Models of deep brain stimulation

## Contents

* 1 What is Deep Brain Stimulation (DBS)
* 2 Parkinson’s disease (PD)
* 2.1 The Basal Ganglia (BG) – the changes in PD
* 2.1.1 Imbalance between BG pathways in PD
* 2.1.2 Synchronized oscillatory activity in BG in PD
* 2.2 DBS in Parkinson’s disease
* 3 Computational models of DBS
* 3.1 BG activity at the neuronal level
* 3.1.1 Tremor reduction
* 3.1.2 Amplitude and frequency dependence of DBS
* 3.1.3 Testing different targets and novel stimulation paradigms
* 3.2 Neural mass models of BG
* 3.2.1 Improving bradykinesia
* 3.2.2 Changes in cognitive function
* 3.3 Modeling local stimulation effects in the 3D brain
* 3.3.1 Decoupling of activity in cell body and axon
* 3.3.2 Selective targeting and field steering
* 3.3.3 Patient-specific modeling to assist DBS programming
* 4 Towards smarter DBS
* 4.1 Closed-loop control of Vim DBS
* 4.2 Closed-loop control of STN DBS
* 4.3 Less-invasive closed-loop control
* 5 Conclusion
* 6 References

## [edit] What is Deep Brain Stimulation (DBS)

Deep brain stimulation (DBS) is a neurosurgical therapy in which a thin lead with a number of electrode contacts at the tip is chronically implanted in deep brain structures, and delivers electrical pulses (60-200 μs duration, 1-5 V amplitude) at high frequency (typically > 130 Hz) to surrounding brain tissue through one or a combination of electrode contacts. These electrical pulses interfere with the pathological neuronal activity patterns of the targeted structures, such that disease symptoms are reduced. DBS has been found to be an effective treatment for movement disorders (especially for Parkinson’s disease, with over 100,000 patients implanted with DBS devices worldwide, but also for essential tremor, dystonia for example), and its use is currently being investigated for a wide variety of other drug-resistant conditions, e.g., epilepsy, depression, obsessive-compulsive disorder, Gilles de la Tourette syndrome, chronic pain, addiction, and Alzheimer’s disease (Benabid et al., 1991, 1993, 2000; Alba-Ferrara et al., 2014; Figee et al., 2014; Huys et al., 2014; Kuhn et al., 2014; Laxpati et al., 2014; Riva-Posse et al., 2014). However, since DBS mechanisms still remain partly unknown and therapeutic parameters are still largely derived by trial-and-error, the potential of DBS cannot be fully exploited yet. A first step in unlocking the fundamental mechanisms of DBS is to gain insight into the activity patterns and function of the involved specific brain circuits under physiological and pathological conditions. In order to address this issue, combining computational modeling at various levels of description (from the cellular to the large-scale brain networks) with in vivo and clinical data appears to be a promising approach.

In this review, a number of computational models are presented, focusing on the use of DBS in Parkinson’s disease (PD). The reason for this focus is the extent of the PD population benefiting from DBS, the time frame within which symptoms respond to stimulation (DBS has an almost immediate effect on PD tremor, while in dystonia it may take several weeks before clinical benefits are observed), and the observability of the effects of DBS on motor control. Let us note that, despite this focus on PD, general concepts can be obtained through these modeling studies that may be applicable to other applications of DBS in the context of other neurological diseases.

## [edit] Parkinson’s disease (PD)

The primary pathophysiological feature of PD is a gradual degeneration of midbrain dopaminergic neurons in the substantia nigra pars compacta (SNc). When the dopamine (DA) deficiency becomes sufficiently severe, i.e. when approximately 80% of the dopaminergic neurons have degenerated, motor symptoms gradually appear (Hornykiewicz and Kish, 1987). Abnormal activity in the basal ganglia (BG) due to the loss of DA is generally considered to be the origin of the characteristic features of PD: resting tremor, muscle rigidity, akinesia (difficulty initiating movement), and bradykinesia (slowness of movement). The BG are a group of subcortical nuclei (including the substantia nigra) that are strongly interconnected with the cortex and thalamus. In addition to their role in motor control, the BG are known to also be involved in cognitive and associative functions (Bar-Gad et al., 2003; Squire et al., 2003). As a consequence, PD patients also may show a variety of cognitive deficits such as impaired procedural learning, memory storage, decision making, and attention (Alexander et al., 1986; Bar-Gad et al., 2003; Frank, 2005).

### [edit] The Basal Ganglia (BG) – the changes in PD

It is widely accepted that the BG play a crucial role in the control of voluntary movement, but the exact role is still under debate. One of the main functions of the basal ganglia is thought to be action selection (Mink, 1996; Gurney et al., 2001a, b), which can be described using the concept of three pathways: the hyperdirect, direct, and indirect pathways (Fig. 1A), each having a specific role.

Figure 1: A) The main structures comprising the basal ganglia (BG) are the striatum (caudate nucleus, putamen, and ventral striatum), pallidum (internal and external segments, GPi and GPe, respectively; and ventral pallidum), substantia nigra (pars compacta [SNc], pars reticularis [SNr], and pars lateralis [SNl]), and subthalamic nucleus (STN) (Alexander et al., 1986; Alexander and Crutcher, 1990; Nambu et al. 2002, 2005; Squire et al., 2003; Brown, 2003). B) Loss of DA in PD and the differential effect of DA in the direct and indirect pathways cause an imbalance between the two, over-inhibiting the thalamus. The thickness of lines indicates relative increase and decrease of average firing rates compared to the physiological situation presented in A).

