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Open in new window. Go to Editor View. The actomyosin network has emerged as a central regulator of the intrinsic axial tension i. In vivo , in embryonic Drosophila , when motor neuron axons are slackened mechanically, axons shorten within 2—4 min and restore their straight configuration Tofangchi et al.
This longitudinal contractility decreases dramatically after myosin II knockdown and inhibition, namely by using ML-7, an inhibitor of myosin light chain kinase Saitoh et al. Further supporting the involvement of myosin II in longitudinal axonal contractility, disruption of actin filaments in embryos treated with cytochalasin D and latrunculin A Spector et al.
Interestingly, the rate of contraction is faster when microtubules are disrupted by either nocodazole or colchicine Tofangchi et al. Using trypsin-mediated detachment in chick embryo DRG neuron cultures, as an alternative model to evaluate axonal longitudinal contraction Mutalik et al.
In contrast to the Drosophila model, in chick DRG neurons nocodazole reduces axonal contraction Mutalik et al. As such, the role of microtubules in longitudinal axonal contraction awaits further clarification. Similarly to axial tension, circumferential tension is also regulated by myosin-II Figure 1. These results suggest that the actomyosin machinery is contractile along the circumferential direction of axons such that the relaxation of tension results in increased axonal diameter.
The authors also raised the hypothesis that circumferential tension applies a compressive force on microtubules, and that the force balance between cortical actin and microtubule results in an equilibrium diameter of the axon. Accordingly, a decrease in axon diameter was observed when microtubules were disrupted with nocodazole or colchicine.
In this context, it is interesting to note that the membrane periodic skeleton is thought to interact with axonal microtubules Zhong et al. Additionally, in Drosophila neurons, formins actin nucleators that crosslink actin and microtubules were shown to contribute to the actin-spectrin network Qu et al.
As such, one cannot exclude that cytoskeleton crosslinkers may also be involved in modulating axon diameter. Overall, a mechanism in which longitudinal and circumferential axonal tension are coupled, is supported by recent data Fan et al.
However, the cytoskeletal arrangement that underlies axial and circumferential axonal contractility, and their coupling, remains to be resolved. Several data support that axon diameter is dynamic and regulated by activity-dependent mechanisms. In fact, nerves and axons from different species including giant axons of squid Iwasa and Tasaki, ; Tasaki and Iwasa, , crayfish Hill et al. In the squid giant axon, swelling starts nearly at the onset of the action potential and the peak of swelling during excitation coincides accurately with the peak of the action potential recorded intracellularly Iwasa and Tasaki, Of note, concurrently with axon swelling, longitudinal shortening of nerve fibers occurs Tasaki and Iwasa, ; Tasaki et al.
This observation points towards the similarities between the mechanical changes that occur in the muscle and those that take place in the nerve Tasaki and Byrne, Recently, the inability of conventional light microscopy to resolve thin unmyelinated axons, which can have diameters well below nm i.
In these settings, whereas synaptic boutons underwent a rapid transient enlargement that decayed, the axon shaft showed a more delayed and progressive increase in diameter, swelling gradually over the duration of the experiment approximately 1 h. This initial phase was followed by a sustained increase in conduction speed when the axon shaft widened. In summary, increasing axon diameters accelerated action potential conduction along the axons.
This finding is in line with cable theory Goldstein and Rall, as axons of increased diameter have less internal electrical resistance, which facilitates the spread of action potential. This study shows that activity-dependent changes in the nanoscale axon morphology modify the speed of action potentials along hippocampal unmyelinated axons, revealing a new layer of complexity for the regulation of axon physiology. Lately, new insights have been provided for the physical mechanisms that may account for the increase in axon diameter during action potential firing Berger et al.
The axon initial segment, located in the proximal axon of multipolar neurons, is the region where action potentials are generated Ogawa and Rasband, The mechanisms that underlie the assembly of the axon initial segment in the proximal axon and its plasticity are still poorly understood. Recently, it has been demonstrated that phosphorylated myosin light chain is highly enriched in the axon initial segment Berger et al. Using STORM nanoscopy, the authors proposed that activated phospho-myosin light chain associates with the periodic actin cytoskeleton forming actomyosin rings at the axon initial segment, and raised the hypothesis that myosin II filaments are oriented parallel to the actin rings.
It is however interesting to speculate the possible existence of an alternative model, in which myosin filaments might not be exclusively oriented in parallel to actin rings, thus providing the additional control of longitudinal axonal contractility Figure 1.
The cytoskeletal arrangement that regulates circumferential axonal contraction and expansion is just starting to be unveiled. Understanding its structure is likely to be a challenging enterprise as it will most probably rely on the simultaneous detection of multiple axonal components by super-resolution microscopy.
Additionally, although axons are generally depicted as straight regular structures, even in vitro their courses can be tortuous with complex twisting that may hamper withdrawing straightforward conclusions. If, as suggested by an emergent body of literature, an actomyosin network participates in the fine control of axon diameter, a possible interplay between the axonal subcortical cytoskeleton and the deep axonal actin filaments, as putative anchors of myosin filaments, may need to be explored.
