Estimating muscle activations from EMG data. The rectified EMG signal is shown in blue, the Bayesian estimate of muscle electrical activity x ( t ) based on [22] is shown in green and activation estimates a ( t ) obtained by feeding in the Bayesian estimate through bilinear activation dynamics are in red. The step-like feature of the Bayesian estimate is apparent. The muscle activation estimate builds up after the EMG bursts on, and lasts well after the EMG turns off. Step to step variations in the EMG signals are seen, as are their effects on the activation estimates. The position of the leg corresponding to the time axis is shown for interpretation. The leg is shaded during stance (between heel strike and toe off) and transparent during swing. doi:10.1371/journal.pcbi.1001107.g002 

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Context 1

... albeit in different ways, by the very joint actuation redundancies they seek to address. Extra sources of information are needed to address this problem. EMG data contains information about muscle activity, and could potentially be used as a source of biologically realistic neural control commands to muscles. This promises to circumvent the above-mentioned difficulties in obtaining optimal muscle activations. Further, having muscle activation profiles could also enable a more systematic study of the effects of MTU structure (design) on the breakdown of joint actuation amongst individual elements. In other words, estimating muscle activations from the data allows a consideration of both neural control and muscle-tendon design, in tandem, on the operation of individual muscles and tendons. Motivated by the above ideas, we have developed a theoretical framework to (a) address how the load of actuating a joint is shared amongst the many MTUs, (b) elucidate features of leg design and neuromuscular control enabling the breakdown and (c) clarify functional advantages arising from the load sharing. As a case study, we examine ankle joint actuation in human walking. We model the three primary leg MTUs contributing to ankle action in walking (Figure 1). Each MTU is characterized by (a) Hill-type muscle dynamics [15], (b) a common non-linear tendon model [16] and (c) a bilinear excitation-activation relation [3] - all of which are assumed to be internally consistent. These relations are parameterized with a minimal set of twelve muscle-tendon morphological features (representing leg MTU design). We conduct a computational exploration of the muscle-tendon design space for correspondence to well-known biological objectives. Specifically, for each set of system parameters, we actuate the model with joint state and muscle activations from healthy human gait data (Methods) to characterize the resulting joint torque and metabolic consumption. An overview of the modeling scheme is presented in Figure 1. Our results are organized into five sections. First we present our estimates of muscle activations from EMGs recorded during human walking. In the second section, we characterize the leg parameter space by ability to produce human-like ankle torques and economy. We show that there is a unique parameter vector that is able to accomplish both, and that this unique vector corresponds to the maximum metabolic economy. Third, we present the optimal leg parameters, compare them with biological values and discuss their influence on metabolic economy. Fourth, we present model plantar flexor muscle and tendon strain predictions, compare them with two sets of independent empirical recordings and use them to evaluate mechanical power breakdown between muscle and tendon within each MTU. In the fifth section, we present metrics regarding the breakdown of ankle actuation amongst the two different plantar flexors. Muscle activation is an indicator of a muscle’s force-generation capability, indicated by the proportion of troponin bound to calcium [17–19]. It is driven by neurally stimulated electrical activity in the muscle. Since EMG data is a qualitative indicator of muscle electrical activity [4], it contains valuable information about individual muscle activity and can be useful in understanding the breakdown of joint actuation. However, quantitative uses of EMG data have been limited by variability in the signal and measurement artifacts. Here we show that considering dominant biophysical characteristics of the muscle activation build-up along with the randomness inherent in the EMG measurement yields repeatable and reasonable activation estimates. Classic EMG analysis involves rectification and low-pass filtering [20,21]. But low-pass filters smear out the filtered signal, leading to loss of both phase and amplitude information, particularly turn-on and turn-off of muscle activity [22]. Recently Sanger proposed a probabilistic method to resolve the signal variability and noise floor related problems in analyzing EMG signals [22]. In this paper, the muscle electrical activity x ( t ) driving the EMG signal was modeled as a jump-diffusion process: where dW is a diffusion process with rate E , dN is a jump process with frequency b and U represents a uniform distribution indicating that x ( t ) is a uniform random variable when there is a jump. The measured EMG signal was modeled as a random process with an exponential density and rate given by 1 = x ( t ) : Propagating the probability densities in a classic