Control and Synchronization of Neuron Ensembles

Abstract

Synchronization of oscillations is a phenomenon prevalent in natural, social, and engineering systems. Controlling synchronization of oscillating systems is motivated by a wide range of applications from neurological treatment of Parkinson's disease to the design of neurocomputers. In this article, we study the control of an ensemble of uncoupled neuron oscillators described by phase models. We examine controllability of such a neuron ensemble for various phase models and, furthermore, study the related optimal control problems. In particular, by employing Pontryagin's maximum principle, we analytically derive optimal controls for spiking single-and two-neuron systems, and analyze the applicability of the latter to an ensemble system. Finally, we present a robust computational method for optimal control of spiking neurons based on pseudospectral approximations. The methodology developed here is universal to the control of general nonlinear phase oscillators.

Design of Charge-Balanced Time-Optimal Stimuli for Spiking Neuron Oscillators

“…Work on considering stochastic effects to neuron systems such as the optimal control of neuronal spiking activity receiving a class of random synaptic inputs has also been investigated [18]. In addition, controllability of an ensemble of uncoupled neurons was explored for various mathematically ideal phase models, where an effective computational optimal control method based on pseudospectral approximations was employed to construct optimal controls that elicit simultaneous spikes of a neuron ensemble [19,20]. The derivation of time-optimal and spike timing controls for spiking neurons has been attempted for limited classes of control functions [21,22], however, a complete characterization of the optimal solutions has not been provided, and an analytical and systematic approach for synthesizing the time-optimal controls has been missing.…”

Optimal Control of an Epileptic Neural Population Model

“…The pseudospectral method has been successfully applied in a variety of applications including guidance for aircraft, quantum control, and neuroscience (Fahroo and Ross [2008], Li et al [2011Li et al [ , 2013). The method employs a relationship between orthogonal Legendre polynomials, which permit spectral accuracy (i.e., only a small number of terms are needed to approximate the function), and interpolating Lagrange polynomials, which enable the collocation nodes of the interpolation approximation to be used directly in the subsequent nonlinear optimization.…”

“…We sample this section of the curve and create an optimal ensemble control problem which seeks to simultaneously drive N Ω systems starting at various points along the trajectory to the stable equilibrium with a single common stimulus. The pseudospectral method provides a natural extension to implement and solve the ensemble optimal control problem (Li et al [2011(Li et al [ , 2013). Figure 5 displays the trajectories corresponding to five (N Ω = 5) different starting locations which are simultaneously driven to a neighborhood of x r by the common ensemble control, plotted on the inset axis.…”

“…Within the control community, several groups, including Moehlis et al [2006] and Li et al [2013], have applied these systematic methods, however, have focused mainly on phase-models, which are highly reduced models of individual neurons. Since most patient data is collected by EEG, which operates at the macroscopic scale, in this work we use a more clinically relevant neural population model.…”

Constrained charge-balanced minimum-power controls for spiking neuron oscillators

“…Synchronization engineering techniques, which utilized a global, nonlinear delayed feedback to induce a pre-selected synchronization structure, have been developed for effective control of the collective behavior of globally coupled nonlinear phase oscillators [17,18]. Controllability of a network of neurons described by phase models has also been examined [19,20]. Recently, unconstrained and constrained minimum-power controls for spiking a neuron at specified time instances have been derived for several phase models with mathematically ideal and practical PRCs [21][22][23]; however, these control designs did not take the charge-balance constraint into account.…”