This is certainly a very exciting field of research that will further the notion that the cytoskeleton in the axon shaft is likely to be much more dynamic than initially expected. The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
We are indebted to Dr. Marco Lampe EMBL, Heidelberg , for all the enthusiastic technical and conceptual support provided to our analysis of the axonal cytoskeleton.
Alfei, L. Cytoskeletal components and calibers in developing fish mauthner axon Salmo-Gairdneri Rich. Benshalom, G. Ultrastructural observations on the cytoarchitecture of axons processed by rapid-freezing and freeze-substitution. Berger, S. Localized myosin II activity regulates assembly and plasticity of the axon initial segment. Neuron 97, Bray, D.
Axonal growth in response to experimentally applied mechanical tension. Superresolution imaging reveals activity-dependent plasticity of axon morphology linked to changes in action potential conduction velocity. U S A , — Dennerll, T. Tension and compression in the cytoskeleton of Pc neurites II: quantitative measurements. Cell Biol. STED nanoscopy reveals the ubiquity of subcortical cytoskeleton periodicity in living neurons. Cell Rep. Subcortical cytoskeleton periodicity throughout the nervous system.
Local modulation of neurofilament phosphorylation, axonal caliber and slow axonal-transport by myelinating schwann-cells.
Cell 68, — Fan, A. Coupled circumferential and axial tension driven by actin and myosin influences in vivo axon diameter. Fields, R. Signaling by neuronal swelling. Friede, R. Axon caliber related to neurofilaments and microtubules in sciatic nerve fibers of rats and mice. Gallo, G. Myosin II activity is required for severing-induced axon retraction in vitro. Ganguly, A. A dynamic formin-dependent deep F-actin network in axons.
Gardner, K. Modulation of spectrin actin assembly by erythrocyte adducin. Nature , — George, E. Axonal shortening and the mechanisms of axonal motility. Cell Motil. Cytoskeleton 9, 48— Gilligan, D. U S A 96, — Goldstein, L. Flying through the Drosophila cytoskeletal genome. Goldstein, S. Changes of action potential shape and velocity for changing core conductor geometry. Gotow, T. Dephosphorylation of the largest neurofilament subunit protein influences the structure of crossbridges in reassembled neurofilaments.
Cell Sci. PubMed Abstract Google Scholar. Greenberg, M. Irregular geometries in normal unmyelinated axons: a 3D serial EM analysis. Hammarlund, M. Han, B. The time constants are voltage-dependent [ 60 ], but for simplicity we assume here that they remain constant throughout the formation of the action potential.
Throughout this article we use Eq 88 to describe the sodium channel dynamics. The time constants are chosen such that the resulting action potential fits best the numerical results for the cortex model in [ 24 ], see Fig 12 for a graphical comparison. Likewise, we can define the potassium current as follows: 89 with Once more we seek to identify the inflection point, i. Action potentials are driven by the ionic currents generated at multiple nodes along the axon. Due to the linear nature of the cable equation, the effect of multiple input currents can be described by linear superposition: 97 where U is the r.
U x , t describes the depolarisation due to the current at a nearby node. The relationship between the firing threshold V thr and the time-to-spike t sp is therefore given by The effect of distant nodes is dampened by the fact that in addition to passing along myelinated segments, currents from distant sources also pass by unmyelinated nodes, and thereby further lose amplitude.
This leads to the updated equation for the membrane potential, Eq 9 in the Results section. As we have shown in Fig 4 , the formation of an action potential is a collective process that incorporates ion channel currents from multiple nearby nodes.
However, as we show in Fig 13 , reducing N can lead to a considerable reduction of the propagation velocity at short internode lengths. 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.
This framework allows us to describe unmyelinated axons as well. Since the internode length is zero in this case, the node length l is now an arbitrary discretisation of the axon. We assume that the conductivity of the axonal membrane scales linearly with the channel density, which implies that the electrotonic length constant of an unmyelinated axon is , and its time constant is. In addition to the correction terms introduced in Eq 9 , we also investigate delays that occur at the nodes due to finite transmission speeds.
We assume that action potentials travel with velocities v determined by Eq 9 along myelinated segments, and with velocities v n inferred from Eq 99 at nodes. The corrected velocity is then given by Eq 12 in the Results section. Here we explain how to solve Eq 29 with non-zero extra-cellular potential. It follows from the electric decoupling of the fibre bundle from the external medium that the sum of longitudinal currents within the fibre bundle is zero [ 31 ]: R ex denotes the axial resistance of the extra-cellular medium, which depends inversely on its cross-sectional area.
Since these equations are linear, they can be decoupled using orthogonalisation into with , , , and , where These equations can be solved as above, and the solutions of the coupled equations can be recovered using and. We explore this case in the Results section. In order to compare the spike-diffuse-spike model with the biophysical model presented in [ 24 ], we generate data points using the biophysical model for the parameters reported therein for the cortex model, and fit our model parameters to these data points.