During any voluntary movement, a multitude of potentially competing motor mechanisms must be inhibited to prevent them from interfering with the desired movement. Therefore, the tonically active BG inhibitory output delivered through the internal part of the globus pallidus (GPi) acts as a brake on motor patterns. At the time when a movement is about to be initiated, the hyperdirect pathway is responsible for even further suppressing activity in large areas of the thalamus and cerebral cortex related to both the selected motor program and other competing programs. When a movement is initiated, GPi neurons projecting to those parts of the thalamus involved in the desired movement decrease their discharge rate, disinhibiting these thalamic areas and therewith reinforcing appropriate patterns of cortical activity and facilitating the desired motor pattern. At the same time, by lateral inhibition, GPi neurons projecting to competing motor patterns increase their firing rate, leading to the suppression of unintended movements. Sequentially, the indirect pathway suppresses the targets in the thalamus and cerebral cortex, extensively resulting in the termination of the selected program (Albin et al., 1989; Alexander and Crutcher, 1990; Nambu et al., 2002).

#### [edit] Imbalance between BG pathways in PD

As a first assumption, PD pathophysiology was suggested to result from abnormalities in the mean discharge rates of BG nuclei. Most specifically, hypokinetic symptoms may be explained by the abnormally high firing rates of inhibitory GPi neurons, resulting in increased inhibition in the thalamic target, interfering with the ability to release a selected motor program by disinhibiting this area. This increased GPi activity may result due to the differential effects of DA on direct and indirect pathways (DA increases direct pathway activity via D1 receptors and decreases indirect pathway activity via D2 receptors). Therefore, DA depletion is thought to cause an imbalance between the two pathways (Fig. 1B), which has indeed been demonstrated in hemiparkinsonian rats (with PD induced by 6-hydroxydopamine [6-OHDA]). In these rats, striatonigral neurons (direct pathway) were inhibited whereas striatopallidal neurons (indirect pathway) were activated as a result of the dopaminergic lesion (Mallet et al., 2006). In addition to the increased firing rates in the BG output nuclei (GPi and SNr), elevated discharge rates in striatum and subthalamic nucleus (STN) were recorded, while reduced activity was found in the external part of the globus pallidus (GPe) in PD animal models as well as PD patients (Miller and DeLong, 1988; Bergman et al., 1994; Schneider and Rothblat, 1996; Elder and Vitek, 2001). For example, Magnin et al. (2000) recorded a mean firing rate of 9.8 ± 3.8 spikes/s in the putamen of PD patients, in contrast to the tonic activity level of 0.1-1 Hz in normal individuals, as found by Squire et al. (2003). The GPi mean neuronal firing rate was found to be increased to 89.9 ± 3.0 spikes/s (Tang et al., 2005) and 91 ±52.5 spikes/s (Magnin et al., 2000) in PD patients compared to a normal level of 60-80 spikes/s (Squire et al., 2003). Physiologic tonic firing rates of STN cells of around 20 spikes/s (Squire et al., 2003) were increased to 42.3 ± 22.0 spikes/s in PD patients suffering from severe akinesia and rigidity (Benazzouz et al., 2002).

Instead of an imbalance between the direct and indirect pathways, Leblois et al. (2006) investigated the idea that an imbalance between the direct and hyperdirect pathways leads to action selection impairment, plus subsequent pathological oscillations (see section 2.1.2). In their model of the BG-thalamus cortex loop, with single neuron dynamics being described by a rate model (see Section 3.1 on modeling BG at the neuronal level), the role of dopamine was hypothesized to be involved in potentiating synaptic transmission from the cortex to the striatum. Interestingly, this high-level model of the BG was based on a significant number of neuroanatomical and neurophysiological findings, and has provided a link between BG organization and function, with implications for the pathophysiology of PD.

In addition to abnormal firing rates, there is ample evidence to suggest that loss of segregation occurs in PD, possibly originating from compensatory mechanisms attempting to counteract DA deficiency, for example by enlarging receptive fields in the striatum, GPi, GPe, STN, and thalamus, and increased corticostriatal transmission (Calabresi et al., 2000; Romanelli et al., 2005; Pessiglione et al., 2005). The result is that the cortical input targeting a particular set of striatal neurons activates not only the desired set of neurons, but also neurons in surrounding motor pathways. Loss of functional segregation may lead to impaired inhibition of competing motor patterns, possibly leading to co-selection of antagonist motor programs, resulting in both akinesia and muscular rigidity. Furthermore, the inability to decorrelate motor sub-circuits may explain why PD patients have difficulty in performing two simultaneous movements (Benecke et al. 1986; Pessiglione et al. 2005).

#### [edit] Synchronized oscillatory activity in BG in PD

Not only are mean firing rates altered in PD patients, but so are the discharge patterns of BG neurons, including a tendency of neurons to discharge in bursts (observed for example in the GPe, GPi, and STN), and oscillatory synchrony within and across BG nuclei (Miller and DeLong, 1988; Bergman et al., 1994; Brown et al., 2001; Wichmann and Soares, 2006; Heimer et al., 2006; Tachibana et al. 2011). Most specifically, increased beta oscillations (11-30 Hz) are observed in cortico-basal ganglia circuits in PD (Brown et al., 2001; Levy et al., 2002). According to Mallet et al. (2008), this enhanced beta activity is likely a consequence of plasticity induced by long-term progressive dopamine depletion rather than an acute network response to a lack of dopamine. In terms of functional role, a correlation between beta band power in STN local field potentials (LFP) and bradykinesia and rigidity has been established (Davidson et al., 2014; McIntyre et al., 2014). In addition, a correlation between BG oscillations in the theta band (3-7 Hz) and limb tremor has been established in PD patients (Davidson et al., 2014). Both theta and beta oscillations are thought to be antikinetic (Brown, 2003; Hutchison et al., 1994; Brown and Williams, 2005; Rubin et al., 2012). In contrast, gamma band oscillatory activities (35-90 Hz), thought to be prokinetic, are less prominent in the BG and cortex of PD patients and in PD animal models (Wang et al., 1999; Brown, 2003; Lalo et al., 2008). It has been shown that administration of levodopa restores gamma band activity while beta band power is suppressed, which is paralleled by improvements in motor symptoms (Brown, 2003; Gatev et al., 2006; Hammond et al., 2007; Kuhn et al., 2006). The experiments of Costa et al. (2006) in DAT knockout mice suggest that DA levels can rapidly modulate the synchronicity and oscillatory behavior of cortical and striatal circuits and are consistent with the observed increase in beta oscillations in untreated PD patients and increased gamma oscillations after treatment with levodopa.

