Applied Mathematics and Mechanics (English Edition) ›› 2026, Vol. 47 ›› Issue (4): 719-740.doi: https://doi.org/10.1007/s10483-026-3371-9
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Haodong WANG1, Ying YU1,†(
), Qingyun WANG1,2
Email Alert
RSSReceived:2025-12-12
Revised:2026-02-05
Published:2026-03-31
Contact:
Ying YU, E-mail: yuyingmath@163.comSupported by:2010 MSC Number:
Haodong WANG, Ying YU, Qingyun WANG. Transcranial electrical stimulation enhancing brain network integration: a dynamic modeling study. Applied Mathematics and Mechanics (English Edition), 2026, 47(4): 719-740.
Fig. 1
Whole-brain network model. (a) Each node is implemented as the Jansen-Rit neural mass model, representing a cortical column that comprises three interconnected populations, i.e., pyramidal neurons (the output population), excitatory interneurons, and inhibitory interneurons. The neuronal populations are interconnected via the constants C, C1, C2, C3, and C4. hE and hI represent the impulse response functions. S denotes the sigmoid function. Pyramidal neurons project to other nodes and receive Gaussian-distributed input p(t) as well as inputs from other nodes. (b) The connectivity matrix extracted from the structural data. (c) Nodes are connected through the structural matrix. (d) The dynamic activity of the node output. (e) Based on the generalized hemodynamic model, simulated blood oxygenation level dependent (BOLD)-like recording for the entire brain region is obtained (color online)"
Table 1
Parameter interpretation and values of the Jansen-Rit model"
| 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 | |
Fig. 2
(a)–(c) Characteristics of changes in the average synchronization level R¯, global efficiency EG, and modularity Q across the γβ-parameter space; (d)–(g) node activities, BOLD signals, and sFC matrices for the parameter pairs (γ, β) of (0.4, 0.05), (1.2, 0.15), (2, 0.25), and (3, 0.45), respectively, where the values in the matrix represent the Pearson correlations of BOLD-like signals. As the parameters increase, neurons shift from low-amplitude random oscillations to stable discharge activities, and then to high-amplitude hyperexcited states. To reflect the specific characteristics of discharges at different scales, the node activity signals are displayed within a 5-second period, while the sFC matrix is constructed based on a 600-second time series (color online)"
Fig. 3
Electric field distributions and modulation of network dynamics under tES. (a)–(b) Whole-brain electric field intensity distributions obtained via ROAST simulation under different electrode configurations: (a) anode at C3 and (b) anode at C3 and cathode at C4; (c) magnitude of Enormal changes under dual-electrode stimulation; (d)–(e) variations in the average regional synchronization level R¯ with the stimulation intensity under anode and dual-electrode configurations, respectively, where the tES leads to a decrease in the mean synchronization level; (g)–(h) changes in the integration level under stimulation. As the stimulus intensity increases, EG exhibits a non-monotonic trend of first increasing and then decreasing; (f)–(i) EG as a function of both the global coupling parameter and tES intensity (color online)"
Fig. 4
(a)–(c) Node under three conditions, i.e., no stimulation, anodal tES at C3, and dual-electrode tES at C3/C4, respectively; (d)–(f) sFC and BOLD signals under three conditions, respectively, when γ=1.6 and β=0.2; (g) and (h) EG as a function of the global coupling parameter γ, with β fixed at 0.2, and the inhibitory gain β, with γ fixed at 1.6, where tES enhances integration within specific parameter ranges u^tES=1; (i) change in the maximum EG, quantified as ΔEG=max(EGtES)−max(EGnormal), across β∈ [0, 0.5] under different γ values in the range [0.8, 2.2] (color online)"
Fig. 5
(a)–(c) Time courses of the phase synchronization level: (a) under no stimulation, (b) stimulation intensity of 1, and (c) stimulation intensity of 5, showing continuous fluctuations between high and low synchronization states, where the corresponding variances are 0.033 6, 0.042 9, and 0.037 0, respectively; (d) raster plot of population firing across brain regions over a 10 s period, where the green box highlights an example of lower synchronization, and the red box indicates a period of higher synchronization; (e) variation in the modularity Q with the stimulation intensity (color online)"
Fig. 6
