Activating function

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.^[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 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 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 $V^{m}$ for each node is:

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

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 $\Delta x\to 0$ then:

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

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

↑ 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. p. 264. ISBN 3-211-82247-X.
↑ 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. p. 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 (4): 329–337. doi:10.1109/TBME.1976.324593.

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