The Activating Function: How Electric Fields Excite Nerve Fibers

When a cochlear implant makes a deaf person hear a tone, when a deep brain stimulation lead calms the tremor of Parkinson's disease, or when a spinal cord stimulator masks chronic back pain, the same physical event sits underneath: current flowing out of a metal contact sets up an electric potential in the tissue, and somewhere along a nearby nerve fiber the membrane voltage is pushed past threshold, so the fiber fires an action potential. Engineers who design these devices need to answer a deceptively simple question again and again. If I put this electrode here and drive this current through it, which fibers will fire, and where along each fiber will the impulse start?

The activating function is the best-known first answer to that question. It is a compact mathematical expression, developed by the Austrian biomedical engineer Frank Rattay in a series of papers beginning in 1986, that estimates how strongly an applied extracellular field drives the membrane of a nerve fiber at each point along its length. Its central message is easy to remember. A nerve fiber does not respond to the extracellular potential itself, and not even to the slope of that potential. To a first approximation it responds to the curvature of the potential along the fiber, that is, to the second spatial derivative. Where that curvature is positive the membrane is pushed toward depolarization and possible firing; where it is negative the membrane is pushed toward hyperpolarization.

This article explains where the idea came from, how it is derived, what it predicts, where it is used in practice, and where it breaks down. It stays with what the published literature actually says, and it labels the one worked numerical example as hypothetical.

The problem it solves

Why stimulation is hard to predict

A nerve fiber is a long, thin tube of conducting fluid (the axoplasm) wrapped in a membrane that behaves electrically like a leaky capacitor studded with voltage-sensitive ion channels. In a myelinated fiber, most of the membrane is covered by an insulating myelin sheath, and the excitable membrane is exposed only at short gaps called nodes of Ranvier. When an electrode outside the fiber injects current into the surrounding tissue, the tissue acts as a volume conductor. Current spreads out in three dimensions, and the potential falls off with distance from the electrode.

Because the fiber is long and its interior conducts, some of that current takes a short cut. It enters the fiber through the membrane at one place, travels along the inside, and leaves through the membrane somewhere else. Where current leaves the fiber outward, the membrane is depolarized. Where current enters the fiber inward, it is hyperpolarized. The question of which fibers fire therefore depends not just on how close a fiber is to the electrode, but on the shape of the potential field along the fiber's path, the fiber's diameter, its internal resistance, its membrane properties and, for myelinated fibers, the spacing of the nodes.

One way to predict the result is to build a full computational model: solve for the potential in the tissue, then feed that potential into a detailed cable model of each fiber with nonlinear ion channel dynamics, and integrate forward in time. That approach is the gold standard today, but it is expensive, and in the 1970s and 1980s it was far more so. A quick, physically interpretable estimate of where a field will excite a fiber was, and still is, very useful.

The ingredients before Rattay

Several earlier developments made the activating function possible. The quantitative description of the action potential itself comes from Alan Hodgkin and Andrew Huxley's 1952 model of the squid giant axon, which treats the membrane as a capacitor in parallel with voltage-dependent sodium and potassium conductances. Bernhard Frankenhaeuser and Andrew Huxley later produced an equivalent description for amphibian myelinated nerve in 1964, and that formulation became the standard node-of-Ranvier model for decades of stimulation research.

The decisive step toward extracellular stimulation came from Donald McNeal. In a 1976 paper in IEEE Transactions on Biomedical Engineering, McNeal pointed out that there was no adequate model that let you compute a nerve fiber's threshold for pulses of finite duration delivered by electrodes not in direct contact with the fiber, and he built one. His model represents a myelinated fiber as a chain of active nodes, using Frankenhaeuser-Huxley membrane kinetics, connected by perfectly insulated resistive segments of internodal axoplasm. McNeal used a cytoplasmic resistivity of 110 ohm-centimeters, the same value Frankenhaeuser and Huxley had used in 1964. The extracellular potential at each node is computed from the electrode, and that potential drives the chain.