### [edit] DBS in Parkinson’s disease

The very first report of the effectiveness of DBS in PD was published by the group of Benabid in 1987 (Benabid et al., 1987), and was a few years later followed by other reports of DBS applications in PD as well as in other types of movement disorders (Benabid et al., 1991; Benabid et al., 1993). The main targets for DBS in PD are the ventral thalamic intermedial nucleus (Vim), STN, and GPi. Recently, GPe and the pedunculopontine nucleus (PPN) have been explored (Davidson et al., 2014). PPN stimulation is often applied in combination with STN DBS to improve gait and locomotion, which are not affected substantially by STN DBS alone (Beuter and Modolo, 2009). Vim-DBS is used to treat PD tremor, which can produce an improvement of up to 80%. DBS of the GPi or STN is used to treat all symptoms of PD, resulting in an average improvement of 80% in tremor and dyskinesia, more than 60% in bradykinesia and rigidity, and approximately 40-50% in gait and postural dysfunction (Benabid et al., 2000). Typically, DBS frequency must be sufficiently high (> 130 Hz) to be clinically effective. DBS applied at frequencies below 60 Hz were found to have no clinical effect, or even to deteriorate PD symptoms and worsen motor performance (Rizzone et al., 2001, Moro et al., 2002, Timmermann et al., 2004, Fogelson et al., 2005, Eusebio et al., 2008).

Initially, since DBS showed similar therapeutic benefits as surgical lesions, it was hypothesized that the effective mechanism of DBS was either based on synaptic inhibition or on depolarization blockade (Breit et al., 2004; Benabid et al., 2002; Dostrovsky and Lozano, 2002; Grill et al., 2001; Heida et al., 2008; Kringelbach et al., 2007; Lozano et al., 2002; McIntyre et al., 2004a). DBS affects indifferently multiple neural elements, including myelinated and unmyelinated axons, dendrites, and cell bodies, which may be differentially activated (see Section 3.3.1). Extracellular stimulation may excite or block axons of passage, and fiber activation will result in both antidromic and orthodromic propagation (Chiken and Nambu, 2015; Hashimoto et al., 2003; Heida et al., 2008; McIntyre et al., 2004a). However, due to the presence of significant stimulation artefacts, it is complicated to identify the activity patterns occurring during STN stimulation from simultaneous recordings in STN or neighbouring structures. Therefore, the responses following short bursts of stimulation are used to analyze local and global stimulation effects. GP LFPs recorded in awake PD patients after neurosurgery showed that therapeutically effective stimulation suppresses beta band activity in the GP, suggesting that DBS may modulate oscillatory activity patterns between the cortex and BG (Heida et al., 2008; Kringelbach et al., 2007; McIntyre et al., 2004a). A new promising method in experimental research is the use of optogenetics: light can be used to target individual neurons that have been genetically modified such that they express light-sensitive ion channels, allowing their precise activation and inactivation. With this technique, Gardinaru et al. (2009) have found by targeting different elements of the cortico-basal ganglia circuit in hemiparkinsonian rodents that the afferents to the STN region are a major target of STN DBS. In conclusion, the effects of DBS appear much more complex than a simple inhibition of the targeted structure, which has become apparent in part through modeling efforts.

## [edit] Computational models of DBS

An abundance of computational models has been developed in an attempt to explain the role of (part of) the BG-thalamo-cortical loop in health and disease, which also may aid in understanding the mechanism(s) of DBS. These models provide pieces of information regarding the seemingly contradictory effect that increased GPi firing associated with PD should require high-frequency stimulation (supposedly enhancing GPi inhibitory output even further) to improve motor symptoms. The starting point for modeling is often based on the interpretation given to BG function and the neuronal mechanism(s) underlying PD symptoms. Since a consensus is still to be reached regarding BG function, inherently different approaches have been chosen, resulting in computational models differing in a number of ways. A selection of models is described in this review.

### [edit] BG activity at the neuronal level

In order to model the changes that are observed in the firing patterns in the pathological BG compared to the physiological patterns, neuronal activity may be investigated at the neuronal level, based on neurophysiological and neuroanatomical data. Such models consist of single neurons that are represented by a system of equations describing membrane dynamics (see the seminal work by Hodgkin and Huxley, 1952, which has been used in hundreds of models since then). In these models, action potentials are generated by the opening and closing of ion channels, as in biological neurons, with each type of ion channel having its own dynamics in terms of the extent and rate of opening and closing. An important issue that has extensively been investigated using these types of models is whether brain rhythms are caused by single cells having pacemaker properties (Lopes da Silva, 1991). This issue is relevant in describing normal as well as pathological neuronal behaviour, such as synchronized oscillations.

