Controlling absence seizures by deep brain stimulus applied on substantia nigra pars reticulata and cortex

Chaos, Solitons & Fractals 80 (2015) 13–23

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Chaos, Solitons & Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos

Controlling absence seizures by deep brain stimulus applied on substantia nigra pars reticulata and cortex Bing Hu, Qingyun Wang ⇑ Department of Dynamics and Control, Beihang University, Beijing 100191, China

a r t i c l e

i n f o

a b s t r a c t Epilepsy is a typical neural disease in nervous system, and the control of seizures is very important for treating the epilepsy. It is well known that the drug treatment is the main strategy for controlling the epilepsy. However, there are about 10–15 percent of patients, whose seizures cannot be effectively controlled by means of the drug. Alternatively, the deep brain stimulus (DBS) technology is a feasible method to control the serious seizures. However, theoretical explorations of DBS are still absent, and need to be further made. Presently, we will explore to control the absence seizures by introducing the DBS to a basal ganglia thalamocortical network model. In particular, we apply DBS onto substantia nigra pars reticulata (SNr) and the cortex to explore its effects on controlling absence seizures, respectively. We can find that the absence seizure can be well controlled within suitable parameter ranges by tuning the period and duration of current stimulation as DBS is implemented in the SNr. And also, as the DBS is applied onto the cortex, it is shown that for the ranges of present parameters, only adjusting the duration of current stimulation is an effective control method for the absence seizures. The obtained results can have better understanding for the mechanism of DBS in the medical treatment. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction Epilepsy can exhibit a chronic recurrent temporary brain dysfunction syndrome, and can be characterized by complex firings of neural function area [1–4]. For example, comprehensive tonic-clonic seizures, simple partial seizures, complex partial seizures, absence seizures have been found from the clinical features of the epilepsy. Absence seizure or ‘‘petit mal’’ seizure is one type of main features for generalized epilepsy, which is with typical 2– 4 Hz spike-and-slow wave discharges (SWDs) [5]. Presently, there are some methods for treating the epilepsy. Although drug treatment is a trivial method [6,7], and some good results can be achieved for most of the patients,

⇑ Corresponding author. E-mail address: [email protected] (Q. Wang). http://dx.doi.org/10.1016/j.chaos.2015.02.014 0960-0779/Ó 2015 Elsevier Ltd. All rights reserved.

it is still difficult to control seizures for about 10–15 percent of the patients. Hence, surgical treatments must be considered for these serious seizures. A number of clinical researches and computational modeling studies have supported the viewpoint that epilepsy seizure can originate from the mutual roles between the cerebral cortex and the thalamus, which form the corticothalamic circuits [8–10]. Additionally, some experiments from the animal and human have also shown that the basal ganglia, which is closely connected with the thalamocortical system, can have a great influence on the activity of the corticothalamic circuit [11–13]. Recently, Chen et al. [14] used a mean-field model to study the precise roles of these direct basal ganglia-thalamus pathways (the pathways from the substantia nigra pars reticulata (SNr) to thalamic reticular nucleus (TRN) and specific relay nuclei (SRN) of thalamus) in controlling absence seizures, and the bidirectional control effect of absence seizures is

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B. Hu, Q. Wang / Chaos, Solitons & Fractals 80 (2015) 13–23

explored by the competitive mechanism between the pathway SNr-TRN and SNr-SRN. Hu et al. have investigated the control of absence seizures induced by the pathways of SRN in corticothalamic system [15]. Volman et al. have shown that the suitable gap junctions among neurons can control the seizure behaviors [16]. These results are from the internal physiological regulation among different functional regions, and can have a good guidance for the drug treatment. In addition, it is shown that epilepsy seizures are also correlated to the neural synchronization and resonance. Hence, effects of the periodic stimulus on stochastic resonance and synchronization have been extensively studied to explore its crucial impact in some literatures [17,18]. Thus, we know that surgical treatment of the epilepsy is still in the early stage. Both parkinson and epilepsy are neural physiological diseases, which are related to discharge activity of the neurons in brain, and they have similar physiological anatomy structure [14,19,20]. Hence, it is expected that the surgical treatment for parkinson’s disease can be applicable for the epilepsy seizures. The DBS method is widely used by neurophysiologists to treat the nerve diseases, which implies that some electrodes are implanted into the brain region of patients with applying the nuclear pulse generator to stimulate certain nerve organizations in deep regions of the brain. This can correct the abnormal brain electrical loop, and finally reduce the neurological pathological symptoms [21]. This method is widely used in the treatment of the parkinson’s disease, and has a clear curative effect [22–26]. In recent years, the DBS technology is gradually developed for treatment of the epilepsy, but the effect of treatment still needs to be witnessed. To do this, it is necessary for theoretical explorations on the effect of seizure treatment [27–29]. In what follows, based on the model presented in [14], we will investigate the control effects of DBS on the epilepsy seizures by varying the period, amplitude and duration of DBS currents. In particular, it is shown that absence seizures induced by some key pathways can be inhibited by suitable DBS currents as the selected stimulus targets are SNr and the cortex.

