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

$f_n=1/c\left( \frac{V^e_{n-1}-V^e_{n}}{R_{n-1}/2+R_{n}/2} + \frac{V^e_{n+1}-V^e_{n}}{R_{n+1}/2+R_{n}/2} + ... \right)$ ,

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]

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:

$\frac{dV^m_n}{dt}=\left[-i_{ion,n} + \frac{d\Delta x}{4\rho_i L} \cdot \left( \frac{V^m_{n-1}-2V^m_n+V^m_{n+1}}{\Delta x^2}+ \frac{V^e_{n-1}-2V^e_{n}+V^e_{n+1}}{\Delta x^2} \right) \right] / c$ ,

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:

$f_n=\frac{d\Delta x}{4\rho_i Lc} \frac{V^e_{n-1}-2V^e_{n}+V^e_{n+1}}{\Delta x^2}$ .

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:

$f=\frac{d}{4\rho_ic}\cdot\frac{\delta^2V^e}{\delta x^2}$ .

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.

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. and Rattay, F. (2011). Electrical stimulation of myelinated axons. Saarbrücken: VDM. p. 92. ISBN 978-3-639-37082-9.
↑ Rattay, F., Greenberg, R.J. and 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.