Ridall, Gareth and Pettitt, Tony and Henderson, Robert and McCombe, Pam
(2013)
*Statistical modelling of neuron degeneration.*
Working Paper.
UNSPECIFIED.
(Submitted)

## Abstract

SUMMARY: Parkinson’s disease, Huntington’s disease, Amyotrophic lateral sclerosis (ALS) and Alzheimer’s disease are all examples of neurodegenerative disorders that result from the premature death of nerve cells or neurons. In order to understand the mechanisms through which these diseases advance, a number of models have been put forward to describe the decline in the numbers of surviving neurons. Such work has been hampered by the poor quality of estimates of the numbers of surviving neurons and also by questionable model selection techniques. Recent work has favoured the adoption of the exponential model to explain neurodegenerative decline. We present in this paper a methodology for challenging this model, using data from patients with ALS. We use a two stage procedure to study motor unit numbers. The first stage involves determining the number of motor units in a muscle on several occasions over a period of time. The method of Ridall et al. (2007) is used which makes use of reversible jump Markov chain Monte Carlo (RJMCMC). The second stage involves the analysis of the RJMCMC output by using a hiddenMarkov process of decline. Two such processes of decline are compared. The first is the exponential where the rate parameter is constant. This is compared to a more general semi-parametric process where the rate parameter is allowed to vary over time. The rate is set to be piecewise constant between recordings where the magnitudes of the change in rate are weakly constrained by the length of the interval between recording occasions. Between model comparisons are based on electrophysiological data collected from a group of ALS patients where motor units (MUs) are gradually lost leading to progressive muscle weakness. By calculating marginal likelihoods, we find the Bayes factor in support of the exponential decline model against the more general alternative. This approach is illustrated with four ALS patients. Prediction of MU numbers lost, which incorporates both models, can also be made. Our methods, we therefore believe, have a role in formulating and evaluating biological models for neural degeneration of the motor system in ALS patients.