Exploiting pallidal plasticity for stimulation in

Journal of Neural Engineering J. Neural Eng. 12 (2015) 026005 (20pp)

doi:10.1088/1741-2560/12/2/026005

Exploiting pallidal plasticity for stimulation in Parkinson’s disease Marcel A J Lourens1,3, Bettina C Schwab1, Jasmine A Nirody2, Hil G E Meijer1 and Stephan A van Gils1 1

MIRA: Institute for Biomedical Technology and Technical Medicine, University of Twente, Enschede, 7500 AE, The Netherlands 2 Biophysics Graduate Group, University of California, Berkeley, CA 94720, USA E-mail: [email protected] and [email protected] Received 13 December 2014 Accepted for publication 9 January 2015 Published 4 February 2015 Abstract

Objective. Continuous application of high-frequency deep brain stimulation (DBS) often effectively reduces motor symptoms of Parkinson’s disease patients. While there is a growing need for more effective and less traumatic stimulation, the exact mechanism of DBS is still unknown. Here, we present a methodology to exploit the plasticity of GABAergic synapses inside the external globus pallidus (GPe) for the optimization of DBS. Approach. Assuming the existence of spike-timing-dependent plasticity (STDP) at GABAergic GPe–GPe synapses, we simulate neural activity in a network model of the subthalamic nucleus and GPe. In particular, we test different DBS protocols in our model and quantify their influence on neural synchrony. Main results. In an exemplary set of biologically plausible model parameters, we show that STDP in the GPe has a direct influence on neural activity and especially the stability of firing patterns. STDP stabilizes both uncorrelated firing in the healthy state and correlated firing in the parkinsonian state. Alternative stimulation protocols such as coordinated reset stimulation can clearly profit from the stabilizing effect of STDP. These results are widely independent of the STDP learning rule. Significance. Once the model settings, e.g., connection architectures, have been described experimentally, our model can be adjusted and directly applied in the development of novel stimulation protocols. More efficient stimulation leads to both minimization of side effects and savings in battery power. Keywords: spike-timing-dependent plasticity, external globus pallidus, neuronal synchrony, GPe–GPe synapses, alternative stimulation protocols (Some figures may appear in colour only in the online journal) bursts, have slightly altered firing rates and exhibit abnormally synchronized oscillatory activity (Hammond et al 2007, Galvan and Wichmann 2008). Single-unit and local field potential recordings have demonstrated that the external part of the globus pallidus (GPe) and the subthalamic nucleus (STN), which are parts of the BG-thalamocortical loop, exhibit a tendency to oscillate and synchronize at low frequencies (3–30 Hz) in the parkinsonian state (Bergman et al 1994, Nini et al 1995, Raz et al 2000, Brown et al 2001, Mallet et al 2008). Beta-frequency oscillations (13–30 Hz) are frequently believed to be responsible for bradykinesia and rigidity in PD patients (Kühn et al 2006, Weinberger et al 2006, Ray et al 2008, Kühn et al 2009), whereas theta

1. Introduction Prevalent motor symptoms of Parkinson’s disease (PD) include muscle rigidity, tremor of the limbs at rest, slowness and impaired scaling of voluntary movement (bradykinesia), loss of voluntary movements (akinesia) and postural instability (Jankovic 2008). Although it is not clear what causes the symptoms, they are often related to altered neural activity in the basal ganglia (BG), induced by dopamine depletion. In PD, neurons in these nuclei tend to discharge in 3

Present address: Medtronic Eindhoven Design Center, High Tech Campus 41, 5656 AE Eindhoven, The Netherlands.

1741-2560/15/026005+20$33.00

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motor function (Tass et al 2012). Recently, Adamchic et al (2014) have demonstrated in six PD patients that CR-stimulation of the STN leads to a significant and cumulative reduction of beta band activity, resulting in a significant improvement of motor function. In this study, we introduce a methodology to exploit the plasticity of intrinsic synapses of the GPe for improving the efficiency of DBS in PD treatment. Intrinsic inhibitory synapses of the GPe seem to have a pivotal role for dynamics of the BG, in particular in terms of synchrony (Miguelez et al 2012, Bugaysen et al 2013). They show short-time depression (Sims et al 2008, Miguelez et al 2012, Bugaysen et al 2013) and changes in synaptic conductances after dopamine depletion (Miguelez et al 2012, Wichmann and Smith 2013). Up till now, spike-timing-dependent plasticity (STDP) at these synapses has not been investigated. However, STDP (having variable learning rules) seems to also be a common feature in GABAergic synapses (Caporale and Dan 2008). Our model describes the network dynamics of both STN and GPe. Cell dynamics are governed by the biophysically plausible single-compartment models as proposed by Rubin and Terman (2004). We connect these cell models together via a sparse structured architecture. Notably, STDP is introduced for the experimentally established inhibitory connections between GPe cells. Quantifying the levels of neural synchrony, we show how DBS can be adapted to optimally use the plasticity. Although the appropriate settings of the model must be determined by experiments, we create a framework that can be used to optimize stimulation protocols.

oscillations (3–10 Hz) have often been associated with tremor (Levy et al 2000, Steigerwald et al 2008, Tass et al 2010, Contarino et al 2012). A common therapy for severe PD motor symptoms is high-frequency (>100 Hz) deep brain stimulation (DBS) of the STN (Hamani et al 2006, Benabid et al 2009). With optimized stimulation parameters, established empirically (Rizzone et al 2001, Moro et al 2002, Volkmann et al 2002, 2006) and confirmed theoretically (Cagnan et al 2009, Dorval et al 2010, Meijer et al 2011), STN–DBS is able to reduce dyskinesia and to improve motor symptoms including tremor, bradykinesia and rigidity (Krack et al 2003, Rodriguez-Oroz et al 2005). However, STN–DBS is less effective for gait disturbance and postural instability, and its therapeutic benefits often decline over time (Krack et al 2003, Rodriguez-Oroz et al 2005). Furthermore, STN–DBS may cause adverse effects including cognitive decline, speech difficulty, instability, gait disorders and depression (Rodriguez-Oroz et al 2005). Still, the working mechanism of DBS is not fully understood yet, impeding the improvement of DBS techniques. It has been both suggested that DBS excites (Garcia et al 2003) or inhibits neural activity (Obeso et al 2000, Levy et al 2001, Wu et al 2001). McIntyre and Hahn (2010) hypothesized that high-frequency stimulation disrupts or desynchronizes the pathological activity by changing the underlying dynamics of the stimulated brain networks, which could be achieved via activation, inhibition, or lesion. Several studies (Paul et al 2000, Windels et al 2000, Anderson et al 2003, Hashimoto et al 2003) could show that synaptic outputs are enhanced in stimulated areas (Rubin and Terman 2004). Recently, Rosenbaum et al (2014) suggested that high-frequency stimulation leads to somatic inhibition of neurons that are close to the electrical field due to failure of axons and synapses. Both cellular and network effects may contribute to the clinical effect of DBS. In the clinic, interesting phenomena have been reported that cannot directly be explained by standard models of DBS. When stimulation is turned off, the symptoms revert back gradually: tremor within minutes, bradykinesia and rigidity within half an hour to an hour, and axial signs within 3–4 h (Temperli et al 2003). When the stimulator is turned on again, the symptoms improve in the same order, but at a faster rate. This observation implies that the DBS-induced dynamical changes have a long-lasting effect, suggesting that different pathophysiological mechanisms underlie the major PD symptoms. Synaptic plasticity could explain such long-lasting effects. Thus, DBS may start a cascade of long term changes, up-regulating some synapses and down-regulating others, slowly reversing when the stimulator is switched off. To exploit the effect of plasticity at excitatory synapses for DBS, Tass and colleagues have proposed coordinated reset (CR) stimulation, a short-duration desynchronizing stimulation protocol that leads to a therapeutic rewiring of neuronal networks (Tass 2003, Hauptmann and Tass 2009, Popovych and Tass 2012). In rat epileptic hippocampal slices, it was shown that the CR-stimulation has long-lasting desynchronizing effects (Tass et al 2009). Parkinsonian nonhuman primates receiving CR-stimulation showed long-lasting after-effects on

