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# AMATH 422/522 Problem set 2 solved

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## I Coding and strategies for simulating markov chains and dwell times.

through the third (and final) section of the MATLAB “lab manual,” posted online.

## II Dwell time distributions: theory.

Consider an ion channel with 4 open states and 2 closed states.
Give a mathematical argument, similar to that in class, that derives the typical functional form of the
dwell time distribution for the channel being in any one of the open states. Note: I am looking for a
derivation here, not just a statement of the answer or a “mantra” or result from class.

## III Simulating Markov chains and dwell times.

Models for stochastic switching among conformational states of membrane channels are somewhat
more complicated than the two-state models with which we started our discussions of Markov chains.
There are usually more than 2 states, and the transition probabilities are state dependent.

Moreover,
in measurements some states cannot be distinguished from others. We can observe transitions from
an open state to a closed state and vice versa, but transitions between open states (or between closed
states) are “invisible”. Here we shall simulate data from a Markov chain with 3 states, collapse that
data to remove the distinction between 2 of the states and then analyze the data to see that it cannot
be readily modeled by a Markov chain with just two states.

Suppose we are interested in a membrane current that has three states: one open state, O, and two
closed states, C1 and C2. As in the kinetic scheme discussed in class, state C1 cannot make a transition
to state O and vice-versa. We assume that state C2 has shorter residence times than states C1 or O.
Here is the transition matrix of a Markov chain that we will use to simulate these conditions. The
state S1 = 1 corresponds to C1, S2 = 2 corresponds to C2, and S3 = 3 corresponds to O:


.98 .1 0
.02 .7 .05
0 .2 .95



You can see from the matrix that the probability 0.7 of staying in state C2 is much smaller than the
probability 0.98 of staying in state C1 or the probability 0.95 of remaining in state O.
Our goal is to compute the distribution of dwell times in the closed state for this system.

website. (2) Modify it to simulate the three-state system above. We discussed how to do
this in class, splitting up the unit interval into more than three segments and using the
rand command. (3) To compute dwell times in the closed state, it’s convenient to make
a “reduced” list of states rstates after you’ve simulated to produce states.

In rstates,
you’ll lump together both closed states – say, giving them both the same numerical value
of 1. (4) Compute a list of the simulated dwell times in the closed state, make a histogram
of the dwell times, and see if it follows is indeed poorly fit by a single exponential, as
expected from class. Note that you might need to simulate for a long time to get a
sufficiently resolved histogram.
Note: Recall that many steps of the above, with code, were discussed in Lab manual part 3, section 3.

## IV Simulating Markov chains and neural spiking.

In electrically active cells, different ion channels
correspond to different membrane currents. These currents can be “inward,” tending to increase the
intracellular potential (such as the Na current), or “outward,” tending to decrease it (such as the K
current). What actually happens to this potential therefore depends on the balance between inward
and outward currents.

Here, we will consider an inward current carried by an ion channel with three states: one open state,
O, and two closed states, C1 and C2. As in the above, state C1 cannot make a transition to state O
and vice-versa. We assume that state C2 has shorter residence times than states C1 or O. Here is the
transition matrix of a Markov chain that we will use to simulate these conditions. The state S1 = 1
corresponds to C1, S2 = 2 corresponds to C2, and S3 = 3 corresponds to O:


.98 .1 0
.02 .7 .05
0 .2 .95



When a single inward channel is open, a current of +1 units flows through the channel.
We will consider an outward current that is carried by an ion channel that also has one open state,
O, and two closed states, C1 and C2. According to the same convention, the corresponding transition
matrix is:


.9 .1 0
.1 .6 .1
0 .3 .9



When a single outward channel is open, a current of −1 units flows through the channel.
Assume there are a total of Ninward = 100 inward channels, each evolving independently under a
realization of the Markov kinetics above. If ninward of these channels are in the open configuration at
timestep t, then the total inward current is +ninward.

Assume there are a total of Noutward = 50 outward channels, each evolving independently under a
realization of the Markov kinetics above. If noutward of these channels are in the open configuration at
timestep t, then the total outward current is −noutward units.

Thus, the net current into the cell at timestep t is ninward − noutward: the number of open inward
channels minus the number of open outward channels. In our model, the cell will produce an action
potential (spike) in a given timestep if this net current is greater than a threshold value T.

Assume that the channels have settled into equilibrium (i.e., that a time has passed that is large enough
since a simulation was initialized). Plot the probability that the cell will produce a spike in a given
timestep, as a function of the spiking threshold T. (That is, T should be on horizontal axis, and
a probability on the vertical axis.)

There are at least two ways of doing this: (1) by computing the
equilibrium state probabilities, and simulating many coin tossings, or (2) by computing the equilibrium
state probabilities, and using the form of the binomial distribution.

Repeat this for Ninward = 10, Noutward = 5 and for Ninward = 1000, Noutward = 500, and for any other
combinations of values you wish. Write a few sentences reporting on any qualitative changes that you
observe.

For AMATH 522, also either (a) give your best mathematical explanation of the trends that
you see, or (b) discuss their implication for the ways that neurons might be built in order to be reliable,
including one reference to the literature on “ion channel noise.”