Nonlinear Dyn (2014) 76:1291–1299 DOI 10.1007/s11071-013-1210-3
ORIGINAL PAPER
Fuzzy adaptive unscented Kalman filter control of epileptiform spikes in a class of neural mass models Xian Liu · Hui-Jun Liu · Ying-Gan Tang · Qing Gao · Zhan-Ming Chen
Received: 9 December 2012 / Accepted: 17 December 2013 / Published online: 12 January 2014 © Springer Science+Business Media Dordrecht 2014
Abstract A new closed-loop control method based on the fuzzy adaptive unscented Kalman filter (FAUKF) is proposed to suppress epileptiform spikes in a class of neural mass models with uncertain measurement noise. The FAUKF is used to estimate the nonlinear system states of the underlying models and amend measurement noise adaptively. The control law is constructed via the estimated states. Numerical simulations illustrate the efficiency of the proposed method. Keywords Neural mass model · Epileptiform spikes · The FAUKF control
1 Introduction The neural mass models play important roles in modeling of electroencephalography (EEG), most notably in dynamic modeling of epilepsy seizures [1,2]. These models describe the activity of neurons at the macroX. Liu · H.-J. Liu · Y.-G. Tang · Q. Gao Key Lab of Industrial Computer Control Engineering of Hebei Province, Institute of Electrical Engineering, Yanshan University, Qinhuangdao 066004, China X. Liu e-mail:
[email protected] Z.-M. Chen (B) School of Economics, Renmin University of China, Beijing 100872, China e-mail:
[email protected]
scopic level, i.e., the activity of populations of neurons. In comparison with the models aiming at the neuronal levels, such as the Hodgkin–Huxley model [3], the neural mass models are empirical priors to emulate realistic EEG because the categories of neurons are manifold and the connections of neurons are complex. It is demonstrated that the class of models can simulate the dynamics of real EEG measured during the transition from interictal to fast ictal activity of epilepsy seizures in the hippocampus. An epileptic neural mass exhibits the features of excitability which is reflected in its use to produce interictal spiking activities in a single population in response to random input [1]. The spiking activity can be propagated between populations in networks of two- and three-coupled neural populations, and the ever-increasing coupling strength may increase the rhythm of synchronized activities for multiple neural populations [1]. A new method for a parameter-driven transition to epilepsy seizure based on the neural mass models is provided by Goodfellow et al. [4] to produce slow spike–wave discharges as well as sinusoidal background oscillations. The effects of local functional heterogeneities, which are considered crucial for the mechanism of epilepsy, are investigated in a spatially extended neural mass formulation with local excitatory and inhibitory circuits [5]. The dynamic evolution of EEG activity during epileptic seizures is characterized as a path through the parameter space of a neural mass model, reflecting gradual changes in underlying physiological mechanisms [6].
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These investigations provide profound insight into understanding the mechanism of epilepsy seizures. However, the model-based closed-loop control for epilepsy seizures is also an important research subject. It can provide a theoretical basis for clinical electrotherapy of seizures, which still depends heavily on the empirical tuning of parameters and protocols until now [7–9]. The closed-loop control has been shown to be able to optimize the whole course of seizures treatment including efficacy of treatment, minimization of side effects, improvement of response by providing intermittent or minimal stimulation, reduction of damage, and minimization of power consumption [10]. A fuzzy PID controller is designed for a class of neural mass models to quench epileptiform spikes and make the output track an expected one [11]. In the closed-loop control strategy, the development of feedback algorithms requires precise metric of system state [12]. Recently, different types of state observers have been designed for different nonlinear systems, such as descriptor observer [13], acceleration observer [14], and adaptive sliding mode observer [15]. A robust circle criterion observer is designed and applied to the neural mass models [16]. The unscented Kalman filtering (UKF) is another powerful tool for the state estimate of