Dan Wilson

Optimal Chaotic Desynchronization for Neural Populations

A procedure is developed for finding an energy-optimal stimulus which gives a positive Lyapunov exponent, and hence desynchronization, for a neural population. The procedure is illustrated for three different neural models. Not only does it achieve desynchronization for each model, but it also does so using less energy than recently proposed methods, suggesting a powerful alternative to pulsatile stimuli for deep brain stimulation. Furthermore, we calculate error bounds on the optimal stimulus which will guarantee a minimum Lyapunov exponent. Also, a related control strategy is developed for desynchronizing neurons based on the populations phase distribution.

Locally optimal extracellular stimulation for chaotic desynchronization of neural populations

We use optimal control theory to design a methodology to find locally optimal stimuli for desynchronization of a model of neurons with extracellular stimulation. This methodology yields stimuli which lead to positive Lyapunov exponents, and hence desynchronizes a neural population. We analyze this methodology in the presence of interneuron coupling to make predictions about the strength of stimulation required to overcome synchronizing effects of coupling. This methodology suggests a powerful alternative to pulsatile stimuli for deep brain stimulation as it uses less energy than pulsatile stimuli, and could eliminate the time consuming tuning process.

Phasic Burst Stimulation: A Closed-Loop Approach to Tuning Deep Brain Stimulation Parameters for Parkinson’s Disease

We propose a novel, closed-loop approach to tuning deep brain stimulation (DBS) for Parkinson’s disease (PD). The approach, termed Phasic Burst Stimulation (PhaBS), applies a burst of stimulus pulses over a range of phases predicted to disrupt pathological oscillations seen in PD. Stimulation parameters are optimized based on phase response curves (PRCs), which would be measured from each patient. This approach is tested in a computational model of PD with an emergent population oscillation. We show that the stimulus phase can be optimized using the PRC, and that PhaBS is more effective at suppressing the pathological oscillation than a single phasic stimulus pulse. PhaBS provides a closed-loop approach to DBS that can be optimized for each patient.

Clustered Desynchronization from High-Frequency Deep Brain Stimulation.

While high-frequency deep brain stimulation is a well established treatment for Parkinson’s disease, its underlying mechanisms remain elusive. Here, we show that two competing hypotheses, desynchronization and entrainment in a population of model neurons, may not be mutually exclusive. We find that in a noisy group of phase oscillators, high frequency perturbations can separate the population into multiple clusters, each with a nearly identical proportion of the overall population. This phenomenon can be understood by studying maps of the underlying deterministic system and is guaranteed to be observed for small noise strengths. When we apply this framework to populations of Type I and Type II neurons, we observe clustered desynchronization at many pulsing frequencies.

Optimal entrainment of heterogeneous noisy neurons

We develop a methodology to design a stimulus optimized to entrain nonlinear, noisy limit cycle oscillators with uncertain properties. Conditions are derived which guarantee that the stimulus will entrain the oscillators despite these uncertainties. Using these conditions, we develop an energy optimal control strategy to design an efficient entraining stimulus and apply it to numerical models of noisy phase oscillators and to in vitro hippocampal neurons. In both instances, the optimal stimuli outperform other similar but suboptimal entraining stimuli. Because this control strategy explicitly accounts for both noise and inherent uncertainty of model parameters, it could have experimental relevance to neural circuits where robust spike timing plays an important role.

Extending Phase Reduction to Excitable Media: Theory and Applications

Phase reduction methods have been tremendously useful for understanding the dynamics of nonlinear oscillators, but have been difficult to extend to systems with a stable fixed point, such as an excitable system. Using the notion of isostables, which measure the time it takes for a given initial condition in phase space to approach a stable fixed point, we present a general method for isostable reduction for excitable systems. We also devise an adjoint method for calculating infinitesimal isostable response curves, which are analogous to infinitesimal phase response curves for oscillatory systems. Through isostable reduction, we are able to implement sophisticated control strategies in a high-dimensional model of cardiac activity for the termination of alternans, a precursor to cardiac fibrillation.

A Hamilton-Jacobi-Bellman approach for termination of seizure-like bursting

We use Hamilton-Jacobi-Bellman methods to find minimum-time and energy-optimal control strategies to terminate seizure-like bursting behavior in a conductance-based neural model. Averaging is used to eliminate fast variables from the model, and a target set is defined through bifurcation analysis of the slow variables of the model. This method is illustrated for a single neuron model and for a network model to illustrate its efficacy in terminating bursting once it begins. This work represents a numerical proof-of-concept that a new class of control strategies can be employed to mitigate bursting, and could ultimately be adapted to treat medically intractible epilepsy in patient-specific models.

An Energy-Optimal Methodology for Synchronization of Excitable Media

We propose a novel energy-optimal methodology to synchronize the cells within an excitable medium during spiral wave activity. The optimal stimulus takes the cells of an excitable medium to a target set where the isostables of the system, which measure the time it takes for a given initial condition in phase space to approach a stable fixed point, are diffuse, thereby synchronizing the activity of the medium. This method is illustrated for a FitzHugh--Nagumo-based model which captures important characteristics of the myocardium and represents a significant first step in the development of energy-optimal defibrillation strategies.

An energy-optimal approach for entrainment of uncertain circadian oscillators.

We develop an approach to find an energy-optimal stimulus that entrains an ensemble of uncertain, uncoupled limit cycle oscillators. Furthermore, when entrainment occurs, the phase shift between oscillators is constrained to be less than a predetermined amount. This approach is illustrated for a model of Drosophila circadian activity, for which it performs better than a standard 24-h light-dark cycle. Because this method explicitly accounts for uncertainty in a given system and only requires information that is experimentally obtainable, it is well suited for experimental implementation and could ultimately represent what is believed to be a novel treatment for patients suffering from advanced/delayed sleep-phase syndrome.

Toward a More Efficient Implementation of Antifibrillation Pacing

We devise a methodology to determine an optimal pattern of inputs to synchronize firing patterns of cardiac cells which only requires the ability to measure action potential durations in individual cells. In numerical bidomain simulations, the resulting synchronizing inputs are shown to terminate spiral waves with a higher probability than comparable inputs that do not synchronize the cells as strongly. These results suggest that designing stimuli which promote synchronization in cardiac tissue could improve the success rate of defibrillation, and point towards novel strategies for optimizing antifibrillation pacing.