^{1}

^{*}

^{1}

^{2}

^{1}

^{3}

^{3}

^{1}

^{2}

^{3}

Edited by: Stephen Ellis Robinson, National Institutes of Health, USA

Reviewed by: Seppo P. Ahlfors, Massachusetts General Hospital, USA; Daniel Baumgarten, Ilmenau Technical University, Germany

*Correspondence: Risto J. Ilmoniemi

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

_{0} is defined; λ_{0} describes the axial decay of the membrane voltage in the case of constant applied electric field. In TMS, however, the induced electric field waveform is typically a segment of a sinusoidal wave, with characteristic frequencies of the order of several kHz.

_{eff}, which governs the spatial decay of the membrane voltage. We model the behavior of a dendrite in an applied electric field oscillating at 3.9 kHz with the complex cable equation and by solving the traditional cable equation numerically.

_{0} = 1.5 mm, the effective length constant at 3.9 kHz is decreased by a factor 10 to 0.13 mm.

In transcranial magnetic stimulation (TMS; Barker et al.,

Typically, the magnetic field is produced with a coil wound in a figure-of-eight form; if such a coil is placed tangentially over the scalp, a current pulse in the coil induces a reasonably focal electric field in the superficial brain. The primary electric field is very homogeneous on the cellular scale; however, the complicated conductivity structure of the neurons, to a large extent defined by cell membranes, changes the local electric current patterns dramatically. Because the microscopic tissue structure is not generally available, precise activation patterns usually cannot be predicted; one has to rely on computing the transmembrane potentials in assumed, generally highly simplified, neuronal geometries. Often, the cable equation (e.g., Roth and Basser,

The cable equation, as described in Section 2.1, describes a passive axon or dendrite as a cylinder defined by a resistive–capacitive membrane and conducting intracellular fluid. The analysis leads to two useful concepts, the length constant and the membrane time constant. The length constant describes the rate of exponential decay of membrane voltage as a function of distance from the location where current is injected (typically, a synapse or site of transmembrane ion flow during an action potential). The classical length constant is defined in the limit of low frequencies or for dc currents. The time constant describes the exponential decay of membrane capacitance via current leakage through the resistive membrane. This leakage has been assumed to weaken the effect of TMS when long-risetime pulses are used.

Here, we also use the cable equation but define the length constant for alternating currents instead of dc currents. It turns out that at the characteristic frequencies in TMS, i.e., several kHz, the length constant is far shorter than at dc. This has consequences on how neurons are activated since the area of the the membrane that is depolarized is proportional to the length constant.

In case of a passive cell, the cable equation is derived as follows:

The axial current _{i} inside the cell (see Figure _{m} is the membrane potential, which equals the potential inside the cell if the extracellular potential is zero, _{x} is the axial component of the applied electric field, and _{i} is the axial resistance of the cytoplasm per unit length. The law of conservation of currents gives the membrane current per unit length
_{x}(_{m} is the membrane capacitance per unit length of the cell and _{m} is the membrane resistance of a segment of the axon times the length of the segment.

Combining Equations (3) and (4) gives the cable equation

However, as we have documented earlier (Saari, _{0} describes the spatial decay of membrane potential only in the steady state. This results from the fact that the cellular membrane has capacitive properties and its conductivity can be described as a frequency-dependent complex value. Although the quasi-static approximation is valid at the macroscopic level at the dominant TMS-pulse frequencies of several kilohertz (Roth et al.,

In the present work, we study the frequency-dependent behavior of a neuron and formulate the effective length constant that governs the spatial decay of the membrane voltage. Preliminary results of this work have been reported in abstract form (Ilmoniemi et al.,

Let us assume that the electric field component parallel to the neuronal axis (_{x}) is constant along the length of the axon or dendrite and oscillating at angular frequency ω:
_{m} as

Combining Equations (10) and (3) gives a complex frequency-dependent cable equation
_{f} = λ_{0}, the steady-state length constant.

