Multi-site Stimulation of Subthalamic Nucleus Diminishes Thalamocortical Relay Errors in a Biophysical Network Model

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1 Multi-site Stimulation of Subthalamic Nucleus Diminishes Thalamocortical Relay Errors in a Biophysical Network Model Yixin Guo a,, Jonathan E. Rubin b a Department of Mathematics, Drexel University, Philadelphia, PA 1914 b Department of Mathematics and Complex Biological Systems Group, University of Pittsburgh, Pittsburgh, PA 1526 Abstract This paper presents results on a computational study of how multi-site stimulation of the subthalamic nucleus (STN), within the basal ganglia, can improve the fidelity of thalamocortical (TC) relay in a parkinsonian network model. In the absence of stimulation, the network model generates activity featuring synchronized bursting by clusters of neurons in the STN and internal segment of the globus pallidus (GPi), as occurs experimentally in parkinsonian states. This activity yields rhythmic inhibition from GPi to TC neurons, which compromises TC relay of excitatory inputs. We incorporate two types of multi-site STN stimulation into the network model. One stimulation paradigm features coordinated reset pulses that are on for different subintervals of each period at different sites. The other is based on a filtered version of the local field potential recorded from the STN population. Our computational results show that both types of stimulation significantly diminish TC relay errors; the former reduces the rhythmicity of the net GPi input to TC neurons and the latter reduces, but does not eliminate, STN activity. Both types of stimulation represent promising directions for possible therapeutic use with Parkinson s disease patients. Keywords: Parkinson s disease, deep brain stimulation, thalamocortical relay, local field potential, multi-site stimulation, basal ganglia model yixin@math.drexel.edu; phone: ; fax: Preprint submitted to Neural Networks February 28, 211

2 1. Introduction Deep brain stimulation (DBS) is an established clinical intervention for Parkinson s disease (PD), essential tremor, and dystonia that is now being explored for use in a variety of disorders (reviewed in McIntyre & Hahn 21; Wichmann & DeLong 26). In the PD case, DBS is implemented by an implanted pulse generator that delivers an ongoing stream of high frequency current pulses. Although this form of therapy has achieved remarkable successes, improvements in DBS would be desirable in order to reduce the associated energy use and need for invasive battery changes, help patients with symptoms that do not respond to current DBS paradigms, and allow for individualized optimization of DBS strategies (Volkmann, 24; Rodriguez-Oroz et al., 25; Deuschl et al., 26; Feng et al., 27a,b; Hauptmann et al., 27). Efforts to develop such improvements are hampered, however, by a lack of theoretical understanding of the mechanisms through which DBS achieves its clinical efficacy. Parkinson s disease (PD) and experimental models of parkinsonism are associated with changes in activity patterns of neurons in the basal ganglia, including increases in synchrony, firing rates, and bursting activity in the subthalamic nucleus (STN) and internal segment of the globus pallidus (GPi) (Bergman et al., 1994; Nini et al., 1995; Boraud et al., 1998; Wichmann et al., 1999; Magnin et al., 2; Raz et al., 2; Brown et al., 21; Levy et al., 23; Hurtado et al., 25; Wichmann & Soares, 26). Since motor outputs from the basal ganglia emanate specifically from the GPi (Alexander et al., 199; Kelly & Strick, 24; Middleton & Strick, 2), it seems likely that changes in GPi activity contribute to the development of parkinsonian motor complications. Furthermore, because motor outputs from GPi target the anterior ventrolateral nucleus of the thalamus (VLa) (DeVito & Anderson, 1982; Kelly & Strick, 24; Yoshida et al., 1972), which serves to relay signals between cortical areas (Haber, 23; Guillery & Sherman, 22a,b,c), we and other authors have hypothesized that pathological GPi outputs may induce parkinsonian signs by changing thalamic activity patterns or information processing (Montgomery & Baker, 2; Grill et al., 24; Garcia et al., 25; Dorval et al., 28; Xu et al., 28; Dorval et al., 21) or, in particular, by compromising thalamocortical (TC) relay (Rubin & Terman, 24; Guo et al., 28; Cagnan et al., 29; Pirini et al., 29). This idea is consistent with the properties of TC neurons, specifically their tendency to fire rebound bursts when exposed to phasic synaptic inhibition, such as that 2

3 induced by burstiness in the GPi under parkinsonian conditions. According to this viewpoint, DBS achieves its therapeutic efficacy for PD by restoring TC relay fidelity. Importantly, computational models and analysis suggest that returning STN, GPi, and TC activity patterns to their non-parkinsonian states is not necessary for achieving this goal, inasmuch as a variety of alternative activity patterns can also be associated with successful relay in computational models or alleviation of bradykinesia in human patients with parkinsonism (Rubin & Terman, 24; Feng et al., 27a,b; Guo et al., 28; Dorval et al., 21). In this paper, we study a form of STN DBS that has been suggested in the literature as an alternative to standard DBS, namely multi-site STN stimulation with delays between stimulation periods at different stimulation sites (Tass, 23; Hauptmann et al., 25, 27). We consider such stimulation with two types of current injection, one using periodic square pulses and another based on a local field potential signal recorded from the STN population (Rosenblum & Pikovsky, 24b; Hauptmann et al., 25, 27; Tukhlina et al., 27). To perform our investigation, we introduce these stimulation paradigms into a computational model based on our earlier work (Terman et al., 22; Rubin & Terman, 24). This model consists of a small network of conductance-based STN, GPe (external segment of the globus pallidus), GPi, and TC neurons that, by design, generates parkinsonian activity patterns in the absence of stimulation. We simulate the delivery of an excitatory input train to the model TC neurons and compare TC relay performance across simulations without stimulation and simulations with coordinated reset or LFP-based delayed feedback stimulation of various amplitudes and periods. Although the forms of stimulation that we study were introduced previously (Tass, 23; Hauptmann et al., 25, 27), this represents the first work in which they are incorporated into a biophysically-detailed basal ganglia network model and in which their impact on TC relay fidelity is explored. We find that both forms of multi-site stimulation, applied to the STN, regularize GPi outputs and significantly diminish TC relay errors. While both can completely suppress STN activity if introduced with a large enough amplitude, both can also restore TC relay performance without such a drastic effect. Moreover, multi-site delayed feedback stimulation based on the LFP in particular requires relatively small currents (see also Hauptmann et al. 25, 27; Tukhlina et al. 27) and, unlike stimulation with a constant current of similar magnitude, maintains STN responsiveness to its own excitatory inputs (e.g., from the hyperdirect pathway). Hence, these results 3