#### [edit] Tremor reduction

As suggested by several research groups, low frequency oscillations (5-10 Hz) associated with PD may originate in the BG (Brown, 2003) and cortex (Terman et al., 2002). The idea of a “tremor pacemaker” in the BG has been proposed to be formed by the GPe-STN network, which was experimentally tested by Plenz and Kital (1999). The suggested mechanism behind these sustained, low-frequency oscillations of the STN-GPe network was that excitatory STN output leads to GPe bursting, sequentially leading first to hyperpolarization and then rebound bursting in the STN, initiating a new cycle by inducing GPe bursting activity (Plenz and Kital, 1999; Holgado et al., 2010). In order to investigate the modulation of BG activity by DBS, Rubin and Terman (2004) developed a computational model including single compartment models of 16 STN neurons, 16 GPe neurons, 16 GPi neurons, and 2 thalamocortical (TC) neurons (Terman et al., 2002; Rubin and Terman, 2004). More specifically, the authors investigated how STN DBS modulates the relay capabilities of TC neurons. Under physiological conditions, GPi output is irregular and uncorrelated, allowing the thalamus to transmit faithfully excitatory sensorimotor inputs to cortical areas. In the model, it was shown that increased striatal input to, and weakened intrapallidal inhibition within, the indirect pathway as included in the model switched BG global activity from irregular to rhythmic, resembling the parkinsonian state consistent with experimental data (DeLong, 1971; Filion and Tremblay, 1991; Wichmann et al., 1999; Raz et al., 2000; Hashimoto et al., 2003; Terman et al., 2002). It was shown that GPi model neurons fired bursts of action potentials at a frequency of 3-8 Hz (i.e., the PD resting tremor frequency range), and bursts are synchronized among subpopulations of GPi neurons. This results in phasic inhibition of TC cells, which are no longer able to transmit cortical input faithfully. Applying DBS at high frequency (167 Hz in the model) as an excitatory input to all STN neurons, resulted in GPi neurons to tonically fire at high frequency, therewith eliminating the oscillatory nature of TC cell inhibition and restoring the ability of TC cells to relay sensorimotor input (Rubin and Terman, 2004; Heida et al., 2008). Using a slightly modified version of the TC cell model, it was tested how different patterns of GPi inhibition, generated from experimental recordings of physiological and parkinsonian monkeys with and without DBS, affected TC relay fidelity (Guo et al., 2008). This study showed that, without DBS, TC relay fidelity was compromised when receiving input recorded from the GPi of a parkinsonian monkey compared to the GPi activity recorded from a healthy monkey. Relay fidelity improved significantly under therapeutic DBS conditions, but not under sub-therapeutic conditions.

#### [edit] Amplitude and frequency dependence of DBS

Clinical observations indicate that PD motor symptoms are still present for therapeutic stimulation frequencies at sub-therapeutic amplitudes. By assuming that DBS does not completely replace oscillatory synchronous activity in the STN, stimulus amplitude effects can be analyzed (Cagnan et al., 2009; Meijer et al., 2011). Meijer et al. (2011) therefore considered thalamic input as a convergent inhibitory input with characteristics of physiological, PD, and stimulation-induced activity patterns according to experimental data. Without additional sensorimotor input (i.e., resting state), TC cells may generate rebound bursts in response to synchronized activity at PD resting tremor frequencies (4-6 Hz), which are transmitted to the cortex. This pathological activity can be stopped by cortical excitatory input, which may be associated with the execution of voluntary movements (resting tremor is usually absent during voluntary movement). Low-frequency DBS enhances rebound bursts, while high-frequency DBS with amplitudes above a certain threshold level may suppress those bursts. Excessive amplitudes, however, may block the relay of excitatory input, suggesting that DBS effectiveness consists of diminishing rebound activity while preserving the relay function. This results in a clinically effective stimulation window that requires high amplitudes for stimulation frequencies below 40 Hz, in comparison to stimulation frequencies above 70 Hz. This is in accordance with the observed inverse relationship between therapeutic stimulation frequency and amplitude (Benabid et al., 1991; Limousin et al., 1995).

The complexity of (multi-compartment) single-cell models requires large amounts of computational power. Cagnan et al. (2009) have therefore tested both a morphologically realistic multi-compartment model and a single-compartment model. Interestingly, they found no mismatch between the two, suggesting that the dendritic structure as included in the multi-compartment model did not have a significant effect on the main functioning of the TC cells with regard to the conditions tested.

A next step in reducing model dimensions of the STN-GPe network models is the use of a mean-field approach (the activity of each structure being described by a distribution of membrane voltages; see also the next section on neural mass models) and incorporating STN-GPe connectivity patterns derived from neuroanatomy. Modolo et al. (2008) have used a neural mass model to investigate STN-GPe activity modulation under DBS. As in the Rubin and Terman model (2004), switching from physiological STN-GPe activity to pathological, low-frequency oscillations characteristic of PD was modelled by increased striatal input to the GPe. DBS was applied to STN neurons as an additional excitatory input: at low frequencies (< 20 Hz), low-frequency pathological oscillations were enhanced, whereas with increasing DBS frequency the oscillations are gradually decreased. It was suggested that, at high DBS frequencies, STN neurons only evoked subthreshold responses due to their inability to follow DBS pulses, thereby leading to a suppression of low-frequency oscillations.

The possibility that DBS provides clinical improvements by restoring functional relay of sensorimotor inputs by TC cells, as described in this and the previous section, may be plausible. However, whether the STN-GPe loop is indeed responsible for tremor generation is questionable. The fact that lesioning as well as application of DBS in the Vim provide clinical improvements in PD tremor may prove that cerebellar instead of BG pathways may be involved in tremor generation since BG does not project to the Vim (Bostan et al., 2010; Helmich et al., 2011). Furthermore, it questions the hypothesis that restoring the thalamic relay of sensorimotor inputs is involved in DBS therapeutic effects.

#### [edit] Testing different targets and novel stimulation paradigms

An adapted version of the model of Rubin and Terman (2004) was used to test DBS network effects using different targets for stimulation: STN, GPi, and GPe (Pirini et al., 2009). Simulation results for STN DBS showed, similar to model simulations from Rubin and Terman, a restoration of thalamic relay function, while GPe and GPi DBS resulted in functional over-activation and inhibition of the TC relay activity, respectively, consistent with experimental and clinical evidence.

