Synchrony-induced modes of oscillation of a neural field model

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.

Figure 1

Schematic representation of the ring network and coupling architecture under study. Panel (a) shows $N$ excitatory (red triangles) and $N$ inhibitory (blue circles) neurons arranged on a ring. The location of neurons is parameterized by the angular variable ${\varphi }_j=\frac{2\pi j}{N}-\pi ,j=1,\cdots ,N$. Red (solid) and blue (dashed) lines indicate synaptic connections between neuron pairs $({\varphi }_j,{\varphi }_k)$. An example of the excitatory and inhibitory space-dependent connectivity kernels Eqs. (1) are shown in panel (b) where the abscissa represents the distance $\left|{\varphi }_k-{\varphi }_j\right|$ between neurons $j$ and $k$. Panel (c) represents an effective model in which pairs of excitatory/inhibitory neurons located at a certain location ${\varphi }_k$ are modeled as single neurons. The effective pattern of synaptic connectivity is obtained subtracting the inhibitory pattern from the excitatory one, as show in panel (d).

Figure 2

Transient episodes of spike synchrony in heterogeneous ring networks of $N=2.5\times {10}^5$ excitatory and $N=2.5\times {10}^5$ inhibitory QIF neurons, Eqs. (2) and (3), as a result of spatially inhomogeneous perturbations applied at time $t=0.05$. In panels (a) and (b) only excitatory neurons are perturbed. In panels (c) and (d) all neurons are perturbed. In panels (a) and (c) the perturbation has wave number $K=1$; in panels (b) and (d) the perturbation has wave number $K=3$. Other parameters are $\mathrm{\Delta }=1,\tau =20$ ms, and $\overline{\eta }=5$. All Fourier components of the connectivity Eq. (1) are $J_K^{e,i}=0$, except $J_0^e=23,J_1^e=10,J_2^e=7.5,J_3^e=-2.5,J_0^i=23$.

Figure 3

(a) Phase diagram of Eqs. (7) (with $J_0=0$) showing the regions of stability of the SHS, determined by the eigenvalues Eq. (10). Spatial perturbations of wave number $K>0$ show oscillatory and nonoscillatory decay to the spatially homogeneous state in the light-shaded and dark-shaded regions of the diagram, respectively. The eigenvalues ${\lambda }_{K\pm }$ associated with the $K\mathrm{th}$ mode are schematically represented in the complex plane (red crosses) for the three qualitatively different regions of the phase diagram. Right panels show the response of the Eqs. (7) with $J_1=10,J_2=7.5,J_3=-2.5$, and $J_K=0$ ($K\ne 1,2,3$), $\overline{\eta }=4.5,\mathrm{\Delta }=1$, and $\tau =20\mathrm{ms}$, to a perturbation of the (b) $K=1$ and (c) $K=3$ spatial modes. Both perturbations produce standing waves with frequency and decay rate described by Eqs. (10). In the white region, limited by the curve Eq. (11), these perturbations grow and lead to a bump state (BS) with $K$ bumps (see Fig. 4).

Figure 4

(a) Phase diagram of the QIF-NFM [Eqs. (7)] with $J_2=7.5,J_3=-2.5,J_K=0$, for $K>3$, and $\mathrm{\Delta }=1$. Solid line: supercritical [red (gray)] and subcritical (black) Turing bifurcation boundary Eq. (12)]. Dashed lines: saddle-node bifurcation of bumps (numerical). (b) Diagram—obtained using a weakly nonlinear analysis—showing the regions where the Turing bifurcation is supercritical or subcritical for $J_1=10,J_3=-2.5$, and $J_K=0$. (c) Bifurcation diagram (rescaled) $\parallel R_{*}{\parallel }_2={\left(2\pi \right)}^{-1}{\int }_{-\pi }^{\pi }{\left|R_{*}\left(\varphi \right)\right|}^{2}d\varphi$ vs. $\overline{\eta }$, for $J_1=10$. Solid/dashed black lines: stable/unstable SHS. Solid/dashed blue (light gray) lines: stable/unstable bump states (BS).

Figure 5

Spectrum [(a) and (b)] and firing rate profiles [(d) and (c)] of an unstable [(a) and (c)] and stable [(b) and (d)] bump states of the QIF-NFM [Eqs. (7)]. In panels (a) and (b) the eigenvalues Eq. (10) are superimposed with red crosses. Panel (e) shows a numerical simulation of the BS of panel (d). At $t=0.05$ s, a perturbation of wave number $K=6$ is applied. Parameters are $J_0=0,J_1=10,J_2=7.5,J_3=-2.5,J_K=0$ for $K>3,\mathrm{\Delta }=1,\tau =20\mathrm{ms}$. Panels (a) and (c): $\overline{\eta }=2.2120$; panels (b), (d), and (e): $\overline{\eta }=2.1828$.

Figure 6

Algorithm used for the Euler integration of the QIF neuron Eq. (2).