Studies on human finger tapping neural networks by phase transition curves

Generally, the phase resetting experiments can be used to investigate the behaviors of the stable biological oscillators (e.g. circadian rhythms, biochemical oscillators, pacemaker neurons, bursting neurons). Winfree found that there are two types of phase transition curves (PTC) in the phase resetting experiments of biochemical oscillations. The one is the curve with an average slope of unity (Type 1) and is obtained for the small magnitude of perturbation. The other curve is that with a zero average slope (Type 0) and is obtained for the large magnitude of perturbation. Previously, we explained these results mathematically by the homotopy theory. In this paper, some properties of the human finger tapping neural network are studied psychologically using PTC on the basis of above theoretical results: assuming that an oscillatory neural network controls the human finger tapping, we performed two kinds of phase resetting experiments on the finger tapping. In the first experiment, we showed that the PTC was available to estimate the degree of functional interaction between the finger tapping neural network and that which controls another task. Three tasks (rapid key-pushing, rapid voicing and pattern discrimination) were chosen as the perturbation of the phase resetting experiment. Analyzing shapes of PTCs, it was found that the interaction with the key-pushing network was the largest, and that with the pattern recognition network was the smallest of the three. In the second experiment, we modified the first task as perturbation of the phase resetting experiments to investigate further the interactions between the left and the right hand motor systems. Consequently the following results were revealed. First, shapes of PTCs are very different according as subject's experiences of finger tapping. Second, the type of PTC for some subjects changes from Type 0 to Type 1 by learning. Third, the PTC tends to become Type 0 for shorter tapping periods. Fourth, neither changes of motor loads (the necessary force to push the key) nor an alternation of the tapping hand and the key-pushing hand affects the shape of PTCs.     

Dynamics and bifurcations of two coupled neural oscillators with different connection types

In this paper we present an oscillatory neural network composed of two coupled neural oscillators of the Wilson-Cowan type. Each of the oscillators describes the dynamics of average activities of excitatory and inhibitory populations of neurons. The network serves as a model for several possible network architectures. We study how the type and the strength of the connections between the oscillators affect the dynamics of the neural network. We investigate, separately from each other, four possible connection types (excitatory→excitatory, excitatory→inhibitory, inhibitory→excitatory, and inhibitory→inhibitory) and compute the corresponding bifurcation diagrams. In case of weak connections (small strength), the connection of populations of different types lead to periodicin-phase oscillations, while the connection of populations of the same type lead to periodicanti-phase oscillations. For intermediate connection strengths, the networks can enter quasiperiodic or chaotic regimes, and can also exhibit multistability. More generally, our analysis highlights the great diversity of the response of neural networks to a change of the connection strength, for different connection architectures. In the discussion, we address in particular the problem of information coding in the brain using quasiperiodic and chaotic oscillations. In modeling low levels of information processing, we propose that feature binding should be sought as a temporally coherent phase-locking of neural activity. This phase-locking is provided by one or more interacting convergent zones and does not require a central “top level” subcortical circuit (e.g. the septo-hippocampal system). We build a two layer model to show that although the application of a complex stimulus usually leads to different convergent zones with high frequency oscillations, it is nevertheless possible to synchronize these oscillations at a lower frequency level using envelope oscillations. This is interpreted as a feature binding of a complex stimulus.     

Neural Control of Interappendage Phase During Locomotion

Interappendage phasing of crayfish swimmeret movements dependsupon a central nervous system network of command, oscillator,and coordinating neurons. The command neurons serve to set thegeneral excitation level in each of the segmental oscillators.The oscillator neurons in each hemi-ganglion generate the rhythmicalternations of powerstroke and returnstroke motor neuron activity.The coordinating neurons transmit the precise timing informationabout the state of one oscillator to other oscillators. Thisinformation can serve to advance or to delay the motor burstsdriven by the other oscillators. Which effect is observed dependsupon the arrival time of the coordinating neuron discharge withinthe cycle period of the modulated oscillator. This type of modulationleads to the prediction that a stable interappendage phase canresult from situations where there is not a fixed excitabilitygradient among the segmental oscillators. This prediction hasbeen verified using a cut command neuron preparation.     

