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

Electrical muscle stimulation demonstrates potential for preventing muscle atrophy and 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 14 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.     

Identification of Hammerstein model for bioreactors with input multiplicities

A closed loop identification method of Hammerstein model for continuous bioreactor with input multiplicity is proposed. Hammerstein model consists of nonlinear steady-state gain followed by a unity gain linear system. The method consists of first getting local first order plus time delay (FOPTD) models around the two input multiplicity values of the substrate feed concentration. The model parameters of the FOPTD is identified by a least square optimization method. The initial guess for the model parameters are obtained from the settling time, the initial delay in the closed loop servo response and using a simple proportional controller formula. From the local process gain values obtained for the several step changes around the two operating conditions, the nonlinear gain portion of the Hammerstein is then obtained. The actual nonlinear gain and the identified nonlinear gain is compared.     

General methodology for nonlinear modeling of neural systems with Poisson point-process inputs

This paper presents a general methodological framework for the practical modeling of neural systems with point-process inputs (sequences of action potentials or, more broadly, identical events) based on the Volterra and Wiener theories of functional expansions and system identification. The paper clarifies the distinctions between Volterra and Wiener kernels obtained from Poisson point-process inputs. It shows that only the Wiener kernels can be estimated via cross-correlation, but must be defined as zero along the diagonals. The Volterra kernels can be estimated far more accurately (and from shorter data-records) by use of the Laguerre expansion technique adapted to point-process inputs, and they are independent of the mean rate of stimulation (unlike their P-W counterparts that depend on it). The Volterra kernels can also be estimated for broadband point-process inputs that are not Poisson. Useful applications of this modeling approach include cases where we seek to determine (model) the transfer characteristics between one neuronal axon (a point-process 'input') and another axon (a point-process 'output') or some other measure of neuronal activity (a continuous 'output', such as population activity) with which a causal link exists.     

Spike discharges of skeletomotor neurons during random noise modulated transmembrane current stimulation and muscle stretch

 Spike discharges of skeletomotor neurons innervating triceps surae muscles elicited by white noise modulated transmembrane current stimulation and muscle stretch were studied in decerebrated cats. The white noise modulated current intensity ranged from 4.3 to 63.2 nA peak-to-peak, while muscle stretches ranged from 100 μm to 4.26 mm peak-to-peak. The neuronal responses were studied by averaging the muscle length records centered at the skeletomotor action potentials (peri-spike average, PSA) and by Wiener analysis. Skeletomotor spikes appeared after a sharp peak in PSA of the injected current, preceded by a longer-lasting smaller wavelet of either depolarizing or hyperpolarizing direction. The PSA amplitude was not related to the injected current amplitude nor showed any differences related to the motor unit type. The PSA amplitudes were virtually independent of the stretching amplitude σ, after an initial increase with stretching amplitudes in the range of 15–40 μm (S.D.), or 100–270 μm peak-to-peak.Analyses of cross-spectra indicated a small or absent increase in gain with frequency in response to injected current, but about 20 dB/decade in the range 10–100 Hz in response to muscle stretch. The peaks of both Wiener kernels in response to current injection appear to decrease with the amplitude of injected current, but this decrease was not statistically significant. The narrow first-order kernels suggest that the transfer function between the current input and spike discharge is lowpass with a wide passband, i.e. there is very little change in dynamics. The values of the second-order kernels appear to be nonzero only along the main diagonal. This is characteristic of a simple Hammerstein type cascade, i.e. a zero memory nonlinearity followed by a linear system. Small values of second-order kernels away from the origin and narrow first-order kernels suggest that the linear cascade contributes very little to the overall dynamic response.In contrast to Wiener kernels found in response to current injection, the Wiener kernels in response to stretch showed a decreasing trend with stretch amplitude. The size of the second-order kernels decreased to a somewhat larger extent with input amplitude than that of the first-order kernels, indicating an amplitude-dependent nonlinearity. Overall, the transformation between length and spike output was described as an LNNL cascade with second-order nonlinearities. Received: 1 April 1993/Accepted in revised form: 24 March 1994     

The identification of nonlinear biological systems: LNL cascade models

Systems that can be represented by a cascade of a dynamic linear (L), a static nonlinear (N) and a dynamic linear (L) subsystem are considered. Various identification schemes that have been proposed for these LNL systems are critically reviewed with reference to the special problems that arise in the identification of nonlinear biological systems. A simulated LNL system is identified from limited duration input-output data using an iterative identification scheme.     

