Global and local synchrony.

In the section ‘The Kuramoto model’ we defined the order parameter r for a single population of phase oscillators with ρ as a measure of the phase synchrony. We will now generalise this to the case of N oscillators grouped into S populations (of variable size) with N_σ oscillators per population. For this system, the order parameter (1) can be rewritten using a double summation (11) with oscillator n of population σ being denoted by θ_σn. The factor of can be brought inside the first summation and rewritten as . Then, with (12) the order parameter for the system can be written as (13) Using the definition of the order parameter (1), Eq (13) can be written as (14) with (15) and (16) Eq (13) shows that the total order parameter r for the system can be written as a superposition of the local order parameters r_σ for each population. To clearly distinguish between the two, we will now refer to the total order parameter r as the global order parameter of the system and r_σ as the local order parameter for population σ. In each case, the polar representation gives an associated amplitude and phase. The polar representation of the local order parameter can be written in terms of a local amplitude ρ_σ and local phase ψ_σ (17) Similarly, we refer to ρ and ψ from (1) as the global amplitude and global phase. The global amplitude (as a measure of total synchrony) is particularly significant since it is correlated to symptom severity in the case of ET and PD. Our objective for stimulation is therefore to maximally reduce the global amplitude ρ.

In practice, the global signal may either be measured directly or constructed from LFP recordings. For ET, it is natural to assume that the tremor itself is a manifestation of the global signal. Hence the global signal can be obtained directly by measuring the tremor. The global amplitude and global phase is then taken to be the amplitude and phase of the tremor, respectively. This is of course an idealisation, with the alternative being to correlate pathological neural activity in the LFP with the symptom itself. The global signal would then be constructed using LFP recordings from multiple contacts. We discuss this in more detail in the section entitled ‘Obtaining local activities through electrode measurements’. This approach would be more appropriate in the case of PD, where motor impairment is a set of symptoms whose severity is correlated to LFP activity, particularly in the beta band [4].

We can also relate (14) to feedback signals we might measure by using (2) and taking the real part. Using this, we obtain (18) We refer to f(t) and {f_σ(t)} as the global and local signals (or population activities), respectively. Using (3), Eq (18) can also be written in terms of the global and local amplitudes and phases (19) Using (13), the Kuramoto Eq (4) can also be generalised to multiple populations (20) where is the now the stimulation intensity at a population σ. The coupling constant k in Eq (4) is now a S × S matrix with elements k_σσ′. The diagonal and off-diagonal elements, denoted by k_diag and k_offdiag, describe the intrapopulation and interpopulation coupling, respectively.

Multi-population response curves.

We now derive an expression describing the change in the global amplitude due to stimulation as a function of the local (population) amplitudes and phases. For now it is assumed that the local quantities (to base the stimulation on) can be measured. We will discuss how these quantities can be measured later. Using the polar form of the order parameter (1), Eq (14) can be written as a summation involving the amplitudes and phases of individual populations (21) Taking the time derivative of (21) leads to (22) which can be written in terms of the real and imaginary components (23) It can be seen that the time derivative of the amplitude is the real part of (23) (24) The quantities dρ_σ/dt and dψ_σ/dt of Eq (24) are the changes in the amplitude and phase of a population with respect to time. If we assume the distribution of phases within a population satisfies the ansatz of Ott and Antonsen [32], we can substitute Eqs (9) and (10) into (24) to obtain the amplitude response due to stimulation in terms of the Fourier coefficients of Z(θ) (25) Eq (25) contains an expansion over the harmonics of Z(θ). In our previous paper, we demonstrated that, for a biologically realistic uPRC, it is reasonable to neglect higher harmonic terms (m > 1) [11], leading to a simpler expression for the instantaneous amplitude response (26) Adaptive coordinated desynchronisation (ACD) is now a strategy that uses Eq (26) to determine when stimulation should be provided. Eq (26) shows the change in the global amplitude due to stimulation can be expressed as a sum of contributions from each population. Each term in the summation can be further split into three terms, the first of which depends only on the global phase with the second and third terms depending on both the global phase and the local quantities. We will refer to these terms as simply the global and local terms, respectively. Eq (26) tells us how the global amplitude (i.e. the symptom severity) should change when stimulation is provided at a particular global phase, local phase and local amplitude. Using this, we can construct a closed-loop strategy for DBS by stimulating only when dρ_stim/dt < 0, i.e. when stimulation is predicted to lead to a suppression in the global amplitude.