Novel targets and stimulation paradigms have been tested by Kumar et al. (2011) using a large-scale network of GPe (2000 cells) and STN (1000 cells), which were implemented as leaky-integrate-and-fire neurons. GPe stimulation was tested by injecting Poisson types of inhibitory input of varying frequencies to a selection of GPe neurons for 20 ms. A stimulation-induced increase in the firing rate of GPe neurons was found to be effective in reducing pathological oscillations. Furthermore, two methods of periodic stimulation were implemented: 1) Excitatory inputs to STN were periodically switched on and off (at a frequency in the range of 10 to 200 Hz), which is equivalent to repeated electrical stimulation of excitatory afferents to the STN resulting in a cessation of axonal spiking due to adaption effects. Only frequencies above 100 Hz appeared to be effective. 2) Periodic inhibition of excitatory afferents to the STN (10-150 Hz) also showed increased efficacy in quenching pathological oscillations (in the beta range) with increased frequency. Finally, aperiodic STN stimulation was applied such that after each pulse the next pulse was delivered after a period γΔt with Δt the minimal interval between pulses (varied between 5 and 15 ms) and γ a randomly selected integer value (n={1,2,3}). This type of stimulation was found to be more effective in quenching pathological oscillations than periodic stimulation.

In addition to the question of whether alternative DBS patterns would be equally, or even more, effective in alleviating PD symptoms, we may also wonder whether certain stimulation patterns would be effective in patients who do not respond effectively to standard, high-frequency DBS. In a computational model of the parkinsonian BG, several non-regular patterns of DBS, all having a mean frequency of 185 Hz, but having non-regular features such as short periods in which pulses were absent or presence of short bursts of pulses, were found to suppress beta band oscillatory activity (Brocker et al. 2013). The degree of suppression in the model was strongly correlated with the clinical efficacy across stimulation patterns as tested during a finger tapping task. Another recent study investigated the possibility of using irregular stimulation patterns instead of the standard high-frequency stimulation pattern classically used in DBS (Summerson et al., 2015). These authors used a stochastic stimulation period, keeping the average stimulation frequency constant. Interestingly, both in their model and in a rat model of PD (6-OHDA), they found that stochastic DBS resulted in an inter-spike interval (ISI) entropy decrease, along with a decrease in beta band power classically linked with an improvement in PD motor symptoms. However, these results are in contradiction with other studies as mentioned and replicated by McConnell et al. (2016).

Instead of using a single target for stimulation, it has been suggested to use multiple sites for stimulation (Hauptmann et al., 2005; Tass, 2006; Tass and Majtanik, 2006; Hauptmann and Tass, 2010; Tass, 2011). Since PD symptoms are associated with hyper-synchronized (pathological) neuronal activity, it may be assumed that effective stimulation paradigms should aim at desynchronizing hyper-synchronized networks. A population of STN neurons was simulated using the Morris-Lecar model (Morris and Lecar, 1981), a simplified version of the Hodgkin-Huxley model, to represent the membrane dynamics of each cell (Hauptmann et al., 2005). The delayed band-pass filtered local field potential (LFP) ‘recorded’ at the center of the network was used as a stimulation signal. Four stimulation sites were used with the LFP fed back with different time delays related to the mean period of neural activity, at each site. The results showed a very robust and effective demand-controlled desynchronization, with reduced energy consumption compared to standard continuous DBS (see also Section 4).

### [edit] Neural mass models of BG

Neural mass models of the BG use a high level of abstraction, in which BG nuclei are decomposed into functional units, each nucleus being modeled as a single equation with relatively few parameters, representing the combined actions of all neurons or a set of neurons within the nucleus. The advantage of neural mass models is that they allow the prediction of large-scale properties of neuronal assemblies and directly assess their dependence on connection strengths between populations. Furthermore, to some extent, these models can provide insights comparable to analytical analyses (e.g., stability of equilibria) regarding the possible dynamical states of the system.

#### [edit] Improving bradykinesia

Experimental evidence indicates that bradykinesia is more pronounced in the execution of complex movements, i.e. sequential or simultaneous motor tasks, than in the execution of simple movements (Suri et al., 1998; Berardelli et al., 2001). To investigate factors contributing to bradykinesia in the control of simple and complex voluntary limb movement in PD patients, the functional scheme of the BG-thalamocortical circuit has been described by a mathematical model based on the mean firing rates of BG nuclei (Moroney et al., 2008). For the control of each muscle group involved in flexion and extension at the elbow joint, a separate BG-thalamocortical circuit was modeled (Fig. 2). The “PD condition” was simulated as a reduction in DA level, producing an increase in activation of D2 receptors and a decrease in activation of D1 receptors, and a loss of functional segregation between two competing motor modules (i.e. those involved in flexion and extension). As an example, the firing rate of the STN, Stn(t), in one module was described as

\\(\tag{1} \frac{d}{dt}Stn(t)=-A_{Stn} Stn(t)+(B_{Stn}-Stn(t))(I_{CorStn}+I_{tonicStn})-(Stn(t)-D_{Stn})10Ge(t-τ_{GeStn}) \\)

with AStn representing the passive decay rate of neural activity in STN; BStn and DStn representing the upper and lower bounds of STN activity, respectively (conforming to tonic firing rates recorded in humans); ICorStn representing the excitatory input from the cortex via the hyperdirect pathway; ItonicStn representing STN tonic activity; and Ge representing a GPe inhibitory input, multiplied by a weighting factor (in this case 10) representing the connection strength, with a delay of τGeStn (Moroney et al., 2008). Seven potential mechanisms of STN DBS were tested by assuming that DBS induces inhibition or excitation of various neuronal elements. For example, direct inhibition of STN neuron somas was simulated by adding an inhibitory input IDBS,inh to GPe input, i.e., the last component of the previous equation was written as \\(-(Stn(t)-D_{Stn})[10Ge(t-τ_{GeStn})+I_{DBS,inh}]\\). DBS-induced excitation of inhibitory afferent axons projecting to the STN was simulated by adding an additional weight factor to GPe input, i.e., the last component of the equation then became \\(-(Stn(t)-D_{Stn})w_{DBS,aff}10Ge(t-τ_{GeStn})\\).

