A feedback control systems view of epileptic seizures

October 10, 2017 | Autor: K. Tsakalis | Categoria: Applied Mathematics, Feedback Control
Share Embed

Descrição do Produto

Cybernetics and Systems Analysis, Vol. 42, No. 4, 2006

A FEEDBACK CONTROL SYSTEMS VIEW OF EPILEPTIC SEIZURES1 K. Tsakalis,a N. Chakravarthy,a Sh. Sabesan,a L. D. Iasemidis,b and P. M. Pardalosc

UDC 519.6

To understand basic functional mechanisms that cause epileptic seizures, the paper discusses some key features of theoretical brain functioning models. The hypothesis is put forward that a plausible reason for seizures is pathological feedback in brain circuitry. The analysis of such circuitry has an interesting physical interpretation and may be used to cure epilepsy. Keywords: feedback control system, epileptic seizure, coupled-oscillator model. 1. INTRODUCTION Epilepsy is the second most common neurological disorder after stroke, and affects at least 50 million people worldwide. The main seizure control methods include the use of antiepileptic drugs (AEDs), surgical removal of the seizure focus and electrical stimulation. Approximately 60% of new onset epilepsy cases respond to existing AEDs but 40% are pharmaco-resistant, with seizures that cannot be fully controlled with available medical therapy or without unacceptable side effects [1]. Surgical removal of the seizure focus is an important and effective therapeutic intervention for some patients with difficulty to control epilepsy, but is not possible in the large majority of patients because of multiple foci, or seizure foci located within non-recommended for resection areas of the brain. Resective surgery is unlikely to ever replace chronic treatment as the primary mode of epilepsy management in the large majority of patients with epilepsy. Currently, AEDs are the principal form of chronic epilepsy treatment. However, in addition to the lack of efficacy for complete seizure control in at least one third of all patients with epilepsy, there also is substantial morbidity associated with the use of AEDs in many patients, especially when polypharmacy is required. Electrical stimulation paradigms as a means of seizure control seem to have the advantage of not producing the systemic and central nervous system side effects, which are seen frequently with AEDs. The vagus nerve stimulator as an antiepileptic device was approved by the US Food and Drug Administration in 1997. Approximately one-third of patients experience at least a 50% reduction of seizure frequency, but fewer than 10% become seizure free. Deep brain stimulation (DBS), principally of thalamic structures, has also been reported to reduce seizure frequency in humans [2–4]. In a parallel route to control, seizure prediction has also attracted great interest. Using new tools from signal processing developed in the 1980s, Iasemidis et al. [8] reported the first application of nonlinear dynamical measures to monitoring of clinical epilepsy. The existence of long-term preictal (before a seizure) periods was shown using nonlinear dynamical analysis of electroencephalographic (EEG) subdural arrays, leading to the development of seizure prediction algorithms by monitoring the temporal evolution of the maximum Short-Term Lyapunov exponents (STLmax), e.g., [9–11]. In these studies, the central concept was that seizures represent transitions of the epileptic brain from its “normal” less ordered (chaotic) interictal (between seizures) state to an abnormal (more ordered) ictal (during a seizure) state and back to a “normal” postictal (after seizures) state along the lines of chaos-to-order-to-chaos transitions. Seizure prediction could then be achieved by monitoring the dynamical behavior of critical brain sites to reveal “entrainment,” or, in other words, a form of 1

The paper is dedicated to the 70th birthday of Academician I. V. Sergienko. a

Department of Electrical Engineering, Arizona State University, USA. bHarrington Department of Bioengineering, Arizona State University, USA. cDepartment of Industrial and Systems Engineering, Department of Biomedical Engineering and McKnight Brain Institute, University of Florida, USA, [email protected]. Published in Kibernetika i Sistemnyi Analiz, No. 4, pp. 26-40, July-August 2006. Original article submitted June 8, 2006. 1060-0396/06/4204-0483


