Biomedical Engineering Reference
In-Depth Information
appears to be one long open time may actually be two or three open times separated
by shut times that are too short to be distinguished. This is known as the problem
of time interval omission (TIO). One way of coping with this problem is to study
the total burst length or the total open time per burst, as these should not be very
sensitive to missing short shut times.
In order to take account of missed events when dealing with the individual open-
ings and shuttings it has been the custom to assume that there is a critical constant
dead-time,
, such that all open times or shut times greater than this are observed ac-
curately but times less than this are missed (note that a safe
value can be imposed
retrospectively on recorded data). We then work with
apparent open times
defined
as periods that start with an open time of duration greater than
which may then be
extended by a number of openings separated by shut periods each of duration less
than
; they are terminated at the start of a shut period of duration greater than
.
Apparent shut times may be similarly defined.
A number of people presented approximate solutions for the distributions of ap-
parent open times and apparent shut times before Ball and Sansom [15, 16] obtained
exact results in the form of Laplace Transforms and also considered the effect of
TIO on correlations between interval durations. Exact expressions for the probabil-
ity density function of apparent open times and shut times were found by Hawkes et
al. [46]. These are fine for small to moderate values of time t, but when trying to
compute them they tend to be numerically unstable for large
t
. These results were
also studied by Ball, Milne and Yeo [9, 10] in a general semi-Markov framework. In
a series of papers, Hawkes et al. [46] and Jalali and Hawkes [50, 51] obtained asymp-
totic approximations that are extremely accurate for values of
t
from very large right
down to fairly small; for small
t
the exact results are readily obtainable, so that the
distributions are obtained over the whole range. Ball [3] studied these approxima-
tions further and showed mathematically why they are so very good. It is interesting
to note that, if the true distribution is a mixture of
k
exponentials, then the approxi-
mation to the distribution of apparent times allowing for TIO is also a mixture of
k
exponentials: the time constants are, however, different.
These methods were subsequently applied by Colquhoun et al. [34] to study the
effect of TIO on joint distributions of apparent open and shut times. They also used it
to calculate the likelihood of a complete series of intervals and used this to estimate
the parameters of any postulated mechanism - thus generalising the work of Ball and
Sansom [17], who used similar methods for the ideal non-TIO case. For a given
model, TIO can induce some indeterminacy in the process of estimating parameters.
For data recorded under fixed conditions there can be two sets of parameters that
seem to fit the data about equally well: typically a
fast solution
and a
slow solution
.
A simple example is given by Colquhoun and Sigworth [35]. These can, however, be
discriminated by observing the same channel under different conditions of voltage
and/or ligand concentration, see [5, 20, 75].
Colquhoun et al. [33] and Merlushkin and Hawkes [62] studied the TIO problem
in the context of recording apparent open and shut intervals elicited by a pulse of
agonist concentration or voltage change.
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