ThreadAlongside the phases

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.

SpineDecoding / signal processing / MLComputational neuroscienceAnalog / mixed-signal hardware

On this pageAlongside the phases

Linear algebra, first and most
Signals and systems, with Phase 2
Probability and estimation, with Phase 3
Control, alongside
Electromagnetism, for Phase 4
Differential equations, throughout
Numerical methods, as needed

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

Linear algebraevery montage, filter, classifier and decoder is a matrixas early as possible

Signals and systemsconvolution, Fourier, sampling, frequency responsewith Phase 2

Probability and estimationBayes, likelihood, chance bounds, and the Kalman filterwith Phase 3

Controlfeedback, stability, and why the DRL loop can oscillateyear two or three

Electromagnetisma current dipole in a conducting headwith Phase 4

Differential equationsHodgkin-Huxley, the cable equation, numerical integrationthroughout

Take linear algebra as early as the prerequisites allow and signals and systems the semester you build the amplifier. The two will teach each other.

Figure 1. Six subjects, in the order the phases need them. Each one is far easier taken next to the project that makes it concrete, which is the only real argument this page is making.

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.

A decoder output is usually a dot product of weights and data:

y = \mathbf{w}^\top \mathbf{x} = \sum_i w_i x_i

Here $y$ is one number out, $\mathbf{w}$ holds the weights, and $\mathbf{x}$ holds the measurements.

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.

Convolution says each output sample is a sum of scaled echoes of the input through the filter:

(x * h)[n] = \sum_k x[k], h[n-k]

Here $x$ is the signal, $h$ is the filter’s impulse response, and $n$ and $k$ count samples.

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.

Bayes’ rule turns a likelihood into a belief about the world:

P(\theta \mid x) = \frac{P(x \mid \theta), P(\theta)}{P(x)}

Here $\theta$ is the thing estimated (a movement direction, a stimulus), $x$ is the measurement, $P(x \mid \theta)$ is how likely the measurement is under each candidate, and $P(\theta)$ is what was believed before.

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.

Closing a loop with forward gain $G$ and feedback gain $K$ gives the closed-loop map from command to output:

T = \frac{KG}{1 + KG}

When the denominator goes small, small disturbances grow: that is instability, and the driven-right-leg loop is analyzed exactly this way.

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.

The potential in the head obeys Poisson’s equation for a conductive medium:

\nabla \cdot (\sigma \nabla \phi) = -s

Here $\phi$ is the electric potential, $\sigma$ is the conductivity (different in brain, skull, and scalp), and $s$ is the source current density from neural activity.

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.

The leaky integrator of Phase 4 fits on one line:

\tau \frac{dV}{dt} = -(V - V_{rest}) + RI

Here $V$ is the membrane voltage, $\tau$ is the membrane time constant, $V_{rest}$ is the resting voltage, and $R I$ is the input current times the membrane resistance.

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.

ExhibitI have completed a linear algebra course (or Strang’s lectures) and can explain what a covariance matrix’s eigenvectors mean for a set of EEG trials.

Recall

Why is linear algebra the first and most important math for this field?

Montages, filters, classifiers, decoders, and source localization are all matrices acting on vectors; covariance eigenvectors, projections, and the pseudoinverse appear in every phase after the second.

Recall

Which two courses should be taken alongside which phases?

Signals and systems with Phase 2 (filters, sampling, frequency response) and probability and estimation with Phase 3 (classifiers, chance bounds, Bayesian decoding).