4 support the idea that multi-site delayed feedback stimulation of STN merits further consideration as a possible alternative to standard forms of DBS for PD. 2. The Network Model We use a network of conductance-based, single-compartment model neurons adapted from earlier work by Rubin & Terman (24) to explore how well different patterns of subthalamic nucleus stimulation can improve thalamic relay responses in parkinsonian conditions. The model includes neurons in the thalamus and several nuclei of the basal ganglia, namely the internal and external segments of the globus pallidus and the subthalamic nucleus. It is known that the subnetwork consisting of the external segment of globus pallidus (GPe) and the subthalamic nucleus (STN) can fire in synchronized clusters when the coupling parameters are appropriately tuned (Terman et al., 22; Best et al., 27). Furthermore, the rhythmic bursting activity of the STN and GPe clusters induces rhythmic bursting of clusters of downstream neurons, in the internal segment of globus pallidus (GPi). We consider this rhythmic clustered regime as the parkinsonian state and refer to the network in this state as the parkinsonian network. Our main goal is to study stimulation patterns that can improve the fidelity with which thalamocortical (TC) relay neurons respond to, or relay, excitatory input signals in a parkinsonian network. For the parkinsonian network described in detail in section 2.2, STN neurons form two synchronized clusters that induce rhythmic and bursty inhibitory GPi activity with strong correlations in burst times among GPi neurons. We show an example of STN clusters, the resulting rhythmic GPi inhibition, and the corresponding TC relay responses to excitatory inputs in Figure 3. We will explore stimulation patterns that can eliminate the synchronized, clustered activity of STN neurons, which consequently will alter GPi firing in a way that facilitates TC relay responses. We focus on the impact of two very different STN stimulation paradigms on TC relay performance in the parkinsonian network. One approach is to use coordinated reset stimulation of the STN neurons. The other is to use the local field potential measured from the population of STN neurons to drive closed-loop, delayed feedback stimulation of the STN neurons. 4

5 2.1. Model Equations for Each Neuron Type We consider a network consisting of model TC, STN, GPe, and GPi neurons. Neurons in these areas are linked by various excitatory and inhibitory synaptic connections and receive certain external inputs, as depicted in Figure 1. Both GPi and GPe receive excitatory inputs from STN, and GPe is subject to an inhibitory striatal input that is assumed to be constant in the model. The details of the architecture of connections between individual neurons within each area are discussed in section 2.2. Next, we describe the conductance-based model equations for each neuron type in the model network in more detail. All specifics of the functions and parameter values used for each type of neuron in the model are given in the Appendix. Figure 1 Thalamocortical (TC) relay neurons: The model for each TC neuron takes the form C m v = I L I Na I K I T I Gi TC + I E (1) h r = (h (v) h)/τ h (v) = (r (v) r)/τ(v) In system (1), v denotes membrane potential, the evolution of which depends on I L = g L (v v L ), I Na = g Na m 3 (v)h(v v Na ), and I K = g K (.75(1 h)) 4 (v v K ), which are leak, sodium, and potassium currents, respectively. Here we use a standard reduction in our expression for the potassium current, which decreases the dimensionality of the model by one variable (Rinzel, 1985). The current I T = g T p 2 (v)r(v v T ) is a low-threshold calcium current, where r is the inactivation and p 2 (v) is the activation. Note that reversal potentials are given in mv, conductances in ms/cm 2, and time constants in msec. In all the neuron models, the membrane capacitance C m is normalized to 1 µf/cm 2. The current I Gi TC in system (1) represents the inhibitory input to the TC neuron model from the GPi, as discussed further in Section 2.4. I E denotes simulated excitatory synaptic signals to the TC neuron. We assume that these are sufficiently strong to induce a spike (in the absence of inhibition) and therefore may represent synchronized inputs from multiple presynaptic cells. We tune the parameters so that the TC cell yields a firing rate of roughly 12 Hz in the absence of inhibitory GPi and excitatory synaptic inputs. The parameter values chosen place the model TC neuron near a transition from silence to spontaneous oscillations. In the model, I E takes 5

