A great variability is found in the velocity of the propagation of action potentials. In fact, the propagation velocity of the action potentials in nerves can vary from meters per second miles per hour to less than a tenth of a meter per second 0. Why do some axons propagate information very rapidly and others slowly? In order to understand how this process works, it is necessary to consider two so-called passive properties of membranes, the time constant and the space or length constant.
Why are these called passive properties? They have nothing to do with any of the voltage-dependent conductances discussed earlier. They have nothing to do with any pumps or exchangers. They are intrinsic properties of all biological membranes. Time Constant. First, consider a thermal analogue. Place a block of metal at 10 o C on a hotplate at o C. How would the temperature change? It will increase from its initial value of 10 o C to a final value of o C.
But the temperature will not change instantly. In fact, it would change as an exponential function of time. An analogous situation occurs in nerve cells, when they receive an instantaneous stimulus. The figure at right represents an idealized nerve cell. The recording electrode initially measures a potential of mV the resting potential. At some point in time time 0 , the switch is closed.
The switch closure occurs instantaneously and as a result of the instantaneous closure, instantaneous current flows through the circuit. This is equivalent to slamming the block of metal on the hotplate.
Note that despite the fact that this stimulus changes instantly, the change in potential does not occur instantaneously. It takes time for the potential to change from its initial value of mV to its final value of mV. There is a total of 10 mV depolarization, but the change occurs as an exponential function of time. There is a convenient index of how rapidly exponential functions change with time. Thus, the time constant is 10 msec. The smaller the time constant, the more rapid will be the change in response to a stimulus.
These fast-response action potentials in non-nodal tissue are altered by antiarrhythmic drugs that block specific ion channels. Sodium-channel blockers such as quinidine inactivate fast-sodium channels and reduce the rate of depolarization decrease the slope of phase 0. Calcium-channel blockers such as verapamil and diltiazem affect the plateau phase phase 2 of the action potential. Potassium-channel blockers delay repolarization phase 3 by blocking the potassium channels that are responsible for this phase.
Once an action potential is initiated, there is a period of time comprising phases 0, 1, 2, 3 and early phase 4 that a new action potential cannot be initiated see figure at top of page. During the ERP, stimulation of the cell by an adjacent cell undergoing depolarization does not produce new, propagated action potentials. This occurs because fast sodium channels remain inactivated following channel closing during phase 1.
Remember, sodium has a positive charge, so the neuron becomes more positive and becomes depolarized. It takes longer for potassium channels to open. When they do open, potassium rushes out of the cell, reversing the depolarization. Also at about this time, sodium channels start to close. This causes the action potential to go back toward mV a repolarization. The action potential actually goes past mV a hyperpolarization because the potassium channels stay open a bit too long.
Gradually, the ion concentrations go back to resting levels and the cell returns to mV. Lights, Camera, Action Potential This page describes how neurons work. Resting Membrane Potential When a neuron is not sending a signal, it is "at rest.
Action Potential The resting potential tells about what happens when a neuron is at rest. And there you have it Do you like interactive word search puzzles? In the following we present the analytical solutions for all the current types. I 0 is chosen such that the amplitude of an action potential is approximately mV , with all the other parameters chosen as for scenario D with standard parameters, see Methods section.
The threshold condition Eq 10 then reads Although this is the simplest scenario, it is not obvious how to invert the r. The spatio-temporal evolution of an action potential is now given by 18 and the threshold condition reads In the Methods section we show how to obtain this solution.
Eq 20 thus represents solutions for ion currents with instantaneous onset and exponential decay. Hence, the spatio-temporal evolution of an action potential is expressed by 21 and the threshold condition to determine t sp is The linearity of the cable equation allows us to recur to the solution for scenario C to describe the response to currents described by multiple exponentials.
We use this formulation to describe both sodium currents and potassium currents with rising and falling phase. The sodium current is expressed as follows: The potassium current is modelled as 26 throughout the manuscript. Hence, the spatio-temporal evolution of an action potential is expressed by 27 with , and C is the problem-specific normalisation constant. The threshold condition to determine t sp is Anticipating results from the next subsection, we found that scenarios A and C yield velocities that are too fast compared with experimental results.
As it is the most realistic and most flexible model for ion channel currents, we decided to select scenario D to study the sensitivity of the propagation speed to structural parameters. There is a wide consensus that the propagation velocity in myelinated axons is proportional to the axon diameter. This is mostly due to the fact that both the internode length as well as the electrotonic length constant increase with the diameter.
One quantity that does not scale linearly with the axonal diameter is the node length, which determines the amount of current that flows into the axon, as well as setting a correction term for the physical and electrotonic distance between two nodes.
We find that the latter introduces a slight nonlinearity at small diameters, although at larger diameters the linear relationship is well preserved, see Fig 5A. 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. Decreasing the ion channel density results in slower action potential propagation. In Fig 5A we compare the four ion channel scenarios with experimental results obtained by Boyd and Kalu [ 53 ]. Scenario A instantaneous ion channel current yields velocities that are about one order of magnitude larger than the experimental results.