recursive Bayesian manner, to estimate the x ( t ) that best describes the observed EMG signal, Sanger reported excellent temporal resolution of EMG turn-on/turn-off during forced maximal contraction tasks. However, the biophysical relevance to analyzing EMG from dynamic tasks is limited by (a) the sharp, near- instantaneous turn-on and turn-off in the Sanger estimates, and (b) the lack of amplitude-buildup when the muscle is on (Figure 2). We attribute this to differences between the modeled jump- diffusion process and the true buildup of muscle active state in normal tasks (Supplementary Text S2). The constant frequency and uniform amplitude of the jump process [22] compromises the history dependence of active-state buildup, causing sudden jumps in the estimates when the EMG signal turns on/off. Further, the Sanger model has the same jump rate for source and sink or for activation and deactivation. This neglects the differences in activation and deactivation time constants that are critical to muscle activation build-up [19]. Thus the Bayesian approach proposed in [22] appears to estimate the times when muscle electrical activity turns on/off, and not the muscle active state because activation dynamics (relating electrical activity to cross bridge formation) are not explicitly included. One way to account for the activation dynamics would be to incorporate them directly into the jump-diffusion model and numerically evaluate a solution. We chose a simpler approximation, and applied the activation dynamics on the muscle electrical activity x ( t ) from Sanger’s model to estimate muscle active state a ( t ) . Activation dynamics was specified by the classic bilinear form [3]: This differential equation models the history dependence in build-up of muscle activation, and captures differences between activation ( t act ) and deactivation ( t deact ) time constants with the ratio c . Notes on the biophysical relevance of our estimation procedure are available in Supplementary Text S2. The muscle activation profiles estimated using our two-step procedure are shown in Figure 2. The intermediate Bayesian estimate x ( t ) has a step-like shape as it primarily captures the turn on and turn off of the muscle electrical activity measured by EMG. The estimated activations a ( t ) have profiles that are qualitatively expected from known temporal features of ankle muscle force buildup [4]. Further, the synergistic soleus and gastrocnemius muscles have similar profiles. Random step to step variations in EMG signals do not drastically change the estimated activation profiles. A repeatable ensemble average was obtained in as few as eight trials in cases of minimal motion artifact. The ensemble average estimates (Supplementary Text S2) show little variability in turn-on/turn-off times, and show greater variability in amplitude features (particularly when activation is high). The method and resulting estimates were found to be quite robust to normal, day-to-day variations in electrode placement for a given subject. We used our estimates of neurally stimulated muscle activations observed in walking to conduct the computational exploration (illustrated in Figure 1) of muscle-tendon morphologies. Using the muscle activations a ( t ) and joint kinematics h joint ( t ) estimated from normal walking data, we actuated the leg muscle- tendon model M parameterized by a set of morphological features m . The parameter vector m comprises the tendon reference strain l ref , the tendon shape factor K sh , the muscle maximum isometric force F max and the tendon slack length l sl for each of the three ankle MTUs. We randomly generated sets of leg muscle-tendon parameter vectors, m (from a uniform distribution with bounds stated in Supplementary Text S1), and computed both the model ankle torque profile, t mod ( t ) and metabolic energy consumed, C , for each set: The resulting errors between model and human ankle torques are plotted against the model metabolic consumption (Figure 3). Notable features of the plot include (a) the overall L shape, (b) a vertical boundary evidently representing the minimum energy that model muscles have to expend given the inputs, regardless of torque match and (c) an evidently systematic horizontal boundary below the population representing the best match between model and data. Each point along this horizontal boundary corresponds to a different metabolic consumption for the same level of error between model and human dynamics. A published empirical estimate of the range of metabolic consumption for ankle actuation in walking [8] is indicated, and is seen to be well-approximated by points exhibiting near-minimal economies, close to the the vertical boundary. Remarkably, this overall parameter-space characterization reveals that the empirically observed realization is among the most economical of the many ways to produce human-like torque. Thus the human leg and the nervous system controlling it resolve the load-sharing redundancies in actuating the ankle most economically. Points that best approximate human-like dynamics and optimal human-like metabolics lie near the bottom horizontal and left vertical boundary respectively. Thus points representing a logical intersection of the model’s ability to best produce both human-like dynamics and metabolics lie in a small region at the lower-left corner (indicated by box in Figure 3). ...

