Picking up from Volume I
In Volume I you met the cosine tuning curve and the population vector as friendly intuitions: a motor-cortex cell fires most for one reach direction, and if you let every cell vote with its preferred direction you recover the movement. That picture is correct as far as it goes — but it is the beginner's snapshot of a field that has since split into two competing languages for describing the same motor cortex. This track is about those two languages and the modern synthesis between them.
The representational view
The classical account, dating to Georgopoulos's discovery of directional tuning, is representational: each neuron encodes a movement parameter — direction, velocity, force — and the population's job is to represent that parameter with redundancy and noise tolerance. Decoding is then read-out: invert the encoding to recover the represented variable.
\lambda_i(\theta) = b_{0i} + m_i\cos(\theta - \theta_i)Cosine tuning: cell i's expected rate rises with the cosine of the angle between reach direction \theta and its preferred direction \theta_i.
A motor-cortex cell fires fastest for one preferred reaching direction and less as the reach angle turns away, following a cosine curve. Its rate is a baseline plus a cosine bump.
- \lambda_i(\theta)
- Expected firing rate of cell i for reach direction \theta.
- \theta_i
- The cell's preferred direction — where it fires hardest.
- b_{0i},\ m_i
- Baseline rate and modulation depth (how strongly it tunes).
- \cos(\theta - \theta_i)
- Falls from 1 (reach at preferred) to -1 (opposite direction).
A cell preferring "up" fires hard for upward reaches, at baseline for sideways, and least for downward — the cosine tuning curve.
The dynamical-systems view
Starting in the 2010s, Churchland, Shenoy and colleagues argued the representational frame asks the wrong question. Motor cortex, they proposed, is better understood as a dynamical system: a recurrent network whose activity evolves under internal rules, producing the time-varying commands muscles need. Neurons are not labels for parameters; they are state variables of a machine that generates movement.
\dot{\mathbf{x}}(t) = A\,\mathbf{x}(t)The low-dimensional population state \mathbf{x} evolving under internal dynamics A — activity that unfolds, rather than a static code that is read.
Instead of reading a fixed code, the dynamical view says the population's state \mathbf{x} evolves over time by its own internal rule A — activity that unfolds like a wound spring releasing, rather than a static message being read off.
- \mathbf{x}(t)
- The low-dimensional population state.
- \dot{\mathbf{x}}(t)
- Its rate of change — velocity in state space.
- A
- The dynamics matrix, the internal rule linking state to how it changes.
A pendulum's position and speed evolve by such a rule; motor activity shows similar lawful flow — the dynamical-systems view.
In this view preparatory activity sets the initial condition, then dynamics roll forward. A striking prediction confirmed in data: much of the variance at movement onset is a condition-independent signal — a trigger shared across all reaches — which makes little sense as representation of direction but is exactly what you expect when a dynamical system is kicked into motion.
Why a BCI engineer should care
This is not philosophy for its own sake — the two views prescribe different decoders. If tuning is the truth, decode the instantaneous represented variable. If dynamics are the truth, estimate a smooth low-dimensional latent state and exploit its temporal structure — exactly what a velocity Kalman filter or LFADS does. Empirically, the neural manifold and its dynamics are more stable across days than any single neuron's tuning, which is why modern decoders lean on population structure.
The road ahead
- Guide 2 — cosine tuning and the population vector done rigorously, and why the optimal linear estimator beats it.
- Guide 3 — neural manifolds and the dimensionality of activity: PCA, factor analysis, GPFA.
- Guide 4 — motor cortex as a dynamical system: preparation, rotational dynamics, fixed points, tangling.
- Guide 5 — the frontier: dynamics-based decoders, cross-day stability, and honest open problems.