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PubMed1995

A model of the neuro-musculo-skeletal system for human locomotion. II Real-time adaptability under various constraints.

Adaptive gaits of humans were produced as a result of emergent properties of a model based on the neurophysiology of the central pattern generator and the biomechanics of the human musculoskeletal system. We previously proposed a neuromusculoskeletal model for human locomotion, in which movements emerged as a stable limit cycle that was generated through the global entrainment among the neural system, composed of neural oscillators, the musculoskeletal system, and the environment. In the present study, we investigated the adaptability of this model under various types of environmental and task constraints. Using a computer simulation, it was found that walking movements were robust against mechanical perturbations, loads with a mass, and uneven terrain. Moreover, the speed of walking could be controlled by a single parameter which tonically drove the neural oscillators, and the step cycle could be entrained by a rhythmic input to the neural oscillators.

Adaptation, Physiological↗
PubMed1995

Vergence eye movement control and multivalent perception of autostereograms.

We introduce a dynamical model for automatic vergence eye movement control. In connection with our dynamical system of binocular model neurons that solves the correspondence problem of stereo-vision, we present a complete model for stereo-vision. Our automatic vergence eye movement control adjusts an image segment, which is of momentary interest to the observer. The adjustment is done in such a way that we only need to define a disparity search range of minimal extension. Recently, a new method of encoding (3D) three-dimensional information in 2D pictures was designed in the form of computer-generated patterns of colored dots. At first glimpse, these so-called autostereograms appear as structured but meaningless patterns. After a certain period of observation, a 3D pattern emerges suddenly in an impressive way. Applying our algorithm to autostereograms, we find a fully satisfactory agreement with the multivalent perception experienced by humans. As in nature, in our model the phase transition between the initial state and the 3D perception state takes place in a very short time. Our algorithm is very robust against noise, and there is no need to interpolate a sparse depth map.

Algorithms↗
PubMed1995

Synchrony measures for biological neural networks.

Synchronous firing of a population of neurons has been observed in many experimental preparations; in addition, various mathematical neural network models have been shown, analytically or numerically, to contain stable synchronous solutions. In order to assess the level of synchrony of a particular network over some time interval, quantitative measures of synchrony are needed. We develop here various synchrony measures which utilize only the spike times of the neurons; these measures are applicable in both experimental situations and in computer models. Using a mathematical model of the CA3 region of the hippocampus, we evaluate these synchrony measures and compare them with pictorial representations of network activity. We illustrate how synchrony is lost and synchrony measures change as heterogeneity amongst cells increases. Theoretical expected values of the synchrony measures for different categories of network solutions are derived and compared with results of simulations.

Action Potentials↗
PubMed1995

The hybrid mass-spring pendulum model of human leg swinging: stiffness in the control of cycle period.

Human leg swinging is modeled as the harmonic motion of a hybrid mass-spring pendulum. The cycle period is determined by a gravitational component and an elastic component, which is provided by the attachment of a soft-tissue/muscular spring of variable stiffness. To confirm that the stiffness of the spring changes with alterations in the inertial properties of the oscillator and that stiffness is relevant for the control of cycle period, we conducted this study in which the simple pendulum equivalent length was experimentally manipulated by adding mass to the ankle of a comfortably swinging leg. Twenty-four young, healthy adults were videotaped as they swung their right leg under four conditions: no added mass and with masses of 2.27, 4.55, and 6.82kg added to the ankle. Strong, linear relationships between the acceleration and displacement of the swinging leg within subjects and conditions were found, confirming the motion's harmonic nature. Cycle period significantly increased with the added mass. However, the observed increases were not as large as would be predicted by the induced changes in the gravitational component alone. These differences were interpreted as being due to increases in the active muscular stiffness. Significant linear increases in the elastic component (and hence stiffness) were demonstrated with increases in the simple pendulum equivalent length in 20 of the individual subjects, with r2 values ranging between 0.89 and 0.99. Significant linear relationships were also demonstrated between the elastic and gravitational components in 22 subjects, with individual r2 values between 0.90 and 0.99.(ABSTRACT TRUNCATED AT 250 WORDS)

Acceleration↗
PubMed1995

On the relationship between the number of negative slope regions in the voltage-current curve of the Hodgkin-Huxley model and its parameter values.