Synchronization in a neural network of phase oscillators with the central element

A neural network model is considered which is designed as a system of phase oscillators and contains the central oscillator and peripheral oscillators which interact via the central oscillator. The regime of partial synchronization was studied when current frequencies of the central oscillator and one group of peripheral oscillators are near to each other while current frequencies of other peripheral oscillators are far from being synchronized with the central oscillator. Approximation formulas for the average frequency of the central oscillator in the regime of partial synchronization are derived, and results of computation experiments are presented which characterize the accuracy of the approximation.     

Entrainment ranges of forced phase oscillators

In the vertebrate spinal cord, a neural circuit called the central pattern generator produces the basic locomotory rhythm. Short and long distance intersegmental connections serve to maintain coordination along the length of the body. As a way of examining the influence of such connections, we consider a model of a chain of coupled phase oscillators in which one oscillator receives a periodic forcing stimulus. For a certain range of forcing frequencies, the chain will match the stimulus frequency, a phenomenon called entrainment. Motivated by recent experiments in lampreys, we derive analytical expressions for the range of forcing frequencies that entrain the chain, and how that range depends on the forcing location. For short intersegmental connections, in which an oscillator is connected only to its nearest neighbors, we describe two ways in which entrainment is lost: internally, in which oscillators within the chain no longer oscillate at the same frequency; and externally, in which the the chain no longer has the same frequency as the forcing. By analyzing chains in which every oscillator is connected to every other oscillator (i.e., all-to-all connections), we show that the presence of connections with lengths greater than one do not necessarily change the entrainment ranges based on the nearest–neighbor model. We derive a criterion for the ratio of connection strengths under which the connections of length greater than one do not change the entrainment ranges produced in the nearest–neighbor model, provided entrainment is lost externally. However, when this criterion holds, the range of entrained frequencies is a monotonic function of forcing location, unlike experimental results, in which entrainment ranges are larger near the middle of the chain than at the ends. Numerically, we show that similar non-monotonic entrainment ranges are possible if the ratio criterion does not hold, suggesting that in the lamprey central pattern generator, intersegmental connection strengths are not a simple function of the connection length.     

The topology of phase response curves induced by single and paired stimuli in spontaneously oscillating chick heart cell aggregates.

The topological properties of the phase resetting of biological oscillators by an isolated stimulus delivered at various phases of the cycle depend on whether the stimulus is "weak" or "strong." When multiple stimuli are delivered to the oscillator, the response to stimulation also depends on the time between the stimuli, and the rate at which the oscillator returns to an underlying limit cycle attractor. If the time between two consecutive "weak" stimuli is sufficiently short, the effects produced by the pair of stimuli may be characteristic of a single "strong" stimulus. These results are demonstrated in a model experimental system, spontaneously beating aggregates of cells derived from embryonic chick heart, and are illustrated by consideration of a simple theoretical model of nonlinear oscillators, the Poincaré oscillator.     

Walking is not like reaching: evidence from periodic mechanical perturbations

The control architecture underlying human reaching has been established, at least in broad outline. However, despite extensive research, the control architecture underlying human locomotion remains unclear. Some studies show evidence of high-level control focused on lower-limb trajectories; others suggest that nonlinear oscillators such as lower-level rhythmic central pattern generators (CPGs) play a significant role. To resolve this ambiguity, we reasoned that if a nonlinear oscillator contributes to locomotor control, human walking should exhibit dynamic entrainment to periodic mechanical perturbation; entrainment is a distinctive behavior of nonlinear oscillators. Here we present the first behavioral evidence that nonlinear neuro-mechanical oscillators contribute to the production of human walking, albeit weakly. As unimpaired human subjects walked at constant speed, we applied periodic torque pulses to the ankle at periods different from their preferred cadence. The gait period of 18 out of 19 subjects entrained to this mechanical perturbation, converging to match that of the perturbation. Significantly, entrainment occurred only if the perturbation period was close to subjects' preferred walking cadence: it exhibited a narrow basin of entrainment. Further, regardless of the phase within the walking cycle at which perturbation was initiated, subjects' gait synchronized or phase-locked with the mechanical perturbation at a phase of gait where it assisted propulsion. These results were affected neither by auditory feedback nor by a distractor task. However, the convergence to phase-locking was slow. These characteristics indicate that nonlinear neuro-mechanical oscillators make at most a modest contribution to human walking. Our results suggest that human locomotor control is not organized as in reaching to meet a predominantly kinematic specification, but is hierarchically organized with a semi-autonomous peripheral oscillator operating under episodic supervisory control.     