Results from the model suggested that primary deficits in movement arise directly from degeneration of DA neurons in the nigrostriatal pathway. DA depletion produces smaller-than-normal pallidothalamic gating signals, failing to sufficiently reinforce cortical input, producing smaller-than-normal movement amplitudes and velocities. Excessive DA depletion was predicted to contribute to the additional delays experienced by PD patients in the execution of complex movements. Abnormal activity in the inactive module due to a loss of segregation could lead to excessive depletion of the available neurotransmitters, which can have severe consequences for subsequent movements if neurotransmitter reserves are not sufficiently replenished before the next movement begins.

Figure 2: Two parallel layers of the cortico-basal ganglia network, each controlling a different motor program/muscle, in this case muscles responsible for flexion and extension at the elbow joint (adapted from Moroney et al. 2008).

Based on the changes in the BG nuclei firing rate, the model demonstrated that the effective mechanism of STN DBS results from stimulation-induced STN inhibition, partial synaptic failure of efferent projections, or excitation of inhibitory afferent axons, even though the underlying methods of action may be quite different for the different mechanisms (Moroney et al., 2008). For example, both DBS simulated as an excitation of afferent axons, or as a partial synaptic failure, produced a decrease in GPi activity. However, stimulation of the afferents was assumed to cause a release of the inhibitory neurotransmitter GABA, resulting in decreased STN activity, while stimulation of efferents was assumed to cause synaptic failure due to an inability of the stimulated neurons to follow the rapid train of electrical stimuli produced by DBS, resulting in increased STN activity.

#### [edit] Changes in cognitive function

STN DBS has been reported to cause cognitive side effects such as impulsivity. Interestingly, circuits linking BG with more cognitive areas of frontal cortex have been found to be rather similar to the circuits playing a role in motor control. Based on this analogy, Frank (2005, 2006) developed a BG model in which the proposed role of (low-level) action selection was extended by including higher-level cognitive decisions. The units of which the different nuclei were built up in the model of Frank (2005, 2006) were implemented using the “Leabra” framework (‘learning in an error-driven and associative, biologically realistic algorithm’). In this model, the direct and indirect pathways were associated to Go and No-Go activities to facilitate the execution of a response or to suppress competing responses, respectively. DA changes have been inferred to occur in humans receiving positive and negative feedback (e.g., subjects were told their responses were correct or incorrect) in cognitive tasks. These changes in DA modulate neuronal excitability, and may therefore act to reinforce the efficacy of recently active synapses, leading to reinforced learning of correct responses. In this model, the STN received direct projections from cortical areas detecting and integrating response conflicts. In the face of conflict, the STN raises decision thresholds by sending a global No-Go signal to BG output nuclei. The model of Frank (2005, 2006) suggests that a large dynamic range of DA release is necessary for learning subtle differences between positive and negative reinforcement values of responses.

PD was simulated by reducing the amount of DA, leading to a reduced DA range and consequently a lower ability to resolve Go/No-Go association differences required for discriminating between subtly different response reinforcement histories. The effect of STN-DBS was simulated in the model either as a lesion (by removing STN processing) or as enhanced STN output (by applying high-frequency excitatory input). In both cases, conflict-induced slowing, which was shown in the intact network, was disrupted. The model thus proposes a mechanism leading to impulsive decision-making in PD patients making use of DBS. DBS parameters were not sensitive to variations, and it was therefore suggested that disruption of physiological STN processing by external high-frequency stimulation to a subset of STN units (3 out of 9), or by adding Gaussian noise to the activity of each STN unit, or by removing the STN altogether, prevented the system from regulating decision times in proportion to decision conflict.

### [edit] Modeling local stimulation effects in the 3D brain

At a physiological level, DBS can have multiple effects on its targets due to the wide range of neuronal elements that may be stimulated by the electrode’s electric field (Breit et al., 2004; Lozano et al., 2002; Grill et al., 2001). It is known that axons are much more excitable than cell bodies, due to their much shorter chronaxie time (100 μs for axons, vs. approximately 10 ms for somas), and that large myelinated fibers are more excitable than unmyelinated axons (Ranck, 1975; Rattay, 1999). Current density decreases with distance from the electrode tip, and axons near the cathode are more likely to be activated than axons near the anode. Furthermore, it is more feasible to activate fibers oriented parallel to the current field than fibers oriented transversely. These qualitative observations have been formalized as the “lambda-E” model (see for example Radman et al., 2009), where lambda is a vector aligned with the fiber. Due to the properties of the dot product, the depolarizing/hyperpolarizing effect is maximal when the angle between lambda and the electric field is zero (electric field parallel to the fiber), and vanishes when the angle is $\pi/2$ (electric field orthogonal to the fiber orientation).

Furthermore, DBS electrodes may be placed in regions with heterogeneous populations of neuronal elements. The applied current may affect several neuronal components near the stimulation electrode, and induce both depolarizing and hyperpolarizing effects. Stimulation may influence afferent (axon or axon terminal) and efferent projection neurons, as well as local interneurons. Extracellular stimulation may also excite or block axons of passage, and fiber activation can result in both antidromic and orthodromic propagation. All of these complex effects, which are mostly neglected in the models previously described, can be addressed in volume conduction models. Notably, the combination of models based on the finite element method (FEM) and multi-compartment neuron and/or axon models provide tools to estimate the volume of activated tissue (VAT), optimize stimulation settings, and to test new stimulation paradigms.