Identification of nonlinear systems using random impulse train inputs

Nonlinear systems that require discrete inputs can be characterized by using random impulse train (Poisson process) inputs. The method is analagous to the Wiener method for continuous input systems, where Gaussian white-noise is the input. In place of the Wiener functional expansion for the output of a continuous input system, a new series for discrete input systems is created by making certain restrictions on the integrals in a Volterra series. The kernels in the new series differ from the Wiener kernels, but also serve to identify a system and are simpler to compute. For systems whose impulse responses vary in amplitude but maintain a similar shape, one argument may be held fixed in each kernel. This simplifies the identification problem. As a test of the theory presented, the output of a hypothetical second order nonlinear system in response to a random impulse train stimulus was computer simulated. Kernels calculated from the simulated data agreed with theoretical predictions. The Poisson impulse train method is applicable to any system whose input can be delivered in discrete pulses. It is particularly suited to neuronal synaptic systems when the pattern of input nerve impulses can be made random.     

Membrane potential changes of skeletomotor neurons in response to random stretches of the triceps surae muscles in decerebrate cats

 The properties of membrane potential changes of skeletomotor neurons (S, FR, and FF) innervating triceps surae muscles during pseudorandom stretching of these muscles were studied in decerebrate cats. Peak amplitudes of pseudorandom muscle stretches ranged from 119 μm to 4.15 mm peak-to-peak. Sequences of ten identical stretching periods were applied for averaging. Shapes of membrane potential changes and probability density distribution of amplitudes of the input and output signals and power spectra suggest that the skeletomotor neuron membrane has nonlinear properties. First- and second-order Wiener kernels were determined by applying the cross-correlation (Lee-Schetzen) method. The results suggest that the transfer function between muscle stretches and subthreshold membrane potentials is a Wiener-type cascade. This cascade is consistent with a linear, second-order, underdamped transfer function followed by a simple quadratic nonlinearity [linear (L) system followed by nonlinear (N) system, or LN cascade]. Including the nonlinear component calculated from the second-order Wiener kernel improved the model significantly over its linear counterpart, especially in S-type motoneurons. Qualitatively similar results were obtained with all types of motoneurons studied. Received: 1 April 1993/Accepted in revised form: 24 March 1994     

Applications of minimum-order Wiener modeling to retinal ganglion cell spatiotemporal dynamics

In a previous paper (Marmarelis et al. 1986) we presented the concept of minimum-order Wiener (MOW) modeling of continuous-input/spike-output (CISO) systems. The associated MOW methodology aims at obtaining low-order Wiener models for CISO systems of practical interest. The assertion was made that many neurophysiological systems that fall in this class can be studied effectively by the use of this method. We have chosen a sensory system to demonstrate the efficacy of the method with actual experimental data. The response of retinal ganglion cells to spatiotemporal visual stimuli was studied with this approach and a second-order MOW model was obtained. The results appear to corroborate the adequacy of this model in terms of predicting the timing of the output spikes.     

Wiener analysis of functionals of a Markov chain: Application to neural transformations of random signals

The Wiener method of nonlinear system identification is extended to systems with a Markov chain input. Multivariate functionals are constructed that are orthonormal with respect to the probability measure of the Markov input. Any system operating on a Markov chain may be represented by an orthogonal expansion in these functionals. The coefficients of the orthogonal expansion may be evaluated by crosscorrelation. Application of this technique to nonlinear neural systems with a Markov actionpotential input are discussed.     

Minimum-order Wiener modelling of spike-output systems

Systems that generate spike outputs in response to continuous inputs abound in neurophysiology. The study of their dynamics with the use of systems analysis methods has been complicated by the difference in modality of the input and output signals. When the problem is placed in the framework of Wiener's theory in discrete time, an infinite functional series is required for the formal representation of the input-output relation. This has given rise to the belief that a large number of Wiener functionals is needed in practice before a model of reasonable accuracy can be obtained. In this paper, we introduce the concept of minimum-order Wiener models for spike-output systems, and we show that a low-order Wiener model is adequate in many cases for predicting fully the timing of the output spikes.     

