article · Chaos An Interdisciplinary Journal of Nonlinear Science
Mathematical analysis reveals the conditions required for modulated waves to form between two ephaptically coupled nerve fibres. Through a multiple scale expansion, coupled Hodgkin-Huxley equations simplify into a single differential-difference nonlinear equation. Linear stability analysis establishes the specific parameters that allow nonlinear wave structures to emerge, demonstrating that these instability traits depend strongly on both the discreteness parameter and the ephaptic coupling strength. Numerical simulations confirm these predictions and track the long-term behaviour of slightly perturbed waves. When a signal is applied to only one fibre, the interaction supports quasi-perfect communication between neurons. Under these conditions, active myelinated fibres can recruit neighbouring non-myelinated or damaged fibres, restoring or initiating electrical conduction along previously inactive neural pathways.
Understanding how nerve impulses spread between adjacent fibres without direct synaptic connections illuminates how neural tissue coordinates signals. Discovering that functional fibres can recruit damaged or unmyelinated pathways offers fundamental insights that could aid broader computational models of nerve repair, signal recovery, and conduction dynamics in injured neural networks.
The research represents early-stage theoretical and numerical modelling with no immediate commercial pathway described in the abstract. While the mechanism of recruiting damaged fibres might eventually offer foundational insights for researchers designing neuromodulation devices or neural repair therapies, practical use cases remain distant from this computational stage.
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We exclusively analyze the condition for modulated waves to emerge in two ephaptically coupled nerve fibers. Through the multiple scale expansion, it is shown that a set of coupled cable-like Hodgkin-Huxley equations can be reduced to a single differential-difference nonlinear equation. The standard approach of linear stability analysis of a plane wave is used to predict regions of parameters where nonlinear structures can be observed. Instability features are shown to be importantly controlled not only by the ephaptic coupling parameter, but also by the discreteness parameter. Numerical simulations, to verify our analytical predictions, are performed, and we explore the longtime dynamics of slightly perturbed plane waves in the coupled nerve fibers. On initially exciting only one fiber, quasi-perfect interneuronal communication is discussed along with the possibility of recruiting damaged or non-myelinated nerve fibers, by myelinated ones, into conduction.
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DOI: 10.1063/1.4919077
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