#### [edit] Decoupling of activity in cell body and axon

An axisymmetric FEM model of the Medtronic 3387 DBS lead (Medtronic, Minneapolis, MN) positioned in a homogeneous isotropic (a hypothesis classically made, even though anisotropies in tissue conductivity can play an important role, but increase model complexity significantly) volume conductor combined with a multi-compartment model of a TC relay neuron showed that, during high-frequency DBS, somatic activity may be decoupled from axonal activity (McIntyre et al., 2004b). As a consequence, a single neuron may simultaneously exhibit suppression of intrinsic activity in the soma while excitation occurs in the axon. This raises the possibility that the main effect of DBS results from stimulation-induced efferent output in neurons located within the vicinity of the active contact(s), and that this soma/axon decoupling somehow “breaks” the transmission of pathological activity throughout the BG-thalamo-cortical loop.

#### [edit] Selective targeting and field steering

DBS may have a low threshold to side effects, especially in STN DBS, since STN size is comparable to DBS lead size and STN is composed of different functional areas, the targeted dorsolateral sensorimotor region being one of them (the other two being associative and limbic). In addition, bundles of fibers (that are easily excitable) surround the STN, such as the internal capsule, and can induce face contractions or dysarthria for example if a sufficiently high current spreads into this region. Current DBS leads consist of four relatively large annular electrode contacts (1.27 mm diameter, 1.5 mm height, 0.5 mm between each contact). The stimulating electric field at the active contact(s) is distributed symmetrically around the contact and, when slightly misplaced, can induce current spread outside targeted regions. For instance, current spread to STN non-motor regions can disrupt the transmission of physiological information from these STN non-motor regions (Frank et al., 2007; Zwartjes et al., 2013; Frankemolle et al., 2010). This disruption may be responsible for DBS-related cognitive-motor declines observed under dual-task conditions.

Furthermore, optimal clinical effects likely require oddly shaped VATs, and possibly activation of different neuronal elements. Novel high-resolution DBS lead designs have been introduced that allow directional steering of the electric field (van Dijk et al., 2015; Martens et al., 2011; Pollo et al., 2014) and address this issue. Another issue is that, in most current DBS applications, a single electrode contact is used for monopolar cathodal stimulation with the implanted pulse generator (IPG) providing voltage-controlled stimulation. However, in the case where multiple contacts are used, current-controlled stimulation is preferable to fully controlled current flow across the contacts, which is not affected by unavoidable impedance changes of the electrode-tissue interface in the months following electrode implantation. Butson and McIntyre (2008) developed a FEM model to determine the voltage distribution in tissue during current-controlled stimulation, in which the total current amplitude was divided between adjacent electrode contacts. Simulations showed that current-steering can be used to “sculpt” the VAT to achieve the desired overlap with target tissue structures, i.e. STN projection neurons, and GPi fibers of passage (Miocinovic et al., 2006; Butson and McIntyre, 2008). Balancing current delivery across two contacts was found to increase VAT size as compared to monopolar stimulation.

In terms of energy consumption and battery lifetime, the response of fibers of passage and local projection neurons to different stimulus waveforms was tested using an electrostatic, axisymmetric FEM model of Medtronic DBS electrodes, model 3387 (Minneapolis, MN) (Foutz and McIntyre, 2010). Results from this study showed that, compared to the standard 100 µs rectangular pulses, non-rectangular waveforms with longer pulse widths provided optimal stimulation with an energy saving of up to 64%.

#### [edit] Patient-specific modeling to assist DBS programming

One constraint in DBS optimization is that clinically effective stimulation settings need to be found within the small stimulation windows discriminating therapeutic settings from those inducing side effects, which is a tedious and time-consuming process. Patient-specific DBS computer models based on the integration of imaging data and volume conduction models providing an estimate of the VAT as a function of stimulation parameters can assist in the DBS programming process (Butson et al., 2007; Chaturvedi et al., 2010; Frankemolle et al., 2010; McIntyre et al., 2014). Quantitative theoretical predictions may be used to define effective stimulation parameter settings, tailored to the patient, maximizing stimulation of determined target areas while minimizing stimulation spread to non-target areas (Frankemolle et al., 2010).

## [edit] Towards smarter DBS

Current DBS systems are open-loop, and therefore not able to automatically optimize stimulation settings in order to respond to motor fluctuations. Closed-loop DBS is a more intelligent way of stimulation in which stimulation settings are continuously adjusted, depending on changes either in monitored neurophysiological variables, or in the clinical condition of the patient as evaluated by pre-determined variables (e.g., tremor amplitude) to optimize treatment benefits (Priori et al., 2013).

With the use of the power spectra of LFPs recorded at the stimulation site, the correlations found for specific features of these spectra with tremor, bradykinesia, and rigidity, and the changes that occur as a consequence of clinically effective therapies, strategies may be developed for automatic adjustment of stimulation settings. Rosin et al. (2011) have shown in the vervet MPTP model of Parkinson’s disease that closed-loop paradigms in which stimulation pulses are delivered based on ongoing activity, can indeed disrupt abnormal cortico-basal ganglia oscillatory activity, resulting in improved PD akinesia. Furthermore, Little et al. (2013) have demonstrated in a proof-of-principle study that beta activity from STN LFPs may provide a faithful biomarker of PD impairment, and improvements in motor deficits could be achieved by delivering stimulation when ongoing beta activity reached a user-defined threshold. It was found that this type of stimulation even has an anti-kindling effect, as it has also been noted in computational studies (Hauptmann and Tass, 2007).

Another strategy consisting of feeding back a delayed and scaled version of recorded LFPs to the network to reduce global oscillatory activity has been explored by Goldobin et al. (2003), along with Rosenblum and Pikovsky (2004) (see section 3.1.3). It was shown using numerical simulations that specific combinations of delay and gain could cause an instability in a stable oscillatory network, and therewith suppression of pathological brain rhythms.

