Skip to main content
SpringerLink
Log in

Identifying critical regions for spike propagation in axon segments

  • Published:
Journal of Computational Neuroscience Aims and scope Submit manuscript

Abstract

Morphological reconstructions of axon segments reveal the abundance of geometrical ultrastructures that can dramatically affect the propagation of Action Potentials (AP). Moreover, deformations and swellings in axons resulting from brain traumas are associated to many neural dysfunctions and disorders. Our aim is to develop a computational framework to distinguish between geometrical enlargements that lead to minor changes in propagation from those that result in critical phenomenon such as reflection or blockage of the original traveling spike. We use a few geometrical parameters to model a prototypical shaft enlargement and explore the parameter space characterizing all possible propagation regimes and dynamics in an unmylienated AP model. Contrary to earlier notions that large diameter increases mostly lead to blocking, we demonstrate transmission is stable provided the geometrical changes occur in a slow manner. Our method also identifies a narrow range of parameters leading to a reflection regime. The distinction between these three regimes can be evaluated by a simple function of the geometrical parameters inferred through numerical simulations. We suggest that evaluating this function along axon segments can detect regions most susceptible to (i) transmission failure due to perturbations, (ii) structural plasticity, (iii) critical swellings caused by brain traumas and/or (iv) neurological disorders associated with the break down of spike train propagation.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig.2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

References

  • Altenberger, R., Lindsay, K.A., Ogden, J.M., Rosenberg, J.R. (2001). The interaction between membrane kinetics and membrane geometry in the transmission of action potentials in non-uniform excitable fibres: a finite element approach. Journal of Neuroscience Methods, 112, 101–117.

    Article  CAS  PubMed  Google Scholar 

  • Antic, S., Wuskell, J.P., Loew, L., Zecevic, D. (2000). Functional profile of the giant metacerebral neuron of Helix aspersa: temporal and spatial dynamics of electrical activity in situ. The Journal of Physiology, 527, 55–69.

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  • Baccus, S.A. (1998). Synaptic facilitation by reflected action potentials: enhancement of transmission when nerve impulses reverse direction at axon branch points. Proceedings of the National Academy of Sciences, 95(14), 8345–8350.

    Article  CAS  Google Scholar 

  • Baccus, S.A., Burrell, B.D., Sahley, C.L., Muller, K.J. (2000). Action potential reflection and failure at axon branch points cause stepwise changes in EPSPs in a neuron essential for learning. Journal of Neurophysiology, 83(3), 1693–1700.

    CAS  PubMed  Google Scholar 

  • Barron, D.H., & Matthews, B.H. (1935). Intermittent conduction in the spinal cord. The Journal of Physiology, 85, 73–103. Neuron, 60(4), 590–597.

    CAS  PubMed Central  PubMed  Google Scholar 

  • Becker, N., Wierenga, C.J., Fonseca, R., Bonhoeffer, T., Nagerl, U.V. (2008). LTD Induction causes morphological changes of presynaptic boutons and reduces their contacts with spines. Neuron, 60(4), 590–597. http://www.sciencedirect.com/science/article/pii/S0896627308007666.

    Article  CAS  PubMed  Google Scholar 

  • Beeman, D., & Bower, J.M. (1998). The book of genesis: exploring realistic neural models with the general neural simulation system (2nd Ed.). New York: Springer-Verlag.

    Google Scholar 

  • Bielefeldt, K., & Jackson, M.B. (1993). A calcium-activated potassium chan- nel causes frequency-dependent action-potential failures in a mammalian nerve terminal. Journal of Neurophysiology, 70, 284–298.

    CAS  PubMed  Google Scholar 

  • Bourque, C.W. (1990). Intraterminal recordings from the rat neurohypophysis in vitro. The Journal of Physiology, 421, 247–262.

    CAS  PubMed Central  PubMed  Google Scholar 

  • Bucher, D., & Goaillard, J.M. (2011). Beyond faithful conduction: short term dynamics, neuromodulation, and long-term regulation of spike propagation in the axon. Progress in Neurobiology, 94, 307–346.

    Article  PubMed Central  PubMed  Google Scholar 

  • Butz, M., Worgotter, F., van Ooyen, A. (2009). Activity-dependent structural plasticity. Brain Research Reviews, 60(2), 287–305.