2006 Springer Science+Business Media, Inc.


dynamical synchronization between sites prior to seizures (i.e., in a period between the interictal and ictal state, called preictal state). The application of this technique to epileptic patients with temporal and frontal lobe focal epilepsy has shown that epileptic seizures can be prospectively anticipated in the range of 70 minutes prior to their occurrence with sensitivity of 85% and false prediction rate of 1 false warning every 8 hours [10]. Other research groups then found marked transitions toward low-dimensional states and reduction of brain’s complexity a few minutes before the occurrence of epileptic seizures [12–16]. Seizures appear to be bifurcations of a neural network that involves a progressive coupling of the focus with the normal brain sites during a preictal period that may last days to tens of minutes. In search of a model and a mechanism to explain the observed behavior of the epileptic brain, [18] followed Freeman’s approach of representing the brain as interconnections of nonlinear oscillators, e.g., [19]. There it was postulated that brain sites (i.e., groups of neurons at each site) might be viewed as chaotic oscillators, connected with each other via diffusive coupling. An increase in the strength of coupling results in progressive dynamical entrainment between the oscillators. Further analysis showed that, in terms of entrainment, this model’s behavior was consistent with the preictal behavior of the epileptic brain. Motivated by the analysis and results of burst phenomena in adaptive systems [20–22], we extended the above model by postulating the existence of a feedback action in the oscillators’ network that enabled the appearance of seizure-like behavior (also see [23]). In general, adaptation bursts are caused by relatively slow parameter drift and recovery cycles. When the adaptation is part of a feedback loop, their effect is exacerbated by simultaneous destabilization of the loop. By incorporating an appropriate feedback structure in the original model by Iasemidis et al., we presented a class of coupled oscillator models that exhibit more key aspects of seizure-like behavior [23]. For example, while changes in coupling do not cause “seizures” in the “normal” brain models, they do bring the “epileptic” brain models in an instability region where “seizures” may occur. Long-term dynamical entrainment is observed during “preictal” periods in the “epileptic” model and is interpreted as an indicator of pathology in the existing internal feedback of the network. At this point, we should emphasize that the models we have proposed are not aimed at reproducing the exact output of the brain (e.g., EEG recordings). Instead, the objective is to capture the essential functional parts of the operation that leads to seizures and incorporate effective compensation strategies to prevent seizures. At this level, the analysis of the oscillator models provides guidance for development of novel control strategies for the suppression and control of epileptic seizures. In distinction with the chaotic dynamical systems modeling approach found in the literature to date, we do not build a detailed model of the brain on the basis of which to try to detect and predict future bifurcations. For example, a detailed model as in [24, 25] is invaluable but could obscure the basic mechanisms of seizure generation. In our approach, the occurrence of the observed seizures, which are related to burst phenomena in adaptive control systems, does not rely upon pathologies in the precise structure of the underlying system, but results from pathologies in the particular implementation of the general operational objectives of the system. At first glance, our analyzed oscillator model with feedback (i.e., with adaptation) may seem as a quite specialized structure. However, biological systems are specialized (optimized), and are the results of adaptation (evolution). This modeling approach is consistent with the concept of Highly Optimized Tolerance (HOT) proposed by Carlson and Doyle [26, 27], since optimization and feedback are the fundamentally impaired mechanisms in our proposed theoretical model for epilepsy. In this paper, in Sec. 2 we present the coupled-oscillator “epilepsy” model and explain its “seizure”-generation mechanism. We then use this model as a guide to develop and test several strategies to control “seizures” in oscillator networks. In particular, we present two main closed-loop control schemes: (a) a discrete closed-loop control (Sec. 3.1) and (b) a continuous closed-loop control with predefined stimulus-based unidirectional and bidirectional stimulation (Sec. 3.2.1), and feedback decoupling (Sec. 3.2.2). In Sec. 4 we present results where discrete closed-loop control and predefined stimulus-based continuous closed-loop control can mitigate “seizures” in certain simple cases, but fail in more complicated cases. These results are consistent with the preliminary results from clinical trials of electrical stimulation based on such simple stimulation strategies [17]. Finally, results from the closed-loop feedback decoupling control scheme are presented. It is shown that the proposed feedback decoupling is the most consistent scheme for “seizure” control. 2. NETWORK OF CHAOTIC OSCILLATORS WITH FEEDBACK The electrical activity at different brain sites has been observed to exhibit patterns of dynamics similar to ones in networks of coupled chaotic oscillators[18]. As an example of this class of models (chaotic oscillators with diffusive coupling), we herein consider a system of N coupled Rossler-like oscillators (i = 1, K , N ) with each oscillator i described by


the following equations: dx i = - w i yi - z i + bi + dt



j =1, j ¹ i

(e i ,


x j - e¢i , j x i ) ,

dy i dz i (1) = w i xi + a i yi , = b i xi + z i ( xi - g i ) , dt dt where the intrinsic parameters a, b, g , w are chosen in the chaotic regime, e.g., 0.4, 0.33, 5, 0.95, respectively; bi are small constant bias terms, different for each oscillator, which ensure that the origin is not an equilibrium point (in our examples, bi ’s have “random” values in [– 0.2, 0.2]); e, e ¢ are the generally asymmetric coupling strengths; in this example, we take e = e ¢. When the e between two oscillators increases, their dynamical behavior progressively becomes similar until it becomes nearly identical at high values of e. As shown in [18], the STLmax of the coupled oscillators begin to converge for values of e above 0.1 and the system then loses its spatio-temporal chaotic behavior (STLmax) for stronger coupling (e ~ 0.25). It is important to realize that synchronization (dynamical entrainment), which is a structural/network property, occurs while the temporal response of each oscillator remains chaotic. For the sake of simplicity of presentation and simulation expedience, in the simulation data we analyze next, we use the correlation coefficient between pairs of oscillator signals (instead of a distance measure between their STLmax profiles) to quantify the observed synchronization between them. 2.1.

Network internal feedback: a natural compensation for changes

Motivated by the adaptation burst paradigm [22], in the general oscillator network we construct feedback around each pair of oscillators with the objective to decorrelate their outputs when excessive coupling occurs as a result of a change (input) to the network. Such inputs are translated into temporal changes of the coupling between the network oscillators. The thus modified oscillator network is now described by the following equations: dx i = - w i yi - z i + bi + dt



j =1, j ¹ i

(e i ,


x j - ei,


xi ) + u i , j ,

dy i dz i = w i xi + a i yi , = b i xi + z i ( xi - g i ) . dt dt The model is solved with a fixed time step 0.01 sec. The internal feedback signals u i , ui ,


= k i , j ( x i - x j ),

k i , j = PI I {r [ x i , x j ] - c * }.

