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Neural Signal Processing
Spring 2025
Neural Signal Processing
Midterm Review Problems
1. Refractory periods.
(a) In class, we introduced the exponential distribution to model inter-spike intervals (ISI). Does the exponential distribution incorporate the notion of a refractory period? Why or why not?
(b) Consider a model neuron, which spikes according to a homogeneous Poisson process. What is the maximum rate at which this model neuron can fire so that no more than 5% of its spikes violate a 1 ms refractory period?
2. A model neuron spikes according to a homogeneous Poisson process with rate λ = 50 spikes.s-1 . Again, the ISIs are exponentially distributed, i.e., T ~ exp(λ).
(a) What is the mean ISI of this neuron? What about the median ISI?
(b) What is the probability that a given ISI is greater than the mean ISI? What about the median ISI?
(c) What is the expected ISI given that no spikes were observed in the first 20 ms?
(d) What is the expected number of spikes that will be fired before one sees an ISI greater than 100 milliseconds?
(e) As you happily observe spikes from your model neuron, your recording equipment starts to mal- function, missing 50% of the spiking events. What is the mean ISI of this neuron now? What about the median ISI?
3. As discussed in class, it is possible (and often common) for spikes from different neurons to appear on the same recording electrode. Say that two neurons are contributing to the spikes seen on a particular electrode. Each neuron is spiking independently according to an homogeneous Poisson process with rate λ 1 and λ2 , respectively.
(a) Show that the raw spike train recorded on the electrode is an homogeneous Poisson process.
(b) What is the rate of this Poisson process?
(c) What is the probability that each neuron will spike once within the same 1ms interval?
4. Biological Background: In the knee-jerk spinal reflex, an extension of the quadriceps muscle (extensor) activates a sensory neuron that in turn activates an extensor motor neuron (leading to the contraction of the extensor muscle), and an inhibitory interneuron (leading to the relaxation of the flexor muscle). For this reflex to occur, the contraction of the extensor muscle must be synchronous with the relaxation of the flexor muscle.
Assume that an extensor motor neuron (Neuron 1) and the inhibitory interneuron (Neuron 2) fire independently according to a homogeneous Poisson processes with rates λ 1 and λ2 , respectively.
(a) What is the probability that the first spike from the motor neuron (Neuron 1) and the first spike from the inhibitory interneuron (Neuron 2) happen no more than τ seconds apart?
(b) What is the expected time between the first spike from the motor neuron (Neuron 1) and the first spike from the inhibitory interneuron (Neuron 2)? Note that the time between the two spikes is always positive.
(c) Assume now that the sensory neuron activates n extensor motor neurons and n inhibitory interneu- rons. Furthermore, assume that all extensor motor neurons and all inhibitory interneurons fire independently according to a Poisson process with rate λ . Determine n such that there’s a 99% chance the first spike from a motor neuron and the first spike from an inhibitory interneuron happen no more than t seconds apart.
5. Consider recording the average firing rates of 5 neurons (xi , i = 1, ...5) from primary motor cortex. Let’s assume that each neuron you record can be either a pyramidal tract neuron (A) or an inhibitory interneuron (B). In other words, each of your neurons belongs to either Class A (CA ) or Class B (CB ). The firing rates of pyramidal tract neurons are modeled by a Gaussian distribution with mean µ = 15 and variance σ = 15, while the firing rates of the firing rates of inhibitory interneurons are modeled by a Gaussian distribution with mean µ = 50 and variance σ = 10; in other words, the class conditional firing rate densities are given by:
Here, for simplicity, we ignore the fact that real firing rates are nonnegative.
(a) Given the following single samples for the firing rates of each of your five neurons, which class label maximizes the likelihood of the observing this data, assuming that the two classes are equally likely?
(b) Visualize your work by plotting the class conditional p.d.fs and class boundary used to fill out the table above. For clarity, the x-axis should vary between 0 and 70hz.
(c) How would your answer change if you were less likely to record from inhibitory neurons (i.e. P(CB ) < 0.5)?
6. A probabilistic generative model for classification comprises class-conditional densities P(x|Ck ) and class priors P(Ck ), k = 1,..., K. In this problem, we will consider the following naive Bayes model:
where xi is the ith element of the D-dimensional vector x. As shown, the spike counts xi ∈ {0,..., m} are independent across the D neurons, given the class Ck . The maximum spike count m is assumed to be constant.
(a) Consider the two class case, where P(C1 ) = π . The training set includes data points {xn , tn }, n = 1,..., N, where tn = 1 if xn ∈ C1 , 0 otherwise. Find the maximum likelihood parameters π and pki.
(b) The equation of the decision boundary can be written as f(x) = 0. Find f(x).
(c) Is the decision boundary linear? Why or why not?
7. Classification accuracy: Say you are recording from a neuron that responds to pictures of animals. It fires X1 spikes when shown dog pictures and X2 spikes when shown cat pictures. Say its firing is well-described as a Gaussian distribution with a fixed standard deviation σ and the mean is µ 1 for dog pictures and µ2 for cat pictures.
(a) Assume the neuron has equal experience with dog and cat pictures (i.e., equal priors), and for simplicity, it prefers cats to dogs (μ2 > μ 1 ). Say you observe X spikes - how will you infer if a dog or a cat picture was shown?
(b) Derive an expression for the accuracy of classification in terms of the parameters of the problem.
Comment on your answer by explaining how accuracy can be improved.