Biomedical Engineering Reference
In-Depth Information
CHAPTER 8
Extracellular Recording and
Stimulation
The theory we have derived so far is largely based upon transmembrane potentials and currents. Recording
V m or I m in a real preparation, however, is nontrivial and nearly impossible in the clinic. In most situations
it is, therefore an extracellular potential that is recorded. In this chapter we will use Maxwell's equations
to derive a relationship between extracellular potentials and membrane currents.
8.1 MAXWELL'S EQUATIONS APPLIEDTONEURONS
Maxwell's equations describe how an electric field creates a distribution of electric potentials that can be
recorded everywhere in space. One mechanism by which an electric field may arise is if a current source is
present in a large conductive bath of fluid. In the physics and electrophysiology literature, a large bath is
called a volume conductor . If we assume a small current, I o , emanates from a point in a uniform conducting
medium of infinite extent, the current will flow radially in all directions. By placing a small sphere of
radius, r , around the current source, we can compute the flux, J , through the sphere as
I o
4 πr 2 a r
J
=
(8.1)
where 4 πr 2 is the surface area of the sphere and a r is a unit vector in the radial direction.
Although an electric field can theoretically change at the speed of light, biological systems are
much slower. Therefore, all analysis of biological system can assume changes in the electric field to be
quasi-static . The practical meaning is that if we record a potential, even if it is far from the source, it will
reflect the current at exactly that time. In reality, this is not true but the field changes much faster than
the biological current can change. Given this assumption
E
=−∇ φ e
(8.2)
where E is the electric field vector and
φ e is the gradient of extracellular potentials in the bath.To relate
the gradient of φ e
to flux, we can use the general form of Ohm's Law:
=−
σ e
J
φ e
(8.3)
where σ e is the conductivity of the bath. Far from any current sources, Eq. (8.3) reduces to Laplace's
Equation.
2 φ e =
0 .
(8.4)
 
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