2. Methods 2.1. Neuron models for STN and GPe

The dynamics of each STN and GPe cell are represented by a single-compartment conductance-based model as proposed by Terman et al (2002) and Rubin and Terman (2004). To produce an action potential, we include a sodium current (INa) in each cell, a potassium current (IK) and a leak current (IL). Each cell model contains also the following types of ionic currents (Iion): a calcium-activated,voltage-independent afterhyperpolarization potassium current (IAHP), a highthreshold calcium current (ICa) and a low-threshold T-type calcium current (IT). In addition to these ionic and leak currents, each STN and GPe cell receives a synaptic current (Isyn) and an applied current (Iapp). The rate of change of the membrane potential (Vm) for each cell is given by Cm

dVm = −IL − Iion − Isyn + Iapp, dt

(1)

where Cm is the membrane capacitance. We use the equations and parameter values for leak and ionic currents as described in Terman et al (2002), but adopt modifications as in Rubin and Terman (2004) and Guo and Rubin (2011) to match in vivo firing patterns more closely (appendix A). 2

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Figure 1. The structured, sparsely connected network architecture adopted from Rubin and Terman (2004). Based on the connectivity, the network is divided into two subpopulations, each consisting of two STN groups and two GPe groups. Each group contains four cells, represented by solid circles and numbered separately for each cell type, projecting to and receiving input from the other groups as illustrated by the lines. A solid line denotes a strong connection, whereas a dashed line denotes a weak connection. Lines ending with arrows and circles indicate excitatory glutamatergic and inhibitory GABAergic synaptic connections, respectively. Each GPe cell receives inhibitory input from two other GPe cells of its own subpopulation, randomly chosen, and receives excitatory input from three STN cells, two of which come from its own subpopulation. Each STN cell receives inhibitory input from two GPe cells of its own subpopulation. In addition to the synaptic inputs, all cells receive direct current injection as indicated with the two double arrowed lines. Note that the connections between individual cells are not shown.

activity in a small network without the tendency to synchronize (Terman et al 2013). As in Rubin and Terman (2004) the synaptic current from j ∈ J presynaptic cells of nucleus α to a postsynaptic cell i of nucleus β is modeled as:

2.2. Network architecture

We use a structured, sparsely connected architecture given in Rubin and Terman (2004) for the synaptic connections between the STN and GPe cells (figure 1). The network consists of two subpopulations of eight STN and eight GPe neurons each, connected to each other via weak synaptic connections. Each subpopulation is further subdivided into four groups (STN 1, STN 2; GPe 1, GPe 2) of four neurons each (figure 1). Neurons within the same group provide synaptic inputs to the same target groups. An STN group sends excitatory input to a GPe group within its own population as well as weaker excitatory input to the corresponding GPe group in the other subpopulation. For example, an STN cell in STN group 1 of subpopulation 1 may provide excitatory input to a cell in GPe group 2 of subpopulation 1 and weaker input to a GPe cell in GPe group 2 of subpopulation 2. In turn, each GPe group inhibits one group of STN neurons of its own subpopulation. Within each subpopulation, there are also local inhibitory connections between GPe neurons. Finally, each cell (GPe and STN) receives a constant applied current input (Iapp in (1)) representing net input from other brain structures. These currents are used to tune the firing rates and the network state, see section 2.3. The use of four cells in each of the eight groups is justified by the fact that simulations with larger number of cells in each group result in the same main findings, see appendix B. Moreover, inhibitory networks such as within the GPe can robustly exhibit irregular

J

(

i Iαi →β = gα→ β V m, β − Eα→β

) ∑w

α, j → β , i

sαj ,

(2)

j=1

where gα → β and Eα → β are the maximal synaptic conductance and reversal potential for connections from presynaptic cells of nucleus α to postsynaptic cells of nucleus β, respectively, with α and β representing STN or GPe. The summation is taken over cells in nucleus α with the synaptic weight (wα, j → β, i ) that projects to cell i of nucleus β. The synaptic weight is set to 1 for all connections from GPe to STN and STN to GPe, except for the weak excitatory connections which are set to 0.2. For the connection within the GPe, the synaptic weight (wGPe, j → GPe, i , from now on abbreviated as wij) is adjusted manually (section 2.3) or under control of plasticity (section 2.4) to obtain different activity patterns. For both STN and GPe cells the kinetics of the rise and decay of the synaptic variable sαj are described by a first order process: dsαj = A α 1 − sαj S∞ V m,j α − Bα sαj , dt

(

) (

)

(3)

where S∞ (x ) = 1 (1 + exp (−(x − θα ) σα )). The kinetic 3

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parameters for STN and GPe are [Aα , Bα , θα , σα ] = (5, 1, −9, 8), (2, 0.04, −37, 2), respectively (Terman et al 2002, Rubin and Terman 2004).

described and observed extensively for excitatory synapses, it has also been observed in inhibitory synapses (Holmgren and Zilberter 2001, Woodin et al 2003, Haas et al 2006). Experimental results show alterations in the coupling strength between the GPe cells in parkinsonian conditions (Stanford and Cooper 1999, Ogura and Kita 2000). In this study, the inhibitory connections within the GPe cells are subject to STDP. The synaptic weight (wij) is updated with an additive nearest-spike pair-based STDP rule:

2.3. Healthy and parkinsonian states of the network

In PD patients and in animal models of PD, electrophysiological changes with respect to healthy conditions have been observed in neurons of the BG, including a tendency of neurons to discharge in bursts, increased interneuronal synchrony and oscillatory activity (see Galvan and Wichmann 2008, and references therein). Terman et al (2002) demonstrated that an STN–GPe network described in the previous section can display rhythmic activity, uncorrelated irregular spiking or propagating waves. The observed activity depends on many characteristics of the network, including its architecture, the strengths of synaptic connections between STN and GPe neurons and among GPe neurons, and the input to the network. Our network, which has a sparse structured pattern of connections, can also display both healthy (uncorrelated firing) and PD (highly correlated clustered firing) activity, depending on the strength of the synaptic connections between the cells and the inputs to them. In section 2.5, we give a quantitative measure of the activity. We determine appropriate coupling and input parameters such that the network model’s activity mimics either the experimentally observed activity in PD or healthy conditions. First, we adjust the synaptic strengths for the connections from GPe to STN, STN to GPe and within the GPe by changing their maximal synaptic conductance (gGPe → STN , gSTN → GPe and gGPe → GPe ), and the value of the applied current for both cell types (Iapp,STN and Iapp,GPe) to model the PD activity. For the connection within the GPe we also change the synaptic weight (wij) to adjust their synaptic strengths. Except for gGPe → GPe and wij, we take the values given by Guo and Rubin (2011): Iapp,STN = 0 μA cm−2, Iapp, −2 −2 and gSTN → GPe GPe = −1.2 μA cm , gGPe → STN = 0.9 mS cm −2 = 0.18 mS cm . The value for gGPe → GPe is set to 1 mS cm−2 and wij is set to 0.1 for all connection within the GPe. Second, we define a different parameter set for the PD situation, letting the network display irregular activity. Following the approach of Rubin and Terman (2004), we vary only Iapp,GPe and wij, leaving the other four parameters (Iapp,STN, gGPe → STN , gSTN → GPe and gGPe → GPe ) unchanged, for this transition. In the next section, we incorporate synaptic plasticity.

