Abstract
Action potentials in neurons are generated on the plasma membrane through depolarization, i.e. exchange of charges through the membrane. Hodgkin and Huxley developed a mathematical model which describes the interaction of ions through an active plasma membrane. In Vossen et al. (Comput Visual Sci 10:107–121, 2007) we developed a passive three-dimensional (3D) model for signal propagation in dendrite. We now combine this model with a generalized Hodgkin-Huxley model to obtain a 3D-model that describes active signal processing on realistic cell morphologies. Time dependent changes of the neuron’s intra- and extracellular potential is regulated by the Ohmic flux of charges. These fluxes are balanced in membrane-near areas by the capacitory and Hodgkin-Huxley flux. The active model we present consists of five non-linear, coupled integro-differential equations which are solved numerically with a finite volume approach, implicit time stepping and Newton’s method for solving the underlying non-linear system of equations with multigrid solver methods. We present numerical results as well as axon behavior in a biological setting. This model can be considered as a three-dimensional expansion of existing state of the art one-dimensional models, with the significant advantage of being able to investigate the morphological influence of neuron cell types on their specific signaling properties.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Wong R.K., Prince D.A.: Participation of calcium spikes during intrinsic burst firing in hippocampal neurons. Brain Res. 159, 385–390 (1978)
Llinas R., Sugimori M.: Electrophysiological properties of in vitro purkinje cell somata in mammalian cerebellar slices. J. Physiol. 305, 171–195 (1980)
Häusser M., Spruston N., Stuart G.J.: Diversity and dynamics of dendritic signaling. Science 290, 739–744 (2000)
Häusser M., Stuart G., Racca C., Sakmann B.: Axonal initiation and active dendritic propagation of action potentials in substantia nigra neurons. Neuron 15, 637–647 (1995)
Chen W.R., Midtgaard J., Shepherd G.M.: Forward and backward propagation of dendritic impulses and their synaptic control in mitral cells. Science 278, 463–467 (1997)
Debanne D.: Information processing in the axon. Nature 5, 304–316 (2004)
Gu X.: Effect of conduction block at axon bifurcations on synaptic transmission to different postsynaptic neurones in the leech. J. Physiol. 441, 755–778 (1991)
Baccus S.A.: Synaptic facilitation by reflected action potentials: enhancement of transmission when nerve impulses reverse direction at axon branch points. Proc. Natl. Acad. Sci. USA 95, 8345–8350 (1998)
Chen W.R., Shen G.Y., Shepherd G.M., Hines M.L., Midtgaard J.: Multiple modes of action potential initiation and propagation in mitral cell primary dendrite. J. Neurophysiol. 88, 2755–2764 (2002)
Hoffmann D.A., Magee J.C., Colbert C.M., Johnston D.: K channel regulation of signal propagation in dendrites of hippocampal pyramidal neurons. Nature 387, 869–875 (1997)
Colbert C.M., Magee J.C., Hoffmann D.A., Johnston D.: Slow recovery from inactivation of na+ channels underlies the activity-dependent attenuation of dendritic action potentials in hippocampal ca1 pyramidal neurons. J. Neurosci. 17, 6512–6521 (1997)
Mickus T., Jung H., Spruston N.: Properties of slow, cumulative sodium channel inactivation in rat hippocampal ca1 pyramidal neurons. Biophys. J. 76, 846–860 (1999)
Rall W.: Branching dendritic trees and motoneuron membrane resistivity. Exp. Neurol. 1, 491–527 (1959)
Rall W.: Theory of physiological properties of dendrites. Ann. N.Y. Acad. Sci. 96, 1071–1092 (1962)
Hines M.L., Carnevale N.T.: The NEURON simulation environment. Neural Comput. 9, 1179–1209 (1997)
Beeman, D., Bower, J.M.: The book of genesis: exploring realistic neural models with the general neural simulation system. (1998)
Vossen C., Eberhard J.P., Wittum G.: Modeling and simulation for three-dimensional signal propagation in passive dendrites. Comput. Visual. Sci. 10, 107–121 (2007)
Petersen C., Grinvald A., Sakman B.: Spatiotemporal dynamics of sensory responses in layer 2/3 of rat barrel cortex measured in vivo by voltage-sensitive dye imaging combined with whole-cell voltage recordings and neuron reconstructions. J. Neurosci. 23, 1298 (2003)
Broser P., Schulte R., Roth A., Helmchen F., Waters J., Lang S., Sakmann B., Wittum G.: Nonlinear anisotropic diffusion filtering of three-dimensional image data from two-photon microscopy. J. Biomedical Opt. 9(6), 1253–1264 (2004)
Queisser G., Wittmann M., Bading H., Wittum G.: Filtering, reconstruction, and measurement of the geometry of nuclei from hippocampal neurons based on confocal microscopy data. J. Biomedical Opt. 13(1), 014009 (2008)
Bastian P., Johannsen K., Lang S., Nägele S., Reichenberger V., Wieners C., Wittum G.: Parallel solutions of partial differential equations with adaptive multigrid methods on unstructured grids. In: Jäger, W., Krause, E. (eds) High Performance Computing in Science and Engineering, Springer, Berlin (2000)
Hodgkin A.L., Huxley A.F.: A quantitative descsription of membrane current and its application to conduction and excitation in nerve. J. Physiol. 117, 500–544 (1952)
Grossmann C.: Numerical Treatment of Partial Differential Equations. Springer, Berlin (2007)
Hackbusch W.: Lösung Großr Schwachbesetzter Gleichungssysteme. Teubner, Stuttgart (1993)
Shepherd, G.M.: Neurobiology. Oxford University Press (1994)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Randy Bank.
Rights and permissions
About this article
Cite this article
Xylouris, K., Queisser, G. & Wittum, G. A three-dimensional mathematical model of active signal processing in axons. Comput. Visual Sci. 13, 409–418 (2010). https://doi.org/10.1007/s00791-011-0155-7
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00791-011-0155-7