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Numerical optimization of coordinated reset stimulation for desynchronizing neuronal network dynamics

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Abstract

Excessive synchronization in neural activity is a hallmark of Parkinson’s disease (PD). A promising technique for treating PD is coordinated reset (CR) neuromodulation in which a neural population is desynchronized by the delivery of spatially-distributed current stimuli using multiple electrodes. In this study, we perform numerical optimization to find the energy-optimal current waveform for desynchronizing neuronal network with CR stimulation, by proposing and applying a new optimization method based on the direct search algorithm. In the proposed optimization method, the stimulating current is described as a Fourier series, and each Fourier coefficient as well as the stimulation period are directly optimized by evaluating the order parameter, which quantifies the synchrony level, from network simulation. This direct optimization scheme has an advantage that arbitrary changes in the dynamical properties of the network can be taken into account in the search process. By harnessing this advantage, we demonstrate the significant influence of externally applied oscillatory inputs and non-random network topology on the efficacy of CR modulation. Our results suggest that the effectiveness of brain stimulation for desynchronization may depend on various factors modulating the dynamics of the target network. We also discuss the possible relevance of the results to the efficacy of the stimulation in PD treatment.

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Acknowledgements

JR received support from US National Science Foundation awards DMS 1516288 and DMS 1612913.

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Authors and Affiliations

  1. Graduate School of Science and Engineering, Yamagata University, 4-3-16 Jonan, Yonezawa, Yamagata, 992-8510, Japan

    Shigeru Kubota

  2. Department of Mathematics, University of Pittsburgh, 301 Thackeray Hall, Pittsburgh, PA, 15260, USA

    Jonathan E. Rubin

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  1. Shigeru Kubota
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Correspondence to Shigeru Kubota.

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Action Editor: Charles Wilson

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Figure S1

Comparison of the energy efficiency of the asymmetric pulse input with that of the symmetric pulse and Fourier inputs. (a) An example of the current waveform of the asymmetric pulse, which is optimized with the weight parameter α = 3.1 nA−2·mS. Each current is delivered by the electrode represented using the same color in Fig. 1(a). (b) The red lines show the values of E (solid) and ρ (dashed) as function of α, obtained by using the optimal asymmetric pulse. The black lines show the case of using the optimal symmetric pulse (the same as shown in Fig. 3b), for comparison. (c) The red line shows the relationship between ρ and E obtained by applying the asymmetric pulse which is optimized with various α values. The black lines represent the cases with the optimal Fourier (solid) and symmetric pulse (dashed) (the same as shown in Fig. 3c), for comparison. (EPS 1589 kb)

High resolution image (PNG 63 kb)

Figure S2

The predicted effects of applying intermittent stimulus control in which the current stimulation is periodically switched on and off. (a and b) Examples of the time courses of order parameter r (top) and one of the four stimulating currents (\( {I}_{stim}^j \) in Eq. (4)) (bottom), when the intermittent control is applied to the Fourier (a) and symmetric pulse (b) inputs. The stimulus waveform is optimized in the absence of intermittent regulation with the weight parameter α = 3.1 nA−2·mS. The dashed vertical lines show the onset and end of the time period for activating stimulation. (c) The relationship between the average order parameter ρ and the energy consumption rate E obtained in the presence (red) and absence (black) of intermittent regulation for the inputs optimized with various α values. The intermittent control is applied to the Fourier (solid) and symmetric pulse (dashed) inputs, which are optimized without this control. In (a)-(c), the time durations of the active and inactive phases for the intermittent regulation are 1000 and 100 ms, respectively. (EPS 2038 kb)

High resolution image (PNG 65 kb)

Movie 1

Network activity in the absence of current stimulation, obtained with the same condition as in Fig. 4a. The colors of the circles and boxes represent the levels of the membrane potentials of neurons and the currents delivered by the electrodes, respectively. (MP4 2601 kb)

Movie 2

Network activity for the case of applying the optimal Fourier input with moderate intensity, obtained with the same condition as in Fig. 4b. (The meanings of the colors of the circles and boxes are the same as Movie 1.) (MP4 2608 kb)

Movie 3

Network activity for the case of applying the optimal Fourier input with strong intensity, obtained with the same condition as in Fig. 4c. (The meanings of the colors of the circles and boxes are the same as Movie 1.) (MP4 2597 kb)

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Kubota, S., Rubin, J.E. Numerical optimization of coordinated reset stimulation for desynchronizing neuronal network dynamics. J Comput Neurosci 45, 45–58 (2018). https://doi.org/10.1007/s10827-018-0690-z

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