6 the form g E s(v v E ), where g E =.18 ms/cm 2, and s satisfies the equation s = α(1 s)exc(t) βs, where α =.8 msec 1 and β =.25 msec 1. The function exc(t) controls the onset and offset of the excitation: exc(t) = 1 during each excitatory input, whereas exc(t) = between excitatory inputs. We used periodic exc(t), which was one of the cases considered in previous work (Guo et al., 28; Rubin & Terman, 24), where similar results were obtained with periodic and Poisson excitation. Specifically, exc(t) = H(sin(2πt/p))(1 H(sin(2π(t + d)/p))), where the period p = 5 msec and duration d = 5 msec, and where H(x) is the Heaviside step function, such that H(x) = if x < and H(x) = 1 if x >. That is, exc(t) = 1 from time up to time d, from time p up to time p +d, from time 2p up to time 2p +d, and so on. A baseline input frequency of 2 Hz is consistent with the high-pass filtering of corticothalamic inputs observed in vivo (Castro-Alamancos & Calcagnotto, 21); at this input rate, the model TC cells rarely fire spontaneous spikes between inputs. STN Neurons: The STN voltage equation that we use, of the form C m v = I L I Na I K I T I Ca I AHP I Ge Sn + I stim, was introduced in (Terman et al., 22). All the currents and corresponding kinetics are the same except that we make some parameter adjustments so that STN firing patterns are more similar to those reported in vivo (Bevan et al., 26; Urbain et al., 2, 22). I Ge Sn is the inhibitory input current from GPe to STN. I stim is the external stimulation applied to STN, which is either multi-site coordinated reset stimulation (CRS) or multi-site feedback stimulation based on the local field potential (LFP). Different types of stimulation will be discussed further in Section 3 and Section 4. GPe Neurons: The voltage of each model GPe neuron obeys the equation C m v = I L I Na I K I T I Ca I AHP I Ge Ge I Sn Ge + I app, where I Ge Ge is the inhibitory input from other GPe cells, I Sn Ge is the excitatory input from STN cells, and I app is a constant external current that represents hyperpolarizing striatal input to all GPe cells. 6

7 GPi Neurons: The voltage equation for each model GPi neuron is similar to that for the GPe neurons, namely C m v = I L I Na I K I T I Ca I AHP I Sn Gi + I Ge Gi + I appi, where I Sn Gi represents the excitatory input from STN to GPi, I Ge Gi is the inhibitory input from GPe to GPi, and I appi is a constant external current that represents hyperpolarizing striatal input to all GPi cells Architecture of Coupling Between Individual Neurons As shown previously (Terman et al., 22), the STN and GPe subnetwork can generate both irregular asynchronous and synchronous activity (Plenz & Kitai, 1999; Terman et al., 22; Best et al., 27). Our model includes 16 STN neurons and 16 GPe neurons. We designed the structure of the STN/GPe loop in the model following the work on clustered rhythms in (Terman et al., 22), so that the STN cells will segregate into two rhythmically bursting clusters, with synchronized activity within each cluster. The detailed structure of connections between STN and GPe neurons is depicted in Figure 2. Ge and Sn represent subpopulations of GPe and STN neurons, respectively. In the two cluster case, we can distinguish two subpopulations within each cluster, such that neurons within the same subpopulation provide synaptic inputs to the same targets. We use K ij, where K=Ge or Sn, i = 1, 2, and j = 1, 2, to denote subpopulation j within the ith cluster of type K neurons. For example, the first subpopulation of STN cluster one, Sn 11, sends excitation to the first subpopulation of GPe cluster two, Ge 21 (Figure 2, +). The same subpopulation of STN neurons are also weakly coupled with the other half of the same GPe cluster, Ge 22 (Figure 2, w+). Each subpopulation of STN neurons is connected with two GPe subpopulations in an analogous way. Each subpopulation of GPe neurons inhibits one group of STN neurons, as is also illustrated in Figure 2. Within each GPe subpopulation Ge ij, there are also local inhibitory connections. The model also includes 16 GPi neurons, each receiving input from a single corresponding STN neuron. Thus, the rhythmic, bursty, synchronized outputs of each STN cluster induce rhythmic, bursty, synchronized activity in a corresponding group of GPi neurons. These GPi activity patterns mimic those seen experimentally in parkinsonian conditions. The network architecture is set up so that members of each such synchronized GPi group 7

8 (Gi 1 or Gi 2 ) send synaptic inhibition to the same TC neuron, and hence each TC neuron receives a rhythmic inhibitory signal in the parkinsonian network (see Figure 2), which disrupts the fidelity of TC relay responses to excitatory inputs, as discussed in more detail in section 2.3. Figure TC Relay Responses and Error Index We first define how we evaluate the TC relay fidelity. In the parkinsonian network described in section 2.2, the synaptic input from GPi to TC (the top trace in Figures 3B and 3C) is rhythmic and bursty. Although the TC neuron responds with a single spike to some of the excitatory inputs that it receives, others elicit either no spikes or multiple spikes (compare middle and bottom traces in Figures 3B and 3C). Figure 3 In this paper, we quantify relay performance of each TC neuron using a simple error index computed by dividing the total number of errors by the total number of excitatory inputs, namely error index = (b + m)/n, (2) where n is the total number of excitatory inputs. In equation (2), b denotes the number of excitatory inputs to which a TC neuron gives a bad response consisting of more than one spike, either a burst response (typically) or a single-spike response followed after a delay, but before the next input, by additional spikes. The number m denotes the count of excitatory inputs that are missed by the TC neuron, in the sense that it fails to fire any spikes during a detection window. This definition of errors guarantees that at most one error is counted for each excitatory input. The detection window we use in this paper extends from the beginning of each excitatory input to 12 msec after each input. This error index was first introduced in (Rubin & Terman, 24) and was used previously with the same error detection algorithm to quantify how different patterns of inhibitory GPi signals obtained from experimental recordings of normal and parkinsonian monkeys, with and without DBS (Hashimoto et al., 23), affect TC relay responses (Guo et al., 28) Averaged GPi Synaptic Input to TC In our network model, the synaptic input from the GPi to a TC neuron, I Gi TC, comes from a subgroup of GPi neurons. As illustrated in Figure 2, the subgroup that sends input to TC 1 is Gi 1, and the subgroup Gi 2 connects 8