( )

wij ( tn + 1) = wij ( tn ) + δΔwij Δtij ,

(4)

where δ is the update rate and Δwij is the synaptic modification, which depends on the temporal difference Δtij = ti − tj between the nearest onsets of the spikes of the pre-synaptic neuron j and post-synaptic neuron i. In experiments, the observed plasticity time windows, describing the relation between the time difference and the synaptic modification, vary substantially (Holmgren and Zilberter 2001, Woodin et al 2003, Haas et al 2006, Caporale and Dan 2008). Following Popovych and Tass (2012) we use an asymmetric time window for STDP of inhibitory synapses, given by

( )

Δwij Δtij

⎧ ⎛ γ Δtij ⎞ ⎪ −β1 exp ⎜ − 1 ⎟ ⎝ τSTDP ⎠ ⎪ ⎪ =⎨ ⎛ γ Δt Δtij ⎪ ⎜ − 2 ij β exp ⎪ 2τ ⎜ τSTDP ⎪ STDP ⎝ ⎩

for Δtij ⩾ 0, (5) ⎞ ⎟ for Δtij < 0. ⎟ ⎠

This is an anti-Hebbian STDP update window (figure 2(a)), i.e., the synaptic strength between the GPe cells is potentiated or depressed depending on whether the postsynaptic spike comes before or after the pre-synaptic spike, respectively (Popovych and Tass 2012). An anti-Hebbian weight modification has been observed for inhibitory synapses (Holmgren and Zilberter 2001, Caporale and Dan 2008). In our model the weight increase and decrease are independent of the present weight of a synapse. Therefore, an upper bound of 0.6 and a lower bound of 0.0001 is placed on each synaptic weight to avoid unbounded growth and negative conductances. In Popovych and Tass (2012) the net effect of STDP, denoted as Δw , resulting from the difference between the time windows for depression and potentiation (Kepecs et al 2002), is calculated for uniformly distributed relative firing times Δt in the interval Δt ∈ [−ϵ, ϵ] by: Δw (ϵ) =

2.4. Synaptic plasticity

1 2ϵ

ϵ

∫−ϵ Δw (ξ)dξ.

(6)

The parameter values for the STDP window can be tuned such that the net effect of STDP results in down- or upregulation of the synaptic weights between GPe cells when they fire in synchrony (Δt is narrowly distributed, i.e., ϵ small) or in an uncorrelated manner (Δt is broadly distributed, i.e., ϵ large), respectively (Popovych and Tass 2012). The parameter values for the STDP window are given in table 1 and gGPe → GPe (maximal synaptic conductance) is set to 1 mS cm−2 in the simulations with plasticity. Note that the coupling strength between the GPe cells is defined by the product of

The model we described above considers the synaptic weight between a presynaptic cell to a postsynaptic cell to be static. However, several experiments have shown that the strength of synaptic connections changes depending on the relative spike timing of the pre-synaptic and post-synaptic neurons within a short time window (Magee and Johnston 1997, Markram et al 1997, Bi and Poo 1998, Zhang et al 1998, Feldman 2000, Sjöström et al 2001). This kind of synaptic plasticity is referred to as STDP. Although STDP has been 4

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Figure 2. Time window for STDP of inhibitory synapses (a) and its effective time window (b) as defined by Popovych and Tass (2012). The

STDP time window shows the prescribed change in synaptic weight changes as a function of the time difference between the pre- and postsynaptic spikes ( Δt = tpost − tpre ). The synaptic weight is potentiated when a post-synaptic spike precedes a pre-synaptic spike, and depressed when a pre-synaptic spike precedes a post-synaptic spike. The effective time window shows whether on average strengthening (uncorrelated spike trains, ϵ large) or weakening (correlated spike trains, ϵ small) of the synapse occur for uniformly distributed Δt ∈ [−ϵ, ϵ].

Best et al (2007) is able to detect the number of synchronous clusters in a population. In short, the method is based on principal component analysis (PCA). We compute the eigenvectors (vi) of the covariance matrix for the meanadjusted voltage traces and sort them in decreasing order by eigenvalue. Projection of the dataset onto the subspace generated by the first n eigenvectors {v1, v2, …, vn−1, vn}, n ⩽ N , where N is the number of STN (or GPe) cells in the network, captures the maximal possible variance within an ndimensional subspace. After a certain linear transformation, n variables are sufficient to describe a certain percentage of the data. For example, in our network, if two variables describe most of the population activity, it strongly suggests the existence of two clusters. The fraction of the variation in the data that is captured by the subspace spanned by the n largest eigenvectors equals ηn, where

Table 1. Parameters used in implementation of STDP.

Parameter

Value

β1 β2 γ1 γ2 τSTDP δ

1 5 4 2 8 0.004

gGPe → GPe and wij. In our simulations gGPe → GPe is always 1 mS cm−2, therefore, wij corresponds to the coupling strength. 2.5. Analysis of network activity

The PD state in our model is characterized by synchronized activity within each cluster and cells firing in a burst-like pattern. These characteristics are readily observed visually in a raster plot of spike times. However, to quantify the level of network synchrony and burstiness, as well as how they are affected by stimulation and network parameters, we used quantitative measures as follows.

ηn =

λ1 + λ 2 + ⋯ + λ n , λ1 + λ 2 + ⋯ + λ N

n ⩽ N.

(7)

We report the number of principal components (eigenvectors) required to capture at least 90% of the variation in the data. Because of memory issues, we store only the spike times of the neurons instead of their membrane voltage traces. To perform the PCA, each spike train i is transformed into a continuous waveform x˜i by convolving each spike with a Gaussian filter with a standard deviation σ = 7.5 ms,

There are several measures that quantify the level of synchrony in the network, i.e., the extent to which cells in a population (GPe or STN) spike at the same points in time, including both firing time measures (Kuramoto 1984, Strogatz and Mirollo 1991, Pinsky and Rinzel 1995) and continuous time measures (Golomb and Rinzel 1994). Most of these measures are not really suitable to assess the level of population synchrony if the population consists of clusters of cells which fire only approximately synchronously within each cluster. The method proposed in

Number of principal components:

⎛ i ⎜ t − tk x˜i (t ) = ∑ exp ⎜ − ⎜ σ2 k=1 ⎝ Si

(

) ⎞⎟, 2

⎟⎟ ⎠

(8)

where Si is the number of spikes in spike train i and tki is the spike time of kth spike in spike train i. 5

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k Figure 3. Stimulation protocol for CR. (a) Schematic representation of the administration of Istim , k = 1 ,..., 4 , via four electrodes, indicted by

k . (b) Time series of the the markers +, to the four STN groups defined in section 2.2. Each STN group receives its own stimulation signal Istim stimulation signals. Within stimulation ON periods of length τCR, the stimulation signal consists of a train of high-frequency pulses, given by (10). The stimulation signal vanishes within stimulation OFF periods of length a2τCR. All stimulation signals are identical, but have a phase shift τCR between any two consecutive numbered stimulation signals. In this example τCR = 45 ms and a2 = 3.

Mean burst rate: The burst rate of individual neurons is

receive exactly the same stimulation signal Istim = P(t). The other protocol is the so-called coordinated reset (CR) stimulation, in which the four STN groups defined in section 2.2 receive pulse trains from four different stimulation electrodes as shown in figure 3(a). During a stimulation interval of length Ts, stimulation at electrode k is turned on and off as specified by the periodic step function:

estimated as the average burst count over the last 10 s simulation period. The mean population burst rate is then obtained by averaging the burst rates of all STN (or GPe) neurons. To detect bursts in the ordered sequences of spike times we use the modified Poisson surprise method of Hahn et al (2008) with a minimum surprise index of 1.5 and a minimum number of 3 spikes per burst. In short, the method identifies bursts by finding consecutive interspike intervals (ISIs) that are less than the mean ISI, and testing whether these ISI sequences would be expected if the spike train follows a Poisson process.

