Biological Cybernetics

, Volume 110, Issue 1, pp 55–71 | Cite as

Closed-loop firing rate regulation of two interacting excitatory and inhibitory neural populations of the basal ganglia

  • Ihab Haidar
  • William Pasillas-Lépine
  • Antoine Chaillet
  • Elena Panteley
  • Stéphane Palfi
  • Suhan Senova
Original Article

Abstract

This paper develops a new closed-loop firing rate regulation strategy for a population of neurons in the subthalamic nucleus, derived using a model-based analysis of the basal ganglia. The system is described using a firing rate model, in order to analyse the generation of beta-band oscillations. On this system, a proportional regulation of the firing rate reduces the gain of the subthalamo-pallidal loop in the parkinsonian case, thus impeding pathological oscillation generation. A filter with a well-chosen frequency is added to this proportional scheme, in order to avoid a potential instability of the feedback loop due to actuation and measurement delays. Our main result is a set of conditions on the parameters of the stimulation strategy that guarantee both its stability and a prescribed delay margin. A discussion on the applicability of the proposed method and a complete set of mathematical proofs is included.

Keywords

Neural oscillations Firing rate models Time-delay systems Basal ganglia Parkinson’s disease Deep brain stimulation Closed-loop stimulation 

1 Introduction

Parkinson’s disease is a movement disorder associated with a loss of dopaminergic neurons in the substantia nigra, a nucleus located in the basal ganglia. In its initial stage, the disease’s symptoms can be relieved using dopaminergic medications. In more advanced stages, surgery is one of the alternatives proposed to some patients (Davie 2008). Even if ablative surgery of the subthalamic nucleus has been considered in the past (Bergman et al. 1990), it has now been replaced by deep brain stimulation (DBS), that is by the implantation of a lead that delivers electrical impulses (Benabid et al. 1991). This procedure has the advantage of being reversible (Limousin et al. 1995). Currently, there is no consensus on the mechanisms that explain the therapeutic effects of DBS (McIntyre et al. 2004a). Nevertheless, some clinicians seem to agree on the fact that the treatment is not always optimal and that there is still room for improvements (Kumar et al. 2003).

One of the directions proposed recently in order to optimize the efficiency of DBS is closed-loop stimulation (Rosin et al. 2011). Indeed, in its present form, DBS is an intrinsically open-loop procedure. The amplitude and frequency of stimulation are adjusted by the surgeon after the operation, but then these parameters remain fixed, with only sporadic adjustments during medical visits (Kumar et al. 2003). It is therefore logical to ask if the use of real-time measurements of brain activity might improve the efficiency of brain stimulation signals. Several works in the literature have explored this direction (for recent surveys, see, for example, Schiff 2010 and Carron et al. 2013). On the one hand, following the observation that in the parkinsonian state, some areas of the basal ganglia exhibit an increased synchronization in the beta-band (Nini et al. 1995; Hammond et al. 2007), some works focus on the desynchronizing aspects of stimulation (see, for example, Hauptmann et al. 2005a, b; Pyragas et al. 2007; Tukhlina et al. 2007; Omel’chenko et al. 2008; Lysyansky et al. 2011; Franci et al. 2011, and Montaseri et al. 2013). While, on the other hand, exploiting a detailed computational model of the basal ganglia such as (Rubin and Terman 2004), some other works try to reduce the impact of the pallidal output on the thalamo-cortical relay function (see, for example, Feng et al. 2007; Agarwal and Sarma 2010; Liu et al. 2010, 2011; Dunn and Lowery 2013). Nevertheless, on both sides, there seems to be a difficulty in reducing the gap between theoretical developments and experimentation.

Several problems might be at the origin of this difficulty. A first one is the microscopic nature of the considered models, which describe the behaviour of individual neurons rather than neural populations in their whole (Pascual et al. 2006). A second problem is the intermittent character of the basal ganglia dynamics (Dovzhenok et al. 2013). One should stress, moreover, that some of these works focus on an electrical stimulation and do not consider other possible ways of stimulating a neural population. In this paper, we consider a mesoscopic model in order to propose an alternative approach that takes into account the first problem, while remaining compatible with the modelling of some intermittent behaviours (see Sect. 5). In addition, in order to remain open to the possibilities offered by non-electric stimulations (see, for example, Strafella et al. 2004; Tufail et al. 2010; Han et al. 2011), we restrict ourselves to an abstract generic closed-loop stimulation.

A certain number of mesoscopic models have been proposed recently in the literature, in order to analyse the possible sources of pathological oscillations (see, for example, Leblois et al. 2006; Coombes and Laing 2009; Nevado-Holgado et al. 2010; Pavlides et al. 2012; Pasillas-Lépine 2013; Haidar et al. 2014). One of the motivations for the development of such models is the tight relationship between dopamine depletion, motor symptoms, and the enhanced oscillatory activity in the beta-band of parkinsonian patients (Jenkinson and Brown 2011). To our knowledge, this kind of mesoscopic models has never been used in order to design closed-loop stimulation strategies. In this paper, we aim at filling this gap. We rely on the basal ganglia model developed by Nevado-Holgado et al. (2010) and show that pathological oscillations can be reduced by decreasing artificially the gain of the subthalamo-pallidal feedback loop. We show that this can be achieved with a simple proportional feedback on the subthalamic nucleus. It can be shown, however, that this strategy is poorly robust with respect to actuation or measurement delays. We therefore complement this proportional closed-loop stimulation with a well-chosen low-pass filter and show that the resulting strategy can cope with these inherent delays.

2 Firing rate dynamics

Recently, a firing rate model of the subthalamo-pallidal loop has been proposed in Nevado-Holgado et al. (2010). We follow the same line and consider the model
$$\begin{aligned} {\begin{array}{l} \tau _{s}\dot{x}_s(t) = -x_s(t) + F_{s}\Big ( -w_{gs} x_g(t-\delta _{gs}) + w_{cs} v_s + u(t) \Big )\\ \tau _{g}\dot{x}_g(t) = -x_g(t) + F_{g}\Big ( w_{sg} x_s(t-\delta _{sg}) - w_{gg} x_g(t-\delta _{gg}) {-} w_{xg}v_g \Big ), \end{array}} \end{aligned}$$
(1)
where \(x_s\) and \(x_g\) denote, respectively, the firing rates of the subthalamic nucleus (STN) and of the external segment of the globus pallidus (GPe). The constants \(v_s\) and \(v_g\) denote, respectively, the external inputs coming from the cortex and the striatum. The main difference between this model and the one considered by Nevado-Holgado et al. (2010) is the introduction of an additional variable u(t) that represents the effect of a generic subthalamic stimulation.