This suggests that the main bottleneck for faster action potential propagation is indeed ion channel dynamics and their associated delays. Introducing a hard delay with scenario B, we find that we can reproduce the experimentally observed range of velocities. With scenarios C and D we introduce temporally distributed ion channel dynamics.
The instantaneous onset and exponential decay of scenario C yields velocities that are slightly faster than experimental results. In scenario D we explore two sets of parameters. The first set of parameters is obtained by using electrophysiological parameters found in the literature. As it is not obvious how to choose the time constants governing the temporal profile of the ion channel currents, we decided to choose them such that the shape of action potentials of our spike-diffuse-spike model match the shape of action potentials of the biophysical model used by Arancibia-Carcamo et al.
The velocities obtained with this set of parameters fall within the range of experimental results. The second set of parameters is obtained by fitting the model parameters to data generated by the same biophysical model see Methods. The latter yields velocities slightly below the experimental range, but it matches well the results from the biophysical model. We find that reducing the ion channel density also decreases the propagation velocity.
Two geometric parameters that are not readily accessible to non-invasive MRI techniques are the length of the nodes of Ranvier, and the length of internodes. Here we examine the effect of the node and internode length on the speed of action potentials. We assume that the channel density in a node is constant, which is in agreement with experimental results [ 52 ]. The channel current that enters the node is proportional to its length, yet the increase of the node length also means that more of this current flows back across the node rather than entering the internodes.
Another effect of the node length is the additional drop-off of the amplitude of axonal currents. The length of internodes is known to increase with the fibre diameter [ 21 , 22 ]. We restrict the analysis to the activation by sodium currents, since potassium currents are slow and only play a minor role in the initial depolarisation to threshold value.
The results are shown graphically for scenario D with standard parameters in Fig 6A , and for parameters fitted to the biophysical model by Arancibia-Carcamo et al. Changing the threshold value did have a small effect on the maximum velocity, but did not change the relative dependence on the other parameters.
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. We find that the propagation velocity varies relatively little with changes in the nodal and internodal length. Interestingly, we find that decreasing node length and internode length simultaneously, the velocity increases steadily. In Fig 6C and 6D we show cross-sections of Fig 6B , and compare these results with numerical results from the cortex model used in [ 24 ].
There is a good agreement between our model and the biophysical model, with the biggest discrepancies occurring at short node and internode lengths. In the Methods section, we show that reducing the number of nodes significantly alters the results at short node and internode lengths Fig 13 in Methods section. The relative thickness of the myelin layer is given by the g-ratio, which is defined as the ratio of inner to outer radius. Hence, a smaller g-ratio indicates a relatively thicker layer of myelin around the axon.
In humans, the g-ratio is typically 0. A classical assumption is that the propagation velocity scales in the same manner [ 1 ]. Our results suggest see Fig 7A that the velocity depends more strongly on the g-ratio. The latter represents the case of an unmyelinated axon.
Parameters: fitted parameters see Table 1 in Methods section. In Fig 8 we present two-parameter plots of the velocity as function of the g-ratio and axon diameter Fig 8A , and g-ratio and fibre diameter Fig 8B. If the axon diameter is held constant, the velocity increases monotonically with decreasing g-ratio. A : Velocity plotted against g-ratio and axon diameter. B : Velocity plotted against g-ratio and fibre diameter. We demonstrate here that it is possible to study the effects of ephaptic coupling on action potential propagation within our framework.
We choose two axonal fibres as a simple test case, but more complicated scenarios could also be considered using our analytical approach.
Ephaptic coupling occurs due to the resistance and finite size of the extra-cellular space. We follow Reutskiy et al. The resulting cable equation for the n th axon reads 29 with V e being the potential of the extra-cellular medium. In the Methods section we describe how to obtain solutions to this set of equations. We explore solutions to Eq 29 in a number of ways, which are graphically represented in Fig 9.
We focus on sodium currents as described by scenario D with standard parameters. First, we study how the coupling could lead to entrainment, i. Next, we asked how the coupling affects the speed of two entrained action potentials. We compare the depolarisation curves of the simultanously active axons with when only one axon is active, and find that the voltages rise more slowly if two action potentials are present, thus increasing t sp and decreasing the speed of the two action potentials, see Fig 9B.
Thirdly, we considered the case when there is an action potential only in one axon, and computed the voltage in the second, passive axon. We find that the neighbouring axon undergoes a brief spell of hyperpolarisation, with a half-width shorter than that of the action potential. This hyperpolarisation explains why synchronous or near-synchronous pairs of action potentials travel at considerably smaller velocities than single action potentials.
The hyperpolarisation is followed by weaker depolarisation. A : Depolarisation curves for a pair of action potentials with initial offset of 0. 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,. We have developed an analytic framework for the investigation of action potential propagation based on simplified ion currents.
