Collective excitability in a mesoscopic neuronal model of epileptic activity

At the mesoscopic scale, the brain can be understood as a collection of interacting neuronal oscillators, but the extent to which its sustained activity is due to coupling among brain areas is still unclear. Here we address this issue in a simplified situation by examining the effect of coupling between two cortical columns described via Jansen-Rit neural mass models. Our results show that coupling between the two neuronal populations gives rise to stochastic initiations of sustained collective activity, which can be interpreted as epileptic events. For large enough coupling strengths, termination of these events results mainly from the emergence of synchronization between the columns, and thus it is controlled by coupling instead of noise. Stochastic triggering and noise-independent durations are characteristic of excitable dynamics, and thus we interpret our results in terms of collective excitability.

Figure 1

A system of two reciprocally coupled Jansen-Rit cortical columns. Each column consists of three neuronal populations: pyramidal neurons (green triangles), excitatory interneurons (blue hexagons), and inhibitory interneurons (red circles). Red (blue) connectors indicate inhibitory (excitatory) connections. Each column is driven by three external signals acting upon the population of pyramidal neurons: (i) output from the interconnected column, (ii) white noise, and (iii) a constant signal. The two latter components stand for background activity of brain areas that are not modeled directly.

Figure 2

Typical dynamics of the two columns, in terms of the net PSP signal $y_1\left(t\right)-y_2\left(t\right)$, marked by thin black and blue (gray in print) lines. Running averages of these signals obtained with a sliding window of length 0.5 s are shown by thick black and blue (gray) lines. The threshold $T$ is shown as a dashed line, and periods classified as prolonged activity are represented by the gray background. Here the coupling strength is $K=10$, the intensity of white noise is $D=0.5{\mathrm{s}}^{-1}$, and $p$ is adjusted to $K$ according to the methodology introduced in the text. The rest of the parameters are given in the text.

Figure 3

Typical time courses obtained for three different values of the coupling strength. 100 s of activity of the model is shown for $K=5$ (a), $K=10$ (b), and $K=15$ (c). Periods classified as prolonged activity are represented by a gray background. In all cases, the noise intensity $D$ was set to $0.5{\mathrm{s}}^{-1}$, and the system was operating $\mathrm{\Delta }p=1{\mathrm{s}}^{-1}$ below the excitability threshold set by the SNIC bifurcation.

Figure 4

Dependence of the initiation and termination rates on coupling strength and noise intensity. Initiation (“init,” filled markers) and termination (“term,” hollow markers) rates for discrete values of the coupling strength $K$ are shown for three different intensities of the driving noise: $D=0.25{\mathrm{s}}^{-1}$ (circles), $D=0.5{\mathrm{s}}^{-1}$ (triangles), and $D=1{\mathrm{s}}^{-1}$ (squares). Dashed and dotted lines are plotted to guide the eye. For the lowest noise intensity $\left(D=0.25{\mathrm{s}}^{-1}\right)$ and the two lowest coupling values $\left(K=0,1\right)$, the initiation rate was too low to gather a sufficient number of episodes to measure the termination rate. Therefore, the two lowest points of the termination rate for the lowest noise intensity are considered outliers and are not plotted.

Figure 5

Termination patterns of prolonged excitation transients. Panels (a) and (b) show typical time courses of two cortical columns (black and blue lines) for $K=5$ and 15, respectively. Periods of prolonged excitation transients are marked in gray. Panels (c)–(e) show averaged time courses from both columns (thick black lines) and from all prolonged excitation transients for $K=5$, 10, and 15, and obtained from averaging $2.5\times {10}^3,11\times {10}^3$, and $18\times {10}^3$ individual time courses, respectively. The thin gray lines in the three panels represent 100 typical time courses. In all five panels, the point of excitation termination has been shifted to the middle of the plot (corresponding to $t=1$ s). The exact location of this point depends on the parametrization of the classification algorithm, and thus it does not match necessarily the synchronization peaks, as observed in (b). The results were obtained for $D=0.5{\mathrm{s}}^{-1}$.

Figure 6

Bifurcation structure of the model around the SNIC bifurcation for the system of two coupled Jansen-Rit modules. The three panels (a), (b), and (c) demonstrate invariant sets for the coupling strength $K$ equal to 0, 10, and 20, respectively. Continuous (dashed) lines mean stable (unstable) equilibria. The filled circle marks the location of the SNIC bifurcation.

Figure 7

(a) Coupling function characterizing the interaction between two columns. (b) Location of the saddle-node bifurcation in the system of two connected Jansen-Rit modules computed with xppaut (solid line) and with the perturbation convergence method. The approximated linear dependency is marked by the dotted line.