**Q1. (total 15 points) **

- Describe what a perceptron is and how it can be used to categorize data (5 points )
- A set of data as an input to a perceptron are shown in Figure 1. The perceptron is initialized with weights w1 = -0.2, w2 = -0.6, and output function h = sgn(w1*S1+w2*S2).

The input data, A to D, is given as {s1, s2} = {(1,1), (1,-1), (-1, 1), (-1, -1)}, with class c = {-1, -1, +1, +1}. Black markers indicate class +1, white ones indicate class -1. The line marks the decision boundary of the perceptron, which classifies the region in the direction of the arrow as +1. Note that here, B and C are misclassified.

If the data is correctly classified no update is performed. If the data is misclassified, the learning rules are w_i(t+1) = w_i(2)-(eta)*s_i(t) if h = +1 and c(t) = -1 w_i(t+1) = w_i(t)+(eta)*s_i(t) if h = -1 and c(t) = +1

Using the sequence of data A,B,C,D,A,B,C,D,A… and learning rate (eta) = 0.5, what is the state of the perceptron after 2 steps? (5 points) How many steps are required until the perceptron converges (finishes learning)? (5 points)

**Q2. (total 20 points)**

- Describe how the periodic coding of the grid cells arises in the continuous attractor network model. (10 points)
- Provide two arguments
__against__the continuous attractor network as a potential model for grid cell generation. (10 points)

**Q3. (total 25 points) **

Describe what (a) tuning curve, (b) noise correlations and (c) signal correlation are (each 5 points )

(d) Pick an example of two cells with positive or negative signal or noise correlations (whichever you want) in response to two stimuli. Describe the consequences of these correlations on stimulus response mutual information (5 points) and optimal decision boundary for decoding the two stimuli (5 points)

**Q4.** **( total 30 points )**

Suppose that 5 stimuli (A, B, C and D) are presented to an animal with equal probabilities. The table below shows the probability of different spike counts recorded in the experiment in response to each of these stimuli.

stimulus/response |
5 spikes |
10 spikes |
15 spikes |
20 spikes |

stimulus A | 1/4 | 2/4 | 1/4 | 0 |

stimulus B | 2/5 | 3/5 | 0 | 0 |

stimulus C | 1/4 | 1/4 | 1/4 | 1 / 4 |

stimulus D | 0 | 0 | 1/2 | 1 / 2 |

stimulus E | 1/3 | 0 | 0 | 2 / 3 |

- What is the entropy of stimulus set? (5 points )
- What is the entropy of the responses? (5 points )
- What is the stimulus response mutual information? (10)
- Assume that after these measurements, someone shows one of the stimuli to the animal. If 20 spikes are recorded in response to this stimulus, what would be your best guess for which stimulus was presented according to the maximum likelihood and maximum a posteriori estimate? (10)

**Q5.** **(total 10 points) **Consider the following 3 layer network. The input values shown in ellipses on top of the figure are propagated through the network. The numbers next to each line show the synaptic weights and each neuronal transfer function is shown in a circle. As shown in the figure, neurons in the second layer are all binary neurons with threshold 1 except for the right most neuron which has a threshold of 0. Neurons in the second layer are threshold-linear neurons with threshold at 1 and a gain of 1.

Calculate input to and the firing rate of each of the 5 neurons. (2 points each)

**Q1. (total 15 points) **

- Describe what a perceptron is and how it can be used to categorize data (5 points )
- A set of data as an input to a perceptron are shown in Figure 1. The perceptron is initialized with weights w1 = -0.2, w2 = -0.6, and output function h = sgn(w1*S1+w2*S2).

The input data, A to D, is given as {s1, s2} = {(1,1), (1,-1), (-1, 1), (-1, -1)}, with class c = {-1, -1, +1, +1}. Black markers indicate class +1, white ones indicate class -1. The line marks the decision boundary of the perceptron, which classifies the region in the direction of the arrow as +1. Note that here, B and C are misclassified.

If the data is correctly classified no update is performed. If the data is misclassified, the learning rules are w_i(t+1) = w_i(2)-(eta)*s_i(t) if h = +1 and c(t) = -1 w_i(t+1) = w_i(t)+(eta)*s_i(t) if h = -1 and c(t) = +1

Using the sequence of data A,B,C,D,A,B,C,D,A… and learning rate (eta) = 0.5, what is the state of the perceptron after 2 steps? (5 points) How many steps are required until the perceptron converges (finishes learning)? (5 points)

**Q2. (total 20 points)**

- Describe how the periodic coding of the grid cells arises in the continuous attractor network model. (10 points)
- Provide two arguments
__against__the continuous attractor network as a potential model for grid cell generation. (10 points)

**Q3. (total 25 points) **

Describe what (a) tuning curve, (b) noise correlations and (c) signal correlation are (each 5 points )

(d) Pick an example of two cells with positive or negative signal or noise correlations (whichever you want) in response to two stimuli. Describe the consequences of these correlations on stimulus response mutual information (5 points) and optimal decision boundary for decoding the two stimuli (5 points)

**Q4.** **( total 30 points )**

Suppose that 5 stimuli (A, B, C and D) are presented to an animal with equal probabilities. The table below shows the probability of different spike counts recorded in the experiment in response to each of these stimuli.

stimulus/response |
5 spikes |
10 spikes |
15 spikes |
20 spikes |

stimulus A | 1/4 | 2/4 | 1/4 | 0 |

stimulus B | 2/5 | 3/5 | 0 | 0 |

stimulus C | 1/4 | 1/4 | 1/4 | 1 / 4 |

stimulus D | 0 | 0 | 1/2 | 1 / 2 |

stimulus E | 1/3 | 0 | 0 | 2 / 3 |

- What is the entropy of stimulus set? (5 points )
- What is the entropy of the responses? (5 points )
- What is the stimulus response mutual information? (10)
- Assume that after these measurements, someone shows one of the stimuli to the animal. If 20 spikes are recorded in response to this stimulus, what would be your best guess for which stimulus was presented according to the maximum likelihood and maximum a posteriori estimate? (10)

**Q5.** **(total 10 points) **Consider the following 3 layer network. The input values shown in ellipses on top of the figure are propagated through the network. The numbers next to each line show the synaptic weights and each neuronal transfer function is shown in a circle. As shown in the figure, neurons in the second layer are all binary neurons with threshold 1 except for the right most neuron which has a threshold of 0. Neurons in the second layer are threshold-linear neurons with threshold at 1 and a gain of 1.

Calculate input to and the firing rate of each of the 5 neurons. (2 points each)

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