    Article  PubMed  Google Scholar 

  • Carnevale, N.T., & Hines, M.L. (2009). The neuron book. Cambridge: Cambridge University Press.

    Google Scholar 

  • Chen, W.R., Shen, G.Y., Shepherd, G.M., Hines, M.L., Midtgaard, J. (2002). Multiple modes of action potential initiation and propagation in mitral cell primary dendrite. Journal of Neurophysiology, 88, 2755–2764.

    Article  PubMed  Google Scholar 

  • Cooley, J.W., & Dodge, F.A. (1966). Digital computer solutions for excitation and propagation of the nerve impulse. Biophysical Journal, 6(5), 583–599.

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  • Dayan, P., & Abbot, F.L. (2001). Theoretical neuroscience. Cambridge: MIT Press.

    Google Scholar 

  • Debanne, D. (2004). Information processing in the axon. Nature Reviews Neuroscience, 5(4), 304–316.

    Article  CAS  PubMed  Google Scholar 

  • Debanne, D., Guerineau, N.C., Gahwiler, B.H., Thompson, S.M. (1997). Action-potential propagation gated by an axonal I(A)-like K?? conductance in hippocampus. Nature, 389, 286–289.

    Article  CAS  PubMed  Google Scholar 

  • Debanne, D., Campanac, E., Bialowas, A., Carlier, E., Alcaraz, G. (2011). Axon physiology. Physiological Reviews, 91, 555–602.

    Article  CAS  PubMed  Google Scholar 

  • De Paola, V., Holtmaat, A., Knott, G., Song, S., Willbrecht, L., Caroni, P., Svoboda, K. (2006). Cell type-specific structural plasticity of axonal branches and boutons in the adult neocortex. Neuron, 49, 861–875.

    Article  PubMed  Google Scholar 

  • Deschenes, M., & Landry, P. (1980). Axonal branch diameter and spacing of nodes in the terminal arborization of identified thalamic and cortical neurons. Brain Research, 191, 538–544.

    Article  CAS  PubMed  Google Scholar 

  • Ducreux, C., Reynaud, J.C., Puizillout, J.J. (1993). Spike conduction properties of T-shaped C neurons in the rabbit nodose ganglion. Pflgers Arch, 424, 238–244.

    Article  CAS  Google Scholar 

  • Dyball, R.E., Grossmann, R., Leng, G., Shibuki, K. (1988). Spike propagation and conduction failure in the rat neural lobe. The Journal of Physiology, 401, 241–256.

    CAS  PubMed Central  PubMed  Google Scholar 

  • Ermentrout, G.B., & Rinzel, J. (1996). Reflected waves in an inhomogeneous excitable medium. SIAM Journal on Applied Mathematics, 56(4), 1107–1128.

    Article  Google Scholar 

  • Evans, C.G., Ludwar, B., Cropper, E.C. (2007). Mechanoafferent neuron with an inexcitable somatic region: consequences for the regulation of spike propagation and afferent transmission. Journal of Neurophysiology, 97, 3126–3130.

    Article  PubMed  Google Scholar 

  • Fiala, J.C., Spacek, J., Harris, K.M. (2002). Dendritic spine pathology: cause or consequence of neurological disorders? Brain Research Reviews, 39, 29–54.

    Article  PubMed  Google Scholar 

  • Fitzhugh, R. (1961). Impulses and physiological states in theoretical models of nerve membrane. Biophysical Journal, 1(6), 445–466.

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  • Gogolla, N., Galimberti, I., Caroni, P. (2007). Structural plasticity of axon terminals in the adult. Current Opinion in Neurobiology, 17, 516–524.

    Article  CAS  PubMed  Google Scholar 

  • Goldfinger, M.D. (2005). Rallian equivalent cylinders reconsidered: comparisons with literal compartments. Journal of Integrative Neuroscience, 4(2), 227–263.

    Article  CAS  PubMed  Google Scholar 

  • Goldstein, S.S., & Rall, W. (1974). Changes of action potential shape and velocity for changing core conductor geometry. Biophysical Journal, 14, 731–757.

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  • Grossman, Y., Parnas, I., Spira, M.E. (1979a). Differential conduction block in branches of a bifurcating axon. The Journal of Physiology, 295, 283–305.

