Nonlinear DynDOI 10.1007/s11071-013-1210-3

ORIGINAL PAPER

Fuzzy adaptive unscented Kalman filter controlof 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© Springer Science+Business Media Dordrecht 2014

Abstract A new closed-loop control method based onthe fuzzy adaptive unscented Kalman filter (FAUKF) isproposed to suppress epileptiform spikes in a class ofneural mass models with uncertain measurement noise.The FAUKF is used to estimate the nonlinear systemstates of the underlying models and amend measure-ment noise adaptively. The control law is constructedvia the estimated states. Numerical simulations illus-trate the efficiency of the proposed method.

Keywords Neural mass model · Epileptiformspikes · The FAUKF control

1 Introduction

The neural mass models play important roles in mod-eling of electroencephalography (EEG), most notablyin dynamic modeling of epilepsy seizures [1,2]. Thesemodels describe the activity of neurons at the macro-

X. Liu · H.-J. Liu · Y.-G. Tang · Q. GaoKey Lab of Industrial Computer Control Engineeringof Hebei Province, Institute of Electrical Engineering,Yanshan University, Qinhuangdao 066004, China

X. Liue-mail: liuxian@ysu.edu.cn

Z.-M. Chen (B)School of Economics, Renmin University of China,Beijing 100872, Chinae-mail: chenzhanming@pku.edu.cn

scopic level, i.e., the activity of populations of neu-rons. In comparison with the models aiming at theneuronal levels, such as the Hodgkin–Huxley model[3], the neural mass models are empirical priors toemulate realistic EEG because the categories of neu-rons are manifold and the connections of neurons arecomplex. It is demonstrated that the class of modelscan simulate the dynamics of real EEG measured dur-ing the transition from interictal to fast ictal activityof epilepsy seizures in the hippocampus. An epilepticneural mass exhibits the features of excitability whichis reflected in its use to produce interictal spiking activ-ities in a single population in response to random input[1]. The spiking activity can be propagated betweenpopulations in networks of two- and three-coupledneural populations, and the ever-increasing couplingstrength may increase the rhythm of synchronizedactivities for multiple neural populations [1]. A newmethod for a parameter-driven transition to epilepsyseizure based on the neural mass models is providedby Goodfellow et al. [4] to produce slow spike–wavedischarges as well as sinusoidal background oscilla-tions. The effects of local functional heterogeneities,which are considered crucial for the mechanism ofepilepsy, are investigated in a spatially extended neuralmass formulation with local excitatory and inhibitorycircuits [5]. The dynamic evolution of EEG activityduring epileptic seizures is characterized as a paththrough the parameter space of a neural mass model,reflecting gradual changes in underlying physiologicalmechanisms [6].

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These investigations provide profound insight intounderstanding the mechanism of epilepsy seizures.However, the model-based closed-loop control forepilepsy seizures is also an important research subject.It can provide a theoretical basis for clinical electrother-apy of seizures, which still depends heavily on theempirical tuning of parameters and protocols until now[7–9]. The closed-loop control has been shown to beable to optimize the whole course of seizures treatmentincluding efficacy of treatment, minimization of sideeffects, improvement of response by providing inter-mittent or minimal stimulation, reduction of damage,and minimization of power consumption [10]. A fuzzyPID controller is designed for a class of neural massmodels to quench epileptiform spikes and make the out-put track an expected one [11]. In the closed-loop con-trol strategy, the development of feedback algorithmsrequires precise metric of system state [12]. Recently,different types of state observers have been designed fordifferent nonlinear systems, such as descriptor observer[13], acceleration observer [14], and adaptive slidingmode observer [15]. A robust circle criterion observeris designed and applied to the neural mass models [16].The unscented Kalman filtering (UKF) is another pow-erful tool for the state estimate of nonlinear systems[17,18]. It is applied to observe state and track dynam-ics from a spatiotemporal computational model of cor-tical dynamics [19]. It is also applied to the parameterestimation and control of epileptiform spikes for theneural mass model [20,21]. The UKF is able to achievegood performance if the complete information of mea-surement noise distribution is assumed to be known.This assumption is inconsistent with practical applica-tions because the environment changes all the time andstatistics of noise cannot be accurately obtained. TheUKF combining with the fuzzy inference system (FIS)is developed to amend measurement noise adaptively[22–24].

