Loading [MathJax]/jax/output/SVG/jax.js
Research article Special Issues

The mechanism of Parkinson oscillation in the cortex: Possible evidence in a feedback model projecting from the globus pallidus to the cortex


  • The origin, location and cause of Parkinson's oscillation are not clear at present. In this paper, we establish a new cortex-basal ganglia model to study the origin mechanism of Parkinson beta oscillation. Unlike many previous models, this model includes two direct inhibitory projections from the globus pallidus external (GPe) segment to the cortex. We first obtain the critical calculation formula of Parkinson's oscillation by using the method of Quasilinear analysis. Different from previous studies, the formula obtained in this paper can include the self-feedback connection of GPe. Then, we use the bifurcation analysis method to systematically explain the influence of some key parameters on the oscillation. We find that the bifurcation principle of different cortical nuclei is different. In general, the increase of the discharge capacity of the nuclei will cause oscillation. In some special cases, the sharp reduction of the discharge rate of the nuclei will also cause oscillation. The direction of bifurcation simulation is consistent with the critical condition curve. Finally, we discuss the characteristics of oscillation amplitude. At the beginning of the oscillation, the amplitude is relatively small; with the evolution of oscillation, the amplitude will gradually strengthen. This is consistent with the experimental phenomenon. In most cases, the amplitude of cortical inhibitory nuclei (CIN) is greater than that of cortical excitatory nuclei (CEX), and the two direct inhibitory projections feedback from GPe can significantly reduce the amplitude gap between them. We calculate the main frequency of the oscillation generated in this model, which basically falls between 13 and 30 Hz, belonging to the typical beta frequency band oscillation. Some new results obtained in this paper can help to better understand the origin mechanism of Parkinson's disease and have guiding significance for the development of experiments.

    Citation: Minbo Xu, Bing Hu, Weiting Zhou, Zhizhi Wang, Luyao Zhu, Jiahui Lin, Dingjiang Wang. The mechanism of Parkinson oscillation in the cortex: Possible evidence in a feedback model projecting from the globus pallidus to the cortex[J]. Mathematical Biosciences and Engineering, 2023, 20(4): 6517-6550. doi: 10.3934/mbe.2023281

    Related Papers:

    [1] Honglei Wang, Wenliang Zeng, Xiaoling Huang, Zhaoyang Liu, Yanjing Sun, Lin Zhang . MTTLm6A: A multi-task transfer learning approach for base-resolution mRNA m6A site prediction based on an improved transformer. Mathematical Biosciences and Engineering, 2024, 21(1): 272-299. doi: 10.3934/mbe.2024013
    [2] Shanzheng Wang, Xinhui Xie, Chao Li, Jun Jia, Changhong Chen . Integrative network analysis of N6 methylation-related genes reveal potential therapeutic targets for spinal cord injury. Mathematical Biosciences and Engineering, 2021, 18(6): 8174-8187. doi: 10.3934/mbe.2021405
    [3] Pingping Sun, Yongbing Chen, Bo Liu, Yanxin Gao, Ye Han, Fei He, Jinchao Ji . DeepMRMP: A new predictor for multiple types of RNA modification sites using deep learning. Mathematical Biosciences and Engineering, 2019, 16(6): 6231-6241. doi: 10.3934/mbe.2019310
    [4] Yong Zhu, Zhipeng Jiang, Xiaohui Mo, Bo Zhang, Abdullah Al-Dhelaan, Fahad Al-Dhelaan . A study on the design methodology of TAC3 for edge computing. Mathematical Biosciences and Engineering, 2020, 17(5): 4406-4421. doi: 10.3934/mbe.2020243
    [5] Atefeh Afsar, Filipe Martins, Bruno M. P. M. Oliveira, Alberto A. Pinto . A fit of CD4+ T cell immune response to an infection by lymphocytic choriomeningitis virus. Mathematical Biosciences and Engineering, 2019, 16(6): 7009-7021. doi: 10.3934/mbe.2019352
    [6] Tamás Tekeli, Attila Dénes, Gergely Röst . Adaptive group testing in a compartmental model of COVID-19*. Mathematical Biosciences and Engineering, 2022, 19(11): 11018-11033. doi: 10.3934/mbe.2022513
    [7] Wenli Cheng, Jiajia Jiao . An adversarially consensus model of augmented unlabeled data for cardiac image segmentation (CAU+). Mathematical Biosciences and Engineering, 2023, 20(8): 13521-13541. doi: 10.3934/mbe.2023603
    [8] Tongmeng Jiang, Pan Jin, Guoxiu Huang, Shi-Cheng Li . The function of guanylate binding protein 3 (GBP3) in human cancers by pan-cancer bioinformatics. Mathematical Biosciences and Engineering, 2023, 20(5): 9511-9529. doi: 10.3934/mbe.2023418
    [9] Xin Yu, Jun Liu, Ruiwen Xie, Mengling Chang, Bichun Xu, Yangqing Zhu, Yuancai Xie, Shengli Yang . Construction of a prognostic model for lung squamous cell carcinoma based on seven N6-methylandenosine-related autophagy genes. Mathematical Biosciences and Engineering, 2021, 18(5): 6709-6723. doi: 10.3934/mbe.2021333
    [10] Tahir Rasheed, Faran Nabeel, Muhammad Bilal, Yuping Zhao, Muhammad Adeel, Hafiz. M. N. Iqbal . Aqueous monitoring of toxic mercury through a rhodamine-based fluorescent sensor. Mathematical Biosciences and Engineering, 2019, 16(4): 1861-1873. doi: 10.3934/mbe.2019090
  • The origin, location and cause of Parkinson's oscillation are not clear at present. In this paper, we establish a new cortex-basal ganglia model to study the origin mechanism of Parkinson beta oscillation. Unlike many previous models, this model includes two direct inhibitory projections from the globus pallidus external (GPe) segment to the cortex. We first obtain the critical calculation formula of Parkinson's oscillation by using the method of Quasilinear analysis. Different from previous studies, the formula obtained in this paper can include the self-feedback connection of GPe. Then, we use the bifurcation analysis method to systematically explain the influence of some key parameters on the oscillation. We find that the bifurcation principle of different cortical nuclei is different. In general, the increase of the discharge capacity of the nuclei will cause oscillation. In some special cases, the sharp reduction of the discharge rate of the nuclei will also cause oscillation. The direction of bifurcation simulation is consistent with the critical condition curve. Finally, we discuss the characteristics of oscillation amplitude. At the beginning of the oscillation, the amplitude is relatively small; with the evolution of oscillation, the amplitude will gradually strengthen. This is consistent with the experimental phenomenon. In most cases, the amplitude of cortical inhibitory nuclei (CIN) is greater than that of cortical excitatory nuclei (CEX), and the two direct inhibitory projections feedback from GPe can significantly reduce the amplitude gap between them. We calculate the main frequency of the oscillation generated in this model, which basically falls between 13 and 30 Hz, belonging to the typical beta frequency band oscillation. Some new results obtained in this paper can help to better understand the origin mechanism of Parkinson's disease and have guiding significance for the development of experiments.