    CAS  PubMed Central  PubMed  Google Scholar 

  • Grossman, Y., Parnas, I., Spira, M.E. (1979b). Mechanisms involved in differential conduction of potentials at high frequency in a branching axon. The Journal of Physiology, 295, 307–322.

    CAS  PubMed Central  PubMed  Google Scholar 

  • Gu, X.N. (1991). Effect of conduction block at axon bifurcations on synaptic transmission to different postsynaptic neurones in the leech. The Journal of Physiology, 441, 755–778.

    CAS  PubMed Central  PubMed  Google Scholar 

  • Hatt, H., & Smith, D.O. (1976). Synaptic depression related to presynaptic axon conduction block. The Journal of Physiology, 259, 367–393.

    CAS  PubMed Central  PubMed  Google Scholar 

  • Hines, M.L., & Carnevale, N.T. (1997). The NEURON simulation environment. Neural Computation, 9, 1179–1209.

    Article  CAS  PubMed  Google Scholar 

  • Hodgkin, A.L., & Huxley, A.F. (1952). A quantitative description of membrane current and its application to conduction and excitation in nerve. The Journal of Physiology, 117(4), 500–545.

    CAS  PubMed Central  PubMed  Google Scholar 

  • Holtmaat, A., & Svoboda, K. (2009). Experience-dependent structural synaptic plasticity in the mammalian brain. Nature Reviews Neuroscience, 10(9), 647–658.

    Article  CAS  PubMed  Google Scholar 

  • Izhikevich, E.M. (2007). Dynamical systems in neuroscience: the geometry of excitability and bursting. Cambridge: MIT Press.

    Google Scholar 

  • Jackson, M.B., & Zhang, S.J. (1995). Action potential propagation and propagation block by GABA in rat posterior pituitary nerve terminals. The Journal of Physiology, 483, 597–611.

    CAS  PubMed Central  PubMed  Google Scholar 

  • Khodorov, B.I., & Timin, E.N. (1975). Nerve impulse propagation along nonuniform fibres. Progress in Biophysics and Molecular Biology, 30(23), 145–184.

    CAS  PubMed  Google Scholar 

  • Krnjevic, K., & Miledi, R. (1959). Presynaptic failure of neuromuscular propagation in rats. The Journal of Physiology, 149, 1–22.

    CAS  PubMed Central  PubMed  Google Scholar 

  • Kutz, J.N. (2013). Data-driven modeling and scientific computing. New York: Oxford Press.

    Google Scholar 

  • Luscher, H.R., & Shiner, J.S. (1990a). Computation of action potential propagation and presynaptic bouton activation in terminal arborizations of different geometries. Biophysical Journal, 58(6), 1377–1388.

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  • Luscher, H.R., & Shiner, J.S. (1990b). Simulation of action potential propagation in complex terminal arborizations. Biophysical Journal, 58(6), 1389–1399.

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  • Luscher, C., Streit, J., Lipp, P., Luscher, H.R. (1994). Action potential propagation through embryonic dorsal root ganglion cells in culture. II. Decrease of conduction reliability during repetitive stimulation. Journal of Neurophysiology, 72, 634–643.

    CAS  PubMed  Google Scholar 

  • Magdesian, M., Sanchez, F.S., Lopez, M., Thostrup, P., Durisic, N., Belkaid, W., Liazoghli, D., Grütter, P., Colman, R. (2012). Atomic force microscopy reveals important differences in axonal resistance to injury. Biophysical Journal, 103(3), 405–414.

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  • Manor, Y., Gonczarowski, J., Segev, I. (1991a). Propagation of action potentials along complex axonal trees. Model and implementation. Biophysical Journal, 60(6), 1411–1423.

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  • Manor, Y., Koch, C., Segev, I. (1991b). Effect of geometrical irregularities on propagation delay in axonal trees. Biophysical Journal, 60, 1424–1437.

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  • Marcuse, D. (1974). Theory of dielectric optical waveguides. New York: Academic Press.

    Google Scholar 

  • Mascagni, M. (1990). The backward Euler method for numerical solution of the Hodgkin–Huxley equations of nerve conduction. SIAM Journal on Numerical Analysis, 27(4), 941–962.

    Article  Google Scholar 

  • Maxwell, W.L., Povlishock, J.T., Graham, D.L. (1997). A mechanistic analysis of nondisruptive axonal injury: a review. Journal of Neurotrauma, 17(7), 419–440.

