Activating function

The activating function is an mathematical formalism that is used to approximate the influence of an exctracellular field on an axon or neurons.^[1]^[2]^[3]^[4]^[5]^[6] It is a useful tool to approximate the influence of functional electrical stimulation (FES) or neuromodulation techniques on target neurons.^[7] It points out locations of high hyper- and depolarization caused by the electrical field acting upon the nerve fiber. As a rule of thumb, the activating function is proportional to the second-order spatial derivative of the extracellular potential along the axon.

Equations

In an compartment model of an axon, the activating function of compartment n, $f_{n}$ , is derived from the driving term of the external potential, or the equivalent injected current

$f_{n}=1/c\left({\frac {V_{n-1}^{e}-V_{n}^{e}}{R_{n-1}/2+R_{n}/2}}+{\frac {V_{n+1}^{e}-V_{n}^{e}}{R_{n+1}/2+R_{n}/2}}+...\right)$ ,

where $c$ is the membrane capacity, $V_{n}^{e}$ the extracellular voltage outside compartment $n$ relative to the ground and $R_{n}$ the axonal resistance of compartment $n$ .

The activating function represents the rate of membrane voltage change if the neuron is in resting state before the stimulation. Its physical dimensions are V/s or mV/ms, respectively. Put into other words it represents the slope of the membrane voltage at the beginning of the stimulation.^[8]

Following McNeal's^[9] simplifications for long fibers of an ideal internode membrane, with both membrane capacity and conductance assumed to be 0 the differential equation determining the membrane potential $V^{m}$ for each node is:

${\frac {dV_{n}^{m}}{dt}}=\left[-i_{ion,n}+{\frac {d\Delta x}{4p_{i}L}}\cdot \left({\frac {V_{n-1}^{m}-2V_{n}^{m}+V_{n+1}^{m}}{\Delta x^{2}}}+{\frac {V_{n-1}^{e}-2V_{n}^{e}+V_{n+1}^{e}}{\Delta x^{2}}}\right)\right]/c$ ,

where $d$ is the constant fiber diameter, $\Delta x$ the node-to-node distance, $L$ the node length $\rho _{i}$ the axomplasmatic resistivity, $c$ the capacity and $i_{ion}$ the ionic currents. From this the activating function follows as:

$f_{n}={\frac {d\Delta x}{4\rho _{i}Lc}}{\frac {V_{n-1}^{e}-2V_{n}^{e}+V_{n+1}^{e}}{\Delta x^{2}}}$ .

In this case the activating function is proportional to the second order spatial difference of the extracellular potential along the fibers. If $L=\Delta x$ and $\Delta x\to 0$ than:

$f={\frac {d}{4\rho _{i}c}}\cdot {\frac {\delta ^{2}V^{e}}{\delta x^{2}}}$ .

Thus $f$ is proportional to the second order spatial differential along the fiber.

Interpretation

Positive values of $f$ suggest a depolarization of the membrane potential and negative values a hyperpolarization of the membrane potential.

References

^ Attention: This template ({{cite doi}}) is deprecated. To cite the publication identified by doi:10.1109/TBME.1986.325670, please use {{cite journal}} (if it was published in a bona fide academic journal, otherwise {{cite report}} with |doi=10.1109/TBME.1986.325670 instead.
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^ Rattay, F. (1990). Electrical Nerve Stimulation: Theory, Experiments and Applications. Wien, New York: Springer. p. 264. ISBN 3-211-82247-X.
^ Attention: This template ({{cite doi}}) is deprecated. To cite the publication identified by doi: 10.1109/10.678611, please use {{cite journal}} (if it was published in a bona fide academic journal, otherwise {{cite report}} with |doi= 10.1109/10.678611 instead.
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^ Danner, S., Wenger, C. and Rattay, F. (2011). Electrical stimulation of myelinated axons. Saarbrücken: VDM. p. 92. ISBN 978-3-639-37082-9.{{cite book}}: CS1 maint: multiple names: authors list (link)
^ Rattay, F., Greenberg, R.J. and Resatz, S. (2003). "Neuron modeling". Handbook of Neuroprosthetic Methods,. CRC Press. ISBN 978-0849311000.{{cite book}}: CS1 maint: multiple names: authors list (link)
^ Attention: This template ({{cite doi}}) is deprecated. To cite the publication identified by doi:10.1109/TBME.1976.324593, please use {{cite journal}} (if it was published in a bona fide academic journal, otherwise {{cite report}} with |doi=10.1109/TBME.1976.324593 instead.

[Rattay1986-1] Attention: This template ({{cite doi}}) is deprecated. To cite the publication identified by doi:10.1109/TBME.1986.325670, please use {{cite journal}} (if it was published in a bona fide academic journal, otherwise {{cite report}} with |doi=10.1109/TBME.1986.325670 instead.

[Rattay1988-2] Attention: This template ({{cite doi}}) is deprecated. To cite the publication identified by doi:1010.1109/10.1362, please use {{cite journal}} (if it was published in a bona fide academic journal, otherwise {{cite report}} with |doi=1010.1109/10.1362 instead.

[Rattay1989-3] Attention: This template ({{cite doi}}) is deprecated. To cite the publication identified by doi:10.1109/10.32099, please use {{cite journal}} (if it was published in a bona fide academic journal, otherwise {{cite report}} with |doi=10.1109/10.32099 instead.

[Rattay1990-4] Rattay, F. (1990). Electrical Nerve Stimulation: Theory, Experiments and Applications. Wien, New York: Springer. p. 264. ISBN 3-211-82247-X.

[Rattay1998-5] Attention: This template ({{cite doi}}) is deprecated. To cite the publication identified by doi: 10.1109/10.678611, please use {{cite journal}} (if it was published in a bona fide academic journal, otherwise {{cite report}} with |doi= 10.1109/10.678611 instead.

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[7] Danner, S., Wenger, C. and Rattay, F. (2011). Electrical stimulation of myelinated axons. Saarbrücken: VDM. p. 92. ISBN 978-3-639-37082-9.{{cite book}}: CS1 maint: multiple names: authors list (link)

[Rattay2003-8] Rattay, F., Greenberg, R.J. and Resatz, S. (2003). "Neuron modeling". Handbook of Neuroprosthetic Methods,. CRC Press. ISBN 978-0849311000.{{cite book}}: CS1 maint: multiple names: authors list (link)

[McNeal1976-9] Attention: This template ({{cite doi}}) is deprecated. To cite the publication identified by doi:10.1109/TBME.1976.324593, please use {{cite journal}} (if it was published in a bona fide academic journal, otherwise {{cite report}} with |doi=10.1109/TBME.1976.324593 instead.

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