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Somatosensory evoked potentials are a well-established tool for assessing volley conduction in afferent neural pathways. However, from a clinical perspective, recording of spinal signals is still a demanding task due to the low amplitudes compared to relevant noise sources. Computer modeling is a powerful tool for gaining insight into signal genesis and, thus, for promoting future innovations in signal extraction. However, due to the complex structure of neural pathways, modeling is computationally demanding. We present a theoretical framework which allows computing the electric potential generated by a single axon in a body surface lead by the convolution of the neural lead field function with a propagating action potential term. The signal generated by a large cohort of axons was obtained by convoluting a single axonal signal with the statistical distribution of temporal dispersion of individual axonal signals. For establishing the framework, analysis was based on an analytical model. Our approach was further adopted for a numerical computation of body surface neuropotentials employing the lead field theory. Double convolution allowed straightforward analysis in the frequency domain. The highest frequency components occurred at the cellular membrane. A bandpass type spectral shape and a peak frequency of 1800 Hz was observed. The volume conductor transmitting the signal to the recording lead acted as an additional bandpass reducing the axonal peak frequency from 200 Hz to 500 Hz. The superposition of temporally dispersed axonal signals acted as an additional low-pass filter further reducing the compound action potential peak frequency from 90 Hz to 170 Hz. Our results suggest that the bandwidth of spinal evoked potentials might be narrower than the bandwidth requested by current clinical guidelines. The present findings will allow the optimization of noise suppression. Furthermore, our theoretical framework allows the adaptation in numerical methods and application in anatomically realistic geometries in future studies.

Somatosensory evoked potentials (SEPs) are a well-established tool for studying the conduction of stimulated peripheral activity in sensory pathways to the somatosensory cortex in clinical routine [

While in [

Besides experimental data, modeling can be used for predicting spectral properties of bioelectric signals. In particular, in cardiac electrophysiology, model-based spectral analysis is a well-established tool [

Clark and Plonsey [

In this study, we present a novel theoretical framework using distributed dipoles along a central axis of a nerve, fascile, or neural tract for efficiently computing the lead field generated by such a structure. Thus, a tool is provided, which allows for multiscale modeling by linking cell models with body surface potentials. Our approach allows using statistical distribution functions for computing the potential field generated by a large number of axons. Furthermore, the use of Fourier transform (FT) allows a straightforward investigation of spectral signal properties.

We aimed at developing our theoretical framework for analytic expressions and, therefore, chose a simplified half-space geometry which approximates longitudinal conduction of a volley within a spinal tract. An early study revealed low-amplitude (

In this section, we present the key features of our analytical framework. All detailed mathematical derivations are listed in the Appendices (available

The concept underlying the lead field two-domain model is depicted in Figure

Concept of the lead field two-domain model. (a) A neural tract of diameter

We aimed at computing the electric potential in the lead field of a neural tract. Here, the distance ^{−1} to 80 ms^{−1} and have a duration of 0.5 ms to 1.0 ms. Thus, the activated segment of each axon has a length of some centimeters. Note that this is much larger than the diameter

Thus, we considered two domains: an active intracellular domain and a passive extracellular or bulk domain. In the first step, we computed the signal generated by a single axon. In the second step, we applied superposition of dispersed axonal signals for obtaining the compound action potential (CAP) of a neural tract.

We considered the signal generated by a single axon located at the central axis of a neural tract. Figure

(a) Geometry of the model. The axon is located at a distance

Dipole strength is proportional to the first temporal derivative of the membrane potential (see Appendix A.1). We translated the movement of the membrane action potential along the axon into a convolution term and obtained the body surface potential

Here,

Applying convolution theory and time continuous Fourier transform (FT), the frequency spectrum of the axonal signal was obtained by the product of the FT of two terms. The first (source) term is the temporal derivative of the membrane potential. The second (volume conductor) term reflects the lead field of a moving dipole source

A neural pathway is formed by a large number of

The spectrum of the CAP was obtained by multiplying the spectrum of a single axon with the spectrum of the statistical distribution of the dispersion

In the time domain, the CAP was obtained by a convolution of the signal of a single axon

The spectrum of the distribution function (

In the time domain, the CAP signal

We observed that our analytical model represents the frequency spectrum of the lead field potential

The membrane potential derivative (source term, obtained from a cell model)

The lead field (moving dipoles)

The dispersion term

Figure

Signal generation model in the frequency domain (see text).

(a) Membrane action potential (dashed) of a sensory axon [

Furthermore, the derivative of an action potential traveling along the axon is a piecewise continuous function. Thus, we concluded from basic theory of Fourier transform that the spectrum approaches zero also with increasing frequency (see Figure

The lead field term yields a function with a single oscillation and point-symmetric shape (see Figure

Here,

Only four biophysically relevant parameters enter this estimation: duration of an activation cycle

For numerical evaluation of axonal potentials (

Schematic drawing of the discrete model underlying the numerical approximation. At the body surface, the potential

Here

We simulated the action potential of a sensory axon applying the ionic current model described in [

In the first step, we varied the axon diameter ^{−1}, 60 ms^{−1}, and 75 ms^{−1}) [

According to equation (

For all combinations of parameters, we computed the axonal potential

For simulating neural potentials, we assigned three values to the standard deviation

^{−1}, 60.0 ms^{−1}, and 75.1 ms^{−1}, respectively. Variation in diameter had only a minor effect on the action potential. For all three simulations, peak-to-peak amplitude (maximum at activation vs. minimum during hyperpolarization) was in the range of 107.9 mV to 108.8 mV. The time span between the two potential extrema was in the range of 480

Results for parameter setting ^{−1}, and

Peak frequency | Bandwidth | Frequency estimates | Amplitude | |
---|---|---|---|---|

(Hz) | (Hz) | (Hz) | (peak-to-peak) | |

AP derivative, | 1830 | 547 to 4545 | 1750 | 3.1 |

Axon potential, _{A} | 300 | 100 to 661 | 338 | 1.24 |

CAP, _{N} | 125 | 50 to 227 | 126 | 0.32 |

The potentials computed for an axon diameter of 10

(a) Axonal potential

Figure ^{−1} for variable depths. As expected, axonal body surface amplitude was inversely proportional to approximately the square of the depth.

