Prediction of Electrically Stimulated Muscle Force Under Isometric Conditions Using Self-Constructing Neural Networks

Abbas Erfanian¹ and Howard J. Chizeck²

1Dept.of Biomed. Eng., Faculty of Electrical Eng., Iran University of Science & Technology, Tehran, IRAN

2Dept.of Electrical Eng., University of Washington Seattle, Washington 98195-2500, USA

erfanian@iust.ac.ir chizeck@ee.washington.edu

Introduction

Improved performance of motor neuroprostheses can, in principle, be obtained through better real-time predictive models of muscle response, either for feed-forward or feedback control. Although a large number of mathematical models of electrically stimulated muscle have appeared, these efforts have not resulted in models that can be used for long-time prediction of muscle force. The Hammerstein form is the most common structure for modeling the electrically stimulated muscle, relating stimulation to the muscle force under isometric conditions [1]. This model assumes that the activation dynamics is independent of activation level. This ignores the nonlinear dynamics due to differences in motor units which are recruited at different activation level. Moreover, time-varying properties such as muscle potentiation and fatigue were not considered in the model.

In [2], a model consisted of two first-order low-pass filters separated by a static nonlinearity was developed for predicting the force produced by an isometric muscle in response to varying interpulse intervals. It was reported a normalized mean-square prediction error of 34% at the best case while the rate constant of the second filter had to vary with the level of the muscle force during prediction. In [3], a mathematical model was developed that predicts the force generated by rat skeletal muscles during short (six-pulse trains) isometric contraction. Although the model predicts accurately isometric force during brief trains stimulation (six-pulse), it failed to predict long-train stimulation.

All of the muscle modeling works described above relates the stimulation signals to muscle output force. There are several possible sources which degrade the predictive capability of the stimulation-to-force model. The stimulation-to-force model cannot capture the effects of the spasticity or activation of motoneurons due to reflex phenomena. It also cannot capture changes in the recruitment properties of the stimulating electrode, due to changes in muscle geometry that are induced by contractions (or other time variations in the recruitment characteristic). The predictive capability of the stimulation-to-torque model is also damaged by any changes in the excitation gain (from stimulation to motor unit action potentials) that might occur. For example, changes in the stimulation electrode ground may provide a source of variability which degrades the stimulation-to-force model predictions. Potentiation, fatigue, and dependency of activation dynamics to stimulation level are other sources of variability which cannot captured by the existing stimulation-to-force model. Due to the fact that the majority of these changes are reflected by the Evoked EMG (EEMG), a Hamerstein model of electrically stimulated muscle was developed in [4], where the EEMG is used as the input of the model. It was demonstrated that the use of the EEMG as the input to a predictive model of muscle force generation is superior to the use of the electrical stimulation signal as the model input.

In this work, we develop a method for estimating the force generated by electrically stimulated muscle during isometric contraction based upon measurements of the EEMG and stimulation by using artificial neural networks. For this purpose, we employ a radial basis function (RBF) neural network with gradient descent leaning algorithm and an enhanced self-constructing neural network for muscle modeling. It is demonstrated that the neural network model of electrically stimulated muscle is capable of predicting the muscle force accurately under a wide range of stimulation patterns.

Neural Network Models

Radial basis function (RBF) neural networks can offer approximation capabilities similar to those of the multi-layer perceptrons. The universal approximation capabilities of neural models suggest the possibility of using neural networks to identify nonlinear dynamical systems. Due to this fact, we use RBF network with gradient descent leaning algorithm for predicting muscle force. In the classical approach to RBF network implementation, the number of hidden units is predetermined. It, usually, results in large or small network. Too small networks are unable to adequately learn the problem well while overly large networks tend to overfit the training data. To overcome this drawback, we employ a learning algorithm for incremental construction of neural network architecture. The algorithm starts with a small network and dynamically grow the network by adding neurons as needed until a satisfactory solution is found.

A. Radial Basis Function (RBF) Neural Network

The architecture of RBF networks is simple and consists of one hidden layer. The hidden layer is composed of a number of kernel nodes with kernel activation functions. The output of the network is simply a weighted linear summation of the kernel functions:

₍₁₎

where is the input vector, M is the number of kernel nodes in the hidden layer, w_i () is the vector of weights from the i-th kernel node to the output nodes, is Euclidean distance, and k is a radial symmetric kernel function. A Gaussian function is normally chosen as kernel function. The vector C_i represents the locations of the kernel functions in Rⁿ . Finally, s_i is the smoothing factor or kernel bandwidth of the i-th kernel node. The degree of localization can be adjusted by kernel bandwidth. Now, the key question is how to determine the parameters of RBF network appropriately for the purpose of functional approximation. One approach is that all the free parameters of the network undergo the gradient descent procedure, i.e. least-mean-square (LMS).

