Inhibition of absence seizures in a reduced corticothalamic circuit via closed-loop control

Yan Xie; Rui Zhu; Xiaolong Tan; Yuan Chai; Yan Xie; Rui Zhu; Xiaolong Tan; Yuan Chai

doi:10.3934/era.2023134

Electronic Research Archive

2023, Volume 31, Issue 5: 2651-2666. doi: 10.3934/era.2023134

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Research article

Inhibition of absence seizures in a reduced corticothalamic circuit via closed-loop control

School of Mathematics and Physics, Shanghai University of Electric Power, Shanghai 201306, China

Academic Editor: Yanxiang Zhao

Received: 18 October 2022 Revised: 16 December 2022 Accepted: 27 February 2023 Published: 09 March 2023

Inhibition of spike-wave discharges (SWD) was thought to be associated with remission of seizure symptoms in absence epilepsy. In the previous studies, pulse stimulation was applied directly to the brain as an effective means to inhibit SWD. However, this method not only failed to provide a real-time tracking control of the disease, but also caused incalculable damage to the patient's brain tissue. To fill the gap, this work studied the mitigation and elimination effects of SWD by applying single-pulse coordinated resetting stimulation (SCRS) to three different neurons, firstly. Secondly, based on the 2I:2O cortical-thalamic model (2I:2O SCT model), four kinds of m:n on-off SCRS with the same period were compared, and the control efficiency and impulse energy consumption were combined to investigate these different stimulations. Finally, we further optimized the regulation strategies, including the weighted structure of stimulation and closed-loop control. Simulation results show that the weighted stimulation and closed-loop control strategy proposed here further improve control performance by reducing energy consumption, which may be more reliable in applications. Moreover, this study provides a new method for optimizing SCRS by the weighted processing and closed-loop control of electrical pulses to alleviate the absence epileptic state.

Keywords:

spike-wave discharges,
weighted stimulation,
single-pulse coordinated reset stimulation,
closed-loop control

Citation: Yan Xie, Rui Zhu, Xiaolong Tan, Yuan Chai. Inhibition of absence seizures in a reduced corticothalamic circuit via closed-loop control[J]. Electronic Research Archive, 2023, 31(5): 2651-2666. doi: 10.3934/era.2023134

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Abstract

1. Introduction

As one of the most common chronic neurological diseases in the world ^[1], epilepsy affects the daily life of tens of millions of people. In severe cases, it can even be life-threatening, which causes the premature death rate of epilepsy patients to be one to two times higher than that of the normal population ^[2,3]. Absence seizure as a branch of many epileptic seizure types ^[4], generally occurs in childhood. Childhood absence seizure (CAE) syndrome is actually a benign syndrome that remits in most CAE patients in early adolescence ^[5,6]. At the onset of the absence epilepsy, bilateral symmetric SWD at 3 Hz ^[7] can be observed in the electroencephalogram (EEG) of the patient ^[8], which provides a basis for our research.

As early as 1873, Jackson in the UK first proposed that epileptic seizures may originate from the cerebral cortex ^[9,10]. Since then, people have begun to explore the cerebral cortex accordingly. Penfield and Jasper proposed the concept of central brain epilepsy in the 1950s, which aroused the attention of neuropathic researchers to the cortical-thalamic circuit ^[11,12]. Such inferences and concepts laid the foundation for subsequent research on neurological diseases. The mean-field model ^[13] with cortex and thalamus as main structures was proposed by Taylor et al. ^[14] in 2014 to simulate and research Parkinson's disease. The cortical-thalamic (CT) model proposed by Taylor is made up of four neurons, among which the cortical network structure consists of excitatory pyramidal neurons (EPN) and inhibitory interneurons (IIN), and the thalamic structure is composed of thalamic reticular nucleus (TRN) and specific relay nuclei (SRN). In the same year, Chen et al. ^[15] found that adding the structure of basal ganglia on the basis of Taylor's CT model has certain research significance in absence epilepsy, so we have the basal ganglia-corticothalamic (BGCT) model. However, simplified studies of the BGCT model have not been carried out, and we do not know which stimulation mode is more suitable for the simplified model.