Alternatively, a model-free adaptive optimization method based on genetic algorithms (GA) may be implemented such that based on a number of instantaneously measurable parameters and with a limited number of trials it converges to the most optimal stimulation setting (Feng et al., 2007a). One drawback is that these models still have a high level of abstraction. Two models that are more closely related to reducing PD symptoms are described below.

### [edit] Closed-loop control of Vim DBS

By combining volume conduction modeling and the simulation of the intrinsic activity of a population of 100 thalamic neurons (ventral intermediate nucleus, Vim) (McIntyre et al., 2004b), Santaniello et al. (2011) have investigated whether LFPs generated by these neurons could be used as a control variable for closed-loop control of DBS amplitude. The independent, non-communicating neurons, described by multi-compartment models, were uniformly distributed within 3 mm of a point source electrode positioned in an infinite homogeneous isotropic medium. Interestingly, LFPs can readily be measured using the implanted DBS electrode, and are highly correlated with tremor in PD (Lenz et al., 1988). Based on in-vivo recordings, four different spiking patterns were implemented to represent physiological thalamic activity, i.e. the reference spectrum, and four different spiking patterns were used to simulate intrinsic tremor-related activity. An autoregressive model with exogenous input (ARX model) was used to describe the relationship between the DBS input and LFP output. Model parameters were fitted based on three conditions: 1) tremor-free, 2) tremor with DBS off, and 3) tremor with DBS on. With a fixed stimulation frequency (130 Hz), the stimulation amplitude was modified using a feedback controller such that tremor-related oscillations in the power spectrum of the LFPs were suppressed and the main features of the tremor-free condition were restored. The ability of closed-loop DBS to restore the neuronal activity present in tremor-free conditions was found to be remarkably better than open-loop DBS (Santaniello et al., 2011).

### [edit] Closed-loop control of STN DBS

Feng et al. (2007a, b) used a GA to identify optimal DBS periodic or aperiodic stimulation currents alleviating Parkinsonian behavior, evaluated in the model by rhythmic, synchronized activity, which was simulated in the BG network model developed by Rubin and Terman (Terman et al., 2002; Rubin and Terman, 2004). They proposed that the algorithm requires four basic properties to be clinically applicable: The algorithm needs to be model-free so that it can directly utilize clinical observables, and it must converge in an acceptable number of trial DBS settings. Furthermore, to fully exploit the full range of DBS parameter settings, the algorithm needs to have global search capability. Lastly, in order to provide maximum flexibility the algorithm must be able to use different types of stimulation waveforms (e.g., periodic and aperiodic waveforms).

GA implementation first involved testing of the effectiveness for a set of DBS waveforms, described by a set of trial parameters, on the ‘physiological responses of the patient’. In terms of the simulated network, this meant that, under effective stimulation conditions, GPi cells should display irregular firing (physiological state), in contrast to bursting spike patterns with a common frequency and characteristic clustering (pathological state). By minimizing a cost function in which practical stimulator constraints were also considered, the most effective waveforms were selected and used to guide the selection of a next set of trial stimulation parameters. This iterative process ended when one or more DBS waveforms provided the desired effect. Although the simulation results cannot directly be transferred to clinical use because of the simplicity of the model, the study provided insights into the hurdles that need to be overcome in order to use adaptive algorithms to “close the loop” in clinical DBS applications.

### [edit] Less-invasive closed-loop control

The idea of using cortical (superficial), as opposed to deep (as in DBS), closed-loop stimulation, has also been proposed and tested using computational models (Modolo et al., 2010; Beuter et al., 2014). In terms of neuroanatomy, arguments for stimulating the cortex include 1) the hyperdirect pathway from the primary motor cortex (M1) to the STN; 2) M1 is the “last output” before sending tremor-frequency signals to muscles. In a neural field model including the effects of closed-loop stimulation, it was shown that a stimulation signal computed from the extracted pathological oscillation was able to drastically reduce the pathological oscillation, without affecting other oscillations which were included to mimic the presence of physiological oscillations (Modolo et al., 2010).

## [edit] Conclusion

Deep brain stimulation (DBS) has become an established intervention for Parkinson's disease. However, the exact mechanism(s) of DBS and the optimal site(s) and parameters of stimulation are still open issues, and therefore most therapies are still largely based on trial-and-error. In an attempt to use a more rationale-driven approach, computational models are becoming increasingly accepted and used. Obviously, each model has its limitations, for example, the actual wiring within the BG network is much more complex, no heterogeneity of neurons is included, and the effects of stimulation are simplified. Despite these simplifications, models have proven useful in suggesting hypotheses about PD pathophysiological mechanisms and potential mechanisms of action of DBS. It has been shown that the neural response to stimulation fields is complex, depending on numerous geometrical, physical, and neurophysiological parameters. Furthermore, model simulations have demonstrated the feasibility and therapeutic potential of closed-loop DBS and suggest that, in the near future, closed-loop DBS should be feasible clinically and provide improved clinical benefits, and may even move from targeting deep brain structures to more superficial targets. The methods developed so far cannot readily be used in clinical practice, but show that optimization strategies can be used to explore the DBS parameters space more efficiently than tuning the settings by hand. In addition, it is suggested that, even if conflicting data is present in the literature, constant high-frequency DBS may not provide optimal clinical benefits, since both rate and pattern of DBS may play an important role in DBS function.

Advances in (functional) imaging technology, experimental and clinical research methods, and computer technology will allow for increasing complexity and realism of computational models. This will further enable us to generate testable predictions and may help to formulate new hypotheses on disease mechanisms, not limited to PD, and (patient-specific) therapeutic paradigms. In addition, being able to exploit these mechanisms of interaction between the stimulation waveform and brain tissue would enable an optimized, mechanism-driven DBS therapy that could be proposed to a much larger number of patients than today.

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Title: Models of deep brain stimulation Page ID: 33311