2. Model description and method In this section, we firstly give the schematic network model as shown in Fig. 1, which contains nine neural populations (also see Ref. [14,15]). For a convenience, they are abbreviated as follows, e = excitatory pyramidal neurons; i = inhibitory interneurons; r = TRN; s = SRN; d1 = striatal D1 neurons; d2 = striatal D2 neurons; p1 = substantia nigra pars reticulata (SNr); p2 = globus pallidus external (GPe) segment; f = subthalamic nucleus (STN). And, there are three types of neural connections in the network, and they are distinguished by different line types and heads (marked in Fig. 1. The excitatory projections mediated by glutamate are represented by lines with arrows, and round heads denote inhibitory projections mediated by GABAA (solid line) and GABAB (dotted line), respectively. In addition, DBS denotes the deep brain stimulation current, which acts on the SNr and cortex.

Fig. 1. Schematic diagram of the studied basal ganglia-corticothalamic network model. Neural populations are denoted as, e = excitatory pyramidal neurons; i = inhibitory interneurons; r = thalamic reticular nucleus (TRN); s = specific relay nuclei (SRN); d1 = striatal D1 neurons; d2 = striatal D2 neurons; p1 = substantia nigra pars reticulata (SNr); p2 = globus pallidus external (GPe) segment; f = subthalamic nucleus (STN). Lines with arrow heads show the excitatory projections. Solid lines with round heads are the inhibitory projections. Dashed line means the inhibitory projection from TRN to SRN. DBS means the deep brain stimulation current, and it is applied onto the SNr and cortex, respectively.

The detailed equations used to describe the neural populations of the network are given as follows [14,19,20]. Firstly, the mean firing rate Q a for each neural population is modeled by a sigmoid curve as a function of the cell-body potential V a [19],

Q a ðtÞ  F½V a ðtÞ ¼

Q max ha 1 þ exp  ppffiffi3

V a ðtÞha

i

ð1Þ

r

where different neural populations are denoted as is the maxia ¼ e; i; r; s; d1 ; d2 ; p1 ; p2 ; f, respectively. Q max a mum firing rate, ha is the mean threshold potential, and r is the standard deviation of firing thresholds. When the membrane potential V a exceeds the threshold potential ha , the neural population can fire with the average firing rate Q a . Generally, the mean cell-body potential of population V a is modeled by the following equations [20],

Dab V a ðtÞ ¼ R Sðv ab Þ  v ab  /b ðtÞ b2A

Dab

" # 1 @2 @ ¼ þ ða þ bÞ þ ab @t ab @t2

ð2Þ

ð3Þ

where the differential operator Dab is a physiologically realistic representation of dendritic and synaptic integration of incoming signals [20]. a and b indicate the decay and rise rates of the cell-body potential, respectively. v ab is the connection strength from the neural population of type b to type a. A is a set of populations projecting to population a. (Smab ) is a positive or negative signal. In particular, if the input from b to a is excitatory, then (Smab ) = 1; otherwise, (Smab ) = 1. /b ðtÞ is the incoming pulse rate from the neural population of type b. A delay parameter s is introduced to the pathway from TRN to SRN to describe

B. Hu, Q. Wang / Chaos, Solitons & Fractals 80 (2015) 13–23

its slow synaptic kinetics induced by the second messenger process [14]. Finally, to describe the propagation effect of field /a , a damped wave equation is introduced [14],

1

"

