6th Annual International IEEE EMBS Conference on Neural Engineering San Diego, California, 6 - 8 November, 2013

Simulation of PID Control Schemes for Closed-Loop Deep Brain Stimulation

Eleanor M. Dunn, Student Member, IEEE and Madeleine M. Lowery, Member, IEEE

ABSTRACT— Recent studies have shown that specific motor symptoms of Parkinson's disease (PD) are correlated with oscillatory beta-band (12-30 Hz) activity in the basal ganglia. Deep brain stimulation (DBS) is an established therapy for PD, but is currently applied in an open-loop configuration. A computational model of the cortico-basal ganglia network was developed to systematically test the performance of five separate linear control schemes for closed-loop DBS. Each controller modulated the amplitude of DBS using the oscillatory activity in the basal ganglia as a biomarker. All controllers yielded a reduction in current and demonstrated a response to the suppression and resurgence of oscillations within the network. The model developed here can be further used to design and test more complex non-linear control schemes.

I. INTRODUCTION

Parkinson's disease (PD) is a progressive neurodegenerative disorder characterized by four primary motor symptoms: resting tremor, rigidity, bradykinesia, and postural instability. Over the past decade, deep brain stimulation (DBS) has become established as a standard surgical therapy for the treatment of medically refractory P D and is being further investigated as a potential treatment for various other neurological disorders. The symptoms of P D may be effectively managed using D B S ; however the mechanisms of action remain a topic of much debate. Recent studies have suggested that D B S disrupts pathological neural synchrony which results in increased oscillatory activity throughout the cortico-basal ganglia network.

Several clinical investigations have linked the power of oscillations in the beta (12-30 Hz) frequency range of the local field potential (LFP) detected using D B S electrodes implanted in the cortico-basal ganglia network to levels of bradykinesia and rigidity in P D [1]. Beta-band power in the subthalamic nucleus (STN) is attenuated during and following S T N - D B S [2], which suggests that the therapeutic benefit of D B S may in part be due to the suppression of pathological beta-band activity. Given this behavior, the level of oscillatory activity in the tremor and beta range has been proposed as a biomarker for the treatment efficacy of D B S therapy [3].

The realization of a biomarker for P D symptoms could enable the development of closed-loop D B S therapy. Closed-loop D B S not only allows for the real time

Research supported by the Higher Education Authority Program for Research in Third Level Institutions (PRTLI) in Ireland.

E. M. Dunn and M. M. Lowery are with the School of Electrical, Electronic & Communications Engineering, University College Dublin, Dublin 4, Ireland. (e-mail: eleanor.dunn@ucdconnect.ie).

optimization of the delivered stimulus, it has the potential to decrease side effects, reduce the power consumption, and take into account fluctuations in symptom severity as well as disease progression [4].

To successfully implement closed-loop D B S , appropriate control schemes must first be identified. Given that the design of suitable control schemes has received limited attention, a simple linear controller may be a logical place to start. There is no shortage of the use of proportional-integral-derivative (PID) controllers in the field of engineering, but more recently they have been gaining notability in biomedical applications [5]. Previous simulation studies have shown that modulation of the D B S amplitude in parkinsonian S T N populations, through closed-loop control, is capable of restoring the L F P power spectrum to approximately physiological conditions. Although only integral control was investigated in the study, the results demonstrate that a simple control scheme could be used to effectively modulate the D B S amplitude [6].

P I D controllers present a potential means of implementing closed-loop D B S . Testing of prospective control schemes in vivo or in vitro is difficult and undesirable. Computational modeling thus provides a means with which to design and tune various types of controllers. The aim of this study was to investigate and quantify the performance of P I D , proportional-integral (PI), proportional (P), integral (I) and integral-derivative (ID) controllers and to evaluate their efficacy and suitability for modulation of D B S parameters using a model of oscillatory neural activity within the cortico-basal ganglia network.

I I . METHODS

A physiologically based model of the cortico-basal ganglia network, incorporating D B S modulated by a selection of P I D controllers, was implemented in order to investigate closed-loop control of D B S . The structure of the network model representing the closed cortico-basal ganglia-thalamo-cortical loop is presented in Fig. 1 and [7]. The major model components include single compartment, conductance based biophysical models of the S T N , globus pallidus external (GPe), globus pallidus internal (GPi), thalamus, and cortical neurons, each of which have been validated and employed in previous modeling studies [8], [9], [10], [11]. Each component is described in greater detail below.

One hundred and fifty cells consisting of thirty S T N , GPe, GPi, thalamic, and cortical neurons were connected by excitatory or inhibitory synapses, glutamergic and G A B A , respectively, as described below [12], [13]. The S T N neurons received direct excitatory input from the cortex via the hyperdirect pathway and inhibitory input from the GPe.