Eqs (25) and (26) both involve summations over populations, with each term being the product of a weight w_σ, a stimulation intensity and a local amplitude response, which we shall denote here by Γ_σ. The local amplitude response Γ_σ is the contribution of a single population to the global amplitude response. Using Eq (26), Γ_σ is given by (27) Plots for Γ_σ together with the corresponding uPRC for Z(θ) = a₀/2 − sin(θ), with a₀ = 0, 2 and 4 are shown in Fig 2A, 2B and 2C, respectively. Regions in blue are areas of amplitude suppression while orange regions predict amplification. In both cases, these regions can be seen to occur in bands. Graphically, the dependence of Γ_σ on the global and local phases can be inferred from the direction of the banding. A purely horizontal band implies the response is independent of the local phase. An example of this can be seen at low amplitudes in Fig 2A. Other plots show diagonal banding, which implies the response is dependent on both the global and local phases. This behaviour can be understood by considering the 3 terms of (27). At low amplitudes, the first term dominates, which is only dependent on the global phase. As the local amplitude increases, the second and third terms depending on local quantities become increasingly more important. For the cases where |a₀| is small, the effect is less apparent. The left panel of Fig 2A shows that stimulation can either increase or reduce the phase (i.e. an uPRC of type II), implying a relatively small |a₀|. For this case, the second term can be neglected, leading to a dominance of the first term at low amplitudes where only a small dependence on the local phase can be seen. Fig 2B and 2C shows that stimulation has the effect of only increasing the phase, which is indicative of Z(θ) with larger |a₀|. For these systems the response can be seen to depend more strongly on the local phase for all amplitudes.

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Fig 2. The local amplitude response Γ_σ for different ρ_σ and Z(θ).

The panels (a), (b) and (c) are for Z(θ) = a₀/2 − sin(θ), with a₀ = 0, 2 and 4, respectively. For each plot, the vertical axis is the global phase (ψ) and the horizontal axis is the local (or population) phase (ψ_σ). The corresponding uPRC Z(θ) is also shown, with zero indicated by a red dashed line. Blue regions indicate areas where stimulation is predicted to suppress amplitude.

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Modelling stimulation.

The goal of modelling closed-loop DBS due to spatially separated electrodes requires us to consider how electric fields behave within the brain, both due to stimulation and the LFP arising from neural activity. We will consider each of these separately, with the latter being presented in the section entitled ‘Obtaining local activities through electrode measurements’.

Detailed computational models of DBS typically consist of two components: the first is a volume conductor model of the electrodes and the second is a multi-compartmental model of neural populations [34, 35]. When the electrode geometry is considered explicitly, the resulting potential due to stimulation can be found by solving Laplace’s equation using finite element methods [34]. Alternatively, a point source approximation for the electrode may be used, leading to a simple analytical expression for the extracellular potential due to current at the electrode [35]. The effects of stimulation are then calculated using a multi-compartmental neuron, where the dendrites and axons are treated explicitly and then discretised into multiple segments. In this subsection, our aim is to connect these ideas with Eq (25) for the amplitude response. We use the following quantities in this analysis: positions p, voltages V and currents I. Each quantity is accented depending on whether it is associated with an electrode or population, which carry no accent and a hat (ˆ), respectively. A full description of our notation can be found in Table 1.

Firstly, and in contrast to the more detailed models, the response of an individual neural unit is not calculated using the multi-compartmental approach in our model, but instead given a simple analytical form using the uPRC Z(θ). Then, we expect that for a system of electrodes and neural populations, should depend on the stimulation provided by all the electrodes in the system in addition to the geometry of the electrode placement and properties of the brain tissue. These ideas can be incorporated into our model by simply interpreting the ‘stimulation intensity’ as the extracellular potential (or voltage) at a population due to stimulation. Since Eq (25) describes the response of neural populations, one assumption here is that this potential does not vary within each population, i.e. each unit of population σ experiences the same intensity of stimulation. This assumption becomes more valid for ‘small populations’, which can be effectively treated as point sources. We expect the small population assumption to be more valid for systems described by larger S. This can be explained by considering a simpler system consisting of 〈N_σ〉 units per population and S populations. The total number of units is then N = 〈N_σ〉S. For fixed N, increasing the number of populations must lead to a reduction in the number of units per population. Since we expect each unit to occupy a volume in space, this therefore leads to smaller populations. Therefore, the small population assumption should be more valid for systems described by larger S.