# @2 @ 2 þ 2 c þ c a a /a ðtÞ ¼ Q a ðtÞ @t @t2

c2a

ð4Þ

where ca ¼ v a =r a is the damping rate, v a is the velocity of the pulse, and ra is the characteristic axonal range. The anatomical structure of brain and physiological experiments have indicated that the axons of inhibitory interneurons, thalamic reticular nucleus, specific relay nuclei, striatal D1 neurons, striatal D2 neurons, substantia nigra pars reticulata, globus pallidus external segment and subthalamic nucleus are very short, which results in large damping rate ca . Hence, we can omit the wave propagation effects from these populations, which implies that, /c ¼ FðV c Þðc ¼ i; r; s; d1 ; d2 ; p1 ; p2 ; fÞ [19,20]. And the propagation effects of the excitatory pyramidal neurons can be described as follows,

1

"

c2e

# @2 @ 2 þ 2ce þ ce /e ðtÞ ¼ Q e ðtÞ @t @t2

ð5Þ

Usually, in order to reduce the dimension of equations, we can further simplify the model by setting V i ¼ V e and Q i ¼ Q e (intracortical connectivities are proportional to the numbers of synapses involved), as appeared in many previous studies [14,19,20]. For more detailed explanations of this field model, one can refer to the related important literatures [14,19,20]. In addition, DBS is described by the common square wave stimulation current [30],

     2p t sinð2pðt þ DÞÞ 1H IDBS ¼ A  H sin P P

ð6Þ

where, H is the Heaviside step function, such that HðxÞ ¼ 0 if x < 0 and HðxÞ ¼ 1 if x > 0. A is the amplitude of DBS current, P is the period of DBS current, D is the duration of positive input. In particular, a typical square stimulation current is shown in Fig. 2.

15

When DBS is applied, we can rewrite the above Eqs. (1)– (5) in the first-order form for all neural populations, which are arranged as follows,

d/e ðtÞ ¼ /_e ðtÞ dt d/_e ðtÞ ¼ c2e ½/e ðtÞ þ FðV e ðtÞÞ  2ce /_e ðtÞ dt dXðtÞ _ ¼ XðtÞ dt XðtÞ ¼ ½V e ðtÞ; V d1 ðtÞ; V d2 ðtÞ; V p1 ðtÞ; V p2 ðtÞ; V f ðtÞ; V r ðtÞ; V s ðtÞT dV_e ðtÞ ¼ abðv ee /e þ v ei FðV e Þ þ v es FðV s Þ  V e ðtÞ þ RIDBS Þ dt  ða þ bÞV_e ðtÞ _ dV d1 ðtÞ ¼ abðv d1 e /e þ v d1 d1 FðV d1 Þ þ v d1 s FðV s Þ  V d1 ðtÞÞ dt  ða þ bÞV_d1 ðtÞ dV_d2 ðtÞ ¼ abðv d2 e /e þ v d2 d2 FðV d2 Þ þ v d2 s FðV s Þ  V d2 ðtÞÞ dt  ða þ bÞV_d2 ðtÞ

dV_p1 ðtÞ ¼ abðv p1 d1 FðV d1 Þ þ v p1 p2 FðV p2 Þ þ v p1 f FðV f Þ dt  V p1 ðtÞ þ RIDBS Þ  ða þ bÞV_p1 ðtÞ dV_p2 ðtÞ ¼ abðv p2 d2 FðV d2 Þ þ v p2 p2 FðV p2 Þ þ v p2 f FðV f Þ dt  V p2 ðtÞÞ  ða þ bÞV_p2 ðtÞ dV_ f ðtÞ ¼ abðv fe /e þ v fp2 FðV p2 Þ  V f ðtÞÞ  ða þ bÞV_ f ðtÞ dt dV_ r ðtÞ ¼ abðv re /e þ v rp1 FðV p1 Þ þ v rs FðV s Þ  V r ðtÞÞ dt  ða þ bÞV_ r ðtÞ

6

5

dV_ s ðtÞ ¼ abðv se /e þ v sp1 FðV p1 Þ þ v srA FðV r Þ þ v Bsr FðV r ðT  sÞÞ dt  V s ðtÞ þ /n Þ  ða þ bÞV_ s ðtÞ

IDBS(pA)

4

3

2

1

0

0

10

20

30

40

50

60

70

80

90

100

Time(ms)

Fig. 2. An illustrative example of the common square wave stimulation current used in this paper. Here, we choose the period P ¼ 25 ms, the amplitude of current A ¼ 5 pA, the duration of positive input D ¼ 5 ms.

where the parameter /n denotes the constant nonspecific subthalamic input onto SRN, and we assume that v srA  v Bsr . Unless otherwise noted, all the parameter values used for the model are listed in appendix. Basically, these values are based on physiological experiments, and taken in the range of the normal physiological significance, which can be found in some previous studies [14,19,20,31–34]. As DBS is applied onto substantia nigra pars reticulata and cortex, IDBS is added into the corresponding equation with multiplying a resistance R for transferring the DBS current into voltage. And also, we set R ¼ 1mX for simplicity.