In the 1980s, neurosurgeons began to implant stimulation electrodes to treat movement disorders ^[16,17]. Benabid implanted thalamic stimulation systems in patients' brains ^[18] to control the trembling phenomenon in 1987. Electrical pulse stimulation ^[19] is an effective treatment for patients with partial seizure or partial secondary generalized seizure, especially for patients with refractory epilepsy ^[20,21] whose epileptogenic focus are located in important functional areas or cannot be located. Despite the risks, the side effects are far smaller than excision of epileptogenic zone ^[22,23]. On account of the electrical nerve stimulator needs to be implanted through minimally invasive surgery ^[24,25] to come into play, it is necessary to reduce energy consumption ^[26,27] so as to achieve prolong stimulator's life in the programming design process. In 2020, Fan et al. applied the closed-loop control principle in the programming process of electrical nerve stimulator to further advance the process of epilepsy control ^[28,29,30]. However, the consumption of stimulus is still a problem, and most pulse stimulation has not realized the real-time control function at present.

The first goal of this study is to reduce computational cost and improve neuromodulation efficiency, and the basal ganglia part of Chen's BGCT model is simplified in this paper. Taking the basal ganglia as a whole, the average discharge rate of substantia nigra pars reticulata (SNr/GPi) was calculated as the fixed input parameter for the simplified model, and the simplified model of 2I:2O was finally obtained. The second goal of this study is to optimize the SCRS by combining energy consumption and therapeutic effect. Firstly, we proposed a constant period m:n on-off SCRS, whose sum of two parameters m and n is guaranteed to be a fixed value. The efficiency and energy consumption of stimulation under different combinations of m and n were compared. Secondly, in order to reduce the energy consumption of electrical stimulator, the concept of the weighted stimulation was proposed, which is to multiply the amplitude and pulse width of original stimulation by the corresponding weight. Finally, the closed-loop control theory was applied to the control of stimulation. The continuous recognition of the mean firing rate of excitatory pyramidal neurons as a feedback signal realized the real-time control function. The main contribution of this work is to create a simplified model and open up new ways to treat epilepsy. The results show that the presentation of the weighted stimulation can not only improve the energy consumption under the premise of ensuring that the stimulation efficacy is not reduced, but also greatly extend the working life of the electrical stimulator and reduce the secondary injury to patients. It is worth noting that the closed-loop treatment of several kinds of stimulation also enables the modulation scheme to achieve a superior effect.

2. Model and plans

Based on Taylor's CT model in 2014, Chen's BGCT model appended the network structure of the basal ganglia ^[15]. The basal ganglia network structure contains five neurons, Λ_BG∈{d₁, d₂, p₁, p₂, ζ}, as shown in purple in Figure 1(a). They represented striatal D₁ neurons, striatal D₂ neurons, substantia nigra pars reticulata (SNr), globus pallidus external (GPe) and subthalamic nucleus (STN), respectively. The pink and yellow structures in Figure 1(a). represent cortical and thalamic structures in the Taylor's CT model. The cortex contains Λ_C∈{e, i}, indicating that e = EPN, I = IIN, and the thalamus structure consists of Λ_T∈{r, s}, representing r = TRN, s = SRN. The blue solid/dashed arrows denote inhibitory projections mediated by GABA_A/GABA_B, and the red arrows mean excitatory synapses mediated by glutamate.

Figure 1. Presentation of models before and after simplification. (a) The referenced BGCT model is composed of a tricolor structure. The pink cortical structures contain excitatory pyramidal neurons (E) and inhibitory interneurons (I), the thalamus in yellow is divided into specific relay nuclei (SRN) and thalamic reticular nucleus (TRN), purple basal ganglia are constructed from striatal D₁ neurons, striatal D₂ neurons, substantia nigra pars reticulata (SNr), globus pallidus external (GPe) and subthalamic nucleus (STN), red and blue arrows represent excitatory or inhibitory projections. (b) In the simplified 2I:2O SCT model, the structure of the cortex and thalamus is unchanged, and all purple structures are divided into a group, and there are only two inhibitory output signals for the cortex and thalamus when modeling.

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To calculate the mean firing rate R_a, we needed to know the values of the following parameters: maximum firing rate $R_a^{max}$ , the threshold θ_a of the mean firing rate, and change rate σ of θ_a. Employing the method of mean field model, the mean membrane potential V_a of each neuron can be expressed by an S-shaped function of R_a ^[31,32].

