A mathematical model of ephaptic interactions in neuronal fiber pathways: could there be more than transmission along the tracts?

Abstract

In the past several decades, there has been numerous experimental and modeling efforts to study ephaptic interactions in neuronal systems. While studies on the matter have looked at either axons of the peripheral nervous system or cortical neuronal structures, no attention has be given to the possibility of ephaptic interactions in the white matter tracts of the brain. Inspired by the highly organized and tightly packed geometry of axons in neuronal fiber pathways, we aim to theoretically investigate the potential effects of ephaptic interactions along these structures that are resilient to experimental probing. For that end, we use axonal cable theory to derive a minimal model of a sheet of N ephaptically coupled axons. We numerically solve the equations and explore the dynamics of the system as the ephaptic coupling parameter is varied. We demonstrate that ephaptic interactions can lead to local phase locking between impulses traveling along adjacent axons. As ephaptic coupling is increased, traveling impulses trigger new impulses along adjacent axons resulting in finite size traveling fronts. For strong enough coupling, impulses propagate laterally and backwards, resulting in complex spatio-temporal patterns. While it is common for large scale brain network models to assume the role of brain fiber pathways to be that of mere transmission of signals between different brain regions, our work calls for a closer re-examination of the validity of such a view. The results suggest that in the presence of significant ephaptic interactions the brain fiber tracts can act as a dynamic active medium.\n\nAuthor summaryStarting from local circuit theory and the Fitzhugh-Nagumo cable model of an axon, we derive a system of nonlinear coupled partial differential equations (PDEs) to model a sheet of N ephaptically coupled axons. We also put forward a continuous limit approximation that transforms the model into a field equation in the form of a two-dimensional PDE that allows for the extension of the model to a 3D domain. We numerically solve the equations and explore the dynamic responses as the ephaptic coupling strength is varied. We observe that ephaptic interaction allows for phase locking of adjacent impulses and coordination of subthreshold dynamics. In addition, when strong enough, ephaptic interaction can lead to the generation of new impulses along the axons as well as lateral and backward propagation in the form of traveling fronts and complex spatio-temporal patterns. The transition between different dynamic regimes happens abruptly at critical values of the parameter. We also compare the dynamics of the two models and find good qualitative correspondence in certain parameter regimes. The results put into question the validity of assuming the role of fiber pathways to be that of mere interneuronal transmission and calls for further investigation of the matter.