The measurement of EEG in clinical practice isquite easily to be contaminated by many factors suchas high-frequency interference and power interference.The measurement noise is of uncertainty and difficultto be measured. In this study, a fuzzy adaptive UKF(FAUKF) control based on the UKF and the FIS isproposed to suppress epileptiform spikes in a classof neural mass models with uncertain measurementnoise. The FAUKF estimates the system states throughthe UKF and adjusts measurement noise adaptivelythrough the FIS. Thus, it enhances the filtering accu-

racy in comparison with the UKF. The efficiency ofthe FAUKF control is demonstrated through plentifulsimulation tests.

2 Model description

As mentioned in the introduction, our results apply to aclass of neural mass models put forward by Jansen andRit [25]. In this model, each neuron population is com-posed of the pyramidal cells that receive excitatory andinhibitory feedbacks from inter-neurons (other pyra-midal cells, stellate, or basket cells) that only receiveexcitatory input. The delay from a connection popu-lation is modeled by a similar postsynaptic potentialblock. This type of model can generate the epilepti-form spikes in EEGs [1]. It can be described by thefollowing differential equations:

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

x1l(t) = x2l(t)x2l(t) = AaS(x3l (t) − x5l (t)) − 2ax2l (t) − a2x1l(t)x3l(t) = x4l (t)

x4l (t) = Aa[pl (t) + C2 S(C1x1l(t)) +j �=l∑

j=1,2,h, N

kl j x7l (t)] − 2ax4l (t) − a2x3l (t)x5l(t) = x6l (t)x6l(t) = BbC4 S(C3x1l(t)) − 2bx6l (t) − b2x5l (t)x7l(t) = x8l (t)x8l(t) = Aad S(x3l (t)−x5l(t)) − 2ad x8l (t) − a2

d x7l(t),

(1)

where N is an integer greater than 1; l is the popula-tion under consideration; the state variables xkl(t), k =1, 3, 5, 7 are the mean membrane potentials of the neu-ronal populations, while xkl(t), k = 2, 4, 6, 8 are theirderivatives; the excitatory input pl(t) is the afferentinfluence from other populations, which is assumed tobe a uniformly distributed signal; and the connectivityconstants C1, C2, C3, and C4 are used to model interac-tions between main cells and interneurons. The outputof population l is y(l)(t) = x3l(t) − x5l(t), which isused to simulate EEG signals. All parameters in (1) areset on a physiological interpretation basis. The standardvalue 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,

C1 = 135, C2 = 108, C3 = 33.75, C4 = 33.75.

(2)

In our simulations, N is set at three, and (1) issolved by using a fourth-order Runge–Kutta differ-

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Fig. 1 The model of threecoupled neural populations

k13

k21 k32

2 31

ential solver. The concrete connection among three-coupled neural populations is shown in Fig. 1. If theparameters are set at standard values and the cou-pling constants k21, k32, k13 are set at 0, then the simu-lated signals (Fig. 2a) resemble the spontaneous EEGrecorded from neocortical structure electrodes duringinterictal periods [6]. Thus, they reflect normal activ-ities resembling those reflected by real SEEG signals.Increasing the value of A of population 1–3.4 mv andkeeping the coupling constants invariant, the first pop-ulation produces sporadic spikes, and the two othersexhibit normal activities, as shown in Fig. 2b. KeepingA of population 1 invariant and increasing the couplingconstants k21, k32, k13 to 100, sustained discharges ofspikes that resemble real SEEG signals during the prop-agation of temporal lobe seizures are observed (seeFig. 2c).

3 Fuzzy adaptive unscented Kalman filter control

The discrete-time filtering form of the neural massmodel is described as{

xk = F(xk−1)

yk = Hk xk + Vk,(3)

where xk is the system state vector at timestep k; yk isthe measurement vector; F(·) is the system transitionfunction; Hk is the measurement matrix; and Vk is themeasurement noise vector and assumed to be Gaussianwhite noise satisfying E(Vk) = 0, E(Vk, V T

j ) =Rkδk j , where δk j is the Kronecker delta function.