9 to TC 2. Using vtc j to denote the membrane potential of neuron TC j, this input takes the form I Gij TC j = g Gi (vtc j vgi) k Ω j s k Gi j, j = 1, 2. (3) Here, each Ω j is an index set for neurons in Gi j, while g Gi is the constant maximal conductance and v Gi is the synaptic reversal potential for inhibition from GPi. Each s k Gi j satisfies the equation s Gi = α Gi (1 s Gi )S (ṽ) β Gi s Gi (4) where S (x) = (1 + e (x+57)/2 ) 1 and ṽ represents the membrane potential of the kth GPi neuron from subgroup Gi j (in fact, the exact half-activation value of -57 mv in S is not essential, as in our exploratory simulations, the GPi resting potential was always far enough below this value to avoid synaptic activation without threshold crossing). We also define sg 1 k Ω 1 s k Gi 1 and sg 2 k Ω 2 s k Gi 2 where sg 1 is the top trace in Figure 3B and sg 2 is the top trace in Figure 3C. Based on the form of equation (4), each s k Gi j is between and 1, and hence sg i [, 8] for each i. In our simulations, we use the variability of the time-average of each sg i as an indicator of GPi rhythmicity. Specifically, we form histograms based on the frequency with which each sg i time course, averaged over 25 msec time windows, takes different values in bins that cover the range [, 8]. We display 6 bins centered at 1 through 6, respectively, and each represents a subinterval of 1 ms/cm 2, except that all values less than 1.5 are placed in the 1 bin and all values greater than 5.5 are sorted into the 6 bin. In the parkinsonian network without stimulation, the average sg i values mostly fall into the 1 and 6 bins, as displayed in Figure 4. This result occurs because GPi firing is rhythmic and bursty (see the top traces in Figure 3B), such that GPi synaptic output is high during each burst and low between bursts. A few values do fall into the middle bins, due to transitions between bursting and quiescent phases. We shall see that very different results emerge when stimulation is applied to the STN neurons (Figure 9 in section 3.2 and Figure 11 in section 4.2). Figure 4 9

10 3. Coordinated Reset Stimulation We first investigate whether coordinated reset stimulation (CRS) can improve TC relay performance in the parkinsonian network. In a previous study, we evaluated the ability of a model TC neuron to relay excitatory inputs under the influence of inhibitory GPi signals generated from experimental data (Guo et al., 28). We found that GPi firing patterns, and corresponding synaptic outputs, produced in parkinsonian conditions without high frequency stimulation of STN switch rhythmically between low and high phases. Under the assumption that all STN neurons receive exactly the same train of high frequency pulses, CRS in which the pulse train is perioidly on significantly improved TC relay by inducing tonic, high frequency GPi activity that yielded approximately constant effective synaptic output levels (Rubin & Terman, 24; Guo et al., 28). Continual stimulation with strong, high frequency pulses has drawbacks in vivo, however, including relatively large energy requirements and potential for damage to surrounding tissue. Thus, we use our parkinsonian network model to explore whether there is a milder and more efficient stimulation technique that can improve TC relay responses. What we mean by milder is a signal with a smaller amplitude (or intensity) a of the pulses delivered to STN cells. Efficient here refers to a form of stimulation, such as coordinated reset stimualtion (CRS) that can be on for a certain time interval and then off for a rest interval, rather than applied continuously Methods The CRS given by the formula I stim k = a h(t)f k (t)f hi (t) (5) is applied to STN neurons through four stimulation sites as shown in Figure 5. The 16 model STN neurons, represented in the figure by solid circles, are arranged in a four by four grid centered at the + sign. The first row on the square grid are STN 1, STN 2, STN 3, and STN 4, from left to right. STN 5 to STN 8, STN 9 to STN 12, and STN 13 to STN 16 are on the second, third and fourth rows, respectively. We assume the distance d between two adjacent horizontal or vertical STN neurons is.1. The four small square boxes in the figure are the stimulation sites, which we number as 1, 2, 3, 4, proceeding clockwise from the upper left. Note that each 1