nonlinear systems [17,18]. It is applied to observe state and track dynamics from a spatiotemporal computational model of cortical dynamics [19]. It is also applied to the parameter estimation and control of epileptiform spikes for the neural mass model [20,21]. The UKF is able to achieve good performance if the complete information of measurement noise distribution is assumed to be known. This assumption is inconsistent with practical applications because the environment changes all the time and statistics of noise cannot be accurately obtained. The UKF combining with the fuzzy inference system (FIS) is developed to amend measurement noise adaptively [22–24]. The measurement of EEG in clinical practice is quite easily to be contaminated by many factors such as high-frequency interference and power interference. The measurement noise is of uncertainty and difficult to be measured. In this study, a fuzzy adaptive UKF (FAUKF) control based on the UKF and the FIS is proposed to suppress epileptiform spikes in a class of neural mass models with uncertain measurement noise. The FAUKF estimates the system states through the UKF and adjusts measurement noise adaptively through the FIS. Thus, it enhances the filtering accu-
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racy in comparison with the UKF. The efficiency of the FAUKF control is demonstrated through plentiful simulation tests. 2 Model description As mentioned in the introduction, our results apply to a class of neural mass models put forward by Jansen and Rit [25]. In this model, each neuron population is composed of the pyramidal cells that receive excitatory and inhibitory feedbacks from inter-neurons (other pyramidal cells, stellate, or basket cells) that only receive excitatory input. The delay from a connection population is modeled by a similar postsynaptic potential block. This type of model can generate the epileptiform spikes in EEGs [1]. It can be described by the following differential equations: ⎧ x˙1l (t) ⎪ ⎪ ⎪ ⎪ x˙2l (t) ⎪ ⎪ ⎪ ⎪ x˙3l (t) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ x˙4l (t) ⎪ ⎪ ⎪ ⎪ ⎪ x˙5l (t) ⎪ ⎪ ⎪ ⎪ x˙6l (t) ⎪ ⎪ ⎪ ⎪ x˙7l (t) ⎪ ⎪ ⎩ x˙8l (t)
= x2l (t) = AaS(x3l (t) − x5l (t)) − 2ax2l (t) − a 2 x1l (t) = x4l (t) j=l = Aa[ pl (t) + C2 S(C1 x1l (t)) + j=1,2,h¯ , N
= = = =
kl j x7l (t)] − 2ax4l (t) − a 2 x3l (t) x6l (t) BbC4 S(C3 x1l (t)) − 2bx6l (t) − b2 x5l (t) x8l (t) Aad S(x3l (t)−x5l (t)) − 2ad x8l (t) − ad2 x7l (t),
(1) where N is an integer greater than 1; l is the population under consideration; the state variables xkl (t), k = 1, 3, 5, 7 are the mean membrane potentials of the neuronal populations, while xkl (t), k = 2, 4, 6, 8 are their derivatives; the excitatory input pl (t) is the afferent influence from other populations, which is assumed to be a uniformly distributed signal; and the connectivity constants C1 , C2 , C3 , and C4 are used to model interactions between main cells and interneurons. The output of population l is y(l) (t) = x3l (t) − x5l (t), which is used to simulate EEG signals. All parameters in (1) are set on a physiological interpretation basis. The standard value of the parameters is given anatomically as [25] A = 3.25 mv, B = 22 mv, a = 100 s−1 , b = 50 s−1 , v0 = 6 mv, e0 = 2.5 s−1 , r = 0.56 mv−1 , ad = 33 s−1 , (2) C1 = 135, C2 = 108, C3 = 33.75, C4 = 33.75.
In our simulations, N is set at three, and (1) is solved by using a fourth-order Runge–Kutta differ-
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Fig. 1 The model of three coupled neural populations
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ential solver. The concrete connection among threecoupled neural populations is shown in Fig. 1. If the parameters are set at standard values and the coupling constants k21 , k32 , k13 are set at 0, then the simulated signals (Fig. 2a) resemble the spontaneous EEG recorded from neocortical structure electrodes during interictal periods [6]. Thus, they reflect normal activities resembling those reflected by real SEEG signals. Increasing the value of A of population 1–3.4 mv and keeping the coupling constants invariant, the first population produces sporadic spikes, and the two others exhibit normal activities, as shown in Fig. 2b. Keeping A of population 1 invariant and increasing the coupling constants k21 , k32 , k13 to 100, sustained discharges of spikes that resemble real SEEG signals during the propagation of temporal lobe seizures are observed (see Fig. 2c).