(2) j

are defined as follows: (3)

Following the adaptive control paradigm, the feedback gains k i , j are themselves produced by a Proportional-Integral (PI) feedback, while r denotes the correlation between two signals and c * is a threshold parameter (here taken as c * = 01 . — see below). The subscript I in the PI I notation indicates that the PI feedback is a part of the internal network of the system. The estimation of the correlation is performed in an exponentially weighted fashion in order to simplify the model’s simulation: (4) r [ x i , x j ] = m 2x x / ( mxi xi mx j x j ) , i j

& xi x j ( t )dt = - amxi x j ( t ) + ax i ( t ) x j ( t ) . m In all of our simulations we used a time constant of 200 sec (a = 0.005). Despite the highly nonlinear nature of this model, it is observed that a simple PI compensator is sufficient to achieve the decorrelation of the oscillators, as long as its bandwidth is not too large. (For the PI tuning we followed [30], although a working solution can easily be obtained by simple trial-and-error.) Further, PI I is restricted to produce signals in the interval [0, 1] and it employs limited integration as an anti-windup mechanism (e.g., see [29]). This guarantees that when the correlation between the two signals is below the threshold c * , no PI feedback is generated. The PI I feedback k i , j can be viewed either as a decoupling compensator or as an estimator of the oscillators’ coupling parameter e i , j . During its operation, PI I , that emulates the internal feedback of the brain, generates an output that attempts to cancel the effect of the excessive coupling in the oscillator network and maintain the correlation between the respective two signals below a given threshold c * . From a feedback point of view, the PI destabilizes the oscillator network to maintain its chaotic behavior and counteract the stabilizing effect of the diffusive coupling. Electrotonic coupling between 485

Fig. 1. Brain emulator as a network of coupled oscillators. Only the non-zero coupling between the respective oscillators is shown with the connecting lines. neuronal axons may also account for coupling changes in the brain, and may thus constitute a better microscopic model for the type of coupling (diffusive) we consider in our networks herein. While results from a simple 3-oscillator network case were reported in [28], here we consider a network model containing N = 10 oscillators. Its topology is shown in Fig. 1. We assume that abnormal feedback can occur in oscillator pairs {3,4}, {4,5}, {3,8}, and {4,8} which can lead to very complex behavior. In the following simulations, we consider different time-varying values for e 3 , 4 , e 4 , 5 , e 3 , 8 , and e 4 , 8 ; e1,2 = e 2 , 3 = e 3 , 1 = c1 = 0.08, e 5 ,6 = e 6 ,7 = e 7 , 5 = c 2 = 0.07, e 8 ,9 = e 9 , 10 = e10 ,8 = c 3 = 0.05. 2.2.

Seizure generation due to pathological internal feedback

Our underlying assumption is that “the pathology of the epileptic brain lies in that its intelligent controller does not provide the necessary feedback action to compensate for the increase of the oscillator network coupling e.” In other words, the feedback correction from an improperly tuned internal feedback controller may get out of phase with an input that caused the coupling change in the oscillator network. This results in a negative effective coupling coefficient of the combined oscillator network and PI I s. If the effective coupling (i.e., the one caused by the coupling change and the corresponding internal feedback compensation) is large enough, it may destabilize the chaotic states of the network and produce high amplitude divergence, like “seizures”. A precursor of this scenario is an abnormal increase in coupling and synchronization that is not removed quickly enough by the internal compensation mechanism. Implicit in this theoretical analysis is the dependence of “seizures” on the variations of the coupling e. Thus, while the “epileptic” network of oscillators is susceptible to “seizures” due to its pathologically high values of the effective coupling, the exact onset of “seizures” depends on the inputs to the network that cause variations to its coupling. Our overall hypothesis also provides a model for the operation of the “normal” brain as follows: (a) system in spatiotemporal chaos, (b) stimulus enters the system, changes e and enables spatial coupling, (c) spatial coupling produces spatial correlations, possibly storing the information about the stimulus and/or initiating action upon this information, (d) spatial correlations also activate an internal compensating feedback mechanism, (e) internal compensation removes (or assimilates) the stimulus effect, (f) the system returns to spatiotemporal chaos. The assumption that in the “normal brain” correlations in the network have to exist, and lie within “normal” range, leads us to assume that the existing PI I s in the “normal brain” should follow changes in e i , j and compensate for them in a short time. On the contrary, in the pathological “epileptic brain”, we expect that the PI I s would not be able to compensate for such e i , j changes, and the corresponding parts of the system will exhibit adaptation bursts. For demonstration purposes, in our simulations we have modeled the normal PI I s to have proportional gain K p = 2.1 and integral gain K i = 0.0315, whereas the pathologic PI I s have an order of magnitude less values (K p = 0.21 and K i = 0.00315, see below for more details). In the implementation of this model we use some additional highlevel logic in order to emulate the pathological “brain’s” recovery after a “seizure:” the model states x i , y i , and z i are reinitialized (x i = y i = z i = 0) as soon as their norm exceeds a large value threshold M xyz (here 500). This reinitialization has a reasonable, quasi-physiological interpretation as high electrical activity might deplete critical neurotransmitters and thus deactivate critical neuroreceptors in a seizure participating neuronal network (passive mechanism). An alternative justification for this reinitialization is the release of neuropeptides in the brain 486

Fig. 2. “Brain” response for oscillators 3–4. “Normal brain network” behavior where signal correlations between sites remain low throughout the operation.

Fig. 3. Uncontrolled “brain“ response for pathological oscillator pair 3–4.