For a system of L electrodes and S populations, we now express the vector of population voltages (with elements ) in terms of currents at the electrode I (28) Both populations and electrodes are now treated as a point sources, with the latter found to be reasonable for modelling stimulation [36]. The currents I would be equivalent to the user-controllable stimulation intensities. The elements of matrix (of dimensions S × L) are coefficients which reflect the medium and geometry of the electrode-neuron system. For example, if we assume the brain tissue to be a homogeneous isotropic medium (one where the conductivity is independent of both position and direction), the coefficients would be (29) where κ_e is a constant. The positions in space of the electrodes and populations are given by p_l and , respectively. In summary, the quantities we consider in our model for stimulation are voltages at population σ due to stimulation which delivers current I_l(t) to electrode l. In the section entitled ‘Obtaining local activities through electrode measurements’ we will also consider a model for measurement, where we use voltages V_l(t) measured at electrode l due to the activity of population σ producing currents .

Optimal stimulation strategy.

We now seek an expression for how much current to deliver across each electrode on the basis of feedback signals. A more compact expression for the amplitude response (26) can be written using linear algebra notation, with Γ equal to the vector of local amplitude responses (27) and equal to the vector of voltages at a population (30) where the weights w_σ have now been absorbed into Fourier coefficients of Γ for simplicity. Inserting (28) into (30) leads to an expression for the amplitude response in terms of the currents at the electrodes, i.e. the control variables (31) Often, concern for tissue damage due to stimulation imposes a limit on how much current can be delivered to a single contact. To account for this, we can also impose a constraint on the current for each contact such that it does not exceed some maximum value I_max (32) For each time step, our objective is to deliver stimulation which maximally suppresses the global amplitude, i.e. to choose I, within the constraints (32), so as to make dρ_stim/dt as negative as possible. A simple optimal solution for achieving this can be obtained by setting the current for the lth contact to I_max if the lth component of is negative (33) It is worth noting that the strategy proposed here is a ‘greedy’ one, where optimality can be locally guaranteed (i.e. per time step), but not overall.

The utility of ACD for ET.

A number of experimental studies on ET patients have demonstrated the benefit of stimulating according to the phase of the tremor, which we will refer to here as phase-locked (PL) stimulation [17, 19]. Eq (26) predicts this benefit but also predicts this strategy would be suboptimal in general, since the complete amplitude response for a multi-population Kuramoto system requires more knowledge of the patient’s state, involving the local amplitudes and phases. However, under certain conditions, the local terms of Eq (26) may become negligible and hence the predicted amplitude response may approximate to a function of only the global phase. In this scenario, we expect the efficacy difference between ACD and PL stimulation to be negligible. We refer to this efficacy difference here as the utility. To analyse the utility of ACD over PL stimulation we therefore have to understand the significance of the local terms relative to the global terms in (26). It is clear that this significance is dependent both on the parameters of the system and the time dependent state.

Eq (26) contains two local terms involving powers of ρ_σ. Since ρ_σ ≤ 1, we expect that the term involving will generally be small relative to the other terms. We may then neglect the local term involving , leading to (34) We therefore expect the utility of ACD to be almost entirely dependent on the local term of (34), namely (35) Hence, we expect the utility of ACD to increase as (35) becomes more significant relative to the other terms in (34). This is likely to occur when the zeroth harmonic of the uPRC is large, implying a type I uPRC. We would expect lower utility for systems where is negligible, i.e. for type II uPRCs. In addition to this, the dependence on sin(ψ_σ − ψ) implies that stimulating on the basis of local quantities would only have an effect if the phases of individual populations differ sufficiently from the mean phase. One situation in which such phase difference may be particularly high are for clustered configurations of oscillators. Examples of different configurations of oscillators are shown in Fig 3.

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Fig 3. Different configurations of oscillators color coded according to population.

Panels (a) and (b) are for a unimodal distribution and multimodal (clustered) distribution, respectively. Configurations were obtained by simulating the multi-population Kuramoto Eq (20).

