Direction Fields

Another example of field computation in the brain is provided by direction fields, in which a direction in space is encoded in the activity pattern over a brain region (Georgopoulos 1995). Such a region is characterized by a vector field ${\bf D}$ in which the vector value at each neural location gives the preferred direction encoded by the neuron at that location. The population code for a direction ${\bf r}$ is proportional to the scalar field given by the inner product of ${\bf r}$ at each point of ${\bf D}$. It will have a peak at the location corresponding to ${\bf r}$, which falls off as the cosine of the angle between this vector and the surrounding neurons' preferred directions. (See MacLennan 1997, section 6.2, for a more detailed discussion.)

Field computation is also used in the brain for modifying direction fields. For example, a direction field representing a remembered location, relative to the retina, must be updated when the eye moves (Droulez & Berthoz 1991a, 1991b), and the peak of the direction field must move in a direction given by the velocity vector of the eye motion. The change in the direction field is given by a differential field equation, in which the change in the value of the direction field is given by the inner product of the eye velocity vector and the gradient of the direction field: $\d\phi / \d t = {\bf v} \cdot \nabla\phi$. Each component (x and y) of the gradient is approximated by a convolution between the direction field and a ``derivative of Gaussian'' (DoG) field, which is implemented by the DoG shape of the receptive fields of the neurons. (See MacLennan 1997, section 6.3, for a more detailed discussion.)

Other examples of field computation in motor control include the control of frog leg position by the linear superposition of convergent force fields generated by spinal neurons (Bizzi & Mussa-Ivaldi 1995), and the computation of convergent vector fields, defining motions to positions in head-centered space, from positions in retina-centered space, as represented by products of simple receptive fields and linear gain fields (Andersen 1995). (See MacLennan 1997, section 6, for more details.)