Objective. Modeling the brain as a white box is vital for investigating the brain. However, the physical properties of the human brain are unclear. Therefore, BCI algorithms using EEG signals are generally a data-driven approach and generate a black-or gray-box model. This paper presents the first EEG-based BCI algorithm (EEG-BCI using Gang neurons, EEGG) decomposing the brain into some simple components with physical meaning and integrating recognition and analysis of brain activity. Approach. Independent and interactive components of neurons or brain regions can fully describe the brain. This paper constructed a relation frame based on the independent and interactive compositions for intention recognition and analysis using a novel dendrite module of Gang neurons. A total of 4,906 EEG data of left-and right-hand motor imagery(MI) from 26 subjects were obtained from GigaDB. Firstly, this paper explored EEGG's classification performance by cross-subject accuracy. Secondly, this paper transformed the trained EEGG model into a relation spectrum expressing independent and interactive components of brain regions. Then, the relation spectrum was verified using the known ERD/ERS phenomenon. Finally, this paper explored the previously unreachable further BCIbased analysis of the brain. Main results. (1) EEGG was more robust than typical "CSP+" algorithms for the lowquality data. (2) The relation spectrum showed the known ERD/ERS phenomenon. (3) Interestingly, EEGG showed that interactive components between brain regions suppressed ERD/ERS effects on classification. This means that generating fine hand intention needs more centralized activation in the brain. Significance. EEGG decomposed the biological EEG-intention system of this paper into the relation spectrum inheriting the Taylor series (in analogy with the data-driven but human-readable Fourier transform and frequency spectrum), which offers a novel frame for analysis of the brain.