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. 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, formula_1, is derived from the driving term of the external potential, or the equivalent injected current
formula_2,
where formula_3 is the membrane capacity, formula_4 the extracellular voltage outside compartment formula_5 relative to the ground and formula_6 the axonal resistance of compartment formula_5.
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 formula_8 for each node is:
formula_9,
where formula_10 is the constant fiber diameter, formula_11 the node-to-node distance, formula_12 the node length formula_13 the axomplasmatic resistivity, formula_3 the capacity and formula_15 the ionic currents. From this the activating function follows as:
formula_16.
In this case the activating function is proportional to the second order spatial difference of the extracellular potential along the fibers. If formula_17 and formula_18 then:
formula_19.
Thus formula_20 is proportional to the second order spatial differential along the fiber.
Interpretation.
Positive values of formula_20 suggest a depolarization of the membrane potential and negative values a hyperpolarization of the membrane potential.