Activating function

{{short description|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|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.{{Cite journal | last1 = Rattay | first1 = F. | doi = 10.1109/TBME.1986.325670 | title = Analysis of Models for External Stimulation of Axons | journal = IEEE Transactions on Biomedical Engineering | issue = 10 | pages = 974–977 | year = 1986 | volume = 33 | pmid = 3770787 | s2cid = 33053720 }}{{Cite journal | last1 = Rattay | first1 = F. | title = Modeling the excitation of fibers under surface electrodes | doi = 10.1109/10.1362 | journal = IEEE Transactions on Biomedical Engineering | volume = 35 | issue = 3 | pages = 199–202 | year = 1988 | pmid = 3350548| s2cid = 27312507 }}{{Cite journal | last1 = Rattay | first1 = F. | title = Analysis of models for extracellular fiber stimulation | doi = 10.1109/10.32099 | journal = IEEE Transactions on Biomedical Engineering | volume = 36 | issue = 7 | pages = 676–682 | year = 1989 | pmid = 2744791| s2cid = 42935757 }}{{cite book|last=Rattay|first=F.|title=Electrical Nerve Stimulation: Theory, Experiments and Applications|url=https://archive.org/details/electricalnerves00ratt|url-access=limited|year=1990|publisher=Springer|location=Wien, New York|isbn=3-211-82247-X|pages=[https://archive.org/details/electricalnerves00ratt/page/n265 264]}}{{Cite journal | last1 = Rattay | first1 = F. | title = Analysis of the electrical excitation of CNS neurons | doi = 10.1109/10.678611 | journal = IEEE Transactions on Biomedical Engineering | volume = 45 | issue = 6 | pages = 766–772 | year = 1998 | pmid = 9609941| s2cid = 789370 }}{{Cite journal | last1 = Rattay | first1 = F. | title = The basic mechanism for the electrical stimulation of the nervous system | doi = 10.1016/S0306-4522(98)00330-3 | journal = Neuroscience | volume = 89 | issue = 2 | pages = 335–346 | year = 1999 | pmid = 10077317| s2cid = 41408689 }} 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.{{cite book|author=Danner, S.M. |author2=Wenger, C. |author3=Rattay, F.|title=Electrical stimulation of myelinated axons|year=2011|publisher=VDM|location=Saarbrücken|isbn=978-3-639-37082-9|pages=92}} 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.{{cite book|author=Rattay, F. |author2=Greenberg, R.J. |author3=Resatz, S.|title=Handbook of Neuroprosthetic Methods|year=2003|publisher=CRC Press|isbn=978-0-8493-1100-0|chapter=Neuron modeling}}

Following McNeal's{{Cite journal | last1 = McNeal | first1 = D. R. | volume=BME-23 |doi = 10.1109/TBME.1976.324593 | title = Analysis of a Model for Excitation of Myelinated Nerve | journal = IEEE Transactions on Biomedical Engineering | issue = 4 | pages = 329–337 | year = 1976 | pmid = 1278925 | s2cid = 22334434 }} 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

Category:Bioelectrochemistry

Category:Computational neuroscience