Page 126 - Advances in Biomechanics and Tissue Regeneration

122 7. MULTISCALE NUMERICAL SIMULATION OF HEART ELECTROPHYSIOLOGY

the rate of change from the closed state to the open state is proportional to the concentration in the closed channels.
Therefore, we obtain

¼ αðVÞ½CŠ βðVÞ½OŠ:
d½OŠ
dt
Dividing this equation by the total density [O]+[C] we obtain
dg
¼ αðVÞð1 gÞ βðVÞg, (7.26)
dt
where g ¼ [O]/([O]+[C]) is the rate of open channels.
As α and β depend on V, it is not possible to use a general solution of Eq. (7.26). So Eq. (7.26) can be written as
dg
¼ðg ∞ ðVÞ gÞ=τ g ðVÞ, (7.27)
dt
with g ∞ ¼ α=ðα + βÞ and τ g ¼ 1/(α + β). Here, g ∞ and τ g are constants. The solution of Eq. (7.28)is
gðtÞ¼ g ∞ + ðg o g ∞ Þ e t=τ g , (7.28)
where g o is the initial value of g, and g ∞ is the value of g in the stable state.

7.2.3.4 Ionic Channels
The current through an ion channel can be computed using Ohm’s Law as the product of the channel conductance
times the potential difference between the membrane potential and the equilibrium potential for the specific ion
defined by the Nernst potential (Eq. 7.23)
I i ¼ gV E i Þ,
ð
where g is the permeability (or channel conductance) of the membrane to the ion i. Depending on the type of ion, g can
be either a constant or a function on the time and on the membrane potential as well as on the ionic concentrations. In
general, the conductance of a given channel is given as
g ¼ G max O,
where G max is the maximum channel conductance, and O is the probability that the channel is open.
As explained in the previous section, the cell membrane behaves as a condenser from an electric point of view due to
its dielectric characteristics. In addition, the proteins dissolved in the cell membrane form specific units that allow the
ionic exchange between the intracellular and extracellular space. Furthermore, some of these units, the ion channels,
possess a large specificity to certain ions. Hence, from an electric standpoint, the electric current flowing across the cell
membrane during activation can be described using a parallel conductance model, as shown in Fig. 7.2. It consists of a
capacitive current plus different currents associated with the different ions under consideration.
In this model, each of these current components is assumed to be independent, that is, each current utilizes its own
channel. The modern notation considers a positive current and potential from the inside to the outside.
From the circuit in Fig. 7.2, we obtain the following expression for the current through the membrane
n
dV X
J m ¼ C m ++ J ex + J pump + g j ðV E j Þ, (7.29)
dt
j¼1
where C m is the membrane capacitance, V is the membrane potential, and g j and E j are the conductance and the Nernst
potential for the ion j, respectively. J ex and J pump are currents associated with ion exchangers and ionic pumps (active
transport elements) present in the membrane.

FIG. 7.2 The equivalent circuit of the cell membrane.

I. BIOMECHANICS