The math track
Linear algebra, then signals and systems, then probability and Bayesian estimation, then control, then enough electromagnetism to reason about dipoles in conductors. In that order, each attached to the phase that needs it.
You are skimming: the title, the first figure, and the short version. Switch to Read in the header for the full page, or Deep to open every deep dive.
The math this field uses is not exotic, but it has to be there when the phase needs it. Linear algebra is everything (every montage, filter, classifier, and decoder is a matrix). Signals and systems is Phase 2. Probability and Bayesian estimation is Phase 3. Control theory is the driven-right-leg loop and every closed-loop device. Electromagnetism is Phase 4’s head model. Here is the order, what each is for, and where to learn it fast.
Linear algebra, first and most
Vectors as signals; matrices as transformations; the dot product as correlation; projection as filtering; eigenvectors and the singular value decomposition as the axes along which data vary. Covariance matrices, which are where Phase 3’s classifiers live. The pseudoinverse, which is the minimum-norm solution of Phase 4.
Learn it from Gilbert Strang’s MIT lectures (free, and the textbook is good) or from 3Blue1Brown’s Essence of Linear Algebra for the geometric intuition first. Take the university’s course as early as the prerequisites allow. Every week of this site after Phase 2 uses it.
Signals and systems, with Phase 2
Convolution, the Fourier transform, sampling, the Laplace and z transforms, filters as systems, frequency response. Oppenheim’s Signals and Systems is the standard text; the MIT OpenCourseWare lectures are free. Everything in the filters and conversion explainers is this course applied. Take it the semester you build the ADS1299 board and the two will teach each other.
Probability and estimation, with Phase 3
Random variables, distributions, expectation and variance, Bayes’ rule, maximum likelihood, the Gaussian and why it appears everywhere, the binomial (your chance bound), hypothesis tests and their limits, and linear regression as estimation. Then Bayesian estimation as belief updating, which makes the Kalman filter obvious. A university statistics course for engineers covers the first half; Kalman comes from a controls or estimation course or from reading the Phase 3 explainer’s equations with a linear algebra background.
Control, alongside
Feedback, stability, gain and phase margin. The driven-right-leg circuit is a feedback loop that can oscillate; the Kalman filter’s cousin is the controller; closed-loop stimulation is a control problem; and co-adaptive BCIs are two controllers learning each other. One controls course, taken in the second or third year, reframes half of this site.
Electromagnetism, for Phase 4
Enough to reason about a current dipole in a conducting medium: Ohm’s law in its field form, the quasi-static approximation (why EEG has no wave propagation), Poisson’s equation for the potential, boundary conditions at interfaces of different conductivity. The physics department’s second electromagnetism course has it; Nunez and Srinivasan’s Electric Fields of the Brain applies it. This is also the background for understanding why MEG sees what EEG cannot.
Differential equations, throughout
Hodgkin–Huxley, the cable equation, the leaky integrator, the Kalman prediction step: all differential equations, mostly first and second order, some nonlinear. The engineering differential equations course plus the habit of integrating things numerically in Python (which Phase 4 forces) is enough.
Numerical methods, as needed
Forward Euler and when it fails (stiff systems, big steps), Runge-Kutta, linear solvers and conditioning, optimization as gradient descent. Picked up in the projects; a numerical methods course makes it systematic.