Realistic activity propagation for mean field models of human cortex

Abstract

Inspired by the great density of neurons in cortex (about 106 per macrocolumn), continuum mean field models (CMFMs) treat the cortex as one continuous neural medium. Interactions between neurons thus become flows of activity spreading in this medium. The most popular propagation model [1] is derived by a two-fold ansatz: a pulse of activity will on one hand spread isotropically with a conduction velocity c, and on the other hand its amplitude will decay exponentially with a distance scale σ. This ansatz reflects action potentials traveling with constant speed through axons and that the number of synapses axons form falls roughly exponentially with distance. However, a pulse in a CMFM means a mass of 105 to 106 neurons pulsing together. It is well known that axons can vary greatly in conduction speed, depending on their myelination and diameter. Hence it is natural to expect that such a mass has a broad conduction velocity distribution f(v), rather than a singular one f(v) = δ(v-c). Furthermore, one still has to expand for large wavelengths in order to derive a manageable partial differential equation (PDE) – a damped wave equation as it turns out. If one calculates the velocity distribution of this approximation, one sees that the Dirac delta peak has softened only towards lower velocities, leaving an unnatural distribution with a hard velocity cut-off. We hence propose a new propagation PDE: where Φ is the activity being propagated, S is a local source (e.g., a firing rate function), Nα the total number of connections, c is the conduction velocity and σ the decay parameter, and n > 0 will usually be chosen as an integer.