Two other bodies of earlier work matter for the context. First, the classic strength-duration description of excitation, associated with Louis Lapicque in the first decade of the twentieth century and with Georges Weiss in 1901, describes how threshold current depends on pulse duration. Lapicque's relation is commonly written I = b(1 + c/d), where b is the rheobase (the threshold for an infinitely long pulse), c is the chronaxie (the pulse duration at which threshold is twice the rheobase), and d is the pulse duration. Historical accounts differ on whether the key Lapicque publication should be dated 1907 or 1909. Second, James Ranck's 1975 review in Brain Research, titled "Which elements are excited in electrical stimulation of mammalian central nervous system," gathered the experimental evidence on what actually responds when the brain is stimulated. Among its widely cited points are that it takes less cathodal than anodal current to stimulate a myelinated fiber passing a monopolar electrode, and that most and perhaps all central myelinated fibers have chronaxies of 50 to 100 microseconds.

Rattay's contribution

The papers

Frank Rattay, working at the Vienna University of Technology, published "Analysis of Models for External Stimulation of Axons" in IEEE Transactions on Biomedical Engineering in 1986. He followed it with "Modeling the excitation of fibers under surface electrodes" (1988) and "Analysis of models for extracellular fiber stimulation" (1989) in the same journal, and gathered the theory into a 1990 Springer monograph, Electrical Nerve Stimulation: Theory, Experiments and Applications. In 1998 he extended the concept from peripheral fibers to neurons of the central nervous system in "Analysis of the electrical excitation of CNS neurons," and in 1999 he published an overview in the journal Neuroscience, "The basic mechanism for the electrical stimulation of the nervous system."

The insight in these papers is that if you take McNeal's compartment model and look only at the moment a stimulus is switched on, before any ion channel has had time to respond, the equations simplify to something remarkably clean. The rate at which the membrane voltage starts to change at each node is set by a single term that depends only on the applied extracellular potential and on passive fiber properties. Rattay called that term the activating function.

Why "the moment the stimulus starts" matters

At rest, the membrane sits at its resting potential, ion currents are balanced, and nothing is changing. When an external field is suddenly applied, the first thing that happens is that the membrane capacitance at each compartment begins to charge or discharge. For a brief instant, the ionic currents have not yet changed from their resting values, and the voltage differences across the internal resistances have not yet built up. In that instant, the slope of the membrane voltage at each point is driven purely by the external potential distribution. The activating function is exactly that initial slope. Rattay and his colleague Christian Wenger, writing in Neuroscience in 2010, described it as equal to the initial slope of the membrane voltage at stimulus onset in a resting neuron, and noted that it is independent of the ion channel composition of the membrane.

This is both the strength and the weakness of the concept. It is powerful because it separates the geometry of the field from the biology of the membrane. It is limited because real excitation unfolds over tens to hundreds of microseconds, during which current redistributes along the fiber and ion channels open and close.

How the activating function is derived

The compartment picture

Imagine a myelinated fiber laid along an x-axis. Number the nodes of Ranvier n = 1, 2, 3 and so on, separated by a distance Δx. Each node has a membrane capacitance, a set of ionic conductances, and a membrane voltage. Neighboring nodes are connected through the axoplasm, which has a resistance that depends on the axoplasmic resistivity ρi, the fiber's inner diameter d, and the internodal length.

Outside the fiber, the electrode produces an extracellular potential V^e_n at each node. Inside, the potential is V^i_n. The membrane voltage is the difference between the two. Applying Kirchhoff's current law at node n, the capacitive current plus the ionic current through that node's membrane must equal the net axial current arriving from the neighbors. When you write this out and rearrange it in terms of the reduced membrane voltage (the deviation from rest), the extracellular potentials appear only in one grouped term. That term is the activating function for node n, written in the Wikipedia formulation as f_n. Its units are volts per second, or equivalently millivolts per millisecond, which tells you immediately that it is a rate of change of membrane voltage.

The McNeal simplification

For a myelinated fiber with McNeal's assumption of an ideal internode (membrane capacitance and conductance of the myelinated segment set to zero), the expression becomes:

f_n = [d·Δx / (4·ρi·L·c)] × [(V^e_(n−1) − 2·V^e_n + V^e_(n+1)) / Δx²]

Here d is the fiber diameter, Δx is the node-to-node distance, L is the length of the node, ρi is the axoplasmic resistivity and c is the membrane capacitance per unit area. The second bracket is the standard finite-difference formula for a second derivative. It compares the extracellular potential at a node with the average of its two neighbors.