11 stimulation site is at the center of the square formed by the four nearest STN cells. Figure 5 There are several components in equation (5). Since H denotes the Heaviside step function, the function h(t) = H(t l 1 )(1 H(t l 2 )) equals 1 on (l 1,l 2 ) and outside of this interval and thus simply specifies that the overall stimulation period starts at time l 1 and stops at time l 2. Within this period, stimulation at electrode k is turned on and off as specified by the function f k (t), k = 1,...,4. Each f k (t) is a periodic step function with period 2.5τ for a constant parameter τ. Within each full period is a time interval of length τ during which f k (t) = 1, followed by an interval of length 1.5τ on which f k (t) =. We call these two intervals the ON period and OFF period, respectively, for electrode k. The function f hi (t) = H(sin(ρ 1 t) a 1 ) introduces a train of high frequency pulses ( 1 Hz). The product f k (t)f hi (t) therefore takes the form of a 2.5τ -periodic function consisting of a train of high frequency pulses delivered for τ time units and equal to during 1.5τ time units, repeated periodically. Finally, a is the amplitude of the stimulation. Since stimulation is distance-dependent, the four neurons directly surrounding site k receive the same stimulation. The stimulation administrated at the four different sites has the same overall period 2.5τ, with the same rate of high-frequency pulse delivery within the ON period, but there are phase shifts between stimulation sites so that the ON periods at the four sites do not coincide. The phase shift between the ON period start times at any two consecutively numbered stimulation sites is fixed at 42.5mssec, which is one fourth of the period of GPi bursting activity in the absence of stimulation. Figure 6 illustrates the relationships among the stimulation times at the four stimulation sites. Figure 6 We explore effects of stimulation over a region in (τ,a ) parameter space. The τ values used are centered around τ = 42.5 msec, which is one fourth of the period of GPi bursting activity in the absence of stimulation. The range of a values was selected empirically, spanning roughly from minimal values that give any change in TC relay performance to maximal values at which it is clear that results have saturated. As discussed below, we place extra focus on an area in the τ -a plane where we find that relatively mild and efficient CRS can give good TC relay performance Results CRS with particular choices of period and amplitude can reduce TC relay error dramatically. One example, generated with τ = 42.5 msec and a = 48, 11

12 is shown in Figure 7. In this case, the STN neurons form two clusters when there is no CRS, from to 5 msec into the simulation. From 5 to 2 msec, when CRS is on, the two clear STN clusters are no longer there, although there is still structure to the STN firing pattern. The stimulation is turned off again and the STN neurons gradually return to a two cluster firing pattern. Figures 7B and 7C show the synaptic input from GPi to TC (the top trace in both B and C) and the TC voltage trace (middle), illustrating the key result for our focus, namely that the TC spikes are more faithful to the the excitatory signals with stimulation on than before or after the stimulation period. Unlike the bimodal distribution of GPi synaptic outputs arising in the parkinsonian case (Figure 4), the sg i values under stimulation cluster in a few consecutive bins in the middle of their possible range (Figure 8). Figure 7 The simulation outcomes illustrated in Figures 7 and 8, as well as results from many other simulations, suggest that CRS can restore TC relay fidelity Figure 8 by changing the firing pattern of STN neurons. These transitions in activity patterns are typical for simulations across a range of τ and a values, although the rate of convergence to a steady clustered state after stimulation seems to speed up with larger τ and a and the precise STN activity patterns observed during and after stimulation depend on the times of onset and offset of stimulation and the stimulation parameter values. Although a systematic exploration of these dependencies is tangential to the investigation presented here, we do observe that a complete desynchronization of STN activity is not necessary; some break-up of clusters suffices to yield improvements in TC relay. The period τ and the amplitude a of the CRS are the two major parameters that can be used to optimize CRS results. First, we discuss the two extreme cases. The ON period τ can be small and very close to the period of the function f hi (t) that delivers the high frequency pulse. In this case, the stimulation is almost the same as the standard deep brain stimulation used clinically; correspondingly, the TC performance is good for all a (see the bottom rows, showing results for τ = 16.5, 18.5 msec, in Figure 9). The other extreme case is when a is strong enough to drive STN neurons to completely overcome their pathological bursting pattern (the right columns when a value is bigger than 65 in Figure 9). When the stimulation amplitude is higher than 65, the stimulation period is not critical, and overall the error is on the low side, unless the ON period τ becomes too large (e.g, 6 msec). For larger τ, the OFF period 1.5τ is long enough to let the STN 12

13 recover its parkinsonian bursting rhythm between stimulation periods, compromising relay. What we are interested in is finding a regime of optimal TC performance with a relatively weak stimulation amplitude. After running simulations of the computational model for various values of τ and a, we found a region that yields low error index values for both TC cells without excessive a values, given by τ values near one-fourth of the period of the GPi bursting and a between 45 and 6 (rectangular box in Figure 9). Figure 9 4. Multi-site Delayed Feedback Stimulation The second type of stimulation that we apply to our parkinsonian network is the multi-site delayed feedback stimulation based on LFP of STN population. Similar stimulation has been studied in other neuron models in previous work (Hauptmann et al., 25; Popovych et al., 26; Rosenblum & Pikovsky, 24b). There is no clear evidence on how the LFP is related to synaptic and ionic currents of a single neuron. Computational models sometimes simulate LFP by summing the membrane potential changes of all neurons of the network (Ursino & Cara, 26). Some authors adopt a simple computation as the sum of the absolute values of AMPA and GABA currents of pyramidal cells (Mazzoni et al., 28), while others consider a + sign for excitatory connections (or currents) and a - sign for inhibitory connections (or currents) (Tsirogiannis et al., 21). For multi-compartmental neuron models, the LFP has been computed as the low-pass filtered extracellular potential generated by the transmembrane currents across all compartments (Pettersen & Einevoll, 28; Pettersen et al., 28; Protopapas et al., 1998). According to current-source density analysis (Holt & Koch, 1998; Leung, 199; Mitzdorf, 1985; Nunez, 1981), the field potential depends on the linear sum of potentials from sources (currents injected into the extracellular medium) and sinks (current removed from the extracellular medium). The extracellular field potential Φ is governed by the Poisson equation (σ Φ) = I v (6) where σ is the conductivity of the extracellular medium, which is assumed to be isotropic and homogeneous, and I v is the current source density (CSD) computed as the sum of the membrane currents of the relevant neuronal population. For a single point source in an infinite homogeneous medium, 13