    The constituent members in a system mainly found in nature can be interacting with each other through cooperation and competition. Demonstrations for such systems involve biological species, countries, businesses, and many more. It's very much intriguing to investigate in a comprehensive manner numerous social as well as biological interactions existent in dissimilar species/entities utilizing mathematical modeling. The predation and the competition species are the most famous interactions among all such types of interactions. Importantly, Lotka [1] and Volterra [2] in the 1920s have announced individually the classic equations portraying population dynamics. Such illustrious equations are notably described as predator-prey (PP) equations or Lotka-Volterra (LV) equations. In this structure, PP/LV model represents the most influential model for interacting populations. The interplay between prey and predator together with additional factors has been a prominent topic in mathematical ecology for a long period. Arneodo et al. [3] have established in 1980 that a generalized Lotka-Volterra biological system (GLVBS) would depict chaos phenomena in an ecosystem for some explicitly selected system parameters and initial conditions. Additionally, Samardzija and Greller [4] demonstrated in 1988 that GLVBS would procure chaotic reign from the stabled state via rising fractal torus. LV model was initially developed as a biological concept, yet it is utilized in enormous diversified branches for research [5,6,7,8]. Synchronization essentially is a methodology of having different chaotic systems (non-identical or identical) following exactly a similar trajectory, i.e., the dynamical attributes of the slave system are locked finally into the master system. Specifically, synchronization and control have a wide spectrum for applications in engineering and science, namely, secure communication [9], encryption [10,11], ecological model [12], robotics [13], neural network [14], etc. Recently, numerous types of secure communication approaches have been explored [15,16,17,18] such as chaos modulation [18,19,20,21], chaos shift keying [22,23] and chaos masking [9,17,20,24]. In chaos communication schemes, the typical key idea for transmitting a message through chaotic/hyperchaotic models is that a message signal is nested in the transmitter system/model which originates a chaotic/ disturbed signal. Afterwards, this disturbed signal has been emitted to the receiver through a universal channel. The message signal would finally be recovered by the receiver. A chaotic model has been intrinsically employed both as receiver and transmitter. Consequently, this area of chaotic synchronization & control has sought remarkable considerations among differential research fields.

    Most prominently, synchronization theory has been in existence for over 30 years due to the phenomenal research of Pecora and Carroll [25] established in 1990 using drive-response/master-slave/leader-follower configuration. Consequently, many authors and researchers have started introducing and studying numerous control and synchronization methods [9,26,27,28,29,30,31,32,33,34,35,36] etc. to achieve stabilized chaotic systems for possessing stability. In [37], researchers discussed optimal synchronization issues in similar GLVBSs via optimal control methodology. In [38,39], the researchers studied the adaptive control method (ACM) to synchronize chaotic GLVBSs. Also, researchers [40] introduced a combination difference anti-synchronization scheme in similar chaotic GLVBSs via ACM. In addition, authors [41] investigated a combination synchronization scheme to control chaos existing in GLVBSs using active control strategy (ACS). Bai and Lonngren [42] first proposed ACS in 1997 for synchronizing and controlling chaos found in nonlinear dynamical systems. Furthermore, compound synchronization using ACS was first advocated by Sun et al. [43] in 2013. In [44], authors discussed compound difference anti-synchronization scheme in four chaotic systems out of which two chaotic systems are considered as GLVBSs using ACS and ACM along with applications in secure communications of chaos masking type in 2019. Some further research works [45,46] based on ACS have been reported in this direction. The considered chaotic GLVBS offers a generalization that allows higher-order biological terms. As a result, it may be of interest in cases where biological systems experience cataclysmic changes. Unfortunately, some species will be under competitive pressure in the coming years and decades. This work may be comprised as a step toward preserving as many currently living species as possible by using the proposed synchronization approach which is based on master-slave configuration and Lyapunov stability analysis.

    In consideration of the aforementioned discussions and observations, our primary focus here is to develop a systematic approach for investigating compound difference anti-synchronization (CDAS) approach in 4 similar chaotic GLVBSs via ACS. The considered ACS is a very efficient yet theoretically rigorous approach for controlling chaos found in GLVBSs. Additionally, in view of widely known Lyapunov stability analysis (LSA) [47], we discuss actively designed biological control law & convergence for synchronization errors to attain CDAS synchronized states.

    The major attributes for our proposed research in the present manuscript are:

    ● The proposed CDAS methodology considers four chaotic GLVBSs.

    ● It outlines a robust CDAS approach based active controller to achieve compound difference anti-synchronization in discussed GLVBSs & conducts oscillation in synchronization errors along with extremely fast convergence.

    ● The construction of the active control inputs has been executed in a much simplified fashion utilizing LSA & master-salve/ drive-response configuration.

    ● The proposed CDAS approach in four identical chaotic GLVBSs of integer order utilizing ACS has not yet been analyzed up to now. This depicts the novelty of our proposed research work.