The continuous limit

For an unmyelinated fiber, or as a conceptual limit, you can let the compartment length shrink. Setting L equal to Δx and letting Δx go to zero turns the finite difference into a true derivative:

f = [d / (4·ρi·c)] × ∂²V^e/∂x²

This is the form most people have in mind when they say the activating function is "the second derivative of the extracellular potential along the fiber." The prefactor d/(4·ρi·c) tells you how strongly a given field curvature drives a particular fiber. It grows with diameter, which is one reason large fibers are generally easier to excite than small ones, a point Rattay and Wenger make explicitly.

Sign conventions

Positive values of f indicate depolarization and negative values indicate hyperpolarization. Where the activating function is positive and large, an action potential is likely to begin if the stimulus is strong enough. Rattay and Wenger describe positive regions as likely spike initiation sites and the negative side lobes as regions that hyperpolarize neighboring parts of the fiber.

Generalizing to real neurons

Real neurons are not infinitely long straight cables. They have a cell body, dendrites, an axon initial segment, branches and curves. Rattay's 1998 and 1999 papers developed what he called the generalized activating function, which applies the same logic compartment by compartment to a neuron of arbitrary shape, with each compartment's own diameter, length and capacitance. In this form, the activating function at a compartment is computed from the extracellular potentials of that compartment and its neighbors, weighted by the axial conductances between them, and divided by the compartment's capacitance. A commercial simulation platform, Sim4Life, offers a generalized activating function tool and describes the classical version as a fast alternative to full neuron simulation that ignores pulse shape, transient membrane dynamics, axial diffusion, leakage currents and activation at fiber terminals.

What the activating function predicts

A point electrode next to a straight fiber

The classic illustration is a tiny electrode, treated as a point current source, in a uniform conducting medium, placed at distance z0 from a long straight fiber. In a homogeneous isotropic medium of resistivity ρe, the potential at distance r from a point source of current I is the standard volume-conductor result V = ρe·I / (4π·r). Along the fiber, r is the square root of x² + z0², where x is the distance along the fiber from the point nearest the electrode.

Taking the second derivative of that potential along x gives a characteristic shape. Directly under the electrode the curvature has one sign, and on either side there are two side lobes of the opposite sign. With a cathodic (negative) current, the central region is depolarized and the flanks are hyperpolarized. With an anodic (positive) current, everything flips: the center is hyperpolarized and the two flanks become depolarized.

The geometry of that shape is fixed. Working through the calculus, the second derivative changes sign where x equals z0 divided by the square root of two. Seen from the electrode, the depolarized central zone therefore subtends an angle of about 70.5 degrees, and its length along the fiber is about 1.41 times the electrode-to-fiber distance. Rattay and Wenger state the same result for a straight fiber in a homogeneous medium: the depolarized region spans a 70 degree angle, giving an excited length of about 1.4 times the distance. The maximum of each side lobe, in the continuous case, is about one fifth of the central peak in magnitude, located at x roughly 1.22 times z0.

Why cathodes usually win on straight fibers

That last fact explains one of the oldest experimental observations in the field. For a cathodic pulse, the strongest depolarization is the large central peak. For an anodic pulse, the depolarization comes from the side lobes, which are only about a fifth as strong. You therefore need several times more anodic current than cathodic current to excite a long straight fiber passing an electrode. Rattay and Wenger report an anodic-to-cathodic threshold ratio of about 4 to 1 for long straight fibers in their simulations, consistent with Ranck's experimental summary that a passing myelinated fiber is excited by less cathodal than anodal current. The simple ratio of lobe heights does not give the exact threshold ratio, because thresholds also depend on time dynamics and on how current redistributes, but it explains the direction and rough size of the effect.