Representing the stimulation of basal ganglia with such a model could be considered, of course, as an abusive oversimplification. Nevertheless, three points should be mentioned in order to understand why the analysis of this model might be interesting. First, it is well known that the basal ganglia have a somatotopic organization (Nambu 2011) and that pathological oscillations do not take place in the whole nuclei, but only in specific subareas (Zaidel et al. 2010). Therefore, even if the previous model cannot represent the whole basal ganglia, one might expect it to describe what happens in a given subregion of the dorsolateral STN, where both beta-oscillations occur and functional organization of neurons is homogeneous (due to somatotopy). Second, even if other basal ganglia models that take into account deep brain stimulation are available (see, for example, Rubin and Terman 2004; Modolo et al. 2008; Santaniello et al. 2011), their complexity makes it difficult to study analytically the stability of control laws that are applied to them. Considering a simplified model is thus a natural option in order to analyse what are the main effects of a closed-loop stimulation. Third, as already said, considering a generic stimulation in the above model has the advantage (at least in theory) to allow the potential consideration of non-electric stimulations. Even though this would also imply the consideration of more detailed models that take into account the specificities of each actuation mechanism in order to be fully analysed. In a few words, despite its simplicity, we expect that this generic stimulation could give some insights on what should be the effects of adding a controlled excitation or inhibition to the considered interconnected excitatory/inhibitory system.

In the model (1) proposed by Nevado-Holgado et al. (2010), the positive gains \(w_{gs}\), \(w_{sg}\), and \(w_{gg}\) denote the weights of the different synaptic interconnections between the two neuronal populations. The constants \(\tau _s\) and \(\tau _g\) describe the synaptic dynamics that is the velocity at which these populations respond to presynaptic inputs. The constants \(\delta _{gs}\), \(\delta _{sg}\), and \(\delta _{gg}\) are assumed to be positive. They describe the axonal time-delays appearing in the interconnection of the STN and GPe populations. Based both on experimental data and on parameters identified in the literature, the authors of Nevado-Holgado et al. (2010) proposed a complete set of parameters for this model. We reproduce them in Table 1.
Table 1

Parameters of Nevado-Holgado’s model (2010)

Parameter

Value

Description

\(\delta _{sg}\)

\(6\,\hbox {ms}\)

Delay from STN to GPe

\(\delta _{gs}\)

\(6\,\hbox {ms}\)

Delay from GPe to STN

\(\delta _{gg}\)

\(4\,\hbox {ms}\)

Internal GPe delay

\(\tau _{s}\)

\(6\,\hbox {ms}\)

STN time constant

\(\tau _{g}\)

\(14\,\hbox {ms}\)

GPe time constant

\(M_s\)

\(300\,\hbox {spk/s}\)

STN maximal firing rate

\(B_s\)

\(17\,\hbox {spk/s}\)

Firing rate at rest for STN

\(M_g\)

\(400\,\hbox {spk/s}\)

GPe maximal firing rate

\(B_g\)

\(75\,\hbox {spk/s}\)

Firing rate at rest for GPe

\(v_{s}\)

\(27\,\hbox {spk/s}\)

Cortical input to STN

\(v_{g}\)

\(2\,\hbox {spk/s}\)

Striatal input to GPe

Parameter

Healthy state

Diseased state

\(w_{sg}\)

19.0

20.0

\(w_{gs}\)

1.12

10.7

\(w_{gg}\)

6.60

12.3

\(w_{cs}\)

2.42

9.2

\(w_{xg}\)

15.1

139.4

The functions \(F_{s}\) and \(F_{g}\) describe, respectively, the activation functions of the STN and GPe. As Nevado-Holgado et al. (2010), we describe them using logistic functions of the form
$$\begin{aligned} F_{i}(x)=\dfrac{B_i M_i}{B_i + (M_i - B_i)\hbox {e}^{-4x/M_i}}, \quad \text{ for } i \in \{s,g\}. \end{aligned}$$
(2)
Moreover, the interconnection gains \(w_{ij}\) from nucleus i to nucleus j are defined by a linear parametrization
$$\begin{aligned} w_{ij} = w^H_{ij} + K \left( w^D_{ij} - w^H_{ij} \right) , \end{aligned}$$
(3)
where K is a parameter that describes the evolution of Parkinson’s disease and \(w^H_{ij}\) and \(w^D_{ij}\) are, respectively, the interconnection gains for the healthy and diseased states.

In the absence of stimulation, in the healthy case, the only equilibrium point of the model is locally asymptotically stable. As a consequence, no STN-GPe endogenous oscillations take place. This is not the case for the pathological case, where the stronger synaptic weights between STN and GPe result in an increase in the pallido-subthalamic loop gain that compromises stability. This generates a limit cycle (for details, see, for example, Nevado-Holgado et al. 2010; Pavlides et al. 2012; Pasillas-Lépine 2013), whose frequency stands in the beta-band, thus correlating with experimental observations (Hammond et al. 2007). It is therefore natural to explore the idea of using the exogenous stimulation signal u(t) in order to restore the system’s stability in the pathological situation.

3 Closed-loop stimulation

One of the simplest strategies that can be explored in order to reduce the STN-GPe loop gain is to apply a stimulation that is proportional to the error between the actual firing rate of the STN and a given reference value \(\bar{x}_s\). More precisely, take
$$\begin{aligned} u(t) = -k_p \Big ( x_s(t) - \bar{x}_s \Big ), \end{aligned}$$
(4)
where \(k_p\) is a proportional gain that prescribes the intensity of stimulation. The impact of this stimulation on pathological oscillations is shown in Fig. 1. In the time domain (on the left), one can see that applying a proportional closed-loop stimulation, with \(k_p=15\), stops the oscillation mechanism. This can be explained in the frequency domain (on the right), where it appears that the stimulation scheme reduces the gain of the subthalamo-pallidal feedback loop (see Sect.4.2) and thus increases the stability of this loop with respect to the delays \(\delta _{sg}\) and \(\delta _{gs}\). Indeed, from the Nyquist criterion (see “Appendix 1”), the critical point in this Nyquist diagram is the point \(-1\). When it is encircled, the STN-GPe becomes unstable and sustained oscillations take place.
Fig. 1

Effect of the proportional stimulation signal (4) on the model (1) in time and frequency domains. The feedback gain is taken as \(k_p = 15\). The (unstable) equilibrium firing rate of the STN is taken as the reference \(\bar{x}_s\). The disease’s advancement is \(K=0.5\). Left the stimulation is OFF during the first second of the simulation. Afterwards, it is turned ON for an additional second. STN-GPe endogenous oscillations are cancelled. After transients, the closed-loop signal goes to zero. Right Nyquist plot of the open-loop transfer function \(\exp \left( -(\delta _{sg}+\delta _{sg}) s \right) H(s)\), associated with the subthalamo-pallidal feedback loop (see Sect. 4.2). The closed-loop signal decreases the STN-GPe loop gain. The critical point is encircled when \(k_p = 0\) (green), but not when \(k_p = 15\) (blue), which stops the oscillation (colour figure online)

At a first glance, the possibility of disrupting the pathological rhythm with a vanishing stimulation seems quite encouraging. But, unfortunately, the robustness of this strategy is quite weak. Indeed, if instead of the desired stimulation u(t) a slightly delayed signal \(u(t-d)\) is applied to the system, the pathological oscillation persists and is even amplified, as it is shown in Fig. 2. Taking into account such issues is important, since delays are systematically present in both measurement and stimulation devices (see Sect. 5.3). The phenomenon observed in Fig. 2 should not be confused with the approach considered in Tukhlina et al. (2007). Indeed, while in both cases a proportional feedback and a delay are taken into account, the analysed models are not the same. In our case, it results from the Nyquist plot in Fig. 2 that there exists a delay \(d_0\) such that the feedback loop is stable for any \(d<d_0\) and unstable for any \(d>d_0\). This situation is completely different from the one analysed in Tukhlina et al. (2007), where the delay plays a positive role in the control law and contributes to its success.
Fig. 2