    This manuscript is outlined as follows: Section 2 presents the problem formulation of the CDAS scheme. Section 3 designs comprehensively the CDAS scheme using ACS. Section 4 consists of a few structural characteristics of considered GLVBS on which CDAS is investigated. Furthermore, the proper active controllers having nonlinear terms are designed to achieve the proposed CDAS strategy. Moreover, in view of Lyapunov's stability analysis (LSA), we have examined comprehensively the biological controlling laws for achieving global asymptotical stability of the error dynamics for the discussed model. In Section 5, numerical simulations through MATLAB are performed for the illustration of the efficacy and superiority of the given scheme. Lastly, we also have presented some conclusions and the future prospects of the discussed research work in Section 6.

    We here formulate a methodology to examine compound difference anti-synchronization (CDAS) scheme viewing master-slave framework in four chaotic systems which would be utilized in the coming up sections.

    Let the scaling master system be

    ˙wm1= f1(wm1), (2.1)

    and the base second master systems be

    ˙wm2= f2(wm2), (2.2)
    ˙wm3= f3(wm3). (2.3)

    Corresponding to the aforementioned master systems, let the slave system be

    ˙ws4= f4(ws4)+U(wm1,wm2,wm3,ws4), (2.4)

    where wm1=(wm11,wm12,...,wm1n)TRn, wm2=(wm21,wm22,...,wm2n)TRn, wm3=(wm31,wm32,...,wm3n)TRn, ws4=(ws41,ws42,...,ws4n)TRn are the state variables of the respective chaotic systems (2.1)–(2.4), f1,f2,f3,f4:RnRn are four continuous vector functions, U=(U1,U2,...,Un)T:Rn×Rn×Rn×RnRn are appropriately constructed active controllers.

    Compound difference anti-synchronization error (CDAS) is defined as

    E=Sws4+Pwm1(Rwm3Qwm2),

    where P=diag(p1,p2,.....,pn),Q=diag(q1,q2,.....,qn),R=diag(r1,r2,.....,rn),S=diag(s1,s2,.....,sn) and S0.

    Definition: The master chaotic systems (2.1)–(2.3) are said to achieve CDAS with slave chaotic system (2.4) if

    limtE(t)=limtSws4(t)+Pwm1(t)(Rwm3(t)Qwm2(t))=0.

    We now present our proposed CDAS approach in three master systems (2.1)–(2.3) and one slave system (2.4). We next construct the controllers based on CDAS approach by

    Ui= ηisi(f4)iKiEisi, (3.1)

    where ηi=pi(f1)i(riwm3iqiwm2i)+piwm1i(ri(f3)iqi(f2)i), for i=1,2,...,n.

    Theorem: The systems (2.1)–(2.4) will attain the investigated CDAS approach globally and asymptotically if the active control functions are constructed in accordance with (3.1).

    Proof. Considering the error as

    Ei= siws4i+piwm1i(riwm3iqiwm2i),fori=1,2,3,.....,n.

    Error dynamical system takes the form

    ˙Ei= si˙ws4i+pi˙wm1i(riwm3iqiwm2i)+piwm1i(ri˙wm3iqi˙wm2i)= si((f4)i+Ui)+pi(f1)i(riwm3iqiwm2i)+piwm1i(ri(f3)iqi(f2)i)= si((f4)i+Ui)+ηi,

    where ηi=pi(f1)i(riwm3iqiwm2i)+piwm1i(ri(f3)iqi(f2)i), i=1,2,3,....,n. This implies that

    ˙Ei= si((f4)iηisi(f4)iKiEisi)+ηi= KiEi (3.2)

    The classic Lyapunov function V(E(t)) is described by

    V(E(t))= 12ETE= 12ΣE2i

    Differentiation of V(E(t)) gives

    ˙V(E(t))=ΣEi˙Ei

    Using Eq (3.2), one finds that

    ˙V(E(t))=ΣEi(KiEi)= ΣKiE2i). (3.3)

    An appropriate selection of (K1,K1,.......,Kn) makes ˙V(E(t)) of eq (3.3), a negative definite. Consequently, by LSA [47], we obtain

    limtEi(t)=0,(i=1,2,3).

    Hence, the master systems (2.1)–(2.3) and slave system (2.4) have attained desired CDAS strategy.

    We now describe GLVBS as the scaling master system:

    {˙wm11=wm11wm11wm12+b3w2m11b1w2m11wm13,˙wm12=wm12+wm11wm12,˙wm13=b2wm13+b1w2m11wm13, (4.1)

    where (wm11,wm12,wm13)TR3 is state vector of (4.1). Also, wm11 represents the prey population and wm12, wm13 denote the predator populations. For parameters b1=2.9851, b2=3, b3=2 and initial conditions (27.5,23.1,11.4), scaling master GLVBS displays chaotic/disturbed behaviour as depicted in Figure 1(a).

    Figure 1.  Phase graphs of chaotic GLVBS. (a) wm11wm12wm13 space, (b) wm21wm22wm23 space, (c) wm31wm32wm33 space, (d) ws41ws42ws43 space.

    The base master systems are the identical chaotic GLVBSs prescribed respectively as:

    {˙wm21=wm21wm21wm22+b3w2m21b1w2m21wm23,˙wm22=wm22+wm21wm22,˙wm23=b2wm23+b1w2m21wm23, (4.2)

    where (wm21,wm22,wm23)TR3 is state vector of (4.2). For parameter values b1=2.9851, b2=3, b3=2, this base master GLVBS shows chaotic/disturbed behaviour for initial conditions (1.2,1.2,1.2) as displayed in Figure 1(b).

    {˙wm31=wm31wm31wm32+b3w2m31b1w2m31wm33,˙wm32=wm32+wm31wm32,˙wm33=b2wm33+b1w2m31wm33, (4.3)

    where (wm31,wm32,wm33)TR3 is state vector of (4.3). For parameters b1=2.9851, b2=3, b3=2, this second base master GLVBS displays chaotic/disturbed behaviour for initial conditions (2.9,12.8,20.3) as shown in Figure 1(c).