14 the solution to equation (6) is given by Φ = R ei v 4πr where r is the distance from the point source, I v is the current from that point source, and R e = 1/σ is the constant extracellular resistance Methods We use the current-source density to compute the local field potential in the center of the extracellular space in which the 16 STN neurons are embedded (Figure 5). The local field potential, which reflects the activity of all 16 STN cells, can be recorded by an electrode at the center of the population (the + sign in Figure 5) (Chen et al., 26; Yoshida et al., 21). Since we have 16 STN neurons, 16 presynaptic GPe cells, and stimulation administered at four sites, the point sources that we sum include the membrane currents from all 16 STN cells, the stimulation currents, and the inhibitory synaptic currents (with - signs) from presynaptic GPe cells. This sum appears in the formula for the LFP, V LFP (t) = R e 4π N j=1 (7) I j r j, (8) which is based on equation (7). In equation (8), the constant R e is the extracellular resistance that is set to 1 and r j is the distance between neuron j and the LFP measuring electrode. I j is the total current source from neuron j, which consists of ionic currents Ij ion and external currents Ij external, including both the stimulation and the synaptic currents from the presynaptic GPe neurons. Hence, we have I j = Ij ion + Ij external where Ij ion Ij external = Ij L + Ij Na + Ij K + Ij AHP + Ij Ca + Ij T = I Gi Sn j + I stim j The simulated LFP signal V LFP (t) is rescaled and filtered by a low-pass harmonic oscillator to generate the stimulation signal. We then apply the stimulation signal via four sites with time delays (Hauptmann et al., 25, 27, 28; Popovych et al., 26; Rosenblum & Pikovsky, 24a,b). The low-pass filtering of V LFP (t) is implemented by the equation 14

15 x + ax + bx = µv LFP (t) (9) where a and b are parameters selected to satisfy the condition a 2 < 4b to guarantee that (9) represents a harmonic oscillator. The parameter µ controls the strength of the stimulation. We first choose the values of a =.25 and b =.136 so that the period of the harmonic oscillator is the same as the natural frequency of the bursts present in the STN clusters. Later, in section 4.3, we use various values of a and b to explore how the frequency of the filter can affect the desynchronization of STN clusters. The stimulation that the jth STN neuron receives from four sites is given by I stim j = h(t) n 4 e 2dist(j,k) x k (t (k 1)τ) (1) k=1 where h(t) defines the stimulus onset and offset times as in equation (5), n is the number of STN cells, dist(j,k) is the distance between the jth neuron and the kth stimulation site, and x k (t (k 1)τ) is the time delayed signal from equation (9) that is delivered at the kth stimulation site. We assume that the STN neurons are arranged in a square grid as shown in Fig (5) with a distance d =.1 between two adjacent horizontal or vertical grid points. We place each stimulation site at the center of a group of four STN cells as seen in Fig (5). Hence we can calculate the dist(j,k) in the two dimensional Euclidean space. For example, the distance between STN 2 and stimulation site 4 is dist(2, 4) = Results We found that, when the parameters are properly tuned, multi-site delayed feedback stimulation (MDFS) can suppress the output of STN neurons in our parkinsonian network. TC relay errors are correspondingly reduced dramatically. Figure 1A shows that under multisite feedback stimulation, the STN neurons do not burst or form synchronized clusters during the stimuliation period, from 5 to 2 msec. Figures 1 B and C show the improved TC relay performance during stimulation. During the MDFS, the averaged values of sg i, i = 1, 2, are low. Histograms of these values show that these GPi output measures mainly lie in the bin centered at 1 (Figure 11) due 15

16 to the suppression of activity in the STN population and hence of synaptic excitation from STN to GPi. The suppression is not due to individual STN neurons or GPe neurons becoming dominant. Rather, it is a population effect. Figures 12A and 12B show the temporal pattern of ionic currents for two representative STN neurons. These currents switch between positive and negative values over time. The sum of ionic currents of STN neurons in one cluster also exhibits positive and negative trends (12C and 12D). The sum of ionic currents over all 16 STN cells (Figure 13A) apparently never becomes positive when the stimulation is on, nor does the external current I external (Figure 13B), which includes both synaptic currents from all GPe cells and the stimulation current (Figures 13C and 13D). Figure Multi-site delayed feedback stimulation (MDFS) with different stimulation strengths and periods Figure 11 Figure 12 We investigated the effectiveness of MDFS with various choices of parameter values in the low-pass filter. Specifically, we fixed a at the small value Figure 13 a =.1 to ensure that the oscillation condition a 2 < 4b would hold for a wide range of b (see equation (9)), and then we varied b. We reasoned that a powerful stimulation signal would be generated most efficiently when the 4π filter period, 4b a 2, is close to the natural period of STN bursting. Thus, we varied b in the interval [.35,.6], chosen because when b =.35 and b =.55, the period of the filter is approximately twice and one half of the natural period of STN bursting, respectively, and examined network performance over a range of µ values for each fixed b. Both of these extreme b values are marked on Figure 14. As illustrated in Figure 14, for each choice of b, there is a corresponding range of µ values for which MDFS gives good TC relay performance. Specifically, the region between the lines in Figure 14 is the area of b and µ in which TC error is between and.3. As the period of the filter decreases, stronger stimulation is required to achieve an error index value of less than.3. The area below the good performance region gives error index values above.3. When parameters are selected in the area above the good performance region, STN activity is completely suppressed. Figure 14 To further explore relay performance under MDFS stimulation, we ran additional simulations for a selection of vectors in the (µ, a, b) parameter space, respecting the bounds.25 < µ <.36,.2 < a <.31, and.128 < b <.142 with a 2 < 4b. When µ is too small, the stimulation does not desynchronize STN clusters well enough to improve TC 16