Virtual cathodes and virtual anodes

The flanking regions are sometimes called virtual electrodes. Under an anode, a fiber can be depolarized not beneath the metal contact but a short distance away, at a "virtual cathode." The same language is used, with a more elaborate theory, in cardiac electrophysiology. Bradley Roth's 1995 bidomain simulations of a unipolar cardiac electrode describe four excitation types: cathode make, anode make, cathode break and anode break. For anode make stimulation, tissue at a virtual cathode is depolarized to threshold. For anode break stimulation, a long anodal pulse hyperpolarizes tissue under the anode and depolarizes a virtual cathode, and when the pulse ends the depolarization spreads into the hyperpolarized region and excites it. In heart tissue the activating function has to be generalized because the intracellular and extracellular spaces have different anisotropies, but the idea that excitation is driven by spatial variation in the field is the same.

Cathodic block

The negative side lobes do more than hyperpolarize. If a cathodic stimulus is strong enough, the impulse that starts under the electrode can be stopped from propagating when it reaches the hyperpolarized flanks. Rattay and Wenger note that negative side lobes can block propagation at higher stimulus strengths, which is called cathodic block, and that reversing the polarity turns those lobes into stimulating regions. The activating function picture therefore predicts a window: too little current fails to excite, the right amount excites and propagates, and too much current excites but blocks.

Fiber diameter and recruitment order

Because the prefactor contains d, thicker fibers see a larger activating function in the same field. In myelinated fibers there is an additional effect. Node spacing grows with diameter, and the second difference across widely spaced nodes captures more of the potential's change. The result is that external stimulation tends to recruit large fibers before small ones, which is the opposite of the size principle seen in natural motor recruitment. This "reverse recruitment" is a well-known property of electrical stimulation and is one of the problems functional electrical stimulation designers try to manage.

Where the activating function is zero

A uniform electric field along a straight, infinitely long fiber has a potential that changes linearly with distance. Its second derivative is zero everywhere, so the activating function predicts no effect on the interior of the fiber. That is correct for the idealized case. In real tissue, though, fibers end, bend, branch and change diameter, and at all of those places the effective driving term is not zero. This is why fiber terminals, bends and the transition between the cell body and the axon are often where excitation actually begins in a uniform field. A user on the NEURON simulator's discussion forum summarized the practical caution bluntly: the activating function assumes an infinitely long structure and fails completely if the second derivative is zero.

A worked example (hypothetical numbers)

To make the formula concrete, here is a calculation with illustrative values. None of these numbers describes a particular device or experiment; they are chosen only to be in a plausible range.

Assume a straight myelinated fiber of diameter 10 micrometers, with nodes 1 millimeter apart and each node 1 micrometer long. Assume an axoplasmic resistivity of 110 ohm-centimeters (McNeal's value, 1.1 ohm-meters in SI), a nodal membrane capacitance of 2 microfarads per square centimeter (0.02 farads per square meter), and an extracellular resistivity of 300 ohm-centimeters (3 ohm-meters, the value Rattay and Wenger used for central tissue). Place a point electrode 1 millimeter from the fiber, directly above one node, and drive a cathodic current of 0.5 milliamperes.

The point-source formula gives the extracellular potential at each node:

The prefactor d/(4·ρi·L·c·Δx) equals 10⁻⁵ divided by (4 × 1.1 × 10⁻⁶ × 0.02 × 10⁻³), which is about 1.14 × 10⁵ per second. Multiplying it by the second difference of the potentials gives the last column. Under the electrode the activating function is strongly positive, so the central node starts depolarizing at nearly 8 millivolts per microsecond. The neighbors are driven negative. Because the nodes are spaced as far apart as the electrode is from the fiber, the discrete samples are coarse, so the strongest hyperpolarization falls at the second node out rather than the first. That is a reminder that in myelinated fibers the node spacing matters, and the exact location of the electrode relative to the nodes changes the numbers.

Two lessons follow. First, the initial depolarization rate at the central node is large enough that, if sustained, it would cross a threshold depolarization of a few tens of millivolts in well under a tenth of a millisecond. In reality it is not sustained, because current redistributes and the membrane leaks, so this is an upper-bound picture. Second, halving the current halves every number in the table, since the activating function is linear in the applied potential. Moving the electrode further away reduces the central value roughly as the inverse cube of distance in the continuous case, because the second derivative of a 1/r potential scales as 1/r³ at the point nearest the electrode.