${R_a}({\mathbf{r}}, t) = \Gamma [{V_a}({\mathbf{r}}, t)] = \frac{{R_a^{max}}}{{1 + \exp [ - \frac{{\pi ({V_a}({\mathbf{r}}, t) - {\theta _a})}}{{\sqrt 3 \sigma }}]}} ,$

(1)

where a∈Λ = {Λ_BG, Λ_C, Λ_T} and Λ including all neurons in Figure 1(a). Until V_a exceeds θ_a, the neural population strikes the action potential at mean firing rate R_a, and the S-shaped function keep R_a below and meet the biological rationality. The variation of the mean membrane potential V_a at position r is caused by the filtered postsynaptic potential of other nerve populations in the dendrites, modeled as ^[33,34]:

${D_{\alpha \beta }}{V_a}({\mathbf{r}}, t) = \sum\limits_{b \in \Lambda } {{C_{ab}} \cdot {\phi _b}({\mathbf{r}}, t)} ,$

(2)

${D_{\alpha \beta }} = \frac{1}{{\alpha \beta }}\left[ {\frac{{{\partial ^2}}}{{\partial {t^2}}} + \left( {\alpha + \beta } \right)\frac{\partial }{{\partial t}} + \alpha \beta } \right] ,$

(3)

where α and β are the decay and rise time of cell body response to input signal respectively, and D_αβ is the differential operator meaning the dendritic filtering of input signal. C_ab describes the coupling strength between neural population b and neural population a. φ_b is the rate of afferent pulses received by other neurons from neuron b. In particular, for GABA_B-mediated inhibitory projection, a delay τ is introduced in the afferent pulse rate of TRN φ_r to simulate its slow synaptic dynamics. V_i = V_e and R_i = R_e are defined, because the intercortical connections are proportional to the number of synapses involved.

Except for excitatory pyramidal neurons, the afferent pulse rate φ_c of remaining neurons c∈{Λ_BG, i, Λ_T} can be directly depicted by the mean firing rate R_c due to their too-short axon characteristics. However, neuron e has a sufficiently long axon structure, its propagation process can be approximated by a damped wave equation ^[35]:

$\frac{1}{{{\gamma _e}^2}}[\frac{{{\partial ^2}}}{{\partial {t^2}}} + 2{\gamma _e}\frac{\partial }{{\partial t}} + {\gamma _e}^2 - {C_e}^2{\nabla ^2}]{\phi _e}({\mathbf{r}}, t) = {R_e}({\mathbf{r}}, t) .$

(4)

In Eq (4), γ_e is the time damping rate of the pulse, and ∇² is the Laplace operator. The generalized spike-wave discharges during absence seizures are synchronous across the brain, so we assumed that the activity space in the system is uniform, which means ∇² = 0. Therefore, the axon field φ_e of neuron e can be further written as ^[36]:

$\frac{1}{{{\gamma _e}^2}}[\frac{{{d^2}}}{{d{t^2}}} + 2{\gamma _e}\frac{d}{{dt}} + {\gamma _e}^2]{\phi _e}(t) = {R_e}(t) \text{,}$

(5)

The dynamic properties of all neural groups were simulated by first-order formal differential equations. The initial BGCT model could be expressed as follows ^[17,37]:

$\frac{{{\text{d}} {\phi _e}(t)}}{{{\text{d}} t}} = {\dot \phi _e}(t) ,$

(6)

$\frac{{{\text{d}} {{\dot \phi }_e}(t)}}{{{\text{d}} t}} = \gamma _e^2[ - {\phi _e}(t) + \Gamma ({V_e}(t))] - 2{\gamma _e}{\dot \phi _e}(t) ,$

(7)

when,

${X_1}(t) = {\left[ {{V_e}(t), {V_{{d_1}}}(t), {V_{{d_2}}}(t), {V_{{p_1}}}(t), {V_{{p_2}}}(t), {V_\zeta }(t), {V_r}(t), {V_s}(t)} \right]^{\mathbf{T}}} ,$

(8)

$\frac{{{\text{d}} {X_1}(t)}}{{{\text{d}} t}} = {\dot X_1}(t) ,$

(9)

$\frac{{{\text{d}} {{\dot X}_1}(t)}}{{{\text{d}} t}} = \alpha \beta {M_1}(t) - (\alpha + \beta ){\dot X_1}(t) ,$

(10)

