Synchrony-induced modes of oscillation of a neural field model

Abstract

We investigate the modes of oscillation of heterogeneous ring networks of quadratic integrate-and-fire (QIF) neurons with nonlocal, space-dependent coupling. Perturbations of the equilibrium state with a particular wave number produce transient standing waves with a specific temporal frequency, analogously to those in a tense string. In the neuronal network, the equilibrium corresponds to a spatially homogeneous, asynchronous state. Perturbations of this state excite the network's oscillatory modes, which reflect the interplay of episodes of synchronous spiking with the excitatory-inhibitory spatial interactions. In the thermodynamic limit, an exact low-dimensional neural field model describing the macroscopic dynamics of the network is derived. This allows us to obtain formulas for the Turing eigenvalues of the spatially homogeneous state and hence to obtain its stability boundary. We find that the frequency of each Turing mode depends on the corresponding Fourier coefficient of the synaptic pattern of connectivity. The decay rate instead is identical for all oscillation modes as a consequence of the heterogeneity-induced desynchronization of the neurons. Finally, we numerically compute the spectrum of spatially inhomogeneous solutions branching from the Turing bifurcation, showing that similar oscillatory modes operate in neural bump states and are maintained away from onset.

Bumps and oscillons in networks of spiking neurons

“…At sufficiently small (large) η only one stable fixed point exists, represented by r 3 (r 1 ); also, there exists an interval in parameter space where the stable solutions r 1 , r 3 coexist with r 2 , which is unstable. The conclusions presented above justify the bifurcation diagram found in References 19,22 , and reported in Figure 2a. Loci of saddle-node bifurcations in the (η, J)-plane can be found by setting dη/dr = 0 in the first equation in (6) which, combined with (7) yields a parameterization in r…”

“…Turing bifurcations A first step towards the construction of heterogeneous steady states is the determination of Turing bifurcations, which mark points in parameter space where a spatially uniform solution becomes unstable to spatially periodic patterns. We remark that it is known that spatiallyextended networks of QIF or θ neurons display this instability 22,23 , and here we present an analytic determination of the loci of such bifurcation in parameter space. Turing bifurcations of a homogeneous steady state (r, v) can be identified by linear stability analysis of the model equations in Fourier space, which results in the following eigenvalue problem:…”

“…To avoid problems with the numerical evaluations of ϕ j for large |x|, we derive an alternative scheme, which uses inner products with weight ρ(x) ≡ 1, as the Fourier Galerkin scheme. We seek a solution to (22) in the form…”

“…Some limitations can be overcome if the microscopic model description is a heterogeneous network of synaptically coupled θ or QIF neurons, subject to random, Cauchy-distributed background currents. Recently, it has been shown that heterogeneous networks of θ-and QIF neurons admit an exact mean field description 18,19 , which has later been extended to spatially-extended networks [20][21][22][23] . In the thermodynamic limit, the network admits an exact mean field description in terms of the network mean rate and voltage 19 , or in terms of a complexvalued order parameter 20,24 Here we study a network of n quadratic integrate-andfire neurons:…”

“…The synaptic currents are regarded as instantaneous for simplicity, but it is straightforward to include synaptic delays via temporal convolutions as well 19 . A neural field model that describes without approximation the average firing rate r(x, t) and the average membrane potential v(x, t) of the spatially-extended networks presented above has been developed recently 22 :…”