# CMPSCI/NSB 691C: Homework #1: Population Codes

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.

# Preferred Directions in Cartesian and Joint Space

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 L1 = L2 = 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?

Question 1.2: At position , what is the cell's apparent preferred direction?

Suppose an MI cell has a real'' preferred direction of .

Question 1.3: At position , what is the apparent preferred direction in joint space?

Question 1.4: At position , what is the cell's apparent preferred direction in joint space?

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:

 dis(diam) = ai + bi Gi(diam - diampref,i).

For cylinder cells:

 dic(diam, len) = ai + bi Gi(diam - diampref,i) + ci Gi(len - lenpref,i) .

For box cells:

 dib(len, width, height) = ai + bi Gi(len - lenpref,i) + ci Gi(width - widthpref,i) + fi Gi(height - heightpref,i) .

Where Gi() is Gaussian in shape:

 Gi(x) =

Another Answer: same as above, but with multidimensional Gaussians.

For sphere cells:

 dis(diam) = ai + bi Gi(diam - diampref,i).

For cylinder cells:

 dic(diam, len) =

For box cells:

 dib(len, width, height) =

Where Gi() is Gaussian in shape:

 Gi(x) =

and where Mi is a scaling matrix.

CMPSCI/NSB 691C:
Homework #1: Population Codes

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The translation was initiated by Andrew H. Fagg on 2001-10-04

Andrew H. Fagg
2001-10-04