Question

Please explain how Rm, Cm, and Ri contribute to the time constant in the length constant of an axon. Can we manipulate Rm, Cm, and Ri? What would we want to change to create the fastest conduction velocity?  

Answer 1

In neurobiology, the length constant (λ) is a mathematical constant used to quantify the distance that a graded electric potential will travel along a neurite via passive electrical conduction. The greater the value of the length constant, the farther the potential will travel. A large length constant can contribute to spatial summation—the electrical addition of one potential with potentials from adjacent areas of the cell.

The length constant can be defined as:

{\displaystyle \lambda \ =\ {\sqrt {\frac {r_{m}}{r_{i}+r_{o}}}}}

where rm is the membrane resistance (the force that impedes the flow of electric current from the outside of the membrane to the inside, and vice versa), ri is the axial resistance (the force that impedes current flow through the axoplasm, parallel to the membrane), and ro is the extracellular resistance (the force that impedes current flow through the extracellular fluid, parallel to the membrane). In calculation, the effects of ro are negligible, so the equation is typically expressed as:

{\displaystyle \lambda \ =\ {\sqrt {\frac {r_{m}}{r_{i}}}}}

The membrane resistance is a function of the number of open ion channels, and the axial resistance is generally a function of the diameter of the axon. The greater the diameter of the axon, the lower the ri.

The length constant is used to describe the rise of potential difference across the membrane

{\displaystyle V(x)\ =\ V_{max}(1-e^{-x/\lambda })}

The fall of voltage can be expressed as:

{\displaystyle V(x)\ =\ V_{max}(e^{-x/\lambda })}

Where voltage, V, is measured in millivolts, x is distance from the start of the potential (in millimeters), and λ is the length constant (in millimeters).

Vmax is defined as the maximum voltage attained in the action potential, where:

{\displaystyle V_{max}\ =\ r_{m}I}

where rm is the resistance across the membrane and I is the current flow.

Setting for x= λ for the rise of voltage sets V(x) equal to .63 Vmax. This means that the length constant is the distance at which 63% of Vmax has been reached during the rise of voltage.

Setting for x= λ for the fall of voltage sets V(x) equal to .37 Vmax, meaning that the length constant is the distance at which 37% of Vmax has been reached during the fall of voltage.

By resistivity[edit]

Expressed with resistivity rather than resistance, the constant λ is (with negligible ro):[1]

{\displaystyle \lambda ={\sqrt {\frac {r\times \rho _{m}}{2\times \rho _{i}}}}}

Where {\displaystyle r} is the radius of the neuron.

The radius and number 2 come from these equations:

• {\displaystyle \rho _{m}={\frac {r_{m}}{2\pi r}}}
• {\displaystyle \rho _{i}={\frac {r_{i}}{\pi r^{2}}}}

Expressed in this way, it can be seen that the length constant increases with increasing radius of the neuron.

Nerve conduction velocity is an important aspect of nerve conduction studies. It is the speed at which an electrochemical impulse propagates down a neural pathway. Conduction velocities are affected by a wide array of factors, including age, sex, and various medical conditions. Studies allow for better diagnoses of various neuropathies, especially demyelinating diseases as these conditions result in reduced or non-existent conduction velocities.

Ultimately, conduction velocities are specific to each individual and depend largely on an axon's diameter and the degree to which that axon is myelinated, but the majority of 'normal' individuals fall within defined ranges.[1]

Nerve impulses are extremely slow compared to the speed of electricity, where the electric field can propagate with a speed on the order of 50–99% of the speed of light; however, it is very fast compared to the speed of blood flow, with some myelinated neurons conducting at speeds up to 120 m/s (432 km/h or 275 mph).

Different sensory receptors are innervated by different types of nerve fibers. Proprioceptors are innervated by type Ia, Ib and II sensory fibers, mechanoreceptors by type II and III sensory fibers, and nociceptors and thermoreceptors by type III and IV sensory fibers.