Activating function explained

The activating function is a mathematical formalism that is used to approximate the influence of an extracellular field on an axon or neurons.^{[1] [2] [3] [4] [5] [6]} It was developed by Frank Rattay and 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 hyperpolarization 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 a 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 currentf _{n=1/c\left(} (V ^{e} _{n -1} - V ^{e} _{n})/(R _{n-1} / 2+ R _{n} / 2) + (V ^{e} _{n +1} - V ^{e} _{n})/(R _{n+1} / 2+ R _{n} / 2) + ... \right) ,

where

c is the membrane capacity, V ^{e} _{n} 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 potential change if the neuron is in resting state before the stimulation. Its physical dimensions are V/s or mV/ms. In other words, it represents the slope of the membrane voltage at the beginning of the stimulation.^[8]

V ^{m} for each node is:(dV ^{m} _{n})/(dt) = \left[ - i _{ion,n} + (d\Delta x)/(4 \rho _{i} L) ⋅ \left( (V ^{m} _{n -1} -2 V ^{m} _{n +1})/(\Delta x ^{2}) + (V ^{e} _{n -1} -2 V ^{e} _{n} + V ^{e} _{n +1})/(\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 = (d\Delta x)/(4 \rho _{i} Lc)} (V ^{e} _{n -1} -2 V ^{e} _{n} + V ^{e} _{n +1})/(\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 then:f = (d)/(⋅ 4 \rho _{ic}) (\delta ^{2V} ^{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.

Notes and References

1. Rattay . F. . 10.1109/TBME.1986.325670 . Analysis of Models for External Stimulation of Axons . IEEE Transactions on Biomedical Engineering . 10 . 974–977 . 1986 . 33053720 .
2. Rattay . F. . Modeling the excitation of fibers under surface electrodes . 10.1109/10.1362 . IEEE Transactions on Biomedical Engineering . 35 . 3 . 199–202 . 1988 . 3350548. 27312507 .
3. Rattay . F. . Analysis of models for extracellular fiber stimulation . 10.1109/10.32099 . IEEE Transactions on Biomedical Engineering . 36 . 7 . 676–682 . 1989 . 2744791. 42935757 .
4. Book: Rattay, F.. Electrical Nerve Stimulation: Theory, Experiments and Applications. limited. 1990. Springer. Wien, New York. 3-211-82247-X. 264.
5. Rattay . F. . Analysis of the electrical excitation of CNS neurons . 10.1109/10.678611 . IEEE Transactions on Biomedical Engineering . 45 . 6 . 766–772 . 1998 . 9609941. 789370 .
6. Rattay . F. . The basic mechanism for the electrical stimulation of the nervous system . 10.1016/S0306-4522(98)00330-3 . Neuroscience . 89 . 2 . 335–346 . 1999 . 10077317. 41408689 .
7. Book: Danner, S.M. . Wenger, C. . Rattay, F.. Electrical stimulation of myelinated axons. 2011. VDM. Saarbrücken. 978-3-639-37082-9. 92.
8. Book: Rattay, F. . Greenberg, R.J. . Resatz, S.. Handbook of Neuroprosthetic Methods. 2003. CRC Press. 978-0-8493-1100-0. Neuron modeling.
9. McNeal . D. R. . BME-23 . 10.1109/TBME.1976.324593 . Analysis of a Model for Excitation of Myelinated Nerve . IEEE Transactions on Biomedical Engineering . 4 . 329–337 . 1976 . 1278925 . 22334434 .