${M_1}(t) = {C_1}{\Gamma _1}(t) - {V_1}(t) ,$

(11)

with,

${C_1} = \left[ {\begin{array}{*{20}{c}} {{C_{ee}}}&{{C_{ei}}}&0&0&0&0&0&{\left( {0, 0} \right)}&{{C_{es}}} \\ {{C_{{d_1}e}}}&0&{{C_{{d_1}{d_1}}}}&0&0&0&0&{\left( {0, 0} \right)}&{{C_{{d_1}s}}} \\ {{C_{{d_2}e}}}&0&0&{{C_{{d_2}{d_2}}}}&0&0&0&{\left( {0, 0} \right)}&{{C_{{d_2}s}}} \\ 0&0&{{C_{{p_1}{d_1}}}}&0&0&{{C_{{p_1}{p_2}}}}&{{C_{{p_1}\zeta }}}&{\left( {0, 0} \right)}&0 \\ 0&0&0&{{C_{{p_2}{d_2}}}}&0&{{C_{{p_2}{p_2}}}}&{{C_{{p_2}\zeta }}}&{\left( {0, 0} \right)}&0 \\ {{C_{\zeta e}}}&0&0&0&0&{{C_{\zeta {p_2}}}}&0&{\left( {0, 0} \right)}&0 \\ {{C_{re}}}&0&0&0&{{C_{r{p_1}}}}&0&0&{\left( {0, 0} \right)}&{{C_{rs}}} \\ {{C_{se}}}&0&0&0&{{C_{s{p_1}}}}&0&0&{\left( {C_{sr}^A, C_{sr}^B} \right)}&0 \end{array}} \right] ,$

(12)

${\Gamma _1} = {\left[ {{\phi _e}, \Gamma \left( {{V_e}} \right), \Gamma \left( {{V_{{d_1}}}} \right), \Gamma \left( {{V_{{d_2}}}} \right), \Gamma \left( {{V_{{p_1}}}} \right), \Gamma \left( {{V_{{p_2}}}} \right), \Gamma \left( {{V_\zeta }} \right), \left( {\Gamma \left( {{V_r}} \right), \Gamma \left( {{V_r}\left( {t - \tau } \right)} \right)} \right), \Gamma \left( {{V_s}} \right)} \right]^{\mathbf{T}}} ,$

(13)

${V_1} = {\left[ {{V_e}(t), {V_{{d_1}}}(t), {V_{{d_2}}}(t), {V_{{p_1}}}(t), {V_{{p_2}}}(t), {V_\zeta }(t), {V_r}(t), {V_s}(t) - {\phi _n}} \right]^{\mathbf{T}}} ,$

(14)

In order to simplify the model, the basal ganglia was regarded as a collective in this paper, so the cortex and thalamus would receive two inhibitory inputs from the basal ganglia and output two excitatory signals. The simplified model was called 2I:2O SCT model, and SCRS were added into neurons e, s and r, respectively. For the simplified model, the expression of the axon field φ_e of neuron e still adopted Eqs (6) and (7), and the dynamic characteristics of other neurons could be calculated according to the following equation:

when,

${X_2}(t) = {\left[ {{V_e}(t), {V_r}(t), {V_s}(t)} \right]^{\mathbf{T}}} ,$

(15)

$\frac{{{\text{d}} {X_2}(t)}}{{{\text{d}} t}} = {\dot X_2}(t) ,$

(16)

$\frac{{{\text{d}} {{\dot X}_2}(t)}}{{{\text{d}} t}} = \alpha \beta {M_2}(t) - (\alpha + \beta ){\dot X_2}(t) ,$

(17)

${M_2}(t) = {C_2}{\Gamma _2}(t) - {V_2}(t) + S(t) ,$

(18)

with,

${C_2} = \left[ {\begin{array}{*{20}{c}} {{C_{ee}}}&{{C_{ei}}}&0&{\left( {0, 0} \right)}&{{C_{es}}} \\ {{C_{re}}}&0&{{C_{r{p_1}}}}&{\left( {0, 0} \right)}&{{C_{rs}}} \\ {{C_{se}}}&0&{{C_{s{p_1}}}}&{\left( {C_{sr}^A, C_{sr}^B} \right)}&0 \end{array}} \right] ,$

(19)

