A unified view on weakly correlated recurrent networks

Abstract

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.

Gaussian Network’s Dynamics Reflected into Geometric Entropy

“…In fact, such a kind of process is known to maintain the Gaussian character of the distributions over time, and it has been already employed in neuronal networks (see, e.g., [16]). Additionally, a large variety of physical processes are described by such a stochastic evolution.…”

“…with g(θ) = [ g ij (θ)] the n × n symmetric matrix whose entries are given by Equation (16). Notice, however, that also in this case, V(W ) might be ill-defined.…”

“…Hence, from Equations (15) and (16), with the network, at the time t, is associated the manifold M t = Θ t , G(θ t ) , where:…”

Correlated neuronal activity and its relationship to coding, dynamics and network architecture

“…Four studies Zhou et al (2013), Grytskyy et al (2013), Barreiro et al (2014), and Jahnke et al (2013) have focused on the first question. Zhou et al (2013) investigated coupled pairs of neurons receiving temporally correlated input currents.…”

“…Temporal correlations in the noise that these neurons receive may also promote synchrony. Grytskyy et al (2013) have addressed how recurrent neural networks can support the generation of pairwise correlations. The authors put forward a unified framework for the generation of pairwise correlations in recurrent networks and hypothesize that many different single model neurons, when coupled to a network, may generate the same pairwise correlation structures.…”