    Article  Google Scholar 

  • Meeks, J.P., & Mennerick, S. (2004). Selective effects of potassium elevations on glutamate signaling and action potential conduction in hippocampus. Journal of Neuroscience, 24, 197–206.

    Article  CAS  PubMed  Google Scholar 

  • Miura, R.M. (1982). Accurate computation of the stable solitary wave for the FitzHugh–Nagumo equations. Journal of Mathematical Biology, 13(3), 247–269.

    Article  Google Scholar 

  • Morris, C., & Lecar, H. (1981). Voltage oscillations in the barnacle giant muscle fiber. Biophysical Journal, 35(1), 193–213.

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  • Nagumo, S., Arimoto, Yoshizawa, S. (1962). An active pulse transmission line simulating nerve axon. Proceedings of the IRE, 50(10), 2061–2070.

    Article  Google Scholar 

  • Parnas, I. (1979). Propagation in nonuniform neurites: form and function in axons. In F.O. Schmitt & F.G. Worden (Eds.), The neurosciences (pp. 499–512). Cambridge: MIT Press.

    Google Scholar 

  • Parnas, I. (1972). Differential block at high frequency of branches of a single axon innervating two muscles. Journal of Neurophysiology, 35, 903–914.

    CAS  PubMed  Google Scholar 

  • Parnas, I., Hochstein, S., Parnas, H. (1976). Theoretical analysis of parameters leading to frequency modulation along an inhomogeneous axon. Journal of Neurophysiology, 39(4), 909–923.

    CAS  PubMed  Google Scholar 

  • Purpura, D.P., Bodick, N., Suzuki, K., Rapin I., Wurzelmann, S. (1982). Microtubule disarray in cortical dendrites and neurobehavioral failure. I. Golgi and electron microscopy studies. Developing Brain Research, 5, 287–297.

    Article  Google Scholar 

  • Ramon, F., Joyner, R.W., Moore, J.W. (1975). Propagation of action potentials in inhomogeneous axon regions. Federation Proceedings, 34, 1357–1363.

    CAS  PubMed  Google Scholar 

  • Rall, W. (1959). Branching dendritic trees and motoneuron membrane resistivity. Experimental Neurology, 1, 491–527.

    Article  CAS  PubMed  Google Scholar 

  • Rall, W. (1962). Theory of physiological properties of dendrites. Annals of the New York Academy of Sciences, 96, 1071–1092.

    Article  CAS  PubMed  Google Scholar 

  • Rall, W., & Shepherd, J. (1968). Theoretical reconstruction of field potentials and dendrodendritic synaptic interactions in olfactory bulb. Journal of Neurophysiology, 173, 884–905.

    Google Scholar 

  • Rinzel, J. (1990). Mechanisms for nonuniform propagation along excitable cables. Annals of the New York Academy of Sciences, 591, 51–61.

    Article  CAS  PubMed  Google Scholar 

  • Scott, A. (2002). Neuroscience: a mathematical primer. New York: Springer.

    Google Scholar 

  • Segev, I., & Rall, W. (1998). Excitable dendrites and spines: earlier theoretical insights elucidate recent direct observations. Trends in Neuroscience, 21(11), 453–460.

    Article  CAS  Google Scholar 

  • Segev, I., & Schneidman, E. (1999). Axons as computing devices: basic insights gained from models. The Journal of Physiology, 93, 263–270.

    CAS  Google Scholar 

  • Segev, I., Rinzel, J., Shepherd, G.M. (Eds.) (1995). The theoretical foundation of dendritic function: selected papers of Wilfred Rall with commentaries. Cambridge: Bradford/MIT Press.

    Google Scholar 

  • Shepherd, G.M.G., & Harris, K. (1998). Three-dimensional structure and composition of CA3 to CA1 axons in rat hippocampal slices: implications for presynaptic connectivity and compartmentalization. Journal of Neuroscience, 18(20), 8300–8310.

    CAS  PubMed  Google Scholar 

  • Smith, D.O. (1980). Mechanisms of action potential propagation failure at sites of axon branching in the crayfish. The Journal of Physiology, 301, 243–259.

