Stochastic Modelling of Single Ion Channels - Computational Neuroscience: A Comprehensive Approach - page 144

where, as usual, the inverse of a matrix is denoted by raising it to the power -1. The
results for the duration of a sojourn in a single state, Equation (5.15), are obviously
obtained from this when Q oo is replaced by a scalar q ii . A very important thing to
notice is that, if we carry out a spectral resolution of the matrix

−

Q oo , we see that,
like the derivation of the relaxation equation 5.11, the probability density function
f o can be expressed as a mixture of exponential components with time constants

given by the m o eigenvalues of

( t )

− Φ

oo . Note that, unlike the matrix

− Q , it will not

normally have a zero eigenvalue.
Similarly, just by interchanging o and c , we get the probability density of shut
times as

f c ( t )= − Φ

c exp

( Q cc t ) Q cc u c

with mean shut time

c Q − 1

µ c = − Φ

cc u c

u c is a column vector of m c 1's and Φ c a row vector of probabilities for the initial

state of the shut time calculated by an expression like Equation (5.17) with the o and
c interchanged. This can be expressed as a mixture of exponential components with
time constants given by the m c eigenvalues of the matrix

Q cc .
So the distributions of open times and shut times tell us something about the num-
bers of open states and shut states, again with the caveat that we might not be able to
distinguish all components from an experimental record.

−

5.5.3 Joint distributions

Useful information about the structure of a channel mechanism may be obtained by
looking at joint distributions of adjacent intervals: for example, does a long open
interval tend to be followed by a short shut interval or vice versa? The problem can
be approached by defining a transition density matrix G oc ( t o ) which has dimension

m o × m c and whose ijth element is a joint probability/probability density for the du-

ration of an open time, T o , and the shut state that the system moves to when the open
time ends, all conditional on starting in the ith open state: i.e.,

g ij ( t o )= lim

0 Prob

( t o ≤ T o ≤ t o + ∆ t o and enter jth closed state

t o →

| starting in ith open state )

This is given simply by

G oc ( t o )= R o ( t o ) Q oc = exp ( Q oo t ) Q oc (5.22)

If we are only interested in the shut state that is entered at the end of the open
time, and not the duration of the open time, we can integrate the above expression
with respect to t o to obtain a transition probability matrix

∞

Q − 1 G oc = G oc ( t o ) dt o = −

oo Q oc

(5.23)

0