Instead of modelling the detailed dynamics of the ion channels and its resulting transmembrane currents, we have adopted a simpler notion by which a threshold value defines the critical voltage for the ion current release.
Below that threshold value the membrane dynamics is passive, and once the threshold value is reached the ion current is released in a prescribed fashion regardless of the exact time-course of the voltage before or after.
We studied four different scenarios, of which the simplest was described by a delta-function representing immediate and instantaneous current release. The three other scenarios incorporated delays in different ways, from a shift of the delta function to exponential currents and, lastly, combinations thereof. The latter seemed most appropriate considering experimental results.
The simplified description of the ion currents permitted the use of analytical methods to derive an implicit relationship between model parameters and the time the ion current would depolarise a neighbouring node up to threshold value. From the length of nodes and internodes and the time to threshold value between two consecutive nodes t sp resulted the velocity of the action potential. We only obtained an implicit relationship between the threshold value V thr and the parameter t sp , which needed to be solved for t sp using root-finding procedures.
However, in comparison to full numerical simulations, our scheme still confers a computational advantage, as the computation time is about three orders of magnitude faster than in the biophysical model by Arancibia-Carcamo et al. In the Methods section we have shown that one can achieve a good approximation by linearising the rising phase of the depolarisation curve.
We did not explore this linearisation further, but in future work it might serve as a simple return-map scheme for action potential propagation, in which parameter heterogeneities along the axon could be explored.
We used our scheme to study the shape of action potentials, and we found that the ion currents released at multiple nearby nodes contribute to the shape and amplitude of an action potential. This demonstrates that action potential propagation is a collective process, during which individual nodes replenish the current amplitude without being critical to the success or failure of action potential propagation.
Specifically, the rising phase of an action potential is mostly determined by input currents released at backward nodes, whereas the falling phase is determined more prominently by forward nodes cf. Fig 4. Our scheme allowed us to perform a detailed analysis of the parameter dependence of the propagation velocity.
We recovered previous results for the velocity dependence on the axon diameter, which were an approximately linear relationship with the diameter in myelinated axons, and a square root relationship in unmyelinated axons. Although the node and internode length are not accessible to non-invasive imaging methods, we found it pertinent since a previous study [ 24 ] looked into this using numerical simulations. Our scheme confirms their results qualitatively and quantitatively, and performing a more detailed screening of the node length and the internode length revealed that for a wide range the propagation velocity is relatively insensitive to parameter variations.
Intuitively, changing the thickness of the myelin sheath of relatively short internodes has a smaller effect than changing the myelin thickness around long internodes relative to the node length.
The main results of our spike-diffuse-spike model were compared with the biophysically detailed model recently presented by Arancibia-Carcamo et al. The latter uses the Hodgkin-Huxley framework and models the myelin sheath in detail, including periaxonal space and individual myelin layers. To enable the comparison between the two models, we fitted parameters of our spike-diffuse-spike model to output of the biophysical model. In spite of the differences in the model setup, we find that the results of the two models agree well.
The framework developed here also allowed us to study the effect of ephaptic coupling between axons on action potential propagation. We found that the coupling leads to the convergence between sufficiently close action potentials, also known as entrainment. It has been hypothesised that the functional role of entrainment is to re-synchronise spikes of source neurons. We also found that ephaptic coupling leads to a decrease in the propagation speed of two synchronous action potentials.
Since the likelihood of two or more action potentials to synchronise in a fibre bundle increases with the firing rate, we hypothesise that a potential effect could be that delays between neuronal populations increase with their firing rate, and thereby enable them to actively modulate delays. In addition, we examined the temporal voltage profile in a passive axon coupled to an axon transmitting an action potential, which led to a brief spell of hyperpolarisation in the passive axon, and subsequent depolarisation.
This prompts the question whether this may modulate delays in tightly packed axon bundles without necessarily synchronising action potentials. The three phenomena we report here were all observed by Katz and Schmitt [ 55 ] in pairs of unmyelinated axons. Our results predict that the same phenomena occur in pairs or bundles of myelinated axons. There are certain limitations to the framework presented here. First of all, we calibrated the ion currents with data found in the literature.
This ignores detailed ion channel dynamics, and it is an open problem how to best match ion currents produced by voltage-gated dynamics with the phenomenological ion currents used in this study. Secondly, we assumed that the axon is periodically myelinated, with constant g-ratio and diameter along the entire axon.
The periodicity ensured that the velocity of an action potential can be readily inferred from the time lag between two consecutive nodes. In an aperiodic medium, the threshold times need to be determined for each node separately, resulting in a framework that is computationally more involved. Here it might prove suitable to exploit the linearised expressions for the membrane potential to achieve a good trade-off between accuracy and computational effort. If individual internodes are homogeneous, then one could probably resort to methods used in [ 36 ] to deal with partially demyelinated internodes.
Thirdly, we studied ephaptic coupling between two identical fibres as a test case. Our framework is capable of dealing with axons of different size too, as well as large numbers of axons.
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