Applied Mathematics and Mechanics (English Edition) ›› 2026, Vol. 47 ›› Issue (4): 719-740.doi: https://doi.org/10.1007/s10483-026-3371-9
收稿日期:2025-12-12
修回日期:2026-02-05
发布日期:2026-03-31
Haodong WANG1, Ying YU1,†(
), Qingyun WANG1,2
Received:2025-12-12
Revised:2026-02-05
Published:2026-03-31
Contact:
Ying YU
E-mail:yuyingmath@163.com
Supported by:中图分类号:
. [J]. Applied Mathematics and Mechanics (English Edition), 2026, 47(4): 719-740.
Haodong WANG, Ying YU, Qingyun WANG. Transcranial electrical stimulation enhancing brain network integration: a dynamic modeling study[J]. Applied Mathematics and Mechanics (English Edition), 2026, 47(4): 719-740.
"
| Parameter | Definition | Value |
|---|---|---|
| A | Maximum amplitude of the EPSP | 3.25 mV |
| B | Maximum amplitude of the inhibitory post-synaptic potential (IPSP) | 22 mV |
| a | Inverse of the time constant in the feedback excitatory circuit | 100 s-1 |
| b | Inverse of the time constant in the feedback inhibitory circuit | 50 s-1 |
| | Number of synapses on the dendrites of excitatory feedback circuit | |
| | Number of synapses on the dendrites of inhibitory feedback circuit | |
| | Parameters of the sigmoid function | |
| [1] | COHEN, J. R. and D’ESPOSITO, M. The segregation and integration of distinct brain networks and their relationship to cognition. The Journal of Neuroscience, 36(48), 12083–12094 (2016) |
| [2] | SHINE, J. M., BREAKSPEAR, M., BELL, P. T., EHGOETZ MARTENS, K. A., SHINE, R., KOYEJO, O., SPORNS, O., and POLDRACK, R. A. Human cognition involves the dynamic integration of neural activity and neuromodulatory systems. Nature Neuroscience, 22(2), 289–296 (2019) |
| [3] | DECO, G., TONONI, G., BOLY, M., and KRINGELBACH, M. L. Rethinking segregation and integration: contributions of whole-brain modelling. Nature Reviews Neuroscience, 16(7), 430–439 (2015) |
| [4] | SHINE, J. M. Neuromodulatory influences on integration and segregation in the brain. Trends in Cognitive Sciences, 23(7), 572–583 (2019) |
| [5] | SHINE, J. M., BISSETT, P. G., BELL, P. T., KOYEJO, O., BALSTERS, J. H., GORGOLEWSKI, K. J., MOODIE, C. A., and POLDRACK, R. A. The dynamics of functional brain networks: integrated network states during cognitive task performance. Neuron, 92(2), 544–554 (2016) |
| [6] | WESTPHAL, A. J., WANG, S. L., and RISSMAN, J. Episodic memory retrieval benefits from a less modular brain network organization. The Journal of Neuroscience, 37(13), 3523–3531 (2017) |
| [7] | REN, S., LI, J. H., TAYA, F., DESOUZA, J., THAKOR, N. V., and BEZERIANOS, A. Dynamic functional segregation and integration in human brain network during complex tasks. IEEE Transactions on Neural Systems and Rehabilitation Engineering, 25(6), 547–556 (2017) |
| [8] | SPORNS, O. Network attributes for segregation and integration in the human brain. Current Opinion in Neurobiology, 23(2), 162–171 (2013) |
| [9] | BREAKSPEAR, M. Dynamic models of large-scale brain activity. Nature Neuroscience, 20(3), 340–352 (2017) |
| [10] | LIU, X. T., YU, Y., HAN, F., ZHOU, J., LIU, Z., LUAN, G. M., and WANG, Q. Y. A data-driven brain network modeling for epileptogenic spread analysis. Biomedical Signal Processing and Control, 105, 107645 (2025) |
| [11] | LIU, X. T., YU, Y., and WANG, Q. Y. Dynamics of epileptic seizure propagation under the regulation of ion mechanisms and synaptic networks. Nonlinear Dynamics, 114, 59 (2026) |
| [12] | CORONEL-OLIVEROS, C., CASTRO, S., COFRÉ, R., and ORIO, P. Structural features of the human connectome that facilitate the switching of brain dynamics via noradrenergic neuromodulation. Frontiers in Computational Neuroscience, 15, 687075 (2021) |
