Abstract
Based on the transcranial magnetic stimulation (TMS) therapy for major depression disorder and anxiety disorder, an abstract form of neuron differential equation with quasi-periodic coefficients is established. Considering the interactions among neurons, electrical effect induced by TMS therapy, and random effect, the probability density function with respect to the firing rates of neurons of the above equation is constructed, which satisfies a static Fokker–Planck equation with quasi-periodic coefficients. Hence, different clinical TMS therapies are discussed numerically by setting different parameters, such as the coupling strengths, the intensity, frequency, initial phase, phase difference of the electric field. In addition, a functional connectivity matrix related to the adjacency matrix with variable coupling strengths based on data from a real brain network is constructed. It is shown that the proper coupling strengths and the electric field parameters are crucial for TMS therapy. The methods include Kolmogorov–Arnold–Moser (KAM) theory, numerical analysis, and data analysis, having interdisciplinary characteristics. The result is original, which may be applied to TMS combined with drug therapy and the cognitive process of the brain.
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