as a result of seizures, which subsequently may contribute to the observed seizure recovery by initiating a feedback compensation intervention (active mechanism). Panel Legends for Figs. 2, 3 (top to bottom): Panel I. Coupling coefficient e 3 , 4 (dash) and its feedback estimate by the internal PI feedback (dash-dot); approximate correlation signal (solid). Panel II. Oscillator outputs for each of the 2 oscillators (solid, dash-dot). For clarity of presentation, their DC-shifted values (x 3 + 30 and x 4 - 30) are plotted. Panel III. Applied external control signal (stimulation). In these examples, only the oscillator pair {3,4} is pathological. Fig. 2 shows the response of the oscillator network with a well-tuned internal PI I feedback (“normal brain”), producing a feedback gain that tracks the changing network coupling coefficient reasonably well. For this simulation the PI I feedback has a transfer function 21 . + 0.0315 / s. Fig. 3 shows a network with detuned PI I internal feedback (“epileptic brain”), its output reduced to 10% of the normal case. With a reduced internal feedback gain for the PI I between 3–4, the internal controller can no longer follow the coupling changes closely. Signal growth due to instability bursts appears soon after the coupling estimate exceeds the actual value of coupling. Notice the significant increase in signal correlation between the pathological sites that precedes the “seizures”, which is similar to the entrainment observed in actual epileptic EEG prior to seizures. The PI I estimates the increase of the network coupling relatively well until the network coupling starts to decrease too rapidly. At that point, the detuned (pathological) internal feedback gain of the PI I attains a large value, as it cannot follow the rapid change in e. This destabilizes the oscillator network and its output grows in time. The “seizures” persist until the wrong value of the feedback gain from PI I is dissipated. Pre-"seizure" entrainment similar to the one observed prior to real seizures is also noticed before the “seizure”. It is hypothesized that this is due to the inability of the detuned internal controller PI I to track the changes in the oscillator network coupling e, thus allowing correlation between two oscillator outputs to increase beyond minimal values (compare correlations in panels I of Figs. 2, 3). Similar behavior is also seen when multiple pathological pairs exist-well tuned PI I can track the changing coupling values and maintain low values synchronization between the respective oscillator pairs, while the detuned PI I s result at “seizures” in the corresponding oscillator pairs. 3. SEIZURE CONTROL IN PATHOLOGICAL OSCILLATOR NETWORKS In addition to generating a functional model for the normal brain operation, the proposed network structure provides a test-bed and insight for implementation of feedback control strategies to suppress seizures in the epileptic brain. This is achieved by testing different control schemes for several pathological oscillator networks that can be generated out of the proposed general network structure. Our current in-vivo experiments with epileptic rats show the ability to affect the epileptic brain’s excessive dynamical entrainment by means of external electrical stimulation and/or drug intervention [17]. At this point, it is not possible to assess whether electrical stimulation may provide complete seizure prevention or be used as an add-on therapy to AED treatment. A 487

natural goal for a seizure control scheme would be the disruption of correlation (synchronization/entrainment) patterns observed prior to seizures with minimal side effects. For example, it would not be helpful at all if seizures are prevented, while the patient is rendered unconscious, in pain, or any other dysfunctional condition. A good parallelism here can be drawn with the treatment of heart attacks (shock therapy) versus arrhythmias (pacemakers). Since seizures are chronic, and typically not terminal for the patient, what is needed for their treatment is the equivalent of a cardiac pacemaker, i.e., an epileptic brain pacemaker. The hypothesis-driven simulation experiments that we present next address this line of research with simulation studies, i.e., successful control of oscillator networks. We believe that such research could eventually guide us on the choice of suitable stimulation schemes for the prevention of seizures with minimal side-effects. In the following sections, we investigate two major closed-loop seizure-control strategies, that is, a) discrete closed-loop control, b) continuous closed-loop control. In the continuous closed-loop control scheme, we studied two broad categories: i) predefined stimuli (unidirectional and bidirectional) and ii) stimuli that depend on the state of the system, what we call feedback decoupling scheme. The efficacy of a control scheme in controlling seizures can be quantified as follows. Suppose there are P oscillators in a network that can exhibit “seizures”. Let x b , i , i = 1, K , P, be their “seizure”-free outputs (baseline), and x c, i be their controlled outputs; d i ( t ) = s 2x ( t ) / s 2x , where s 2x ( t ) is the variance of a moving window (fixed-duration) of data till time c, i

instant t, and s 2x


b ,i


is the baseline variance; d ( t ) = (1 / P )



i =1

d i ( t ) . Thus d ( t ) can be considered as a measure of “seizure

intensity” over time across all oscillators with respect to a “seizure”-free oscillator network. Furthermore, the average value of d ( t ) over the whole simulation period, say D, can be used as a measure of the efficacy of a control strategy. For D ~ 1, the oscillator network is close to normal behavior (i.e., “seizure”-free). For example, the value of D for the non-pathological case (Fig. 2) is 0.98, while for the pathological uncontrolled case (in Fig. 3), D = 2.11 . In the following, Dn and D p denote the values of D for the non-pathological and pathological oscillator network, respectively. 3.1.

Discrete closed-loop control

One strategy for an external controller to control the route of the epileptic brain toward seizures is to stimulate the brain in an open-loop mode, e.g., with impulsive, sinusoidal, square or other waveforms, as it has been proposed in the literature and already applied in clinical trials, e.g., [2]. This strategy is called open-loop discrete-time control because the external controller generates a predefined stimulation sequence independent of the state of the brain, and the stimulation sequence is applied at discrete time intervals. It attempts to suppress seizures by a “shock” type of therapy and the hope is to be able to also reset the brain operation and decrease the abnormally high synchronization between critical brain sites. Our simulations with this type of control of our models above show that a “seizure” can be prevented provided that the stimulation has sufficient power and is delivered early enough prior to the “seizure”. This points to a better strategy for an external controller, that is a “shock” therapy combined with a long-term early seizure prediction scheme, referred to as closed-loop discrete-time control. Our simulation results with this control scheme (see Sec. 4) illustrate that seizure suppression is possible with early warning (seizure prediction), but fails when the stimulation is applied close to the seizure onset (seizure detection). 3.2.