17 relay performance (top row in Figure 15). We find that effective desynchronization can be obtained, without complete cessation of STN activity, for µ values from.28 to.33, for a range of values of a, b and fixed τ = 42.5 msec. For example, when µ =.3, for.128 < b <.132 and.2 < a <.31, the TC error is low (see the right figure in the second row in Figure 15). Figure 15 The use of stimulation derived from the LFP signal recorded from STN neurons forms a closed-loop feedback control mechanism. When the stimulation amplitude µ is increased, the suppression of STN neurons is strengthened, and the STN activity can be shut down (such that none of the STN neurons fire any spikes) for a short period of time, after which isolated spikes emerge. In theory, there is no complete shutdown of STN neurons because the LFP is pulled upward toward zero when STN activity drops off, and therefore the stimulation current at each site moves toward zero. This effect results in progressively less suppression, until the STN neurons are released to resume firing. In Figure 16, all STN neurons are completely shut down from the beginning of the stimulation to more than half way through the stimulation period (about 14 msec into the simulation, with the stimulation on from 5 to 2 msec). The corresponding µv LFP (t) and stimulation current increase smoothly during the same period until they are very close to zero, releasing the STN cells from suppression. Since we have a finite stimulation duration, a larger µ can prolong the suppression of STN neurons sufficiently that it lasts throughout the whole stimulation period. Correspondingly, the TC relay performance is perfect (e.g., the right figure on the bottom row in Figure 15) because TC neurons can respond to their excitatory inputs without interference from GPi inhibition. Although we assume that perfect TC relay is desirable, it seems unlikely that elimination of STN and GPi activity would represent an optimal state. Figure Comparison with constant negative current stimulation Given that MDFS restores TC relay fidelity by reducing STN neuron firing, it is reasonable to consider the simpler intervention of applying a constant negative current to STN neurons, without recording and feeding back an LFP signal. We used our model to investigate whether a constant negative stimulation current would work the same as delayed feedback stimulation. We find that there is a narrow range of constant negative current strengths that can induce fairly good TC relay performance without eliminating STN activity. Once the negative current is outside of that range, it either does not have 17

18 a significant effect on TC relay or it completely suppresses STN neurons. Thus, if extreme STN suppression is to be avoided, then constant negative current stimulation (CNCS) must be much more carefully tuned than MDFS to achieve significantly improved TC relay. There are also other advantages of MDFS, relative to constant negative current stimulation. First, the MDFS that we describe is a closed-loop feedback control mechanism. Any changes in STN population activity will lead to automatic adjustment of the LFP signal, as discussed in section 4.3, eliminating the need for manual retuning. Second, we can consider how the suppressed STN neurons will respond to excitatory cortical inputs, such as from the hyperdirect pathway, with constant negative current stimulation (CNCS) and with MDFS, as well as in the absence of DBS, with standard HFS, and with CRS for comparison. To do so, we add an extra term, I cor Sn, representing excitatory cortical input, to the STN voltage equation, such that it becomes C m v Sn = I L I Na I K I T I Ca I AHP I Ge Sn + I stim + I cor Sn. I cor Sn consists of a sequence of square pulses, which we generated using a Poisson process since we are not aware of any data to suggest that other fast time scale structure is present in this input stream. We find that this excitatory input to the STN neurons overwhelms the negative constant stimulus and causes bursty STN activity (with a different frequency compared to the bursts arising in the parkinsonian network). This STN activity significantly compromises TC relay fidelity. In the MDFS case, this excitatory input to the STN induces isolated spikes in STN cells and maintains good TC relay performance. To compare across all of the stimulation (or non-stimulation) types listed above, we performed 4 independent trials of TC relay responses in each case. In each trial with stimulation, the stimulation was turned on from 1 msec to 25 msec. We performed trials with periodic excitatory inputs to TC cells, to match earlier simulations in the paper, as well as with Poisson inputs to TC cells, to allow for a statistical similarity between cortical inputs to different areas. We find that the TC relay error index is always high, near non-stimulation levels, with negative constant stimulation, while it becomes lower for HFS and lower still for CRS and MDFS; see Figure 17. Although the high frequency pulse train used in HFS is the same as the one used in CRS (function f hi (t) in Section 3.1, with the same parameter values for ρ 1 and a 1 ), CRS yields a better relay performance, as seen in Figure 18, 18

19 with the same strength a of stimulation current. To achieve this better performance with HFS, a must be raised to a higher level, which would increase energy consumption and could possibly damage tissue. Figure Multi-site delayed feedback stimulation for heterogeneous TC cells As a final step, we verified that the effectiveness of MDFS at restoring TC neuron relay fidelity is not specific to the baseline parameter values that we use for our model TC neuron. To do so, we generated a population of 4 model TC neurons with heterogeneity in their parameter values by independently selecting g L,g Na, and g T from normal distributions of standard deviations.1,.5, and.8, respectively centered around their baseline values, as was done in previous work (Guo et al., 28). We ran simulations in which time courses of GPi inhibition to the TC neurons were produced by the upstream STN-GPe loop and these fixed time courses were used as synaptic inputs to all members of the heterogeneous TC population. This was done with no applied stimulation as well as with CRS and MDFS forms of stimulation considered in Figure 17. All members of the TC population showed significant decreases in error index in the parkinsonian network with CRS or MDFS stimulation, compared to the case without stimulation, as illustrated in Figure 18. The baseline parameter values used for the TC neurons are given in the Appendix and the stimulation parameters are given in the caption of Figure 18. Figure Discussion In this paper, we consider a network of synaptically-connected, conductancebased model neurons from the STN, GPe and GPi in the basal ganglia, based on previous modeling work (Terman et al., 22; Rubin & Terman, 24; Guo et al., 28). The model is tuned to generate activity patterns featuring synchronized, rhythmic bursts fired by clusters of neurons, with different clusters bursting in alternation, which we take to represent a parkinsonian state. Inhibitory outputs from the GPi are rhythmic (Fig. 3 and Fig. 4) and target model TC neurons that also receive excitatory input trains. We find that the TC neurons are unable to respond reliably to these inputs, in agreement with earlier theory and simulations (Rubin & Terman, 24; Guo et al., 28). Earlier computational studies on alternatives to standard DBS paradigms 19