Distance, current and threshold

How threshold current grows with distance is one of the most practical questions in stimulation, because it sets how far an electrode reaches. Simple activating-function reasoning suggests that, for a fixed threshold value of f, the required current should grow steeply with distance, since the peak curvature falls off as the cube of distance. Measured and simulated thresholds follow a gentler rule, because a whole region of fiber, not just one point, contributes to excitation and because the depolarized length itself grows with distance.

Rattay and Wenger's 2010 simulations of cortical neurons give concrete figures. They found that thresholds follow a quadratic rule, current proportional to distance squared, at distances above about 100 micrometers, consistent with experimental work by Tehovnik in 1996. Below about 50 micrometers the relation becomes roughly linear. They also cite Stoney and colleagues' 1968 estimate that 200-microsecond pulses at 10 and 100 microamperes activate cells within about 100 and 450 micrometers of the electrode tip.

The same paper shows how differently the parts of a neuron respond. In their model, the distal axon initial segment and the nodes of Ranvier were the most excitable structures. Thick dendrites needed roughly two to three times the current of initial segment or nodal positions, and thin branches needed more. With an electrode 50 micrometers away, the soma threshold for a monophasic pulse was about 36 microamperes while the distal initial segment threshold was about 17. A thin axon of 0.2 micrometers passing 10 micrometers from the tip could be activated by 100-microsecond pulses of 2.6 microamperes. Spike initiation was usually close to the electrode but could occur about 200 micrometers away in the low-threshold initial segment.

These results also line up with Rattay's 1999 abstract, which concludes that in most cases artificial excitation starts in the axon and that the soma is much harder to excite, that positive and negative threshold currents in the central nervous system depend on each neuron's position and orientation relative to the field, and that in several cases positive (anodic) currents are easier to use for stimulation than negative ones.

Orientation: passing fibers versus terminating fibers

That last point, that anodic stimulation is sometimes more effective, seems to contradict the straight-fiber result. The resolution is geometry. A fiber that passes by an electrode sees a central lobe and two side lobes, so cathodic stimulation wins. A fiber or neuron that points toward or away from the electrode sees a very different potential profile, and for it the anodic polarity can produce the larger depolarization at a sensitive spot such as the axon initial segment or a terminal.

McIntyre and Grill reported in 1999, in the Biophysical Journal, that in a cable model of a central neuron with an axon, initial segment, axon hillock, soma and simplified dendrites, the site of action potential initiation depended on electrode position, stimulus duration and polarity. At threshold, the axon or initial segment was always the initiation site. They also found that the site of maximum depolarization did not always predict where initiation occurred, because the time course of membrane changes mattered too.

A 2018 study in the Journal of Neural Engineering by Anderson, Duffley, Vorwerk, Dorval and Butson, titled "Anodic stimulation misunderstood," applied this reasoning to deep brain stimulation. Using field models coupled to multicompartment neuron models, they found that cathodic stimulation preferentially activates axons passing the lead, while anodic stimulation preferentially activates axons oriented orthogonally, approaching or leaving the electrode, and that in a clinical scenario anodic stimulation reached pathways associated with therapeutic benefit at amplitudes about 50 percent lower. Their conclusion was that the stimulus waveform could be used to select fibers by orientation, which might also save battery power.

Magnetic stimulation and the activating function

The same reasoning extends to stimulation by changing magnetic fields, as in transcranial magnetic stimulation and peripheral magnetic stimulation. A time-varying magnetic field induces an electric field in tissue. For a long straight axon, the driving term in the cable equation becomes the spatial derivative, along the fiber, of the component of the induced electric field that points along the fiber. Since the electric field is itself the negative gradient of a potential in the electrostatic case, this is the same second-derivative idea expressed in terms of the field rather than the potential.