    The slave system, represented by similar GLVBS, is presented by

    {˙ws41=ws41ws41ws42+b3w2s41b1w2s41ws43+U1,˙ws42=ws42+ws41ws42+U2,˙ws43=b2ws43+b1w2s41ws43+U3, (4.4)

    where (ws41,ws42,ws43)TR3 is state vector of (4.4). For parameter values, b1=2.9851, b2=3, b3=2 and initial conditions (5.1,7.4,20.8), the slave GLVBS exhibits chaotic/disturbed behaviour as mentioned in Figure 1(d).

    Moreover, the detailed theoretical study for (4.1)–(4.4) can be found in [4]. Further, U1, U2 and U3 are controllers to be determined.

    Next, the CDAS technique has been discussed for synchronizing the states of chaotic GLVBS. Also, LSA-based ACS is explored & the necessary stability criterion is established.

    Here, we assume P=diag(p1,p2,p3), Q=diag(q1,q2,q3), R=diag(r1,r2,r3), S=diag(s1,s2,s3). The scaling factors pi,qi,ri,si for i=1,2,3 are selected as required and can assume the same or different values.

    The error functions (E1,E2,E3) are defined as:

    {E1=s1ws41+p1wm11(r1wm31q1wm21),E2=s2ws42+p2wm12(r2wm32q2wm22),E3=s3ws43+p3wm13(r3wm33q3wm23). (4.5)

    The major objective of the given work is the designing of active control functions Ui,(i=1,2,3) ensuring that the error functions represented in (4.5) must satisfy

    limtEi(t)=0for(i=1,2,3).

    Therefore, subsequent error dynamics become

    {˙E1=s1˙ws41+p1˙wm11(r1wm31q1wm21)+p1wm11(r1˙wm31q1˙wm21),˙E2=s2˙ws42+p2˙wm12(r2wm32q2wm22)+p2wm12(r2˙wm32q2˙wm22),˙E3=s3˙ws43+p3˙wm13(r3wm33q3wm23)+p3wm13(r3˙wm33q3˙wm23). (4.6)

    Using (4.1), (4.2), (4.3), and (4.5) in (4.6), the error dynamics simplifies to

    {˙E1=s1(ws41ws41ws42+b3w2s41b1w2s41ws43+U1)+p1(wm11wm11wm12+b3w2m11b1w2m11wm13)(r1wm31q1wm21)+p1wm11(r1(wm31wm31wm32+b3w2m31b1w2m31wm33)q1(wm21wm21wm22+b3w2m21b1w2m21wm23),˙E2=s2(ws42+ws41ws42+U2)+p2(wm12+wm11wm12)(r2wm32q2wm22)+p2wm12(r2(wm32+wm31wm32)q2(wm22+wm21wm22)),˙E3=s3(b2ws43+b1w2s41ws43+U3)+p3(b2wm13+b1w2m11wm13)(r3wm33q3wm23)+p3wm13(r3(b2wm33+b1w2m31wm33)q3(b2wm23+b1w2m21wm23)). (4.7)

    Let us now choose the active controllers:

    U1= η1s1(f4)1K1E1s1, (4.8)

    where η1=p1(f1)1(r1wm31q1wm21)+p1wm11(r1(f3)1q1(f2)1), as described in (3.1).

    U2= η2s2(f4)2K2E2s2, (4.9)

    where η2=p2(f1)2(r2wm32q2wm22)+p2wm12(r2(f3)2q2(f2)2).

    U3= η3s3(f4)3K3E3s3, (4.10)

    where η3=p3(f1)3(r3wm33q3wm23)+p3wm13(r3(f3)3q3(f2)3) and K1>0,K2>0,K3>0 are gaining constants.

    By substituting the controllers (4.8), (4.9) and (4.10) in (4.7), we obtain

    {˙E1=K1E1,˙E2=K2E2,˙E3=K3E3. (4.11)

    Lyapunov function V(E(t)) is now described by

    V(E(t))= 12[E21+E22+E23]. (4.12)

    Obviously, the Lyapunov function V(E(t)) is +ve definite in R3. Therefore, the derivative of V(E(t)) as given in (4.12) can be formulated as:

    ˙V(E(t))= E1˙E1+E2˙E2+E3˙E3. (4.13)

    Using (4.11) in (4.13), one finds that

    ˙V(E(t))= K1E21K2E22K3E23<0,

    which displays that ˙V(E(t)) is -ve definite.

    In view of LSA [47], we, therefore, understand that CDAS error dynamics is globally as well as asymptotically stable, i.e., CDAS error E(t)0 asymptotically for t to each initial value E(0)R3.

    This section conducts a few simulation results for illustrating the efficacy of the investigated CDAS scheme in identical chaotic GLVBSs using ACS. We use 4th order Runge-Kutta algorithm for solving the considered ordinary differential equations. Initial conditions for three master systems (4.1)–(4.3) and slave system (4.4) are (27.5,23.1,11.4), (1.2,1.2,1.2), (2.9,12.8,20.3) and (14.5,3.4,10.1) respectively. We attain the CDAS technique among three masters (4.1)–(4.3) and corresponding one slave system (4.4) by taking pi=qi=ri=si=1, which implies that the slave system would be entirely anti-synchronized with the compound of three master models for i=1,2,3. In addition, the control gains (K1,K2,K3) are taken as 2. Also, Figure 2(a)(c) indicates the CDAS synchronized trajectories of three master (4.1)–(4.3) & one slave system (4.4) respectively. Moreover, synchronization error functions (E1,E2,E3)=(51.85,275.36,238.54) approach 0 as t tends to infinity which is exhibited via Figure 2(d). Hence, the proposed CDAS strategy in three masters and one slave models/systems has been demonstrated computationally.

    Figure 2.  CDAS synchronized trajectories of GLVBS between (a) ws41(t) and wm11(t)(wm31(t)wm21(t)), (b) ws42(t) and wm12(t)(wm32(t)wm22(t)), (c) ws43(t) and wm13(t)(wm23(t)wm13(t)), (d) CDAS synchronized errors.