We consider the problem of the existence of a negative slope region (NSR) in the voltage-current curve of the neuronal membrane and the relationship between this phenomenon and the membrane parameters. For the Hodgkin-Huxley model it is proposed to determine the dependence of the number of NSR on the values of the maximal sodium (gNa) and potassium (gk) conductances. The method is suggested for constructing the boundaries on the (gNa, gk) plane, in passing through which the number of NSR changes to 1. Using the method we partition the (gNa, gk) plane into the regions corresponding to the curves with the different number of NSR. This number can be changed from 0 to 2 in changing the values of gNa and gk over the physiologically possible range.

Animals↗
PubMed1995

Characterization, scaling, and partial representation of neural junctions and coordinated firing patterns by dynamic similarity.

This paper presents a dynamic-similarity-based system for mathematically characterizing the functional connectivity and information flow of neural junctions. This approach allows for quantitative comparison of operations of neural junctions across systems, and an interpretation of their connectivity parameters in terms of the flow of multiunit firing patterns. The paper further uses this characterization to show how to rationally construct reduced operational models of neural junctions. Both uniformly proportional scaling and partial fragmentary representations are developed. The uniformly scaled models are better adapted to overall capacities and broader theoretical conceptualizations; the partial representations are better adapted to direct comparison with microelectrode experimentation. The characterization of information flow is based on coordinated multiunit patterns such as synfire chains or sequential configurations. The system can be applied to component parts of large composite networks including junctions with topographical patchiness and other irregularities. The characterization should be of use to anatomists, physiologists, modelers, and theorists. The theory predicts that the necessity for cooperative confluence of synaptic potentials in sending and receiving sequential configurations across topographically constrained projection fields requires the existence of functional 'pattern modules' within the topographical synaptology of the junction.

Animals↗
PubMed1995

Characterization, scaling, and partial representation of diffuse and discrete input junctions to CA3 hippocampus.

This paper applies a general mathematical system for characterizing and scaling functional connectivity and information flow across the diffuse (EC) and discrete (DG) input junctions to the CA3 hippocampus. Both gross connectivity and coordinated multiunit informational firing patterns are quantitatively characterized in terms of 32 defining parameters interrelated by 17 equations, and then scaled down according to rules for uniformly proportional scaling and for partial representation. The diffuse EC-CA3 junction is shown to be uniformly scalable with realistic representation of both essential spatiotemporal cooperativity and coordinated firing patterns down to populations of a few hundred neurons. Scaling of the discrete DG-CA3 junction can be effected with a two-step process, which necessarily deviates from uniform proportionality but nonetheless produces a valuable and readily interpretable reduced model, also utilizing a few hundred neurons in the receiving population. Partial representation produces a reduced model of only a portion of the full network where each model neuron corresponds directly to a biological neuron. The mathematical analysis illustrated here shows that although omissions and distortions are inescapable in such an application, satisfactorily complete and accurate models the size of pattern modules are possible. Finally, the mathematical characterization of these junctions generates a theory which sees the DG as a definer of the fine structure of embedded traces in the hippocampus and entire coordinated patterns of sequences of 14-cell links in CA3 as triggered by the firing of sequences of individual neurons in DG.

Animals↗
PubMed1995

A ternary logic model for recurrent neuromime networks with delay.

In contrast to popular recurrent artificial neural network (RANN) models, biological neural networks have unsymmetric structures and incorporate significant delays as a result of axonal propagation. Consequently, biologically inspired neural network models are more accurately described by nonlinear differential-delay equations rather than nonlinear ordinary differential equations (ODEs), and the standard techniques for studying the dynamics of RANNs are wholly inadequate for these models. This paper develops a ternary-logic based method for analyzing these networks. Key to the technique is the realization that a nonzero delay produces a bounded stability region. This result significantly simplifies the construction of sufficient conditions for characterizing the network equilibria. If the network gain is large enough, each equilibrium can be classified as either asymptotically stable or unstable. To illustrate the analysis technique, the swim central pattern generator (CPG) of the sea slug Tritonia diomedea is examined. For wide range of reasonable parameter values, the ternary analysis shows that none of the network equilibria are stable, and thus the network must oscillate. The results show that complex synaptic dynamics are not necessary for pattern generation.