    CAS  PubMed Central  PubMed  Google Scholar 

  • Smith, D.H., Wolf, J.W., Lusardi, T.A., Lee, V.M.Y., Meaney, D.F. (1999). High tolerance and delayed elastic response of cultured axons to dynamic stretch injury. The Journal of Neuroscience, 19(11), 4263–4269.

    CAS  PubMed  Google Scholar 

  • Soleng, A.F., Chiu, K., Raastad, M. (2003). Unmyelinated axons in the rat hippocampus hyperpolarize and activate an H current when spike frequency exceeds 1 Hz. The Journal of Physiology, 552, 459–470.

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  • Tang-Schomer, M.D., Patel, A.R., Bass, P.W., Smith, D.H. (2010). Mechanical breaking of microtubules in axons during dynamic stretch injury underlies delayed elasticity, microtubule disassembly, and axon degeneration. The FASEB Journal, 24(5), 1401–1410.

    Article  CAS  PubMed Central  Google Scholar 

  • Tang-Schomer, M.D., Johnson, V.E., Baas, P.W., Stewart, W., Smith, D.H. (2012). Partial interruption of axonal transport due to microtubule breakage accounts for the formation of periodic varicosities after traumatic axonal injury. Experimental Neurology, 233, 364–372.

    Article  PubMed  Google Scholar 

  • Tóth, T.I., & Cruneli, V. (1998). Effects of tapering geometry and inhomogeneous ion channel distribution in a neuron model. Neuroscience, 84(4), 1223–1232.

    Article  PubMed  Google Scholar 

  • Trefethen, L.N. (2000). Spectral methods in matlab. Philadelphia: SIAM.

    Book  Google Scholar 

  • Van Essen, D.C. (1973). The contribution of membrane hyperpolarization to adaptation and conduction block in sensory neurons of the leech. The Journal of Physiology, 230, 509–534.

    PubMed Central  PubMed  Google Scholar 

  • Vladimirescu, A., Zhang, K., Newton, A.R., Pederson, D.O., Sangiovanni-Vincentelli, A. (1981). SPICE version 2G user’s guide. Berkeley: Department of Electrical Engineering and Computer Sciences, University of California.

    Google Scholar 

  • Wang, J., Hamm, R.J., Povlishock, J.T. (2011). Traumatic axonal injury in the optic nerve: evidence for axonal swelling, disconnection, dieback and reorganization. Journal of Neurotrauma, 28(7), 1185–1198.

    Article  CAS  PubMed Central  PubMed  Google Scholar 

  • Xiong, Y., Mahmood, A., Chopp, M. (2013). Animal models of traumatic brain injury. Nature Reviews Neuroscience, 14(2), 128–142.

    Article  CAS  PubMed  Google Scholar 

  • Xylouris, K., Queisser, G., Wittum, G. (2010). A three-dimensional mathematical model of active signal processing in axons. Computing and Visualization in Science, 13(8), 409–418.

    Article  Google Scholar 

  • Yau, K.W. (1976). Receptive fields, geometry and conduction block of sensory neurones in the central nervous system of the leech. The Journal of Physiology, 263, 513–538.

    CAS  PubMed Central  PubMed  Google Scholar 

  • Zhang, S.J., & Jackson, M.B. (1993). GABA-activated chloride channels in secretory nerve endings. Science, 259, 531–534.

    Article  CAS  PubMed  Google Scholar 

  • Zhou, Y., & Bell, J. (1994). Study of propagation along nonuniform excitable fibers. Mathematical Biosciences, 119(2), 169–203.

    Article  CAS  PubMed  Google Scholar 

Download references

Acknowledgments

We wish to thank F. Rieke, A. Fairhall, E. Shea-Brown, N. Cayco-Gajic, and E. Shlizerman for insightful discussions related to the work produced here. J. N. Kutz acknowledges support from the National Science Foundation (DMS-1007621).

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, University of Washington, Seattle, WA, 98195-2420, USA

    Pedro D. Maia & J. Nathan Kutz

Authors
  1. Pedro D. Maia
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. Nathan Kutz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pedro D. Maia.

Additional information

Action Editor: James M. Bower

Rights and permissions

Reprints and permissions

About this article

Cite this article

Maia, P.D., Kutz, J.N. Identifying critical regions for spike propagation in axon segments. J Comput Neurosci 36, 141–155 (2014). https://doi.org/10.1007/s10827-013-0459-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10827-013-0459-3

Keywords

Search

Navigation