)) ( ( × H ( sin ( −2π ( t − (k − 2) τ + 1) ρ) ),

Fk (t ) = H∞ sin 2π ( t − (k − 1) τCR ) ρ ∞

k = 1, … , 4,

2.6. DBS

dVm = −IL − Iion − Isyn + Iapp + Istim . dt

(9)

As in Guo and Rubin (2011) the external stimulation signal consists of a train of high-frequency pulses, P (t ) = a 0 H∞ ( sin ( ω 0 t ) − a1) ,

(11)

where the period of the step function equals ρ = τCR + a 2 τCR, such that τCR and a2τCR are the ON period (Fk(t) = 1) and the OFF period (Fk(t) = 0) of the electrode, respectively (figure 3(b)). We assume that each of the four neurons in an STN group receives exactly the same stimuk (t ) = Fk (t ) P (t ), originating only from its lation signal Istim own electrode k. Although the stimulation administered at the four different electrodes has the same period ρ, with the same pulse train supply within the ON period, the ON periods at the four electrodes do not coincide because of the constant phase shift between the activation of any two consecutive stimulation electrodes (figure 3(b)). This phase shift is set equal to the ON period of the electrode (τCR) in (11), implying that a 2 ⩾ 1 as the off period cannot be shorter than the phase shift. In our CR-stimulation protocol, we set τCR = 45 ms, which is approximately one fourth of the STN burst period of the neuron model in PD state.

One of our goals is to investigate how high-frequency stimulation reshapes the synaptic conductances in our network model with STDP, thereby changing the firing pattern. We apply an external stimulation signal Istim to the STN neurons, such that the equation governing the voltage of an STN cell (see (1)) becomes: Cm

CR

(10)

where H∞ (x ) = 1 (1 + exp (−1000x )) is a smooth approximation of the Heaviside step function, a0 is the amplitude of the injected current and t is the time in milliseconds. Parameter values are ω0 = 0.93 and a1 = 0.7, resulting in a pulse train with a frequency of 148 Hz (1000ω0/2π) and a pulse width of 1.7 ms ((π − 2arcsin(a1 )) ω 0 ). In this study, the pulse train P(t) is administered according to two different protocols. We apply the standard DBS, i.e., continuous stimulation, in which all STN cells

2.7. Simulation

We simulate the STN–GPe network model using the Euler method with a fixed step size of 0.01 ms in Matlab (Mathworks, Inc, Natick, MA, USA). To explore the effect of the coupling strength between the GPe cells (wij or wij,initial) and 6

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Iapp,GPe on the network dynamics, we simulate the network without plasticity for 12 s and the network with plasticity for 1000 s, for each setting. Stimulation is only administered to a clustered solution of the network with STDP where the GPe cells are weakly coupled to each other. Therefore, we set wij,initial to 0.1 and Iapp,GPe to −1.2 μA cm−2 in simulations with DBS and run for 1600 s with stimulation starting at t = 1020 s. To exclude effects of initial transients, only the last 10 s of each simulation are used to compute the mean burst rate and the number of principal components.

GPe network can only emerge when STN cells become sufficiently active and the spontaneous firing of the GPe cells cannot counterbalance it. The strength of the applied current to the GPe neurons, Iapp,GPe, representing the net input from other brain structures, controls its spontaneous firing rate. Thus, increasing the input Iapp,GPe and thereby lowering the level of inhibition to GPe or increasing the value of wij can lead to the transition from a PD state to a more healthy state where the activity is irregular and uncorrelated. Due to striatal dopamine depletion as observed in PD, the level of inhibition to GPe is higher in the PD situation than in the healthy situation, which motivated the former parameter change. Rubin and Terman (2004) based the change in intra-GPe inhibitory synaptic conductance on experimental results in rats (Stanford and Cooper 1999, Ogura and Kita 2000). However, changes in conductances can either be the result (via plasticity) or cause of altered activity (Kumar et al 2011). In figure 4(a), wij is increased from 0.1 to 0.45 and Iapp, −2 GPe is increased from −1.2 to −0.6 μA cm . As can be seen from the spike time raster plot of all 16 STN and 16 GPe cells, all cells display irregular firing times that are only weakly correlated. The activity in figure 4(a) represents the healthy state of our network. For the GPe cells 11 PCA components (figure 4(c)) are needed to capture 90% of the activity. For the STN cells 14 components are needed (figure 4(d)), meaning that there is slightly more correlation between the GPe cells than between STN cells. The firing pattern of both cell types in the healthy state is less bursty (mean burst rate of 1–2 Hz) than in the PD state (mean burst rate of 5 Hz). Figures 4(c) and (d) show how the level of synchrony in the GPe and STN changes as a function of Iapp,GPe and wij. For both types of cells we distinguish clearly between two regions: one where almost all the components (10–14) are needed and one where only two components are needed. The region where we need 2 PCA components represents a synchronized state, whereas the other region represents a desynchronized state. The transition from the synchronized state to the desynchronized state is sharp when we increase wij and keep Iapp,GPe constant. The applied current Iapp,GPe has a minor effect on the level of synchrony. Figures 4(e) and (f) show the mean burst rate of the GPe and STN population as function of Iapp,GPe and wij. For both populations, the mean burst rate depends mainly on wij. The STN and GPe bursting reduces quickly when the intra-GPe inhibitory synaptic weight reaches a certain threshold. This threshold is around 0.25 and depends only mildly on Iapp,GPe. When the number of cells is increased from 4 to 16 per group, a comparable dependency of the network state, i.e., the level of network synchrony and burstiness, on the coupling strength between the GPe cells (wij) and Iapp,GPe is obtained (appendix B, figure B1 (a)).

3. Results 3.1. The STN–GPe network without plasticity

To obtain PD-like activity in the structured sparse connected network of 16 STN and 16 GPe cells without synaptic plasticity, we adopted the parameters for coupling and the applied currents parameters from Guo and Rubin (2011), except for gGPe → GPe and wij which were set to 1 mS cm−2 and 0.1, respectively. When these values are used, the STN cells in our network segregate into two rhythmically bursting clusters (checkerboard pattern), with synchronized activity within each cluster. In particular, the STN neurons within the same group of subpopulation 1 (figure 1) and neurons in the corresponding group of subpopulation 2 synchronize their (bursting) activity. GPe cells in our network show similar clustering and bursting, see figure 4(b). This is in accordance with the results reported in Guo and Rubin (2011). We need 2 PCA components for each population to capture 90% of the activity, reflecting the checkerboard activity (figures 4(c) and (d)). In both STN and GPe populations the mean burst rate is approximately 5 Hz (figures 4(e) and (f)). This clustering may be understood directly from the network architecture. Suppose that cells of group STN 1 excite cells of group GPe 2 to initiate firing (figure 1). As a result, the cells of GPe 2 inhibit the cells of STN 2 and prevent them from firing if the inhibition is strong enough. We are now in a situation where only cells of cluster ‘1’ (STN 1 and GPe 2) are active. Once the cells of STN 2 escape from this suppression, their firing excites cells of GPe 1, which in turn represses cells of STN 1. The cells of cluster ‘1’ become silent and the cells of cluster ‘2’ (STN 2 and GPe 1) are active. The cycle repeats with the roles of the clusters reversed. The mechanism underlying the escape of the STN cells from their suppressed state is the de-inactivation of the inward IT current (Terman et al 2002, Best et al 2007). Persistent inhibition from the GPe cells results in an increase of the availability of IT current in the STN cells, and allows the alternation of the activity clusters. On the one hand, the amount of inhibition from GPe cells to the STN cells is crucial for the generation of clustered rhythmic activity. If, for example, the inhibitory connections within the GPe cells, wij, is high, then the GPe cells of cluster ‘1’ will not be able to sufficiently inhibit the STN cells in the other cluster, causing them to escape and fire rebound bursts before the first cluster has completed. On the other hand, oscillatory activity within the reciprocally connected STN–

3.2. The STN–GPe network with plasticity

The above results were obtained with a network where the synaptic connections between the cells were static. This network exhibits a healthy state characterized by desynchronized activity of both cell types and strong coupling between the GPe 7

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Figure 4. Behavior of the network without STDP for different choices for Iapp,GPe and wij. (a) and (b) show the spike times for all 16 STN and GPe cells, illustrating the activity of the network in the two points indicated by ⋆ and • in (c)–(f). These points represent the healthy and PD state, respectively. In the healthy state the firing patterns of both types of cells are irregular and uncorrelated. In the PD state, cells fire in a bursty and clustered pattern around 5 Hz. The cell numbers correspond to the numbers in figure 1. (c) and (d) are 3D contour plots of the number of PCA components required to capture at least 90% of the GPe and STN activity, respectively. (e) and (f) are contour plots of the mean burst rate of GPe and STN, respectively. See sections 2.5 and 2.7 for computational details.

cells, and a PD state characterized by synchronized clusters of both cell types and weak coupling between the GPe cells. Thus, crucial for the network state is the synaptic strength of the inhibitory connections within the GPe. The regulation of these connections by an STDP rule may contribute to the stabilization of these states. The network states may coexist for the same value Iapp,GPe so that STDP leads to multiple stable synchronized and desynchronized states (Popovych and Tass 2012). Here, STDP supports desynchronized dynamics in the healthy state and synchronized dynamics in the PD state. This result is present if in the healthy state potentiation of the synaptic weights between GPe cells is favored, whereas in the PD state depression of these weights is favored.