Bradley Roth and Peter Basser described this in 1990 in IEEE Transactions on Biomedical Engineering with a cable model of magnetic stimulation of an axon. Basser, Wijesinghe and Roth showed in 1992 that a three-dimensional volume-conductor model reduces to the one-dimensional cable equation forced by the activating function, supporting the earlier assumptions. Experimental work by Maccabee and colleagues is reported to have confirmed that excitation sites lie at or near the peaks of the negative-going spatial derivative of the induced field. Later studies added caveats: the activating function is not the whole story in magnetic stimulation, particularly near bends and terminations where other mechanisms contribute.

Applications

Cochlear implants

Cochlear implants place an array of electrode contacts inside the spiral cochlea to stimulate auditory nerve fibers. The central design goal is selectivity: each contact should excite a distinct population of fibers so that different contacts produce different pitches. Current spread defeats that goal by activating fibers far from the intended place.

The activating function has long been used as a quick way to visualize how a given electrode configuration will shape excitation along the curved paths of auditory nerve fibers. A 2023 study in Frontiers in Audiology and Otology by Cakmak, Pal and Sahin at the New Jersey Institute of Technology built a finite-element model of a realistic cochlear geometry from micro-CT images and used both Rattay's activating function and a full compartmental nerve model. The authors reported that the activating function closely agreed with the lowest-threshold fibers predicted by the full model, while also stating that it only provides an approximation of neural thresholds. Their model compared electrode configurations. The lowest thresholds on one contact were 941 microamperes for monopolar stimulation and 979 for a distant bipolar configuration, against 1,365 for tripolar. A distant bipolar return improved threshold margins over monopolar on 9 of 16 electrodes, by about 31 percent on those electrodes. Removing the insulating electrode carrier from the model raised thresholds by 36 to 67 percent, a reminder of how much the insulating geometry shapes the field. The authors listed their own limitations frankly: no electrophysiological validation, isotropic tissue, assumed fiber paths and a single cochlear geometry.

Deep brain stimulation

Deep brain stimulation leads deliver current to small brain targets through cylindrical contacts. Which fiber tracts are activated depends on their distances and orientations relative to the contacts. Activating-function maps, computed from finite-element models of the field around the lead, are commonly used to estimate which nearby axons are likely to be excited, and to compare contact choices or polarities. The 2018 anodic-stimulation study above is one example of how orientation-dependent predictions feed directly into clinical programming ideas.

Spinal cord stimulation and peripheral nerve stimulation

Spinal cord stimulators place contacts in the epidural space over the dorsal columns. Peripheral nerve stimulators wrap cuff electrodes around nerves. In both, designers want to recruit certain fibers while sparing others. Coupled models, in which a three-dimensional finite-element solution for the potential in the spinal cord or nerve is fed into neuron models, are standard. The activating function serves as the fast screening step in that pipeline, and generalized versions of it are used in commercial tools for optimizing multipolar electrode settings.

Functional electrical stimulation

Rattay's original motivation, and the context in which the Wikipedia summary places the concept, is functional electrical stimulation: restoring movement or other function by stimulating motor nerves after spinal cord injury or stroke. Here the activating function helps explain why surface and implanted electrodes recruit large motor fibers first, why the electrode's distance from the nerve matters so much, and how electrode shape changes the field curvature.

Teaching and intuition

Perhaps the activating function's most lasting use is as a teaching tool. It lets students see, with a pencil sketch of a potential curve, why a cathode excites under itself, why anodes excite at a distance, why bends and terminals are sensitive, why large fibers go first and why uniform fields along straight fibers do nothing. Few other single equations in neural engineering carry that much intuition.

Limitations and extensions

It is an initial-rate estimate

The activating function gives the slope of membrane voltage at the instant the stimulus starts, in a resting fiber. It says nothing directly about what happens next. During a stimulus pulse, current flows along the inside of the fiber from depolarized regions to hyperpolarized ones, smoothing the membrane voltage profile. Ion channels respond. For short pulses the activating function pattern is a fair guide; for long pulses the redistributed pattern can look quite different.

