Activating function
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, [math]\displaystyle{ f_n }[/math], is derived from the driving term of the external potential, or the equivalent injected current
[math]\displaystyle{ f_n=1/c\left( \frac{V^e_{n-1}-V^e_{n}}{R_{n-1}/2+R_{n}/2} + \frac{V^e_{n+1}-V^e_{n}}{R_{n+1}/2+R_{n}/2} + ... \right) }[/math],
where [math]\displaystyle{ c }[/math] is the membrane capacity, [math]\displaystyle{ V^e_n }[/math] the extracellular voltage outside compartment [math]\displaystyle{ n }[/math] relative to the ground and [math]\displaystyle{ R_n }[/math] the axonal resistance of compartment [math]\displaystyle{ n }[/math].
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 [math]\displaystyle{ V^m }[/math] for each node is:
[math]\displaystyle{ \frac{dV^m_n}{dt}=\left[-i_{ion,n} + \frac{d\Delta x}{4\rho_i L} \cdot \left( \frac{V^m_{n-1}-2V^m_n+V^m_{n+1}}{\Delta x^2}+ \frac{V^e_{n-1}-2V^e_{n}+V^e_{n+1}}{\Delta x^2} \right) \right] / c }[/math],
where [math]\displaystyle{ d }[/math] is the constant fiber diameter, [math]\displaystyle{ \Delta x }[/math] the node-to-node distance, [math]\displaystyle{ L }[/math] the node length [math]\displaystyle{ \rho_i }[/math] the axomplasmatic resistivity, [math]\displaystyle{ c }[/math] the capacity and [math]\displaystyle{ i_{ion} }[/math] the ionic currents. From this the activating function follows as:
[math]\displaystyle{ f_n=\frac{d\Delta x}{4\rho_i Lc} \frac{V^e_{n-1}-2V^e_{n}+V^e_{n+1}}{\Delta x^2} }[/math].
In this case the activating function is proportional to the second order spatial difference of the extracellular potential along the fibers. If [math]\displaystyle{ L = \Delta x }[/math] and [math]\displaystyle{ \Delta x \to 0 }[/math] then:
[math]\displaystyle{ f=\frac{d}{4\rho_ic}\cdot\frac{\delta^2V^e}{\delta x^2} }[/math].
Thus [math]\displaystyle{ f }[/math] is proportional to the second order spatial differential along the fiber.
Interpretation
Positive values of [math]\displaystyle{ f }[/math] suggest a depolarization of the membrane potential and negative values a hyperpolarization of the membrane potential.
References
- ↑ 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.
- ↑ 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.
- ↑ 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.
- ↑ Rattay, F. (1990). Electrical Nerve Stimulation: Theory, Experiments and Applications. Wien, New York: Springer. pp. 264. ISBN 3-211-82247-X. https://archive.org/details/electricalnerves00ratt.
- ↑ 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.
- ↑ 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.
- ↑ Danner, S.M.; Wenger, C.; Rattay, F. (2011). Electrical stimulation of myelinated axons. Saarbrücken: VDM. pp. 92. ISBN 978-3-639-37082-9.
- ↑ Rattay, F.; Greenberg, R.J.; Resatz, S. (2003). "Neuron modeling". Handbook of Neuroprosthetic Methods. CRC Press. ISBN 978-0-8493-1100-0.
- ↑ 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.
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