Context 2

... design space for correspondence to well-known biological objectives. Specifically, for each set of system parameters, we actuate the model with joint state and muscle activations from healthy human gait data (Methods) to characterize the resulting joint torque and metabolic consumption. An overview of the modeling scheme is presented in Figure 1. Our results are organized into five sections. First we present our estimates of muscle activations from EMGs recorded during human walking. In the second section, we characterize the leg parameter space by ability to produce human-like ankle torques and economy. We show that there is a unique parameter vector that is able to accomplish both, and that this unique vector corresponds to the maximum metabolic economy. Third, we present the optimal leg parameters, compare them with biological values and discuss their influence on metabolic economy. Fourth, we present model plantar flexor muscle and tendon strain predictions, compare them with two sets of independent empirical recordings and use them to evaluate mechanical power breakdown between muscle and tendon within each MTU. In the fifth section, we present metrics regarding the breakdown of ankle actuation amongst the two different plantar flexors. Muscle activation is an indicator of a muscle’s force-generation capability, indicated by the proportion of troponin bound to calcium [17–19]. It is driven by neurally stimulated electrical activity in the muscle. Since EMG data is a qualitative indicator of muscle electrical activity [4], it contains valuable information about individual muscle activity and can be useful in understanding the breakdown of joint actuation. However, quantitative uses of EMG data have been limited by variability in the signal and measurement artifacts. Here we show that considering dominant biophysical characteristics of the muscle activation build-up along with the randomness inherent in the EMG measurement yields repeatable and reasonable activation estimates. Classic EMG analysis involves rectification and low-pass filtering [20,21]. But low-pass filters smear out the filtered signal, leading to loss of both phase and amplitude information, particularly turn-on and turn-off of muscle activity [22]. Recently Sanger proposed a probabilistic method to resolve the signal variability and noise floor related problems in analyzing EMG signals [22]. In this paper, the muscle electrical activity x ( t ) driving the EMG signal was modeled as a jump-diffusion process: where dW is a diffusion process with rate E , dN is a jump process with frequency b and U represents a uniform distribution indicating that x ( t ) is a uniform random variable when there is a jump. The measured EMG signal was modeled as a random process with an exponential density and rate given by 1 = x ( t ) : Propagating the probability densities in a classic recursive Bayesian manner, to estimate the x ( t ) that best describes the observed EMG signal, Sanger reported excellent temporal resolution of EMG turn-on/turn-off during forced maximal contraction tasks. However, the biophysical relevance to analyzing EMG from dynamic tasks is limited by (a) the sharp, near- instantaneous turn-on and turn-off in the Sanger estimates, and (b) the lack of amplitude-buildup when the muscle is on (Figure 2). We attribute this to differences between the modeled jump- diffusion process and the true buildup of muscle active state in normal tasks (Supplementary Text S2). The constant frequency and uniform amplitude of the jump process [22] compromises the history dependence of active-state buildup, causing sudden jumps in the estimates when the EMG signal turns on/off. Further, the Sanger model has the same jump rate for source and sink or for activation and deactivation. This neglects the differences in activation and deactivation time constants that are critical to muscle activation build-up [19]. Thus the Bayesian approach proposed in [22] appears to estimate the times when muscle electrical activity turns on/off, and not the muscle active state because activation dynamics (relating electrical activity to cross bridge formation) are not explicitly included. One way to account for the activation dynamics would be to incorporate them directly into the jump-diffusion model and numerically evaluate a solution. We chose a simpler approximation, and applied the activation dynamics on the muscle electrical activity x ( t ) from Sanger’s model to estimate muscle active state a ( t ) . Activation dynamics was specified by the classic bilinear form [3]: This differential equation models the history dependence in build-up of muscle activation, and captures differences between activation ( t act ) and deactivation ( t deact ) time constants with the ratio c . Notes on the biophysical relevance of our estimation procedure are available in Supplementary Text S2. The muscle activation profiles estimated using our two-step procedure are shown