Continuous closed-loop control

Another control strategy is to employ closed-loop continuous-time control, which would involve continuous feedback, at least during the high seizure-susceptibility periods. In this strategy, the controller produces a stimulation continuously as long as correlation measures of the brain states exceed a threshold. If this external stimulation, here denoted by u iE , enters the oscillator network in an additive manner, dx i = - w i yi - z i + bi + dt



j =1 j¹i

( e i , j ( x j - x i ) + u iI, j ) + u iE ,


where y i and z i are as in (1). Fig. 4 shows the conjectured functional block diagram of the brain with its internal feedback circuit and the continuous closed-loop external controller used to compensate for pathologies of the brain’s internal feedback. 488

Fig. 4. Functional block diagram of the proposed internal feedback structure of the brain and the proposed closed-loop control mechanisms. The continuous closed-loop control schemes can be classified according to the type of external stimulation u iE . In our simulations with this control strategy, the level of correlations r (see Eqn. (4)) is used as a measure of the dynamical entrainment (synchronization) of “brain” sites. Other options include the T-index of STLmax between different sites. External stimulation is applied when excessive level of correlations between “brain” sites is sensed. 3.2.1. Continuous closed-loop control with predefined external stimuli. In this case, u iE is a predefined stimulus, for example, biphasic pulse inputs with a pre-determined power and duty cycle. Continuous closed-loop control with predefined stimulus can be further classified based on the mode of application of the stimulus. In unidirectional stimulation, u iE is applied only to one of the pathological oscillators of a pair of oscillators i and j that exhibits high correlation ( u iE ¹ 0, u Ej = 0) . In bidirectional stimulation, opposite control inputs are applied to the oscillators in a pathological pair, i.e., u Ej = - u iE per pathological pair {i, j}. The terminology of unidirectional and bidirectional stimulation arises from a state space view of the stimulation signal. In the unidirectional scheme the state of the stimulated oscillator is perturbed in the direction of the stimulus vector field, while in the bidirectional scheme the two stimulated oscillators are perturbed in opposite directions (one along and the other opposite to the stimulus vector field). For simulations presented in this paper, we use biphasic pulse stimulation waveforms. This is motivated by physiological considerations to minimize the total stimulation charge over a time interval. Furthermore, we tune the stimulation parameters (amplitudes, frequency and duty cycle) via a simple combinatorial search of the parameter space in order to achieve the best possible “seizure” suppression. 3.2.2. Continuous closed-loop control via feedback decoupling. Feedback decoupling control is inspired from adaptive control. In this closed-loop continuous-time scheme, the controller is turned automatically on as needed (intelligent). Equally important, its output is not a predetermined sequence of values but it is a function of the “brain” state. In this scheme, the PI E feedback signal has the form C i , j ( x i - x j ) , that is the same form as the hypothesized internal feedback PI I . The external controller PI E gains C i , j are the manipulated variables and are updated using a PI control/ estimation strategy. This controller takes advantage of the hypothesized structural information about the system and theoretically, it could completely decouple the two oscillators. The success of this strategy depends on how realistic our postulated internal feedback structure is. However, a general advantage of continuous feedback is that, since the control input is continuously updated, the requirements on model accuracy are less stringent than the ones for discrete (open-loop) control. 4. RESULTS 4.1.

Discrete closed-loop control

Results from the discrete closed-loop control in the oscillator network with one pathological pair {3,4} (e 3 , 4 same as that in Fig. 2) are shown in Figs. 5, 6. Emulating an early seizure warning (in reality issued by a prospective seizure prediction algorithm), the square pulse external stimulation is switched on in a discrete fashion as the coupling and the 489

Fig. 5. Discrete closed-loop control of the “epileptic brain network” by early pulse-train stimulation.

Fig. 6. Discrete closed-loop control of the “epileptic brain network” by late pulse-train stimulation.

correlation between pathological sites increases. After early activation of the stimulation (a “seizure” would have occurred at about 2500 seconds, see Fig. 3), the site correlation returns to low values and remains low throughout the range of change of coupling. However it appears that this kind of intervention, while it prevented the “seizure”, has a considerable effect on the amplitude of the “brain” responses too (middle panel). Thus, implications of this “treatment” on an actual brain must be carefully evaluated. Panel Legends as in Fig. 2. External stimulation, after an issue of a late warning (typically given by a seizure detection algorithm at the onset of a seizure), is insufficient to suppress the “seizure” even with large stimulation amplitudes. Panel Legends as in Fig. 2. The stimulus is a biphasic pulse waveform (period = 1 second, 50% duty cycle). The stimulation is applied to the pathological oscillator pair bidirectionally (i.e., u 4E = - u 3E ). “Seizure” suppression is achieved with an early delivery of stimulation, but fails when the stimulation is applied as late as at the “seizure” onset. Systematic analysis with the stimulation delivery times indicates that delaying the initiation of stimulation decreases the ability to suppress the upcoming “seizure”. Also, the earlier the stimulation delivery, the less power is required to suppress a “seizure”, although there is a lower threshold below which the “seizure” cannot be suppressed. This emphasizes the need for an early seizure warning (and not simply a seizure detection) scheme as an essential component in an effective and efficient discrete closed-loop seizure control system. 4.2.

Continuous closed-loop control

We show that the predefined stimulus-based control based on continuous monitoring of the correlation between the “brain” states can be successful in mitigating “seizures” only in certain simple cases and we compare its performance with the feedback decoupling control scheme. One such example is the oscillator network with the aforementioned localized pathology (a single pathological pair {3,4}, with e 3 , 4 same as in Fig. 2.) For this configuration, it is possible to mitigate “seizures” with unidirectional stimulation (amplitude = 16, period = 1 second, 50% duty cycle, see Fig. 7) or bidirectional stimulation (amplitude = 9.5, period = 1 second, 50% duty cycle, see Fig. 8). In Fig. 7, the biphasic stimulation is activated by a feedback signal when the correlation between the two sites exceeds a threshold of “normality” (0.1). However, it appears that this kind of intervention, while it prevented the “seizure”, has considerable effect on the amplitude of the oscillator outputs too (see Fig. 10). Panel Legends as in Fig. 2. In Fig. 8, the biphasic stimulation is activated by a feedback signal when the correlation between the two sites exceeds a threshold of “normality” (0.1). This kind of intervention prevented the “seizure” and in comparison with the unidirectional stimulation (of the same pathology) has less interference with the amplitude of the oscillator outputs (see also Fig. 10). Panel Legends as in Fig. 2. Fig. 9 shows the control of the same network pathology with feedback decoupling. Notice that in comparison with the predefined stimulus-based controllers, the feedback decoupling controller a) requires lesser control effort and b) is less interfering with the system.