(a) Axonal signals ^{−1}. Color coding blue, green, and red correspond to a depth of 35 mm, 50 mm, and 65 mm, respectively. (b) Absolute amplitude obtained for the frequency spectrum of the three axonal signals. Color coding is identical as in the time domain. The spectral amplitude was normalized by the peak value obtained at

In our simulations, the volume conductor (lead field term) reduced the peak frequency of axons signals on the body surface. For the chosen parameter range, axonal peak frequency was between 200 Hz and 500 Hz. All axonal signals displayed spectra with a single peak as predicted by our model analysis. For this reason, both at low near DC frequencies and at high frequencies above several kHz, the amplitudes in the spectrum were very low. With increasing depth, the peak frequency decreased, in accordance with the prediction obtained from (

Figure

(a) Signals ^{−1}. Negative amplitude was plotted upwards. Color coding is the same as in Figure

Table

Table

Simulation parameters and characteristic frequencies.

Depth | Conduction velocity | Peak frequency | Bandwidth | Frequency estimates |
---|---|---|---|---|

(mm) | (ms^{−1}) | (Hz) | (Hz) | (Hz) |

35 | 75 | 170 | 69 to 302 | 169 |

50 | 60 | 125 | 50 to 227 | 126 |

65 | 45 | 90 | 36 to 173 | 94 |

With the increasing distance from a source to an observation electrode, spatial smoothing of the lead field reduced the high-frequency content in the axonal signal. The frequency band was in the range of several hundred Hz. This reduction was caused by the Laplace term which underlies each biopotential computation [

Any model is based on idealized assumptions and this is also true for an analytic model. In order to allow for an analytic treatment, we restricted our analysis to axons of equal diameter and a simplified half-space geometry. Thus, we limited our analysis in the first step to the traveling wave along the spinal cord—a waveform which is scientifically well described [

CAP amplitudes were in the order of some tens of microvolts [^{−1} which is a frequently used value for average tissue conductivity [

For single axons, the “quadrupole” signal morphology qualitatively agreed with early theoretical predictions [

Our analytical framework predicted that the width of the temporal dispersion function is the main factor determining the width of the evoked CAP on the body surface. By selecting

The simplifications described above did not allow using our model for studying signals generated by neurophysiologically relevant structures such as the spinal gray matter or subcortical signal generators [

Our analysis on the 50% bandwidth of the signal suggested that most of the signal was contained in a relatively narrow band of 40 Hz to 300 Hz. Bandwidth was significantly more narrow than suggested in the guidelines (20 Hz to 3000 Hz [

We developed a theoretical framework which allows the computation of lead field potentials generated by a neural tract, nerve, or fascicle beyond considering each single axon or fiber. Our approach allows linking membrane potentials obtained from ionic current models with body surface potentials for multiscale modeling. The neural structure was modeled by its central axis, and individual membrane action potential pulses were considered by their statistical distribution in time. Our results suggest that the signals generated by individual axons are sufficiently small and that the number of axons is sufficiently large for obtaining a reasonable approximation by statistical analysis.

Similarly, as for cardiac tissue [

The treatment in the frequency domain allowed gaining insight into the spectral properties of the generated signals. Highest frequencies occur directly at the cell membrane with peak frequencies in the order of kilohertz. The volume conductor transmitting the signal to a lead field electrode acts as a bandpass reducing frequency content with increasing distance between the source and sensor. Furthermore, temporal dispersion of individual activation pulses acts as an additional low-pass filter reducing peak or central frequency in the signal to approximately 100 Hz to 200 Hz. This suggests that the clinical bandwidth of 20 Hz to 3000 Hz [

For establishing the concept of our theory, we applied analytic treatment. This limits analysis to a model of a simple half-space geometry. The results apply to the low-amplitude propagating wave in spinal evoked potentials.

The insight obtained from developing an analytical framework for a lead field two-domain model of a neural tract or nerve provides a sound basis for using numerical tools in the near future for generalizing the approach. This will allow the investigation of a broader spectrum of anatomically and electrophysiologically realistic modeling applications:

Numerical field computation schemes such as the finite element method (FEM) and the boundary element method (BEM) [

Our approach allows numerical computation of axonal signals for fibers of varying diameter (and thus, varying conduction velocity). The model can be generalized to neural tracts containing fibers of varying diameter, by applying numerical tools for summation of many axonal signals (instead of statistical methods as used here). This will allow predicting dispersion at the stimulation site of a peripheral nerve applying computer modeling as described in [

Dispersion functions may be a powerful tool for computing also the lead field potentials of stationary signal generators by direct convolution of a membrane potential derivative with a convolution function.

Thus, the lead field two-domain model has the potential for providing an important novel model-based research tool for evoked or stimulated neurophysiological signals.

The data contained in the article were obtained from simulations which were encoded in Matlab based on the analytical derivations described in the article. All data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.

The supplementary file is named R1_Appendix_CompMathMethMed.