On this grid we determine the action potential velocity of the biophysical model, which is treated as data for the fitting procedure. We use this fitting procedure because there is no direct correspondence between our model and the biophysical model. The latter implements a Hodgkin-Huxley formalism, as well as a detailed model of the myelin sheath that models each membrane individually and includes periaxonal space.
We used the code made available on github by the authors of [ 24 ]. Abstract With the advent of advanced MRI techniques it has become possible to study axonal white matter non-invasively and in great detail. Author summary 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.
Introduction Neurons communicate via chemical and electrical signals, and an integral part of this communication is the transmission of action potentials along their axons. Results For the mathematical treatment of action potential propagation along myelinated axons, we consider active elements periodically placed on an infinitely long cable. In mathematical terms, the governing equation is an inhomogeneous cable equation, which describes the membrane potential V x , t of a leaky cable in space x scalar, longitudinal to the cable and time t in response to input currents: 1 Here, C m and R m are the radial capacitance and resistance of a myelinated fibre, and R c is its axial resistance.
Download: PPT. Ion channel dynamics The classical Hodgkin-Huxley model is described by a set of nonlinear equations which need to be solved numerically. With scenario D we aim to approximate the ion currents as measured in mammals such as the rabbit [ 51 ] and in the rat [ 52 ], which can be described by a superposition of exponential currents: 6 A sketch of all these scenarios is shown in Fig 2 , alongside typical depolarisation curves of the membrane potential.
Fig 2. Sketch of ion channel currents considered here, with representative profiles of membrane potential in nearby nodes. Fig 3. Channel currents divide into a current entering the axon and a current flowing back across the node of Ranvier. Influence of nearby nodes During the propagation of an action potential, ion channel currents are released at multiple nearby nodes that affect the shape and amplitude of the action potential.
Contribution of ion currents from nearby nodes to action potential profile. The resulting velocity then reads 12 We use Eq 12 throughout the manuscript. Scenario A—Fast current. The threshold condition Eq 10 then reads 16 Although this is the simplest scenario, it is not obvious how to invert the r. Scenario B—Delayed fast current. Scenario C—Exponential current. Scenario D—Combination of exponentials.
The threshold condition to determine t sp is 28 Anticipating results from the next subsection, we found that scenarios A and C yield velocities that are too fast compared with experimental results. Sensitivity to parameters Axon diameter.
Fig 5. Propagation velocity as function of fibre diameter and axon diameter. Node and internode length. Fig 6. Velocity dependence on node length and internode length. Myelin thickness. Fig 8. Effect of diameter and g-ratio on propagation velocity. Ephaptic coupling and entrainment We demonstrate here that it is possible to study the effects of ephaptic coupling on action potential propagation within our framework.
Fig 9. Ephaptic coupling reduces AP speed and leads to AP synchronisation. Discussion We have developed an analytic framework for the investigation of action potential propagation based on simplified ion currents.
Methods The cable equation To model action potential propagation along myelinated axons, we consider a hybrid system of active elements coupled by an infinitely long passive cable. Cable parameters. Analytical solution. The homogeneous part of Eq 40 has the solution 41 The inhomogeneous solution in t can be found by the method of variation of the constant, which yields the following convolution integral in t : 42 The inverse Fourier transform of Eq 42 then yields the following double convolution integral in x and t : 43 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.
Current influx and separation. Approximations and analytical solutions 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. 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: 52 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: 53 Without loss of generality we set the time of the current, t 0 , to zero. Differentiating 54 twice yields 56 We multiply all terms by t 4 such that the lowest order term in t is of order zero.
The linear equation for the time-to-spike and the firing threshold is then given by 59 The quantities V t i and can be approximated to be 60 and 61 A comparison of the full nonlinear solution with the linear approximation is shown in Fig 11A.
If we denote by t 0 the time of the threshold crossing, then the ionic current is given by 62 However, by simple linear transformation we may also use t 0 to denote the time of the spike. Differentiating V t twice yields 76 with 77 Since the inflection point occurs at small t , the terms in P t dominate the curvature of the rising phase of V t. Eq 83 can be recast in the form 84 The maximum current is reached at 85 and has the amplitude 86 To construct realistic action potentials, we include both sodium and fast potassium channels.
Fig Comparison of action potentials in the spike-diffuse-spike model and the biophysical model. Influence of distant nodes Action potentials are driven by the ionic currents generated at multiple nodes along the axon.
The relationship between the firing threshold V thr and the time-to-spike t sp is therefore given by 98 The effect of distant nodes is dampened by the fact that in addition to passing along myelinated segments, currents from distant sources also pass by unmyelinated nodes, and thereby further lose amplitude.
Ephaptic coupling and entrainment Here we explain how to solve Eq 29 with non-zero extra-cellular potential. Fitting parameters to biophysical model In order to compare the spike-diffuse-spike model with the biophysical model presented in [ 24 ], we generate data points using the biophysical model for the parameters reported therein for the cortex model, and fit our model parameters to these data points.
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