Page 112 - YORAM RUDY BOOK FINAL

P. 112
the action potential V . Similarly, if the cell is in an extensive medium then ϕ << ϕ , ϕ ~V and
i
i
0
m
m
1 σ  1 
φ = i ∫ V ∇  •  d s
° 4 σ m  r 
π
o
s
The term in equation (4.8) is a solid angle element dΩ (The solid angle is the area inter-

cepted on a unit sphere by a cone connecting all points on the boundary of the double layer, S, to
the field point P; see Figure 4.2). Therefore, an equivalent expression to (4.8) is:

1
φ = 4πσ O o ∫ ( φσ i i −σ o φ ) Ωd (4.9)
o
°
πσ
s

For a resting (unexcited) cell, V = ϕ – ϕ is constant (the rest potential) over the entire cell
0
i
m
surface, which constitutes a closed surface. Because σ and σ are constants, σ ϕ – σ ϕ (the
° °
i
°
i.
i
double-layer strength) is also uniform and constant over the entire cell surface and can be
removed from under the integral operation:
1 1
∫
φ = ( φσ −σ φ ) d Ω = ( φσ −σ φ ) =Ω 0 (4.10)
o
4πσ i i o o 4πσ πσ i i o o
πσ
O o O o
The integral is zero because the solid angle, Ω , at an external field point, subtended by a
closed surface (the whole-cell membrane surface in this case) is zero (the solid angle viewed from
an internal field point within the cell is 4π ). Thus, as expected, a cell at rest does not contribute to
the extracellular potential field. Similarly, for cardiac cells at the plateau phase of the action

potential, V is constant (to a good approximation) over the entire cell surface and the cell docs
m
not contribute to ϕ . Therefore, a cell makes substantial contributions to ϕ only when V varies
0
0
m
during depolarization and repolarization. In the electrocardiogram, the QRS reflects the
depolarization phase and the T-wave reflects repolarization. The ST segment between them
returns to baseline (in the normal heart), reflecting the “silent” period during the action potential
plateau, when cells do not contribute to the ECG despite being in an excited state.