| [13] | YU, Y., FAN, Y. B., HAN, F., LUAN, G. M., and WANG, Q. Y. Transcranial direct current stimulation inhibits epileptic activity propagation in a large-scale brain network model. Science China Technological Sciences, 66(12), 3628–3638 (2023) |
| [14] | YU, Y., WANG, X. M., WANG, Q. S., and WANG, Q. Y. A review of computational modeling and deep brain stimulation: applications to Parkinson’s disease. Applied Mathematics and Mechanics (English Edition), 41(12), 1747–1768 (2020) https://doi.org/10.1007/s10483-020-2689-9 |
| [15] | SUN, Z. K., LIU, Y. Y., YANG, X. L., and XU, W. Control of epileptic activities in a cortex network of multiple coupled neural populations under electromagnetic induction. Applied Mathematics and Mechanics (English Edition), 44(3), 499–514 (2023) https://doi.org/10.1007/s10483-023-2969-9 |
| [16] | SHINE, J. M., ABURN, M. J., BREAKSPEAR, M., and POLDRACK, R. A. The modulation of neural gain facilitates a transition between functional segregation and integration in the brain. eLife, 7, e31130 (2018) |
| [17] | LI, M. K., HAN, Y. N., ABURN, M. J., BREAKSPEAR, M., POLDRACK, R. A., SHINE, J. M., and LIZIER, J. T. Transitions in information processing dynamics at the whole-brain network level are driven by alterations in neural gain. PLoS Computational Biology, 15(6), e1006957 (2019) |
| [18] | CORONEL-OLIVEROS, C., COFRÉ, R., and ORIO, P. Cholinergic neuromodulation of inhibitory interneurons facilitates functional integration in whole-brain models. PLoS Computational Biology, 17(3), e1008737 (2021) |
| [19] | AERTS, H., FIAS, W., CAEYENBERGHS, K., and MARINAZZO, D. Brain networks under attack: robustness properties and the impact of lesions. Brain, 139, 3063–3083 (2016) |
| [20] | LEVIN, M. F., SELLES, R. W., VERHEUL, M. H. G., and MEIJER, O. G. Deficits in the coordination of agonist and antagonist muscles in stroke patients: implications for normal motor control. Brain Research, 853(2), 352–369 (2000) |
| [21] | LI, Y. X., WU, P., LIANG, F. R., and HUANG, W. H. The microstructural status of the corpus callosum is associated with the degree of motor function and neurological deficit in stroke patients. PLoS One, 10(3), e0122615 (2015) |
| [22] | ALSTOTT, J., BREAKSPEAR, M., HAGMANN, P., CAMMOUN, L., and SPORNS, O. Modeling the impact of lesions in the human brain. PLoS Computational Biology, 5(6), e1000408 (2009) |
| [23] | VÁŠA, F., SHANAHAN, M., HELLYER, P. J., SCOTT, G., CABRAL, J., and LEECH, R. Effects of lesions on synchrony and metastability in cortical networks. NeuroImage, 118, 456–467 (2015) |
| [24] | WANG, H. D., JIA, W. L., ZHOU, Y. J., LI, Z. X., YU, Y., and WANG, Q. Y. Unveiling the role of excitation-inhibition homeostasis in stroke recovery from individualized whole-brain dynamical modeling. Communications in Nonlinear Science and Numerical Simulation, 151, 109114 (2025) |
| [25] | HELLYER, P. J., SCOTT, G., SHANAHAN, M., SHARP, D. J., and LEECH, R. Cognitive flexibility through metastable neural dynamics is disrupted by damage to the structural connectome. Journal of Neuroscience, 35(24), 9050–9063 (2015) |
| [26] | LORD, L. D., STEVNER, A. B., DECO, G., and KRINGELBACH, M. L. Understanding principles of integration and segregation using whole-brain computational connectomics: implications for neuropsychiatric disorders. Philosophical Transactions Series A, Mathematical, Physical, and Engineering Sciences, 375(2096), 20160283 (2017) |
| [27] | FLÖEL, A. tDCS-enhanced motor and cognitive function in neurological diseases. NeuroImage, 85, 934–947 (2014) |
