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Proceedings of the 2007 American Control Conference Marriott Marquis Hotel at Times Square New York City, USA, July 11-13, 2007 WeC18.1 Nonlinear...

Please may I know how these inequalities are defined / generated.
i.e. equation 26 and equation 27
Thanks a lot
Nonlinear Control of Synaptic Plasticity Model for Constraining Bursting Activity in Epileptic Seizures Yaozhang Pan, Shuzhi Sam Ge * and Abdullah Al Mamun Abstract — In this paper, we address the problem of con- trolling the synaptic plasticity to constraint bursting activity in epileptic seizures by a direct drug injection. With a good understanding of dynamical changes in seizures onset and the mechanisms that cause these changes, we design a control based on a model of interaction between synaptic strength and the intracellular calcium concentration [ Ca 2+ ] i . By exploring the dynamic properties of the system, the original system is Frst regrouped into two subsystems, the Frst one is a self- stable system; and the second one is a second order system in strict feedback form. Then, nonlinear control for the second subsystem is presented through backstepping design. The closed-loop system is shown to be globally stable. Simulation studies are conducted to show the effectiveness of the control for stopping the bursting activity in brain with epilepsy. Index Terms – Epilepsy, Synaptic Plasticity Model, Nonlinear control, backstepping I. INTRODUCTION Epilepsy is one of the most common neurological dis- orders affecting almost 1% of the population worldwide. Among the treatment methods, neurosurgery is suitable for patients for whom a deFned resectable seizure focus can be identiFed, and vagal nerve stimulation for a small percentage of patients who are not adequately controlled by existing antiepileptic drugs (AED) [1]. However, for most patients, pharmacotherapy is still considered to be the mainstay of treatment. Since the introduction of the Frst antiepileptic drug, bro- mides, in 1857, many effective antiepileptic drugs have been found. Unfortunately, such systemic drug therapy is not an ideal treatment because systemic side effects prohibit use of very high concentrations of the antiepileptic drug at the seizure focus. This problem, however, can be surmounted by local injection of an antiepileptic drug directly into a seizure focus. Many previous studies have demonstrated the ability to stop seizures in animal models with injection of antiepileptic medication directly into the seizure focus [2]– [4]. These novel direct delivery may ensure better drug * To whom all correspondences should be addressed, E-mail: [email protected] The authors are with the Department of Electrical and Computer Engi- neering, National University of Singapore efFcacy and also avoid potential problems of whole-brain and systemic toxicity. Recently, a variety of new strategies have been employed or proposed to reduce the frequency and severity of neo- cortical seizures [2], [5]–[8]. Among the most promising are implantable devices that deliver local therapy, such as direct electrical stimulation or chemical infusions, to affected regions of the brain [9]. Among these therapies, most are open loop systems that operate based on the amplitude and duration of stimulation set by the medical doctor. It therefore strongly suggests that more intelligent “closed loop” seizure prevention based upon understanding of mechanisms under- lying seizure generation and thoroughly investigation on the dynamic changes in the brain with seizures are urgently needed. Motivated by previous works on both modeling of synap- tic plasticity in neuronal network [10], [11] and nonlinear control [12]–[14], we investigate the dynamics of synaptic plasticity in neural network and design a closed loop drug deliver strategy by model-based feedback control design to obtain maximum efFciency while keeping a low toxicity level. By exploring the dynamic properties of the system, the model of synaptic plasticity is divided into two subsystems. The stability of the Frst subsystem is thoroughly analyzed and the control law for the second subsystem is designed via backstepping. As a conclusion, provided a pertinent model of the involved biological mechanism is available, feedback control should help in tuning such therapeutic schemes. To the best of our knowledge, there are few works dealing with epilepsy treatment using such a kind of control therapy in the literature. II. DYNAMIC MODEL AND PROBLEM ±ORMULATION A. Dynamics of the Synaptic Plasticity System In this paper, we shall investigate the problem of control- ling the synaptic plasticity model to restore disordered states in the brain to treat seizure. The simpliFed mathematical model describing the dynamic interaction between intracel- lular calcium concentration and synaptic plasticity has been proposed in [11] and drawn from [10]. ±or the convenience of control design, we can rewrite the Proceedings of the 2007 American Control Conference Marriott Marquis Hotel at Times Square New York City, USA, July 11-13, 2007 WeC18.1 1-4244-0989-6/07/$25.00 ©2007 IEEE. 2012
