Collective Synchronous Spiking in a Brain Network of Coupled Nonlinear Oscillators

A network of propagating nonlinear oscillatory modes (waves) in the human brain is shown to generate collectively synchronized spiking activity (hypersynchronous spiking) when both amplitude and phase coupling between modes are taken into account. The nonlinear behavior of the modes participating in the network are the result of the nonresonant dynamics of weakly evanescent cortical waves that, as shown recently, adhere to an inverse frequency–wave number dispersion relation when propagating through an inhomogeneous anisotropic media characteristic of the brain cortex. This description provides a missing link between simplistic models of synchronization in networks of small amplitude phase coupled oscillators and in networks built with various empirically fitted models of pulse or amplitude coupled spiking neurons. Overall the phase-amplitude coupling mechanism presented in the Letter shows significantly more efficient synchronization compared to current standard approaches and demonstrates an emergence of collective synchronized spiking from subthreshold oscillations that neither phase nor amplitude coupling alone are capable of explaining.

Figure 1

Linear and nonlinear oscillations, spiking, and the nonoscillatory regime of the solution of Eqs. (4) and (5) at different levels of excitation ($\gamma =0.25$, 1, 1.98, and 2: blue, green, red, and magenta). Time is in the units of $2\pi /\omega$ and the amplitude $a$ is in the arbitrary units. The rest of the parameters are the same for all plots: $\omega =1$, ${\beta }_a=2$, ${\beta }_{a^{†}}=1$, $\alpha =3$, ${\delta }_{a^{†}}=\pi /4$, and ${\delta }_a=-\pi /4$.

Figure 2

Comparison of synchronization of phase-only constant amplitude harmonic oscillators [(a),(b)] with amplitude-phase coupling of the nonlinear Eqs. (13) and (14) model [(c),(d)]. Network of 100 ring connected oscillators ($R_{ij}=0.025$, $|i-j|\le 2$ and $R_{ij}=0$, $|i-j|>2$) shows only a transient weekly synchronized state (the order parameter $r=0.32$) in the phase-only system [(a),(b)] vs strongly synchronized spiking (the order parameter $r=0.9999$ and still gradually increasing) in (c) and (d). Single frequency ($\omega \equiv {\omega }_0=1$) was used for both cases and ${\beta }_a=2$, ${\beta }_{a^{†}}=1$, $\alpha =3$, $\gamma =1.875$, ${\delta }_{a^{†}}=\pi /4$, ${\delta }_a=-\pi /4$ were the parameters for the nonlinear Eqs. (13) and (14) system.

Figure 3

Emergence of hypersynchronized spiking in a network of multiple frequency (${\omega }_i<{\omega }_0\equiv 1$, ${\omega }_i$ is uniformly distributed between 0.1 and 1, $\overline{{\omega }_i}=0.54$, ${\sigma }_{\omega }=0.25$) weakly forced (${\gamma }_i=0.25$) nonlinear oscillators. (a) and (b) show low amplitude nonlinear oscillations of an uncoupled set. (c) and (d) show the emergence of a relatively weakly synchronized state with three distinct frequency groups at roughly 0, 0.3, and 0.6 plus some single elements at intermediate frequencies for a nearest neighbor random coupling ($R_{ij}=0,|i-j|>1$, $\overline{R_{ij}}=1$, ${\sigma }_R=0.415$). (e) and (f) show the emergence of a strongly synchronized state with mean frequency at roughly $\sim 0.5$ and with the order parameter $r=0.9882$ for all-to-all random coupling $\overline{R_{ij}}=0.01$, ${\sigma }_R=0.0083$.

Figure 4

(a) Network of $N=10032$ nodes split in 48 local cortical regions with every node connected to 10–11 nodes from every other region. (b) The plot of pairwise correlations averaged across all nodes showing a peak at the 300–500 Hz range in agreement with Ref. [16]. (c) Plot of synchronized oscillations.