Page 111 - YORAM RUDY BOOK FINAL

P. 111
(4.1) and (4.2) in terms of ψ gives:

ψ −ψ = σ φ −σ φ ≠ 0
i ° i i ° ° (4.4)

and

∂ψ i = ∂ψ °

n ∂ n ∂
(4.5)

Stated in words, (4.4) means that ψ is discontinuous and (4.5) that its normal derivative is
continuous at the cell membrane (surface S in Figure 4.1). These properties of ψ identify the cell
membrane as the location of a double layer 244 (a layer of dipolar sources) of strength τ equal to the
discontinuity inψ :

→ ∧
τ = ( ψ i −ψ ° )η (4.6)

The potential field, ψ, generated by this double layer source is 245 :

1  1 
ψ = ∫ ( ψ i −ψ )  •  ds ds
∇
4π °  r 
s (4.7)

∆ is the gradient operator, r is the distance from source to field point; integration is over the entire
cell surface S.
∆

Substituting σ ϕ for ψ in (4.7) according to (4.3) and restricting all field points of interest
to the extracellular space (i.e., σ = σ at these points) provides an expression for the extracellular
°
potential field, ϕ , that is generated by an excitable cell:
°
1
 1 
∇
φ = ∫ ( φσ i i −σ o φ )  •  ds dS (4.8)
o
o
4πσ πσ
O o s  r 

)
The term ( φσ i i − σ o φ o ds is a dipole element associated with a small area ds of the
cell membrane. Integration over all dipole elements in the entire cell membrane, S, yields the

potential generated by whole-cell excitation. The double layer (dipolar) nature of the membrane
source is consistent with the process of excitation, namely the flow of ionic current across the
membrane. Each element of current involves a source and a sink, hence constituting a dipole.
Note that if σ = σ , equation (4.8) reduces to φ = 1 ∫ V ∇  1  •  d s and ϕ can be related directly to

0
i
0
o
4π m  r 
s