The UKF is summarized as the following recursiveequations:

X j = xk−1 ± (√

n Pxx ) j , j = 1, 2, . . . , 2n (4)

X j = F(X j ), Y j = Hj (X j ) (5)

xk = 1

2n

2n∑

j=1

X j , yk = 1

2n

2n∑

j=1

Y j (6)

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Fig. 2 The output of each population. a All parameters keepstandard for all populations, k21 = k32 = k13 = 0. b All parame-ters keep standard except for A of population 1 which is 3.4 mv,

k21 = k32 = k13 = 0. c All parameters keep standard except forA of population 1 which is 3.4 mv, k21 = k32 = k13 = 100

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Pxx = 1

2n

2n∑

j=1

(X j − xk)(X j − xk)T (7)

Pxy = 1

2n

2n∑

j=1

(X j − xk)(Y j − yk)T (8)

Pyy = 1

2n

2n∑

j=1

(Y j − yk)(Y j − yk)T + Rk (9)

K = Pxy P−1yy , xk = xk + K (yk − yk) (10)

Pxx = Pxx − K PTxy, (11)

where n is the number of system states.When the information of measurement noise distri-

bution is known completely, the UKF is able to achievegood performance. In practical applications, the statis-tics of noise can not be accurately obtained, and thefiltering performance of UKF is affected seriously. Sowe combine the UKF with FIS to amend measurementnoise adaptively. Define the residual vector as

rk = yk − yk . (12)

It is obvious that the value of rk reflects the degree towhich the model fits the data, and the covariance of rk

determines the degree of divergence. The theoreticalresidual variance is Pr = Pyy . The actual variance isdefined as Cr [26]

Cr = 1

M

k∑

i=k−M+1

ri ∗ r Ti , (13)

where M denotes the number of the latest residual vec-tors. A ratio of trace between the actual variance andthe theoretic variance is

qk = T r(Cr )

T r(Pr ), (14)

where T r(·) means the matrix trace.As seen in the previous chapter, the value of qk

should be around 1 if the model is accurate. Whenthe actual measurement noise suddenly increases, Cr

increases and qk > 1. Hence, Pr should increase toreduce the influence from the measurement noise. Onthe contrary, when the measurement noise reduces, Cr

reduces and qk < 1. Hence, Pr should decrease. Theadjustment of Pr can be converted to the adjustmentof Rk . From this point of view, Rk can be adjusted asfollows

R′k = sv Rk, (15)

where s is the regulation factor which is calculated byFIS, and v ≥ 0 indicates the extent of the amplificationof s. If v = 0 is chosen, Rk keeps invariant. If v < 1,the adjustment action of s decreases and Rk is slowlychanged. On the contrary, if v > 1, the adjustmentaction of s is amplified and Rk is greatly changed. Thus,the FAUKF algorithm is derived by replacing (9) inthe recursive equations of the UKF with the followingformulas:

Pyy = 1

2n

2n∑

j=1

(Y j − yk)(Y j − yk)T + Rk

R′k = sv Rk

Pyy = 1

2n

2n∑

j=1

(Y j − yk)(Y j − yk)T + R′

k .

(16)

It is noted that if v = 0, then the FAUKF is equivalentto the regular UKF.

The regulation factor s is calculated by FIS. Thespecific process is as follows. FIS contains four partsincluding fuzzification of input qk , fuzzy databases,fuzzy inference, and defuzzification of output s. Thefuzzy 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 mainlyin the premise which is specified by three parameters:(α, β, γ )

μ = f (x, α, β, γ ) =

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

0 x ≤ α

x−αβ−α

α ≤ x ≤ βγ−xγ−β

β ≤ x ≤ γ

0 x ≥ γ,

(17)

where α ≤ β ≤ γ, μ is the membership grade, β is themembership function’s center, and α and γ determinethe endpoints. The membership functions for qk and sare shown in Fig. 3. The fuzzy rule is usually knowl-edge bases constituted by the if–then rule. It is given asfollows:

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 forfuzzy logic inference, and the centroid method is usedfor the defuzzification.