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where xk is the system state vector at timestep k; yk is the measurement vector; F(·) is the system transition function; Hk is the measurement matrix; and Vk is the measurement noise vector and assumed to be Gaussian white noise satisfying E(Vk ) = 0, E(Vk , V jT ) = Rk δk j , where δk j is the Kronecker delta function. The UKF is summarized as the following recursive equations:
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2n 1 ˜ ( X j − x˜k )( X˜ j − x˜k )T P˜x x = 2n
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where n is the number of system states. When the information of measurement noise distribution is known completely, the UKF is able to achieve good performance. In practical applications, the statistics of noise can not be accurately obtained, and the filtering performance of UKF is affected seriously. So we combine the UKF with FIS to amend measurement noise adaptively. Define the residual vector as rk = yk − y˜k .
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It is obvious that the value of rk reflects the degree to which the model fits the data, and the covariance of rk determines the degree of divergence. The theoretical residual variance is Pr = Pyy . The actual variance is defined as Cr [26] 1 Cr = M
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where s is the regulation factor which is calculated by FIS, and v ≥ 0 indicates the extent of the amplification of s. If v = 0 is chosen, Rk keeps invariant. If v < 1, the adjustment action of s decreases and Rk is slowly changed. On the contrary, if v > 1, the adjustment action of s is amplified and Rk is greatly changed. Thus, the FAUKF algorithm is derived by replacing (9) in the recursive equations of the UKF with the following formulas: 2n 1 ˜ (Y j − y˜k )(Y˜ j − y˜k )T + Rk Pyy = 2n j=1
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It is noted that if v = 0, then the FAUKF is equivalent to the regular UKF. The regulation factor s is calculated by FIS. The specific process is as follows. FIS contains four parts including fuzzification of input qk , fuzzy databases, fuzzy inference, and defuzzification of output s. The fuzzy sets for qk and s include less, equal, and more. “less” denotes the set for qk and s are less than 1, “equal” denotes the set for qk and s are equal to 1, “more” denotes the set for qk and s are more than 1. The triangulares membership functions are used mainly in the premise which is specified by three parameters: (α, β, γ ) ⎧ 0 x ≤α ⎪ ⎪ ⎪ ⎨ x−α α ≤ x ≤ β (17) μ = f (x, α, β, γ ) = β−α γ −x ⎪ β ≤ x ≤ γ ⎪ γ −β ⎪ ⎩ 0 x ≥ γ, where α ≤ β ≤ γ , μ is the membership grade, β is the membership function’s center, and α and γ determine the endpoints. The membership functions for qk and s are shown in Fig. 3. The fuzzy rule is usually knowledge bases constituted by the if–then rule. It is given as follows: If qk ∈ less, then s ∈ less. If qk ∈ equal, then s ∈ equal. If qk ∈ more, then s ∈ more. The Mamdani minimax reasoning method is used for fuzzy logic inference, and the centroid method is used for the defuzzification. A closed-loop control strategy based on the FAUKF is given here to suppress the epileptiform spikes in the
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4 Simulation In this section, simulations are provided to demonstrate the filtering effect of the FAUKF and the ability of the FAUKF-based closed-loop control (Fig. 4) to suppress epileptiform spikes in the neural mass models.
4.1 Filtering results The filtering effect of the FAUKF is conducted in the model of three coupled neural populations. The parameters of each population are kept at standard values, and the coupling constants k21 , k32 , k13 are set at 100. The variance of measurement noise is chosen to guarantee that qk is around 1 from the beginning to 5 s.
It is increased to four times from 5 to 10 s and eight times from 10 to 15 s, and then allowed to recover to the original value. Figure 5a presents the output without the interference of measure noise (the black trace) and the noisy measurements (the blue trace). An inset is given there to show the zoom-in on data. Figure 5b–f present the comparison of the application of FAUKF with different values of v (v = 0, 1.5, 3, 11, 12). The black trace is the output without the interference of measure noise, and the red trace is the estimated output. Observation shows that tshe filtering accuracy has a meaningful improvement as v is varied from 0 to 3. However, no meaningful improvement of the filtering accuracy is verified when v is varied from 3 to 11. The estimated output is divergent seriously from the output without the interference of measurement noise when v is increased to 12 or a larger value, as shown in Fig. 5f. Furthermore, it is known that the FAUKF is equivalent to the regular UKF when v = 0 is chosen. One can observe from Fig. 5b and d that the filtering accuracy of the UKF and the FAUKF has no obvious differences at the beginning and last 5 s. However, the filtering accuracy of FAUKF is higher than that of the UKF from the 5 to 15 s. This is due to the adaptive adjustment of the FAUKF for the measurement noise covariance.