in Figure 2. The intermediate Bayesian estimate x ( t ) has a step-like shape as it primarily captures the turn on and turn off of the muscle electrical activity measured by EMG. The estimated activations a ( t ) have profiles that are qualitatively expected from known temporal features of ankle muscle force buildup [4]. Further, the synergistic soleus and gastrocnemius muscles have similar profiles. Random step to step variations in EMG signals do not drastically change the estimated activation profiles. A repeatable ensemble average was obtained in as few as eight trials in cases of minimal motion artifact. The ensemble average estimates (Supplementary Text S2) show little variability in turn-on/turn-off times, and show greater variability in amplitude features (particularly when activation is high). The method and resulting estimates were found to be quite robust to normal, day-to-day variations in electrode placement for a given subject. We used our estimates of neurally stimulated muscle activations observed in walking to conduct the computational exploration (illustrated in Figure 1) of muscle-tendon morphologies. Using the muscle activations a ( t ) and joint kinematics h joint ( t ) estimated from normal walking data, we actuated the leg muscle- tendon model M parameterized by a set of morphological features m . The parameter vector m comprises the tendon reference strain l ref , the tendon shape factor K sh , the muscle maximum isometric force F max and the tendon slack length l sl for each of the three ankle MTUs. We randomly generated sets of leg muscle-tendon parameter vectors, m (from a uniform distribution with bounds stated in Supplementary Text S1), and computed both the model ankle torque profile, t mod ( t ) and metabolic energy consumed, C , for each set: The resulting errors between model and human ankle torques are plotted against the model metabolic consumption (Figure 3). Notable features of the plot include (a) the overall L shape, (b) a vertical boundary evidently representing the minimum energy that model muscles have to expend given the inputs, regardless of torque match and (c) an evidently systematic horizontal boundary below the population representing the best match between model and data. Each point along this horizontal boundary corresponds to a different metabolic consumption for the same level of error between model and human dynamics. A published empirical estimate of the range of metabolic consumption for ankle actuation in walking [8] is indicated, and is seen to be well-approximated by points exhibiting near-minimal economies, close to the the vertical boundary. Remarkably, this overall parameter-space characterization reveals that the empirically observed realization is among the most economical of the many ways to produce human-like torque. Thus the human leg and the nervous system controlling it resolve the load-sharing redundancies in actuating the ankle most economically. Points that best approximate human-like dynamics and optimal human-like metabolics lie near the bottom horizontal and left vertical boundary respectively. Thus points representing a logical intersection of the model’s ability to best produce both human-like dynamics and metabolics lie in a small region at the lower-left corner (indicated by box in Figure 3). Points in this region not only have similar values of the torque and metabolic cost but also have similar values for the morphological parameters defining them. The coefficients of variance amongst parameter values in the corner region, listed in the caption of Figure 3, are low for most of the parameters (details in Supplementary Text S3). Further, all points outside the corner region compromise on either torque match, or economy, or both. Thus, parameter vectors defining the corner region points can be identified computationally by encoding the simultaneous realization of two objectives (torque match and optimal economy) into a multi-objective problem. Solutions for such problems are generally sets of points that simultaneously realize both objectives as best as possible. These solutions, known as Pareto solutions, typically form a frontier along which the two objectives can be traded off against each other to varying degrees. In the special case that both objectives logically intersect at a mathematically sharp corner, there is a single strong Pareto optimal solution that best fulfils both objectives without any tradeoffs. As demonstrated above, our problem resembles this special case - within systematic limits of experimental precision, data variability and functional relevance. Thus it is possible to interpret our problem within the strong Pareto optimal framework, and simplify standard multi- objective optimization methods (such as Aggregate Objective Functions, Pareto ranking, evolutionary algorithms, or cost- constraint techniques [23]) to solve for the biologically realistic parameter vectors. Our simplified approach relies on the ...