This indicates that the axon cross-sectional area decreases relatively more than the total fibre area. The ratio of the inner axon perimeter to the outer myelin perimeter remains constant at or near the optimal value of 0.
Furthermore, the thickness of the myelin remains constant for a given perimeter over the entire period of atrophy studied. This suggests that the number of turns of myelin and the length of each turn remain unchanged during peripheral nerve atrophy. 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. In larger axon bundles, however, it might be necessary to compute the ephaptic coupling from the local field potentials, as the lateral distance between axons may no longer allow for the distance-independent coupling we used here.
Nevertheless, it would be interesting to extend our framework to realistic axon bundle morphologies, and test if the predictions we make here, i. If yes, then there may also be the possibility that delays are modulated by the firing rates of neuronal populations.
To model action potential propagation along myelinated axons, we consider a hybrid system of active elements coupled by an infinitely long passive cable. The latter represents the myelinated axon and is appropriately described by the cable equation, whereas the active elements represent the nodes of Ranvier whose dynamics are governed by parametrically reduced, phenomenological dynamics. In general, a myelinated axon can be described by the following cable equation: 30 where V x , t is the trans-membrane potential, I chan V , t represents the ionic currents due to the opening of ion channels, and x represents the spatial coordinate longitudinal to the cable.
C m and R m are the capacitance and resistance of myelinated segments of the cable. All model parameters are listed in Table 1. The capacitance of a cylindrical capacitor such as a myelin sheath, or the insulating part of a coaxial cable can be found by considering the following relationship, 32 with g being the g-ratio, i.
The radial resistance of the cylinder is given by: The values for k 1 and k 2 correspond to the following values for permittivity and resistivity: With these constants at hand, we can now define the parameters of Eq 31 : The homogeneous part of Eq 40 has the solution The inhomogeneous solution in t can be found by the method of variation of the constant, which yields the following convolution integral in t : The inverse Fourier transform of Eq 42 then yields the following double convolution integral in x and t : Since we assume the nodes of Ranvier to be discrete sites described by delta functions in x , this integral becomes ultimately a convolution integral in time only.
A graphical representation of G x , t is given in Fig 10A for various values of x. We use the following values for R n [ 20 ] and C n [ 57 ]: 47 where , i. This value is striking, since typical time constants for neurons at dendrites and the soma range from 10ms to ms. This can be explained by the higher density of sodium channels at the nodes of Ranvier than at the soma.
Thus, the ratio of ion channel densities between node and soma is nearly The channel current that flows into the axon, I chan t is counter-balanced by currents flowing axially both ways along the axon, I cable t , and a radial current that flows back out across the membrane of the node, I node : This relationship yields It is, in general, not possible to find closed-form solutions to the Hodgkin-Huxley model due to the nonlinear dependence of the gating variables on the voltage.
We therefore focus here on idealisations of the currents generated by the ion channel dynamics, which is described by a function I chan t. In mathematical terms, the depolarisation of the neighbouring node is a convolution of the current entering the cable with the solution of the homogeneous cable equation G x , t , which describes the propagation of depolarisation along the myelinated axon: In the following we present the mathematical treatment for the scenarios introduced in the Results section, and we focus here on an input current at a single site.
The in mathematical terms simplest scenario is the one in which the ion current is described by the Dirac delta function: Without loss of generality we set the time of the current, t 0 , to zero. If only one current is injected into the cable, the time t sp when the threshold value V thr is reached is given implicitly by Eq 55 yields an implicit relation for t sp and the model parameters.
There is no obvious way of solving 55 for t sp explicitly. However, we explore here the possibility to derive an approximate solution for t sp , and consequently for the axonal propagation speed v , by linearisation of A suitable pivot for the linearisation is the inflection point on the rising branch, i. This ensures that the linearisation around this point is accurate up to order , and error terms are of order and higher.
It also provides an unambiguous pivot for the linearisation. Differentiating 54 twice yields We multiply all terms by t 4 such that the lowest order term in t is of order zero. The resulting quadratic equation for the inflection point, t i , yields two positive roots, the smaller of which is The linear equation for the time-to-spike and the firing threshold is then given by The quantities V t i and can be approximated to be 60 and A comparison of the full nonlinear solution with the linear approximation is shown in Fig 11A.
A : Depolarisation curve for instantaneous input current scenario A. If we denote by t 0 the time of the threshold crossing, then the ionic current is given by However, by simple linear transformation we may also use t 0 to denote the time of the spike. The speed of a propagating action potential is then given by 63 neglecting finite transmission speeds at nodes. However, if multiple neighbours are taken into account, the velocity can be faster than this estimate.
At this point, we make the assumption that the channel current rises infinitely fast, and drops off exponentially. In mathematical terms, the currents generated by an action potential at a particular node have the following form: 65 where I 0 denotes the amount of current generated by the channel dynamics, and t 0 denotes the time the spike is generated.
Here we use. We now briefly sketch how to solve this integral. Disregarding prefactors, the integral I to be solved here is of the form Using the substitution yields In addition, we define a second integral of the form Next, we apply the substitution to these two integrals, which yields 70 and The two integrals can be combined as follows: The integral on the right is straightforward to evaluate: And axons with a myelin sheath on them also conduct action potentials faster.
So first let's consider the diameter of an axon and how that affects the speed of action potential conduction. An axon with a larger diameter offers less resistance to the movement of ions down the axon, causing ions to move faster down the axon and causing the action potential to be conducted faster. Let me show you the way I think about this by just considering a single sodium ion that's entering an axon through a voltage-gated sodium channel.
So we'll say we just have a single sodium ion in here we're going to look at, even though there's many sodium ions flowing in through these voltage-gated sodium channels. And let's consider this for both-- the large diameter axon and the small diameter axon.
Here's our sodium ion in the large diameter axon. Now, both of these sodium ions, once they're inside the axon, could move in, really, an infinity of directions. They could kind of go in any direction and any degree of direction so that there's an infinity of pathways these ions could travel. Now, if they happened to travel backward in the axon, back toward the soma, or if they happen to travel perpendicular to the long axis of the axon, that won't really contribute much to the action potential other than the effects they'll have on the other sodium ions that are coming in.
But if they're heading in any of these directions down the length of the axon, that's going to contribute to moving the action potential down the axon. And the same is going to be true for this sodium ion in the small diameter axon. It could go in an infinity of directions as well, just like in a larger diameter axon.
But now let's consider the obstacles to this sodium ion moving down the axon. First, there's the membrane of the axon. And then there are all sorts of structures in the cytoplasm of the axon, such as vesicles or large proteins. And then, there are filaments, and there are tubules. And there are all sorts of structures in the cytoplasm that would pose an obstacle to the movement of this sodium ion. And the concentration of these obstacles would be the same in the smaller diameter axon.
But of course, there's less cytoplasm. So there would be fewer of these obstacles, but the same number for any given volume of cytoplasm. So if we consider this to represent the obstacles in the way of these sodium ions moving down the axon, what we see is that there are fewer potential pathways for the sodium ion to move down the axon in the smaller diameter axon before it runs into something in a fairly short distance.
So let's say this sodium ion heads this way. That gets it a pretty good ways until it collides into the axon membrane. Or if it goes this way, it also gets it a pretty good ways until it collides with the cell membrane. But if it heads this way, it collides into one of these cytoplasmic structures pretty quickly. Or if it heads in any of these directions, it runs in the axon membrane very quickly. Now, if we compare that to the sodium ion in the large diameter axon, it has more potential pathways to travel before it collides into something.
I mean, it certainly has the probability to collide into things in a pretty short distance.
0コメント