Fig 1.
Computing the extracellular potential (EP) generated by a volley of spikes.
A: An action potential, as expressed by the membrane potential Vm along the axial dimension z, generates an EP that varies with z and the distance from the axon d. B: An action potential in an active axon perturbs the membrane potential of a passive axon via the EP. C: We consider spike volleys travelling along axonal fibre bundles, and D: infer from the EP the cumulative effect on the membrane potential of a passive axon.
Fig 2.
Spatial profiles of action potentials and their EPs.
Shown are A: the piecewise linear profile, B: the piecewise quadratic profile, and C: the profile of an action potential generated with the biophysical model. D-F: EPs corresponding to action potential profiles in A-C. G-I: Log-log plots of the EPs (absolute values) at z = 0. Black lines indicate decay with d−3. (The notch at d ≈ 0.3mm is due to a change of sign).
Fig 3.
EP at the centre of a circular axon bundle due to concentric spike volleys.
A: Microscopic cross-section of a fibre bundle, with spike-carrying axons marked in blue. B: Macroscopic extension of (a), with the active area (i.e. where axons carry spikes) marked in blue. C: Waveform of a spike (top), and the resulting spatial (axial) profile of the EP at the centre of the fibre bundle. D: Cross-sections of C.
Fig 4.
EP in fibre bundle with synchronous spike volley, subject to position of reference point.
A: The reference point is moved from the centre of the fibre bundle to a position outside of the fibre bundle. B: Waveform of a spike (top), and the resulting EP plotted against the longitudinal coordinate z and the distance of the reference point from the centre. C: Cross-sections of B.
Fig 5.
Increasing the length of a spike volley attenuates the amplitude of an EP.
The EP is shown for varying bundle diameters and z. We steadily increase the width Δz of the spike volley from A: Δz = 0mm, to F: Δz = 50mm.
Fig 6.
Illustration of properties of the computational model.
A: Distribution of axon diameters sampled from a shifted alpha distribution to match experimental data [35]. B: Rastergram of spike volley generated at proximal end of fibre bundle. C: Rastergram of spike volley reaching the distal end of the fibre bundle. D: Distribution of delay times. E: Snapshot of the longitudinal profile of EP generated by a spike volley. F: The EP modulates the spiking threshold (Vthr) and therefore the delay Δt of action potential generation between two reference points (e.g. two consecutive nodes of Ranvier).
Fig 7.
Comparison of the spike propagation model with a biophysical model.
A synchronous spike volley slows down as a result of ephaptic coupling in fibre bundles with identical axons. The relative change of the propagation velocity varies with the bundle diameter.
Fig 8.
Increasing the stimulus intensity, i.e. the number of spikes in a volley, decreases axonal transmission times and the latency of stimulus response.
A-C: Mean axonal delay with ephaptic coupling (solid) and without ephaptic coupling (dashed) for A: 1ms, B: 2ms, and C: 3ms stimulus duration. D-F: Standard deviation from the mean of axonal delay with ephaptic coupling (solid) and without ephaptic coupling (dashed) for D: 1ms, E: 2ms, and F: 3ms stimulus duration. Mean and standard deviation are computed from the distribution of delay times (cf. Fig 6D). G-I: Latency from stimulus onset to first maximum in neural mass model at G: 1ms, H: 2ms, and I: 3ms stimulus duration. Lines (shaded areas) indicate mean (1σ confidence interval) across 5 simulations. Colours indicate different bundle diameters.
Table 1.
List of parameters used for the spike propagation model.
Table 2.
List of parameters used for the biophysical model.
Table 3.
List of parameters used for the Jansen-Rit model.