for one action over another is a nonlinear process on the systemic level. On the cellular level, spiking neurons are inherently nonlinear. Whether an action potential is elicited depends on the membrane potential to exceed a threshold \citep{Hodgkin1952, Koch1995}. Because of such nonlinearities, understanding and predicting neuronal responses to sensory stimuli is in general a difficult task. -The transfer function that describes the linear properties of a system, is the first-order term of a Volterra series. Higher-order terms successively approximate nonlinear features of a system \citep{Rieke1999}. Second-order kernels have been used in the time domain to predict visual responses in catfish \citep{Marmarelis1972}. In the frequency domain, second-order kernels are known as second-order response functions or susceptibilities. Nonlinear interactions of two stimulus frequencies generate peaks in the response spectrum at the sum and the difference of the two. Including higher-order terms of the Volterra series, the nonlinear nature of spider mechanoreceptors \citep{French2001}, mammalian visual systems \citep{Victor1977, Schanze1997}, locking in chinchilla auditory nerve fibers \citep{Temchin2005}, and bursting responses in paddlefish \citep{Neiman2011} have been demonstrated. +The transfer function that describes the linear properties of a system, is the first-order term of a Volterra series. Higher-order terms successively approximate nonlinear features of a system \citep{Rieke1999}. Second-order kernels have been used in the time domain to predict visual responses in catfish \citep{Marmarelis1972}. In the frequency domain, second-order kernels are known as second-order response functions or susceptibilities. Nonlinear interactions of two stimulus frequencies generate peaks in the response spectrum at the sum and the difference of the two. Including higher-order terms of the Volterra series, the nonlinear nature of mammalian visual systems \citep{Victor1977, Schanze1997}, auditory responses in the Torus semicircularis of frogs \citep{Aertsen1981}, locking in chinchilla auditory nerve fibers \citep{Temchin2005}, spider mechanoreceptors \citep{French2001}, and bursting responses in paddlefish \citep{Neiman2011} have been demonstrated. Noise in nonlinear systems, however, linearizes the system's response properties \citep{Yu1989, Chialvo1997}. Also, in the limit to small stimuli, nonlinear systems can be well described by linear response theory \citep{Roddey2000, Doiron2004, Rocha2007, Sharafi2013}. With increasing stimulus amplitude, the contribution of the second-order kernel of the Volterra series becomes more relevant. For these weakly nonlinear responses analytical expressions for the second-order susceptibility have been derived for leaky-integrate-and-fire (LIF) \citep{Voronenko2017} and theta model neurons \citep{Franzen2023}. In the suprathreshold regime, in which the LIF generates a baseline firing rate in the absence 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 one of two stimulus frequencies equals or both frequencies 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. @@ -478,7 +476,7 @@ Not only the intrinsic noise but also the stimulation with external white-noise In order to characterize weakly nonlinear responses of the cells in the limit to vanishing stimulus amplitudes we utilized the Furutsu-Novikov theorem \citep{Novikov1965, Furutsu1963}. Following \citet{Lindner2022}, a substantial part of the intrinsic noise of a P-unit model \citep{Barayeu2023} is treated as signal. Performing this noise-split trick we can estimate the weakly nonlinear response without the linearizing effect of an additional external white noise stimulus. 41\,\% of the model cells then show the full nonlinear structure (\figref{fig:modelsusceptcontrasts}\,\panel[i]{E}) known from analytical derivations and simulations of basic LIF and theta models driven with pairs of sine-wave stimuli \citep{Voronenko2017,Franzen2023}. Previous studies on second-order nonlinearities have not observed the weakly nonlinear regime, probably because of the linearizing effects of strong noise stimuli \citep{Victor1977, Schanze1997, Neiman2011}. \subsection{Characterizing nonlinear coding from limited experimental data} -Estimating the Volterra series from limited experimental data is usually restricted to the first two or three orders, which might not be sufficient for a proper prediction of the neuronal response \citep{French2001}. As we have demonstrated, a proper estimation of just the second-order susceptibility