| [28] | ORRÙ, G., CONVERSANO, C., HITCHCOTT, P. K., and GEMIGNANI, A. Motor stroke recovery after tDCS: a systematic review. Reviews in the Neurosciences, 31(2), 201–218 (2020) |
| [29] | YUN, G. J., CHUN, M. H., and KIM, B. R. The effects of transcranial direct-current stimulation on cognition in stroke patients. Journal of Stroke, 17(3), 354–358 (2015) |
| [30] | KUNZE, T., HUNOLD, A., HAUEISEN, J., JIRSA, V., and SPIEGLER, A. Transcranial direct current stimulation changes resting state functional connectivity: a large-scale brain network modeling study. NeuroImage, 140, 174–187 (2016) |
| [31] | MERLET, I., BIROT, G., SALVADOR, R., MOLAEE-ARDEKANI, B., MEKONNEN, A., SORIA-FRISH, A., RUFFINI, G., MIRANDA, P. C., and WENDLING, F. From oscillatory transcranial current stimulation to scalp EEG changes: a biophysical and physiological modeling study. PLoS One, 8(2), e57330 (2013) |
| [32] | YU, Y., WANG, H. D., LIU, X. T., and WANG, Q. Y. Closed-loop transcranial electrical stimulation for inhibiting epileptic activity propagation: a whole-brain model study. Nonlinear Dynamics, 112(24), 21369–21387 (2024) |
| [33] | DONG, Y. Q., WEI, J., PENG, S. J., WU, X. R., XU, Y. R., FENG, J. F., ZHANG, J., JIRSA, V., and XIANG, J. Effects of tDCS of the DLPFC on brain networks: a hybrid brain modeling study. PLoS Computational Biology, 21, e1013486 (2025) |
| [34] | GIORDANO, J., BIKSON, M., KAPPENMAN, E. S., CLARK, V. P., COSLETT, H. B., HAMBLIN, M. R., HAMILTON, R., JANKORD, R., KOZUMBO, W. J., MCKINLEY, R. A., NITSCHE, M. A., REILLY, J. P., RICHARDSON, J., WURZMAN, R., and CALABRESE, E. Mechanisms and effects of transcranial direct current stimulation. Dose-Response, 15(1), 1559325816685467 (2017) |
| [35] | IORDAN, A. D., RYAN, S., TYSZKOWSKI, T., PELTIER, S. J., RAHMAN-FILIPIAK, A., and HAMPSTEAD, B. M. High-definition transcranial direct current stimulation enhances network segregation during spatial navigation in mild cognitive impairment. Cerebral Cortex, 32(23), 5230–5241 (2022) |
| [36] | VECCHIO, F., MIRAGLIA, F., RODELLA, C., ALÙ, F., MINIUSSI, C., ROSSINI, P. M., and PELLICCIARI, M. C. tDCS effects on brain network properties during physiological aging. Pflügers Archiv-European Journal of Physiology, 473(5), 785–792 (2021) |
| [37] | JANSEN, B. H. and RIT, V. G. Electroencephalogram and visual evoked potential generation in a mathematical model of coupled cortical columns. Biological Cybernetics, 73(4), 357–366 (1995) |
| [38] | ŠKOCH, A., REHÁK BUČKOVÁ, B., MAREŠ, J., TINTĚRA, J., SANDA, P., JAJCAY, L., HORÁČEK, J., ŠPANIEL, F., and HLINKA, J. Human brain structural connectivity matrices-ready for modelling. Scientific Data, 9(1), 486 (2022) |
| [39] | STEPHAN, K. E., WEISKOPF, N., DRYSDALE, P. M., ROBINSON, P. A., and FRISTON, K. J. Comparing hemodynamic models with DCM. NeuroImage, 38(3), 387–401 (2007) |
| [40] | HUANG, Y., DATTA, A., BIKSON, M., and PARRA, L. C. Realistic volumetric-approach to simulate transcranial electric stimulation: ROAST: a fully automated open-source pipeline. Journal of Neural Engineering, 16, 056006 (2019) |
| [41] | PION-TONACHINI, L., HSU, S. H., MAKEIG, S., JUNG, T. P., and CAUWENBERGHS, G. Real-time EEG source-mapping toolbox (REST): online ICA and source localization. 2015 37th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBC), IEEE, Milan, 4114–4117 (2015) |
| [42] | RAHMAN, A., REATO, D., ARLOTTI, M., GASCA, F., DATTA, A., PARRA, L. C., and BIKSON, M. Cellular effects of acute direct current stimulation: somatic and synaptic terminal effects. The Journal of Physiology, 591(10), 2563–2578 (2013) |