A closed-loop control strategy based on the FAUKFis given here to suppress the epileptiform spikes in the

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Fig. 3 The membershipfunctions of FIS. a Themembership functions ofinput of FIS. b Themembership functions ofoutput of FIS

0 0.5 1 1.5 2

0

0.2

0.4

0.6

0.8

1

(a)D

egre

e of

mem

bers

hip

less equal more

3 3.5 4 4.5 5

0

0.2

0.4

0.6

0.8

1

(b)

Deg

ree

of m

embe

rshi

p

less equal more

Fig. 4 The FAUKF controlscheme

ku 1kx Unitdelay

kx kyH

F

L

r

r

C

PFIS UKFskq

ˆky

ku

considered neural mass models, as shown in Fig. 4,where yk is the noisy measurement influenced by therandom noise η; yk is the estimated output from theFAUKF; qk is the ratio of trace between the actual vari-ance and the theoretic variance; s is the regulation fac-tor; L is the feedback gain matrix; and uk is the controllaw with the form of

uk = L ∗ yk . (18)

4 Simulation

In this section, simulations are provided to demonstratethe filtering effect of the FAUKF and the ability of theFAUKF-based closed-loop control (Fig. 4) to suppressepileptiform spikes in the neural mass models.

4.1 Filtering results

The filtering effect of the FAUKF is conducted in themodel of three coupled neural populations. The para-meters 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 guar-antee that qk is around 1 from the beginning to 5 s.

It is increased to four times from 5 to 10 s and eighttimes from 10 to 15 s, and then allowed to recover tothe original value.

Figure 5a presents the output without the interfer-ence of measure noise (the black trace) and the noisymeasurements (the blue trace). An inset is given there toshow the zoom-in on data. Figure 5b–f present the com-parison of the application of FAUKF with different val-ues of v (v = 0, 1.5, 3, 11, 12). The black trace is theoutput without the interference of measure noise, andthe red trace is the estimated output. Observation showsthat tshe filtering accuracy has a meaningful improve-ment as v is varied from 0 to 3. However, no mean-ingful improvement of the filtering accuracy is verifiedwhen v is varied from 3 to 11. The estimated output isdivergent seriously from the output without the inter-ference of measurement noise when v is increased to12 or a larger value, as shown in Fig. 5f. Furthermore,it is known that the FAUKF is equivalent to the regularUKF when v = 0 is chosen. One can observe fromFig. 5b and d that the filtering accuracy of the UKF andthe FAUKF has no obvious differences at the beginningand last 5 s. However, the filtering accuracy of FAUKFis higher than that of the UKF from the 5 to 15 s. Thisis due to the adaptive adjustment of the FAUKF for themeasurement noise covariance.

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Fig. 5 The output and its estimation from the FAUKF with different values of v. a The outputs with and without interference ofmeasurement noise. b v = 0, c v = 1.5, d v = 3, e v = 11, f v = 12

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Fuzzy adaptive unscented Kalman filter control

It is possible to make the following conjecturesbased on above results: the increase in v significantlyimproves the filtering accuracy up to a limit. Theincrease of v beyond this limit has no further meaning-ful effect of improving the filtering accuracy or evendeteriorates the filtering accuracy. The adaptive adjust-ment for the uncertain measurement noise results in thehigher filtering accuracy of the FAUKF in comparisonwith the UKF.