    In this work, the investigated CDAS approach in similar four chaotic GLVBSs using ACS has been analyzed. Lyapunov's stability analysis has been used to construct proper active nonlinear controllers. The considered error system, on the evolution of time, converges to zero globally & asymptotically via our appropriately designed simple active controllers. Additionally, numerical simulations via MATLAB suggest that the newly described nonlinear control functions are immensely efficient in synchronizing the chaotic regime found in GLVBSs to fitting set points which exhibit the efficacy and supremacy of our proposed CDAS strategy. Exceptionally, both analytic theory and computational results are in complete agreement. Our proposed approach is simple yet analytically precise. The control and synchronization among the complex GLVBSs with the complex dynamical network would be an open research problem. Also, in this direction, we may extend the considered CDAS technique on chaotic systems that interfered with model uncertainties as well as external disturbances.

    The authors gratefully acknowledge Qassim University, represented by the Deanship of Scientific Research, on the financial support for this research under the number 10163-qec-2020-1-3-I during the academic year 1441 AH/2020 AD.

    The authors declare there is no conflict of interest.



    [1] E. Wressle, C. Engstrand, A. K. Granérus, Living with Parkinson's disease: elderly patients' and relatives' perspective on daily living, Aust. Occup. Ther. J., 54 (2007), 131–139. https://doi.org/10.1111/j.1440-1630.2006.00610.x doi: 10.1111/j.1440-1630.2006.00610.x
    [2] P. Mahlknecht, A. Gasperi, P. Willeit, S. Kiechl, H. Stockner, J. Willeit, et al., Prodromal Parkinson's disease as defined per MDS research criteria in the general elderly community, Mov. Disord., 31 (2016), 1405–1408. https://doi.org/10.1002/mds.26674 doi: 10.1002/mds.26674
    [3] M. Politis, K. Wu, S. Molloy, P. G. Bain, K. R. Chaudhuri, P. Piccini, Parkinson's disease symptoms: the patient's perspective, Mov. Disord., 25 (2010), 1646–1651. https://doi.org/10.1002/mds.23135 doi: 10.1002/mds.23135
    [4] C. Váradi, Clinical features of Parkinson's disease: the evolution of critical symptoms, Biology, 9 (2020), 103. https://doi.org/10.3390/biology9050103 doi: 10.3390/biology9050103
    [5] D. J. Surmeier, Determinants of dopaminergic neuron loss in Parkinson's disease, FEBS J., 285 (2018), 3657–3668. https://doi.org/10.1111/febs.14607 doi: 10.1111/febs.14607
    [6] C. Raza, R. Anjum, Parkinson's disease: Mechanisms, translational models and management strategies, Life Sci., 226 (2019), 77–90. https://doi.org/10.1016/j.lfs.2019.03.057 doi: 10.1016/j.lfs.2019.03.057
    [7] A. B. Holt, E. Kormann, A. Gulberti, M. Pötter-Nerger, C. G. McNamara, H. Cagnan, et al., Phase-dependent suppression of beta oscillations in Parkinson's disease patients, J. Neurosci., 39 (2019), 1119–1134. https://doi.org/10.1523/JNEUROSCI.1913-18.2018 doi: 10.1523/JNEUROSCI.1913-18.2018
    [8] A. Singh, R. C. Cole, A. I. Espinoza, D. Brown, J. F. Cavanagh, N. S. Narayanana, Frontal theta and beta oscillations during lower-limb movement in Parkinson's disease, Clin. Neurophysiol., 131 (2020), 694-702. https://doi.org/10.1016/j.clinph.2019.12.399 doi: 10.1016/j.clinph.2019.12.399
    [9] M. H. Trager, M. M. Koop, A. Velisar, Z. Blumenfeld, J. S. Nikolau, E. J. Quinn, et al., Subthalamic beta oscillations are attenuated after withdrawal of chronic high frequency neurostimulation in Parkinson's disease, Neurobiol. Dis., 96 (2016), 22–30. https://doi.org/10.1016/j.nbd.2016.08.003 doi: 10.1016/j.nbd.2016.08.003
    [10] C. Hammond, H. Bergman, P. Brown, Pathological synchronization in Parkinson's disease: networks, models and treatments, Trends Neurosci., 30 (2007), 357–364. https://doi.org/10.1016/j.tins.2007.05.004 doi: 10.1016/j.tins.2007.05.004
    [11] Z. Wang, B. Hu, W. Zhou, M. Xu, D. Wang, Hopf bifurcation mechanism analysis in an improved cortex-basal ganglia network with distributed delays: An application to Parkinson's disease, Chaos, Solitons Fractals, 166 (2023), 113022. https://doi.org/10.1016/j.chaos.2022.113022 doi: 10.1016/j.chaos.2022.113022
    [12] B. Hu, X. Diao, H. Guo, et al., The beta oscillation conditions in a simplified basal ganglia network, Cogn Neurodynamics, 13(2019), 201-217. https://doi.org/10.1007/s11571-018-9514-0 doi: 10.1007/s11571-018-9514-0
    [13] A. B. Holt, T. I. Netoff, Origins and suppression of oscillations in a computational model of Parkinson's disease, J. Comput. Neurosci., 37 (2014), 505–521. https://doi.org/10.1007/s10827-014-0523-7 doi: 10.1007/s10827-014-0523-7
    [14] A. B. Holt, E. Kormann, A. Gulberti, M. Pötter-Nerger, C. G. McNamara, H. Cagnan, et al., Phase-dependent suppression of beta oscillations in Parkinson's disease patients, J. Neurosci., 39 (2019), 1119–1134. https://doi.org/10.1523/JNEUROSCI.1913-18.2018 doi: 10.1523/JNEUROSCI.1913-18.2018