Effect of the proportional stimulation signal (4) on the model (1) in time and frequency domains in the presence of a delay \(d=5\,\hbox {ms}\) in the stimulation feedback loop. Left despite its small value, the delay pulls down the stability of the proportional stimulation strategy, hence showing that robustness with respect to measurement and stimulation delays is fundamental for any practical implementation. Right Nyquist plot of the open-loop transfer function \(G_s = \sigma ^*_s C(s)/(\tau _s s + 1)\), associated with the subthalamic nucleus part of the model (see Sect. 4.1). The critical point is not encircled when \(d = 0\), but is encircled when \(d = 5\,\hbox {ms}\), which explains the lack of stability

In order to increase the robustness of the control law with respect to loop delays, we propose a twofold solution. The first ingredient is to replace the simple proportional stimulation (4) with its filtered version:
$$\begin{aligned} \dot{u}(t) = \omega _p \Big ( -u(t) - k_p (x_s-\bar{x}_s) \Big ), \end{aligned}$$
(5)
where \(\omega _p\) is the filter’s bandwidth. The second ingredient is a careful choice of the proportional gain \(k_p\), of the filter’s bandwidth \(\omega _p\), and of the reference firing rate \(\bar{x}_s\). Indeed, it will be shown in the next section that an adequate choice of these parameters can increase arbitrarily the robustness of the stimulation scheme, with respect to feedback loop delays. The effect of this stimulation on pathological oscillations is shown in Fig. 4.

4 Stability analysis

The aim of this section is to develop the mathematical arguments that lead to the stimulation scheme (5) proposed in the previous section. In particular, it is shown how the parameters of this scheme must be tuned in order to reduce the STN-GPe loop gain and avoid the generation of oscillations.

4.1 Error dynamics in the frequency domain

When the filtered stimulation (5) is included in the STN-GPe dynamics (1), a closed-loop system is obtained. It appears that independently of the values of the gain \(k_p\) and of the reference \(\bar{x}_s\), this system has always a unique equilibrium point \((x_s^{\star },x_g^{\star },u^{\star })\), which does not depend on the filter’s frequency \(\omega _p\). Indeed, in the absence of filter, this is a consequence of (Pasillas-Lépine 2013, Theorem 1), and a simple calculation shows that adding a filter does not change equilibrium points. Therefore, in order to study the local stability properties of this closed-loop system, it can be linearized around its unique equilibrium point. Defining \(e_s = x_s-x_s^{\star }\) and \(e_g = x_g-x_g^{\star }\) and \(e_u = u-u^{\star }\), one can compute the linearized error dynamics
$$\begin{aligned} \begin{aligned} \tau _{s}\dot{e}_s(t)&= \sigma ^{\star }_s\big (-w_{gs}e_g(t-\delta _{gs})+e_u(t-d)\big )-e_s(t)\\ \tau _{g}\dot{e}_g(t)&= \sigma ^{\star }_g\big ( w_{sg}e_s(t-\delta _{sg})-w_{gg}e_g(t-\delta _{gg})\big )-e_g(t)\\ \dot{e}_u(t)&= \omega _p ( -e_u(t) + k_p e_s(t) ), \end{aligned} \end{aligned}$$
(6)
with
$$\begin{aligned} \begin{aligned} \sigma ^{\star }_s&= F_s^\prime \left( -w_{gs} x_g^\star + k_p (\bar{x}_s - x_s^\star ) + w_{cs} v_s \right) \\ \sigma ^{\star }_g&= F_g^\prime \left( w_{sg} x_s^\star - w_{gg} x_g^\star - w_{xg} v_g \right) . \end{aligned} \end{aligned}$$
(7)
By taking the Laplace transform of (6), the linearized error dynamics can be written as a feedback interconnection (Fig. 3). In this feedback loop, the two following closed-loop transfer functions appear
$$\begin{aligned} \begin{aligned} H_s(s)&= \dfrac{\sigma ^{\star }_s}{\tau _{s}s + 1 + \sigma ^{\star }_s \,C(s) \, \hbox {e}^{-ds}}\\ H_g(s)&= \dfrac{\sigma ^{\star }_g}{\tau _{g}s + 1 + \sigma ^{\star }_g \, w_{gg}\, \hbox {e}^{-\delta _{gg}s}}, \end{aligned} \end{aligned}$$
(8)
where \(C(s) = k_p / \left( 1 + s / \omega _p \right) \) is the transfer function of the stimulation scheme.
Fig. 3

Feedback loop associated with the error dynamics (6)

4.2 Reducing the gain of the main feedback loop

Define the transfer function \(H = w_{sg} w_{gs} H_s H_g\), which represents the open-loop transfer function associated with the feedback loop in Fig. 3. In the case where both transfer functions \(H_s\) and \(H_g\) are stable, a natural approach in order to ensure the stability of this loop is to reduce the value of its gain until \(\Vert H\Vert _{\infty }<1\), where
$$\begin{aligned} \Vert H\Vert _{\infty }=\sup _{\omega \in \mathbb {R}}\left| H(j\omega )\right| . \end{aligned}$$
Indeed, in this case, the stability of the feedback loop is guaranteed by the small-gain theorem (see “Appendix 1”). Indeed, if \(\Vert H\Vert _{\infty }<1\), then the Nyquist plot of H entirely lies within the unit circle, which prevents the encirclement of the critical point. Since, moreover, we have
$$\begin{aligned} \Vert H \Vert _{\infty } \le w_{sg} w_{gs} \Vert H_s \Vert _{\infty } \Vert H_g \Vert _{\infty }, \end{aligned}$$
reducing the gains of \(\Vert H_s \Vert _{\infty }\) and \(\Vert H_g \Vert _{\infty }\) seems a good initial objective. In order to reduce \(\Vert H_s \Vert _{\infty }\), the key step is to understand how this gain depends on \(\omega _p\) and on \(\sigma ^{\star }_s\). This point is clarified by the following result.

Proposition 1

For each delay \(d>0\), each firing rate reference \(\bar{x}_s \in [0,M_s]\), and each gain \(k_p>0\), there exists a filter frequency \(\omega _a>0\) such that for every \(\omega _p<\omega _a\), we have
$$\begin{aligned} \Vert H_s\Vert _{\infty }<Q(\omega _p)\sigma ^{\star }_s, \hbox { with } \lim _{\omega _p \rightarrow 0}Q(\omega _p)=1. \end{aligned}$$
(9)

Proposition 1 thus states that the gain of the transfer function \(H_s\) can be reduced arbitrarily close to \(\sigma _s^\star \) by sufficiently reducing the filter bandwidth of the closed-loop stimulation.

It is worth stressing that the expression of the function \(Q(\omega _p)\) is given explicitly in the proof, see Eqs. (21), (22) and (24), together with \(\omega _a\), the maximum value of \(\omega _p\) for which it holds, see Eq. (20).

The case of \(\Vert H_g \Vert _{\infty }\) is simpler than that of  \(\Vert H_s \Vert _{\infty }\). Indeed, it is easy to find conditions for which this gain is bounded and does not depend on the stimulation parameters \(k_p\) and \(\omega _p\). This point is explained by the following result.