${\Gamma _2} = {\left[ {{\phi _e}, \Gamma \left( {{V_e}} \right), \overline {\Gamma \left( {{V_{{p_1}}}} \right)} , \left( {\Gamma \left( {{V_r}} \right), \Gamma \left( {{V_r}\left( {t - \tau } \right)} \right)} \right), \Gamma \left( {{V_s}} \right)} \right]^{\mathbf{T}}} ,$

(20)

${V_2} = {\left[ {{V_e}(t), {V_r}(t), {V_s}(t) - {\phi _n}} \right]^{\mathbf{T}}} ,$

(21)

$S = {\left[ {{S_e}(t), {S_r}(t), {S_s}(t)} \right]^{\mathbf{T}}} ,$

(22)

here, S_e(t), S_r(t), S_s(t) represent single-pulse coordinated resetting signals SCRS applied to neurons e, r and s, respectively in Eq (22). Pulse signals were obtained from the following formula ^[38,39]:

$\begin{gathered} S(t) = {A_0} \times H\left( {\sin \left( {\frac{{2\pi t}}{{{f_0}}}} \right)} \right)\left( {1 - H\left( {\sin \left( {\frac{{2\pi (t + \delta )}}{{{f_0}}}} \right)} \right)} \right), \hfill \\ {S_x}(t) = \sum\limits_{x = 1}^3 {{\rho _x}(t)S(t)} . \hfill \\ \end{gathered}$

(23)

In the stimulus expression, A₀ represents the amplitude of the pulse, δ₀ denotes the pulse width of the stimulus signals, T₀ determines the cycle of the stimulus, from which we can calculate the frequency of the stimulus as T₀ = 1/ f₀. x equal 1, 2, or 3 correspond to electrode inputs in neurons e, s, r, respectively.

Figure 2. Coordinated release of periodic impulses on three neurons. (a) on-on SCRS (b) 3:2 on-off SCRS. In this paper, the following values are used for stimulus parameters: A₀ = 200 mA, f₀ = 50 Hz, δ₀ = 3.5 ms.

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The process of stimulation was equivalent to a process of epilepsy treatment. Electrical stimulators needed to be implanted into patients' cerebral cortex in a minimally invasive way, so the research needed to reduce the energy consumption of electrical pulses as much as possible to ensure the life of electrical stimulators. The calculation method of energy consumption was as follows ^[40]:

$Q_{S A R S} = \frac{1}{\sqrt{N}}\left\|S_x(t)\right\|_2,$

(24)

where N is the total time step, and is the two-norm of S_x(t). To quantify the reduction rate of SWD, the parameter plane (−C_sr, C_es) is divided into 21×21 grid points. In particular, the stimulus efficacy (E) is quantitatively calculated by the grid points Before (B) and After (A) stimulus ^[41]:

$E = \left(1-\frac{A}{B}\right) \times 100 \%.$

(25)

In the MATLAB R2018b environment, the fourth-order Runge-Kutta method was used to solve the differential equations, and the total duration of simulation was set as 15 seconds to complete the simulation process. In order to ensure the accuracy and stability of the simulation, we set up the temporal resolution to 0.00005s, and conducted dynamic analysis of neurons from the 5^ths. If there were no special instructions in the text, the data referred to previous studies ^{[31,40,41,42]}.

3. Results and discussion

3.1. Model analysis before and after simplification

Considering the objective of SCRS, we selected the coupling coefficients −C_sr and C_es of the two pathways related to these three neurons. Initially, we plotted the dynamic state frequency diagram of BGCT and 2I:2O SCT model, as shown in Figures 3(a)–(d). There were several relationships between the excitatory cortical firing rate and time under certain coupling strength. According to the characteristics of the excitatory cortical discharge diagram shown in Figure 3(f), states can be divided into two categories, non-oscillation and oscillation. The non-oscillation waveform had no obvious movement trend after stabilization, and was mainly divided into low firing (IV) and high saturation (I). The way to distinguish these two states was to observe the position of discharge rate after stabilization (i.e., 5s). The waveform characteristic of oscillations was periodic. By combining the states (III) and (II) in bifurcation diagram Figure 3(e), it can be determined that one or two sets of peaks and troughs within a unit period was a straightforward way to distinguish simple oscillation (III) from SWD (II).