The mathematical modeling of entrained biological oscillators

In this paper perturbation methods are used for the mathematical analysis of coupled relaxation oscillators. This study covers entrainment by an external periodic stimulus as well as mutual entrainment of coupled oscillators with different limit cycles. The oscillators are of a type one meets in the modeling of biological oscillators by chemical reactions and electronic circuits. Special attention is given to entrainment different from 1∶1. The results relate to phenomena occurring in physiological experiments, such as the periodic stimulation of neural and cardiac cells, and in the non-regular functioning of organs and organisms, such as the AV-block in the heart.     

Estimation of the transmission time of stimulus-locked responses: modelling and stochastic phase resetting analysis

A model of two coupled phase oscillators is studied, where both oscillators are subject to random forces but only one oscillator is repetitively stimulated with a pulsatile stimulus. A pulse causes a reset, which is transmitted to the other oscillator via the coupling. The transmission time of the cross-trial (CT) averaged responses, i.e. the difference in time between the maxima of the CT averaged responses of both oscillators differs from the time difference between the maxima of the oscillators' resets. In fact, the transmission time of the CT averaged responses directly corresponds to the phase difference in the stable synchronized state with integer multiples of the oscillators' mean period added to it. With CT averaged responses it is impossible to reliably estimate the time elapsing, owing to the stimulus' action being transmitted between the two oscillators.     

Transient and steady state phase response curves of limit cycle oscillators

The experiment of phase shifts resulting from discrete perturbations of stable biological rhythms has been carried out to study entrainment behavior of oscillators. There are two kinds of phase response curves, which are measured in experiments, according to as one measures the phase shifts immediately or long after the perturbation. The former is the first transient phase response curve and the latter is the steady state phase response curve. We redefine both curves within the framework of dynamical system theory and homotopy theory. Several topological properties of both curves are clarified. Consequently, it is shown that we must compare the shapes of both two phase response curves to investigate the inner structures of biological oscillators. Moreover, we prove that a single limit cycle oscillator involving only two variables cannot simulate transient resetting behavior reported by Pittendrigh and Minis (1964). In other words, the circadian oscillator of Drosophila pseudoobscura does not consist of a single oscillator of two variables. Finally we show that a model which consists of two limit cycle oscillators is able to simulate qualitatively the phase response curves of Drosophila.     

Amplitude model for the effects of mutations and temperature on period and phase resetting of the Neurospora circadian oscillator.

This paper analyzes published and unpublished data on phase resetting of the circadian oscillator in the fungus Neurospora crassa and demonstrates a correlation between period and resetting behavior in several mutants with altered periods: As the period increases, the apparent sensitivity to resetting by light and by cycloheximide decreases. Sensitivity to resetting by temperature pulses may also decrease. We suggest that these mutations affect the amplitude of the oscillator and that a change in amplitude is responsible for the observed changes in both period and resetting by several stimuli. As a secondary hypothesis, we propose that temperature compensation of period in Neurospora can be explained by changes in amplitude: As temperature increases, the compensation mechanism may increase the amplitude of the oscillator to maintain a constant period. A number of testable predictions arising from these two hypotheses are discussed. To demonstrate these hypotheses, a mathematical model of a time-delay oscillator is presented in which both period and amplitude can be increased by a change in a single parameter. The model exhibits the predicted resetting behavior: With a standard perturbation, a smaller amplitude produces type 0 resetting and a larger amplitude produces type 1 resetting. Correlations between period, amplitude, and resetting can also be demonstrated in other types of oscillators. Examples of correlated changes in period and resetting behavior in Drosophila and hamsters raise the possibility that amplitude changes are a general phenomenon in circadian oscillators.     