Proposition 2

For each delay \(d>0\), each firing rate reference \(\bar{x}_s \in [0,M_s]\), and each gain \(k_p>0\), if \(w_{gg}\sigma _g<1\), then
$$\begin{aligned} \Vert H_g\Vert _{\infty }<\dfrac{\sigma _g}{1-w_{gg}\sigma _g}, \end{aligned}$$
(10)
where \(\sigma _g\) is the maximal slope of the activation function \(F_g\).
As a consequence of Propositions 1 and 2, for each delay \(d>0\), firing rate reference \(\bar{x}_s\), and gain \(k_p>0\), we can find a maximal frequency \(\omega _a\) such that for each \(\omega _p<\omega _a\)
$$\begin{aligned} \Vert H\Vert _{\infty }<\dfrac{w_{sg} w_{gs} \sigma _g}{ 1-w_{gg} \sigma _g } Q(\omega _p) \sigma ^{\star }_s. \end{aligned}$$
(11)
The objective should thus now be to reduce \(\sigma _s^{\star }\), in order to obtain a small gain \(\Vert H\Vert _{\infty }\) for the complete feedback loop.

As it can be seen in Eq. (7), the quantity \(\sigma _s^{\star }\) depends explicitly on \(k_p\) and \(\bar{x}_s\), but it does not depend on the filter frequency \(\omega _p\). Indeed, we have the following result, which exploits the sigmoid shape of the STN activation function and, in particular, its saturation for small and large firing rates.

Proposition 3

For every \(\epsilon >0 \), there exist a firing rate bound \(\bar{x}_\epsilon >0\) and a minimal gain \(k_\epsilon >0\) such that if \(k_p>k_\epsilon \), and \(\bar{x}_s\in \left( 0,\bar{x}_\epsilon \right) \cup \left( M_s-\bar{x}_\epsilon ,M_s\right) \), then \(\sigma _s^\star <\epsilon \).

In other words, by increasing the stimulation gain \(k_p\), and taking the reference \(\bar{x}_s\) close either to 0 or to \(M_s\) (the maximum value of the activation function \(F_s\)), we can reduce as we want the loop gain \(\Vert H\Vert _{\infty }\).

4.3 Stability of the internal feedback loops

The main consequence of the previous section is that the gain \(\Vert H\Vert _{\infty }\) can be made as small as we want. But in order to reach this goal, one has to increase the gain \(k_p\), which acts against the internal stability of the transfer function \(H_s\). Nevertheless, by reducing the filter’s frequency \(\omega _p\), one can always restore the stability of \(H_s\). Indeed, we have the following result.

Proposition 4

Consider the transfer function \(H_s\). For each delay \(d>0\), each firing rate reference \(\bar{x}_s\), and each gain \(k_p>0\), there exists a frequency \(\omega _b>0\) such that the transfer function \(H_{s}\) is stable for every filter frequency \(\omega _p>0\) such that \(\omega _p<\omega _b\).

For the internal stability of the transfer function \(H_g\), the analysis is simpler. Indeed, the condition \(w_{gg}\sigma _g<1\) of Proposition 2 implies the stability of \(H_g\), independently of the value of the internal delay \(\delta _{gg}\). This comes from the fact that under this condition, the Nyquist plot of \(H_g\) lies entirely within the unit circle, regardless of the value of \(\delta _{gg}\) (see, for example, Pasillas-Lépine 2013).

4.4 Robustness

The previous set of propositions define precisely how the stimulation parameters must be tuned. First, one has to compute a bound on \(\Vert H_g\Vert _{\infty }\) using Proposition 2. Second, one has to choose \(k_p\) and \(\bar{x}_s\), using Propositions 1 and 3, in order to have \(\Vert H_s\Vert _{\infty } < \left( 1 / (w_{sg} w_{gs}) \right) / \Vert H_g\Vert _{\infty }\). Third, the internal stability of \(H_s\) must be guaranteed by a sufficiently low value of the filter bandwidth \(\omega _p\), with the help of Proposition 4. This choice must also respect the constraint imposed by Proposition 1. In other words, the filter frequency must satisfy the constraint \(\omega _p<\min (\omega _a,\omega _b)\).

Nevertheless, when one has to choose concrete stimulation parameters, a critical issue is the question of robustness. Indeed, for a given set of model parameters, one can easily check what are the constraints imposed by the previous propositions. But, in a practical stimulation experiment, those parameters might change and be different from those of the nominal model. This point is not necessarily a problem in our approach, since the constraints imposed by the different propositions are inequalities. One can thus consider (among all possible values of the model parameters) the worst case and then choose the parameters of the control law in such a way that they respect the constraints imposed by the worst case. In other words, what we need in order to check our conditions is not the actual value of the different parameters, but some bounds on their possible values. In this context, the small-gain approach introduced in Sect. 4.2 plays a key role in what concerns robustness issues. Indeed, it automatically provides a design that is robust with respect to variations of the delays \(\delta _{sg}\) and \(\delta _{gs}\).

Of course, our method is probably over-conservative. Indeed, another approach would be to represent all the system’s uncertainties using an appropriate template in the frequency domain (Doyle et al. 1990, Chapt. 4.1). For the delay in the STN/GPe loop, it can easily be modelled as a multiplicative uncertainty. But it is not obvious for us how to find a weighting function (and an appropriate uncertainty model) that combines both the STN/GPe loop delay uncertainty and the actuator delay uncertainty in a tractable way. Moreover, one of our objectives is to propose explicit analytic constraints on the controller parameters that guarantee the closed-loop stability of the system. Even if it would be possible to formulate our robust stability problem in the form of an \(H_\infty \) problem, it is unclear (at least for us) how this problem could be solved analytically. Of course, the price to pay in order to obtain our explicit analytic constraints is some level of conservativeness. In contrast, we believe that an \(H_\infty \) approach might lead to less conservative constraints, but obtained after the numerical resolution of a set of LMI’s (see, for example, Doyle et al. 1989; Gahinet et al. 1994).

Another issue concerns our choice of considering in this paper only constant discrete delays. This choice was dictated by the desire to keep the mathematical complexity of the paper as reduced as possible and, additionally, by the fact that both axonal delays and electronic acquisition delays are usually assumed to be constant. For control theory techniques that are able to handle time-varying delays, the interested reader will find in Fridman and Shaked (2002) and He et al. (2007) two classical references that consider this problem (see also Seuret et al. (2013), for more recent techniques).

5 Discussion

In this paper, a strategy for disrupting sustained beta-band oscillations in the STN/GPe network has been described, using a closed-loop stimulation of the STN in the parkinsonian state. This approach is based on a mesoscopic model of the basal ganglia and takes into account both actuation and measurement delays. For that purpose, a filtered stimulation has been proposed with precise criteria for the choice of the filter bandwidth and of the reference STN firing rate. The encouraging theoretical results presented here need, however, to be confronted to experiments in order to evaluate their ability to handle some important physiological and technological questions. First, although evidence that increased motor symptoms are linked to increased beta frequencies in the STN (Hammond et al. 2007), the precise relation between beta-band oscillations and bradykinesia is still debated. Second, fine modelling of the specificity of the employed stimulation device, including limitations in terms of reactivity and maximum amplitude or frequency of the delivered stimulation signal, was neglected in the present analysis. Physiological limitations in the ability of the neuronal population to be influenced by the stimulation signal also need to be carefully addressed. Only animal experimentation will validate the efficacy of the proposed stimulation strategy. The aim of this section is thus to discuss the compatibility of our approach with an actual experimental setting in order to evaluate its potential significance.

