CMPSCI/NSB 691C:
Homework #2: Prey Selection in Frog

Andrew H. Fagg

This is a programming exercise. It is due at 5 pm on Thursday, October 25th.

When a hungry frog is presented with a fly, he will snap at (and eat) the fly. When presented with multiple flies or stimuli that look like flys, the frog is faced with the problem of selecting one of the discrete objects before initiating a snap. As it turns out, frogs are also quiet good at taking into account the ``flyness'' of the stimuli in making the target selection (Ingle, 1968).

Didday (1970, 1976) suggested a winner-take-all circuit that selected one target from many. Amari and Arbib (1977) proposed a a biologically-inspired, distributed model of this circuit (figure 1). We assume a one-dimensional retina which delivers a value for each retinal position that corresponds to the ``flyness'' of a stimulus at that position (with a zero corresponding to no fly). This retinal input (I) is fed to a selection layer (L) which has the same dimensionality as the retina. These selection cells drive both a global inhibitor (S) and a motor output (M) that determines the direction of tongue snap.

  

Figure 1: Model of frog prey selection.

The dynamics of these cells are defined as follows:

$$\tau_L \frac{d L^{net}_i} {d t}$$ = - L^net_i + I_i + L_i - S

$$\left\{ \begin{array}{ll} 1 & \mbox{if~} L^{net}_i > thresh_L \\ 0 & \mbox{otherwise} \end{array}\right.$$ L_i =

$$\tau_S \frac{d S^{net}}{d t} -S^{net} + \sum_{i=1}^N{L_i}$$ =

$$\left\{ \begin{array}{ll} S^{net} - thresh_S & \mbox{if~} S^{net} > thresh_S \\ 0 & \mbox{otherwise} \end{array}\right.$$ S =

$$\tau_M \frac{d M}{d t}$$ = -M + T

$$\left\{ \begin{array}{ll} \frac{\sum^N_{i=1}{L_i * X_i}}{\sum^N_{... ... & \mbox{if~} \sum^N_{i=1}{L_i} > 0 \\ 0 & \mbox{otherwise} \end{array}\right.$$ T =

I is the vector of retinal inputs; L_i^net and L_i are the membrane potential and activation level of the i'th selection cell; S^net and S are the membrane potential and activation level of the inhibitor neuron; M is the motor output (indicating direction of snap); and X_i expresses the direction of snap given a selected stimulus at the i'th retinal location.

Note that once the global inhibitor is activated (by one or more of the selection units), it generates an inhibitory signal that suppresses the activation of all of the selection cells.

An experimental trial is conducted as follows:

• Initialize I with the visual input, and set L^net_i = 0, S^net = 0.

• Integrate the differential equations until an equilibrium is reached OR 5seconds has passed.

• Read the activation of the motor unit and consider the snap to have happened in that direction.

Assume the following parameters:

$$N \mbox{(number of retinal/selection units)}$$ = 21

$$10\;ms$$ timestep =

$$\tau_L 300\;msec$$ =

$$\tau_S 300\;msec$$ =

$$\tau_M 300\;msec$$ =

thresh_S = 0.1

thresh_L = 0.5

X_i = i-1 - (N-1)/2

Model Behavior

Assume a stimulus at location 4 and ``flyness'' of 1 (i.e. I₄ = 1).

Question 1.1: How long before our ``frog'' snaps? Show the evolution of the relevant state variables in a figure (i.e. L, L^net, and M).

Question 1.2: Explain the occurrence of the peak in L^net₄ - i.e. why does the level drop after the peak?

Question 1.3: If the flyness is only 0.55, how long before the snap occurs? Explain the difference?

Assume a stimulus of 1 at position 8 and a stimulus of 0.6 at position 17.

Question 1.4: Where does the frog snap and when?

Assume that these stimuli are swapped.

Question 1.5: where does the frog snap and when?

Suppose there is a stimulus of 1 at 17 and one of 0.8 at 8.

Question 1.6: Where and when does the frog snap?

Suppose there is a stimulus of 1 at 17 and one of 0.95 at 8.

Question 1.7: Where and when does the frog snap?

Suppose there is a stimulus of 1 at 17 and one of 0.96 at 8.

Question 1.8: Where and when does the frog snap? Describe the time-course of the relevant selection unit state variables.

Suppose there is a stimulus of 1 at 17 and one of 0.999 at 8.

Question 1.9: Where and when does the frog snap?

Hysteresis

Suppose a fly of flyness level 0.8 appears at retinal position 13, and then a second fly of level 1.0 appears $1\;sec$ later at position 2.

Question 2.1: When and where does the frog snap? Explain.

Question 2.2: What is the longest delay at which the second stimulus can be presented such that it wins the competition? When does the frog snap?

Return to a presentation delay of $1\;sec$.

Question 2.3: How high would the second stimulus have to be before it wins the competition? When does the frog snap in this case?

Avoiding Deadlock

As you have observed, there are stimulus conditions under which the frog circuit cannot make a snap decision before he is forced to snap - even though there are suitable stimuli at which to snap.

Question 3.1: Suggest and implement a simple, distributed mechanism that reduces the probability of our frog choosing the average fly. Be convincing in your demonstration that your fix works well; it does not have to work correctly every time (just often enough for our frog not to starve).

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CMPSCI/NSB 691C:
Homework #2: Prey Selection in Frog

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