Discrete impulses in ephaptically coupled nerve fibers

Modeling the excitation of nerve axons under transcutaneous stimulation

Computational models enable a safe and convenient way to study the excitation of nerve fibers under external stimulation. Contemporary models calculate the electric field distribution from transcutaneous stimulation and the resulting neuronal response separately. This study uses finite element methods to develop a multi-scale model that couples electric fields within macroscopic tissue layers and microscopic nerve fibers in a single-stage computational framework. The model included a triaxial myelinated nerve fiber bundle embedded within a volume conductor of tissue layers to represent the median nerve innervating the forearm muscles. The model captured the excitability of nerve fibers under transcutaneous stimulation and their nerve-tissue interactions to a transient external stimulus. The determinants of the strength-duration curve, rheobase, and chronaxie for the proposed model had close correlations with in-vivo experimentation on human participants. Additionally, the excitability indices for the triaxial myelinated nerve fiber implemented using the finite element method agreed well with experimental data from the literature. The validity of the proposed model encourages its use for applications involving transcutaneous stimulation. Capable of capturing field distribution across realistic morphologies, the model can serve as a testbed to improve stimulation protocols and electrode designs with subject-level specificity.

Knösche T. Modelling the effect of ephaptic coupling on spike propagation in peripheral nerve fibres

Experimental and theoretical studies have shown that ephaptic coupling leads to the synchronisation and slowing down of spikes propagating along the axons within peripheral nerve bundles. However, the main focus thus far has been on a small number of identical axons, whereas realistic peripheral nerve bundles contain numerous axons with different diameters. Here, we present a computationally efficient spike propagation model, which captures the essential features of propagating spikes and their ephaptic interaction, and facilitates the theoretical investigation of spike volleys in large, heterogeneous fibre bundles. We first lay out the theoretical basis to describe how the spike in an active axon changes the membrane potential of a passive axon. These insights are then incorporated into the spike propagation model, which is calibrated with a biophysically realistic model based on Hodgkin-Huxley dynamics. The fully calibrated model is then applied to fibre bundles with a large number of axons and different types of axon diameter distributions. One key insight of this study is that the heterogeneity of the axonal diameters has a dispersive effect, and that a higher level of heterogeneity requires stronger ephaptic coupling to achieve full synchronisation between spikes.

Ephaptic entrainment in hybrid neuronal model

In recent decades, there has been a growing interest in the impact of electric fields generated in the brain. Transmembrane ionic currents originate electric fields in the extracellular space and are capable of affecting nearby neurons, a phenomenon called ephaptic neuronal communication. In the present work, the Quadratic Integrated-and-Fire model (QIF-E) underwent an adjustment/improvement to include the ephaptic entrainment behavior between neurons and electric fields. Indeed, the aim of our study is to validate the QIF-E model, which is a model to estimate the influence of electric fields on neurons. For this purpose, we evaluated whether the main properties observed in an experiment by Anastassiou et al. (Nat Neurosci 14:217-223, 2011), which analyzed the effect of an electric field on cortical pyramidal neurons, are reproduced with the QIF-E model. In this way, the analysis tools are employed according to the neuronal activity regime: (i) for the subthreshold regime, the circular statistic is used to describe the phase differences between the input stimulus signal (electrode) and the modeled membrane response; (ii) in the suprathreshold regime, the Population Vector and the Spike Field Coherence are used to estimate phase preferences and the entrainment intensity between the input stimulus and Action Potentials. The results observed are (i) in the subthreshold regime the values of the phase differences change with distinct frequencies of the input stimulus; (ii) in the supra-threshold regime the preferential phase of Action Potentials changes for different frequencies. In addition, we explore other parameters of the model, such as noise and membrane characteristic-time, in order to understand different types of neurons and extracellular environment related to ephaptic communication. Such results are consistent with results observed in empirical experiments based on ephaptic phenomenon. In addition, the QIF-E model allows further studies on the physiological importance of ephaptic communication in the brain, and its simplicity may open a door to simulate the ephaptic response in neuronal networks and assess the impact of ephaptic communication in such scenarios.

Ephaptic coupling in white matter fibre bundles modulates axonal transmission delays

Axonal connections are widely regarded as faithful transmitters of neuronal signals with fixed delays. The reasoning behind this is that extracellular potentials caused by spikes travelling along axons are too small to have an effect on other axons. Here we devise a computational framework that allows us to study the effect of extracellular potentials generated by spike volleys in axonal fibre bundles on axonal transmission delays. We demonstrate that, although the extracellular potentials generated by single spikes are of the order of microvolts, the collective extracellular potential generated by spike volleys can reach several millivolts. As a consequence, the resulting depolarisation of the axonal membranes increases the velocity of spikes, and therefore reduces axonal delays between brain areas. Driving a neural mass model with such spike volleys, we further demonstrate that only ephaptic coupling can explain the reduction of stimulus latencies with increased stimulus intensities, as observed in many psychological experiments.