It misses current redistribution between nodes

Warman, Grill and Durand addressed this directly in a 1992 paper in IEEE Transactions on Biomedical Engineering on modeling the effects of electric fields on nerve fibers and determining excitation thresholds. Their analysis showed that the polarization of a node has two drivers: a source term, which is the activating function at that node, and an ohmic term arising from current redistributed by sources at other nodes. They combined both into a total equivalent driving function, which they reported predicted excitation thresholds for arbitrary applied fields and eliminated the errors associated with using the activating function alone. A related 1991 conference paper compared an expanded activating function with the traditional form and catalogued the common errors that the traditional version produces.

Moffitt, McIntyre and Grill later tested linear threshold-prediction methods of this kind for myelinated fibers in 2004. They found that the prediction error was low, under 3 percent, under the conditions Warman and colleagues had examined, but could reach 23.6 percent under more general conditions. In other words, even improved linear methods have a validity range.

It assumes a particular cable structure

The McNeal-based activating function treats the internode as a perfect insulator. Later fiber models are more detailed. McIntyre, Richardson and Grill's model of mammalian nerve fibers, published in the Journal of Neurophysiology, uses a double cable structure with explicit nodes of Ranvier, paranodal and internodal sections, and a myelin sheath of finite impedance. Related work from the same group compared a perfectly insulating single cable, a finite-impedance single cable and a finite-impedance double cable, and concluded that the simpler models were adequate for low-frequency stimulation below roughly 25 hertz, while the double cable was needed for high-frequency stimulation. The activating function concept still applies to such models, but the compartment-level coefficients change.

It ignores the field's interaction with the fiber

The standard approach computes the extracellular potential as if the fiber were not there, then applies it to the fiber. That is usually a good approximation because fibers are thin, but in densely packed nerves, in fascicles surrounded by resistive perineurium, and near electrode contacts, the presence of tissue structures changes the field. Most modern models therefore solve the field in a detailed anatomical geometry first.

It can fail at structural irregularities

As noted, a uniform field along a straight fiber produces no activating function in the fiber's interior, yet real fibers do respond to uniform fields at their ends, bends and branch points. The generalized activating function handles these cases better because it is computed compartment by compartment on the real morphology, but simple textbook use of the continuous second derivative misses them. Rattay and Wenger note that spike initiation can appear wherever the field is irregular, either from field inhomogeneity or fiber curvature.

It is qualitative for thresholds

The Frontiers cochlear study and the Sim4Life description both make the same point in different words. The activating function is good at telling you where excitation is most likely and which fibers are favored. It is not a reliable substitute for a full nonlinear model when you need an actual threshold current. In practice, engineers use it to narrow down designs and to understand results, then confirm with full simulations.

Common misconceptions

"Fibers respond to the electric potential"

They do not, at least not directly. A uniform shift of the potential everywhere, as you get by touching a fiber to a battery terminal with no current path, changes nothing about excitation. What matters is how the potential varies along the fiber, specifically its curvature.

"Fibers respond to the electric field"

Closer, but still not right for a straight fiber in a uniform field. The electric field is the first derivative of the potential. A field that is the same everywhere along the fiber produces no net transmembrane current in the fiber's interior. It is the change of the field along the fiber, the second derivative of potential, that matters. The field itself becomes important at ends, bends and branches.

"Anodes never excite"

Anodes excite readily at a distance, through the flanking virtual cathodes, and for neurons oriented toward the electrode they can be more effective than cathodes. The rule that cathodes have lower thresholds holds for long straight fibers passing the electrode, not for all geometries.

"The activating function predicts threshold"

It predicts the initial rate of membrane polarization and therefore the likely site and relative ease of excitation. It does not, by itself, give threshold current, which depends on pulse duration, membrane dynamics and current redistribution.

"Big activating function means certain excitation"

A large positive activating function at one node can be accompanied by large negative values nearby that block propagation. The overall pattern, not the peak alone, determines whether an impulse starts and escapes.

Key terms in brief

Frequently asked questions

Who invented the activating function?

Frank Rattay of the Vienna University of Technology introduced the concept in papers published from 1986 onward, building on Donald McNeal's 1976 compartment model of a myelinated nerve fiber stimulated by an external electrode.

What are its units?

Volts per second, often written as millivolts per millisecond. It is a rate of change of membrane voltage.

Why the second derivative and not the first?

