  Extracellular local field potentials (LFP) are usually modeled as arising
from a set of current sources embedded in a homogeneous extracellular medium.
Although this formalism can successfully model several properties of LFPs, it
does not account for their frequency-dependent attenuation with distance, a
property essential to correctly model extracellular spikes. Here we derive
expressions for the extracellular potential that include this
frequency-dependent attenuation. We first show that, if the extracellular
conductivity is non-homogeneous, there is induction of non-homogeneous charge
densities which may result in a low-pass filter. We next derive a simplified
model consisting of a punctual (or spherical) current source with
spherically-symmetric conductivity/permittivity gradients around the source. We
analyze the effect of different radial profiles of conductivity and
permittivity on the frequency-filtering behavior of this model. We show that
this simple model generally displays low-pass filtering behavior, in which fast
electrical events (such as Na$^+$-mediated action potentials) attenuate very
steeply with distance, while slower (K$^+$-mediated) events propagate over
larger distances in extracellular space, in qualitative agreement with
experimental observations. This simple model can be used to obtain
frequency-dependent extracellular field potentials without taking into account
explicitly the complex folding of extracellular space.