5.1 Measuring firing rates

When recording electrophysiological activity of brain matter, most signals can be divided into two kinds of activity. On the one hand, after band-pass filtering between 500 and 2000 Hz, multiunit or single-unit activities can be recorded, typically with 800 k\(\Omega \) or 2 M\(\Omega \) electrodes, respectively, standing for spiking activity from a few axons in the vicinity of the electrode, typically 50–350 microns (Legatt et al. 1980). On the other hand, after low-pass filtering the acquired signal under 200 Hz, local field potentials can be extracted, standing for somato-dendritic activities of neuronal volumes larger than those in which multiunit activities can be recorded, especially when resorting to electrodes with impedances as low as 300 k\(\Omega \) (typically 0.5–3 mm in the vicinity of the electrode Mitzdorf 1987).

The method proposed in this paper resorts to a mesoscopic firing rate, defined for a functionally homogeneous population of neurons (Dayan and Abbott 2001). It is the average, over a specified time interval and over a certain number of neurons that are recorded, of all the spikes measured for this population of neurons. At least for in vitro preparations, there are examples in the literature where such measurements have been obtained in real time (see Wagenaar et al. 2005, for neuron cultures; and Carron et al. 2014, for rodent slices).

The mesoscopic firing rate could be experimentally determined in at least two ways in animal models of Parkinson’s disease. A first option would be to record inside the STN with multiplot electrodes, each plot having impedances around 800 k\(\Omega \). It could then be possible to record from several units in the vicinity of each plot and extrapolate the mesoscopic firing rate based on the recording of these neurons. This may be a good and directly testable option for laboratory experiments but would be less realistic for chronic recordings because of the mechanical instability of the recording setup. The second option for experimental translation of our theoretical work in the setting of chronic closed-loop stimulation would be to perform LFP recordings and then determine spiking activity of the area of neurons recorded. To this end, we may rely on the recent advances in the understanding of the link between single-unit activity and LFP (Teleńczuk and Destexhe 2014) and resort to identification techniques to extract spikes from LFP recordings such as the Hammerstein–Wiener model proposed by Michmizos et al. (2012). The advantage of resorting to LFP is that it enables screening the electrophysiological activity of a larger volume of neurons, and is less sensitive to mechanical instability and gliosis than multiunits recordings.

5.2 Which kind of stimulation?

In this paper, we only considered the effects of a generic stimulation that either excites or inhibits the subthalamic nucleus. Now, when considering specific stimulation technologies, optogenetics (Boyden 2011) would probably be easiest to analyse, because of the advantages this technique offers (in the closed-loop realm) in terms of stimulation artifacts, stimulation selectivity, and ease of modelling. Indeed, as opposed to electrical DBS, the inclusion of optogenetics stimulation in the considered model is rather straightforward as, depending on the nature of the employed photosensitive ionic channel (called more generally opsin), its effect on the targeted population boils down to either inhibition or excitation. This translates into a signal u(t), the sign of which accounts for this inhibitory or excitatory effect, and the amplitude of which depends on the amplitude and frequency of the illumination and on the number of recruited neurons (Han and Boyden 2007). Observe that in Fig. 4, the stimulation signal is negative at all times and thus may require, in order to be implemented, the use of inhibitory opsins (Tye and Deisseroth 2012).
Fig. 4

Simulation showing the impact on pathological oscillations of our filtered proportional stimulation (5) with a gain \(k_p = 45\). We take a reference firing rate \(\bar{x}_s=4\,\hbox {Hz}\) and a filter frequency \(\omega _p=0,15\,\hbox {Hz}\). The delay in the feedback loop \(d=100\,\hbox {ms}\) is 20 times larger than in Fig. 2, which clearly shows the improved robustness of this new stimulation scheme. Top left the pathological oscillations are effectively reduced despite severe acquisition and actuation delays. Observe that the stimulation signal is negative at all times. Right Nyquist plot of the open-loop transfer function \(\exp \left( -(\delta _{sg}+\delta _{sg}) s \right) H(s)\), associated with the subthalamo-pallidal feedback loop (see Sect. 4.2). The gain \(k_p = 45\) makes the whole plot lie in the unit circle, thus fulfilling the conditions of the small-gain theorem, which ensures the stability of the subthalamo-pallidal feedback loop independently of the value of the delays \(\delta _{sg}\) and \(\delta _{sg}\). Bottom left Nyquist plot of the open-loop transfer function \(G_s = \sigma ^*_s C(s)/(\tau _s s + 1)\), associated with the subthalamic nucleus part of the model (see Sect. 4.1). Right Nyquist plot of the open-loop transfer function \(G_g = \sigma ^*_g /(\tau _g s + 1)\), associated with the globus pallidus part of the model (see Sect. 4.1)

Nevertheless, since electrical stimulation is the standard for clinical applications, it would be of interest to adapt the model to account for stimulations of this nature. We emphasize, however, that this is a challenging question in view of the complexity of the impact of electrical DBS on each neuron, which may vary according to the stimulated compartment (soma, dendrite, or axon) and to the orientation of the neuron (McIntyre et al. 2004b).

Other stimulation techniques, including Strafella et al. (2004), Tufail et al. (2010), or Han et al. (2011), could also be considered as possible candidates for experimental testing of the proposed stimulation strategy, provided a fine modelling of their dynamical impact on the targeted neuronal population.

5.3 Delays in the feedback loop

Two kinds of delays can be considered in the proposed STN closed-loop stimulation model: the acquisition and processing delay on the one hand, and the actuation delay on the other hand. Both these delays are embedded in the delay d considered in (6). The acquisition and processing delay combines the delay between the occurrence of a spike and its counting as an event by the electrophysiological recording system and the computation time required for the signal to be sent to the actuating system. It includes the time necessary to compute the firing rate (including filtering and possible spike sorting) and the resulting value of the control law. Nowadays, the order of magnitude of this process is of a few milliseconds. Regarding actuation delays, unlike electrical stimulation, optogenetics neuromodulation techniques offer in deep brain structures both millimetric-spatial and millisecond-time resolution for selective inhibition or excitation resorting to both inhibitory or excitatory photosensitive ionic channels (called opsins) and fibre-coupled lasers delivering the appropriate wavelength. For example, the photocurrent rise time for photostimulable inhibitory ionic channels such as ArchT is 7.4 ms (Zhang et al. 2007), a delay that is compatible with those considered in our approach. Recently, some attempts of disrupting neural oscillations using optogenetics (Han et al. 2009) have been carried out with success (Gradinaru et al. 2009).

5.4 Intermittence of pathological oscillations

Once the cortical and striatal inputs are fixed, the model we use here generates undamped oscillations in STN and GPe. This constitutes a simplified view of what is reported for in vitro (Plenz and Kital 1999) and in vivo (Park et al. 2010) experiments, in which these oscillations appear in a more transient way. This intermittence of pathological oscillations may compromise the efficacy of closed-loop stimulation policies, as reported in Dovzhenok et al. (2013).