Figure 3. Dynamic analysis of original model and simplified model. (a)-(b), state and frequency diagrams for ϕ_e as a function of BGCT model; (c)-(d), state and frequency diagrams of 2I:2O SCT model, and the boundary position of epileptic state is marked by white lines in the state diagram (c), white lines are used to divide the boundary position of epileptic state. The bifurcation diagram (e) shows the number of peaks and troughs in waveform charts under different coupling strength −C_sr with C_es = 1.8mVs, corresponding to the four states in discharge charts (f) respectively. When −C_sr = 0.4mVs, the state is high saturation (I); when −C_sr = 0.8mVs, the state is SWD (II); when −C_sr = 1.4mVs, the state is simple oscillation (III); when −C_sr = 1.8mVs, the state is low firing (IV).

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It can be seen that the white line in Figure 3(c) refers to the boundary position of epileptic state. In Figure 4, we plot the triggering mean discharge rate (TMFR) of different neurons under Figure 3(c) critical position. Furthermore, we completed the fitting of the high/low TMFR so as to obtain the orange/purple line. The simplified model accelerated the running speed of the mean field model and facilitated the stimulation prioritization in the following two chapters.

Figure 4. TMFRs of different neurons. (a), (b) and (c), the AMFRs of EPN(a), SRN(b), TRN(c) as a function of C_es. The position of orange/purple hex star is determined by the coupling strength of −C_sr and C_es that trigger absence seizure. L_1TMFR and L_2TMFR are fitting curves of high and low TMFRs.

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3.2. Major parameter selection of constant period m:n on-off SCRS

To test the adaptation of the stimulus to different neurons, single-pulse stimulus with the same intensity but opposite amplitudes were added to each of EPN, SRN and TRN neurons, respectively. The blue line in Figure 5(a) indicates that the amplitude is negative, and the red line indicates that the amplitude is positive. It can be shown from Figure 5(a) that adding negative pulses on neuron EPN or SRN can 100% control the occurrence of epilepsy. Similarly, when positive amplitude single-pulse stimulus was introduced into neuron TRN, SWD could be completely inhibited. Thus, the coordinated stimulus of three neurons was proposed. In other words, positive pulse stimulation was added to EPN and SRN, negative pulse stimulation was added to TRN, and the duration of stimulation to each neuron was controlled through coordinated reset.

Figure 5. Amplitude and coordinated resetting period of single-pulse stimulation. The three indicators in the radar diagram (a) refer to the efficiency of the stimulation when it is released on the three neurons in the diagram. Select stimulation parameters |A₀| = 200 mA, f₀ = 50 Hz, δ₀ = 3.5 ms. The blue line is a negative amplitude stimulation, and the red line is a positive amplitude stimulation. In diagram (b) of m:n on-off SCRS with constant period, m:n from top to bottom are 4:1, 3:2, 2:3 and 1:4, respectively. In the dual coordinate diagram (c), we combine stimulation efficacy and power consumption to analyze the above four m:n on-off SCRS. The purple bar shows the efficiency of the stimulation, and the blue line shows the energy consumption of four kinds of stimulation.

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Four kinds of m:n on-off SCRS in Figure 5(b) were selected, which were 4:1 on-off SCRS, 3:2 on-off SCRS, 2:3 on-off SCRS, 1:4 on-off SCRS respectively. All four stimuli had the same period 5T₀, and the effective stimulus duration in a unit period was mT₀. In Figure 5(c), the two indexes of efficacy and energy consumption are combined to investigate the constant period stimulation of different m and n. As can be seen from the purple bar chart (efficacy E), when the effective stimulus duration of SCRS was not lower than 3T₀, the complete inhibition effect could be achieved. However, once the effective stimulus duration was lower than 3T₀, the epilepsy control effect showed a trend of continuous decline. Blue broken line graph (energy consumption Q) as the effective stimulus duration decreased, the energy consumption also continuously decreased. Considering the working effect and operating life of electrical stimulator comprehensively, 3:2 on-off SCRS was selected from the four SCRS, which was sufficient to inhibit epilepsy and can extend the life of electrical stimulator relatively. In order to make the 3:2 on-off SCRS play a better role and improve the problem of energy consumption, we would have explored the weighted and closed-loop stimulation in depth in the next chapter.