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It is possible to make the following conjectures based on above results: the increase in v significantly improves the filtering accuracy up to a limit. The increase of v beyond this limit has no further meaningful effect of improving the filtering accuracy or even deteriorates the filtering accuracy. The adaptive adjustment for the uncertain measurement noise results in the higher filtering accuracy of the FAUKF in comparison with the UKF.
is increased to 15 times from the 5 to 10 s, and 30 times from the 10 to 15 s , and then allowed to recover to the original value at the last 5 s. Figure 6a shows the noisy measurements. The feedback gain matrix L is chosen as L = [L 1 L 2 L 3 ]T , where L 1 0 0 0) L 1 = (0 0 0 0
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L 3 at 0, and v is set at 2.5 in simulations. The control T energy is defined as u k ∗ u k . Several simulations are provided to demonstrate the effectiveness of the proposed FAUKF control. Figure 6b and c present the results with the application of the UKF control and the FAUKF control with
L 1 = −0.95 which is the least value that suppresses the epileptiform spikes for the FAUKF. The black trace is the output, and the red trace is its estimation. An inset of the output is given in Fig. 6b to show the zoom-in on data. An inset of the output and its estimation is also given in Fig. 6c to show the zoom-in on data. It is
4.2 Controlling results In this section, the FAUKF control given in Fig. 4 is used to control epileptiform spikes in the model of three coupled neural populations. Populations 2, 3 are assumed to exhibit normal activity by keeping all parameters standard, and population 1 is assumed to be hyperexcitable by keeping all parameters standard except for A which is 3.4 mv. The coupling constants k21 , k32 , k13 are set at 100. The corresponding outputs are shown in Fig. 2c, where the epileptiform spikes are observed. The variance of measurement noise is chosen to guarantee that qk is around 1 at the beginning 5 s. It
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Table 1 The total control energy (mv2 ) over 20-s simulation realizations for different control methods The varied times of measurement noise
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observed that the sustained spikes do not occur under the FAUKF control (comparing Figs. 2c, 6c), while they still occur under the UKF control (comparing Fig. 2c and the inset of Fig. 6b). Meanwhile, the estimated output from the UKF control has a serious deviation from the output. When the magnitude of
L 1 is increased to 1.08, the sustained spikes are suppressed just enough by using the UKF control. The FAUKF control can suppress the epileptiform spikes certainly at the same gain condition. However, the total control energy required in the latter control is less than that required in the former one, as shown in Table 1. Table 1 presents the critical value of
L 1 and the total control energy over 20 s simulation realizations for different measurement noises, for the UKF control and the FAUKF control. The variance of measurement noise is chosen to guarantee that qk is around 1 at the beginning 5 s and the last 5 s, and is varied for different times from 5 to 10 s and from the 10 to 15 s in simulations. It is observed that the magnitude of
L 1 and the total control energy in the FAUKF control are less than those in the UKF control under identical level of measurement noise. The gaps of the magnitude of
L 1 and the total control energy between two control methods are increased with the increasing level of measurement noise. As is expected, suppression of the epileptiform spikes in the underlying neural mass models with uncertain measurement noise can be reached using the FAUKF control. In comparison with the UKF control, the adaptive adjustment of the FAUKF makes the total control energy reduce greatly, especially when the varied amplitude of measurement noise is large.
5 Conclusion The problem of suppression of the epileptiform spikes in the neural mass models with uncertain measurement
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noise has been studied by using the FAUKF closed-loop control strategy. It has been shown by simulations that the epileptiform spikes in the underlying models can be suppressed using the FAUKF control. In comparison with the UKF control, the application of the FAUKF control can reduce the total control energy. This is due to the role of the adaptive adjustment to the uncertain measurement noise that the FAUKF plays. The FAUKF that we described is generic and can be applied in any setting where there is a need to estimate the states of systems with uncertain measurement noise. Acknowledgments This research was supported by the National Natural Science Foundation of China (61004050, 61273260, 61172095, 51207144), the Specialized Research Fund for the Doctoral Program of Higher Education (20101333110006), and the Humanities and Social Sciences Fund, the Ministry of Education Foundation of China (12YJCZH021).
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