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We now investigate which spatial configurations of electrodes and populations should make the effects of stimulation depend only on the global phase. Central to this investigation is an identity which can be obtained by multiplying both sides of Eq (21) by e^−iψ and taking the imaginary part, leading to (36) The similarity between this expression and Eq (35) allows us to use (36) to understand the conditions for (35) to be small/negligible. Eq (35) allows for the uPRC type to differ between populations, since the zeroth harmonic is also indexed by σ. If we consider the less general case of homogeneous uPRC type, i.e. for all σ, then a₀ can be removed from the summation, leading to (37) For such systems, we can see that if the stimulation intensity at a population is identical across populations, i.e. , then (38) which equals zero due to (36). Eq (28) relates the current at a particular contact to the stimulation intensity at a population through a matrix which reflects the geometric and electrical properties of the electrode-neuron system. Using (28) for and (37) we can obtain an expression in terms of the current at a contact (39) Eq (39) indicates that, for homogeneous uPRC types, the utility of ACD is dependent on the geometric and electrical properties of the electrode-neuron system. If we consider the case where, for a given electrode, does not vary across populations, then this leads to (40) which also equals zero due to (36). For a system where the matrix elements are given by (29), this condition would be equivalent to one where, for each electrode, the distances between all the populations and the electrode are equal.

In this section we have investigated the utility of ACD with respect to PL stimulation for ET. In summary, we expect the utility to depend on a combination of geometric, uPRC and state related factors. Firstly, we would expect only a small utility for homogeneous isotropic systems where for each electrode the distances between all the populations and the electrode are equal. Secondly, we expect the utility to strongly depend on the uPRC type. For heterogeneous type I systems, we expect the most utility. For type II systems, we expect marginal utility originating from the local term depending on . Finally, we expect greater utility when ACD is applied to more clustered states of oscillators.

Simulated systems.

We now test the ACD method and compare its efficacy with CR and PL stimulation. To perform this testing, we simulate the multi population Kuramoto model and use (19) to produce oscillations that are similar to those found in tremor from ET patients. These equations depend on a number of parameters, which together define a particular system in terms of its dynamics, electrode-population configuration and intrinsic response to stimulation Z(θ). In this subsection, we will describe how systems are generated for our testing.

The dynamics of the system are determined by the parameters of the multi population Kuramoto model, with additional stimulation and noise terms (41) where is a constant reflecting the amplitude of noise and W is a Wiener process. For simplicity, we consider a system of S = 3 populations throughout with . In this form, the uPRC type is determined by a single parameter . It is worth emphasising that the results we later present would be the same for any uPRC of the form , where ϕ is some arbitrary phase shift. The analytical expressions for the response curves are for an infinite system, so we require that N_σ is large. For each population, we choose the number of oscillators N_σ = 600 to satisfy this and to remain computationally feasible.

In order for (41) to produce tremor frequency oscillations, we need to choose an appropriate distribution for the set of natural frequencies {ω_σn}. We obtain this by fitting to tremor data from the study of Cagnan et al [17]. In this study, phase-locked DBS was delivered according to tremor in ET patients. This tremor was measured using an accelerometer attached to the patient’s hand. Data was collected from 6 ET patients and 3 dystonic tremor patients. All patients gave their informed consent to take part in the study, which was approved by the local ethics committee in accordance with the Declaration of Helsinki. The data from this study can be obtained through an online repository [37]. The tremor data was filtered using a non-causal Butterworth filter of order 2 with cut-off frequencies at ±2 Hz around the tremor frequency. Stimulation was delivered over a set of trials (typically 9), with each trial consisting of 12 blocks of 5 second phase-locked stimulation at a randomly chosen phase from a set of 12. Each block of phase-locked stimulation was also separated by a 1 second interblock of no stimulation [11]. We assume a Lorentzian form (7) for the distribution of {ω_σn} and find the parameters ω₀ and γ by fitting to the power spectrum of a single block from Patient 5 of this study. The coefficient of determination for this was found to be R² = 0.95, indicating an excellent fit to the data. Using this Lorentzian for the distribution {ω_σn}, the multi population Kuramoto model can generate tremor-like output with varying characteristics according to the choice of k_diag and . Fig 4 shows that, for each value of the noise parameter we consider in our testing, we can find a value of k_diag which reproduces the power spectrum of Patient 5. The R² for these fits were found to be 0.90 and 0.89 for and , respectively. Our goal is to test our methods on a variety of systems so we subsequently test across a range of k_diag.

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Fig 4. Power spectra of output from the multi population Kuramoto model for different values of .

The parameters of the model (41) were chosen to reproduce tremor data from the study of Cagnan et al [17]. In each case, the value of k_diag was chosen to give the best fit to the power spectrum of a single block from Patient 5’s data.

