On convergence of formula for forward problem in brain physics
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摘要: 脑物理学中,正向问题是指:已知脑电信号源的位置, 通过脑电位分布计算式来预测头皮上任一点处产生的电压.首次对三层同心球壳模型单偶极子源头皮电位分布计算式进行坐标变换, 得到任意偶极子在头皮上任一点处产生的电压值计算公式, 从数学角度给出了其收敛性的分析和证明.同时代入数据, 用数据进一步对它的收敛性作出验证, 并判断了其收敛的级数.因此, 可以用有限级数在一定误差范围内代替无穷级数, 从而求出头皮上的电位分布.为公式的实际应用提供了理论支持, 也为通过头皮电压来预测信号源在人脑中的定位问题提出了有效的解决方法, 具有较大的实际应用价值.Abstract: In brain physics, the forward problem is the process of predicting scalp potentials distribution from known brain electrical source. The convergence of a forward problem algorithm was analyzed and proved through mathematic method for the first time. This formula calculated the scalp potentials distribution of the three layers concentric sphere head model caused by single dipole located at any point within the model. The convergence was further confirmed by numeric simulation. Thus the finite progression could be used to calculate the scalp potentials distribution in stead of the infinitive progression in a certain error range. A study on the convergence speed was conducted too. The result gives a theoretic guarantee for the finite approximation of the formula that is an infinite series. The approximation provides a useful tool for the backward problem, which is the process of predicting brain electrical source from known scalp potentials.
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Key words:
- electrocortical potentials /
- electrophysiology /
- convergence /
- formulas
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[1] James P A, Stanley A K, Derek H F. Location of sources of evoked scalp potentials:corrections for skull and scalp thicknesses[J] IEEE Transaction Biomedical Engineering, 1981, BME-28(6):447~452 [2] Zhang G L. Source localization in quantitative electrophysiology . Singapore:School of Electrical and Electronic Engineering, Nanyang Technological University,2000 [3] 吴殿鸿,郭立文,贾 诚,等.脑物理学[M] 哈尔滨:哈尔滨工业大学出版社,1995.39~48 Wu Dianhong, Guo Liwen, Jia Cheng, et al. Brain physics[M] Harbin:Harbin Technology Institute Press, 1995. 39~48(in Chinese) -
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