3.3. Continual improvement of 3:2 on-off SCRS

3.3.1. The weighted calculation of SCRS

Firstly, we plotted the time change diagram of the axon field φ_e under different oscillation states of 2I:2O SCT model. In the case of −C_sr = 0.8mvs and C_es = 1.4mvs, the axon field φ_e in Figure 6(a₁) presented a simple oscillation state. Similarly, −C_sr remains unchanged, C_es = 1.8mvs is set, and SWD appears in the axon field φ_e in Figure 6(b₁). Figures 6(a₁) and (b₁) correspond to simple oscillations. SWD attractors across EPN, SRN and TRN phase space were shown in Figures 6(a₂) and (b₂). It was noteworthy that the stimulus amplitudes and pulse width of the 3:2 on-off SCRS originally transmitted to neurons EPN, SRN and TRN were identical. As can be seen from Figures 6(a) and (b), the reduction of SWD was related to the change of attractor shape. Therefore, we hypothesized that EPN, SRN and TRN would have different effects with different stimulus intensity.

Figure 6. Diagram of oscillating attractors and the weighted 3:2 on-off SCRS. (a₁)–(a₂), in the 2I:2O SCT model, the firing rate φ_e waveform and the simple oscillating attractor across three neurons are shown with coupling strength −C_sr = 0.8mVs, C_es = 1.4 mVs. (b₁)–(b₂), the waveform of axon field φ_e and SWD attractor, when −C_sr = 0.8mVs, C_es = 1.8 mVs. (c), weighted regulation of 3:2 on-off SCRS amplitude and pulse width.

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We hoped to obtain 3:2 on-off SCRS with lower energy consumption on the premise of ensuring the therapeutic effect. Here, the stimulation intensity on EPN, SRN and TRN of neurons was weighted, and the weights meet following two conditions: ${w_e} + {w_s} + {w_r} = 1$ and ${w_e} \gt 0$ , ${w_s} \gt 0$ , ${w_r} \gt 0$ . We got the pulse amplitude $\left({A_0^e, A_0^s, A_0^r} \right) = {A_0}\left({{w_e}, {w_s}, {w_r}} \right)$ , the pulse width $\left({{\delta ^e}, {\delta ^s}, {\delta ^r}} \right) = \delta \left({{w_e}, {w_s}, {w_r}} \right)$ . During the simulation , 55 random points on the plane $X + Y + Z = 1$ are selected as 55 groups of random weights. The color bar in the figure represented the number of SWD grid points existing after stimulation. It can be seen from that SWD grid points may disappear with the increase of pulse weight ${w_r}$ obtained by TRN. After calculation, the energy consumption was reduced by 2/3 compared with the unweighted stimulation, and the loss of the electric stimulator was greatly reduced. However, the problem was that patients could still receive electrical stimulation during non-seizure periods, which caused unnecessary damage to the patient. In order to improve the effectiveness of stimulation, closed-loop control pulse stimulation would be applied to the model in the following chapters to achieve the regulation of stimulation.

3.3.2. Realization of closed-loop control

In fact, the patient's brain does not need to input electrical pulses all the time, and closed-loop control can effectively solve this problem. The first step was to determine the cortical firing state of epileptic patients. When the recognition result was epileptic state, the system will continuously output pulse stimulation for 60ms. As soon as the patient's brain waves returned to normal, the pulse input was stopped, which greatly reduced the damage to the patient's brain tissue. Referring to previous studies ^[28], TMFR of cortical neuron EPN could be used as a critical criterion for epileptic seizures in mean-field model simulation. Once the MFR jumped into the range of high/low TMFR, the closed-loop system determined that the network was in a state of disease and activated the stimulus device. According to the above requirements, we designed the following closed-loop control principles:

$\begin{gathered} SCR{S^{CL}}\left( e \right) = H\left( e \right) \cdot \left[ {\begin{array}{*{20}{c}} {SCR{S_e}} \\ {SCR{S_s}} \\ {SCR{S_r}} \end{array}} \right], \hfill \\ e\left( t \right) = MF{R_e} - TMF{R_{{y_d}}}. \hfill \\ \end{gathered}$

(26)

According to the above principle, we plot a closed-loop control diagram in Figure 7(a). Here, the fitting curve of the trigger average discharge rate in Figure 4(a) is used as the expected value of the closed-loop control. Real-time feedback MFR_e and expected value (TMFR_yd) were subtracted to calculate the error e(t) at this moment. Here, when designing the controller of the feedback closed-loop system, H is heaviside function, which was used to judge whether the error e(t) was positive.

