Pulse desynchronization of neural populations by targeting the centroid of the limit cycle in phase space

Abstract

The synchronized activity of neuronal populations can lead to pathological over-synchronization in conditions such as epilepsy and Parkinson disease. Such states can be desynchronized by brief electrical pulses. But when the underlying oscillating system is not known, as in most practical applications, to determine the specific times and intensities of pulses used for desynchronizaton is a difficult inverse problem. Here we propose a desynchronization scheme for neuronal models of bi-variate neural activity, with possible applications in the medical setting. Our main argument is the existence of a peculiar point in the phase space of the system, the centroid, that is both easy to calculate and robust under changes in the coupling constant. This important target point can be used in a control procedure because it lies in the region of minimal return times of the system.

Figures (3)

• Figure 1: FitzHugh--Nagumo (Bonhoeffer--van der Pol) system. a) Example configurations of $N = 1000$ oscillators in the $(x,y)$ phase space at a fixed time, shown for three different values of the coupling $\varepsilon$. All other parameters are fixed to $\xi = 1,\ \nu = -1,\ \delta = -\tfrac{1}{3},\ \mu = 0.6,\ \alpha = 0.1,\ \beta = 0.8,\ \gamma = 0.7,\ \sigma = 0.1.$ Crosses indicate the position of the ensemble mean $(X, Y)$ averaged in time. b) Temporal evolution of the ensemble mean $X(t)$. c) Temporal evolution of the ensemble mean $Y(t)$.
• Figure 2: Interpolated heatmap of return times. The mean-field limit cycle is shown as a solid gray line; nullclines as solid red lines; the unstable fixed point as a red marker. The geometric center of the mean-field limit cycle is shown in green, while the averaged-in-time $(X,Y)$ position for $\epsilon = 0$ is shown in orange. All geometric curves and points are derived from the averages of $X$ and $Y$. Simulations were performed with $N=1000$ oscillators and $1479$ perturbation points on a grid with spacing $\mathrm{d}x=\mathrm{d}y=0.1$. Return times were linearly interpolated within the grid. For each perturbation point, $100$ trials were run for a simulation time of $100$ with timestep $dt=0.05$.
• Figure 3: Estimation of the return times from Eq. \ref{['eq:return_time_roots']}. a) Limit cycle of $(X,Y)$ and key points: trajectory of the limit cycle (dashed gray) with the nullclines $y = f(x)$ and $y = g(x)$ shown in firebrick. Key points include the baricenter (orange X), geometric center (cyan X), unstable points (red X), an example of a return trajectory from the initial condition $(X_0, Y_0)$ in green, and the control point (dark green X). The roots of the the cubic polynomial Eq. \ref{['eq:cubic_standard_form']} are shwon in purple. b) Discriminant of the cubic nullcline: The discriminant $\Delta$ of the cubic equation as a function of the initial condition $Y_0$ for different values of $\epsilon \in \{ 0.1, 0.2, 0.3, 0.4, 0.5\}$. c) Approximate estimation of return times obtained by solving Eq. \ref{['eq:return_time_roots']}, displayed as a heatmap with the nullcline $y = f(x)$ overlaid. d) Numerical results for the return times obtained from extensive simulations for $\epsilon = 0.3$, displayed as a heatmap. The nullcline $dx/dt = 0$ is shown in red.

Theorems & Definitions (2)

• Theorem A.1: Propagation of chaos
• proof