Andrew H. Fagg
This assignment focuses on the population code interpretation of neural activity. It is a pen-and-paper exercise and should require at most 2 hours to complete (including writing time). It is due at 5 pm on Tuesday, October 2nd.
Figure 1 depicts a two-joint robot (or monkey) arm. The forward kinematics define the relationship between the arm's position in joint space ( ) and the endpoint of the arm ( ). Specifically:
x | = | ||
y | = |
The Jacobian, describes the local linear transformation from joint velocities to Cartesian velocities:
= | |||
= |
where:
= | |||
= | |||
= | |||
= |
We will assume that L_{1} = L_{2} = 1 and that a movement from a starting point to a target is very small. The latter allows us to assume that a movement can be expressed as an instantaneous velocity (i.e. as and ).
Suppose that a cell in MI has a ``real'' preferred direction in joint space of .
Question 1.1:
At position
, what is the apparent preferred direction in Cartesian space?
Answer:
Question 1.2:
At position
, what is the cell's apparent preferred direction?
Answer:
Suppose an MI cell has a ``real'' preferred direction of .
Question 1.3:
At position
, what is the apparent preferred direction in joint space?
Answer:
Question 1.4:
At position
, what is the cell's apparent preferred direction in joint space?
Answer:
The model for cell discharge used by Georgopolous et al. was:
(1) |
where is the cell discharge rate, is the direction of movement in Cartesian space and a, b, and are parameters.
Question 1.5:
Construct an equivalent model for a cell that encodes movement in
joint coordinates. In other words, give an expression for the cell
discharge rate as a function of
and
.
Note that there is not a unique answer.
= | (2) |
Georgopolous' ``cosine tuning function'' is but one way to describe the transformation of a distance metric (in this case ) into a cell discharge rate. It happens to be very convenient because relative orientation and the cosine function are both periodic in nature. But - it does not have to be this way...
Suppose that instead of coding movement direction, we would like to encode a description of object shape and size (e.g., as extracted by the visual system). The shapes and parameters that we would like our population of cells to encode are:
Shape | Parameters |
sphere | diameter |
cylinder | diameter, length |
box | length, width, height |
Question 1.6:
Give one possible population coding scheme for this set of objects.
In other words, for the set of cells, write an expression (or
expressions) representing the cells' discharge rate as a function of
the exact object being coded.
One Answer: Assume that there are three separate populations of neurons -
one for each of the different shapes. For any particular shape, only
those cells that encode that shape will be active.
For sphere cells:
d_{i}^{s}(diam) | = | a_{i} + b_{i} G_{i}(diam - diam_{pref,i}). |
For cylinder cells:
d_{i}^{c}(diam, len) | = | a_{i} + b_{i} G_{i}(diam - diam_{pref,i}) + c_{i} G_{i}(len - len_{pref,i}) . |
For box cells:
d_{i}^{b}(len, width, height) | = | a_{i} + b_{i} G_{i}(len - len_{pref,i}) + c_{i} G_{i}(width - width_{pref,i}) | |
+ f_{i} G_{i}(height - height_{pref,i}) . |
Where G_{i}() is Gaussian in shape:
G_{i}(x) | = |
Another Answer: same as above, but with multidimensional Gaussians.
For sphere cells:
d_{i}^{s}(diam) | = | a_{i} + b_{i} G_{i}(diam - diam_{pref,i}). |
For cylinder cells:
d_{i}^{c}(diam, len) | = |
For box cells:
d_{i}^{b}(len, width, height) | = |
Where G_{i}() is Gaussian in shape:
G_{i}(x) | = |
and where M_{i} is a scaling matrix.
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