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Recently, it was found that rhythmic movements (e.g. locomotion, swimmeret beating) are controlled by mutually coupled endogeneous neural oscillators (Kennedy and Davis, 1977; Pearson and Iles, 1973; Stein, 1974; Shik and Orlovsky, 1976; Grillner and Zangger, 1979). Meanwhile, it has been found out that the phase resetting experiment is useful to investigate the interaction of neural oscillators (Perkel et al., 1963; Stein, 1974). In the preceding paper (Yamanishi et al., 1979), we studied the functional interaction between the neural oscillatory which is assumed to control finger tapping and the neural networks which control some tasks. The tasks were imposed on the subject as the perturbation of the phase resetting experiment. In this paper, we investigate the control mechanism of the coordinated finger tapping by both hands. First, the subjects were instructed to coordinate the finger tapping by both hands so as to keep the phase difference between two hands constant. The performance was evaluated by a systematic error and a standard deviation of phase differences. Second, we propose two coupled neural oscillators as a model for the coordinated finger tapping. Dynamical behavior of the model system is analyzed by using phase transition curves which were measured on one hand finger tapping in the preivous experiment (Yamanishi et al., 1979). Prediction by the model is in good agreement with the results of the experiments. Therefore, it is suggested that the neural mechanism which controls the coordinated finger tapping may be composed of a coupled system of two neural oscillators each of which controls the right and the left finger tapping respectively.

Model calculations are presented for the several properties of the development of the retinotectal projection in amphibians and fishes, using the Gierer-Meinhardt equations. One of these properties is the maintenance of topographic mapping between the retina and the tectum during their development despite the fact that the two tissues grow in morphologically different ways. Another is the existence of a "critical period", at which the coordinates of the retina with respect to the tectum are irrevocably determined. It is assumed that the connections between the retinal and the tectal cells are made on the correspondence of positional markers which are given as a form of the distribution of a specific activator, the dynamics of which is described by the Gierer-Meinhardt equations. The monotonic distributions of the activator and the existence of the critical period are shown by a computer simulation of the proliferating retina. Several changes of the retinotectal projection after surgical operations on the retina or the tectum are also explained.

A skeleton photoperiod consists of two short pulses which are applied on the circadian oscillator at times corresponding to the beginning and to the end of a continuous light stimulus. To study several problems in entrainment of circadian rhythms by skeleton photoperiods, we develop a simple diagrammatic solution of the steady state entrainment making use of phase transition curves which are directly gotten from phase response curves. The graphical method is simple and systematic to study entrainment by light cycles with various day lengths. As the method is also intuitive, we can easily examine three problems. (1) In Drosophila the phase relation (psi) between rhythm and light cycle is a continuous function of day length of skeleton photoperiods up to about 12 h, but a marked discontinuity (psi-jump) sets in between 13 and 14 h. By the diagrammatic method we find that psi-jump is mathematically a bifurcation phenomenon. (2) The action of photoperiods up to about 12 h is fully simulated by two 15-min skeleton pulses. Do 3-min skeleton pulses imitate the complete photoperiods? We find that pulse width is arbitrary to some extent. (3) Why skeleton photoperiods up to about 12 h are good models of complete photoperiods? The reason is the small amplitude and the nearly symmetrical form of phase response curves in the subjective day.

The human finger tapping was studied psychologically by the phase transition curves. We assumed that some oscillatory neural network controls the tapping. One of our future works on the human finger tapping is to devise a method to change the magnitude of perturbation continuously in phase resetting experiments of a constant tapping interval with a same kind of task. If this is achieved, we can obtain a phase transition surface (i.e. the new phase as a function of the old phase and the magnitude of perturbation). From the shape of a phase transition surface, the intrinsic properties of the oscillatory network which controls the tapping can be studied (Kawato et al., 1978).