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Similar to the previous studies [14,15], the standard fourth-order Runge–Kutta method is used to solve the studied equations. And then, the delay is handled in the 4th-order Runge–Kutta routine by the iterative methods. In particular, we store the previous data for each step in one dimensional vector with the whole simulation time length. And then, we fetch the history data at the corresponding position from the stored vector, which can be calculated according to the delay and integration step (the fetched position should be: t  sd =dt, where t is the current simulation step, sd is the delay and dt is the integration step). Finally, we substitute such fetched data into our equations to handle the delay term. Still, by recording the local minimum and maximum values of the /e , we obtain the bifurcation diagram and identify different state transition with changing the corresponding parameter. In addition, to investigate the dominant frequency of neural oscillations, we use the typical power spectral analysis of the time series. 3. Main results Firstly, we observe transitions between different dynamical states as TRN-SRN inhibitory coupling strength v sr and the delay parameter s are changed. Results presented in Fig. 3(a) and (b) show the state transitions of four

(a)

different firings in the two-parameter space. Combined Fig. 3(a) with Fig. 3(b), we can clearly see that there exist an island of the typical 2–4 Hz SWD oscillation state as denoted by ‘‘SWD’’ in Fig. 3(b). In particular, we illustrate four typical time series of /e in Fig. 3(c)–(f), which correspond to the above four dynamical states by choosing different values of v sr . It can be seen from the Fig. 3(d) that the typical shock wave of absence epilepsy can be exhibited with two pairs of maximum and minimum values in one period. For more detailed description on results of Fig. 3, one can refer to the Ref. [14]. Here, we mainly wonder how to control SWD oscillation states, which have occurred as shown in Fig. 3 when DBS is applied on the SNr and cortex, respectively. 3.1. The deep brain stimulation current IDBS acted on SNr STN is the most commonly used stimulate target in clinical treatment, and many related results have been reported in previous studies [22,23,25,26,28]. However, some recent experiments show that as DBS is introduced to the main output nucleus of the basal ganglia (Gpi or SNr), a good control effect can also be realized for the parkinson’s disease [35–38]. Especially, the results in [14] have shown that the epileptic seizure activity can be effectively controlled by turning the activation level of the population

(b)

State region

Frequency (Hz) 60

60

6

(1) 45

(2)

4

τ(ms)

τ(ms)

SWD SWD 45

(3)

2

(4) 30 0.2

(d)

Saturation(4)

SWD oscillation(3)

50

0 0

2

4

6

Simple oscillation(2)

40

(f)

20

0

10

0.5

1

Low firing(1)

5

e

φe(Hz)

2.6

e

100

(e)

0 1.4 −vsr(mV s)

φ (Hz)

260 200

0

30 0.2

2.6

φ (Hz)

φe(Hz)

(c)

1.4 −vsr(mV s)

0

0 0

1 Time(s) 2

3

0

1 Time(s)

2

Fig. 3. Absence seizure induced by the TRN-SRN inhibitory coupling strength v sr and the delay parameter s. (a), (b): The different dynamical states and their frequencies are obtained in the parameter plane ðv sr ; sÞ, respectively. In particular, four different colors represent four different dynamical states (1)– (4). The ‘‘SWD’’ region surrounded by dotted lines in Fig. 3(b) represents the typical 2–4 Hz frequency SWDs oscillation region. Different types of extreme points in a period represent different dynamical states, which are denoted as: Saturation (4), SWD oscillation (3), Simple oscillation (2) and Low firing (1), respectively. From (c) to (f), time series of /e of four typical dynamical states are shown. Here we set s = 45 ms and choose v sr = 0.3 mV s (c), v sr = 0.7 mV s (d), v sr = 1.3 mV s (e), v sr = 2 mV s (f), respectively (also see the descriptions of Fig. 2 of Ref. [14].) (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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B. Hu, Q. Wang / Chaos, Solitons & Fractals 80 (2015) 13–23