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The response of the system to stimulation (31) also depends on the parameters , which depend on the geometry of the electrode placement. We assume a homogeneous isotropic system throughout our testing, where the elements of are given according to the population and electrode positions (42) where we have omitted the constant κ_e for simplicity. Generating relevant configurations also requires an appropriate choice of L, which we choose according to a sensible estimate (L = 3) of how many contacts can be implanted in the VIM with present generation DBS leads [38–40]. To generate a particular electrode-population configuration, we first approximate the shape of the VIM to be a sphere of unit radius. We then place the coordinates of each electrode to lie on a line across the diameter of this sphere, thus simulating a collinear configuration of contacts commonly found on DBS leads. This VIM-electrode geometry is kept fixed throughout our testing. We place the coordinates of each population according to a ‘configuration parameter’ η, which we define in the following way. For each electrode indexed by l, we first define η_l as the smallest distance between the electrode and a population divided by the average distance between electrode l and a population (43) The parameter η_l takes values between 0 and 1 and describes how localised the effects of stimulation are from a single electrode. To obtain the configuration parameter for the system, we average over electrodes (44) A small η means the effects of stimulation are localised to a particular population, as shown in Fig 5A. When η approaches one, the smallest distance between an electrode and a population becomes equal to the average, meaning the effects of stimulation become delocalised. This is shown in Fig 5B and 5C.

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Fig 5. Visualisations of 3 electrode 3 population systems for different values of the configuration parameter η.

The 3 electrodes lie on a straight line across a sphere of unit diameter. Panel (a) shows a configuration for η = 0.1, where each population is placed very close to an electrode. Panels (b) and (c) show systems for η = 0.4 and η = 0.8, respectively, where the populations are more dispersed relative to the electrodes. In this scenario, stimulation from electrodes may affect multiple populations.

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Due to (28), the stimulation intensity is dependent on both the geometry of the system and the electrode current I_l. This can lead to large values for if, for example, the separation between populations and electrodes becomes small. To avoid this, we can define the stimulation intensity in terms of the maximum perturbation to a single oscillator Δθ_max. Using Eq (41), the change in θ_σn due to stimulation over a time step in simulation Δt can be expressed as (45) Inserting (28) into (45) leads to (46) If we assume a system where the uPRC are the same across populations and normalised to 1, then Δθ_max can be expressed in terms of the maximum current at an electrode (47) The S × S coupling constant matrix can be simplified by focussing only on the diagonal and off-diagonal components, which we denote by k_diag and k_offdiag, respectively. We fix k_offdiag = 0, so that k_diag can be used to control the level of clustering for a particular configuration of oscillators, with increasing k_diag leading to increasingly multi-modal distributions of oscillators.

Running the simulation.

Our aim is to compare the efficacies of CR, PL and ACD for a variety of test systems. We define a system according to a set of parameters , which can be used to change both the dynamics of the system and its response to stimulation. The high dimensionality of this space prevents a full exploration so we instead investigate the efficacy of each method as a function of a single parameter, leaving the remaining parameters fixed. In our testing we define the efficacy of a particular DBS strategy to be its desynchronising effect on a system of coupled oscillators. We measure this by averaging ρ for a period of time after stimulation is applied and then averaging this over n_trials trials to obtain . For each trial, we randomly generate a system with a given value of η and then integrate (41) using Euler’s method with a time step of Δt = 0.0025 for a total of 15 seconds. After 5 seconds, we turn on the stimulation and then after 10 seconds we begin averaging ρ for the remaining 5 seconds. A summary of the parameters common to all our simulations, unless stated otherwise, are provided in Table 2. The error bars in subsequent plots are the standard errors when averaged over n_trials trials.

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Table 2. Summary of parameters common to all simulations, unless stated otherwise.

https://doi.org/10.1371/journal.pcbi.1009281.t002

We use the time-shifted variant of CR neuromodulation [8, 41] in our testing where stimulation is delivered in bursts of HF pulse trains. The stimulation pattern is time-shifted across each electrode indexed by l by (48) where is the mean of the natural frequencies, which we take to be ω₀. The number of bursts per second, the burst frequency f_burst, was chosen to be equal to and the HF pulse train frequency f_train was chosen to be 130 Hz. The width of each burst t_burst was chosen to be 0.1 seconds. We implement PL stimulation by neglecting the local terms of Eq (26), leading to (49) Stimulation is then provided when , with the resulting strategy depending only on the global phase ψ.

