Solving the two-dimensional Fokker-Planck equation for strongly correlated neurons

Taşkın, Deniz; Rotter, Stefan

doi:10.1103/physreve.95.012412

Cited by 11 publications

(27 citation statements)

References 40 publications

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“…First of all, the theory may become the starting point of analytical approaches beyond the existing ones that are limited to IF models with weak or shortcorrelated Ornstein-Uhlenbeck noise. Second, with more efficient algorithms (efficient tools for the solution of multidimensional FPEs [102,103] might be useful here) also higher-dimensional situations, e.g., an adapting neuron with narrow-band noise input (corresponding to a system of four stochastic differential equations) might be tractable; possible candidates are eigenfunction expansions [104,105] and the matrix-continued-fraction method [13]. Third, similar to the one-dimensional white-noise case [30,31], the calculation of the firing-rate modulation in response to a time-dependent stimulus (other than noise) will follow a very similar mathematical framework as presented here for the calculation of the power spectrum.…”

Section: Summary and Open Problemsmentioning

confidence: 99%

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

Vellmer

Lindner

2019

Phys. Rev. Research

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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.

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Section: Summary and Open Problemsmentioning

confidence: 99%

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

Vellmer

Lindner

2019

Phys. Rev. Research

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“… 2 For the simpler but still formidable problem of how neuron pairs respond to cross-correlated Gaussian white noise sources, see, for instance, Doiron et al ( 2004 ); de la Rocha et al ( 2007 ); Shea-Brown et al ( 2008 ); Ostojic et al ( 2009 ); Vilela and Lindner ( 2009b ); Deniz and Rotter ( 2017 ). …”

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confidence: 99%

Self-Consistent Scheme for Spike-Train Power Spectra in Heterogeneous Sparse Networks

Pena

Vellmer

Bernardi

et al. 2018

Front. Comput. Neurosci.

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Recurrent networks of spiking neurons can be in an asynchronous state characterized by low or absent cross-correlations and spike statistics which resemble those of cortical neurons. Although spatial correlations are negligible in this state, neurons can show pronounced temporal correlations in their spike trains that can be quantified by the autocorrelation function or the spike-train power spectrum. Depending on cellular and network parameters, correlations display diverse patterns (ranging from simple refractory-period effects and stochastic oscillations to slow fluctuations) and it is generally not well-understood how these dependencies come about. Previous work has explored how the single-cell correlations in a homogeneous network (excitatory and inhibitory integrate-and-fire neurons with nearly balanced mean recurrent input) can be determined numerically from an iterative single-neuron simulation. Such a scheme is based on the fact that every neuron is driven by the network noise (i.e., the input currents from all its presynaptic partners) but also contributes to the network noise, leading to a self-consistency condition for the input and output spectra. Here we first extend this scheme to homogeneous networks with strong recurrent inhibition and a synaptic filter, in which instabilities of the previous scheme are avoided by an averaging procedure. We then extend the scheme to heterogeneous networks in which (i) different neural subpopulations (e.g., excitatory and inhibitory neurons) have different cellular or connectivity parameters; (ii) the number and strength of the input connections are random (Erdős-Rényi topology) and thus different among neurons. In all heterogeneous cases, neurons are lumped in different classes each of which is represented by a single neuron in the iterative scheme; in addition, we make a Gaussian approximation of the input current to the neuron. These approximations seem to be justified over a broad range of parameters as indicated by comparison with simulation results of large recurrent networks. Our method can help to elucidate how network heterogeneity shapes the asynchronous state in recurrent neural networks.

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“…When input correlations are weak, covariance transfer for integrate-and-fire neurons is approximately linear so linear response approaches are justified [8,53,54], but transfer can become nonlinear (but still O(1)) when correlations are stronger [54,79,80]. Our results show that, even when the transfer of input covariance to spike train covariance is nonlinear, the mean-field relationship between X, X and S, S is still linear to highest order in N and is the same relationship one would obtain if transfer were linear.…”

Section: Summary and Discussionmentioning

confidence: 99%

Correlated states in balanced neuronal networks

et al. 2019

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Understanding the magnitude and structure of inter-neuronal correlations and their relationship to synaptic connectivity structure is an important and difficult problem in computational neuroscience. Early studies show that neuronal network models with excitatory-inhibitory balance naturally create very weak spike train correlations, defining the "asynchronous state." Later work showed that, under some connectivity structures, balanced networks can produce larger correlations between some neuron pairs, even when the average correlation is very small. All of these previous studies assume that the local network receives feedforward synaptic input from a population of uncorrelated spike trains. We show that when spike trains providing feedforward input are correlated, the downstream recurrent network produces much larger correlations. We provide an in-depth analysis of the resulting "correlated state" in balanced networks and show that, unlike the asynchronous state, it produces a tight excitatory-inhibitory balance consistent with in vivo cortical recordings.

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Solving the two-dimensional Fokker-Planck equation for strongly correlated neurons

Cited by 11 publications

References 40 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

Self-Consistent Scheme for Spike-Train Power Spectra in Heterogeneous Sparse Networks

Correlated states in balanced neuronal networks

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