in the weakly nonlinear regime is challenging in electrophysiological experiments (\figrefb{fig:modelsusceptlown}). Our minimum of 100 FFT segments corresponds to just 26\,s of stimulation and thus is not a challenge. However, the estimates of the second-order susceptibilities start to converge only beyond 10\,000 FFT segments, corresponding to 43\,min of stimulation. Often, however, more than one million segments were needed which requires 71 hours of recording, which is clearly out of reach. We have demonstrated that even in non-converged estimates, the presence of a anti-diagonal ridge is sufficient to predict a triangular pattern in a converged estimate. +Estimating the Volterra series from limited experimental data is usually restricted to the first two or three orders, which might not be sufficient for a proper prediction of the neuronal response \citep{French2001}. As we have demonstrated, a proper estimation of just the second-order susceptibility in the weakly nonlinear regime is challenging in electrophysiological experiments (\figrefb{fig:modelsusceptlown}). Our minimum of 100 FFT segments corresponds to just 26\,s of stimulation and thus is not a challenge. However, the estimates of the second-order susceptibilities start to converge only beyond 10\,000 FFT segments, corresponding to 43\,min of stimulation. Often, however, more than one million segments (corresponding to 71 hours of recording), were needed, which is clearly out of reach. We have demonstrated that even in non-converged estimates based on short recordings, the presence of a anti-diagonal ridge is sufficient to predict a triangular pattern in a converged estimate. Making assumptions about the nonlinearities in a system also reduces the amount of data needed for parameter estimation. In particular, models combining linear filtering with static nonlinearities \citep{Chichilnisky2001} have been successful in capturing functionally relevant neuronal computations in visual \citep{Gollisch2009} as well as auditory systems \citep{Clemens2013}. Linear methods based on backward models for estimating the stimulus from neuronal responses, have been extensively used to quantify information transmission in neural systems \citep{Theunissen1996, Borst1999, Wessel1996, Machens2001}, because backward models do not need to generate action potentials that involve strong nonlinearities \citep{Rieke1999}. @@ -617,7 +615,7 @@ Values larger than one indicate a sharp ridge in the susceptibility matrix close \subsection{Leaky integrate-and-fire models for P-units}\label{lifmethods} -Modified leaky integrate-and-fire (LIF) models were constructed to reproduce the specific firing properties of P-units \citep{Chacron2001, Sinz2020, Barayeu2023}. 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 +Modified leaky integrate-and-fire (LIF) models were constructed to reproduce the specific firing properties of P-units \citep{Chacron2001, Sinz2020, Barayeu2023}. Its basic components (static non-linearity, low-pass filtering and spike generation) are equivalent to models of hair cells in auditory systems \citep{Eggermont1983}. 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{eq:eod} y(t) = y_{EOD}(t) = \cos(2\pi f_{EOD} t) diff --git a/references.bib b/references.bib index f52f017..7ced69b 100644 --- a/references.bib +++ b/references.bib @@ -22,7 +22,7 @@ publisher={Optical Society of America} } -@article{Aertsen1981, +@article{Aertsen1981b, title={{The spectro-temporal receptive field: a functional characteristic of auditory neurons}}, author={Aertsen, Ad M H J and Johannesma, P I M}, journal={Biological Cybernetics}, @@ -7085,3 +7085,31 @@ microelectrode recordings from visual cortex and functional implications.}, volume={392}, pages={321–-424}, } + +@article{Johannesma1981, + title={The phonochrome: a coherent spectro-temporal representation of sound.}, + author={Peter Johannesma and Ad Aertsen and Bert Cranen and Leon van Erning}, + journal={Hearing Research}, + volume={5}, + pages={123--145}, + year={1981}, +} + +@article{Eggermont1983, + title={Reverse-correlation methods in auditory research.}, + author={J. J. Eggermont and P. I. M. Johannesma and A. M. H. J. Aertsen}, + journal={Quarterly Reviews of Biophysics}, + volume={16}, + number={3}, + pages={341--414}, + year={1983}, +} + +@article{Aertsen1981, + title={A comparison of the spectro-temporal sensitivity of neurons to tonal and natural stimuli.}, + author={A. M. H. J. Aertsen and P. I. M. Johannesma}, + journal={Biol. Cybern.}, + volume={42}, + pages={145--156}, + year={1981}, +}