SNr. Therefore, we try to study the control effect of IDBS on the epileptic seizures as it is acted on SNr. Particularly, we investigate the effects of the period, duration time of positive input and amplitude of DBS on the seizure activities. Initially, we study the control effect of DBS on the absence seizures by varying the period P of DBS current. Fig. 4(a) shows the bifurcation diagram of state transitions in the two-dimensional parameter plane ðv sr ; PÞ. It can be observed that by tuning the period P of DBS, we can control the seizure activities as indicated by double arrows in Fig. 4(a). In particular, as the v sr is relatively small, the seizure state (3) can be adjusted to state (4) as the period P changes. Interestingly, we can find that there is a narrow band of v sr , where the absence seizure can not be controlled by variation of the period P. Moreover, the corresponding dominant frequency is also shown in Fig. 4(b), from which we clearly find that the frequency of absence seizures is almost 2–4 Hz. For a clearer vision, two typical

(a)

bifurcation processes of /e are given in Fig. 4(c) (v sr ¼ 0:6 mV s) and Fig. 4(d) (v sr ¼ 1:2 mV s) as the period P changes. It is clearly shown that the SWDs oscillation can be controlled with suitable value of P. Next, we wonder if the seizure activities can be controlled by varying the duration of DBS stimulation current. Similarly, we calculate the bifurcation diagram of state transition in the parameter plane(v sr ; D). It is shown in Fig. 5(a) that SWDs oscillation can be controlled with pushing it into either the high firing state (4) or low firing state (1) as the duration of DBS D is changed to some suitable range. Additionally, the corresponding frequency analysis is also shown in Fig. 5(b), from which we see that the considered seizure is the absence seizure with 2–4 Hz. Furthermore, from Fig. 5(c), a typical transition process can be shown by setting v sr ¼ 1:6 mV s, where the alternate transition phenomena can be found between SWDs oscillation and low firing state as D changes. However,

(b)

State region 2

Frequency (Hz)

2

4

(1)

1

(2) (3) (4)

0.1 0.2

φ (Hz) e

(c)

1.8 −v (mV s) sr

3.4

P(ms)

P(ms)

3 2

1

1 0

0.1 0.2

1.8 −v (mV s)

3.4

sr

300 200 100 0

φ (Hz) e

(d)

0

0.2

0.4

0.6

0

0.2

0.4

0.6

0.8

1 P(ms)

1.2

1.4

1.6

1.8

2

150 100 50 0 0.8

1 P(ms)

1.2

1.4

1.6

1.8

2

Fig. 4. (a) and (b): The transition of different states and dominant frequency analysis in the parameter plane (v sr ; P). v sr is the inhibitory coupling strength of the TRN-SRN pathway, P is the period of stimulation current. Here, A ¼ 50 pA, D ¼ 0:0006 ms, v p1 f = 0.1 mV s. Four different states are represented by different colors, which are similar to Fig. 3. The seizure states can be controlled by tuning the period P to some appropriate intervals as shown by double arrows. Combined Fig. 4(a) with Fig. 4(b), the typical 2–4 Hz SWD region can be found in (b). (c): A typical bifurcation diagram of /e as a function of P. It is obvious that with increasing P, transition between states SWD oscillation and high firing can appear, which implies the realization of controlling of absence seizures with suitable P. We set v sr ¼ 0:6 mV s. (d): A typical bifurcation diagram of /e as a function of P. The obvious transition phenomenon between SWD oscillation and simple firing can appear with increasing of the period P. Here, we set v sr ¼ 1:2 mV s.

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B. Hu, Q. Wang / Chaos, Solitons & Fractals 80 (2015) 13–23

(a)

(b)

State region

(a) 2

Frequency (Hz)

(b) 2

4

(1)

1

(2)

D(ms)

D(ms)

3 2

1

1

(3) (4)

0.1

0.1 0.2

(c)

1.8 −vsr(mV s)

3.4

0 0.2

1.8 −vsr(mV s)

3.4

(c) 100 80 (3)

φe(Hz)

(3) 60 (1)

(1)