Depending on the initial coupling strength between the GPe cells (wij,initial), the network reaches either a stable healthy state (figure 5(a), wij,initial = 0.35) or a stable PD state (figure 5(b), wij,initial = 0.1). As expected, the synchronized dynamics of the PD state results in depression of almost all synaptic weights, thereby stabilizing the PD state (figure 5(b)). The synaptic weights of connections between GPe cells of different clusters are not depressed, because the differences between their spike times are too large for the STDP rule to have an effect. Although in the healthy state a number of weights strongly decay from their initial value, the mean synaptic weight is higher than initially (figure 5(a)). Therefore, STDP has a positive effect by stabilizing the healthy state. The 8

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Figure 5. Behavior of the network with STDP for different choices for Iapp,GPe and wij,initial. (a) and (b) illustrate the activity of the network in the two points indicated by ⋆ and • in (c) and (d), representing the healthy and PD state, respectively. The top panels show the raster plot of both cell types and the bottom panels show the time courses of the synaptic weights and the mean synaptic weight (red dashed line). The healthy state is characterized by asynchronous activity of both cell types and strong coupling between the GPe cells, whereas the PD state is characterized by synchronized clusters of both cell types and weak coupling between the GPe cells. The cell numbers correspond to the numbers in figure 1. (c) and (d) are contour plots of the number of PCA components required to capture at least 90% of the GPe activity and the mean burst rate of GPe, respectively. See sections 2.5 and 2.7 for computational details.

observed bimodal distribution of the weights in the healthy state is characteristic for uncorrelated spike trains subject to an additive STDP rule as we used here (Morrison et al 2008). The effect of the initial coupling strength (wij,initial) and Iapp,GPe on the final synchrony level (figure 5(c)) and the mean burst rate (figure 5(d)) of GPe cells are comparable to the corresponding result in the network without STDP where the coupling strength between the GPe cells is fixed (figures 4(c) and (e), respectively). A major difference is that for high Iapp,GPe and low wij,initial the network with STDP approaches the healthy state, whereas the network without STDP displays PD activity here. Thus, starting with wij,initial low and Iapp,GPe high results in network dynamics that is not perfectly clustered at the

beginning of the simulation. Due to STDP, this leads to potentiation of the weights, which in turn, further supports the de-clustering, which finally results in a stable desynchronized healthy state. On the other hand, the nearly clustered dynamics in a network without STDP is in the basin of attraction of the stable clustered PD state. STN cells show similar behavior (data not shown) and the network size has a minor effect on the results presented here (appendix B, figure B1(b)). 3.3. Continuous stimulation versus CR-stimulation

Depending on the original firing pattern, the network with STDP ‘learns’ either pathological or healthy dynamics by 9

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that the healthy state in figure 7 corresponds to regions where at least 10 PCA components are needed to capture 90% of the activity and where STN has a mean burst rate of lower than 2 Hz. For the original CR-protocol as proposed by Tass and colleagues the stimulation is effective in the long duration, high amplitude region of the parameter space, see figure 7(a). As expected, the PCA and mean burst rate show that the stimulation duration and the amplitude used in figure 6(b) belongs to the effective region of parameter choices, see figure 7(a), marker +. Figure 8 depicts four examples of network dynamics from the region of parameter choices in figure 7(a) (markers*) where CR-stimulation is not effective. For very low amplitude the clustering and bursting remains after stimulation (figure 8(a), right), whereas for slightly higher amplitude the bursting but not the clustering disappears after stimulation (figure 8(b), right). However, in both cases there is a phase shift between the checkerboard pattern of subpopulation 1 and the checkerboard pattern of subpopulation 2, such that we have a four-cluster state. This phase shift explains the reason why we need 4 PCA components to describe the STN activity in figure 7(a). For both cases the corresponding time course of the mean synaptic weight remains low (figures 8(a) and (b), left). Interestingly, the weights approach two different stable plateaus after stimulation with an amplitude of 60 μA cm−2 and a duration of 75 s (figure 8(b), left). This may explain the disappearance of the bursts, that is, two spikes are observed instead of three. Similar weight distribution and spike behavior is obtained when stimulation is applied with a higher amplitude (100 μA cm−2), but a shorter duration (10 s), see figure 8(c). When we increase the duration from 10 s to 30 s, the CR-stimulation is only effective for subpopulation 2 (figure 8(d)). For the CR-protocol with OFF period 1.5 τCR, the ON periods of the stimulation sites partially overlap with each other, leading to stimulation that is less desynchronous and closer to continuous. As can be seen from figure 7(b) the stimulation is less effective compared to the original CRprotocol. Remarkable is the observation that one of the low amplitudes is effective and that it becomes ineffective when the amplitude is too high. When the OFF period is long, e.g., 3.8 τCR, the stimulation does not work across the range of amplitude and duration we consider here. This is explained by the fact that for a2 > 3, there is a period in which the whole network receives no stimulation. If this period is long, the weights may return to their original values of the PD state.

adapting its pattern of the synaptic couplings between the GPe cells. As stimulation can change the firing pattern of the stimulated cells, it can be used to ‘teach’ the network with STDP to display more healthy activity. To illustrate the effect of continuous stimulation and CR-stimulation we first consider the STN–GPe network with STDP in a clustered solution where the GPe cells are weakly coupled to each other, see top panel of figure 5(b). For continuous stimulation all STN cells are locked to half of the stimulation frequency, which in turn, drives the GPe cells in the same tonic mode at 74 Hz (figure 6(a), bottom left). During continuous stimulation, the synaptic weights between the GPe cells quickly decay as a result of the synchronized activity of the GPe cells (figure 6(a), top). When the continuous stimulation is removed, the weights remain low. Continuous stimulation shifts the network from the two-cluster state to an even more synchronized state where all cells are perfectly synchronized and fire periodically at a rate of 21 Hz (figure 6(a), bottom right). We stress that this frequency is characteristic for the model set-up and can be tuned by different settings. For the CR-stimulation we set a2 to 3, such that it corresponds to the CR-protocol proposed by Tass and colleagues. Now, the first stimulation site is turned on again when the last stimulation is turned off, i.e., during a cycle of length ρ = 4τCR each stimulation site is turned on once (figure 3). The bottom left panel in figure 6(b) shows the spike times of both cell types during CR-stimulation. Both cell types are entrained to the CR inputs, however, they sometimes fire single spikes in the OFF period of their stimulation site. This causes some desynchronization within the clusters, which in turn, leads to a gradually rise of mean synaptic weight during stimulation (figure 6(b), top). When CR-stimulation is switched off, the mean synaptic weight keeps rising until it reaches a value around 0.35. The distribution of the weights shows a similar bimodal distribution as we observed in the healthy state of the network (figure 5(a), bottom) and seems stable. Accordingly, the CR-stimulation steers the network away from a stable PD state to a stable healthy state (figure 6(b), bottom right). In a larger network, only an appropriate CR-stimulation is able to steer the network away from a stable PD state to a stable healthy state, see appendix B (figures B2 and B3). 3.4. Robustness of CR-stimulation