Fig. 7. Continuous closed-loop predefined stimulus-based unidirectional control of the oscillator network configuration with 1 pathological oscillator pair {3, 4} and e 3 , 4 same as in Fig. 2.

Fig. 8. Continuous closed-loop predefined stimulus-based bidirectional control of the oscillator network configuration with 1 pathological oscillator pair {3, 4} and e 3 , 4 same as in Fig. 2.

Fig. 9. Continuous closed-loop control of the “epileptic brain network” with PI feedback decoupling compensation. This is much closer to normal operation with respect to all measures (see also Fig. 10), while utilizing lower power stimulation signal. Panel Legends as in Fig. 2.

Fig. 10. The measure of d i ( t ) “seizure intensity” over time for the pathological oscillators {3, 4} (d 3 : solid, d 4 : dash).

From the above simulations, the control efficacy is also evident from the plots of d i ( t ) in Fig. 10 and the value of D, . , DLDCL = 2. 65, where DEDCL and DLDCL represent the values of D for early and late Dn = 0.98, D p = 2.11 , DEDCL = 121 application of discrete closed-loop control, respectively. In Fig. 10, the panels clockwise from top-left: uncontrolled, controlled through early discrete closed-loop simulation, controlled through late discrete closed-loop simulation, under continuous closed-loop unidirectional simulation, under continuous closed-loop bidirectional stimulation, under continuous closed-loop feedback decoupling. For this network configuration several strategies are successful in suppressing “seizures”, but impulsive stimuli lead to an increase in d i ( t ) due to their interference with the oscillator outputs. Notice that although early stimulation can suppress a “seizure”, the stimulus interferes with the output (also seen from DEDCL > Dn in Fig. 10) and thus may temporarily change the state of the “brain”. For the continuous control schemes, the respective D values are: 491

Fig. 11. Unsuccessful continuous closed-loop unidirectional feedback control of the network with 3 pathological oscillator pairs — {3,4}, {4,5}, and {4,8}.

Fig. 12. Unsuccessful continuous closed-loop bidirectional feedback control of the network with 3 pathological oscillator pairs — {3,4}, {4,8}, and {3,8}.

Fig. 13. Successful continuous closed-loop feedback decoupling control of the network with 3 pathological oscillator pairs. Du = 1.17, Db = 104 . , D fb = 0.97, where Du , Db , and D fb stand for the values of D using unidirectional, bidirectional and feedback decoupling stimulation respectively. Both unidirectional and bidirectional stimulation show a increase in the measure D, albeit small, indicating a potential interference with the state of the “brain,” while feedback decoupling does not. These values are representative for other pathological configurations as well. Next, we study the efficacy of the closed-loop controllers in more complicated network settings. To test the limitations of unidirectional and bidirectional feedback stimulation, we consider the oscillator network with three pathological oscillator pairs — {3,4}, {4,5}, {4,8}, resulting in “seizures.” It was not possible to completely control all “seizures” in all oscillators with the “best” combination of stimulation parameters (period = 1 second, duty cycle = 50%, stimulation amplitudes of 14, 11.2, and 9 for pathological pairs {3,4}, {4,5}, and {4,8} respectively, see Fig. 11). In Fig. 11, instabilities occur in each of the oscillator pairs due to the reduction in their respective internal controller gains. Biphasic stimulation is used and its parameters are tuned to obtain the best possible “seizure” suppression. Panel Legends (top to bottom): I — Coupling coefficients. II — Oscillator outputs for each of the pathological oscillators. III— Applied external 492