Example output showing the desynchronising effects of ACD and CR are shown in Fig 6 for a system with parameters (. Here, the system is simulated for a longer period of T = 40 seconds and stimulation is turned on at T = 15 seconds. Fig 6A and 6B shows output from ACD while Fig 6C and 6D shows output from CR. To demonstrate the effect of CR we used a higher stimulation intensity of Δθ_max = 0.01π compared to Δθ_max = 0.001π for ACD. Comparing the outputs from ACD and CR, it is clear that the stimulation pattern from ACD is significantly different from that produced by CR, with the latter pattern being simply time-shifted across electrodes. The stimulation pattern from ACD allows for the possibility that multiple electrodes may be stimulated simultaneously. Fig 6 also shows that ACD achieves a similar desynchronising effect to CR whilst using a much lower stimulation intensity.

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Fig 6. Output from numerical simulations showing the effects of adaptive coordinated desynchronisation (ACD) and coordinated reset (CR).

The top panels (a)-(b) show the model output for ACD and the bottom panels (c)-(d) show the model output for CR. The left column shows model output together with the stimulation (averaged across contacts) as a function of time. The right column shows model output and stimulation for each contact for the respective shaded segment of the left column. The shaded portion of the stimulation for the left column is simply the average across contacts of the stimulation shown in the right column.

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Efficacy and efficiency vs stimulation intensity.

Figs 7 and 8 show the efficacy and efficiency of each stimulation strategy as a function of the stimulation intensity Δθ_max, respectively. The uPRC was assumed to be homogeneous across populations, i.e. Z_σ(θ) = a₀/2 − sin(θ). Each sub plot shows a set of simulations performed with a particular zeroth harmonic of the uPRC a₀. We fixed η = 0.1, and k_diag = 55 in all cases and then performed our investigations using 3 values of a₀ (a₀ = 0, a₀ = 2 and a₀ = 4).

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Fig 7. The average amplitude of simulated data as a function of the stimulation intensity Δθ_max.

The strategies tested were: no stimulation (no stim), adaptive coordinated desynchronisation (ACD), phase-locked (PL) and coordinated reset (CR). The average amplitude is a measurement of efficacy, where lower amplitudes indicate higher efficacy. The maximum stimulation frequencies used for ACD and PL are also given in the legend. Each sub plot shows a set of simulations performed with a particular zeroth harmonic of the uPRC a₀.

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Fig 8. The total energy used by different stimulation strategies, when applied to a simulated system, as a function of the stimulation intensity Δθ_max.

The strategies tested were: adaptive coordinated desynchronisation (ACD), phase-locked (PL) and coordinated reset (CR). The maximum stimulation frequencies used for ACD and PL are also given in the legend. Each sub plot shows a set of simulations performed with a particular zeroth harmonic of the uPRC a₀.

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Fig 7A shows results for a₀ = 0, corresponding to a type II uPRC. For PL and ACD, a regime exists where the efficacy of each strategy is seen to increase with increasing Δθ_max. The utility of ACD is found to be marginally higher than PL, with the exception being at low intensities (approximately Δθ_max ≤ 0.0005π), where both efficacies are approximately equal. The efficacy of CR is shown to increase across the range of Δθ_max tested, leading to a reduced utility of ACD over this method with increasing Δθ_max. Increasing a₀ leads to a reduced efficacy for CR, which is clearly shown in Fig 7B and 7C. The increased utility of ACD with increasing a₀, as predicted in the section ‘The utility of ACD for ET’, can also be seen, although this utility is also seen to diminish with increasing Δθ_max.

The total amount of stimulation delivered by each strategy (the sum of all stimulation pulses) can be likened to the ‘total energy’ used by a strategy. We denote this quantity by . It is expected that for open loop methods, such as CR, should increase linearly with Δθ_max. However, for closed-loop methods, where the stimulation pattern is dependent on the signal itself, a more complex relationship is likely to exist. Plots for as a function of Δθ_max are shown in Fig 8. A linear relationship for is found for the case of CR (as expected). In this case of PL stimulation, is shown to be approximately linear across Δθ_max. For ACD, a linear regime is observed for small Δθ_max, where for ACD and PL stimulation are approximately equal. For a₀ = 0 and a₀ = 2, then begins to plateau. In these cases, for ACD is predicted to be lower than both PL and CR for larger Δθ_max. We find a qualitatively different relationship for a₀ = 4, where for ACD is found to be approximately linear across the range. For this case, for ACD is predicted to be higher than both PL and CR at larger Δθ_max.

