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A sigmoid function is the best fit for the ascending limb of the Hoffmann reflex recruitment curve

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Abstract

The Hoffmann (H)-reflex has been studied extensively as a measure of spinal excitability. Often, researchers compare the H-reflex between experimental conditions with values determined from a recruitment curve (RC). An RC is obtained experimentally by varying the stimulus intensity to a nerve and recording the peak-to-peak amplitudes of the evoked H-reflex and direct motor (M)-wave. The values taken from an RC may provide different information with respect to a change in reflex excitability. Therefore, it is important to obtain a number of RC parameters for comparison. RCs can be obtained with a measure of current (HCRC) or without current (HMRC). The ascending limb of the RC is then fit with a mathematical analysis technique in order to determine parameters of interest such as the threshold of activation and the slope of the function. The purpose of this study was to determine an unbiased estimate of the specific parameters of interest in an RC through mathematical analysis. We hypothesized that a standardized analysis technique could be used to ascertain important points on an RC, regardless of data presentation methodology (HCRC or HMRC). For both HCRC and HMRC produced using 40 randomly delivered stimuli, six different methods of mathematical analysis [linear regression, polynomial, smoothing spline, general least squares model with custom logistic (sigmoid) equation, power, and logarithmic] were compared using goodness of fit statistics (r-square, RMSE). Behaviour and robustness of selected curve fits were examined in various applications including RCs generated during movement and somatosensory conditioning from published data. Results show that a sigmoid function is the most reliable estimate of the ascending limb of an H-reflex recruitment curve for both HCRC and HMRC. Further, the parameters of interest change differentially with respect to the presentation methodology and the analysis technique. In conclusion, the sigmoid function is a reliable analysis technique which mimics the physiologically based prediction of the input/output relation of the ascending limb of the recruitment curve. Therefore, the sigmoid function should be considered an acceptable and preferable analytical tool for H-reflex recruitment curves obtained with reference to stimulation current or M-wave amplitude.

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Acknowledgments

This work was supported by grants to EPZ from the Natural Sciences and Engineering Council of Canada (NSERC), the Heart and Stroke Foundation of Canada (BC & Yukon), and the Michael Smith Foundation for Health Research. MK was supported by a Focus on Stroke award from the Heart and Stroke Foundation of Canada (BC& Yukon). The authors would also like to thank Dr. Timothy Carroll, Dr. Kelvin Jones and Eduardo Villaseñor for their assistance.

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Correspondence to E. Paul Zehr.

Appendix: Details of mathematical analysis techniques

Appendix: Details of mathematical analysis techniques

Linear regression

For the linear fit, the ascending limb of the HRC was defined as all points from the foot of the curve to the current value that occurred at M TH (approximately 10% below the peak of the HRC) by the manual placement of a cursor on the computer display (Funase et al. 1994a). For the cases where the M TH occurred past the peak of the HRC the cursor was placed at approximately 10% below the peak of HRC. This was done to ensure that the slope of the line was not contaminated by H-waves near the peak of the ascending limb that could be potentially affected by collision along the motor axons (Funase et al. 1994a). For the linear curve fit the data from both the calc and the chos methods were fit using the least squares method which assumes that the variability in the reflex amplitude is Gaussian distributed. Equation 2 represents the linear fit model.

$$ y = mx + b $$
(2)

where x is the stimulus intensity, m is the slope, and b is the intercept. The least squares linear regression finds m and b that best fit the sampled data by minimizing the value obtained using the least squares procedure. For detail of the least squares procedure and other possible linear optimization criteria see the National Instruments website (link above).

Polynomial

The polynomial fit technique finds the polynomial equation of the line that minimizes the mean square error from the fitted curve to the sampled data. Equation 3 gives the general form of the polynomial fit.

$$ fi = {\sum\limits_{j = 0}^m {a_{j} x_{i} ^{j} } } $$
(3)

where f represents the output sequence, x represents the input sequence, a represents the polynomial coefficients, and m is the polynomial order. The polynomial order is an arbitrarily chosen parameter that is under investigator control. Systematic increases in the polynomial order can cause random changes in the goodness of fit statistics. Also, the high order polynomials may obtain the best values of the goodness of fit statistics and yet be very noisy. Therefore, it is important to determine a method that defines the ideal polynomial order that produces an optimum combination of lowest mean square error while still producing a smooth curve. The technique used in this experiment is called “error fitting” and will be described below with reference to setting boundary conditions for both the polynomial and smoothing spline curve fits.

Smoothing spline

A cubic spline is a curve constructed of piecewise third-order polynomials which pass through a defined set of points. Analogous to a drawing device, a spline can be thought of as a flexible strip of material that may be bent into a curve and used to draw smooth curves between points. The HRC data was fit with a piecewise cubic spline (smoothing spline) function. This fits the sampled data by minimizing the following function:

$$ p{\sum\limits_{i = 0}^{n - 1} {w_{i} } }(y_{i} - f(x_{i} ))^{2} + (1 - p){\int\limits_{x_{0} }^{x_{{n - 1}} } {\lambda (x)({f}\ifmmode{''}\else$''$\fi (x))^{2} {\text{d}}x} } $$
(4)

where p is the balance parameter, w i is the ith element of weight. y i is the ith element of the set of all normalized H-wave amplitudes. x i is the ith element of the set of all normalized current amplitudes. f′′(x) is the second order derivative of the cubic spline function, f(x). λ(x) is the piecewise constant function. The balance parameter (p) specifies the balance between the smoothness of the curve fit and the accuracy with which it fits the observations. If p = 0, the fitted model is equivalent to a linear model. If p = 1, the fitting is equivalent to cubic spline interpolation where all the data points are connected. The smoothing spline function parameters of data point weight and balance can be subjectively set by the investigator. A weight parameter can be set for each data point to define the relative importance of each data point towards the resultant curve fit. For the purpose of this experiment the weight of each point was considered equivalent, as each data point must have equal probability of defining the recruitment curve. Similar to determining the polynomial order, the balance parameter can be set by the investigator and therefore requires a method to determine the optimal combination of both smoothness and measures of goodness of fit. Increasing the balance parameter would both increase the correlation coefficient and decrease the root mean square error signifying an appropriate fit to the data. However, the fitted curve with a p of 1 would pass though every data point and therefore be very noisy. The balance parameter was adjusted to conform to two predetermined parameters to obtain the optimal combination of both smoothness and measure of MSE. The first penalty feature was the evaluation of the first and second derivatives of the developed fit. This measure would allow a determination of any rapid changes in the slope of the curve that would signify that the curve is too erratic. The second penalty feature was a variation of an idea first described by Hayes et al. (1979) described as “error fitting” in this manuscript.

Error fitting

During error fitting the adjustable parameter in either the polynomial or the smoothing spline technique is increased until the furthest point away from the fitted curve to any individual data point is no greater than the deviation value determined from the recruitment curve. The deviation value is determined using Eq. 5.

$$ {\sqrt {\frac{1} {{N - 1}}{\sum\limits_{i = 1}^{n - 1} {(y_{{i + 1}} - y_{i} )^{2} } }} } $$
(5)

Error fitting is thus sensitive to the greatest variability in the developed RC.

Power

The following equation represents the power fit model:

$$ f = ax^{b} $$
(6)

where x is the input sequence, a is the amplitude, and b is the power. This curve fit finds a and b that minimizes the least square fit to the experimental observations.

Logarithmic

The following equation represents the logarithmic fit model:

$$ f = a\log _{c} (bx) $$
(7)

where x is the input sequence, c is the base, a is the amplitude, and b is the scale. This fit finds a and b that minimizes the least squares fit to the experimental observations.

Sigmoid

A general least squares model of a custom three-parameter sigmoid function similar to one developed in TMS Research was used to fit the ascending limb of all recruitment curves (Carroll et al. 2001; Devanne et al. 1997).

$$ H(s) = \frac{{H_{{{\text{MAX}}}} }} {{1 + e^{{{\text{m(s50 - s)}}}} }} $$
(8)

where H MAX is the upper limit of the curve, m is the slope parameter of the function, s50 is the stimulus at 50% of the H MAX value, and H(s) is the H-reflex amplitude at a given stimulus value (s). Average H MAX was calculated from the 5 largest peak-to-peak H-reflexes. The average H MAX value (defined above) was used to define the upper limits of the sigmoid curve. The ascending limb of the recruitment curve was chosen as all points from zero current to a manually chosen peak of the recruitment curve.

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Klimstra, M., Zehr, E.P. A sigmoid function is the best fit for the ascending limb of the Hoffmann reflex recruitment curve. Exp Brain Res 186, 93–105 (2008). https://doi.org/10.1007/s00221-007-1207-6

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