control signals (stimuli). For clarity of presentation in Panels II and III, the DC-shifted signals are plotted. The optimal stimulation parameters were chosen from a given set through combinatorial search procedure. For the bidirectional feedback stimulation, we consider a network with three pathological feedbacks: {3,4}, {4,8}, and {3,8}, resulting in seizures. Bidirectional feedback stimulation is applied between each of the pathological oscillator pairs. We used the straightforward tuning technique to choose biphasic pulse waveform parameters (period = 1 second, 50% duty cycle and stimulation amplitudes of 14.2, 14.5, and 11.2 corresponding to pathological pairs {3,4}, {4,8}, and {3,8} respectively). Here again, it is not possible to control all “seizures” in all oscillators. In Fig. 12, biphasic stimulation is used and its parameters are tuned to obtain the best possible “seizure” suppression. Panel Legends as in Fig. 11. On the other hand, the feedback decoupling controller was successful in completely controlling “seizures” in all cases and all pathological oscillator pairs, requiring lesser control effort and interfering less with the oscillator outputs. In Fig. 13, left two figures: oscillator outputs and external control signal for the case of pathologies at pairs {3,4}, {4,5}, and {4,8} (same network pathology as in Fig. 11). Right two figures: oscillator outputs and external control signal for the case of pathologies at pairs {3,4}, {4,8}, and {3,8} (same network pathology as in Fig. 12). Notice that in comparison with the corresponding unidirectional and bidirectional stimulation cases, less control effort is required and the “seizures” are completely controlled. Notice that the unidirectional and bidirectional feedback may only partially control all “seizures”. This is evident from the D values for the above simulations. For the case of the failing unidirectional feedback: Dn = 107 . , . , D p = 4.19, Db = 137 . , D p = 2.47 , Du = 1.37, D fb = 1.07. For the case of the failing bidirectional feedback: Dn = 103 D fb = 1.01 . The above two cases are representative of a larger number of cases we tested. In all cases, closed-loop control using pre-defined stimuli was not sufficient to control “seizures,” while a similar feedback decoupling controller (PI E ) was successful in all cases. This can also be seen from the consistent trend in respective D values: Dn ~ D fb ~ 1£ Du , Db < D p , i.e., seizure mitigation using predefined stimuli is not always as efficacious as the complete seizure control using feedback decoupling. This is the result of extracting and using more information for control. Furthermore, in comparison with control using predefined stimuli, the necessary decoupling control input from PI E to prevent seizures (a) is less interfering with the “brain’s” operation (less “side-effects”), and (b) uses lower amplitude/power control signal. 5. CONCLUSIONS Motivated by recent advances in the early detection of a preictal state and, consequently, the prediction of epileptic seizures, we investigated theoretical aspects of the problem of controlling or suppressing seizures by means of feedback control. Towards this goal, we first improved previously proposed networks of chaotic oscillators as functional models of brain operation. We showed that by including internal feedback terms in such networks, many qualitative similarities with the observed dynamical behavior of the epileptic brain exist. In particular, when pathology causes de-tuning of the postulated internal feedback, the network exhibits “seizures” (high amplitude oscillations), preceded by entrainment periods, similar to the ones observed prior to actual epileptic seizures. Although such a model should be considered only as an approximation to what really happens in the epileptic brain, it is developed on fundamental engineering principles and exhibits striking similarities with the observed dynamics in the EEG before, during, and after seizures. Resolving brain signals at the level of neuron firing is a highly nontrivial undertaking. In addition, analysis of large-scale neuronal networks involves interconnected nonlinear systems with complex dynamics. Clearly, modeling with such networks is very complicated and depends on many factors both internal and external to the system (brain). For example, the state of the subject (wake/asleep), sensory inputs, anatomy and physiology, all play a role on the exact long-term brain behavior. However, a theoretical approach that addresses the basic dynamics of such physiological networks appears to be feasible. Such an approach, consistent with the experimental observations, suggests interesting applications to the control of the epileptic brain by adjusting the brain’s information processing mechanisms. The most important aspect of such a modeling approach is that it allows the testing and refinement of control strategies, and suggests alternative ones for seizure control. In particular, our hypothesis for the normal and pathological operation of the brain suggests a method to suppress seizures by supplying external compensation that fixes the pathology of the biological internal feedback. Based on the proposed theoretical model, different seizure control strategies were tested through simulations: discrete closed-loop and continuous closed-loop control. The discrete closed-loop and open-loop strategies have been tried in clinical trials. Our simulation results illustrate the inefficiency and potential side effects of such strategies. Preliminary clinical trial results appear to corroborate our simulation results. While continuous closed-loop stimulation with predefined stimulus was


successful in a few simple networks (limited pathology), the best results on control of “seizures” were achieved by our novel method of continuous closed-loop feedback, that we have called “feedback decoupling control”, that uses a brain state-dependent stimulus. In addition to achieving the best results on seizure control, this method requires considerably less stimulation power to achieve the disentrainment or decorrelation of the brain-network sites, and with minimal side effects (reduced interference with brain states). The in-vivo validation of this model is currently actively pursued on several animal models of epilepsy in our Laboratories at Arizona State University and collaborating sites at Barrow Neurological Institute, Phoenix, Arizona. 6. ACKNOWLEDGEMENTS This research is supported by the American Epilepsy Research Foundation and the Ali Paris Fund for LKS Research and Education, and National Institutes of Health (R01NS396871). Professor Pardalos’ research was partially supported by NIH and CRDF grants.



4. 5. 6. 7. 8. 9.



12. 13.




