Closed-loop deep brain stimulation by pulsatile delayed feedback with increased gap between pulse phases

Computationally it was shown that desynchronizing delayed feedback stimulation methods are effective closed-loop techniques for the control of synchronization in ensembles of interacting oscillators. We here computationally design stimulation signals for electrical stimulation of neuronal tissue that preserve the desynchronizing delayed feedback characteristics and comply with mandatory charge deposit-related safety requirements. For this, the amplitude of the high-frequency (HF) train of biphasic charge-balanced pulses used by the standard HF deep brain stimulation (DBS) is modulated by the smooth feedback signals. In this way we combine the desynchronizing delayed feedback approach with the HF DBS technique. We show that such a pulsatile delayed feedback stimulation can effectively and robustly desynchronize a network of model neurons comprising subthalamic nucleus and globus pallidus external and suggest this approach for desynchronizing closed-loop DBS. Intriguingly, an interphase gap introduced between the recharging phases of the charge-balanced biphasic pulses can significantly improve the stimulation-induced desynchronization and reduce the amount of the administered stimulation. In view of the recent experimental and clinical studies indicating a superiority of the closed-loop DBS to open-loop HF DBS, our results may contribute to a further development of effective stimulation methods for the treatment of neurological disorders characterized by abnormal neuronal synchronization.


S2 Pulsatile NDF and LDF with increased stimulation intensity
We illustrate in more detail how the order parameter R of STN neurons stimulated by pulsatile NDF behaves when parameter of the stimulation intensity K increases. As has also been reported for other models 6-8 the order parameter decays ∼ K γ , as illustrated in Fig. S1. For the considered model containing N=200 STN neurons, a direct fit of the curves in Fig. S1A gives γ ≈ −0.3 for an intermediate range of K, see the straight dashed line in Fig. S1A. The situation can differ for large stimulation intensity, especially, for interphase gaps of a moderate width. We found that for such a range of parameters, the order parameter can decay somewhat faster with smaller exponent γ. For example, γ ≈ −0.41 for the gap width GW = 1 ms, or γ ≈ −0.51 for GW = 2 ms, as illustrated in Fig. S1B.
The same scaling also holds for the amplitude of the filtered LFP, as illustrated in Fig. S1C, which is used to construct the NDF stimulation signal S of the form (7). Since the LFP enters to the NDF stimulation signal in the power of 3, the amplitude of the stimulation signal S will increase (decrease) with increasing K if the exponent γ > −1/3 (γ < −1/3), whereas the order parameter R always decays [ Fig. S1]. Therefore, as the order parameter decreases for increasing K, the amount of the administered stimulation |S| grows for an intermediate range of R, whereas both R and |S| may decay for smaller R and for a moderate width of the interphase gap, as illustrated in Fig. 9B of the paper.
For the pulsatile LDF stimulation the situation is somewhat similar, where the decay rate of the order parameter may slightly increase for large stimulation intensity and for interphase gaps of a moderate width, see Fig. S1D. This however may not suffice for a consistent decay of the amount of the administered stimulation as reported in Fig. 9A of the paper.

S3 High-frequency deep brain stimulation
We stimulate the considered model by a high-frequency (HF) pulse train of the considered charge-balanced pulses with constant amplitude, which corresponds to a constant modulating signal S(t) = K in Fig. 3 and models the standard HF deep brain stimulation (DBS).  Examples of the time courses of the order parameter R(t) of the STN neurons stimulated by the HF DBS are illustrated in Fig. S2A for fixed stimulation intensity K = 2.1 and width of the interphase gap GW = 0 ms (red curve), 1 ms (green curve) and 5 ms (black curve). After the onset of the stimulation at t = 20 s with the considered stimulation intensity, the synchronized collective dynamics of the STN neurons is perturbed, and the order parameter fluctuates either at still relatively large values for zero gap or decays for non-zero gaps and fluctuates at smaller values than that of the initial, pre-stimulation synchronized regime. In the latter cases, this indicates a stimulation-induced desynchronization. Introducing the interphase gap of a finite width can thus improve the desynchronizing impact of HF DBS. In summary, in the considered model strong HF DBS can suppress the abnormal neuronal synchronization. Introducing an interphase gap in the stimulation biphasic charge-balanced pulses may improve the desynchronizing effect for intermediately strong stimulation. This may potentially allow to reduce the stimulation intensity of HF DBS without a significant worsening of its therapeutic effects. The extent of the stimulation-induced desynchronization may, however, strongly fluctuate as reflected by