Quantum-like model of behavioral response computation using neural oscillators

In this paper we propose the use of neural interference as the origin of quantum-like effects in the brain. We do so by using a neural oscillator model consistent with neurophysiological data. The model used was shown elsewhere to reproduce well the predictions of behavioral stimulus-response theory. The quantum-like effects are brought about by the spreading activation of incompatible oscillators, leading to an interference-like effect mediated by inhibitory and excitatory synapses.     

Object selection by an oscillatory neural network

We describe a new solution to the problem of consecutive selection of objects in a visual scene by an oscillatory neural network with the global interaction realised through a central executive element (central oscillator). The frequency coding is used to represent greyscale images in the network. The functioning of the network is based on three main principles: (1) the synchronisation of oscillators via phase-locking, (2) adaptation of the natural frequency of the central oscillator, and (3) resonant increase of the amplitudes of the oscillators which work in-phase with the central oscillator. Examples of network simulations are presented to show the reliability of the results of consecutive selection of objects under conditions of constant and varying brightness of the objects.     

Fast and robust image segmentation by small-world neural oscillator networks

Inspired by the temporal correlation theory of brain functions, researchers have presented a number of neural oscillator networks to implement visual scene segmentation problems. Recently, it is shown that many biological neural networks are typical small-world networks. In this paper, we propose and investigate two small-world models derived from the well-known LEGION (locally excitatory and globally inhibitory oscillator network) model. To form a small-world network, we add a proper proportion of unidirectional shortcuts (random long-range connections) to the original LEGION model. With local connections and shortcuts, the neural oscillators can not only communicate with neighbors but also exchange phase information with remote partners. Model 1 introduces excitatory shortcuts to enhance the synchronization within an oscillator group representing the same object. Model 2 goes further to replace the global inhibitor with a sparse set of inhibitory shortcuts. Simulation results indicate that the proposed small-world models could achieve synchronization faster than the original LEGION model and are more likely to bind disconnected image regions belonging together. In addition, we argue that these two models are more biologically plausible.     

Multiple pulse interactions and averaging in systems of coupled neural oscillators

Oscillators coupled strongly are capable of complicated behavior which may be pathological for biological control systems. Nevertheless, strong coupling may be needed to prevent asynchrony. We discuss how some neural networks may be designed to achieve only simple locking behavior when the coupling is strong. The design is based on the fact that the method of averaging produces equations that are capable only of locking or drift, not pathological complexity. Furthermore, it is shown that oscillators that interact by means of multiple pulses per cycle, dispersed around the cycle, behave like averaged equations, even if the number of pulses is small. We discuss the biological intuition behind this scheme, and show numerically that it works when the oscillators are taken to be composites, each unit of which is governed by a well-known model of a neural oscillator. Finally, we describe numerical methods for computing from equations for coupled limit cycle oscillators the averaged coupling functions of our theory.Research partially supported by the National Science Foundation under grants DMS 8796235 and DMS 8701405 and the Air Force Office of Scientific Research under University Research Contract F 49620-C-0131 to Northeastern University     

An oscillator network exhibiting a long-lasting response to an external signal

This study proposes an oscillator network to model the long-lasting responses observed in neural circuits. The responses of the proposed network model are represented by the temporal synchronization of the oscillators. The response duration does not depend on the natural frequency of the oscillators, which allows the responses to last much longer than the oscillation period of the oscillators. We can control the response duration by tuning the connection strengths between the oscillators and the external signal that triggers the responses. It is possible to break and restart the responses regardless of the way in which the oscillators are connected.