Snapshot 1: the action potential propagating down the axon at a sufficiently short inter-node length.
Snapshot 2: at low time constants and high space constants the cable becomes rigid.
Snapshot 3: the external current may be a pulse or a constant step, with the latter case allowing for successive signals to propagate down the axon.
The speed of signal propagation is of critical importance to information processing in the nervous system. Since the axons of neurons can extend to over one meter in length, biological mechanisms have evolved that increase the velocity of the action potential's propagation.
The axons of typical neurons are ensheathed in layers of a dielectric material called myelin. The width of the sheath is approximately 30-40% of the total diameter of the axon, and is interrupted periodically by nodes, termed the nodes of Ranvier. The
layers of myelin both decrease the capacitance of the membrane and increase its resistance, leading to an increase in the speed of signal propagation and a reduction of radial current leakage [1].
The present simulation follows a classical model of the membrane potential in myelinated axons proposed by Fitzhugh in 1962 [2].
At the nodes of Ranvier, the membrane potential is modeled by Hodgkin-Huxley (HH) dynamics, with parameters given in [4]. In the present work, since the length of the node is short (c.a. 0.1% [1]) relative to the myelinated section, the nodes are modeled by a single HH compartment with no spatial dependence. Along the myelin sheath sections, the lack of voltage-gated ionic channels allows for the potential to be described by the linear cable equation [2]. This may be written as
,
where
is the membrane potential (outside minus inside),
mV is the resting potential,
is the space constant and
is the time constant, which may be expressed as
,
τ =
,
where
is the membrane resistance per unit length,
is the intracellular resistance per unit length along the cable, and
is the capacitance per unit length. These are further related physical properties of the axon and the myelin sheath by
,
,
,
where the parameters are the specific intracellular resistivity
, specific membrane resistance
, specific membrane capacitance
, and fiber diameter
[1].
An external current is applied at the cell body end of the axon [4] as either a pulse or a constant step of adjustable magnitude.
The boundary conditions for the simulation between the node and inter-node dynamics are given explicitly in [2].
The simulation above allows for direct control of the space and time constants, which in turn represent changes in the physical property of the myelin sheath. Additionally, the length of the myelin segments and the strength of the stimulating currents may be adjusted.
Using the autoplay feature, the speed of signal propagation can be observed in real time. Anatomically it has been observed that the inter-node distance and thus the velocity of the signal is proportional to the diameter of the axon [1]. Therefore, the speed may be increased by increasing the space constant or decreasing the time constant given above.
[1] C. Koch,
Biophysics of computation: information processing in single neurons, New York: Oxford University Press, 1999.
[2] R. Fitzhugh, "Computation of Impulse Initiation and Saltatory Conduction in a Myelinated Nerve Fiber,"
Biophysical Journal, 2(1), 1962 pp. 11-21.
[3] L. Hodgkin and A. Huxley, A., "A quantitative description of membrane current and its application to conduction and excitation in nerve,"
J Physiology, 117(4), 1952 pp. 500–544.
[4] Image in modified form: By Nicolas.Rougier (Own work) [GFDL (http://www.gnu.org/copyleft/fdl.html), CC-BY-SA-3.0 (http://creativecommons.org/licenses/by-sa/3.0/) or CC BY-SA 2.5-2.0-1.0 (http://creativecommons.org/licenses/by-sa/2.5-2.0-1.0)], via Wikimedia Commons.