4.2 Controlling results

In this section, the FAUKF control given in Fig. 4is used to control epileptiform spikes in the modelof three coupled neural populations. Populations 2, 3are assumed to exhibit normal activity by keeping allparameters standard, and population 1 is assumed tobe hyperexcitable by keeping all parameters standardexcept for A which is 3.4 mv. The coupling constantsk21, k32, k13 are set at 100. The corresponding outputsare shown in Fig. 2c, where the epileptiform spikes areobserved. The variance of measurement noise is chosento guarantee that qk is around 1 at the beginning 5 s. It

is increased to 15 times from the 5 to 10 s, and 30 timesfrom the 10 to 15 s , and then allowed to recover to theoriginal value at the last 5 s. Figure 6a shows the noisymeasurements. The feedback gain matrix L is chosenas L = [L1 L2 L3]T , where

L1 = (0 0 0 0 L1 0 0 0)

L2 = (0 0 0 0 L2 0 0 0)

L3 = (0 0 0 0 L3 0 0 0)

the superscript T denotes the transpose of a matrix,L1, L2, and L3 are constant numbers. The controlaction is imposed on population 1 by setting L2 andL3 at 0, and v is set at 2.5 in simulations. The controlenergy is defined as

∑uT

k ∗ uk . Several simulationsare provided to demonstrate the effectiveness of theproposed FAUKF control.

Figure 6b and c present the results with the applica-tion of the UKF control and the FAUKF control withL1 = −0.95 which is the least value that suppressesthe epileptiform spikes for the FAUKF. The black traceis the output, and the red trace is its estimation. An insetof the output is given in Fig. 6b to show the zoom-inon data. An inset of the output and its estimation isalso given in Fig. 6c to show the zoom-in on data. It is

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Fig. 6 The output and its estimation with the application of different control methods. a The noisy measurement output. b The UKFcontrol. c The FAUKF control

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Table 1 The total control energy (mv2) over 20-s simulationrealizations for different control methods

The varied timesof measurementnoise

L1 The total controlenenrgy

UKF FAUKF UKF FAUKF

4.8 −1.058 −0.94 87729.49 73748.53

15.30 −1.08 −0.95 151026.92 74651.21

20.40 −1.30 −0.955 315081.23 75184.51

observed that the sustained spikes do not occur underthe FAUKF control (comparing Figs. 2c, 6c), while theystill occur under the UKF control (comparing Fig. 2cand the inset of Fig. 6b). Meanwhile, the estimated out-put from the UKF control has a serious deviation fromthe output. When the magnitude of L1 is increased to1.08, the sustained spikes are suppressed just enoughby using the UKF control. The FAUKF control can sup-press the epileptiform spikes certainly at the same gaincondition. However, the total control energy required inthe latter control is less than that required in the formerone, as shown in Table 1.

Table 1 presents the critical value of L1 and the totalcontrol energy over 20 s simulation realizations for dif-ferent measurement noises, for the UKF control and theFAUKF control. The variance of measurement noise ischosen to guarantee that qk is around 1 at the begin-ning 5 s and the last 5 s, and is varied for differenttimes from 5 to 10 s and from the 10 to 15 s in simu-lations. It is observed that the magnitude of L1 and thetotal control energy in the FAUKF control are less thanthose in the UKF control under identical level of mea-surement noise. The gaps of the magnitude of L1 andthe total control energy between two control methodsare increased with the increasing level of measurementnoise.

As is expected, suppression of the epileptiformspikes in the underlying neural mass models withuncertain measurement noise can be reached using theFAUKF control. In comparison with the UKF control,the adaptive adjustment of the FAUKF makes the totalcontrol energy reduce greatly, especially when the var-ied amplitude of measurement noise is large.

5 Conclusion

The problem of suppression of the epileptiform spikesin the neural mass models with uncertain measurement

noise has been studied by using the FAUKF closed-loopcontrol strategy. It has been shown by simulations thatthe epileptiform spikes in the underlying models can besuppressed using the FAUKF control. In comparisonwith the UKF control, the application of the FAUKFcontrol can reduce the total control energy. This is dueto the role of the adaptive adjustment to the uncertainmeasurement noise that the FAUKF plays. The FAUKFthat we described is generic and can be applied in anysetting where there is a need to estimate the states ofsystems with uncertain measurement noise.

Acknowledgments This research was supported by the Nation-al Natural Science Foundation of China (61004050, 61273260,61172095, 51207144), the Specialized Research Fund for theDoctoral Program of Higher Education (20101333110006), andthe Humanities and Social Sciences Fund, the Ministry of Edu-cation Foundation of China (12YJCZH021).

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