Biomedical Engineering Reference
In-Depth Information
and these ion currents used to adjust the intra- and local extracellular concentra-
tions of
X
at each step of the simulation.
The approach given in Eq. [1] can be applied in an even more detailed man-
ner when it is necessary to deal with the fact that a neuron does not necessarily
have the same membrane potential across its entire surface area. In this situation,
the cell may be treated as a continuous structure, leading to differential equa-
tions for
V
m
as a function of position as well as time (e.g. (4,5-8)). Alternatively,
Eq. [1] can be combined with the passive cable equations developed by Rall (9)
to form what is known as a "multicompartment" model (10). The basic idea is to
divide the neuron into subdivisions known as compartments. Each compartment
has its own
V
m
and conductance variables, which may be treated as Hodgkin-
Huxley channels or by some other formalism. Neighboring compartments are
assumed to be connected by cytoplasmic resistances that carry Ohm's law cur-
rents proportional to the voltage differences between those compartments. These
currents are included as
I
ext
variables in Eq. [1] and the entire system of equa-
tions is solved iteratively by computer. The number of compartments depends
on the problem to be solved and may range from just two (e.g., dendritic and
somatic) to many thousands. In the latter case, compartment geometry may be
derived from micrographs of real neurons (10). A typical application is the study
of the cerebellar Purkinje cell by De Schutter and Bower (11,12), in which
channel parameters were adjusted until a reasonable simulacrum of Purkinje cell
function was obtained. However, even with the large number of parameters used
in that study, certain aspects of Purkinje cell responses (e.g., the increasing fir-
ing rate seen for a time after a stimulating current is turned off) could not be
replicated (Figure 1). Thus, the added complexity of multicompartment model-
ing does not guarantee perfect reproduction of complex neuronal responses, and
for many purposes single-compartment models may be considered "good
enough."
We now consider in more detail methods that can be applied to single-
compartment models or to the individual compartments of multicompartment
models. While the Hodgkin-Huxley equations have been extremely influential in
showing the way accurately to simulate neuronal membrane function, it must be
said that the actual form of the equations for
G
Na
and
G
K
cannot be justified from
first principles. Furthermore, the equations as they stand are incomplete in that,
having been derived for the squid giant axon, they do not incorporate the nu-
merous varieties of voltage- and ion-dependent channels found in CNS neurons,
and they are computationally burdensome, which renders them impractical for
use in large-scale network simulations.
To address the latter limitation, a long line of work has sought simplified
versions of the Hodgkin-Huxley treatment that might afford a more rapid, if
somewhat less exact, computation of neuronal responses. An early example is
the FitzHugh-Nagumo model (13,14), which is analytically tractable and has
only two variables. The endpoint of this line of work is the "leaky integrate and