Animals↗
PubMed1995

Discharge patterns of tonically firing human motoneurones.

We have attempted to reconcile the different patterns of distribution of interspike intervals that are found in motoneurones made to discharge by intracellular injection of constant current in reduced animal preparations and by voluntary control in human subjects. We recorded long spike trains from single motor units in three human muscles made to discharge at constant mean frequencies with the help of auditory and visual feedback. The distribution of interspike intervals in each spike train was analysed quantitatively. We found that the different pattern of discharge of the human motor units could be accounted for when due allowance was made for the variability of the drive to the human motoneurone which arose because of the feedback process used to maintain the target frequency. A model testing this hypothesis gave results that were qualitatively consistent with the human data.

Acoustic Stimulation↗
PubMed1988

Kinematic networks. A distributed model for representing and regularizing motor redundancy.

Motor control in primates relates to a system which is highly redundant from the mechanical point of view--redundancy coming from an imbalance between the set of independently controllable variables and the set of system variables. The consequence is the manifestation of a broad class of ill-posed problems, problems for which it is difficult to identify unique solutions. For example (i) the problem of determining the coordinated patterns of rotation of the arm joints for a planned trajectory of the hand; (ii) the problem of determining the distribution of muscle forces for a desired set of joint torques. Ill-posed problems, in general, require regularization methods which allow to spell acceptable, if not unique, solutions. In the case of the motor system, we propose that the basic regularization mechanism is provided by the potential fields generated by the elastic properties of muscles, according to an organizational principle that we call "Passive Motion Paradigm". The physiological basis of this hypothesis is reviewed and a "Kinematic Network" (K-net) model is proposed that expresses the kinematic transformations and the causal relations implied by elasticity. Moreover, it is shown how K-nets can be obtained from a kinematic "Body Model", in the context of a specific task. Two particularly significant results are: (i) the uniform treatment of closed as well as open kinematic chains, and (ii) the development of a new method for the automatic generation of kinematic equations with arbitrary topology. Moreover, the model is akin to the concept of "motor equivalence" in the sense that it provides families of motor equivalent trajectories parametrized by tunable motor impedances.

Biomechanical Phenomena↗
PubMed1988

Pattern recognition and associative memory as dynamical processes in a synergetic system. I. Translational invariance, selective attention, and decomposition of scenes.

We consider a model for associative memory and pattern recognition which was devised by Haken (1987b). This model treats the activity of the neurons as continuous variables and exploits an analogy with pattern formation in synergetic systems. The capability of such a system to act as associative memory is demonstrated by the reconstruction of faces which are partially offered to the system, and which are restored by the corresponding dynamical process. We demonstrate how this model can be cast into a form which is translation invariant and how partially hidden faces in scenes can be recognized by means of the control of attention parameters of specific patterns.

Cybernetics↗
PubMed1988

A two dimensional field theory for motion computation. First order approximation; translatory motion of rigid patterns.

