Fig 1.
Beta-based feedback stimulation policies.
(row 1) Simulated LFP. (row 2, 3) Power and phase calculated from the LFP using the αSWIFT algorithm. The dotted lines indicate the manually set power threshold and phase trigger for stimulation. (row 4) Power-based stimulation: high frequency stimulation is turned on when the power is above threshold. (row 5) Phase-based stimulation: individual pulses are delivered when the phase crosses the trigger. (row 6) Combined phase/power-based stimulation: individual pulses are delivered when the phase crosses the trigger, but only if the power is above threshold.
Fig 2.
Basal ganglia-thalamocortical system (BGTCS) mean-field model structure.
Black arrows represent excitatory connections, red circles represent inhibitory connections. Simulated DBS was applied to the STN, and local field potentials (LFPs) were recorded from the GPi. Adapted from van Albada and Robinson, 2009 [26, 27].
Fig 3.
(a) time-series data and (b) PSD analysis in three conditions: naïve, DD, and DD with cDBS in the STN. The model produced a spectral peak at 29 Hz, which increased and widened in the DD state. When cDBS was applied to the STN of the model, the spectral power in beta band decreased.
Fig 4.
Adaptive dual controller (ADC) for DBS.
The ADC has dual goals (exploitation and exploration), and is composed of two loops: an inner parameterized stimulator and an outer parameter adjustment loop. The inner loop may incorporate feedback from the patient to alter stimulation. The outer loop is composed of an estimator and a design block, and is given a specification. The estimator builds a model of the relationship between stimulation parameters and some measure of patient outcome, which it passes on to the design block. The design block then incorporates this information with the specification to select new parameters for the inner loop. The inner loop operates on a much shorter timescale than the outer loop.
Fig 5.
Bayesian optimization example.
Three iterations of Bayesian optimization minimizing a 1D function. The figure shows a Gaussian process (GP) approximation (solid black line and blue shaded region) of the underlying objective function (dotted black line). The figure also shows the acquisition function (green). The acquisition function (GP-LCB) is the difference of the mean and variance of the GP (multiplied by a constant), which Bayesian optimization minimizes to determine where to sample next.
Fig 6.
(a) Bayesian ADC control diagram. The Bayesian ADC’s inner loop was composed of a phase/power based feedback stimulator. The outer Bayesian optimization loop was composed of a Gaussian process (GP), and acquisition function. The Gaussian process builds a model of how the stimulation parameters affect the feedback signal, and the acquisition function uses this information to select the next parameter set. (b) Overview of the Bayesian ADC’s cyclic operation. The Bayesian ADC sets the stimulator parameters and applies phase/power based stimulation to the BGTCS for 20s. It then estimates the effect of those parameters on beta power, and updates its GP with the new observation. Finally, it optimizes its acquisition function, and selects the next parameter set.
Fig 7.
Beta power as a function of stimulation parameters.
Feedback stimulator parameter sweep over stimulus phase trigger, power threshold, and amplitude. The sweep revealed a global minimum of -28.6 dB at 〈2.24 rad, 2.37 mA, -28.6 dB〉, denoted with dashed black lines. The sweep revealed a complex underlying landscape with flat regions (in response to power threshold), nonlinearities (in response to stimulation amplitude), and shallow local minima (high power thresholds). The red and yellow lines indicate the isoclines of the beta power with DBS OFF and cDBS, respectively.
Fig 8.
Bayesian ADC optimizing stimulus phase trigger.
Example 1D optimization of stimulus phase trigger. The simulation was run for 25 iterations in which Bayesian optimization was used to select the stimulus phase trigger while holding stimulus amplitude and power threshold constant (2.37 mA, -28.6 dB). (top) Gaussian process built from observations. (bottom) Power as a function of iteration, and minimum value found. The color of each dot represents the iteration at which each parameter setting was visited during the simulation.
Fig 9.
Minimum beta power found by each algorithm as a function of iteration.
BayesOpt (blue) is compared against the Nelder-Mead (orange) and DIRect (green) algorithms, with the shaded region indicating the standard deviation. Each algorithm was run 1000 times in all 7 parameter combinations, and compared for their ability to find the global minimum in as few function evaluations as possible. BayesOpt and DIRect perform comparably in all cases, while NM falls behind in cases where power threshold is optimized. The dotted lines represent the global minimum beta power, as well as the beta power with DBS OFF and cDBS for comparison.
Fig 10.
Histograms of the parameters selected by each algorithm in 1D.
Histograms of the parameters selected by each algorithm (rows 2-4) over 1000 trials of 100 function evaluations are show, as well as the underlying response surfaces (top row). Each row shows the sampling patterns of an algorithm as it attempted to minimize beta power in each of the 1D cases (columns). BayesOpt clustered most tightly on the optimum parameter values in all cases. The NM algorithm explored the space the least and was easily trapped in flat regions or in a local minimum. The DIRect algorithm continually explored the space, and never transitioned to exploitation.
Fig 11.
Mean average regret and noise tolerance.
(a) The mean of the average regret (RT/T) across 1000 trials for each algorithm in the 3D case. BayesOpt asymptotes to the lowest regret, while NM asymptotes fastest but to higher regret. (b) Asymptotic constant, α, under increasingly noisy conditions. As the SNR degrades, each algorithms’ asymptotic performance deteriorates. BayesOpt continues to outperform the other algorithms at moderate to high SNRs and performs similarly at poor SNRs. The horizontal dotted lines indicate the regret incurred with DBS OFF and cDBS, while the vertical line represents the baseline SNR.
Table 1.
Asymptote and time constant of each algorithm in the 3D case.