40 20 0

0

0.2

0.4

0.6

0.8

1 D(ms)

1.2

1.4

1.6

1.8

2

Fig. 5. (a) and (b): The state transitions and dominant frequency analysis in the parameter plane (v sr ; D). v sr is the inhibitory coupling strength of the TRN-SRN pathway, D is the duration of positive input of the DBS current. Here, A ¼ 50 pA, P ¼ 1 ms, v p1 f = 0.1 mV s. Four different states are shown by different colors, which are consistent with Figs. 3 and 4. It is shown that the states of absence seizure can be controlled by tuning D as shown by the double arrows. We also find the typical 2–4 Hz SWD region in Fig. 5(b). (c): A typical bifurcation diagram of /e between SWD oscillation and low firing is illustrated as D varies. Here, we set v sr ¼ 1:6 mV s. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

(a)

(b)

State region

Frequency (Hz)

(b) 65

(a) 65

4 3.5 3

A(pA)

A(pA)

2.5 32.5

32.5

2

(1)

1.5

(2)

1

(3) (4)

0 0.2

1.8 −v (mV s) sr

3.4

0.5 0 0.2

1.8 −v (mV s)

3.4

0

sr

Fig. 6. (a) and (b): The state transitions and dominant frequency analysis in the parameter plane (v sr ; A). v sr is the inhibitory coupling strength of the TRN-SRN pathway, A is the amplitude of stimulation current. We take P ¼ 1 ms, D ¼ 0:8 ms, and v p1 f = 0.1 mV s. Descriptions of four different states are as the above. It is shown that by changing A, we cannot effectively control the absence seizures.

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B. Hu, Q. Wang / Chaos, Solitons & Fractals 80 (2015) 13–23

(a)

(b)

State region

(a) 4.6

Frequency (Hz) 8

(b) 4.6

(1) (2)

2.4

0.2 0.48

(c) (c)

−vei(mV s)

−vei(mV s)

6

(3) (4) 1.24 −vsr(mV s)

2.4

4

SWD

2 0.2 0.48

2

0 1.24 −vsr(mV s)

2

φe(Hz)

300

200 (4)

(2)

(3)

(1) 100

0 1

1.5

2

2.5 3 −vei(mV s)

3.5

4

4.5

Fig. 7. (a) and (b): The state transitions and dominant frequency analysis in the parameter spaces (v sr ; v ei ). v sr is the inhibitory coupling strength of the TRN-SRN pathway, v ei is the inhibitory coupling strength from inhibitory interneurons to excitatory pyramidal neurons, v p1 f = 0.3 mV s. Four different states (1)–(4) can appear and the seizure state (3) can be controlled by suitably increasing or decreasing v ei as indicated by the double arrows. (c): By fixing v sr ¼ 1:2 mV s, a bifurcation diagram show the transition process between seizure and other three states as the inhibitory coupling strength v ei varies.

(a)

(b)

State region

Frequency (Hz)

(b) 4.6

(a) 4.6

4

(1) 2.4

(2) (3)

3

vse(mV s)

vse(mV s)

SWD 2.4

2 1

(4) 0.2 0.48

(c) (c)

0.2 0.48

2

0 1.24 −vsr(mV s)

2

270 250 200

φe(Hz)

1.24 −vsr(mV s)

(1)

(3)

(2)

(4)

150 100 50 0 0

1

2

v (mV s)

3

4

5

se

Fig. 8. (a) and (b): The state transitions and dominant frequency analysis in the parameter spaces (v sr ; v se ). v sr is the inhibitory coupling strength of the TRN-SRN pathway, v se is the excitatory coupling strength from excitatory pyramidal neurons to SRN. We set v p1 f = 0.3 mV s. It is found that four different states (1)–(4) can appear, and the seizure state (3) can be controlled by increasing or decreasing v se . (c): A specific bifurcation diagram is plotted to illustrate the transition process between seizure and other three states (we fix v sr ¼ 1 mV s).

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B. Hu, Q. Wang / Chaos, Solitons & Fractals 80 (2015) 13–23

(a)

(b)

State region

(a) 2

Frequency (Hz)

(b) 2

4

1

(1) (2) (3) (4)

0.1 0.2

1.8 −v (mV s)

D(ms)

D(ms)

3 2

1

1 0.1

3.4

0 0.2

sr

1.8 −v (mV s)

3.4

sr

(c) (c) 100 0

φ (Hz) e

80 60 40 50 0 0

0.5

1 D(ms)

1.5

2

Fig. 9. (a) and (b): The transition of states and dominant frequency analysis in the plane (v sr ; D). v sr is the inhibitory coupling strength of the TRN-SRN pathway, D is the duration of positive input of stimulation current. It is found that the seizure state can be controlled by tuning the D to some appropriate intervals as denoted by the double arrows. Here, we fix A ¼ 50 pA, P ¼ 1 ms, and v p1 f = 0.3 mV s. (c): A typical bifurcation diagram of /e as a function of D is shown. It is clearly seen that with increasing D, transition phenomena between SWD oscillation and low firing can appear, which implies that absence seizures can be controlled by adjusting the value of D. We set v sr ¼ 0:6 mV s.