Figure 7. Closed-loop system diagram and effect display. (a), the MFR_e of neuron EPN in 2I:2O SCT model is used as the real-time feedback signal of the system, and the controller allocated the stimulus to the corresponding receptors. (b), first, the red/blue lines fitted by high/low TMFR are used as the critical position of the trigger controller, then the purple circle is set as the therapeutic target, and different markers represent different stimuli. Finally, the changes of MFR_e after stimulation will be shown in the figure.

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L_1TMFR and L_2TMFR were fitting curves of high/low TMFR_e. The purple circles (•) in Figure 7(b) corresponding to −C_sr = 0.8mVs are all within the range of L_1TMFR and L_2TMFR. At this point, the brain was seen as a pathogenetic state in the system, MFR under this coupling strength was taken as the target. SCRS, 3:2 on-off SCRS, the weighted 3:2 on-off SCRS in Figure 7(b), i.e. darker markers (♦)(★)(►), suppress cortical MFR to a near-resting state. After filtration by closed-loop control system, the MFR of the corresponding stimulus increased, that was, the three brighter markers (♦)(★)(►). The results indicated that closed-loop stimulation not only keeps the MFR beyond the critical value, but also reduced the stimulator loss again. Therefore, the closed-loop control successfully realized the real-time monitoring of the patient's status without affecting the therapeutic effect.

The above results showed that whether it was open-loop or closed-loop controlled SARS, 3:2 SARS and TW-3:2 SARS could pull the state of the target back to the normal state. In this case, energy consumption was used as the criterion. Relatively speaking, the comparison of the energy consumption of several electrical stimulations is as follows: SARS > 3:2 SARS > TW-3:2 SARS, the energy consumption of closed-loop excitation was also less than that of open-loop excitation. Therefore, the weighted stimulation and closed-loop control methods were adopted to solve the redundancy in energy consumption, and TW-3:2SCRS was finally selected.

4. Conclusions

In this work, a simplified 2I:2O SCT network was constructed based on the existing BGCT model. By adjusting the coupling strength −C_sr and C_es generated firing patterns in normal and absence epileptic states. The purpose of this study was to obtain optimal control of patients' motor disorders by changing the stimulus pattern.

There are two main contributions to this work. Firstly, it improves the effectiveness of stimulation. In this paper, we selected neurons EPN, SRN and TRN as stimulation targets and considered the effect of positive/negative amplitude pulse stimulation on each of the above neurons, and proposed income constant-period m:n on-off SCR. The results show that the neural network has ideal performance when negative amplitude pulses are added to EPN and SRN, and positive amplitude pulses are added to TRN, and the stimulation is adjusted to 3:2 on-off SCRS. Secondly, a stimulus model based on the weighted method is proposed, which solves the problem of energy consumption in pulse design. Furthermore, a feedback closed-loop control system is designed to compensate for the deficiency of non-real-time open-loop control network without affecting the therapeutic efficacy. The simulation results indicate that both the proposed the weighted method and the feedback closed-loop control strategy can effectively suppress pathological oscillation and have lower energy consumption. Therefore, the results of this investigation can be concluded that the weighted 3:2 on-off SCRS, including feedback closed-loop control strategies, will be a more suitable alternative for regulating absence epileptic states in the future.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant Nos. 12172210 and 11502139). The authors would like to thank the anonymous referees for their efforts and valuable comments.

Conflict of interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

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Inhibition of absence seizures in a reduced corticothalamic circuit via closed-loop control

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Abstract

1. Introduction

2. Model and plans

3. Results and discussion

3.1. Model analysis before and after simplification

3.2. Major parameter selection of constant period m:n on-off SCRS

3.3. Continual improvement of 3:2 on-off SCRS

3.3.1. The weighted calculation of SCRS

3.3.2. Realization of closed-loop control

4. Conclusions

Acknowledgments

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Inhibition of absence seizures in a reduced corticothalamic circuit via closed-loop control

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Abstract

1. Introduction

2. Model and plans

3. Results and discussion

3.1. Model analysis before and after simplification

3.2. Major parameter selection of constant period m:n on-off SCRS

3.3. Continual improvement of 3:2 on-off SCRS

3.3.1. The weighted calculation of SCRS

3.3.2. Realization of closed-loop control

4. Conclusions

Acknowledgments

Conflict of interest

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