Equation (13) can be divided into real and imaginary parts:
_{f}).

From Equation (14), we see that the real behavior of the membrane voltage can be expressed as:
_{eff} is the effective length constant:
_{k} and ϕ_{k} are the complex Fourier coefficient and phase of frequency component ω_{k}, respectively, the membrane voltage can be written as

To solve the constant _{end} and _{end} are the resistance and capacitance of the end of the cable. Substituting Equation (13) into Equation (22) gives

The variation of the neuronal membrane potential when influenced by the electric field induced in TMS can also be determined by numerically solving a discretized version of the cable equation, Equation (5) (Nagarajan et al.,

That was done by employing a similar approach to the one described in a previous study (Salvador et al.,

The resulting set of non-linear equations was solved using the Crank-Nicholson method, with a staggered time step approach (Hines,

The apical dendrite was represented as a 6-mm-long cylinder with properties described in Table

Radius ( |
4 μm |

Axoplasmic resistivity (ρi) | 0.33 Ωm |

Ohmic membrane conductance per unit area (_{m}) |
2.73 S/m^{2} |

Membrane capacitance per unit area (_{m}) |
0.028 F/m^{2} |

The temporal waveform of the electric field along the neuron was sinusoidal with frequency

_{0} = 61.2 V/m, obtained from Equations (15) and (23).

All simulations took less than a minute to solve in a computer with a quad-core CPU clocked at 2 GHz and 8 Gb of RAM.

The length constant of the numerically calculated membrane voltage was estimated with exponential curve fitting in Matlab: nonlinear least-squares fitting with the trust-region algorithm was applied to the membrane voltage curve near the end of the dendrite.

For the model dendrite with properties described in Table _{i}, _{m}, and _{m} in the following way:

The numerical results agreed with the analytical ones: The decay of the membrane voltage at the end of the dendrite (0…0.23 mm) at the time of maximal ^{2} = 1) with two exponentials, one with a length constant of 0.13 mm [corresponding to the exponential part of Equation (19)] and the other one with a length constant of 4.9 mm [corresponding to the sinusoidal part of Equation (19) and the DC component, i.e., resting membrane voltage].

We have shown how the neuronal length constant decreases with increasing frequency of the stimulation pulse waveform. In TMS, the applied electric field typically has characteristic frequencies of the order of kilohertz, in which case the length constant can be an order of magnitude smaller than the steady-state value. As a consequence, the segment of a neuron where the TMS pulse can trigger voltage-dependent sodium channels is much narrower than one might have previously thought, influencing the efficacy of the initial inflow of Na^{+} current in initiating the action potential. This has potentially two major consequences. First, the threshold voltage of sodium channels is reached with less transferred charge with short than with longer pulses; thus, shorter pulses are more energy-efficient (Barker et al.,

In our analysis, we made the simplifying assumption that the extracellular potential is zero or, equivalently, that the extracellular resistivity is vanishingly small. In the brain, however, the extracellular resistance per unit length cannot be assumed much smaller than the intracellular resistance per unit length. Thus, the extracellular potential is nonzero (Equation 1); qualitative understanding of TMS-induced effects, however, remains unchanged even if the extracellular potential is taken into account (Nagarajan and Durand,

One should note that the traditional cable equation (Equation 5) is also correct (which is why the numerical and analytical results coincide). The frequency dependency of the length constant can be seen by Fourier-transforming the traditional cable equation. This is what Meffin and Kameneva (

All authors listed, have made substantial, direct and intellectual contribution to the work, and approved it for publication.

RI, HM, and JS were supported by grants from the Academy of Finland (decision numbers 121167, 256525, and 283105). RS and PM were supported in part by the Foundation for Science and Technology (FCT), Portugal, via FCTIBEB Strategic Project UID/BIO/00645/2013.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The authors want to thank Panu Vesanen for helpful discussions.