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Calcium-dependent nonlinearity describes the after-effects of different patterns of theta-burst TMS.

Ke Ma,S. Chung,J. C. Rothwell,Masashi Hamada,Stephan M. Goetz

Published 2025 in Clinical Neurophysiology

ABSTRACT

OBJECTIVE Optimisation of transcranial magnetic stimulation protocol parameters can potentially improve their efficacy through theoretical mathematical models. This study aims to develop an elaborate but still parsimonious quantitative model for understanding the dosage-dependency of Theta-Burst Stimulation (TBS). METHODS We propose a calcium-dependent nonlinear model that uses Michaelis-Menten kinetics to represent competing molecular pathways in postsynaptic neurons during TBS intervention. The model integrates four subsystems: calcium dynamics, modulation substances, MEP learning, and metaplasticity, with a particular focus on the interaction between facilitatory and inhibitory signalling cascades. RESULTS Our model successfully reproduces experimental findings from various TBS protocols, including the reversal of after-effects observed in prolonged stimulation. The simulation results suggest that the competing molecular pathways with distinct calcium sensitivities and temporal dynamics determine the direction of after-effects. CONCLUSIONS This proposed model represents a substantial advancement in understanding TBS-induced neuromodulation by providing testable predictions for untested TBS protocols and offers complementary insights into the underlying neural mechanisms. SIGNIFICANCE This model offers a basic framework for potentially incorporating complex, nonlinear relationships between TBS parameters and after-effects under various neurological and psychiatric conditions. It also provides a valuable tool for later protocol optimisation and guiding future experimental designs.

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