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Each STN neuron in the model received excitatory input from three cortical neurons and inhibitory input from one GPe neuron. Each GPe neuron received inhibitory input

1 ,- 1

l>[ GPe !_-' 1 ThalamUS 1

J s™ ux « i Figure 1. Schematic diagram of the cortico-basal ganglia model. Excitatory connections are indicated with a solid line and inhibitory connections are shown with a dashed line.

from two other GPe neurons, in addition to excitatory input from a single STN neuron. Each GPi neuron received excitatory input from a single STN neuron and inhibitory input from a single GPe neuron. Each thalamic neuron received inhibitory input from a single GPi neuron. Cortical neurons received excitatory input from a single thalamic neuron. To simulate dopamine depletion under parkinsonian conditions, synaptic gains within the network were increased to incorporate the finding that dopamine has an overall damping effect and that depletion results in excitability and synchronization within the network [14].

A. Subthalamic nucleus neurons The STN model incorporates a physiological

representation of STN neurons developed by Otsuka et al. [8]. The membrane potential of an STN neuron is described by

dVrn dt -IK IK I A h h

' — Y lk l t-tk Isyn

I Ca-K (1)

where Cm is the membrane capacitance, V is membrane potential, INa is Na+ current, IK is the Kv3-type K* current, IA is the voltage dependent A-type K* current, IL is an L-type long lasting Ca2+ current, IT is a low threshold Ca2+ current, Ica-K is a Ca2+ activated K* current, and 7/ is a leak current. Further details regarding the parameter values used can be found in [8].

Individual synaptic currents, Isy„, were determined by the equation

Rk * (Vm ,) (2)

where 7^yn is the k'h synaptic current, Rk represents the kinetics of the onset and decay of current following a presynaptic spike for synapse k and Erev is the reversal potential for the appropriate type of synapse. Further details regarding the parameter values used in the synaptic models can be found in [12] and [13].

B. Globus pallidus neurons The models used to simulate GPe and GPi neurons are

based on those presented by Rubin and Terman in [9] and [10]. The membrane potential of a GPe neuron is described by

dVrn dt -I, - h . ' INO IT - I Ca I AHP (3)

where Cm is the membrane capacitance, Ii is the leak current, IK is the K+ current, INa is Na+ current, IT is a low-threshold T-type Ca2+ current, ICa is a high-threshold Ca2+ current, and IAHP is a voltage-independent "afterhyperpolarization" /^current. GPi neurons were modeled similarly to the GPe neurons. Synaptic potentials were modeled as expressed by (2). Details regarding the parameters used for both the GPe and GPi models are as in [9] and [10].

C. Thalamic neurons The model used to simulate thalamic neurons is based on

the model presented in Rubin and Terman in [10]. The membrane potential of a thalamic neuron is described by

dVm — —1—1 — l — l — Y l k : d t — 'l 'Na 'K 'T Z J / C ' S J syn

(4)

where Cm is the membrane capacitance, 7/ is the leak current, INa is Na+ current, IK is the K+ current, and IT is a low-threshold Ca2+ current. Synaptic potentials were modeled as expressed by (2). Further details regarding the parameters used can be found in [10].

D. Cortical neurons The model used to simulate cortical neurons is based on

the model developed by Pospischill et al. in [11]. The membrane potential of a cortical neuron is described by

dVrn ■ dt 11 IN a IK IM l^khyn (5)

where Cm is the membrane capacitance, Ii is the leak current, INa is Na+ current, IK is the K+ current, and IM is a slow, voltage-dependent K+ current. Synaptic potentials were modeled as expressed by (2). Further details regarding the parameters used can be found in [11].

E. Biomarker estimation The biomarker used by the proposed control schemes is

based on the hypothesis that beta-band oscillatory activity in the basal ganglia may be a suitable candidate to use for closed-loop control of DBS [3], [15]. The average rectified value (ARV) of the membrane potential of the STN neuron population was calculated using a moving average window to provide an indicator of the level of oscillatory activity within the STN. The average membrane potential was band-pass filtered from 12 to 30 Hz using a Butterworth filter, full wave rectified, and then averaged by low-pass filtering at 2 Hz. The governing equation for ARV of the membrane potential, b(t), is

b(t)= ftj(t) * \i(t) * hb(t)\ (6)

k lsyn

where hb and hi describe the impulse responses of the band­pass and low-pass, respectively.

F. Controller In order to implement closed-loop control of the DBS

waveform, a PID controller was designed to modulate the stimulus amplitude. The reference signal for the controller was calculated by estimating the ARV of the average STN membrane potential prior to the application of DBS, as described in (6). The target value for the controller, br, was set to 30% of the magnitude of the reference signal as it allows for a substantial, but not total, reduction in beta-band activity. A healthy level of beta-band activity may be

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desirable in clinical practice [16]. The error signal, eft), is defined as

e(t) = b(t) - br (7) where b(t) is is the biomarker reflecting the magnitude of beta-band oscillations. The governing equation of the controller is

a(t) - fep e(t) + ktfe(t)dt + kd—e(t) dt

(8)

where kp, k, and kd are the proportional, integral, and derivative gains, respectively. The gain values were initially set to one to test the PID control and then subsequently set to zero as necessary to examine PI, P, I, and ID control (eg. kd set to 0 for PI control/ The amplitude of the DBS waveform, amp(t), is calculated as

amp(t) — amp(t — 1) + step * a(t) (9)

where step is a user-defined step-size. The frequency of the DBS waveform was set to 136 Hz.