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model in [11] as follows ˙ x 1 = f 1 ( x 3 ) g 1 ( x 3 ) x 1 (1) ˙ x 2 = f 2 ( x 2 )+ p 1 g 2 ( x 2 ) x 3 (2) ˙ x 3 = f 3 ( x 2 ,x 3 ) u (3) where f 1 ( x 3 )= e 80( x 3 - 0 . 55) 1+ e 80( x 3 - 0 . 55) +0 . 25 p 1 e 80( x 3 - 0 . 35) 1+ e 80( x 3 - 0 . 35) P 1+ 10 0 . 001+ x 3 3 (4) g 1 ( x 3 )= 1 1+ 10 0 . 001+ x 3 3 (5) f 2 ( x 2 )= p 2 x 2 (6) g 2 ( x 2 )=1 x 2 (7) f 3 ( x 2 ,x 3 )= p 2 p 3 x 2 p 1 p 3 x 3 (1 x 2 ) p 4 x 3 x 3 + p 5 (8) x 1 , x 2 and x 3 are the state variables, p i ( i =1 , 2 , ··· , 5) are constant parameters, u is the control variable. TABLE II.1 DESCRIPTION OF VARIABLES AND PARAMETERS Variables/Parameters Description x 1 synaptic strength x 2 fraction of buffer that are occupied by Ca 2+ x 3 intracellular calcium concentration p 1 forward binding rates p 2 backward binding rates p 3 total concentration of the buffer p 4 rate of calcium removal by Ca 2+ pump p 5 modi±er of calcium removal by Ca 2+ pump u control signal, u = - αI Ca + I NMDA The de±nitions according to the model in [10], [11] are given in Table II.1, and their chemical properties and interpretation are described as follows: (i) In this synaptic plasticity model, synaptic strength x 1 is controlled by the intracellular calcium concentration x 3 ; (ii) Intracellular calcium dynamics are described by (2) and (3), and p 3 is the total concentration of the buffer, x 2 represents the fraction of buffer that are occupied by Ca 2+ , p 1 and p 2 are forward and backward binding rates respectively; (iii) I NMDA and I Ca are currents carried by Ca 2+ ions through the N-methyl-D-aspartate (NMDA) Receptor channels and by voltage dependent calcium channels (VDCC); and (iv) p 4 represents rate of calcium removal by Ca 2+ pump, and p 5 is the modi±er in the Ca 2+ pump expression. Seizure is a disturbance of the neuronal electrochemical activity that a set of neurons suddenly produce a repeti- tive, synchronous discharge. It has been proved that non- synaptic neural plasticity (i.e. calcium dependent afterhy- perpolarization in neurons) can regulate the frequency of the dominant rhythm in EEG, while synaptic potentiation may be responsible for irregular bursting prior to seizure termination [11], [15]. The underlying hypotheses is that synaptic potentiation and afterhyperpolarization (which reg- ulate patterns of neuronal bursting) play signi±cant roles in alteration of neural rhythmic activity during seizures. Therefore, synaptic strength is a critical factor in regulating seizure. By decreasing the synaptic strength, we aim to restore the neural rhythmic activity, and further modeling must take into account the way in which the synaptic strength acts on regulate neural rhythmic. From the models, it is reasonable to think of modifying calcium concentration by NMDA receptor channels current I NMDA and voltage dependent calcium channels current I ca , which can be chosen as control signals and become negative and positive feedback signal respectively. However, I NMDA and I ca can be acted upon by more conventional means: drugs, such as calcium channel blocker (CCBs) and antiepileptic drugs etc., which are known to have an in²uence on I Ca and I NMDA [2]. According to this, we do not consider the underlying mechanism of the relationship between drugs and these two kinds of currents, and suppose a parameter u that we are able to vary as a control signal u = αI Ca + I NMDA . From a strict automatic control point of view, we are now dealing with a nonlinear system with three state variables x 1 , x 2 , x 3 and one control variable u = αI Ca + I NMDA . III. CONTROL DESIGN METHODS Let x 0 =[ x 10 x 20 x 30 ] T R 3 0 denote the health value. Consequently, the control objective is to force x to converge to x 0 . We introduce the external control agent u to reduce the strength of synaptic plasticity for preventing the seizures. Before proceeding further, we need to study the properties of the system. A. Nonnegativity For the synaptic plasticity system, it is easy to prove that the states of this system are nonnegative. From the chemical property, we know the variables x 1 ( t ) > 0 , x 2 ( t ) > 0 , x 3 ( t ) > 0 , t , which will also be proved mathematically later. Therefore, we can assume that the initial values x i (0) > 0 , i =1 , 2 , 3 . It is obvious that e 80( x 3 - 0 . 35) 1+ e 80( x 3 - 0 . 35) < 1 . And it is easy to show that if x 3 > 0 , we have that f 1 ( x 3 ) > 0 (9) g 1 ( x 3 ) > 0 (10) WeC18.1 2013
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This question was asked on Jan 08, 2013.

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