Weakly nonlinear response of noisy neurons

Voronenko, Sergej O.; Lindner, Benjamin

doi:10.1088/1367-2630/aa5b81

Cited by 17 publications

(40 citation statements)

References 49 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Under the crucial assumption that neurons are mainly subject to temporally uncorrelated noise, in particular, the asynchronous state with its weak cross correlations among neurons can be well described by linear response theory [20][21][22][23] (for a related problem in an Ising-type spin system, see [24]). For white-noise-driven neurons, there exist also a number of exact results for the firing rate [5], the power spectrum [25], and the linear [12,[26][27][28] and nonlinear [15,29] response functions to periodic stimulation. Moreover, an efficient numerical scheme, the threshold-integration method [30,31], has been developed for the swift computation of these statistics for IF models with arbitrary voltage dependence.…”

Section: Introductionmentioning

confidence: 99%

Theory of spike-train power spectra for multidimensional integrate-and-fire neurons

Vellmer

Lindner

2019

Phys. Rev. Research

Self Cite

View full text Add to dashboard Cite

Multidimensional stochastic integrate-and-fire (IF) models are a standard spike-generator model in studies of firing variability, neural information transmission, and neural network dynamics. Most popular is a version with Gaussian noise and adaptation currents that can be described via Markovian embedding by a set of d + 1 stochastic differential equations corresponding to a Fokker-Planck equation (FPE) for one voltage and d auxiliary variables. For the specific case d = 1, we find a set of partial differential equations that govern the stationary probability density, the stationary firing rate, and, central to our study, the spike-train power spectrum. We numerically solve the corresponding equations for various examples by a finite-difference method and compare the resulting spike-train power spectra to those obtained by numerical simulations of the IF models. Examples include leaky IF models driven by either high-pass-filtered (green) or low-pass-filtered (red) noise (surprisingly, already in this case, the Markovian embedding is not unique), white-noise-driven IF models with spike-frequency adaptation (deterministic or stochastic) and models with a bursting mechanism. We extend the framework to general d and study as an example an IF neuron driven by a narrow-band noise (leading to a three-dimensional FPE). The many examples illustrate the validity of our theory but also clearly demonstrate that different forms of colored noise or adaptation entail a rich repertoire of spectral shapes. The framework developed so far provides the theory of the spike statistics of neurons with known sources of noise and adaptation. In the final part, we use our results to develop a theory of spike-train correlations when noise sources are not known but emerge from the nonlinear interactions among neurons in sparse recurrent networks such as found in cortex. In this theory, network input to a single cell is described by a multidimensional Ornstein-Uhlenbeck process with coefficients that are related to the output spike-train power spectrum. This leads to a system of equations which determine the self-consistent spike-train and noise statistics. For a simple example, we find a low-dimensional numerical solution of these equations and verify our framework by simulation results of a large sparse recurrent network of integrate-and-fire neurons.

show abstract

Section: Introductionmentioning

confidence: 99%

Theory of spike-train power spectra for multidimensional integrate-and-fire neurons

Vellmer

Lindner

2019

Phys. Rev. Research

Self Cite

View full text Add to dashboard Cite

show abstract

“…An analytical description of the transition between the resting state and the stimulus induced state remains to be investigated. Recent results on the nonlinear transfer function of the LIF model will be useful in this approach [50].…”

Section: Discussionmentioning

confidence: 99%

How the connectivity structure of neuronal networks influences responses to oscillatory stimuli

Bos¹,

Schücker²,

Helias³

2017

Preprint

View full text Add to dashboard Cite

Propagation of oscillatory signals through the cortex is shaped by the connectivity structure of neuronal circuits. The coherence of population activity at specific frequencies within and between cortical areas has been linked to laminar connectivity patterns. This study systematically investigates the network and stimulus properties that shape network responses. The results show how input to a cortical column model

show abstract

“…where ξ(t) is Gaussian white noise with a delta correlation function ( ξ(t)ξ(t ) = δ(t − t )); the impact of the noise on the voltage dynamics is determined by the noise intensity D. This model is analytically tractable to a large extent: the probability density and the stationary firing rate can be found in terms of integral expressions (involving the nonlinear function f (v), the base current μ and the strength of the noise D) [6,[29][30][31]; the ISI density [32], or at least its Laplace transform [29,33], is known for particularly simple choices of f (v) (perfect and leaky IF models); and also the linear [11,12,[34][35][36] and weakly nonlinear response [34,37,38] with respect to an additional signal can be calculated. These solutions have been also used in stochastic mean-field theories of recurrent neural networks [12,34] in which it is assumed that the spiking in the network is temporally uncorrelated (equivalent to a Poisson process).…”

Section: Neurons Driven By White Noisementioning

confidence: 99%

Fokker–Planck approach to neural networks and to decision problems

Vellmer

Lindner

2021

Eur. Phys. J. Spec. Top.

Self Cite

View full text Add to dashboard Cite

We review applications of the Fokker–Planck equation for the description of systems with event trains in computational and cognitive neuroscience. The most prominent example is the spike trains generated by integrate-and-fire neurons when driven by correlated (colored) fluctuations, by adaptation currents and/or by other neurons in a recurrent network. We discuss how for a general Gaussian colored noise and an adaptation current can be incorporated into a multidimensional Fokker–Planck equation by Markovian embedding for systems with a fire-and-reset condition and how in particular the spike-train power spectrum can be determined by this equation. We then review how this framework can be used to determine the self-consistent correlation statistics in a recurrent network in which the colored fluctuations arise from the spike trains of statistically similar neurons. We then turn to the popular drift-diffusion models for binary decisions in cognitive neuroscience and demonstrate that very similar Fokker–Planck equations (with two instead of only one threshold) can be used to study the statistics of sequences of decisions. Specifically, we present a novel two-dimensional model that includes an evidence variable and an expectancy variable that can reproduce salient features of key experiments in sequential decision making.

show abstract

Weakly nonlinear response of noisy neurons

Cited by 17 publications

References 49 publications

Theory of spike-train power spectra for multidimensional integrate-and-fire neurons

Theory of spike-train power spectra for multidimensional integrate-and-fire neurons

How the connectivity structure of neuronal networks influences responses to oscillatory stimuli

Fokker–Planck approach to neural networks and to decision problems

Contact Info

Product

Resources

About