G. Simulation details

The model and controller were implemented using NEURON v7.3 with Python as an alternative interpreter [17]. All simulations were run for 80 seconds using a 0.01 millisecond time step and step-size set to 50 nA/cm2 for (9). Simulations were initially executed for two seconds to allow network activity to reach a steady state before calculating the reference biomarker signal. Two iterations were run for each of the five types of controllers: one with the DBS amplitude updated every 0.5 seconds and one with the DBS amplitude updated every 1.5 seconds.

III. RESULTS

A. Generation of pathological oscillations With the simulated dopamine depletion conditions that

were implemented in the model, both the STN and the GPe exhibited burst firing within the beta-band range at approximately 21 Hz. The simulated firing patterns of the STN neurons were similar to experimental results reported from rats under parkinsonian conditions [8]. The power spectrum of the averaged STN membrane potential is presented in Fig. 2 along with pulse trains illustrating the firing activity from a representative STN neuron.

5

< ■

3 ■

I -

■

1

^ I f e «

III III III

> nd

Figure 2. The power spectrum of a representative STN neuron exhibiting a peak frequency at 21 Hz. A pulse train depicting the firing times and bursting activity of a representative STN neuron is shown inset.

B. Application of DBS The modulation of DBS amplitude obtained using the five

methods of control examined, based on the error between the target beta-band oscillation amplitude and the biomarker of beta-band activity, is shown in Fig. 3. The amplitude response of the closed-loop DBS controller when updated every 0.5 seconds is shown in (a) and the response when updated every 1.5 seconds is shown in (b). The pure proportional controller yielded the slowest response rate in both simulations, while the PI and PID controllers were found to have the fastest response rate in both scenarios.

(a)„

3D 40 50

Time (si

(b)

30 40 50

Time(s)

Figure 3. DBS amplitude responses for each of the five controllers from the simulations with DBS getting updated every (a) 0.5 seconds and (b) 1.5 seconds.

The current consumption for each controller during both simulations is given in Table 1. The PI controller had the lowest overall current consumption during the 0.5 second updating simulation, while the PID controller had the lowest consumption during the 1.5 second updating simulation. Updating the DBS amplitude more frequently, however, required less stimulation current, regardless of the controller type.

TABLE I. CURRENT CONSUMPTION DURING THE 80 SECOND STIMULATION

Controller Type " " U p d a t e 0.5 s Update (nA/cm')

PID

PI

P

ID

I

686.8

694.5

1253.3

981.5

987.92

446.2

408.3

842.2

453.4

534.9

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IV. DISCUSSION

The aim of this study was to use a computational model to investigate the performance of several types of linear controllers for implementation of closed-loop D B S . The controllers, which modulated the amplitude of a high-frequency D B S waveform applied to the STN, used a biomarker that was representative of the beta-band oscillatory activity in the STN.

The dopamine depleted network was allowed to reach steady-state before D B S was applied, in which it exhibited beta-band oscillations at approximately 21 Hz, with parkinsonian bursting evident in the STN. The power spectrum of the averaged membrane potential from the S T N exhibited a peak in the beta-band, indicating increased synchronization within the network.

Each controller that was tested during the simulations was able to track changes in the oscillatory activity of the S T N using the calculated biomarker. The proportional controller resulted in the slowest rate of change of the stimulator amplitude, while the PI and P I D controllers tracked the changes at a faster rate. When there were sudden changes in oscillatory activity in the STN, however, the P I D controller exhibited a much more serrated response compared to the PI controller. The lack of derivative action in the P I controller may make the system steadier in the presence of noisy data, which may be advantageous in real applications in vivo.

The frequency of amplitude updates also impacts the rate of change of each controller. All five controllers were able to reach the 'off' state, corresponding to zero stimulation amplitude, within 80 seconds using the controller that was updated every 0.5 seconds, whereas only the PI and P I D controllers reached the 'off state when the amplitude was updated every 1.5 seconds. Although there was an overall lower current consumption when the amplitude was changed every 0.5 seconds, it may not be desirable from a clinical standpoint to update the parameters more frequently as there may be side effects associated with altering the stimulation amplitude back and forth.

As previously stated, the oscillatory activity in the S T N prior to D B S was used as a biomarker to modulate the amplitude of the stimulation current. In practice, however, the stimulation current is both applied and the neural activity recorded extracellularly. Due to the simplicity of the single compartment models employed, D B S was simulated as the application of an intracellular current to the STN. Furthermore, the membrane potential was used to calculate the biomarker to provide an estimate of oscillatory activity. In a clinical setting, however, this information would not be available. Alternative signals, such as local field potentials could be used as a biomarker in practice.

The systematic comparison made in this study indicates that D B S paradigms based on linear control schemes provide a reduction in current and can respond to the suppression and resurgence of oscillations within the cortico-basal ganglia network. In addition to the simple control schemes implemented in this study, the model developed here can be used to design and test more complex control schemes that would not be possible to assess in vivo. Advanced nonlinear

control schemes may provide a more effective therapy and could be tested in a systematic approach similar to that used in this study.

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