Mammalian Motor Nerve Fibers, Models of
A model of a mammalian motor nerve fiber is a mathematical description of the electrochemical events taking place during the initiation and propagation of an action potential in a peripheral nerve fiber. This description accounts for the movement of ions through protein channels (conductances) embedded in the axon membrane. These channels are differentially distributed along different regions of the nerve fiber (e.g., node of Ranvier vs. internode) and have different conduction properties (e.g., fast sodium, persistent sodium, or slow potassium conductances).
The flow of ions (charges) involved in the initiation and propagation of an action potential can be calculated with electrical circuit equivalent models that account for the membrane capacitance (phospholipid bilayer) and the resistance or conductance of individual ionic channels (Fig. 1b, c). A short length or “patch” of nerve fiber membrane constitutes a compartment, which can be assembled in series with other compartments to represent a section or an entire fiber. In a myelinated fiber, each node of Ranvier can be represented as a single nodal compartment. In addition, if myelin acts as an electrical insulator, then consecutive nodal compartments can be coupled with single resistors. These resistors represent the intracellular conduction between nodes along the length of axon under the myelin. However, more comprehensive representations include a finite impedance myelin sheath attached to internodal compartments (Halter and Clark 1991). Furthermore, since the fiber morphology appears to play an important role in the threshold fluctuations following an action potential (recovery cycle), specific segments including the belt of myelin attachment and the paranodal region can be explicitly represented. Hence, a double cable model has been proposed in which the internodal segments are composed of the myelin sheath and the internodal axolemma (Fig. 1b) (Stephanova and Bostock 1995; Richardson et al. 2000; McIntyre et al. 2002).
Each compartment of the cable model represents the electrochemical properties of the associated membrane patch. First, since the phospholipid bilayer membrane acts as a dielectric separating two conductive media, it can be modeled as a capacitor. Secondly, the differences in concentrations of each ion between the intracellular and extracellular spaces (i.e., across the membrane) give rise to a transmembrane voltage difference that can be modeled with batteries representing the Nernst potential of each of the contributing ions. Finally, the protein channels embedded in the membrane that allow passage of ions can be modeled with resistors. Under low-amplitude stimulation (below threshold), the ionic conductances (resistors) are represented with linear models. In this case, the membrane passively conducts current and no action potential occurs. However, larger stimuli (above threshold) can drive the membrane to conduct actively ionic currents. In this situation, the ionic conductances are better described by nonlinear models. According to the Hodgkin and Huxley (1952) formulation, the conductance of an ionic channel equals the maximum conductance multiplied by one or more gating variables. It follows that each gating variable, u, evolves according to the differential equation du/dt = α(1−u)−βu, where the parameters α and β are estimated empirically and usually vary with the transmembrane voltage.
In mammalian peripheral fibers, different types of sodium channels as well as slow voltage-gated potassium channels tend to concentrate in the node of Ranvier, whereas other potassium channels, cation channels, and calcium channels, as well as certain pumps and exchangers, concentrate in the paranode and internode (Krishnan et al. 2009). Consequently, models of mammalian motor fibers have represented the nodal compartments with fast (Naf) and persistent (Nap) sodium and slow potassium (Ks) conductances (e.g., Schwarz et al. 1995; McIntyre et al. 2002; Howells et al. 2012, who also added a fast potassium conductance), although models with a single sodium conductance have also been proposed (e.g., Chiu et al. 1979; Sweeney et al. 1987).
Following an action potential, axons exhibit absolute and relative refractory periods, where they are less susceptible to re-excitation, followed by periods of decreased (supernormality) and increased (subnormality) thresholds that are associated with oscillations in the transmembrane voltage, termed depolarizing (DAP) and hyperpolarizing (AHP) afterpotentials, respectively. This sequence of changes in excitability following an action potential constitutes the recovery cycle of axons. Models of motor fibers have illuminated the mechanism of the recovery cycle by incorporating more accurate geometrical and electrical descriptions of the axon. For instance, the double cable model of McIntyre et al. (2002) suggested that both persistent sodium current and passive discharge through the paranodal seal resistance contributed to the DAP, whereas AHP was mediated by slow potassium channels.
Difference with Sensory Fibers
Motor and sensory fibers differ in action potential shape, recovery cycle, and excitability properties (Howells et al. 2012). The mechanisms underlying these differences may be determined using fiber-type-specific models. For example, McIntyre et al. (2002) argued that the smaller supernormality of sensory fibers could be attributed to the kinetics of the sodium conductances rather than the density of these channels in the node. Specifically, they found that slowing the fast sodium channels resulted in reduced supernormality, which is consistent with experimental findings in sensory fibers. Similarly, both the duration and amplitude of the supernormal period decreased with faster sodium channel inactivation. Moreover, Howells et al. (2012) proposed that an increased expression of the fast isoform of the hyperpolarization-activated cyclic-nucleotide gated (HCN) channel could participate in limiting hyperpolarization in sensory fibers. Additionally, their model suggested that a reduced expression of slow potassium channels in the node of sensory fibers and the concomitant depolarization of resting membrane potential could be responsible for the broader action potential in these fibers. Taken together, these findings emphasize that dynamical and structural considerations can lead to more accurate models of functionally distinct fibers.
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