A possible way to take into account this intermittence in our model is through the variation of cortical and striatal inputs. Such variations would modify the equilibrium of the STN-GPe network, which would in turn modify the value of the considered slopes of the activation functions (7). Thus, striatal and cortical inputs corresponding to lower slopes would not lead to oscillations, whereas those corresponding to higher slopes would favour oscillations onset. Such an analysis has been conducted in Pasillas-Lépine (2013).

We believe that the closed-loop stimulation policy proposed here is likely to cope with the oscillation intermittence produced by this kind of external inputs. The reason for this is that our stimulation policy essentially boils down to artificially decreasing the STN-GPe loop gain, as explained in Sect. 4.2. We expect this gain reduction to be beneficial in terms of oscillation generation during both intense busts of activity and more quiet periods. A simulation that illustrates this point is shown in Fig. 5.

5.5 Originality and potential significance

The first originality of our computational approach aiming at performing a stimulation disrupting oscillations in the beta-band frequency in the STN-GPe network is to consider a mesoscopic and not a microscopic model (see, for example, Hauptmann et al. 2005b; Franci et al. 2011; Rubin and Terman 2004; Agarwal and Sarma 2010; Liu et al. 2011). This mesoscopic approach may be more robust and realistic from an experimental point of view due to in vivo electrophysiological recordings technical requirements in rodents, non-human primates, or parkinsonian patients. The second originality is to build a method that is robust with respect to experimental constraints such as the delays occurring into a closed-loop system due to measurement, computational processing, and finally actuation.

Although there is a clear link between the intensity of beta-oscillations in STN and the strength of parkinsonian motor symptoms (Hammond et al. 2007; Pogosyan et al. 2010; Little et al. 2012), the nature of this link is still not fully understood. To the best of our knowledge, there is yet no clear evidence on whether these beta-oscillations are the cause, the consequence, or a remote correlation with these motor symptoms. We believe that the selective disruption of these pathological oscillations, in animal models of the disease (Sharott et al. 2005), and the observation of the resulting behaviour of the animal could contribute to the understanding of this link. Therefore, developing methods that show how to decrease or amplify oscillations in the beta-band may help answering this question.
Fig. 5

Effect of an intermittent cortical input on oscillations, with and without closed-loop stimulation. All parameters are the same as in Fig. 4, with the exception of the time-varying cortical input that has a 1.5 Hz frequency. In the absence of stimulation (first second), the cortical input generates a wave of oscillations in the beta-band each time it has an elevated value. The closed-loop stimulation (last 2 s) clearly counters this instability

From a more remote perspective, this work may provide some insights into how to design strategies for efficient and less energy-consuming clinical neuromodulation of parkinsonian patients. And, more generally, this stimulation strategy may be used to disrupt synchronization observed in physiological systems, may they be noticed in pathological conditions like epilepsy (Bragin et al. 2002) or schizophrenia (Gonzalez-Burgos and Lewis 2008), or in healthy conditions (Gray et al. 1989) enabling researchers to probe the role of synchronization in both healthy and pathological conditions.

Notes

Acknowledgments

This work was financially supported by the European Commission through the FP7 NoE HYCON2 and by the region Ile-de-France through the Neurosynch project (RTRA Digiteo). The work of the fourth author was supported by the Government of the Russian Federation (grant 074-U01).

