Optimal control of oscillations and synchrony in nonlinear models of neural population dynamics

We adapt nonlinear optimal control theory (OCT) to control oscillations and network synchrony and apply it to models of neural population dynamics. Conventionally, OCT requires a target trajectory. This requirement may be overly restrictive for oscillatory targets since the exact trajectory shape including amplitude and phase might not be relevant. To overcome this limitation, we introduce three alternative cost functionals to target oscillations and synchrony without specifying a reference trajectory.

A controlled nonlinear dynamical system with state $x$ and control $u$ is defined by
$\dot{x}=h(x,u)=0.$
OCT provides methods to compute the most efficient OC for a particular purpose [1]. A cost functional $F$ trades accuracy against input strength and is conventionally defined as
F=w_P\underbrace{\frac{1}{2}\int_0^T(x(t)-\tilde{x}(t))^2~dt}_{=F_P~\text{("precision cost")}}+\frac{1}{2}\int_0^Tu(t)^2~dt,
where $F_P$ measures the closeness to the target state $\tilde{x}(t)$, $F_E$ the total control strength, and $T$ denotes the simulation duration. The OC $u^*=\arg\min_uF$ minimizes the cost. The adjoint method enables to compute the gradient $\frac{dF}{du}$ of the cost with respect to the control. We approach a cost minimum by gradient descent.

We target oscillations or synchrony without specifying $\tilde{x}$ by replacing $F_P$ in above Equation. To enforce oscillations at a frequency $\tilde{f}$, we replace $F_P$ by the the squared Fourier component of $x$ corresponding to $\tilde{f}$, the Fourier cost
$F_F=-\frac{1}{NT^2}\mathcal{C}_{\tilde{f}}^2(x(t)) =-\frac{1}{NT^2}\bigg\vert\int_0^T x(t)\cdot e^{-2\pi i\tilde{f}t}~dt\bigg\vert^2$.
To synchronize an oscillating $N$-node network, we replace $F_P$ by either the cross-correlation cost [2]
$F_{cc}=-\frac{2}{N(N-1)T}\int_0^T\sum_{n=1}^N\sum_{m=n+1}^N \frac{(x_n(t)-\bar{x}_n)(x_m(t)-\bar{x}_m)}{\sigma(x_n)\sigma(x_m)}~dt$
with the temporal mean $\bar{x}=\frac{1}{T}\int_0^Tx(t)~dt$ and its variance $\sigma^2(x) = \frac{1}{T} \int_0^T(x(t)-\bar{x})^2~dt$, or the variance cost
$F_{var}=\frac{1}{NT}\sum_{n=1}^N\int_0^T(x_n(t)-\bar{x}(t))^2~dt$
with the network mean $\bar{x}=\frac{1}{N}\sum_{n=1}^Nx_n(t)$.

We apply these cost functionals to a Wilson-Cowan (WC) [3,4] model and a mean-field model of excitatory and inhibitory EIF neurons [5]. $F_F$ drives oscillations at a particular frequency, while $F_{cc}$ and $F_{var}$ force any asynchronously oscillating network to synchronize (see Fig. 1).