Axonal fibre bundles that connect distant cortical areas contain millions of densely packed axons. When synchronous spike volleys travel through such fibre bundles, the extracellular potential within the bundles is perturbed. We use computer simulations to examine the magnitude and shape of this perturbation, and demonstrate that it is sufficiently strong to affect axonal transmission speeds. Since most spikes within a spike volley are positioned in an area where the extracellular potential is negative (relative to a distant reference), the resulting depolarisation of the axonal membranes accelerates the spike volley on average. This finding is in contrast to previous studies of ephaptic coupling effects between axons, where ephaptic coupling was found to slow down spike propagation. Our finding has consequences for information transmission and synchronisation between cortical areas.

Action potential propagation and synchronisation in myelinated axons

With the advent of advanced MRI techniques it has become possible to study axonal white matter non-invasively and in great detail. Measuring the various parameters of the long-range connections of the brain opens up the possibility to build and refine detailed models of large-scale neuronal activity. One particular challenge is to find a mathematical description of action potential propagation that is sufficiently simple, yet still biologically plausible to model signal transmission across entire axonal fibre bundles. We develop a mathematical framework in which we replace the Hodgkin-Huxley dynamics by a spike-diffuse-spike model with passive sub-threshold dynamics and explicit, threshold-activated ion channel currents. This allows us to study in detail the influence of the various model parameters on the action potential velocity and on the entrainment of action potentials between ephaptically coupled fibres without having to recur to numerical simulations. Specifically, we recover known results regarding the influence of axon diameter, node of Ranvier length and internode length on the velocity of action potentials. Additionally, we find that the velocity depends more strongly on the thickness of the myelin sheath than was suggested by previous theoretical studies. We further explain the slowing down and synchronisation of action potentials in ephaptically coupled fibres by their dynamic interaction. In summary, this study presents a solution to incorporate detailed axonal parameters into a whole-brain modelling framework.

With more and more data becoming available on white-matter tracts, the need arises to develop modelling frameworks that incorporate these data at the whole-brain level. This requires the development of efficient mathematical schemes to study parameter dependencies that can then be matched with data, in particular the speed of action potentials that cause delays between brain regions. Here, we develop a method that describes the formation of action potentials by threshold activated currents, often referred to as spike-diffuse-spike modelling. A particular focus of our study is the dependence of the speed of action potentials on structural parameters. We find that the diameter of axons and the thickness of the myelin sheath have a strong influence on the speed, whereas the length of myelinated segments and node of Ranvier length have a lesser effect. In addition to examining single axons, we demonstrate that action potentials between nearby axons can synchronise and slow down their propagation speed.

Heart Rate and Extracellular Sodium and Potassium Modulation of Gap Junction Mediated Conduction in Guinea Pigs

Background: Recent studies suggested that cardiac conduction in murine hearts with narrow perinexi and 50% reduced connexin43 (Cx43) expression is more sensitive to relatively physiological changes of extracellular potassium ([K⁺]_o) and sodium ([Na⁺]_o).

Purpose: Determine whether similar [K⁺]_o and [Na⁺]_o changes alter conduction velocity (CV) sensitivity to pharmacologic gap junction (GJ) uncoupling in guinea pigs.

Methods: [K⁺]_o and [Na⁺]_o were varied in Langendorff perfused guinea pig ventricles (Solution A: [K⁺]_o = 4.56 and [Na⁺]_o = 153.3 mM. Solution B: [K⁺]_o = 6.95 and [Na⁺]_o = 145.5 mM). Gap junctions were inhibited with carbenoxolone (CBX) (15 and 30 μM). Epicardial CV was quantified by optical mapping. Perinexal width was measured with transmission electron microscopy. Total and phosphorylated Cx43 were evaluated by western blotting.

Results: Solution composition did not alter CV under control conditions or with 15μM CBX. Decreasing the basic cycle length (BCL) of pacing from 300 to 160 ms decreased CV uniformly with both solutions. At 30 μM CBX, a change in solution did not alter CV either longitudinally or transversely at BCL = 300 ms. However, reducing BCL to 160 ms caused CV to decrease more in hearts perfused with Solution B than A. Solution composition did not alter perinexal width, nor did it change total or phosphorylated serine 368 Cx43 expression. These data suggest that the solution dependent CV changes were independent of altered perinexal width or GJ coupling. Action potential duration was always shorter in hearts perfused with Solution B than A, independent of pacing rate and/or CBX concentration.

Conclusions: Increased heart rate and GJ uncoupling can unmask small CV differences caused by changing [K⁺]_o and [Na⁺]_o. These data suggest that modulating extracellular ionic composition may be a novel anti-arrhythmic target in diseases with abnormal GJ coupling, particularly when heart rate cannot be controlled.