The above results show that CR-stimulation applied to a network with STDP in the PD state can reinforce the mean synaptic weight between GPe cells during stimulation. This results in a long-lasting redistribution of the weights and desynchronization corresponding to the healthy state. We test the robustness of the long-lasting effects of CR-stimulation with respect to the stimulation duration and amplitude as well as variation in the OFF period of the stimulation sites, which is controlled by a2 (see (11)). Figure 7 illustrates the dependence of the effectiveness of the CR protocol on stimulation duration and amplitude for three OFF periods via the number of PCA components and the mean burst rate measures for STN activity as described above. The CR-stimulation is considered effective when it induces the healthy state. Note

4. Discussion In this article, we have introduced a novel method to use the inherent plasticity of synapses in the GPe to improve DBS in the treatment of PD. We simulated inhibitory GPe–GPe synapses with STDP in a realistic network model of both STN and GPe, showing the stabilizing roles of STDP for firing patterns. Different stimulation protocols highly vary in their ability to make use of pallidal plasticity. In particular, in contrast to standard high-frequency DBS, alternative protocols such as CR-stimulation, proposed by Tass (2003), can 10

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Figure 6. Effect of continuous stimulation (a) and CR-stimulation with a2 = 3 (b) on the network activity. Stimulation is on from 1020 to 1095 s indicated by a green and a red arrow. Pulse train stimulation (10) is applied at 148 Hz with amplitude 100 μA cm−2 and a pulse width of 1.7 ms. In both (a) and (b), the top panel shows the time courses of the synaptic weights and the mean synaptic weight (red dashed line). The bottom panels show spike times of both cell types during (left) and after (right) stimulation. See sections 2.5 and 2.7 for computational details.

which is hard to measure due to stimulation artifacts. The choice of STDP learning rule should be guided by experimental evidence. Moreover, our model does not include synaptic delays which could effectively change the STDP time window and learning function (Babadi and Abbott 2010). However, in our simulations, the choice of STDP time window (e.g., Hebbian or anti-Hebbian; symmetric or asymmetric) was not crucial, as long as the effective time window was comparable to figure 2. Experimental

profit from pallidal plasticity. Still, a lot of settings in our model are not known yet. These settings include especially the synaptic connection architecture, learning rules for STDP as well as its effective window and firing rates and patterns during stimulation. Once more experimental data is available, our model should be up-scaled to a size of several hundred neurons connected with the desired architecture. Firing dynamics should be fitted to reproduce the results observed in electrophysiology experiments, especially during stimulation, 11

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in the Parkinsonian state, the model cells typically burst at low frequency (around 5 Hz), which may be a characteristic of this model. In particular, the model relies on the assumption that standard high-frequency DBS entrains stimulated cells. Experiments of DBS in Parkinsonian monkeys have shown that DBS regularizes neural activity in the output structures of the BG (Anderson et al 2003, Hashimoto et al 2003, McCairn and Turner 2009), and this regularization is at least partly responsible for alleviated bradykinesia (Dorval et al 2010). While the exact effect of DBS on neural activity is still debated, the assumption of entrainment may be challenged. It has also been suggested that high-frequency DBS desynchronizes oscillating neurons (Wilson et al 2011). Current studies indicate that one of the most relevant effects of DBS may be short term depression of synapses, induced by activation of axons (Shen and Johnson 2008, Erez et al 2009, Ammari et al 2011, Zhang et al 2011, Moran et al 2012, Rosenbaum et al 2014). This depression could suppress the transfer of information, especially low frequency oscillations and synchrony (Rosenbaum et al 2012, Chiken and Nambu 2013, Rosenbaum et al 2014) and thereby reduce PD motor symptoms similar to a lesion of the STN. Furthermore, we use synchrony as the critical parameter to optimize stimulation techniques. Although it seems plausible that highly synchronized neural activity in the BG fails to control motor activities, the relationship between neural synchronization and motor symptoms is far from being clear. Standard high-frequency DBS is effective despite, or because of, regularization of the BG output, probably at an altered firing rate (Rubin and Terman 2004, Dorval et al 2010, McConnell et al 2012). Synchronized beta oscillations occur independent of the onset of motor symptoms in animal models of increasing dopamine depletion (Leblois et al 2007, Dejean et al 2012). Other factors such as firing patterns, rates or amplitudes, that are not considered in our model, might play additional roles.

Figure 7. Effectiveness of CR-stimulation, measured via the number of PCA components (color coded and explicitly indicated in each circle) and mean burst rate (circle outline coded) for STN activity, for three OFF periods of the stimulation sites over a range of stimulation durations and amplitudes. (a) a2 = 3: original CRprotocol; (b) a2 = 1.5: ON periods partially overlap; (c) a2 = 3.8: contains a stimulus free period after each stimulation cycle. Healthy activity corresponds to regions where at least 10 PCA components are needed and where the mean burst rate is lower than 2. Pulse train (10) is applied at 148 Hz with pulse width of 1.7 ms. The markers + and * indicate the stimulation settings used in figures 6(b) and 8, respectively. See sections 2.5 and 2.7 for computational details.

4.2. Intrinsic GPe–GPe coupling

Synchrony in our model was most dependent on the strength of GPe–GPe and STN–GPe synapses, whereas the strengths of GPe–STN synapses in a physiological range played a minor role (data not shown, but see also Terman et al 2002). Changes in intra-GPe inhibitory synaptic conductance as a result of dopamine depletion have been observed (Stanford and Cooper 1999, Miguelez et al 2012). Therefore, we chose to implement STDP only at GPe–GPe synapses. However, STDP at other synapses may have additional effects that are not captured in the current model. Coupling inside the GPe also had a much higher effect on synchrony than inhibition from the striatum (figures 4 and B1(a)). Whether inhibition from stratum to GPe is enhanced in PD is debated controversially in experimental literature. DeLong (1990) described an increase of the indirect pathway leading to reduced activity in the GPe in parkinsonian monkeys. After dopamine depletion, Ingham et al (1997) reported enlargements of striatum–GPe synapses and Kita and Kita (2011) found that striatal neurons projecting to the GPe have higher firing rates. In contrast, Miguelez et al (2012) showed

validation regarding the choice of the effective time window is thus essential.

4.1. Neural activity under healthy, parkinsonian and DBS condititons and relation to motor symptoms

Our model is based on the work of Terman et al (2002) and is therefore subject to all assumptions made in this previous model. Although GABAergic interneurons have been described in the human STN (Levesque and Parent 2005), they are not included in the model due to lacking information on electrophysiological properties and connectivity. Furthermore, 12

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Figure 8. Response of the network in PD to different ineffective CR-stimulation amplitudes a0 (μA cm−2) and durations Ts (s). (a) a0 = 20 and

Ts = 75; (b) a0 = 60 and Ts = 75; (c) a0 = 100 and Ts = 10; (d) a0 = 100 and Ts = 30. For each setting the time courses of the synaptic weights (left, red dashed line corresponds to the mean) and the raster plot of both cell types after stimulation (right) are shown. The other CRstimulation parameters are as in figure 6(b).

that inhibition from the striatum to the GPe is not significantly altered after dopamine depletion in a rat 6-OHDA model. These discrepancies indicate that looking solely at the strength of inhibition from striatum might not suffice to explain synchrony in pallidal regions. Also patterns and synchrony of striatal inhibition could be important. In our

simulations, the strength of the inhibitory input from the striatum also plays a minor role. However, GPe–GPe synapses may be highly relevant for PD and neural activity in the BG (Miguelez et al 2012, Terman et al 2013, Wilson 2013). In our simulation, GPe–GPe inhibition decreases synchrony (figures 4, 5 and B1), which has also been 13

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Figure B1. Behavior of the 64×64 cells network without STDP (a) and with STDP (b) for different choices for Iapp,GPe and wij or wij,initial. The left panels show contour plots of the number of PCA components required to capture at least 90% of the STN activity and the right panels show contour plots of the mean burst rate of STN. The same computational settings as for the 16×16 cells network are used.

suggested recently by Wilson (2013) and Terman et al (2013). However, our knowledge on the properties and especially connection architecture of GPe–GPe synapses is deficient. A morphological study of the GPe indicates that the connectivity may be sparse, and has an organization from the outer to the inner parts of the GPe (Sadek et al 2007). Recently, an electrophysiological study (Bugaysen et al 2013) described GPe–GPe synapses as sparse but highly efficacious, enabling them to continuously modulate postsynaptic firing rates despite substantial short time depression. Further studies of GPe–GPe coupling will be necessary to fully understand their role in healthy and PD situations.