S4 Slowly varying parameters
To further verify the robustness of the considered stimulation methods with respect to parameter variation, we simulate the considered model under continuous stimulation by pulsatile LDF or NDF and slowly vary the delay parameter τ. Since the optimal stimulation delay for desynchronization relates to the oscillation period of the mean field 6-10 , such varying τ can model a slow variation of the firing frequency of the stimulated neurons. In our simulations, for example, the stimulation delay changed by 2 ms every 100 s such that the initial conditions of the model for the next value of τ were the last state of the system for the previous value of τ. The results of such a continuation by parameter τ are illustrated in Fig. S3, where the time-averaged order parameter R of the STN neurons is depicted versus the stimulation delay τ for pulsatile LDF [Fig. S3A, B] and NDF [ Fig. S3C, D]. The variation of τ does not cause any problem with respect to the stimulation-induced desynchronization. Indeed, the pulsatile LDF stimulation with slowly varying delay [ Fig. S3A, B, red circles and blue squares] demonstrates the same desynchronizing effects as for the case where the stimulation is administered to an initially synchronized population 4/7 [ Fig. S3A, B, black solid curves]: The size and location of desynchronization regions are well preserved. For the pulsatile NDF stimulation the situation can even be improved, where the variation of parameters can significantly extend the desynchronization regions [ Fig. S3C, D, red circles and blue squares]. With such an approach of slowly varying stimulation parameters, the NDF stimulation can desynchronize the stimulated neurons for practically all values of the stimulation delay as has also been reported for other models [6][7][8] . For the model studied in the present paper, the considered slow variation of the delay parameter allows for the NDF stimulation to explore other coexisting stable desynchronized regimes to which the stimulated neuronal population can be shifted by the pulsatile NDF, in such a way extending the desynchronization parameter regions.

S5 Weakly and intermittently synchronized neurons
We also consider the case of weak coupling g G→S = 1.28 nS/µm 2 where the STN neurons exhibit a weak and intermittent synchronization, and the order parameter fluctuates around small values as illustrated in Fig. S4A. The local field potential (LFP) oscillates with the mean period T ≈ 100 ms and with small and varying amplitude as illustrated in Fig. S4B. We consider such a regime of weak and intermittent synchronization as an initial state for stimulation by pulsatile delayed feedback.
The effect of the stimulation by pulsatile LDF and NDF is illustrated in Figs. S4C and S4D, respectively, for two interphase gaps GW = 0 ms and 5 ms as indicated in the legend. Since neurons were initially only weakly synchronized, the desynchronizing stimulation only slightly changes the extent of synchronization, where the order parameter of the stimulated neurons still fluctuates at similarly small values as without stimulation. We observe that introducing an interphase gap can improve the desynchronizing effect of the stimulation also in the case of weakly and intermittently synchronized neurons.  Fig. S4A]. The observed differences of the order parameter are statistically significant as revealed by the Wilcoxon rank sum test applied to all pairs of the gap widths for LDF and NDF including the stimulation-free case. The maximal p-value for all pairs was obtained to be smaller than 0.001 even for time series of the order parameter downsampled to the sampling rate of 5 Hz.
To illustrate the desynchronizing impact of the pulsatile LDF and NDF stimulations administered to weakly coupled and weakly synchronized STN neurons, we scan the parameter space (τ, K) of the stimulation delay τ and the stimulation intensity K and depict the time-averaged order parameter R of the stimulated neurons in color in Figs. S5A and S5B for the pulsatile LDF and NDF stimulations, respectively. We found that the desynchronizing effect of both pulsatile LDF and NDF stimulations is robust with respect to the variation of the extent of synchronization and frequency in the neuronal population. Indeed, for the same stimulation parameters as for the initially strongly coupled and synchronized neurons, the pulsatile LDF stimulation does not cause any enhancement of synchronization in the ensembles of initially weakly coupled and weakly synchronized neurons, see Fig. S5A and compare with Fig. 6B in the paper. Also for the pulsatile NDF stimulation, the desynchronization regions obtained for initially strongly coupled and strongly synchronized neurons shown in Fig. 7B in the paper overlap well with the desynchronization regions for initially weakly coupled and weakly synchronized neurons, see Fig. S5B (blue domain).  Figure S5. Impact of the pulsatile LDF and NDF stimulations on the weakly coupled and weakly synchronized neuronal ensemble (1) - (3). The time-averaged order parameter R(t) of the stimulated STN neurons is depicted in color ranging from 0 (blue) to 1 (red) versus the feedback delay τ and the stimulation intensity K for a weakly coupled regime with g G→S = 1.28 nS/µm 2 for (A) pulsatile LDF and (B) pulsatile NDF. The width of the interphase gap GW = 5 ms.
Therefore, for an appropriate selection of the stimulation parameters causing a pronounced desynchronization of initially strongly synchronized neurons, the stimulation by pulsatile LDF and NDF preserves desynchronization when, e.g., due to variations of system parameters the neuronal population runs into a regime of weak or intermittent synchronization with a moderate variation of the firing frequency.