Fig 1.
Action potential propagation in a myelinated axon.
A: The axon is made of myelinated segments (internodes), with the nodes of Ranvier forming periodic gaps in the myelin sheath. B: The nodes of Ranvier constitute active sites at which threshold-triggered ion channel currents are released. C: The currents entering nearby nodes of Ranvier determine the membrane potential at each node, thus forming an action potential. D: The velocity of an action potential is determined by the distance L between two consecutive nodes, and the time difference tsp it takes to reach a given threshold value.
Fig 2.
Sketch of ion channel currents considered here, with representative profiles of membrane potential in nearby nodes.
After the membrane potential V reaches the threshold value Vthr, the current I is released. A: The instantaneous current is described by a delta-peak at t0, when the threshold value is reached. B: The simplest way to accommodate delays or refractoriness is to introduce a refractory period Δ, after which the instantaneous current is released. C: Exponential current with characteristic time scale τsp. D: A combination of exponential currents describes a realistic current profile.
Fig 3.
Channel currents divide into a current entering the axon and a current flowing back across the node of Ranvier.
A: Sketch of currents entering and leaving a node of Ranvier. B: Plot of currents as function of node length. Since we assume constant channel density, the channel current increases linearly with the node length.
Fig 4.
Contribution of ion currents from nearby nodes to action potential profile.
A: Sodium currents contributing to action potential, and B: same for potassium. Depolarising effect is color-coded by node index, larger indices are lumped. Total effect is indicated by black line. C: Action potential composed of both currents. D: Contribution of sodium currents to reaching threshold value. Standard parameters are used here (Table 1 in Methods).
Fig 5.
Propagation velocity as function of fibre diameter and axon diameter.
A: In myelinated axons, the relationship between velocity and fibre diameter is nearly linear, with a slightly supralinear relationship at small diameters. Here we compare the different scenarios with experimental results (grey-shaded area). B: In unmyelinated axons, the propagation speed increases approximately with the square root of the axon diameter. Here, ρ indicates the relative ion channel density compared with a node of Ranvier. Decreasing the ion channel density results in slower action potential propagation.
Fig 6.
Velocity dependence on node length and internode length.
A: Propagation velocity plotted against node length and internode length. Contours indicate percentages of maximum velocity. (Scenario D with standard parameters.) B: Same as A, with fitted parameters. C: Propagation velocity as function of internode length (scenario D with fitted parameters), and comparison with numerical results from biophysical model. D: Propagation velocity as function of node length, and comparison with the model by Arancibia-Carcamo et al. [24].
Fig 7.
Relative propagation velocity as function of g-ratio.
A: Result of our spike-diffuse-spike model, and v = κ(ln(1/g))α fitted to this result (first with α = 0.5 fixed, and then with κ and α fitted). B: Fitted α changes with the ratio of internode length to node length in the spike-diffuse spike model (lines), and in the biophysical model (dots). Parameters: fitted parameters (see Table 1 in Methods section).
Fig 8.
Effect of diameter and g-ratio on propagation velocity.
A: Velocity plotted against g-ratio and axon diameter.B: Velocity plotted against g-ratio and fibre diameter.
Fig 9.
Ephaptic coupling reduces AP speed and leads to AP synchronisation.
A: Depolarisation curves for a pair of action potentials with initial offset of 0.02ms converge, reducing the time difference between action potentials. B: Depolarisation of a synchronous pair of action potentials is slower than for a single action potential. C: An action potential induces initial hyperpolarisation and subsequent depolarisation in an inactive neighbouring axon. Parameters: standard parameters, .
Table 1.
List of model parameters used in this manuscript.
Fig 10.
Green’s function of the cable equation.
A: Green’s function for various distances x. B: Green’s function for x = 1mm, showing the slow (dotted) and fast (dash-dotted) approximation.
Fig 11.
Depolarisation curves and their linear approximation.
A: Depolarisation curve for instantaneous input current (scenario A). B: Depolarisation curve for exponential input current (τs = 100μs).
Fig 12.
Comparison of action potentials in the spike-diffuse-spike model and the biophysical model.
We chose the time scales τm = 20μs and τh = 40μs such that the profile, and in particular the rising phase of the action potential in the spike-diffuse-spike model matches well the action potential of the cortical axon model by Arancibia-Càrcamo et al. [24].
Fig 13.
Effect of number of nearest nodes on velocity.
We demonstrate here that considering only a small number of nodes can lead to considerable discrepancies in the computed velocity at small node and internode lengths. A: N = 1000, as in Fig 6A. B: N = 30. C: N = 10.