Whole-brain electric field distributions under different stimulation targets and the corresponding changes in the maximum global efficiency ΔEG: (a) anode at F3 and cathode at F4; (b) anode at FC3 and cathode at FC4; (c) anode at CP5 and cathode at CP6; (d) anode at P3 and cathode at P4, where under different stimulus targets, the maximum value of EG shows a trend of initially increasing and subsequently decreasing with the change in γ, and reaches its peak within the middle range of γ (color online)"
Fig. 7
Effects of focal node lesions on network synchronization and integration. (a) Changes in R¯ and EG (ΔR¯=R¯max,lesion−R¯max,normal, ΔEG=EGmax,lesion−EGmax,normal) after sequentially lesioning each of the 45 left-hemisphere nodes, where error bars indicate the standard deviation of changes; (b) change in the maximum EG following single-node lesions, plotted as a function of the inhibitory gain β (ranging from 0.15 to 0.35), where error bars represent the standard deviation; (c) EG as a function of γ under normal conditions and after lesioning node 7 (selected from the red box in (b), representing the lesion causing the largest decrease in integration), with β fixed at 0.25; (d) EG as a function of γ under normal conditions and after lesioning node 61 (selected from the red box in (b), representing the lesion causing the largest increase in integration), with β fixed at 0.35; (e) sFC matrices at the value of γ yielding the maximum change in EG, where upper panel represents sFC for the normal state versus lesion at node 7 (γ=2), and lower panel represents sFC for the normal state versus lesion at node 61 (γ=2.2) (color online)"
Fig. 8
Recovery of the integration level under tES. (a)–(b) Changes in the integration level under C3/C4 dual-electrode tES at varying stimulation intensities, using node 7 lesion as an example, where red dot line denotes normal level and black dot line denotes lesion level; in (a), the stimulation intensity ranges from 0 to 5; in (b), the range is refined to 0.25–2 to identify the optimal intensity; (c) curves of EG under different conditions, with C3/C4 dual-electrode tES intensity set to 0.25; (d)–(e) changes in the integration level under C3 anodal tES at varying stimulation intensities, ranging from 0 to 5 in (d) and refined to 0.25–2 in (e); (f) curves of EG under different conditions, with C3 anodal tES intensity set to 0.75, where the black, green, and red curves represent the situations of normal, lesion, and tES effects, respectively (color online)"
Fig. 9
(a) Changes in EG under different conditions, where the red dots represent the variation in EG after lesioning individual nodes, and the blue and green symbols denote the changes in EG under the C3/C4 dual-electrode tES and C3 anodal tES, respectively; (b) distribution of EG levels across the 45 left-hemisphere nodes (odd numbers from 1 to 89) under individual node lesions is plotted, with the increase in the stimulus intensity u^tES as the vertical axis; (c) impact of node lesions on EG, where the histogram shows the difference in the integration level after lesion relative to the normal state (ΔEG=EGmax,lesion−EGmax,normal); (d)–(e) restorative effect of dual-electrode and single-electrode tES, quantified as the change in EG from the lesioned to stimulated state (ΔEG1=EGmax,tES−EGmax,lesion) (color online)"
Fig. 10
Effects of diffuse connection loss on network integration and nodal dynamics. (a) Changes in the maximum EG as a function of the global coupling strength γ under 2%, 5%, and 10% connection loss, where for each lesion level, 20 simulations are performed and error bars represent the variance across these runs; (b) representative γ-EG curves under different lesion levels; (c)–(d) nodal firing frequency as a function of γ under different conditions, where the red shaded area indicates regular spiking activity at approximately 4.8 Hz, and the lesion causes this bifurcation point to shift towards a higher value of γ (the red dotted line), indicating that the lesion system requires stronger global coupling to achieve the same dynamic state (color online)"
Fig. 11
(a)–(c) Changes in the integration level with the stimulation intensity in the range [0, 2] under the connection loss of (a) 2%, (b) 5%, and (c) 10%, where the red dashed line indicates the integration level without stimulation, and for each lesion level, 20 simulations are performed, in which error bars represent the variance across these runs; (d) difference in the maximum global efficiency EG relative to the normal level under 10% connection loss, comparing conditions without stimulation, with C3/C4 dual-electrode tES, and with C3 single-electrode tES; (e) changes in the integration level with the stimulation intensity in the range [0, 5] under 10% connection loss (color online)"
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