    [15] L. L. Grado, M. D. Johnson, T. I. Netoff, Bayesian adaptive dual control of deep brain stimulation in a computational model of Parkinson's disease, PLoS Comput. Biol., 14 (2018), e1006606. https://doi.org/10.1371/journal.pcbi.1006606 doi: 10.1371/journal.pcbi.1006606
    [16] J. E. Fleming, E. Dunn, M. M. Lowery, Simulation of closed-loop deep brain stimulation control schemes for suppression of pathological beta oscillations in Parkinson's disease, Front. Neurosci., 14 (2020), 166. https://doi.org/10.3389/fnins.2020.00166 doi: 10.3389/fnins.2020.00166
    [17] A. B. Holt, D. Wilson, M. Shinn, J. Moehlis, T. I. Netoff, Phasic burst stimulation: a closed-loop approach to tuning deep brain stimulation parameters for Parkinson's disease, PLoS Comput. Biol., 12 (2016), e1005011. https://doi.org/10.1371/journal.pcbi.1005011 doi: 10.1371/journal.pcbi.1005011
    [18] K. Kumaravelu, D. T. Brocker, W. M. Grill, A biophysical model of the cortex-basal ganglia-thalamus network in the 6-OHDA lesioned rat model of Parkinson's disease, J. Comput. Neurosci., 40 (2016), 207–229. https://doi.org/10.1007/s10827-016-0593-9 doi: 10.1007/s10827-016-0593-9
    [19] A. J. N. Holgado, J. R. Terry, R. Bogacz, Conditions for the generation of beta oscillations in the subthalamic nucleus-globus pallidus network, J. Neurosci., 30 (2010), 12340–12352. https://doi.org/10.1523/JNEUROSCI.0817-10.2010 doi: 10.1523/JNEUROSCI.0817-10.2010
    [20] A. Pavlides, S. J. Hogan, R. Bogacz, Improved conditions for the generation of beta oscillations in the subthalamic nucleus-globus pallidus network, Eur. J. Neurosci., 36 (2012), 2229–2239. https://doi.org/10.1111/j.1460-9568.2012.08105.x doi: 10.1111/j.1460-9568.2012.08105.x
    [21] Gillies, D. Willshaw, Z. Li, Subthalamic-pallidal interactions are critical in determining normal and abnormal functioning of the basal ganglia, Proc. R. Soc. London, Ser. B, 269 (2002), 545–551. https://doi.org/10.1098/rspb.2001.1817 doi: 10.1098/rspb.2001.1817
    [22] J. E. Rubin, D. Terman, High frequency stimulation of the subthalamic nucleus eliminates pathological thalamic rhythmicity in a computational model, J. Comput. Neurosci., 16 (2004), 211–235. https://doi.org/10.1023/B:JCNS.0000025686.47117.67 doi: 10.1023/B:JCNS.0000025686.47117.67
    [23] D. Terman, J. E. Rubin, A. C. Yew, C. J. Wilson, Activity patterns in a model for the subthalamopallidal network of the basal ganglia, J. Neurosci., 22 (2002), 2963–2976. https://doi.org/10.1523/JNEUROSCI.22-07-02963.2002 doi: 10.1523/JNEUROSCI.22-07-02963.2002
    [24] B. Hu, M. Xu, L. Zhu, J. Lin, Z. Wang, D. Wang, et al., A bidirectional Hopf bifurcation analysis of Parkinson's oscillation in a simplified basal ganglia model, J. Theor. Biol., 536 (2022), 110979. https://doi.org/10.1016/j.jtbi.2021.110979 doi: 10.1016/j.jtbi.2021.110979
    [25] S. R. Cole, R. van der Meij, E. J. Peterson, C. de Hemptinne, P. A. Starr, B. Voytek, Nonsinusoidal beta oscillations reflect cortical pathophysiology in Parkinson's disease, J. Neurosci., 37 (2017), 4830–4840. https://doi.org/10.1523/JNEUROSCI.2208-16.2017 doi: 10.1523/JNEUROSCI.2208-16.2017
    [26] B. Pollok, V. Krause, W. Martsch, C. Wach, A. Schnitzler, M. Südmeyer, Motor‐cortical oscillations in early stages of Parkinson's disease, J.Physiol., 590 (2012), 3203–3212. https://doi.org/10.1113/jphysiol.2012.231316 doi: 10.1113/jphysiol.2012.231316
    [27] S. J. van Albada, P. A. Robinson, Mean-field modeling of the basal ganglia-thalamocortical system. I: Firing rates in healthy and parkinsonian states, J. Theor. Biol., 257 (2009), 642–663. https://doi.org/10.1016/j.jtbi.2008.12.018 doi: 10.1016/j.jtbi.2008.12.018
    [28] S. J. van Albada, R. T. Gray, P. M. Drysdale, P. A. Robinson, Mean-field modeling of the basal ganglia-thalamocortical system. Ⅱ: dynamics of parkinsonian oscillations, J. Theor. Biol., 257 (2009), 664–688. https://doi.org/10.1016/j.jtbi.2008.12.013 doi: 10.1016/j.jtbi.2008.12.013
    [29] G. W. Arbuthnott, M. Garcia-Munoz, Are the symptoms of parkinsonism cortical in origin?, Comput. Struct. Biotechnol. J., 15 (2017), 21–25. https://doi.org/10.1016/j.csbj.2016.10.006 doi: 10.1016/j.csbj.2016.10.006
    [30] C. F. Underwood, L. C. Parr-Brownlie, Primary motor cortex in Parkinson's disease: Functional changes and opportunities for neurostimulation, Neurobiol. Dis., 147 (2021), 105159. https://doi.org/10.1016/j.nbd.2020.105159 doi: 10.1016/j.nbd.2020.105159
    [31] M. D. Bevan, P. J. Magill, D. Terman, J. P. Bolam, C. J Wilson, Move to the rhythm: oscillations in the subthalamic nucleus-external globus pallidus network, Trends Neurosci., 25 (2002), 525–531. https://doi.org/10.1016/S0166-2236(02)02235-X doi: 10.1016/S0166-2236(02)02235-X