A new algorithm for the identification of multiple input Wiener systems

Multiple-input Wiener systems consist of two or more linear dynamic elements, whose outputs are transformed by a multiple-input static non-linearity. Korenberg (1985) demonstrated that the linear elements of these systems can be estimated using either a first order input-ouput cross-covariance or a slice of the second, or higher, order input-output cross-covariance function. Korenberg's work used a multiple input LNL structure, in which the output of the static nonlinearity was then filtered by a linear dynamic system. In this paper we show that by restricting our study to the slightly simpler Wiener structure, it is possible to improve the linear subsystem estimates obtained from the measured cross-covariance functions. Three algorithms, which taken together can identify any multiple-input Wiener system, have been developed. We present the theory underlying these algorithms and detail their implementation. Simulation results are then presented which demonstrate that the algorithms are robust in the presence of output noise, and provide good estimates of the system dynamics under a wide set of conditions.     

Parallel cascade identification and its application to protein family prediction

Parallel cascade identification is a method for modeling dynamic systems with possibly high order nonlinearities and lengthy memory, given only input/output data for the system gathered in an experiment. While the method was originally proposed for nonlinear system identification, two recent papers have illustrated its utility for protein family prediction. One strength of this approach is the capability of training effective parallel cascade classifiers from very little training data. Indeed, when the amount of training exemplars is limited, and when distinctions between a small number of categories suffice, parallel cascade identification can outperform some state-of-the-art techniques. Moreover, the unusual approach taken by this method enables it to be effectively combined with other techniques to significantly improve accuracy. In this paper, parallel cascade identification is first reviewed, and its use in a variety of different fields is surveyed. Then protein family prediction via this method is considered in detail, and some particularly useful applications are pointed out.     

Input-output behaviour of a model neuron with alternating drift

The input-output behaviour of the Wiener neuronal model subject to alternating input is studied under the assumption that the effect of such an input is to make the drift itself of an alternating type. Firing densities and related statistics are obtained via simulations of the sample-paths of the process in the following three cases: the drift changes occur during random periods characterised by (i) exponential distribution, (ii) Erlang distribution with a preassigned shape parameter, and (iii) deterministic distribution. The obtained results are compared with those holding for the Wiener neuronal model subject to sinusoidal input.     

Output regulation of a class of unstructured models of continuous bioreactors: steady-state approaches

This article deals with the output regulation of continuous bioreactors in the face of constant disturbances and inverse dynamics. Nonlinear controllers developed on the basis of approximate equilibrium manifolds can almost attenuate measurable or unmeasurable disturbances on the output. This nonlinear feed-forward/feedback control framework without any tuning parameters can be directly implemented to strictly nonlinear systems. Under dynamic actuator constraints and the availability of only output signals for use in the control law, closed-loop simulations demonstrate that the proposed control techniques are superior to a nonlinear PI control scheme based on the identified Hammerstein model.     

The dynamic nonlinear behavior of fly photoreceptors evoked by a wide range of light intensities.

Fly photoreceptor cells were stimulated with steps of light over a wide intensity range. First- and second-order Volterra kernels were then computed from sequences of combined step responses. Diagonal values of the second-order Volterra kernels were much greater than the off-diagonal values, and the diagonal values were roughly proportional to the corresponding first-order kernels, suggesting that the response could be approximated by a static nonlinearity followed by a dynamic linear component (Hammerstein model). The amplitudes of the second-order kernels were much smaller in light-adapted than in dark-adapted photoreceptors. Hammerstein models constructed from the step input/output measurements gave reasonable approximations to the actual photoreceptor responses, with light-adapted responses being relatively better fitted. However, Hammerstein models could not account for several features of the photoreceptor behavior, including the dependence of the step response shape on step amplitude. A model containing an additional static nonlinearity after the dynamic linear component gave significantly better fits to the data. These results indicate that blowfly photoreceptors have a strong early gain control nonlinearity acting before the processes that create the characteristic time course of the response, in addition to the nonlinearities caused by membrane conductances.     

Ions, energy and axonal injury: towards a molecular neurology of multiple sclerosis

It is now clear that effective treatments for nervous system disorders such as multiple sclerosis (MS) represent achievable objectives. Molecular mechanisms of axonal degeneration - a major pathological substrate for disability in MS - have been identified, pointing to the possibility of neuroprotection. Although previous studies were mainly carried out in laboratory models, recent analyses of human MS tissue have identified molecular targets that are related to mitochondrial function and specific isoforms of ion channels as contributors to axonal degeneration in MS. Taken together, the observations in model systems and in human tissue converge on the identification of a group of molecules that are related to ion fluxes and energetics as significant actors in the axonal-injury cascade, and suggest a set of molecular targets that might be useful in the development of new therapies.