2. A spike train, commonly used to study neural activity, is a sequence of recorded times at which a neuron fires an action potential (spike). The time in between consecutive spikes is called the interspike interval (ISI). Answer the questions below for an experiment in which the firing rate for a neuron is 10 per second. (a) Let X represent a single interspike interval (ISI) having an exponential distribution. State the distribution of X and give its parameter value(s). (2 points) (b) Give the expected value and variance for an interspike interval (ISI). (4 points) (c) What is the probability that an ISI is less than 0.07 seconds

a) X ~ exp ( 10 )

b) E(X) = 0.1 , Var (X) = 0.01

c)  P ( X < 0.07 ) = 0.00698

Step-by-step explanation:

Solution:-

- The spike train, used to study neural activity, the given time in between consecutive spikes (ISI) where the firing rate = 10 neurons per seconds.

- Denote a random variable "X"represent a single interspike interval (ISI) having an exponential distribution.

- Where X follows exponential distribution defined by event rate parameter i.e λ.

                               X ~ Exp ( λ )

- The event rate (λ) is the number of times an event occurs per unit time. Since we are studying a single interspike interval (ISI) - which corresponds to the firing rate. So, event rate (λ) = firing rate = 10 neurons per second. Hence, the distribution is:

                               X ~ Exp ( 10 )

- The expected value E(X) denotes the amount of time in which a single an event occurs; hence, the time taken for a single neuron.

                               E(X) = 1 / λ

                               E(X) = 1 / 10

                               E(X) = 0.1 s per neuron.

- The variance is the variation in the time taken by a single neuron to be emitted. It is defined as:

                               Var (X) = 1 / λ^2

                               Var (X) = 1 / 10^2

                               Var (X) = 0.01 s^2

- The probability that ISI is less than t = 0.07 seconds: P ( X < t = 0.07 s):

- The cumulative distribution function for exponential variate "X" is:

                                P ( X < t ) = 1 - e^(-λ*t)

- Plug the values and the determine:

                                P ( X < 0.07 ) = 1 - e^(-0.1*0.07)

                                                      = 1 - 0.99302

                                                      = 0.00698