    [32] A. Pavlides, S. J. Hogan, R. Bogacz, Computational models describing possible mechanisms for generation of excessive beta oscillations in Parkinson's disease, PLoS Comput. Biol., 11 (2015), e1004609. https://doi.org/10.1371/journal.pcbi.1004609 doi: 10.1371/journal.pcbi.1004609
    [33] Y. Chen, J. Wang, Y. Kang, M. B. Ghori, Emergence of beta oscillations of a resonance model for Parkinson's disease, Neural Plast., 2020 (2020), 1–15. https://doi.org/10.1155/2020/8824760 doi: 10.1155/2020/8824760
    [34] M. M. McGregor, A. B. Nelson, Circuit mechanisms of Parkinson's disease, Neuron, 101 (2019), 1042–1056. https://doi.org/10.1016/j.neuron.2019.03.004 doi: 10.1016/j.neuron.2019.03.004
    [35] M. D. Humphries, J. A. Obeso, J. K. Dreyer, Insights into Parkinson's disease from computational models of the basal ganglia, J. Neurol., Neurosurg. Psychiatry, 89 (2018), 1181–1188. https://doi.org/10.1136/jnnp-2017-315922 doi: 10.1136/jnnp-2017-315922
    [36] B. C. M. van Wijk, H. Cagnan, V. Litvak, V. Litvak, A. A. Kühn, K. J. Friston, et al., Generic dynamic causal modelling: An illustrative application to Parkinson's disease, NeuroImage, 181 (2018), 818–830. https://doi.org/10.1016/j.neuroimage.2018.08.039 doi: 10.1016/j.neuroimage.2018.08.039
    [37] M. C. Chen, L. Ferrari, M. D. Sacchet, L. C. Foland-Ross, M. Qiu, I. H. Gotlib, et al., Identification of a direct GABA ergic pallidocortical pathway in rodents, Eur. J. Neurosci., 41 (2015), 748–759. https://doi.org/10.1111/ejn.12822 doi: 10.1111/ejn.12822
    [38] A. Saunders, I. A. Oldenburg, V. K. Berezovskii, C. A. Johnson, N. D. Kingery, H. L. Elliott, et al., A direct GABAergic output from the basal ganglia to frontal cortex, Nature, 521 (2015), 85–89. https://doi.org/10.1038/nature14179 doi: 10.1038/nature14179
    [39] P. R. Castillo, E. H. Middlebrooks, S. S. Grewal, L. Okromelidze, J. F. Meschia, A. Quinones-Hinojosa, et al., Globus pallidus externus deep brain stimulation treats insomnia in a patient with Parkinson disease, in Mayo Clinic Proceedings, Elsevier, 95 (2020), 419–422. https://doi.org/10.1016/j.mayocp.2019.11.020
    [40] J. Dong, S. Hawes, J. Wu, W. Le, H. Cai, Connectivity and functionality of the globus pallidus externa under normal conditions and Parkinson's disease, Front. Neural Circuits, 15 (2021), 8. https://doi.org/10.3389/fncir.2021.645287 doi: 10.3389/fncir.2021.645287
    [41] T. Tsuboi, M. Charbel, D. T. Peterside, M. Rana, A. Elkouzi, W. Deeb, et al., Pallidal connectivity profiling of stimulation-induced dyskinesia in Parkinson's disease, Mov. Disord., 36 (2021), 380–388. https://doi.org/10.1002/mds.28324 doi: 10.1002/mds.28324
    [42] R. G. Burciu, D. E. Vaillancourt, Imaging of motor cortex physiology in Parkinson's disease, Mov. Disord., 33 (2018), 1688–1699. https://doi.org/10.1002/mds.102 doi: 10.1002/mds.102
    [43] G. Foffani, J. A. Obeso, A cortical pathogenic theory of Parkinson's disease, Neuron, 99 (2018), 1116–1128. https://doi.org/10.1016/j.neuron.2018.07.028 doi: 10.1016/j.neuron.2018.07.028
    [44] A. Guerra, D. Colella, M. Giangrosso, A. Cannavacciuolo, G. Paparella, G. Fabbrini, et al., Driving motor cortex oscillations modulates bradykinesia in Parkinson's disease, Brain, 145 (2022), 224–236. https://doi.org/10.1093/brain/awab257 doi: 10.1093/brain/awab257
    [45] Z. Wang, B. Hu, L. Zhu, J, Lin, M. Xu, D. Wang, Hopf bifurcation analysis for Parkinson oscillation with heterogeneous delays: A theoretical derivation and simulation analysis, Commun. Nonlinear. Sci., 114 (2022), 106614. https://doi.org/10.1016/j.cnsns.2022.106614 doi: 10.1016/j.cnsns.2022.106614
    [46] T. P. Vogels, K. Rajan, L. F. Abbott, Neural network dynamics, Annu. Rev. Neurosci., 28 (2005), 357–376. https://doi.org/10.1146/annurev.neuro.28.061604.135637 doi: 10.1146/annurev.neuro.28.061604.135637
    [47] H. Kita, Y. Tachibana, A. Nambu, S. Chiken, Balance of monosynaptic excitatory and disynaptic inhibitory responses of the globus pallidus induced after stimulation of the subthalamic nucleus in the monkey, J. Neurosci., 25 (2005), 8611–8619. https://doi.org/10.1523/JNEUROSCI.1719-05.2005 doi: 10.1523/JNEUROSCI.1719-05.2005
    [48] J. T. Paz, J. M. Deniau, S. Charpier, Rhythmic bursting in the cortico-subthalamo-pallidal network during spontaneous genetically determined spike and wave discharges, J. Neurosci., 25 (2005), 2092–2101. https://doi.org/10.1523/JNEUROSCI.4689-04.2005 doi: 10.1523/JNEUROSCI.4689-04.2005
    [49] H. Kita, S. T. Kitai, Intracellular study of rat globus pallidus neurons: membrane properties and responses to neostriatal, subthalamic and nigral stimulation, Brain Res., 564 (1991), 296–305. https://doi.org/10.1016/0006-8993(91)91466-E doi: 10.1016/0006-8993(91)91466-E
    [50] M. A. Lebedev, S. P. Wise, Oscillations in the premotor cortex: single-unit activity from awake, behaving monkeys, Exp. Brain Res., 130 (2000), 195–215. https://doi.org/10.1007/s002210050022 doi: 10.1007/s002210050022
