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 ( $\theta = \left[{\theta_s \atop \theta_e}\right]$) and the endpoint of the arm ( $X = \left[{x \atop y}\right]$). Specifically:

x = $\displaystyle L_1 \cos \theta_s + L_2 \cos(\theta_s + \theta_e)$  
y = $\displaystyle L_1 \sin \theta_s + L_2 \sin(\theta_s + \theta_e) .$  

The Jacobian, $J(\theta)$ describes the local linear transformation from joint velocities to Cartesian velocities:

$\displaystyle \dot{X}$ = $\displaystyle J(\theta)\dot{\theta}$  
  = $\displaystyle \left[
\frac{\partial x}{\partial \theta_s} &
\dot{\theta_s} \\


$\displaystyle \frac{\partial x}{\partial \theta_s}$ = $\displaystyle -L_1 \sin \theta_s - L_2 \sin(\theta_s+\theta_e)$  
$\displaystyle \frac{\partial x}{\partial \theta_e}$ = $\displaystyle - L_2 \sin(\theta_s+\theta_e)$  
$\displaystyle \frac{\partial y}{\partial \theta_s}$ = $\displaystyle L_1 \cos \theta_s + L_2 \cos(\theta_s+\theta_e)$  
$\displaystyle \frac{\partial y}{\partial \theta_e}$ = $\displaystyle L_2 \cos(\theta_s+\theta_e).$  

Figure: Kinematic model of a two-link arm. $\theta_s$ and $\theta_e$ are the joint angles for the shoulder and elbow, respectively. L1 and L2 are the link lengths. The origin of the Cartesian coordinate system is rooted at the shoulder.
\epsfig{file=arm.eps, width = 3in}\par\end{center}

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 $\dot{X}$ and $\dot{\theta}$).

Suppose that a cell in M1 has a ``real'' preferred direction in joint space of $\dot{\theta} = \left[ {1 \atop -1} \right]$.

Question 1.1: At position $\theta = \left[ {\frac{\pi}{4} \atop \frac{\pi}{2}}
\right]$, what is the apparent preferred direction in Cartesian space?

Question 1.2: At position $\theta = \left[ {0 \atop \frac{\pi}{2}}
\right]$, what is the cell's apparent preferred direction?

Suppose an M1 cell has a ``real'' preferred direction of $\dot{X} = \left[ {1 \atop 0} \right]$.

Question 1.3: At position $\theta = \left[ {\frac{\pi}{4} \atop \frac{\pi}{2}}
\right]$, what is the apparent preferred direction in joint space?

Question 1.4: At position $\theta = \left[ {0 \atop \frac{\pi}{2}}
\right]$, what is the cell's apparent preferred direction in joint space?

The model for cell discharge used by Georgopolous et al. was:

$\displaystyle d(\psi) = a + b \cos\left(\psi - \psi_{pref}\right),$     (1)

where $d(\psi)$ is the cell discharge rate, $\psi$ is the direction of movement in Cartesian space and a, b, and $\psi_{pref}$ 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 $\dot{\theta}_e$ and $\dot{\theta}_s$. Note that there is not a unique answer.

Georgopolous' ``cosine tuning function'' is but one way to describe the transformation of a distance metric (in this case $\theta_{movement} - \theta_{pref}$) 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.

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Homework #1: Population Codes

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Andrew H. Fagg