Fatigue and non-fatigue mathematical muscle models during functional electrical stimulation of paralyzed muscle

Abstract

Electrical muscle stimulation demonstrates potential for preventing muscle atrophy and for restoring functional movement after spinal cord injury (SCI). Control systems used to optimize delivery of electrical stimulation protocols depend upon the algorithms generated using computational models of paralyzed muscle force output. The Hill-Huxley-type model, while being highly accurate, is also very complex, making it difficult for real-time implementation. In this paper, we propose a Wiener-Hammerstein system to model the paralyzed skeletal muscle under electrical stimulus conditions. The proposed model has substantial advantages in identification algorithm analysis and implementation including computational complexity and convergence, which enable it to be used in real-time model implementation. Experimental data sets from the soleus muscles of fourteen subjects with SCI were collected and tested. The simulation results show that the proposed model outperforms the Hill-Huxley-type model not only in peak force prediction, but also in fitting performance for force output of each individual stimulation train.

Fig. 1

Force responses (thick curves) to the stimulation pulse (thin pulses) for subject 1. The muscle was stimulated by a sequence of trains with duration of 2 seconds for total 124 trains. Each train is composed of 10-pulses with 15 Hz frequency followed by a resting period of 1337ms. The muscle fatigue effect is clearly shown through the reduced peak force response of the 62^nd and 124^th train.

Fig. 2

(a): Wiener-Hammerstein muscle model. (b): The middle nonlinear block can be decomposed into three parts. (c): The simplified Wiener-Hammerstein muscle model.

Fig. 3

The cost function value vs. b̂₁ with given â₁ = 0.9.

Fig. 4

The cost function J(â₁ (p), h(â₁ (p)) vs. â₁. The right plot is the zoom of the left plot with â₁ ∈ [0.9, 1].

Fig. 5

Schematic representation of the limb fixation and force measurement system

Fig. 6

The force response of the first (upper plots) and the last (bottom plots) three stimulation trains for actual output (solid) and the predicted output (dashed) for subject 1.

Fig. 7

Wiener-Hammerstein model parameters a₁, b₁ and b₂ for each stimulation train of the subject 1. x's denote the predicted parameter value and dots denote parameter value identified using non-fatigue model.

Fig. 8

Peak force for each stimulation train of subject 1. Dots denote the actual peak force and stars denote the predicted peak force.

Fig. 9

Box plots of the gof_ave, R, and gof_pk of fourteen subjects for Wiener-Hammerstein model (W-H) and Hill-Huxley-type (H-H) model, respectively.