In subsequent testing we ensure that, for a given trial, the amount of stimulation delivered by a particular strategy is approximately equal. To achieve this, we restrict our testing to small Δθ_max, where is approximately equal for both ACD and PL. Smaller Δθ_max is also more relevant from a practical standpoint, since lower stimulation intensities are better for power consumption and reducing the side effects of DBS. We also apply a correction to Δθ_max for CR, using the relationships found in Fig 8. This amounts to using a Δθ_max for CR of ≃ 1.3 times that of ACD.

Efficacy vs coupling.

Fig 9 shows the efficacy of various stimulation strategies as a function of k_diag. The uPRC was assumed to be homogeneous across populations, i.e. Z_σ(θ) = a₀/2 − sin(θ). Each sub plot shows a set of simulations performed with a particular zeroth harmonic of the uPRC a₀ and value of noise . We fixed η = 0.1 and Δθ_max = 0.001π in all cases. We performed our investigations using 2 values of ( and ) and 3 values of a₀ (a₀ = 0, a₀ = 2 and a₀ = 4). ACD was tested at maximum frequencies of 130 Hz and 50 Hz.

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Fig 9. The average amplitude of simulated data as a function of the coupling constant k_diag.

The strategies tested were: no stimulation (no stim), adaptive coordinated desynchronisation (ACD), phase-locked (PL) and coordinated reset (CR). The average amplitude is a measurement of efficacy, where lower amplitudes indicate higher efficacy. The maximum stimulation frequencies used for ACD and PL are also given in the legend. Solid lines are for the ACD method. Each sub plot shows a set of simulations performed with a particular zeroth harmonic of the uPRC a₀ and value of noise .

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Significant improvements with ACD at 130 Hz over PL and CR can be observed when a₀ = 2 and a₀ = 4 (i.e. type I uPRCs) for both noise parameters tested. The utility of ACD over other methods is also shown to be greatest when k_diag is larger, which corresponds to larger local amplitudes ρ_σ and increased clustering. The increased utility of ACD with increasing a₀ and clustering agree with our predictions in the section ‘The utility of ACD for ET’. For systems where a₀ = 0, ACD is found to be marginally more effective than PL, as predicted. ACD at 50 Hz is generally found to have similar efficacy to PL at 130 Hz but with stimulation delivered at half the frequency, translating to considerably less energy consumption.

Efficacy vs configuration parameter.

We now investigate the efficacy of each stimulation strategy for different values of the configuration parameter η. In the section ‘The utility of ACD for ET’ we predicted that the efficacy of ACD should depend explicitly on the geometry of the electrode-population configuration and that the utility of ACD relative to PL stimulation should diminish as η increases. We also predicted that this effect could be mitigated if the uPRCs for the system of populations were heterogeneous in type, i.e. by using a form .

To test these predictions, we investigated the efficacy of each stimulation strategy as a function of η by simulating (41) with k_diag = 55, and Δθ_max = 0.0014π. To investigate the effects of heterogeneous we generate each system of 3 populations with sampled from a normal distribution with mean 2 and standard deviation s_a. We performed our investigations at 3 values of s_a (s_a = 0, s_a = 0.5 and s_a = 1), corresponding to the homogeneous case and then increasing heterogeneity, respectively. For each η, we randomly generate a set of 480 systems, calculate the efficacy of each strategy and then average across all systems. The results from these simulations are shown in Fig 10, with each sub plot showing a set of simulations performed with a particular s_a. Fig 10 shows that the utility of ACD relative to PL stimulation reduces with increasing η, but that this effect is reduced as the level of heterogeneity in the uPRC type increases. This is in agreement with our predictions from the section ‘The utility of ACD for ET’.

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Fig 10. The average amplitude of simulated data as a function of the configuration parameter η.

The strategies tested were: no stimulation (no stim), adaptive coordinated desynchronisation (ACD), phase-locked (PL) and coordinated reset (CR). The average amplitude is a measurement of efficacy, where lower amplitudes indicate higher efficacy. The maximum stimulation frequencies used for ACD and PL are also given in the legend. Each sub plot shows a set of simulations performed with a particular value for the standard deviation of the zeroth harmonic of the uPRC s_a.

https://doi.org/10.1371/journal.pcbi.1009281.g010