Activating function

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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[edit]

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

,

where is the membrane capacity, the extracellular voltage outside compartment relative to the ground and the axonal resistance of compartment .

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]

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 for each node is:

,

where is the constant fiber diameter, the node-to-node distance, the node length the axomplasmatic resistivity, the capacity and the ionic currents. From this the activating function follows as:

.

In this case the activating function is proportional to the second order spatial difference of the extracellular potential along the fibers. If and then:

.

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

Interpretation[edit]

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

References[edit]

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