P. Kwan and M. J. Brodie, “Early identification of refractory epilepsy,” New Engl. J. Medicine, 342, 314–319 (2000). J. F. Kerrigan, B. Litt, R. S. Fisher, S. Cranstoun, J. A. French, D. E. Blum, M. Dicher, A. Shetter, G. Baltuch, J. Jaggi, S. Krone, M. A. Brodie, M. Rise, and N. Graves, “Electrical stimulation of the anterior nucleus of the thalamus for the treatment of intractable epilepsy,” Epilepsia, 45(4), 346–354 (2004). E. H. Kossoff, E. K. Ritzl, J. M. Politsky, A. M. Murro, J. R. Smith, R. B. Duckrow, D. D. Spencer, and G. K. Bergey, “Effect of an external responsive neurostimulator on seizures and electrographic discharges during subdural electrode monitoring,” Epilepsia, 45(12), 1560–1567 (2004). I. Osorio, M.G. Frei, S. Sunderam, J. Giftakis, N. C. Bhavaraju, S. F. Schaffner, and S. B. Wilkinson, “Automated seizure abatement in humans using electrical stimulation,” Annals of Neurology, 57(2), 258–268 (2005). J. G. Milton, J. Gotman, G. M. Remillard, and F. Andermann, “Timing of seizure recurrence in adult epileptic patients: a statistical analysis,” Epilepsia, 28, 471–478 (1987). W. Penfield, “The evidence for a cerebral vascular mechanism in epilepsy,” Ann. Int. Med., 7, 303–310 (1933). W. G. Lennox, Science and Seizures, Harper, New York (1946). L. D. Iasemidis, H. P. Zaveri, J. C. Sackellares, W. J. Williams, and T. W. Hood, “Nonlinear dynamics of electrocorticographic data,” J. Clinical Neurophysiology, No. 5 (1988). L. D. Iasemidis and J. C. Sackellares, “The temporal evolution of the largest Lyapunov exponent on the human epileptic cortex,” in: D. W. Duke and W. S. Pritchard (eds.), Measuring Chaos in the Human Brain, World Scientific, Singapore (1991), pp. 49–82. L. D. Iasemidis, D. S. Shiau, W. Chaovalitwongse, J. C. Sackellares, P. M. Pardalos, J. C. Principe, P. R. Carney, A. Prasad, B. Veeramani, and K. Tsakalis, “Adaptive epileptic seizure prediction system,” IEEE Trans. Biomedical Engineering, 50, 616–627 (2003). L. D. Iasemidis, D. S. Shiau, J. C. Sackellares, P. M. Pardalos, and A. Prasad, “Dynamical resetting of the human brain at epileptic seizures: application of nonlinear dynamics and global optimization techniques,” IEEE Trans. Biomedical Engineering, 51, 493–506 (2004). K. Lehnertz, G. Widman, and C. E. Elger, “Predicting seizures of mesial temporal and neocortical origin,” Epilepsia, 39 (1998). K. Lehnertz, R. Andrzejak, J. Arnhold, T. Kreuz, F. Morman, C. Rieke, G. Widman, and C. E. Elger, “Nonlinear EEG analysis in epilepsy: Its possible use for interictal focus localization, seizure anticipation and prevention,” J. Clinical Neurophysiology, 18, 209–222 (2001). H. Stoegbauer, L. Yang, P. Grassberger, R. G. Andrzejak, T. Kreuz, A. Kraskov, C. E. Elger, and K. Lehnertz, “Lateralization of the focal hemisphere in mesial temporal lobe epilepsy using independent component analysis,” Epilepsia, 43 (2002). M. Le Van Quyen, C. Adam, M. Baulac, J. M. Martinerie, and F. J. Varela, “Nonlinear interdependencies of EEG signals in human intracranially recorded temporal lobe seizures,” Brain Research, 792, 24–40 (1998).

16. 17. 18.

19. 20. 21. 22. 23. 24. 25.

26. 27. 28. 29. 30.

M. Le Van Quyen, J. M. Martinerie, V. Navarro, M. Baulac, and F. J. Varela, “Characterizing neurodynamic changes before seizures,” J. Clinical Neurophysiology, 18, 191–208 (2001). L. B. Good, S. Sabesan, L. D. Iasemidis, and D. M. Treiman, “Real-time control of epileptic seizures,” Proc. 3rd European Medical and Biological Engineering Conf., Prague, Czech Republic (2005). L. D. Iasemidis, A. Prasad, J. C. Sackellares, P. M. Pardalos, and D. S. Shiau, “On the prediction of seizures, hysteresis and resetting of the epileptic brain: insights from models of coupled chaotic oscillators,” in: T. Bountis and S. Pneumatikos (eds.), Order and Chaos, Publ. House of K. Sfakianakis, Thessaloniki, Greece, p. 283–305, Proc. 14th Summer School on Nonlinear Dynamics: Chaos and Complexity, Patras, Greece (2001). A. Skarda and W. J. Freeman, “How brains make chaos in order to make sense of the world?” Behav. Brain Sci., 10, 161–195 (1987). B. D. O. Anderson, “Adaptive systems, lack of persistency of excitation and bursting phenomenon,” Automatica, 21, 247–258 (1985). W. A. Sethares, Jr., C. R. Johnson, and C. E. Rohrs, “Bursting in adaptive hybrids,” IEEE Trans. Commun., C-35, 791–799 (1989). K. S. Tsakalis, “Performance limitations of adaptive parameter estimation and system identification algorithms in the absence of excitation,” Automatica, 32, No. 4, 549–560 (1996). K. S. Tsakalis, N. Chakravarthy, and L. D. Iasemidis, “Control of epileptic seizures: Model of chaotic oscillator networks,” Proc. IEEE CDC2005, Seville, Spain (2005). R. D. Traub and A. Bibbig, “A model of high-frequency ripples in the hippocampus, based on synaptic coupling plus axon-axon gap junctions between pyramidal neurons,” J. Neurosci., 20, 2086–2093 (2000). F. L. Da Silva, W. Blanes, S. N. Kalitzin, J. Parra, P. Suffczynski, and D. N. Velis, “Epilepsies as dynamical diseases of brain systems: basic models of the transition between normal and epileptic activity,” Epilepsia, 44 (Suppl. 12), 72–83 (2003). J. M. Carlson and J. Doyle, “Highly optimized tolerance: a mechanism for power laws in designed systems,” Physical Review E., 60, No. 2, 1412–1427 (1999). J. Doyle and J. M. Carlson, “Power laws, highly optimized tolerance, and generalized source coding,” Physical Review Letters, 84, No. 24, 5656–5659 (2000). K. Tsakalis, “Prediction and control of epileptic seizures,” in: Proc. Intern. Conf. and Summer School Complexity in Sci. and Society European Advanced Studies Conference V, July 14–26, Patras and Ancient Olympia, Greece (2004). K. J. Astrom and L. Rundqwist, “Integrator windup and how to avoid it,” Proc. American Control Conf., Pittsburgh (1989), pp. 1693–1968. E. Grassi, K. Tsakalis, S. Dash, S. V. Gaikwad, W. MacArthur, and G. Stein, “Integrated identification and PID controller tuning by frequency loop-shaping,” IEEE Trans. Contr. Systems Technology, 9, No. 2, 285–294 (2001).


Lihat lebih banyak...


Copyright © 2017 DADOSPDF Inc.