Page 8 - Cardiac Electrophysiology | A Modeling and Imaging Approach

P. 8
dV 1
dV

m m = − Ι • ion (2.1)
dt dt C m

In this equation, space-clamp conditions are assumed (V is the same at all locations of the
m
cell membrane) and no external stimulation is being applied. I is the total transmembrane ionic
ion
current through ion-selective protein channels, pumps and exchange mechanisms. Equation (2.1)
states that the rate of change of V is proportional to I . According to convention, a negative I
m ion ion
represents the flow of positive ions into the cell, producing a positive dV /dt to elevate (depolarize)
m
V . Conversely, a positive I is an outward current that reduces (repolarizes) the membrane
m ion
potential.

In general, I is the balance of several inward and outward currents that are carried by different
ion
ions and have different magnitudes and time course during the action potential; their dynamic
interplay determines the action potential amplitude, shape and duration. Following the
Hodgkin-Huxley formulation of the nerve axon action potential (the first computational model of

an action potential, published in 1952), each ion channel current is represented by its conductance
11
(g) times a driving force (V - E):
i m i
= g (
)
)
Ι = g i i i (V − ) (2.2)
E E E
V
g (V −
=
−
Ι Ι
m m m
i i i
i i i
E, the reversal potential at which I reverses direction, is determined by the transmembrane
i i
concentration gradients of permeant ions through the channel and their relative permeabilities.
The conductance may depend on membrane potential, time, intracellular and extracellular
concentrations of certain ions, and other factors such as ligand binding. The Hodgkin Huxley
formalism developed for the axon, was adapted for cardiac cells and used extensively for
computing the cardiac action potential.

The Hodgkin-Huxley scheme computes an ionic current, I , that flows through a large
i
ensemble of ion channels; it therefore constitutes a macroscopic approach. I is typically
i
expressed as current density through a unit area of membrane. The conductance, g , is computed
i
as a function of parameters (hypothetical “gates”) that provide voltage and time dependence. Each
gate can transition from a “closed” to an “open” position or from an open to a closed position as

V changes. The position of a given gate and its rate of transition are assumed independent of the
m
positions of all other gates. We use the sodium current, I , as an example:
Na

g •
E
Ι I Na Na = g Na Na (V − E ) ) (2.3)
m
Na Na