Computational Neuroscience Homework 1 Solved

Question 1. [50 points] Assume that a neural population computes weighted linear combinations of its input x, characterized by a system of equations Ax = b. Here A is the transfer function and b is the output vector.


0
1
0
1
2
1
x1


0
1
1








x3












B
x

C





A single output measurement is recorded, given by

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3
0
9




=

9

.

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2
1
1
5
0
x2
1

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4
A







A B

4
C











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A





Answer the questions below. Derive the results by hand rst and then con rm them on the computer.

    a) Find all solutions xn to Ax = 0.

    b) Find a particular solution xp to Ax = b.

    c) Find all solutions to Ax = b.

    e) Find the pseudo-inverse of A.

    e) Find the sparsest solution to Ax = b (the solution with the least number of non-zero entries).

    f) Find the least-norm solution to Ax = b (the solution with the minimum Euclidean norm).
Question 2. [50 points]

‘Reverse inference’ is a common, albeit poorly exercised method in neuroscience. It refers to the practice of inferring that a cognitive process is engaged on the basis of activation in some brain area. For example, if Broca’s area was found to be activated in some task, researchers might infer that the subjects were using language. After a comprehensive search of the literature, we nd that Broca’s area was reported to be activated in 103 out of 869 fMRI tasks involving engagement of language, but this area was also active in 199 out of 2353 tasks not involving language.


    a) Assume that the conditional probability of activation given language and activation given no language, each follow a Bernoulli distribution (i.e., active with some probability p, or not with probability 1 p). Compute the likelihoods of observed frequencies of activation in literature, as functions of the possible values of their respective Bernoulli probability pa-rameters p = xl and p = xnl. Compute these functions at the values x =[0:.001:1] and plot them as separate bar charts.

    b) Find the values of xl and xnl that maximize their respective discretized likelihood func-tions.

    c) Using the likelihood functions computed for discrete x, compute and plot the discrete pos-terior distributions P (Xjdata) and the associated cumulative distributions P (X xjdata) for both processes (language and no language cases). To do this, assume a uniform prior P (x) / 1 and note that it will be necessary to compute (rather than ignore) the normalizing constant for Bayes’ rule. Use the cumulative distributions to compute (discrete approxima-tions to) upper and lower 95% con dence bounds on each proportion (xl;nl).

    d) Consider the joint posterior distribution P (Xl; Xnljdata) over xl and xnl, the Bernoulli probability parameters for the language and non-language contrasts. Given that these two frequencies are independent, the (discrete) joint distribution is given by the outer product of the two marginals. Plot it (with imagesc). Compute (by summing the appropriate entries in the joint distribution) the posterior probabilities that P (Xl > Xnljdata) and conversely that P (Xl Xnljdata).

    e) Using the estimates from part b as the relevant conditional probabilities, and assuming the prior that a contrast engages language, P (language) = 0:5, compute the probability P (languagejactivation) that observing activation in this area implies engagement of lan-guage processes. Is the critique on ‘reverse inference’ correct? How con dent should you be in implicating language if you observe activity in Broca’s area?