B. Self-Constructing Neural Network (SCNN)

The architecture of SCNN consists of one hidden layer. The hidden layer is composed of a number of RNF nodes and, in addition, non-RBF units. The output of the network is simply a weighted linear summation of the RBF and non-RBF outputs [5]:

where G is a radially symmetric function and g is a known real function of x. The function can take a variety of forms [5]. A linear function has been used in this work. he learning of the network involves generation of new RBF units as well as adaptation of network parameters. The hidden units are generated incrementally, in stages, by random partitioning the embedding space into regions in each of which it is possible to approximate the dynamics with a kernel function. A stage is specified by a parameter that specifies the width of kernel units which are generated at the stage k. At any stage, a vector x is randomly selected from training set and search for all other training vectors within the -neighborhood of x. The training vectors in the -neighborhood are used to define a kernel unit and then removed from the training set. To generate the next unit, another input vector is randomly selected from remaining training set and its -neighborhood similarly searched for other vectors. The process of generating new unit is repeated unit the remaining set to be empty. RBF units of various widths can be generated by repeating this whole process for various . can be initially set to the standard deviation of the training set and then it is reduced at a fixed rate, , at the next stages.

As the new kernel units are generated at each stage, the network parameters are estimated by using the standard recursive least-squares (RLS) algorithm instead of linear programming which is used in [5]. Whenever, the training and testing set error becomes small or over fitting occurs, the learning process stops. It can be seen that self-constructing learning method has the interesting feature of combining both local and global modeling capabilities.

C. Identification of Muscle Torque Models

Three types of models are considered here: (1) EEMG-to-torque model relating the EEMG to measured torque; (2) stimulation-to-torque model relating the stimulation to the muscle torque; and (3) stimulation & EEMG-to-torque model. In the third model, the stimulation as well as measured EEMG constitutes the input of the muscle model as follows:

where y denotes the muscle torque, s is the stimulation signals, e is the measure EEMG. From the universal approximation capabilities of the neural networks, it follows that a neural network can be constructed to approximate the f.

Experiments

Experiments were conducted on a two complete level T7 spinal cord injury paraplegics. Percutaneous intramuscular electrodes were implanted near the motor points of the major lower limbs. During the experiments reported here, only the lower limb vastus lateralis muscle was stimulated, by activating merely the corresponding intramuscular electrode. The muscle was stimulated using pulsewidth modulation at a constant frequency (20 Hz) and constant amplitude (10 mA), under isometric conditions.

EMG information was collected from surface electrodes through use of a differential amplifier with a common mode rejection ratio of 120 dB and bandwidth of 250 kHz. This data was then sampled at a rate of 1200 Hz. Isometric knee torque was measured using a Cybex II dynamometer. The subject was seated on the bench of Cybex machine, with his hip flexed at approximately 90° and his thigh held against the seat with a restraining strap. An instrumented torque arm (having a set of strain gauges) was used to measure the isometric muscle torque. Measured values of the knee torque were low-pass filtered (cut-off frequency 100 Hz), and sampled at 1200 Hz.

Two stochastic patterns were used for dynamic model parameter identification. The first pattern consisted of a succession of one second long sets of twenty pulses, where the first ten pulsewidths were increased from a fixed minimum value to a randomly determined maximum value, and then the next ten pulses symmetrically decreased. The maximum pulsewidth (for the tenth pulse in each set) was randomly chosen to vary between 0 and , according to a uniform distribution. In the second stimulation pattern, sets of 10 pulses were delivered, where these pulses monotonically increased or decreased in pulsewidth. Both the initial and final pulsewidth values were randomly selected (uniformly distributed between 0 and ). The increment between each successive pulsewidth was the same in this second stimulation pattern. Results A. Radial Basis Function (RBF) Neural Network RBF networks with different hidden units were considered. Model order (l, m, n) determination was accomplished by comparing the average predication-error for different model orders. Table 1 summarizes the results for different predictive models of muscle force and different stochastic patterns after 500 training epochs. It was found that the error was minimized by the model order (3,3) for EMG-to-torque model, by the order (4,4) for stimulation-to-torque model, and by the order (3,3,3) for stimulation & EEMG-to-torque, when using the first stochastic pattern for model identification and validation. Almost the same results were obtained for the second stimulation pattern. Fig.1 shows the muscle force prediction, obtained using different predictive models for trails that were not used to train the network. Comparing these figures, it is observed that the EEMG-to-torque is far more accurate than the stimulation-to-torque model. Moreover, stimulation & EEMG-to-torque provides better prediction of the output torque than the EEMG-to-torque model during different level of stimulation and activation speed.

