Axonal Velocity Distributions in Neural Field Equations

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.Author Summary: Due to the sheer number of neurons and the complexity of their interactions, the modelling of brain activity is particularly challenging. How can computationally tractable models of brain function be developed that are nevertheless biologically plausible? The “mean field” approach, borrowed from statistical physics, is to model the average activity of populations of neurons rather than the behaviour of individual neurons. While a large number of promising theories have been developed with this approach, they fall short of biological fidelity in the way interactions between distant populations have been modelled. In particular, it is often assumed that all neurons interact via connections of very similar conduction velocity, when in fact experiment suggests quite the opposite: populations of neurons are connected by axonal fibres with a broad range of velocities. We develop here activity propagators that provide for the first time the ability to realistically and efficiently simulate connectivity in mean field theories, and demonstrate how to use them to fit successfully experimental data from both human and rat. With our novel propagators, one can thus study on an empirical basis the role of activity propagation in both healthy and diseased mammalian brains.