A Stochastic Model of Muscle Mass Potential

Abstract

A stochastic model of muscle mass potential, which may give the basis of the interpretation of its frequency analysis, is presented. Muscle mass potential is supposed to be the superposition of a number of single NMU activities in the extra-cellular current field, and it is natural to associate the statistics of mass potential with that of single NMU activity.
From this point of view, we propose a constructive model of the mass potential e (t) as sum of filtered point processes s_i (t) corresponding to the NMU activities, that is,
s_i (t) =∞∑j=1h_i (t-τ_ij) j=1, 2, …, N (1)
e (t) =N∑i=1s_i (t) =N∑i=1∞∑j=1h_i (t-τ_ij) (2)
Here N, τ_ij and h_i (t) denote the number of NMUs contributing to the superposed process, j^th spike occurrence time in the i^th NMU and observed spike wave form of i^th NMU. Under the assumption, valid for the normal moderate voluntary contraction, that component processes are almost renewals and mutually independent, and N is large enough, theoretical formula of the autocorrelation function c_e (τ) and power spectrum Ψ_e (ω) of the muscle mass potential e (t) are derived.
c_e (τ) = {a_i>N∑i=11/μ_i} ²∫^∞_-∞h (τ) ⊗h (-τ) dτ+<a_i²> (N∑i=11/μ_i) h (τ) ⊗h (-τ) (3)
Ψ_e (τ) =δ (ω) [{a_i>N∑i=11/μ_i} ²∫^∞_-∞h (τ) ⊗h (-τ) dτ] +<a_i²> (N∑i=11/μ_i) H (ω) (4)
Here we assume h_i (t) =a_ih (t) for simplicity, μ_i, H (ω) and < > denote mean interval of i^th NMU activity, power spectrum of h (t) and ensemble average respectively.
The formulae (3) and (4) show that the spike wave form is greatly reflected in the spectrum pattern of the muscle mass potential, and that the experiment on normal biceps muscle at constant voluntary contraction has proved the appropriateness of the proposed model. The results suggest also that estimation of average wave form of NMU discharges in the muscle would be possible from the surface record of mass EMG.