References

  1. Agarwal R, Sarma SV (2010) Restoring the basal ganglia in Parkinson’s disease to normal via multi-input phase-shifted deep brain stimulation. In: IEEE/EMBS conference on neural engineering (NER)Google Scholar
  2. Aström KJ, Murray RM (2010) Feedback systems: an introduction for scientists and engineers. Princeton University Press, PrincetonGoogle Scholar
  3. Benabid AL, Pollak P, Gervason C, Hoffmann D, Gao DM, Hommel M, Perret JE, de Rougemont J (1991) Long-term suppression of tremor by chronic stimulation of the ventral intermediate thalamic nucleus. Lancet 337:403–406CrossRefPubMedGoogle Scholar
  4. Bergman H, Wichmann T, Delong MR (1990) Reversal of experimental parkinsonism by lesions of the subthalamic nucleus. Science 249:1436–1438CrossRefPubMedGoogle Scholar
  5. Boyden ES (2011) A history of optogenetics: the development of tools for controlling brain circuits with light. F1000 Biol Rep 3:1–12. Art ID 11Google Scholar
  6. Bragin A, Wilson CL, Staba RJ, Reddick M, Fried I, Engel J (2002) Interictal high-frequency oscillations (80–50 Hz) in the human epileptic brain: entorhinal cortex. Ann Neurol 52(4):407–415CrossRefPubMedGoogle Scholar
  7. Carron R, Chaillet A, Filipchuk A, Pasillas-Lépine W, Hammond C (2013) Closing the loop of deep brain stimulation. Front Syst Neurosci 7:1–18. Art ID 112Google Scholar
  8. Carron R, Filipchuk A, Nardou R, Singh A, Michel FJ, Humphries MD, Hammond C (2014) Early hypersynchrony in juvenile pink1(-)/(-) motor cortex is rescued by antidromic stimulation. Front Syst Neurosci 8:1–12. Art ID 95Google Scholar
  9. Coombes S, Laing C (2009) Delays in activity-based neural networks. Philos Trans R Soc A Math Phys Eng Sci 367(1891):1117–1129CrossRefGoogle Scholar
  10. Curtain RF, Zwart H (1995) An introduction to infinite-dimensional linear systems theory, volume 21 of texts in applied mathematics, 1st edn. Springer, New YorkCrossRefGoogle Scholar
  11. Davie CA (2008) A review of Parkinson’s disease. Br Med Bull 86(1):109–127CrossRefPubMedGoogle Scholar
  12. Dayan P, Abbott LF (2001) Theoretical neuroscience: computational and mathematical modeling of neural systems. The MIT Press, CambridgeGoogle Scholar
  13. Dovzhenok A, Park C, Worth RM, Rubchinsky LL (2013) Failure of delayed feedback deep brain stimulation for intermittent pathological synchronization in Parkinson’s disease. PLoS ONE 8(3):e58264PubMedCentralCrossRefPubMedGoogle Scholar
  14. Doyle J, Francis B, Tannenbaum A (1990) Feedback control theory. Macmillan Publishing Co., LondonGoogle Scholar
  15. Doyle JC, Glover K, Khargonekar PP, Francis B (1989) State-space solutions to standard \(h_2\) and \(h_\infty \) control problems. IEEE Trans Autom Control 34(8):831–847CrossRefGoogle Scholar
  16. Dunn EM, Lowery MM (2013) Simulation of PID control schemes for closed-loop deep brain stimulation. In: IEEE/EMBS conference on neural engineering (NER), San Diego (California)Google Scholar
  17. Feng X, Greenwald B, Rabitz H, Shea-Brown E, Kosut R (2007) Toward closed-loop optimization of deep brain stimulation for parkinson’s disease: concepts and lessons from a computational model. J Neural Eng 4:L14CrossRefPubMedGoogle Scholar
  18. Franci A, Chaillet A, Pasillas-Lépine W (2011) Existence and robustness of phase-locking in coupled Kuramoto oscillators under mean-field feedback. Automatica 47(6):1193–1202CrossRefGoogle Scholar
  19. Fridman E, Shaked U (2002) An improved stabilization method for linear time-delay systems. IEEE Trans Autom Control 47(11):1931–1937CrossRefGoogle Scholar
  20. Gahinet P, Nemirovskii A, Laub AJ, Chilali M (1994) The LMI control toolbox. In: Proceedings of the IEEE conference on decision and control, pp 2038–2041Google Scholar
  21. Gonzalez-Burgos G, Lewis DA (2008) GABA neurons and the mechanisms of network oscillations: implications for understanding cortical dysfunction in schizophrenia. Schizophr Bull 34(5):944–961PubMedCentralCrossRefPubMedGoogle Scholar
  22. Gradinaru V, Mogri M, Thompson KR, Henderson JM, Deisseroth K (2009) Optical deconstruction of parkinsonian neural circuitry. Science 324(5925):354–359CrossRefPubMedGoogle Scholar
  23. Gray CM, König P, Engel AK, Singer W (1989) Oscillatory responses in cat visual cortex exhibit inter-columnar synchronization which reflects global stimulus properties. Nature 338(6213):334–337CrossRefPubMedGoogle Scholar
  24. Haidar I, Pasillas-Lépine W, Panteley E, Chaillet A, Palfi S, Senova S (2014) Analysis of delay-induced basal ganglia oscillations: the role of external excitatory nuclei. Int J Control 87(9):1936–1956CrossRefGoogle Scholar
  25. Hammond C, Bergman H, Brown P (2007) Pathological synchronization in Parkinson’s disease: networks, models and treatments. Trends Neurosci 30(7):357–364 INMED/TINS special issue–Physiogenic and pathogenic oscillations: the beauty and the beastCrossRefPubMedGoogle Scholar
  26. Han X, Boyden ES (2007) Multiple-color optical activation, silencing, and desynchronization of neural activity, with single-spike temporal resolution. PLoS ONE 2(3):299CrossRefGoogle Scholar
  27. Han X, Qian X, Stern P, Chuong AS, Boyden ES (2009) Informational lesions: optical perturbation of spike timing and neural synchrony via microbial opsin gene fusions. Front Mol Neurosci 2:1–9. Art ID 12Google Scholar
  28. Han X, Chow BY, Zhou H, Klapoetke NC, Chuong A, Rajimehr R, Yang A, Baratta MV, Winkle J, Desimone R et al (2011) A high-light sensitivity optical neural silencer: development and application to optogenetic control of non-human primate cortex. Front Syst Neurosci 5:18PubMedCentralCrossRefPubMedGoogle Scholar
  29. Hauptmann C, Popovych O, Tass PA (2005a) Delayed feedback control of synchronization in locally coupled neuronal networks. Neurocomputing 65:759–767CrossRefGoogle Scholar
  30. Hauptmann C, Popovych O, Tass PA (2005b) Effectively desynchronizing deep brain stimulation based on a coordinated delayed feedback stimulation via several sites: a computational study. Biol Cybern 93(6):463–470CrossRefPubMedGoogle Scholar
  31. He Y, Wang QG, Lin C, Wu M (2007) Delay-range-dependent stability for systems with time-varying delay. Automatica 43(2):371–376CrossRefGoogle Scholar
  32. Jenkinson N, Brown P (2011) New insights into the relationship between dopamine, beta oscillations and motor function. Trends Neurosci 34(12):611–618CrossRefPubMedGoogle Scholar
  33. Kumar R, Lozano AM, Sime E, Lang AE (2003) Long-term follow-up of thalamic deep brain stimulation for essential and parkinsonian tremor. Neurology 61(11):1601–1604CrossRefPubMedGoogle Scholar
  34. Leblois A, Boraud T, Meissner W, Bergman H, Hansel D (2006) Competition between feedback loops underlies normal and pathological dynamics in the basal ganglia. J Neurosci 26(13):3567–3583CrossRefPubMedGoogle Scholar
  35. Legatt AD, Arezzo J, Vaughan HG Jr (1980) Averaged multiple unit activity as an estimate of phasic changes in local neuronal activity: effects of volume-conducted potentials. J Neurosci Methods 2(2):203–217CrossRefPubMedGoogle Scholar
  36. Limousin P, Pollak P, Benazzouz A, Hoffmann D, Le Bas J-F, Perret J-E, Benabid A-L, Broussolle E (1995) Effect on parkinsonian signs and symptoms of bilateral subthalamic nucleus stimulation. Lancet 345(8942):91–95CrossRefPubMedGoogle Scholar
  37. Little S, Pogosyan A, Kuhn AA, Brown P (2012) Beta band stability over time correlates with Parkinsonian rigidity and bradykinesia. Exp Neurol 236(2):383–388PubMedCentralCrossRefPubMedGoogle Scholar
  38. Liu J, Oweiss KG, Khalil HK (2010) Feedback control of the spatiotemporal firing patterns of neural microcircuits. In: IEEE conference on decision and controlGoogle Scholar
  39. Liu J, Khalil HK, Oweiss KG (2011) Model-based spatiotemporal analysis and control of a network of spiking basal ganglia neurons. In: IEEE/EMBS conference on neural engineering (NER)Google Scholar