| [43] | LANCASTER, G., IATSENKO, D., PIDDE, A., TICCINELLI, V., and STEFANOVSKA, A. Surrogate data for hypothesis testing of physical systems. Physics Reports, 748, 1–60 (2018) |
| [44] | BENJAMINI, Y. and HOCHBERG, Y. Controlling the false discovery rate: a practical and powerful approach to multiple testing. Journal of the Royal Statistical Society. Series B (Methodological), 57(1), 289–300 (1995) |
| [45] | LATORA, V. and MARCHIORI, M. Efficient behavior of small-world networks. Physical Review Letters, 87(19), 198701 (2001) |
| [46] | RUBINOV, M. and SPORNS, O. Complex network measures of brain connectivity: uses and interpretations. NeuroImage, 52(3), 1059–1069 (2010) |
| [47] | ACEBRÓN, J. A., BONILLA, L. L., PÉREZ VICENTE, C. J., RITORT, F., and SPIGLER, R. The Kuramoto model: a simple paradigm for synchronization phenomena. Reviews of Modern Physics, 77(1), 137–185 (2005) |
| [1] | . [J]. Applied Mathematics and Mechanics (English Edition), 2026, 47(5): 1157-1176. |
| [2] | . [J]. Applied Mathematics and Mechanics (English Edition), 2025, 46(1): 37-62. |
| [3] | . [J]. Applied Mathematics and Mechanics (English Edition), 2025, 46(1): 123-138. |
| [4] | Xiaofeng LIU, Qishuai WANG, Haiquan LI, Guoping CAI. Dynamics and control of variable geometry truss manipulator[J]. Applied Mathematics and Mechanics (English Edition), 2017, 38(2): 243-262. |
| [5] | Haiyan SONG, Lifu LIANG. Investigation of power-type variational principles in liquid-filled system[J]. Applied Mathematics and Mechanics (English Edition), 2015, 36(12): 1651-1662. |
| [6] | Wenan JIANG, Guoce ZHANG, Liqun CHEN. Forced response of quadratic nonlinear oscillator: comparison of various approaches[J]. Applied Mathematics and Mechanics (English Edition), 2015, 36(11): 1403-1416. |
| [7] | Fengxiang MEI, Jinchao CUI. Skew-gradient representations of constrained mechanical systems[J]. Applied Mathematics and Mechanics (English Edition), 2015, 36(7): 873-882. |
| [8] | A. M. SIDDIQUI, H. ASHRAF, A. WALAIT, T. HAROON. On study of horizontal thin film flow of Sisko fluid due to surface tension gradient[J]. Applied Mathematics and Mechanics (English Edition), 2015, 36(7): 847-862. |
| [9] | A. FARROKHABADI;A. KOOCHI;A. KAZEMI;M. ABADYAN. Effects of size-dependent elasticity on stability of nanotweezers[J]. Applied Mathematics and Mechanics (English Edition), 2014, 35(12): 1573-1590. |
| [10] | 吴惠彬;梅凤翔. Form invariance and conserved quantity for weakly nonholonomic system[J]. Applied Mathematics and Mechanics (English Edition), 2014, 35(10): 1293-1300. |
| [11] | 高芳;张晓波;傅景礼. Application of canonical coordinates for solving single-freedom constraint mechanical systems [J]. Applied Mathematics and Mechanics (English Edition), 2014, 35(8): 1029-1038. |
| [12] | K. REGER R. A. VAN GORDER. Lane-Emden equations of second kind modelling thermal explosion in infinite cylinder and sphere[J]. Applied Mathematics and Mechanics (English Edition), 2013, 34(12): 1439-1452. |
| [13] | 李凤明;刘春川. Parametric vibration stability and active control of nonlinear beams[J]. Applied Mathematics and Mechanics (English Edition), 2012, 33(11): 1381-1392. |
| [14] | 黄卫立;蔡建乐. Conformal invariance for nonholonomic system of Chetaev’s type with variable mass[J]. Applied Mathematics and Mechanics (English Edition), 2012, 33(11): 1393-1402. |
| [15] | 武秀根;郑百林;贺鹏飞;刘曙光. Equilibrium equations for 3D critical buckling of helical springs[J]. Applied Mathematics and Mechanics (English Edition), 2012, 33(8): 1049-1058. |
| 阅读次数 | ||||||
|
全文 |
|
|||||
|
摘要 |
|
|||||

Email Alert
RSS