Invariance of covariances arises out of noise

Grytskyy, Dmytro; Tetzlaff, Tom; Diesmann, Markus; Helias, Moritz

doi:10.1063/1.4776531

Cited by 10 publications

(31 citation statements)

References 9 publications

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“…(27) of Ref. [59]. More importantly, this systematic calculation reveals the assumption that is underlying the Gaussian approximation for the input field, namely, that cumulants higher than order two are ignored on the level of individual neuronal activities.…”

Section: Gaussian Approximationmentioning

confidence: 78%

Correlated Fluctuations in Strongly Coupled Binary Networks Beyond Equilibrium

2016

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Randomly coupled Ising spins constitute the classical model of collective phenomena in disordered systems, with applications covering glassy magnetism and frustration, combinatorial optimization, protein folding, stock market dynamics, and social dynamics. The phase diagram of these systems is obtained in the thermodynamic limit by averaging over the quenched randomness of the couplings. However, many applications require the statistics of activity for a single realization of the possibly asymmetric couplings in finite-sized networks. Examples include reconstruction of couplings from the observed dynamics, representation of probability distributions for sampling-based inference, and learning in the central nervous system based on the dynamic and correlation-dependent modification of synaptic connections. The systematic cumulant expansion for kinetic binary (Ising) threshold units with strong, random, and asymmetric couplings presented here goes beyond mean-field theory and is applicable outside thermodynamic equilibrium; a system of approximate nonlinear equations predicts average activities and pairwise covariances in quantitative agreement with full simulations down to hundreds of units. The linearized theory yields an expansion of the correlation and response functions in collective eigenmodes, leads to an efficient algorithm solving the inverse problem, and shows that correlations are invariant under scaling of the interaction strengths.

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Section: Gaussian Approximationmentioning

confidence: 78%

Correlated Fluctuations in Strongly Coupled Binary Networks Beyond Equilibrium

2016

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“…This finding can readily be understood by the linearization procedure, presented in the current work, that takes into account the network- generated fluctuations of the total input. The amplitude σ of these fluctuations scales linearly in J and the effective susceptibility depends on J /σ in the case β → ∞, explaining the invariance (Grytskyy et al, 2013). In the current manuscript we generalized this procedure to finite slopes β and to other models than the binary neuron model.…”

Section: Discussionmentioning

confidence: 93%

“…Even in the absence of local noise (β → ∞), the above mentioned linearization is applicable and yields a finite effective slope 〈ϕ′〉 (27). In the latter case the resulting effective synaptic weight is independent of the original synapse strength (Grytskyy et al, 2013). …”

Section: Binary Neuronsmentioning

confidence: 99%

A unified view on weakly correlated recurrent networks

Grytskyy

Tetzlaff

Diesmann

et al. 2013

Front. Comput. Neurosci.

131

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The diversity of neuron models used in contemporary theoretical neuroscience to investigate specific properties of covariances in the spiking activity raises the question how these models relate to each other. In particular it is hard to distinguish between generic properties of covariances and peculiarities due to the abstracted model. Here we present a unified view on pairwise covariances in recurrent networks in the irregular regime. We consider the binary neuron model, the leaky integrate-and-fire (LIF) model, and the Hawkes process. We show that linear approximation maps each of these models to either of two classes of linear rate models (LRM), including the Ornstein–Uhlenbeck process (OUP) as a special case. The distinction between both classes is the location of additive noise in the rate dynamics, which is located on the output side for spiking models and on the input side for the binary model. Both classes allow closed form solutions for the covariance. For output noise it separates into an echo term and a term due to correlated input. The unified framework enables us to transfer results between models. For example, we generalize the binary model and the Hawkes process to the situation with synaptic conduction delays and simplify derivations for established results. Our approach is applicable to general network structures and suitable for the calculation of population averages. The derived averages are exact for fixed out-degree network architectures and approximate for fixed in-degree. We demonstrate how taking into account fluctuations in the linearization procedure increases the accuracy of the effective theory and we explain the class dependent differences between covariances in the time and the frequency domain. Finally we show that the oscillatory instability emerging in networks of LIF models with delayed inhibitory feedback is a model-invariant feature: the same structure of poles in the complex frequency plane determines the population power spectra.

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“…E Fluctuating input h E averaged over the excitatory population (black), separated into contributions from excitatory synapses h EE (gray) and from inhibitory synapses h EI (light gray). F Distribution of time averaged activity obtained by direct simulation (symbols) and analytical prediction (17) using the numerically evaluated self-consistent solution for the first m E ≃ m I ≃ 0.11 and second moments q E ≃ 0.019, q I ≃ 0.018 (19). Duration of simulation T = 100 s, mean activity m X = 0.1, other parameters as in Figure 3.…”

Section: Resultsmentioning

confidence: 99%

The Correlation Structure of Local Neuronal Networks Intrinsically Results from Recurrent Dynamics

2014

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Correlated neuronal activity is a natural consequence of network connectivity and shared inputs to pairs of neurons, but the task-dependent modulation of correlations in relation to behavior also hints at a functional role. Correlations influence the gain of postsynaptic neurons, the amount of information encoded in the population activity and decoded by readout neurons, and synaptic plasticity. Further, it affects the power and spatial reach of extracellular signals like the local-field potential. A theory of correlated neuronal activity accounting for recurrent connectivity as well as fluctuating external sources is currently lacking. In particular, it is unclear how the recently found mechanism of active decorrelation by negative feedback on the population level affects the network response to externally applied correlated stimuli. Here, we present such an extension of the theory of correlations in stochastic binary networks. We show that (1) for homogeneous external input, the structure of correlations is mainly determined by the local recurrent connectivity, (2) homogeneous external inputs provide an additive, unspecific contribution to the correlations, (3) inhibitory feedback effectively decorrelates neuronal activity, even if neurons receive identical external inputs, and (4) identical synaptic input statistics to excitatory and to inhibitory cells increases intrinsically generated fluctuations and pairwise correlations. We further demonstrate how the accuracy of mean-field predictions can be improved by self-consistently including correlations. As a byproduct, we show that the cancellation of correlations between the summed inputs to pairs of neurons does not originate from the fast tracking of external input, but from the suppression of fluctuations on the population level by the local network. This suppression is a necessary constraint, but not sufficient to determine the structure of correlations; specifically, the structure observed at finite network size differs from the prediction based on perfect tracking, even though perfect tracking implies suppression of population fluctuations.

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Invariance of covariances arises out of noise

Cited by 10 publications

References 9 publications

Correlated Fluctuations in Strongly Coupled Binary Networks Beyond Equilibrium

Correlated Fluctuations in Strongly Coupled Binary Networks Beyond Equilibrium

A unified view on weakly correlated recurrent networks

The Correlation Structure of Local Neuronal Networks Intrinsically Results from Recurrent Dynamics

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