4.3. STDP in models of DBS

Our study is not the first one to describe effects of STDP in the BG. Hauptmann and Tass (2009) introduced a model of STN and GPe including STDP of excitatory STN–STN synapses. The model exhibits a stable healthy state characterized by desynchronized STN bursts and weak connectivity within the STN, and a stable pathological state characterized by synchronized STN bursts and strong connectivity within the STN. The authors (Hauptmann and Tass 2009) show that stimulation of the STN neurons according to their proposed CR-stimulation protocol reduces 14

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the mean synaptic weight and shifts the network to a healthy state. In contrast to our work, their model relies on the presence of all-to-all connections within the STN that become weakened by desynchronizing stimuli. The existence of such dense connections within the STN, if present at all, might however be doubted (Wilson et al 2004). Although CR-stimulation has been shown to have long-lasting effects on motor signs in parkinsonian nonhuman primates (Tass et al 2012) and PD patients (Adamchic et al 2014), it is not clear whether these effects can be led back to the original theory. In a later publication of the same group (Popovych and Tass 2012), indirect, synaptic-mediated CR-stimulation was also considered for a network of spiking Hodgkin– Huxley neurons. Each neuron was connected to all other neurons via STDP-controlled excitatory and inhibitory synapses. Popovych and Tass (2012) concluded that both direct electrical and indirect, synaptic-mediated CR-stimulation applied to strongly coupled and synchronized population of spiking neurons can lead to long-lasting redistribution of the synaptic weights and desynchronization. Pfister and Tass (2010) and Popovych et al (2013) further investigated the influence of DBS in oscillatory networks with STDP. In particular, Popovych et al (2013) could show that stimulation with independent noise rather synchronizes than desynchronizes oscillatory neural populations with STDP. Our study differs substantially from the ones mentioned in this paragraph by introducing a realistic BG network with plasticity of established GPe–GPe connections.

both populations (Terman et al 2002, Rubin and Terman 2004, Guo and Rubin 2011). Here we describe the equations and parameters of the membrane currents of (1) for both the STN and the GPe cell model. In both cell models, time is in ms, voltages are in mV, ion concentrations are in mM, currents are in μA cm−2, conductances are in mS cm−2 and the membrane capacitance is in μF cm−2. The membrane capacitance for each cell is assumed to be unity. Membrane currents

In the STN and GPe cell model it is assumed that the instantaneous current–voltage relation for IL, IK, INa, IT, ICa and IAHP is linear. The leak current and the voltage-dependent currents (i.e., except IAHP) are given by Hodgkin–Huxley formalism and are identical for both STN and GPe cells:

IL (v) = gL ( v − E L ), INa (v) = gNa m∞3 (v) h ( v − E Na ), IK (v) = gK n4 ( v − EK ), ICa (v) = gCa c∞2 (v) ( v − E Ca ), except for IT: STN : IT (v) = gT a∞3 (v) b∞2 (r ) ( v − E Ca ), GPe : IT (v) = gT a∞3 (v) r ( v − E Ca ),

where v is the membrane potential; m, h, n, a, b, r and c are (in)activation variables (gating variables); EL, ENa, EK and ECa are the reversal potentials of the leak, sodium, potassium and calcium current, respectively; and gL, gNa, gK, gT and gCa are maximal conductance. The reversal potential is defined by the Nernst equation. We assume the ionic concentrations are constant during our simulations. The maximal conductances and reversal potentials for both population models are listed in table A1 . The steady state of the gating variable ( X∞) depends on the voltages as follows

4.4. Clinical significance

The clinical relevance (Tass et al 2012) and the effect on neural activity (Adamchic et al 2014) of CR-stimulation have been demonstrated. Undoubtedly, refining DBS for treatment of PD with the help of plasticity has large therapeutic impacts. If the mean frequency of stimulation and therefore its power can be reduced, less energy is needed, and batteries can be replaced after longer intervals. Most importantly, a reduced stimulation power will likely result in less severe side effects. Such improvements can result in a highly improved quality of life for patients undergoing DBS, due to significant decrease in depression, cognitive decline, or similar effects during treatment.

X∞ (v) =

1

(

1 + exp −( v − θ X ) qX

)

X ∈ {m , h , n , r , a , c},

(A.1)

where θX and kX are the half (in)activation voltage and slopes, respectively. For the T-current inactivation variable b, we used

Acknowledgments

b∞ (r ) =

We received financial support from the Dutch Organization for Scientific Research (NWO), on grant 635.100.019, From spiking neurons to brain waves and Desynchronization of Parkinsonian Oscillations in the Subthalamic Nucleus.

1 1 + exp

( ( r − θ b ) qb)

−

1

(

1 + exp −θ b qb

)

. (A.2)

The gating variables h, n and r for both cell models are treated as slowly varying variables, whereas m, c, a and b for both cell models are treated as fast varying variables. For the slow variables we have first order kinetics of the form:

Appendix A. STN and GPe cell model

dX = ϕX ( X∞ (v) − X ) τ X (v) dt X ∈ {h , n , r},

We consider a GPe–STN network model in which each cell is represented as a single-compartment conductance-based model, based on voltage-clamp and current-clamp data of

(A.3)

where ϕX is a time scaling constant of X. The voltage15

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Table A1. Maximal conductances (gx) and reversal potentials (Ex) of the membrane currents for STN and GPe models.

gx (mS cm−2)

STN GPe

Ex (mV)

L

K

Na

T

Ca

AHP

L

K

Na

Ca

2.25 0.1

45 30

37.5 120

0.5 0.5

0.5 0.1

9 30

−60 −55

−80 −80

55 55

140 120

Table A2. STN kinetic parameters.

m h n r a b c

θX

qX

τX0

−30 −39 −32 −67 −63 0.25 −39

15 −3.1 8 −2 7.8 −0.07 8

— 1 1 7.1 — — —

τX1

θ Xτ

qXτ

ϕX

— 500 100 17.5 — — —

— −57 −80 68 — — —

— −3 −26 −2.2 — — —

— 0.75 0.75 0.5 — — —

the gating kinetics of the ionic channels of the STN and GPe models are given in tables A2 and A3 . The afterhyperpolarization potassium current (IAHP) also depends on the calcium concentration instead of membrane potential and is given by:

(

Calcium dynamics

θX

qX

τX0

τX1

θ Xτ

qXτ

ϕX

−37 −58 −50 −70 −57 −35

10 −12 14 −2 2 2

— 0.05 0.05 30 — —

— 0.27 0.27 0 — —

— −40 −40 — — —

— −12 −12 — — —

— 0.05 0.1 1 — —

The intracellular concentration of calcium ions (Ca2 + ) available for the IAHP depends on the calcium currents (IT and ICa) and is governed by the first-order differential equation: d[Ca] = ϵCa ( −ICa − IT − k Ca [Ca] ), dt

STN GPe

kCa

k1

5e−5 1e−4

22.5 15

15 30

Appendix B. Effect of network size To investigate the effect of network size, we expand the model from 16 cells to 64 cells in each population of STN and GPe neurons, i.e., each group in figure 1 contains now sixteen cells instead of four cells, thereby preserving the structured, sparsely connected architecture. Moreover, the number of synaptic connections that each cell type makes is kept the same and all cells receive direct current injection. Using the parameter values given in the main text, the effect of the coupling strength between the GPe cells (wij) and Iapp,GPe on the STN activity (figure B1(a)) in the 64×64 cells network without STDP is comparable to the corresponding result in the 16×16 cells network without STDP (figure 4). A difference is that the threshold for wij at which the transition occurs from PD to healthy activity is higher for the 64×64 cells network than for the 16×16 cells network. Moreover, there is a jump in the transition threshold value around Iapp, −2 GPe = −0.6 μA cm . Before subjecting the GPe–GPe

dependent (in)activation time constant (τX) of X is given: τ X = τ X0 +

τ X1

(

1 + exp −( v − θ Xτ ) qXτ

X ∈ {h , n , r}.