Current enters or leaves the fiber where the axial current inside it changes. The axial current is proportional to the slope of the potential along the fiber. The change of that slope along the fiber is the second derivative. Where the slope changes, current must cross the membrane, and that crossing current charges the membrane capacitance.

Does it work for unmyelinated fibers?

Yes. The continuous form, with the second derivative multiplied by d/(4·ρi·c), applies to uniform unmyelinated cables. Myelinated fibers use the discrete, node-by-node form.

Why are large fibers excited first?

The activating function scales with fiber diameter, and in myelinated fibers the larger node spacing of thick fibers samples more of the potential change. Both effects favor large fibers.

Is it still used now that full models are cheap?

Yes, as a screening tool, an explanatory tool and a design aid. Commercial and research software packages implement both the classical and generalized versions. Detailed models remain necessary for thresholds and for long or complex pulses.

Does it apply to the heart?

A generalized activating function is used in cardiac electrophysiology, where bidomain models account for the different anisotropies of intracellular and extracellular spaces. The virtual electrode concept is central to understanding how the heart responds to pacing and defibrillation shocks.

Can it explain magnetic stimulation?

Yes, in a modified form. For magnetic stimulation, the driving term is the spatial derivative along the fiber of the induced electric field component parallel to the fiber, as described by Roth and Basser in 1990.

Summary

The activating function compresses a complicated three-dimensional problem into one line: the rate at which an applied field starts to polarize a nerve fiber is proportional to the curvature of the extracellular potential along that fiber, scaled by the fiber's diameter, internal resistivity and membrane capacitance. From that line follow most of the textbook facts about electrical stimulation, from the lower cathodal threshold of passing fibers to virtual cathodes, cathodic block, reverse recruitment and the sensitivity of fiber ends and bends. Its limits are equally well documented. It is an initial-rate estimate that ignores current redistribution and membrane dynamics, it can mislead in uniform fields and complex geometries, and it does not on its own predict threshold currents. Modern neural engineering uses it in that spirit: as the quick first look and the explanation, followed by detailed simulation when numbers matter.

Sources

• Activating function (Wikipedia)
• Rattay, F. (1986). Analysis of Models for External Stimulation of Axons. IEEE Trans. Biomed. Eng. 33(10):974–977
• Rattay, F. (1989). Analysis of models for extracellular fiber stimulation. IEEE Trans. Biomed. Eng. 36(7):676–682
• Rattay, F. (1999). The basic mechanism for the electrical stimulation of the nervous system. Neuroscience 89(2):335–346 (PubMed)
• Rattay, F. & Wenger, C. (2010). Which elements of the mammalian central nervous system are excited by low current stimulation with microelectrodes? Neuroscience 170:399–407 (PMC)
• McNeal, D.R. (1976). Analysis of a Model for Excitation of Myelinated Nerve. IEEE Trans. Biomed. Eng. BME-23(4):329–337
• McIntyre, C.C. & Grill, W.M. (1999). Excitation of central nervous system neurons by nonuniform electric fields. Biophysical Journal 76(2):878–888 (PMC)
• Anderson, D.N. et al. (2018). Anodic stimulation misunderstood: preferential activation of fiber orientations with anodic waveforms in deep brain stimulation. J. Neural Eng. 16(1):016026 (PMC)
• Cakmak, O., Pal, S. & Sahin, M. (2023). Improving the stimulation selectivity in the human cochlea by strategic selection of the current return electrode. Frontiers in Audiology and Otology
• Warman, Grill & Durand (1992) and related work, Duke Scholars record
• Sim4Life: Predicting Neural Responses with the Generalized Activating Function
• Ranck, J.B. Jr. (1975). Which elements are excited in electrical stimulation of mammalian central nervous system: a review. Brain Research 98(3):417–44090364-9)
• Chronaxie (Wikipedia)
• Rheobase (Wikipedia)
• NEURON forum: Extracellular stimulation and recording
• Roth, B.J. & Basser, P.J. (1990). A model of the stimulation of a nerve fiber by electromagnetic induction. IEEE Trans. Biomed. Eng. 37(6):588–597