further investigation shows that by changing the amplitude of stimulation current A, the seizure activity can not be controlled as shown in Fig. 6(a) and (b), from which, we can find that the state transitions are almost independent of the value A. Hence, this implies that changing the amplitude of DBS A can not be an effective method for controlling the epilepsy.

3.2. The deep brain stimulation current IDBS acted on the cortex Because epileptic shocks (e.g. SWDs) are mainly reflected in the pyramidal cortex, we try to control the absence seizures by using DBS current to the cortex. Firstly, two main parameters v sr and v ei (v sr is the inhibitory coupling strength of pathway TRN-SRN and v ei is the inhibitory coupling strength from inhibitory interneurons to excitatory pyramidal neurons) are used to investigate the effect of internal regulations on the absence seizures. Bifurcation diagram of state transitions is shown in Fig. 7(a) in the parameter plane ðv sr ; v ei Þ, where it can be found that the SWD state (3) can appear within the regions of suitable parameters. And then, it can be controlled by suitably increasing or decreasing v ei as indicated by the double

arrows of Fig. 7(a). Furthermore, the corresponding dominant frequency is also given in Fig. 7(b), where the typical absence seizure with 2–4 Hz can be found within the region surrounded by the dotted lines. In addition, when we fix v sr ¼ 1:2 mV s, and change v ei , a typical bifurcation diagram is illustrated in Fig. 7(c) to exhibit obvious transition process among four states. Similarly, by suitably tuning the coupling strength v se from excitatory pyramidal neurons to SRN, the seizure activities can also occur, and then be controlled as presented in Fig. 8(a)–(c). Hence, it can be inferred that both inhibitory and excitatory pyramidal neurons can control absence seizures with the implement of a suitable coupling strength. Inspired by the above results, we want to apply inhibitory and excitatory deep brain stimulus currents on the cortex to investigate if the absence seizures can be controlled. For this purpose, we choose T ¼ 1 ms, A ¼ 50 pA and v p1 f = 0.3 mV s as inhibitory deep brain stimulus currents. The obtained results are presented in Fig. 9(a) and (b), where it can be found that the seizure activities can be controlled by tuning D to some appropriate intervals as marked by the double arrows. We can further understand this from a specific bifurcation diagram as shown in Fig. 9(c), where the alternating transition phenomenon

21

B. Hu, Q. Wang / Chaos, Solitons & Fractals 80 (2015) 13–23

(a)

(b)

State region

Frequency

(b) 2

(a) 2

4

(1) 1

(2)

D(ms)

D(ms)

3 2

1

1

(3) (4) 0.1 0.2

1.8 −v (mV s)

0.1 0.2

3.4

sr

0 1.8 −v (mV s)

3.4

sr

(c) (c) 260

e

φ (Hz)

200

100

0 0

0.5

1 D(ms)

1.5

2

Fig. 10. (a) and (b): The transition of states and dominant frequency analysis in the plane (v sr ; D). v sr is the inhibitory coupling strength of the TRN-SRN pathway, D is the duration of positive input of stimulation current. We can find that by adjusting the D to some appropriate intervals, the seizure state can be controlled. Here, we fix A ¼ 50 pA, P ¼ 1 ms, v p1 f = 0.3 mV s. (c): A typical bifurcation diagram of /e is illustrated as D varies, where the transition process between SWD oscillation and saturation state can be exhibited. We set v sr ¼ 0:6 mV s.

between the state seizure and low firing can appear as D varies. Similarly, A is taken as 50 pA to study the effect of the excitatory deep brain stimulus currents on the absence seizure. We can find from Fig. 10 that the absence seizure can be also controlled by adjusting D because there exist the alternative transitions between the saturation state and SWD. Furthermore, we investigate the effects of the parameters P and A on controlling absence seizures in the considered regions of the parameters of this paper. Unfortunately, their variations cannot have clear impact on the absence seizures. Hence, it can be concluded that the absence seizure can only be inhibited by the value D as we introduce the inhibitory and excitatory deep brain stimulus currents onto the cortex.