(a) (b) © Fig. 1. The measured (solid line) and predicted (dotted line) muscle torque obtained using RBF network with 40 hidden units: (a) stimulation-to-torque model, (b) EEMG-to-torque model, (c) stimulation&EEMG-to-model.

B. Self-Constructing Neural Network

Table 2 shows the performance of the constructive learning algorithm on the muscle force prediction using stimulation-to-torque and stimulation&EEMG-to-torque models. It shows that a prediction error of 0.0430 was achieved with only 13 kernel units when using stimulation&EEMG-to-torque models.

Table 3 summarizes the results for different predictive models of muscle force using two networks. Once again, it is observed that stimulation&EEMG-to-torque and EEMG-to-torque models provides more accurate prediction than the stimulation-to-torque. Moreover, the results show that the self-constructing neural network method can improve the predictability of muscle force with a suitable number of hidden units compared to the RBF network.

Table 1

Average Muscle Force Prediction Error Using RBF Network

	First Stochastic Pattern			Second Stochastic Pattern
Predictive Model	(l,m,n)	network size	prediction Error	(l,m,n)	network size	prediction Error
Stimulation-to-torque	(4,4)	(8,40,1)	0.152	(4,4)	(8,20,1)	0.188
EEMG-to-torque	(3,3)	(6,60,1)	0.0341	(3,2)	(5,60,1)	0.0578
Stimulation&EEMG-to-torque	(3,3,3)	(9,60,1)	0.0422	(4,4,4)	(12,60,1)	0.0546

Table 2

Average Muscle Force Prediction Error Using Constructive Learning Algorithm

with 80% Delta Reduction Rate.

	Stimulation-to-torque			Stimulation&EEMG-to-torque
Pass No.	Delta	Cumulative No. of Kernel units	Prediction Error	Delta	Cumulative No. of Kernel units	Prediction Error
1	0.450	13	0.197	0.430	13	0.0430
2	0.360	33	0.131	0.344	32	0.0342
3	0.288	52	0.108	0.275	58	0.0324
4	0.230	73	0.109	0.220	93	0.0331

Table 3

Average Prediction Error of different Muscle Force Predictor by Using

Radial Basis Function and Self-Constructing Neural Network

Predictive Model	Network	network size	prediction Error
Stimulation-to-torque	RBFNN	(8,201,1)	0.187
	SCNN	(8,30,1)	0.109
EEMG-to-torque	RBFNN	(5,60,1)	0.0578
	SCNN	(5,57,1)	0.0360
Stimulation&EEMG-to-torque	RBFNN	(12,60,1)	0.0546
	SCNN	(12,64,1)	0.0322

Reference

[1] T. L. Chia, P. Chow and H. J. Chizeck, "Recursive parameter identification of constrained systems: An application to electrically stimulated muscle", IEEE Trans. Biomed. Eng.,vol. 38, no. 5, 429-442, May 1991.

[2] J. Bobet and R. Stein, “A simple model of force generation by skeletal muscle during dynamic isometric contractions,” IEEE Trans. Biomed. Eng., vol. 45, no. 8, 1998.

[3] A.S. Wexler, J. Ding, and A Binder-Macleod, “A mathematical model that predicts skeletal muscle force,” IEEE Trans. Biomed. Eng., vol. 44, no. 5, 1997.

[4] A. Erfanian, H.J. Chizeck, and R. M. Hashemi, "Using evoked EMG as a synthatic force sensor of isometric electrically stimulated muscle," IEEE Trans. Biomed. Eng., vol. 45, no. 2 pp. 188-202, 1998.

[5] A. Roy, S. Govil, R. Miranda, “ A neural-network learning theory and a polynomial time RBF algorithm,” IEEE Trans. Neural Network, vol. 8, no. 6, 1997.

Acknowledgment: The data for this study was collected at the Cleveland VA Medical Center by the authors in 1994, as part of a larger set of research studies at that institution, directed by Dr. E.B. Marsolais.