The local extraction of motion information from brightness patterns by individual movement detectors of the correlation-type is considered in the first part of the paper. A two-dimensional field theory of movement detection is developed by treating the distance between two adjacent photoreceptors as a differential. In the first approximation of the theory we only consider linear terms of the time interval between the reception of a contrast element and its delayed representation by the detector and linear terms of the spatial distances between adjacent photoreceptors. As a result we may neglect terms of higher order than quadratic in a Taylor series development of the brightness pattern. The responses of pairs of individual movement detectors are combined to a local response vector. In the first approximation of the detector field theory the response vector is proportional to the instantaneous pattern velocity vector and linearly dependent on local properties of the moving pattern. The linear dependence on pattern properties is represented by a two by two tensor consisting of elements which are nonlinear, local functionals of the moving pattern. Some of the properties of the tensor elements are treated in detail. So, for instance, it is shown that the off-diagonal elements of the tensor disappear when the moving pattern consists of x- and y-dependent separable components. In the second part of the paper the tensor relation leading to the output of a movement detector pair is spatially integrated. The result of the integration is an approximation to a summation of the outputs of an array of detector pairs. The spatially integrated detector tensor relates the translatory motion vector to the resultant output vector. It is shown that the angle between the motion vector and the resultant output vector is always smaller than +/- 90 degrees whereas the angle between the motion vector and local response vectors, elicited by detector pairs, may cover the entire angular range. In the discussion of the paper the limits of the field theory for motion computation as well as its higher approximations are pointed out in some detail. In a special chapter the dependence of the detector response on the pattern properties is treated and in another chapter questions connected with the so called aperture problem are discussed. Furthermore, properties for compensation of the pattern dependent deviation angle by spatial physiological integration are mentioned in the discussion.

Animals↗
PubMed1988

Locking, intermittency, and bifurcations in a periodically driven pacemaker neuron: Poincaré maps and biological implications.

Slowly adapting stretch receptor (SAO) pacemaker neurons, driven with periodic tugs, were analyzed by way of Poincaré mappings (Appendix). Two behaviors were apparent. i) Intermittency characterized previously unclear situations: discharges shifted irregularly between prolonged epochs where spike phases (relative to tugs) and intervals barely changed (slid), and brief bursts with marked variations (skipped). ii) Locking was well-known: phases and intervals remained almost fixed, regardless of the initiation. Changing frequencies, map domains with locking (ordered according to spikes/tugs ratios), alternated with intermittent ones. The best fit for any experimental map was a curve, not straight but certainly unidimensional, continuous and monotonic; it varied characteristically with frequency. This suggested relations called diffeomorphisms, implying periodicity and quasi-periodicity. Outcomes, expanding previous knowledge and meaningful biologically, were i) a precise, exhaustive behavior list (including between behavior transitions) and ii) a thorough understanding or model. This, in turn, provides norms for more specific models (single-variable ones suffice), constraints upon basic mechanisms (one variable, reflecting several real ones combined, should behave as the phase), and forecasts for future experimentation (e.g., unexamined tug frequencies and amplitudes).

Animals↗
PubMed1988

The effect of temperature on a simulated systematically paranodally demyelinated nerve fiber.

The temperature dependence (from 10 degrees to 50 degrees C) of the intracellular action potentials' parameters as well as of the ionic currents' kinetics in normal and demyelinated nerve fiber is studied. The simulation of the conduction in the normal fiber is based on the Frankenhaeuser and Huxley (1964) and Goldman and Albus (1968) equations, while in the case of a demyelinated fiber according to the same equations modified by Stephanova (1988). The temperature coefficients (Q10) for the rate constants as well as for the sodium and potassium permeabilities are introduced. It is shown that increased temperature blocks conduction in the simulated demyelinated fiber at temperatures much lower than the blocking temperature for the normal fiber. When temperature is increased, the amplitude as well as the wavelength and the asymmetry of the potential decrease. The relationship between conduction velocity and temperature is non-linear. The velocity increases when the temperature approaches the blocking temperature, after which abruptly drops. At a given degree of demyelination with increasing temperatures, the ionic currents' flow and the membrane conduction respectively increase, but, at lower temperatures, when the degree of the demyelination is increased, the conduction is blocked.

Action Potentials↗
PubMed1994

A modified radial isochron clock with slow and fast dynamics as a model of pacemaker neurons. Global bifurcation structure when driven by periodic pulse trains.

A simple mathematical model of living pacemaker neurons is proposed. The model has a unit circle limit cycle and radial isochrons, and the state point moves slowly in one region and fast in the remaining region; regions can correspond to the subthreshold activity and to the action potentials of pacemaker neurons, respectively. The global bifurcation structure when driven by periodic pulse trains was investigated using one-dimensional maps (PTC), two-dimensional bifurcation diagrams, and skeletons involving stimulus period and intensity. The existence of both the slow and the fast dynamics has a critical influence on the global bifurcation structure of the oscillator when stimulated periodically.