(A.6)

where ϵCa is a constant that accounts for the effects of cell volume, buffers, and molar charge of calcium. The constant kCa is the calcium pump rate. The constants for the calcium dynamics for the STN and GPe models are given in table A4. Note that the calcium reversal potential, that is used in the equations for IT and ICa, is not affected by the calcium dynamics.

Table A4. Calcium dynamic parameters.

ϵCa

(A.5)

where gAHP is the maximal conductance and is given in table A1. The constant k1 is the dissociation constant of the calcium-dependent afterhyperpolarization potassium current and is given in table A4 .

Table A3. GPe kinetic parameters.

m h n r a c

)

IAHP = gAHP ( v − EK ) [Ca] ( [Ca] + k1) ,

) (A.4)

Time for (in)activation are then given by a sigmoidal function, with τX0 ϕX as the minimum, (τX0 + τX1 ) ϕX as the maximum, θ Xτ the voltage at which the time constant is midway between the maximum and minimum values, and σ Xτ is the slope factor which determines the level of voltagedependence of the time constant. For the fast variables, activation are taken instantaneous and are determined by (A.1) and in the case for b by (A.2). The parameter values used for 16

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Figure B2. Effect of continuous stimulation (a) and CR-stimulation with a2 = 3 (b) on the network activity. Stimulation is on from 1020 to 1095 s indicated by a green and red arrow. Pulse train stimulation (10) is applied at 148 Hz with amplitude 100 μA cm−2 and a pulse width of 1.7 ms. In both (a) and (b), the top panel shows the time courses of the synaptic weights and the mean synaptic weight (red dashed line). The other panels show spike times of the STN cells during (middle) and after (bottom) stimulation. Spike times of the GPe cells are similar (data not shown). The same computational settings as for the 16×16 cells network are used, except that the simulations run for 2000 s.

synapses in the 64×64 cells network to STDP, the upper bound for each synaptic weight (wij) is increased from 0.6 to 1.2, because of the higher transition threshold. As in the 16×16 cells network, the addition of STDP for GPe–GPe synapses in the large network only affects the dependency of STN activity on wij,initial and Iapp,GPe in the region where high

Iapp,GPe and low wij,initial (figure B1(b)). GPe cells show similar behavior (data not shown). In the 64×64 cells network only an appropriate CR-stimulation is able to steer the network away from a stable PD state to a stable healthy state, see figures B2 and B3 . The same conclusion was drawn for the 16×16 cells network.

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Chiken S and Nambu A 2013 High-frequency pallidal stimulation disrupts information flow through the pallidum by GABAergic inhibition J. Neurosci. 33 2268–80 Contarino M F, Bour L J, Bot M, van den Munckhof P, Speelman J D, Schuurman P R and de Bie R M 2012 Tremorspecific neuronal oscillation pattern in dorsal subthalamic nucleus of parkinsonian patients Brain Stimul. 5 305–14 Dejean C, Nadjar A, le Moine C, Bioulac B, Gross E C and Boraud T 2012 Evolution of the dynamic properties of the cortex–basal ganglia network after dopaminergic depletion in rats Neurobiol. Dis. 46 402–13 DeLong M R 1990 Primate models of movement disorders of basal ganglia origin Trends. Neurosci. 13 281–5 Dorval A D, Kuncel A M, Birdno M J, Turner D A and Grill W M 2010 Deep brain stimulation alleviates parkinsonian bradykinesia by regularizing pallidal activity J. Neurophysiol. 104 911–21 Erez Y, Czitron H, McCairn K, Belelovsky K and Bar-Gad I 2009 Short-term depression of synaptic transmission during stimulation in the globus pallidus of 1-methyl-4-phenyl1,2,3,6-tetrahydropyridine-treated primates J. Neurosci. 29 7797–802 Feldman D E 2000 Timing-based LTP and LTD at vertical inputs to layer II/III pyramidal cells in rat barrel cortex Neuron 27 45–56 Galvan A and Wichmann T 2008 Pathophysiology of parkinsonism Clin. Neurophysiol. 119 1459–74 Garcia L, Audin J, D’Alessandro G, Bioulac B and Hammond C 2003 Dual effect of high-frequency stimulation on subthalamic neuron activity J. Neurosci. 23 8743–51 Golomb D and Rinzel J 1994 Clustering in globally coupled inhibitory neurons Physica D 72 259–82 Guo Y and Rubin J E 2011 Multi-site stimulation of subthalamic nucleus diminishes thalamocortical relay errors in a biophysical network model Neural Netw. 24 602–16 Haas J S, Nowotny T and Abarbanel H D I 2006 Spike-timingdependent plasticity of inhibitory synapses in the entorhinal cortex J. Neurophysiol. 96 3305–13 Hahn P J, Russo G S, Hashimoto T, Miocinovic S, Xu W, McIntyre C C and Vitek J L 2008 Pallidal burst activity during therapeutic deep brain stimulation Exp. Neurol. 211 243–51 Hamani C, Neimat J and Lozano A M 2006 Deep brain stimulation for the treatment of Parkinson’s disease J. Neural Transm. Suppl. 70 393–9 Hammond C, Bergman H and Brown P 2007 Pathological synchronization in Parkinson’s disease: networks, models and treatments Trends Neurosci. 30 357–64 Hashimoto T, Elder C M, Okun M S, Patrick S K and Vitek J L 2003 Stimulation of the subthalamic nucleus changes the firing pattern of pallidal neurons J. Neurosci. 23 1916–23 Hauptmann C and Tass P A 2009 Cumulative and after-effects of short and weak coordinated reset stimulation: a modeling study J. Neural Eng. 6 016004 Holmgren C D and Zilberter Y 2001 Coincident spiking activity induces long-term changes in inhibition of neocortical pyramidal cells J. Neurosci. 21 8270–7 Ingham C A, Hood S H, Mijnster M J, Baldock R A and Arbuthnott G W 1997 Plasticity of striatopallidal terminals following unilateral lesion of the dopaminergic nigrostriatal pathway: a morphological study Exp. Brain Res. 116 39–49 Jankovic J 2008 Parkinson’s disease: clinical features and diagnosis J. Neurol. Neurosurg. Psychiatry 79 368–76 Kepecs A, van Rossum M C W, Song S and Tegner J 2002 Spiketiming-dependent plasticity: common themes and divergent vistas Biol. Cybern. 87 446–58 Kita H and Kita T 2011 Role of striatum in the pause and burst generation in the globes pallidus of 6-OHDA-treated rats Front. Syst. Neurosci. 5 1–11

Figure B3. Effectiveness of CR-stimulation, measured via the

number of PCA components (color coded and explicitly indicated in each circle) and mean burst rate (circle outline coded) for STN activity in the 64×64 cells network with STDP, for the original CRprotocol (a2 = 3) over a range of stimulation durations (Ts) and amplitudes (a0). Healthy activity corresponds to regions where at least 45 PCA components are needed and where the mean burst rate is lower than 2. Pulse train (10) is applied at 148 Hz with a pulse width of 1.7 ms. The same computational settings as for the 16×16 cells network are used, except that the simulations run for 2000 s.

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