20 for Parkinson s disease have identified several promising approaches involving the delivery of stimulation at multiple sites within the STN (reviewed in Hauptmann et al. (27)). We test two such approaches, one involving time-shifted coordinated reset stimulation with a pre-determined pattern and the other involving feedback of filtered LFP signals recorded from within the STN (Rosenblum & Pikovsky, 24b; Hauptmann et al., 25, 27; Tukhlina et al., 27). Both approaches significantly improve TC relay fidelity, the former by reducing the rhythmicity of the net inhibitory input from GPi to each TC neuron and the latter by reducing STN activity. Thus, both do appear to be worthy of additional consideration for possible therapeutic use with PD patients. A trivial way to achieve reliable TC relay in our model is to eliminate STN activity. Although LFP-based delayed feedback stimulation does suppress STN firing and prevent bursting, it does not cut out STN activity completely. Indeed, as STN activity wanes, a reduction of the LFP signal results, as noted by previous authors (Hauptmann et al., 25; Tukhlina et al., 27), until a balance between the activity and the stimulation signal is achieved. Importantly, the LFP signal is generated directly by the STN network and its inputs, and hence would not need to be fine-tuned by a clinician to achieve its effects, unlike a prescribed suppressive current. Moreover, although the elimination of STN activity and restoration of TC relay can be achieved by an imposed inhibitory stimulation of a similar magnitude to the time-averaged LFP signal, such stimulation yields abnormal STN responses to cortical inputs, as might arrive through the hyperdirect pathway, with an associated compromise of relay, whereas effective relay persists despite cortical inputs under MDFS. Previous authors have assessed the performance of STN stimulation based on recorded LFP signals in terms of its desynchronizing effect on model neurons. In particular, Hauptmann et al. (Hauptmann et al., 27) review results showing that, in a network of STN neurons modeled using the conductance-based Morris-Lecar equations, such stimulation greatly reduces an order parameter quantifying phase synchronization without significantly lowering the STN burst rate. Although abnormally high synchronization within various basal ganglia nuclei is correlated with the presence of parkinsonian symptoms, however, a complete understanding of how DBS works requires the elucidation of a causal connection from synchrony to motor outputs. Our work suggests one possibility, namely that synchrony within STN translates into synchrony of GPi activity and outputs to TC neurons, com- 2

21 promising TC relay to motor cortex (Rubin & Terman, 24; Guo et al., 28). Part of the novelty of this study lies in the assessment of the effectiveness of LFP-based stimulation in terms of its impact on TC relay, and this metric leads us to the conflicting prediction that effective LFP-based delayed feedback stimulation does alter the STN burst rate; specifically, this stimulation must be strong enough to reduce synchronized STN bursting in order to achieve therapeutic benefit. Importantly, this prediction does not represent a challenge to the practical utility of LFP-based delayed feedback stimulation, since our results indicate that the STN can still respond to inputs in the presence of this stimulation. While additional insights may be gained from future simulations involving larger network models, a thorough assessment of the relative merits of MCRS, LFP-based delayed feedback stimulation, and other forms of high frequency DBS will require a better understanding of the role of STN activity patterns in motor processing. Meanwhile, combining our results with the finding that simulated standard DBS becomes more effective when it reaches a larger portion of the STN population (Hahn & McIntyre, 21), we can at least conjecture that the use of multiple stimulation sites will be advantageous as long as bursting is sufficiently reduced or desynchronized in a large enough STN subpopulation. This perspective is also supported by data and simulations showing that therapeutic STN-DBS reduces bursting in GPi (Hahn et al., 28; Hahn & McIntyre, 21) and by computational findings that the temporal profile of the inhibitory conductance from GPi to TC neurons is a key predictor of TC rebound burst firing and relay performance (Cagnan et al., 29; Pirini et al., 29; Dorval et al., 21). One possible conclusion from these and other studies is that the regularity of DBS is essential to its success (Dorval et al., 28, 21). In particular, standard STN-DBS more effectively relieved bradykinesia in PD patients when it was regular than when it was irregular, and the introduction of aperiodicity into standard DBS stimulation signals to STN compromised its improvement of TC relay performance (Dorval et al., 21). Our findings, however, suggest that classifying stimulation patterns by regularity alone may be insufficient for predicting their therapeutic utility. Irregular patterns, such as the MDFS that we consider, may still achieve improved relay, and hence represent candidates for therapeutic application, as long as they change GPi output in a way that allows TC neurons to respond to excitatory inputs reliably. The pathway from GPi to VLa thalamus is part of an anatomically-distinct motor circuit that projects to motor cortical areas (Baron et al., 22; DeLong and Wichmann, 27; Samuel et al., 1997), which 21