Biological Clocks↗
PubMed1994

A nonlinear regularization approach to early vision.

We propose a new class of approaches to smooth visual data while preserving significant transitions of these data as clues for segmentation. Formally, the given visual data are represented as a noisy (image) function g, and we present a class of continuously formulated global minimization problems to smooth g. The resulting function u can be characterized as the minimizer of a specific nonquadratic functional or, equivalently, as the result of an associated nonlinear diffusion process. Our approach generalizes the well-known quadratic regularization principle while retaining its attractive properties: For any given g, the solution u to the proposed minimization problem is unique and depends continuously on the data g. Furthermore, convergence of approximate solutions obtained by finite element discretization holds true. We show that the nodal variables of any chosen finite element subspace can be interpreted as computational units whose activation dynamics due to the nonlinear smoothing process evolve like a globally asymptotically stable network. A corresponding analogue implementation is thus feasible and would provide a real time processing stage for the transition preserving smoothing of visual data. Using artificial as well as real data we illustrate our approach by numerical examples. We demonstrate that solutions to our approach improve those obtained by quadratic minimization and show the influence of global parameters which allow for a continuous, scale-dependent, and selective control of the smoothing process.

Computer Simulation↗
PubMed1994

Topography of alpha and theta oscillatory responses upon auditory and visual stimuli in humans.

Brain resonance phenomena and induced rhythms in the brain recently gained importance in electroencephalographic, magnetoencephalographic and cellular studies (Başar and Bullock 1992). It was hypothesized that evoked potentials are superpositions of induced rhythms caused by resonance phenomena in neural populations (Başar et al. 1992). According to Başar (1972), such resonance phenomena are reflected in the main peaks of the amplitude frequency characteristics computed from EEG responses. The present study is based on a frequency domain approach for the evaluation of topography- and modality-dependent properties of oscillatory brain responses. EEG and evoked potentials were recorded from vertex, parietal and occipital scalp locations in 24 volunteers. Two combined methods were applied: (1) amplitude frequency characteristics were computed from the transient evoked responses, and (2) frequency components of the transient responses were obtained by adaptive digital filtering. Our main goal was to investigate theta (4-7 Hz) and alpha (8-15 Hz) response components. (1) Amplitude frequency characteristics. Auditory stimuli elicited theta-alpha compound responses in the 4-11 Hz frequency band (e.g. typical peaking frequency around 7 Hz for vertex recordings). Visual stimuli elicited alpha responses (e.g. typical peaking frequency for vertex recordings around 9-12 Hz). Frequency maxima for visual stimuli thus had main peaks at higher frequency values than frequency maxima for auditory stimuli. (2) Digital filtering confirmed these results: for vertex recordings, theta vs. alpha response amplitudes were 9 muV vs 6 muV for auditory stimuli and 5 muV vs 5 muV for visual stimuli, thus confirming a shift towards higher frequencies, i.e. a more prominent contribution of the alpha range, in the case of visual stimulation. We hypothesize that these properties might reflect site- and modality-specific features of stimulus encoding in the brain in which resonance properties of neuron populations are involved. Furthermore we emphasize the utility of the systems theory approach for a better understanding of brain function by means of EPs.

Biological Clocks↗
PubMed1991

A model for neural control of gradation of muscle force.

A mathematical muscle model is presented that relates neural control signals linearly to muscle force without violating important known physiological constraints, such as the size-principle (Henneman and Mendell 1981) and non-linear twitch summation (Burke et al. 1976). This linearity implies that the neural control signals (defined as a weighted sum of activities in a nerve bundle) can be interpreted as the internal representation of total muscle force. The model allows for different relative contributions from the two force-grading mechanisms, i.e. the recruitment of motor units and the modulation of their firing frequency. It can therefore be applied to a variety of (distal and proximal) muscles. Furthermore, it permits simple mechanisms for controlling muscle force, e.g. in superposed motor tasks. The model confirms our intuitive notion that a weighted sum of activities in a nerve bundle can directly represent an external controlled variable, which in this case is exerted muscle force.

Cybernetics↗