Axonal Velocity Distributions in Neural Field Equations

Ingo Bojak,David T J Liley,Lyle J Graham

doi:10.1371/journal.pcbi.1000653

Abstract

By modelling the average activity of large neuronal populations, continuum mean field models (MFMs) have become an increasingly important theoretical tool for understanding the emergent activity of cortical tissue. In order to be computationally tractable, long-range propagation of activity in MFMs is often approximated with partial differential equations (PDEs). However, PDE approximations in current use correspond to underlying axonal velocity distributions incompatible with experimental measurements. In order to rectify this deficiency, we here introduce novel propagation PDEs that give rise to smooth unimodal distributions of axonal conduction velocities. We also argue that velocities estimated from fibre diameters in slice and from latency measurements, respectively, relate quite differently to such distributions, a significant point for any phenomenological description. Our PDEs are then successfully fit to fibre diameter data from human corpus callosum and rat subcortical white matter. This allows for the first time to simulate long-range conduction in the mammalian brain with realistic, convenient PDEs. Furthermore, the obtained results suggest that the propagation of activity in rat and human differs significantly beyond mere scaling. The dynamical consequences of our new formulation are investigated in the context of a well known neural field model. On the basis of Turing instability analyses, we conclude that pattern formation is more easily initiated using our more realistic propagator. By increasing characteristic conduction velocities, a smooth transition can occur from self-sustaining bulk oscillations to travelling waves of various wavelengths, which may influence axonal growth during development. Our analytic results are also corroborated numerically using simulations on a large spatial grid. Thus we provide here a comprehensive analysis of empirically constrained activity propagation in the context of MFMs, which will allow more realistic studies of mammalian brain activity in the future.

Highlights

Since the introduction of continuum formulations for the dynamics of neural masses in cortical tissue [1,2,3,4,5,6], the interest in this class of neural mean field models (MFMs) has been steadily growing
Most modelling approaches (e.g., [25,26]) follow here the lead of the seminal paper by Jirsa and Haken [27], who employed several simplifying assumptions to describe long-range activity propagation with a partial differential equation (PDE). Their ansatz still assumes a single value for the cortico-cortical axonal conduction velocity, and conduction delays between neural masses are exactly proportional to their distance with one uniform constant
We will show below that approximations made in deriving the actual propagation PDE result in an implicit velocity distribution, which due to its origin remains strongly peaked at maximum conduction velocity and is onesided, i.e., there is an infinitely sharp cut-off at maximum speed

Summary

Introduction

Since the introduction of continuum formulations for the dynamics of neural masses in cortical tissue [1,2,3,4,5,6], the interest in this class of neural mean field models (MFMs) has been steadily growing. In order to incorporate the effects of such distributed activity a number of assumptions are typically made, the most important being a single value for the activity propagation delay between distant neural masses. Most modelling approaches (e.g., [25,26]) follow here the lead of the seminal paper by Jirsa and Haken [27], who employed several simplifying assumptions to describe long-range activity propagation with a partial differential equation (PDE) Their ansatz still assumes a single value for the cortico-cortical axonal conduction velocity, and conduction delays between neural masses are exactly proportional to their distance with one uniform constant. This may indicate the influence of natural selection optimizing information transmission in cortex

Objectives

Results

Discussion

Conclusion