Biomedical Engineering Reference
In-Depth Information
Fig. 3.
Hodgkin-Huxley and Markov-type models of
I
Kr
channel gating.
a
State diagram of
classical Hodgkin-Huxley-type model. The conductive state of the channel is controlled by two
independent activation and inactivation gates (
x
and
y
, respectively), resulting in four different
channel states. C
00
, C
01
, and C
10
are the closed states of the channel and O is the open state.
b
State diagram of Markov-type scheme [56]. C
l
, C
2
, and C
3
are closed states, O is the open state,
and I the inactivated state. All transition rates, except
K
f
and
K
b
, are a function of membrane
potential. Ȍ is defined as a function of other transition rates to satisfy the microscopic
reversibility condition.
conformation of the ion channel proteins. A disadvantage is that Markov models often
add several differential equations (and parameters) to the model, thus increasing the
model complexity and computation time. In the recent model of a mouse ventricular
cell by Shannon et al. [77], Markov channel models were therefore generally avoided
unless they were determined to be necessary.
Markov models should not be confused with stochastic models. Cardiac cells
models employing Markov-type channel gating are as deterministic as models
employing Hodgkin-Huxley channel gating. In either case, the number of ion
channels occupying a certain channel state is represented as a fraction (between 0 and
1) that changes with time as a continuous number. Stochastic models are non-
deterministic (probabilistic) models in which the stochastic 'random' openings and
closures of individual ion channels are taken into account. The number of ion
channels occupying a certain channel state is then represented as a discrete number
that changes 'randomly' and in discrete steps with time. As a consequence, action
potentials show 'channel noise' and beat-to-beat fluctuations, like experimental
recordings [88, 92]. Stochastic ionic models of SA nodal pacemaker cells have been
developed by Guevara and Lewis [29] and Wilders et al. [88], whereas Greenstein and
Winslow have developed an ionic model of the canine ventricular myocyte
incorporating stochastic gating of sarcolemmal L-type calcium channels and
ryanodine-sensitive sarcoplasmic reticulum (SR) calcium release channels [28].
2.6 Computational Aspects
The original computations by Hodgkin and Huxley [34] were done by hand. As
memorized by Huxley in his Nobel lecture, “this was a laborious business: a
membrane action took a matter of days to compute, and a propagated action potential
took a matter of weeks.” For his initial Purkinje fibre action potential simulations,
Noble [59, 60] could make use of the university computer for which he had to write a
program in machine code. It took 2 h of CPU time to simulate a single action potential
model (with only five variables). In the 1970s, Beeler and Reuter [1] used a
mainframe computer to solve the differential equations of their 8-variable ventricular
