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Activating function

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Approximation of the effect of an electric field on neurons For the function that defines the output of a node in artificial neuronal networks according to the given input, see Activation function.

The activating function is a mathematical formalism that is used to approximate the influence of an extracellular field on an axon or neurons. 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. 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 {\displaystyle f_{n}} , is derived from the driving term of the external potential, or the equivalent injected current

f n = 1 / c ( V n − 1 e − V n e R n − 1 / 2 + R n / 2 + V n + 1 e − V n e R n + 1 / 2 + R n / 2 + . . . ) {\displaystyle 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 {\displaystyle c} is the membrane capacity, V n e {\displaystyle V_{n}^{e}} the extracellular voltage outside compartment n {\displaystyle n} relative to the ground and R n {\displaystyle R_{n}} the axonal resistance of compartment n {\displaystyle 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.

Following McNeal's 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 {\displaystyle V^{m}} for each node is:

d V n m d t = [ − i i o n , n + d Δ x 4 ρ i L ⋅ ( V n − 1 m − 2 V n m + V n + 1 m Δ x 2 + V n − 1 e − 2 V n e + V n + 1 e Δ x 2 ) ] / c {\displaystyle {\frac {dV_{n}^{m}}{dt}}=\left/c} ,

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

f n = d Δ x 4 ρ i L c V n − 1 e − 2 V n e + V n + 1 e Δ x 2 {\displaystyle 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 = Δ x {\displaystyle L=\Delta x} and Δ x → 0 {\displaystyle \Delta x\to 0} then:

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

Thus f {\displaystyle f} is proportional to the second order spatial differential along the fiber.

Interpretation

Positive values of f {\displaystyle f} suggest a depolarization of the membrane potential and negative values a hyperpolarization of the membrane potential.

References

  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. S2CID 33053720.
  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. S2CID 27312507.
  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. S2CID 42935757.
  4. Rattay, F. (1990). Electrical Nerve Stimulation: Theory, Experiments and Applications. Wien, New York: Springer. pp. 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. S2CID 789370.
  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. S2CID 41408689.
  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. BME-23 (4): 329–337. doi:10.1109/TBME.1976.324593. PMID 1278925. S2CID 22334434.
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