    [51] W. Schultz, R. Romo, Neuronal activity in the monkey striatum during the initiation of movements, Exp. Brain Res., 71 (1988), 431–436. https://doi.org/10.1007/BF00247503 doi: 10.1007/BF00247503
    [52] N. E. Hallworth, C. J. Wilson, M. D. Bevan, Apamin-sensitive small conductance calcium-activated potassium channels, through their selective coupling to voltage-gated calcium channels, are critical determinants of the precision, pace, and pattern of action potential generation in rat subthalamic nucleus neurons in vitro, J. Neurosci., 23 (2003), 7525–7542. https://doi.org/10.1523/JNEUROSCI.23-20-07525.2003 doi: 10.1523/JNEUROSCI.23-20-07525.2003
    [53] H. Kita, Globus pallidus external segment, Prog. Brain Res., 160 (2007), 111–133. https://doi.org/10.1016/S0079-6123(06)60007-1 doi: 10.1016/S0079-6123(06)60007-1
    [54] H. Kita, A. Nambu, K. Kaneda, Y. Tachibana, M. Takada, Role of ionotropic glutamatergic and GABAergic inputs on the firing activity of neurons in the external pallidum in awake monkeys, J. Neurophysiol., 92 (2004), 3069–3084. https://doi.org/10.1152/jn.00346.2004 doi: 10.1152/jn.00346.2004
    [55] H. Nakanishi, H. Kita, S. T. Kitai, Intracellular study of rat substantia nigra pars reticulata neurons in an in vitro slice preparation: electrical membrane properties and response characteristics to subthalamic stimulation, Brain Res., 437 (1987), 45–55. https://doi.org/10.1016/0006-8993(87)91525-3 doi: 10.1016/0006-8993(87)91525-3
    [56] K. Fujimoto, H. Kita, Response characteristics of subthalamic neurons to the stimulation of the sensorimotor cortex in the rat, Brain Res., 609 (1993), 185–192. https://doi.org/10.1016/0006-8993(93)90872-K doi: 10.1016/0006-8993(93)90872-K
    [57] A. Gillies, D. Willshaw, Membrane channel interactions underlying rat subthalamic projection neuron rhythmic and bursting activity, J. Neurophysiol., 95 (2006), 2352–2365. https://doi.org/10.1152/jn.00525.2005 doi: 10.1152/jn.00525.2005
    [58] H. Kita, S. T. Kitai, Intracellular study of rat globus pallidus neurons: membrane properties and responses to neostriatal, subthalamic and nigral stimulation, Brain Res., 564 (1991), 296–305. https://doi.org/10.1016/0006-8993(91)91466-E doi: 10.1016/0006-8993(91)91466-E
    [59] Y. Hirai, M. Morishima, F. Karube, Y. Kawaguchi, Specialized cortical subnetworks differentially connect frontal cortex to parahippocampal areas, J. Neurosci., 32 (2012), 1898–1913. https://doi.org/10.1523/JNEUROSCI.2810-11.2012 doi: 10.1523/JNEUROSCI.2810-11.2012
    [60] Y. H. Tanaka, Y. Tanaka, F. Fujiyama, T. Furuta, Y. Yanagawa, T. Kaneko, Local connections of layer 5 GABAergic interneurons to corticospinal neurons, Front. Neural Circuits, 5 (2011), 12. https://doi.org/10.3389/fncir.2011.00012 doi: 10.3389/fncir.2011.00012
    [61] M. H. Higgs, S. J. Slee, W. J. Spain, Diversity of gain modulation by noise in neocortical neurons: regulation by the slow afterhyperpolarization conductance, J. Neurosci., 26 (2006), 8787–8799. https://doi.org/10.1523/JNEUROSCI.1792-06.2006 doi: 10.1523/JNEUROSCI.1792-06.2006
    [62] A. L. Barth, J. F. A. Poulet, Experimental evidence for sparse firing in the neocortex, Trends Neurosci., 35 (2012), 345–355. https://doi.org/10.1016/j.tins.2012.03.008 doi: 10.1016/j.tins.2012.03.008
    [63] A. P. Prudnikov, Y. A. Brychkov, O. I. Marichev, Integrals and series: direct laplace transforms, Routledge, 2018. https://doi.org/10.1201/9780203750643 doi: 10.1201/9780203750643
    [64] K. Udupa, N. Bahl, Z. Ni, C. Gunraj, F. Mazzella, E. Moro, et al., Cortical plasticity induction by pairing subthalamic nucleus deep-brain stimulation and primary motor cortical transcranial magnetic stimulation in Parkinson's disease, J. Neurosci., 36 (2016), 396–404. https://doi.org/10.1523/JNEUROSCI.2499-15.2016 doi: 10.1523/JNEUROSCI.2499-15.2016
    [65] M. Dagan, T. Herman, R. Harrison, J. Zhou, N. Giladi, G. Ruffini, et al., Multitarget transcranial direct current stimulation for freezing of gait in Parkinson's disease, Mov. Disord., 33 (2018), 642–646. https://doi.org/10.1002/mds.27300 doi: 10.1002/mds.27300
    [66] E. Lattari, S. S. Costa, C. Campos, A. J. Oliveira, S. Machado, G. A. M. Neto, Can transcranial direct current stimulation on the dorsolateral prefrontal cortex improves balance and functional mobility in Parkinson's disease?, Neurosci. Lett., 636 (2017), 165–169. https://doi.org/10.1016/j.neulet.2016.11.019 doi: 10.1016/j.neulet.2016.11.019
  • This article has been cited by:

    1. Muhammad Zubair Mehboob, Arslan Hamid, Jeevotham Senthil Kumar, Xia Lei, Comprehensive characterization of pathogenic missense CTRP6 variants and their association with cancer, 2025, 25, 1471-2407, 10.1186/s12885-025-13685-0
  • Reader Comments
  • © 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(2088) PDF downloads(108) Cited by(0)

Figures and Tables

Figures(22)  /  Tables(2)

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog