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McKeone, James
&
Pettitt, Anthony N.
(2013) Bayesian P-splines with a
multiplicative term in EMG trace data. In Muggeo, Vito M.R., Capursi,
Vin-cenza, Boscaino, Giovanni, & Lovison, Gianfranco (Eds.)
Proceedings of
the 28th International Workshop on Statistical Modelling
, Statistical
Mod-elling Society, Palermo, Italy, pp. 275-280.
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Bayesian P-splines with a multiplicative term
in EMG trace data
James P. McKeone
1, Anthony N. Pettitt
11
Queensland University of Technology, Australia E-mail for correspondence:[email protected]
Abstract: A method is proposed to describe force or compound muscle action potential (CMAP) trace data collected in an electromyography study for motor unit number estimation (MUNE). Experimental data was collected using incre-mental stimulation at multiple durations. However, stimulus information, vital for alternate MUNE methods, is not comparable for multiple duration data and therefore previous methods of MUNE (Ridall et al., 2006, 2007) cannot be used with any reliability. Hypothesised firing combinations of motor units are mod-elled using a multiplicative factor and Bayesian P-spline formulation. The model describes the process for force and CMAP in a meaningful way.
Keywords: Bayesian P-splines; MUNE; Multiple duration data
1
Introduction
Reliable motor unit number estimates (MUNE) are of key interest in clin-ical and experimental neurophysiology. MUNE has been a popular area of research since the seminal work of McComas et al. (1971) who pioneered the method of incremental stimulation in electromyography (EMG). MUNE is efficacious in assessing the severity or tracking the progression of diseases characterised by muscle weakness resulting from the death or inaction of motor units (Baumann et al., 2012). An exciting application of MUNE is be-ing pursued by the Miami project to cure paralysis, who research methods to replace dead nervous system cells and promote and guide the regrowth of axons with the aims of muscle re-innervation and ultimately motor unit function for patients suffering spinal injury or damage to the central ner-vous system (Casella et al., 2010). MUNE may be used to investigate the success of a cell transplant, specifically, whether a treatment has produced new, active motor units.
MUNE techniques involve analysis of compound muscle action potential (CMAP) or force data resulting from electromyography (EMG) studies. In a clinical setting, a surface EMG study involves applying an electrical stimulus at the nerve for a fixed duration to observe a CMAP or force
2 Bayesian P-splines with a multiplicative term in EMG trace data
0 10 20 30 40 50
−0.6
0.0
0.4
CMAP trace plot
Time CMAP response 0 100 200 300 400 500 600 700 0.00 0.05 0.10 0.15
Force trace plot
Time
F
orce response
FIGURE 1. Electromyography (EMG) responses to stumulus collected on an anesthetised rat with stimuli applied at three durations, 10, 20 and 50µs.(Top)
Compound muscle action potential (CMAP) trace data.(Bottom) Force trace data.
response through electrodes or transducers attached to the skin. The tech-nique of incremental stimulation involves setting the stimulus intensity such that no units are firing initially, and gradually incrementing the stimulus intensity, invoking more motor units until all are believed to be firing. Clin-icians tend to adopt a relatively long duration (usually 50µs) and therefore smaller stimulus intensities to maintain a level of comfort for the patient. The trace responses for each stimulus intensity can be plotted against time for CMAP and force, see Figure 1.
In an experimental setting, specifically the case considered here of ane-thetised rats, it is more common to apply the incremental stimulation tech-nique at multiple durations. It is believed that this allows the process of alternation and therefore individual motor units to be investigated more thoroughly (Casella et al., 2010). However, it is not immediately obvious how existing MUNE techniques, all of which rely on stimulus intensity in-formation as input data, may be adjusted to account for multiple durations. Suppose an experienced clinician applied the incremental stimulation tech-nique and identified the combinations of motor units thought to be active at each particular stimulus. Such information is labelled as a firing pattern. A potential firing pattern for investigation could be the result of a na´’ive approach as the former suggestion or a statistically based approach, such as MUNE techniques using Bayesian hierarchical modelling (Ridall et al., 2006) combined with reversible jump Markov chain Monte Carlo sampling (Ridall et al., 2007). These MUNE techniques are henceforth referred to as Bayesian MUNE. An example of a firing pattern for an assumed 9 unit model is presented in Figure 2.
McKeone and Pettitt 3 0 100 200 300 400 500 600 700 0.00 0.05 0.10 0.15
9 Unit firing pattern
Time F orce response 100000000 110000000 100010000 110010000 111010000 111110000 111001000 111101000 111111000 111111100 111111110 111111111
FIGURE 2. A 9 motor unit firing pattern identified by Bayesian MUNE. The set of units active within each group are identified in the legend, 1 is active, 0 is inactive for each particular motor unit.
The contribution of this work is to investigate the plausibility of different firing patterns −and usually different numbers of total motor units−in multiple duration EMG data, without using stimulus information to inform the model.
Consider a set of plausible firing patterns objectively identified by Bayesian MUNE. The stimulus data is adjusted using an approximation based on the strength-duration relationship (Hill, 1936) that assumes the stimulus multiplied by duration is constant before the algorithm is implemented. The output from Bayesian MUNE is a relatively small number of plausible firing patterns corresponding to values of the total number of units. The posterior model probabilities from Bayesian MUNE may not be relied upon due to the uncertainty in making the varying stimulus duration correction. Here, trace data and each potential firing pattern are modelled using Bayesian P-splines with a multiplicative scale factor. The scale term shifts the spline to the mean of each pre-determined group.
2
Bayesian P-splines and Multiple Duration Data
Leti be a location on curvej in groupk. A group is a set of motor units active at a particular stimulus intensity. Bayesian MUNE suggests seven potential firing patterns for subsequent investigation. Consider the model
yijk=φkg(xi) +ijk (1)
that allows the trace datayijk to be represented as a splineg(xi), that is
scaled to each cluster by a coefficientφk and errorijk. The spline may be
4 Bayesian P-splines with a multiplicative term in EMG trace data
vector of coefficients,β. The spline is fitted as a reference shape across all curves.
A multivariate normal likelihood, yj,k ∼ N(φkXβ, 2I), is assumed for
these data . The model variance parameter 2 is assigned the prior dis-tribution2 ∼Inverse-Gamma(α
, β) and assumed fixed across all curves
and groups.
The splineXβ is formulated as a Bayesian P-spline with a B´ezier basis in the method set out by Eilers and Marx (1996) and extended by Lang and Brezger (2004). A random walk difference prior is placed on the coefficients β such that, p(β|τ2)∝ 1 τ2 rank2(K) exp − 1 2τ2β TKβ , (2)
for the first-order difference matrix K, with the variance parameter defined as τ2 ∼ Inverse-Gamma(α
τ, βτ) a priori. It is important that the prior
on τ2 is not too diffuse, the values α
τ = 1, βτ = 0.005 were adequate,
but ultimately the prior was found to have little effect on the posterior. It remains to describe how the scale parameters φk will be dealt with, two
prior formulations are considered.
In keeping with the assumptions applied in Bayesian MUNE (Ridall et al. 2006), the largest observed group of firing motor units is assumed to be the sum of all units in the firing pattern. The result of this assump-tion is a data dependent prior on units, µn, n= 1, . . . , N, parameters to
represent individual unit size. Each scale coefficient,φk, is assumed to be
defined by the unique sum φk = Pµm for units identified by the firing
pattern. A Dirichlet prior is a natural formulation for coefficientsµn. With
N the total number of units andKthe group with the largest members in terms of amplitude,PN
n=1µn=φK = 1 is satisfied under a Dirichlet prior
specification. Therefore for individual units write, µn ∼ Dirichlet(αµ) for
some concentration parameterαµ. The implicit constraintP N
n=1µn= 1 is
sufficient for identifiability of the spline and the scale coefficients.
2.1 Posterior Distribution and Algorithm
The full-conditional distributions for parameters2, βandτ2may be found after some algebra. The posterior distribution for the µn has no closed
form density and so we use a Metropolis-Hastings within Gibbs approach to sampling from the posterior distribution in tandem with a systematic approach to sampling and updating parameters from the full-conditional distributions.
3
Results
Figure 3 presents the data and fitted models for each group of curves ac-cording to an assumed eleven unit firing pattern. The model describes the
McKeone and Pettitt 5 smooth force curves very well though produces what may seem like a less convincing fit for the CMAP traces. For CMAP data, the region of the curve thought to be most important in action potential investigations is the first peak and while CMAP curves of different groups have heteroge-neous shape, the model does indeed fit the first peak quite well in each group. 0 20 40 −0.6 −0.2 0.2 0.6 CMAP Group 1 Time CMAP 0 20 40 −0.6 −0.2 0.2 0.6 CMAP Group 2 Time CMAP 0 20 40 −0.6 −0.2 0.2 0.6 CMAP Group 3 Time CMAP 0 20 40 −0.6 −0.2 0.2 0.6 CMAP Group 4 Time CMAP 0 20 40 −0.6 −0.2 0.2 0.6 CMAP Group 5 Time CMAP 0 20 40 −0.6 −0.2 0.2 0.6 CMAP Group 6 Time CMAP 0 20 40 −0.6 −0.2 0.2 0.6 CMAP Group 7 Time CMAP 0 20 40 −0.6 −0.2 0.2 0.6 CMAP Group 8 Time CMAP 0 20 40 −0.6 −0.2 0.2 0.6 CMAP Group 9 Time CMAP 0 20 40 −0.6 −0.2 0.2 0.6 CMAP Group 10 Time CMAP 0 20 40 −0.6 −0.2 0.2 0.6 CMAP Group 11 Time CMAP 0200 500 0.00 0.04 0.08 Force Group 1 Time F orce 0200 500 0.00 0.04 0.08 Force Group 2 Time F orce 0200 500 0.00 0.04 0.08 Force Group 3 Time F orce 0200 500 0.00 0.04 0.08 Force Group 4 Time F orce 0200 500 0.00 0.04 0.08 Force Group 5 Time F orce 0200 500 0.00 0.04 0.08 Force Group 6 Time F orce 0200 500 0.00 0.04 0.08 Force Group 7 Time F orce 0200 500 0.00 0.04 0.08 Force Group 8 Time F orce 0200 500 0.00 0.04 0.08 Force Group 9 Time F orce 0200 500 0.00 0.04 0.08 Force Group 10 Time F orce 0200 500 0.00 0.04 0.08 Force Group 11 Time F orce
FIGURE 3. Model fit for an eleven unit firing pattern.(Top)CMAP response.
(Bottom)Force response. Individual group data (solid line) and the fitted mul-tiplicative spline (dashed line).
4
Discussion
The force trace data are used as illustration for the model, as seen in Figure 1. However, CMAP data is far more prevalent in clinical and experimental
6 Bayesian P-splines with a multiplicative term in EMG trace data
investigations involving MUNE. A spline based approach was adopted so as not to exclude the possibility of analysing the more erratic CMAP traces. The model considered is a method to describe different firing patterns for the force or CMAP trace data with stimulus applied at the nerve at mul-tiple durations. Future work involves deciding between different suggested firing patterns in an automatic way. Model fit calculations such as DIC and BIC perform poorly so we propose a model choice approach to target the marginal likelihood or statistical evidence, though such a presentation is beyond the scope of this short paper.
Acknowledgments: The authors would like to thank Christine Thomas, Gareth Ridall and Chris Drovandi for assistance. James McKeone is grate-ful for the support of an Australian Postgraduate Award (APA). Tony Pettitt is supported by an Australian Research Council Discovery Grant.
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