  40. Lysyansky B, Popovych OV, Tass PA (2011) Desynchronizing anti-resonance effect of m: n on-off coordinated reset stimulation. J Neural Eng 8:036019CrossRefPubMedGoogle Scholar
  41. McIntyre C, Savasta M, Kerkerian-Le Goff L, Vitek JL (2004a) Uncovering the mechanism(s) of action of deep brain stimulation: activation, inhibition, or both. Clin Neurophysiol 115(6):1239–1248CrossRefPubMedGoogle Scholar
  42. McIntyre CC, Grill WM, Sherman DL, Thakor NV (2004b) Cellular effects of deep brain stimulation: model-based analysis of activation and inhibition. J Neurophysiol 91(4):1457–1469CrossRefPubMedGoogle Scholar
  43. Michmizos KP, Sakas D, Nikita KS (2012) Prediction of the timing and the rhythm of the parkinsonian subthalamic nucleus neural spikes using the local field potentials. IEEE Trans Inf Technol Biomed 16(2):190–197CrossRefPubMedGoogle Scholar
  44. Middleton RH, Miller DE (2007) On the achievable delay margin using LTI control for unstable plants. IEEE Trans Autom Control 52(7):1194–1207CrossRefGoogle Scholar
  45. Mitzdorf U (1987) Properties of the evoked potential generators: current source-density analysis of visually evoked potentials in the cat cortex. Int J Neurosci 33(1–2):33CrossRefPubMedGoogle Scholar
  46. Modolo J, Henry J, Beuter A (2008) Dynamics of the subthalamo-pallidal complex in parkinson’s disease during deep brain stimulation. J Biol Phys 34(3–4):251–266CrossRefPubMedGoogle Scholar
  47. Montaseri G, Yazdanpanah MJ, Pikovsky A, Rosenblum M (2013) Synchrony suppression in ensembles of coupled oscillators via adaptive vanishing feedback. Chaos: an interdisciplinary. J Nonlinear Sci 23(3):033122Google Scholar
  48. Nambu A (2011) Somatotopic organization of the primate basal ganglia. Front Neuroanat 5:1–9. Art ID 26Google Scholar
  49. Nevado-Holgado AL, Terry JR, Bogacz R (2010) Conditions for the generation of beta oscillations in the subthalamic nucleus–globus pallidus network. J Neurosci 30(37):12340–12352CrossRefGoogle Scholar
  50. Nini A, Feingold A, Slovin H, Bergman H (1995) Neurons in the globus pallidus do not show correlated activity in the normal monkey, but phase-locked oscillations appear in the MPTP model of parkinsonism. J Neurophysiol 74(4):1800–1805PubMedGoogle Scholar
  51. Omel’chenko OE, Hauptmann C, Maistrenko YL, Tass PA (2008) Collective dynamics of globally coupled phase oscillators under multisite delayed feedback stimulation. Phys D 237(3):365–384CrossRefGoogle Scholar
  52. Park C, Worth RM, Rubchinsky LL (2010) Fine temporal structure of beta oscillations synchronization in subthalamic nucleus in Parkinson’s disease. J Neurophysiol 103(5):2707–2716PubMedCentralCrossRefPubMedGoogle Scholar
  53. Pascual A, Modolo J, Beuter A (2006) Is a computational model useful to understand the effect of deep brain stimulation in Parkinson’s disease? J Integr Neurosci 5(04):541–559CrossRefPubMedGoogle Scholar
  54. Pasillas-Lépine W (2013) Delay-induced oscillations in Wilson and Cowan’s model: an analysis of the subthalamo-pallidal feedback loop in healthy and parkinsonian subjects. Biol Cybern 107(3):289–308CrossRefPubMedGoogle Scholar
  55. Pavlides A, Hogan J, Bogacz R (2012) Improved conditions for the generation of beta oscillations in the subthalamic nucleus–globus pallidus network. Eur J Neurosci 36(2):2229–2239CrossRefPubMedGoogle Scholar
  56. Plenz D, Kital ST (1999) A basal ganglia pacemaker formed by the subthalamic nucleus and external globus pallidus. Nature 400(6745):677–682CrossRefPubMedGoogle Scholar
  57. Pogosyan A, Yoshida F, Chen CC, Martinez-Torres I, Foltynie T, Limousin P, Zrinzo L, Hariz MI, Brown P (2010) Parkinsonian impairment correlates with spatially extensive subthalamic oscillatory synchronization. Neuroscience 171(1):245–257CrossRefPubMedGoogle Scholar
  58. Pyragas K, Popovych OV, Tass PA (2007) Controlling synchrony in oscillatory networks with a separate stimulation-registration setup. Europhys Lett 80(4):40002-p1–40002-p6Google Scholar
  59. Rosin B, Slovik M, Mitelman R, Rivlin-Etzion M, Haber SN, Israel Z, Vaadia E, Bergman H (2011) Closed-loop deep brain stimulation is superior in ameliorating parkinsonism. Neuron 72(2):370–384CrossRefPubMedGoogle Scholar
  60. Rubin JE, Terman D (2004) High frequency stimulation of the subthalamic nucleus eliminates pathological thalamic rhythmicity in a computational model. J Comput Neurosci 16(3):211–235CrossRefPubMedGoogle Scholar
  61. Santaniello S, Fiengo G, Glielmo L, Grill WM (2011) Closed-loop control of deep brain stimulation: a simulation study. IEEE Trans Neural Syst Rehabil Eng 19(1):15–24CrossRefPubMedGoogle Scholar
  62. Schiff SJ (2010) Towards model-based control of Parkinson’s disease. Philos Trans R Soc A Math Phys Eng Sci 368(1918):2269–2308Google Scholar
  63. Seuret A, Gouaisbaut F, Fridman E (2013) Stability of systems with fast-varying delay using improved Wirtinger’s inequality. In: IEEE conference on decision and control (CDC), pp 946–951, Florence (Italy)Google Scholar
  64. Sharott A, Magill PJ, Harnack D, Kupsch A, Meissner W, Brown P (2005) Dopamine depletion increases the power and coherence of \(\beta \)-oscillations in the cerebral cortex and subthalamic nucleus of the awake rat. Eur J Neurosci 21(5):1413–1422CrossRefPubMedGoogle Scholar
  65. Sipahi R, Niculescu SI, Abdallah CT, Michiels W, Gu K (2011) Stability and stabilization of systems with time delay. IEEE Control Syst Mag 31(1):38–65CrossRefGoogle Scholar
  66. Strafella AP, Vanderwerf Y, Sadikot AF (2004) Transcranial magnetic stimulation of the human motor cortex influences the neuronal activity of subthalamic nucleus. Eur J Neurosci 20(8):2245–2249CrossRefPubMedGoogle Scholar
  67. Teleńczuk B, Destexhe A (2014) Local field potential, relationship to unit activity. In: Jaeger D, Jung R (eds) Encyclopedia of computational neuroscience. Springer, New York, pp 1–6CrossRefGoogle Scholar
  68. Tufail Y, Matyushov A, Baldwin N, Tauchmann ML, Georges J, Yoshihiro A, Tillery SIH, Tyler WJ (2010) Transcranial pulsed ultrasound stimulates intact brain circuits. Neuron 66(5):681–694CrossRefPubMedGoogle Scholar
  69. Tukhlina N, Rosenblum M, Pikovsky A, Kurths J (2007) Feedback suppression of neural synchrony by vanishing stimulation. Phys Rev E 75(1):011918CrossRefGoogle Scholar
  70. Tye KM, Deisseroth K (2012) Optogenetic investigation of neural circuits underlying brain disease in animal models. Nat Rev Neurosci 13(4):251–266CrossRefPubMedGoogle Scholar
  71. Wagenaar DA, Madhavan R, Pine J, Potter SM (2005) Controlling bursting in cortical cultures with closed-loop multi-electrode stimulation. J Neurosci 25(3):680–688PubMedCentralCrossRefPubMedGoogle Scholar
  72. Zaidel A, Spivak A, Grieb B, Bergman H, Israel Z (2010) Subthalamic span of oscillations predicts deep brain stimulation efficacy for patients with Parkinson’s disease. Brain 133(7):2007–2021CrossRefPubMedGoogle Scholar
  73. Zhang F, Wang L-P, Brauner M, Liewald JF, Kay K, Watzke N, Wood PG, Bamberg E, Nagel G, Gottschalk A et al (2007) Multimodal fast optical interrogation of neural circuitry. Nature 446(7136):633–639CrossRefPubMedGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Ihab Haidar
    • 1
  • William Pasillas-Lépine
    • 1
  • Antoine Chaillet
    • 1
  • Elena Panteley
    • 1
    • 5
  • Stéphane Palfi
    • 2
    • 3
    • 4
  • Suhan Senova
    • 2
    • 3
    • 4
  1. 1.Laboratoire des signaux et systèmesCNRS – CentraleSupélec – Univ. Paris SudGif-sur-YvetteFrance
  2. 2.AP-HP, Hôpital H. Mondor, Service de NeurochirurgieCréteilFrance
  3. 3.IMRB, Inserm, U955, Equipe 14CréteilFrance
  4. 4.Faculté de médecineUniversité Paris EstCréteilFrance
  5. 5.ITMO UniversitySaint PetersburgRussia

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