4. Conclusion In this paper, based on theoretical network model, we have studied the effects of various DBS parameters (the period, the duration of positive input and amplitudes of stimulation current) on the epilepsy seizures by introducing DBS into the SNr and cortex. We have shown that by tuning periods and durations of positive input to some appropriate intervals, absence seizures can be effectively controlled as DBS is applied onto the SNr. Interestingly, it

is found that the amplitude of DBS can not be a key factor in controlling absence seizures, and it is probably less commonly used in clinical and experimental study. However, as DBS is applied onto the cortex, the absence seizures can be controlled by only D within the considered parameter ranges of the present paper. These results show that for different stimulus target regions, DBS can have different influence on the neural function areas. Although the mean field network model of this study is a simplified mathematical model, it can help to well understand the impact for controlling absence seizure in realistic neuronal systems. Epilepsy is a chronic disease, and can be persistent for several years, even decades, which brings into serious adverse effects of the patient’s body, spirit, marriage and social economic status. Therefore, the treatment of epilepsy patients should be paid more attention on. We hope that the present results can have a good guidance to treat the related diseases.

Acknowledgments Author B. Hu thanks Dr. D.Q. Guo for his kind help about program. This research was supported by the National Science Foundation of China (Grant Nos. 11172017 and 11325208).

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B. Hu, Q. Wang / Chaos, Solitons & Fractals 80 (2015) 13–23

Appendix A (continued)

Appendix A Unless otherwise noted, we use these parameter values for simulations as follows [14,19,20,26,28,30,32],

Coupling strength

Source

Target

Value

vd e

Excitatory pyramidal neurons Striatal D2 neurons SRN

Striatal D2 neurons

0.7 mV s

Striatal D2 neurons Striatal D2 neurons SNr

0.3 mV s

SNr SNr GPe

0.03 mV s 0–0.6 mV s 0.3 mV s

GPe GPe STN Excitatory pyramidal neurons SRN

0.075 mV s 0.45 mV s 0.04 mV s 1.8 mV s

STN

0.1 mV s

SRN TRN

0.035 mV s 0.035 mV s

2

Parameter Mean

Value

Cortical maximum firing rate ; Q max Q max e i max Striatum maximum firing rate Q d1 ; Q max d2 SNr maximum firing rate Q max p

250 Hz 65 Hz

vd d

250 Hz

vd s

Q max p2

GPe maximum firing rate

300 Hz

Q max f

STN maximum firing rate

500 Hz

Q max s Q max r he ; hi

SRN maximum firing rate TRN maximum firing rate Mean firing threshold of cortical populations Mean firing threshold of striatum Mean firing threshold of SNr Mean firing threshold of GPe Mean firing threshold of STN Mean firing threshold of SRN Mean firing threshold of TRN Cortical damping rate Time delay due to slow synaptic kinetics of GABAB Synaptodendritic decay time constant Synaptodendritic rise time constant Threshold variability of firing rate Nonspecific subthalamic input onto SRN

250 Hz 250 Hz 15 mV

1

hd1 ; hd2 hp1 hp2 hf hs hr

ce s a b

r /n

2 2

2

vp d

1 1

vp p vp f vp d

1 2 1

2 2

19 mV 10 mV 9 mV 10 mV 15 mV 15 mV 100 Hz 50 ms

vp p vp f v fp v es

50 s1

v fe

2 2 2

2

v se

200 s1 6 mV

v sp v rp

1

1

Striatal D1 neurons GPe STN Striatal D2 neurons GPe STN GPe STN

Excitatory pyramidal neurons Excitatory pyramidal neurons SNr SNr

0.05 mV s 0.1 mV s

2.2 mV s

2 mV s

References Coupling strength

Source

Target

Value

v ee

Excitatory pyramidal neurons Inhibitory interneurons

Excitatory pyramidal neurons Excitatory pyramidal neurons TRN

1 mV s

TRN SRN

0.5 mV s (3.8)  (0.2) mV s 1 mV s

v ei v re v rs v A;B sr vd e 1

vd d

1 1

vd s 1

Excitatory pyramidal neurons SRN TRN Excitatory pyramidal neurons Striatal D1 neurons SRN

Striatal D1 neurons Striatal D1 neurons Striatal D1 neurons

1.8 mV s

0.05 mV s

0.2 mV s 0.1 mV s

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