Cosine tuning curve
Georgopoulos and colleagues observed that a motor-cortex neuron's mean firing rate during reaching varies approximately as a cosine of the angle between the reach direction and a single direction the cell prefers: f(theta) = b0 + b1 cos(theta - theta_pd). Equivalently the modulation is the projection of the movement direction (a unit vector) onto the cell's preferred-direction vector, so the response is linear in the direction cosines. The preferred direction theta_pd is where firing peaks; b1 sets the depth of directional modulation and b0 the baseline. The model extends naturally to velocity and hand position as regressors.
Cosine tuning is deliberately broad: any single neuron is only weakly informative because it responds to a wide swath of directions, which is exactly why direction must be read from a population. It is best understood as a first-order, trial-averaged description rather than a mechanism. Real motor-cortex responses are strongly time-varying and multiphasic, preferred directions rotate over the course of a movement and shift with arm posture and workspace, and a fitted cosine explains only a fraction of single-trial variance.
A cell fires ~40 spikes/s for reaches to the right (its preferred direction), ~10 spikes/s to the left, and ~25 spikes/s for up or down reaches — a cosine profile centred on 0 degrees with a baseline of 25.
The peak locates the preferred direction; the depth (40 minus 10, halved) is the directional modulation b1.
A single preferred direction summarizes only the epoch-averaged response. In the dynamical-systems view, apparent tuning is largely an epiphenomenon of underlying population dynamics rather than an explicit code for direction.