Delay, noise and phase locking in pulse coupled neural networks.

This paper studies the effect of several delay times and noise on the stability of the phase-locked state in the lighthouse model and the integrate and fire model of a pulse coupled neural network. The coupling between neurons may be arbitrary. In both models the increase of delay times leads to a weakening of the stability and to the occurrence of relaxation oscillations.

The cardiovascular system as coupled oscillators?

Based on physiological knowledge, and on an analysis of signals related to its dynamics, we propose a model of the cardiovascular system. It consists of coupled oscillators. Each of them describes one of the subsystems involved in the regulation of one passage of blood through the circulatory system. The flow of blood through the system of closed tubes-the blood vessels-is described by wave equations.

Noise in the brain: a physical network model.

In the brain as in any other open physical systems, noise is inevitable. We present an explicit model of an active physical system that is borrowed from laser physics and allows us to establish the properties of the fluctuating forces that cause noise in the system. It is shown how the cooperation of the individual parts of a system (atoms or neurons) can considerably reduce the noise level. In particular we determine the correlation function between the individual parts. The basic equations can be transformed in such a way that a close analogy with typical equations of neural nets are obtained. In particular, the nonlinear properties of neurons described by the sigmoid function are well captured. Propagation of excitation in axons and dendrites is represented by a linear equation, where we consider both a bandwidth filter and more or less free propagation. In the latter case, a close analogy with an equation for neural activity in the sense of Nunez and established by Jirsa and Haken is pointed out.

Information compression in biological systems.

In biological systems a high coordination between their individual parts occurs. The concept of coordination can be given a rigorous mathematical basis by the concept of order parameters and the slaving principle. We calculate the information of the total system in terms of order parameters and slaved subsystems. When qualitative changes of a system happen, the information change is given by that of the order parameters alone and may amount to few bits only.

Stereo vision by self-organization.

We propose a new algorithm for stereoscopic depth perception, where the depth map is the momentary state of a dynamic process. To each image point we assign a set of possible disparity values. In a dynamic process with competition and cooperation, the correct disparity value is selected for each image point. Therefore, we solve the correspondence problem by a dynamic, self-organizing process, the structure of which shows analogies to the human visual system. The algorithm can be implemented in a massive parallel manner and yields good results for either artificial or natural images.

Synchronized oscillations in the visual cortex--a synergetic model.

We present an oscillator network model for the synchronization of oscillatory neuronal activity underlying visual processing. The single neuron is modeled by means of a limit cycle oscillator with an eigenfrequency corresponding to visual stimulation. The eigenfrequency may be time dependent. The mutual coupling strengths are unsymmetrical and activity dependent, and they scatter within the network. Synchronized clusters (groups) of neurons emerge in the network due to the visual stimulation. The different clusters correspond to different visual stimuli. There is no limitation of the number of stimuli. Distinct clusters do not perturb each other, although the coupling strength between all model neurons is of the same order of magnitude. Our analysis is not restricted to weak coupling strength. The scatter of the couplings causes shifts of the cluster frequencies. The model's behavior is compared with the experimental findings. The coupling mechanism is extended in order to model the influence of bicucullin upon the neural network. We additionally investigate repulsive couplings, which lead to constant phase differences between clusters of the same frequency. Finally, we consider the problem of selective attention from the viewpoint of our model.

The impact of fluctuations on the recognition of ambiguous patterns.

The recognition of ambiguous patterns by humans is modelled by coupled differential equations which describe the formation of percepts by means of order parameters which in turn are determined by the saturation of attention parameters. We study the impact of fluctuations on the attention parameters and thus indirectly on the recognition of ambiguous patterns. Excellent agreement with psychophysical experimental results by Price on the transient behaviour of switching times and by Borsellino et al. on the distribution function of switching times as function of the size of the visual field is obtained. Our model allows us to deal also with the shift of width and position of the distribution function with respect to slow and fast observers in the sense of Borsellino.

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.

Pattern formation in morphogenesis. Analytical treatment of the Gierer-Meinhardt model on a sphere.

We first treat the Gierer-Meinhardt equations by linear stability analysis to determine the critical parameter, at which the homogeneous distributions of activator and inhibitor concentrations become unstable. We find two types of instabilities: one leading to spatial pattern formation and another one leading to temporal oscillations. We consider the case where two instabilities are present. Using the method of generalized Ginzburg-Landau equations introduced earlier we then analyze the nonlinear equations. As we are mainly interested in spatial pattern formation on a sphere we consider the problem under an appropriate constraint. Combining the two occurring solutions we find patterns well-known in biology, such as a gradient system and temporal oscillations.

Co-operative dynamics in organelles.

Some organelles produce elementary life phenomena which are characterized by the spontaneous formation and/or maintenance of ordered macroscopic dynamics like e.g. the shortening of sarcomeres in striated muscle and the transmission of electrical impulses in an axon. It has been widely accepted that such organelles are organized molecular systems where molecular elements work independently under constraint of a more or less rigid and regular structure of the system. On the other hand, such organelles should be regarded as self-organizing systems if the ordered macroscopic dynamics are self-organized. As the macroscopic dynamics gradually emerge, the microscopic dynamics of its elements become linked to each other through a feedback loop. It is crucial for the feedback loop to operate that the macroscopic dynamics are "free" in their behavior. In the present paper, it is pointed out that the traditional view of independent molecular elements has been obtained from experiments in which, by means of external constraint, the macroscopic dynamics is "clamped". Under such conditions, the self-organizing system may behave as an organized one. Based on synergetics we propose criterions for proving self-organizing systems, and, by applying the criterions, we conclude that skeletal muscle actomysin is a co-operative element in the sense of self-organization.

A stochastic theory of phase transitions in human hand movement.

The order parameter equation for the relative phase of correlated hand movements, derived in a previous paper by Haken et al. (1985), is extended to a time-dependent stochastic differential equation. Its solutions are determined close to stationary points and for the transition region. Remarkably good agreement between this theory and recent experiments done by Kelso and Scholz (1985) is found, and new predictions are offered.

A theoretical model of phase transitions in human hand movements.

Earlier experimental studies by one of us (Kelso, 1981a, 1984) have shown that abrupt phase transitions occur in human hand movements under the influence of scalar changes in cycling frequency. Beyond a critical frequency the originally prepared out-of-phase, antisymmetric mode is replaced by a symmetrical, in-phase mode involving simultaneous activation of homologous muscle groups. Qualitatively, these phase transitions are analogous to gait shifts in animal locomotion as well as phenomena common to other physical and biological systems in which new "modes" or spatiotemporal patterns arise when the system is parametrically scaled beyond its equilibrium state (Haken, 1983). In this paper a theoretical model, using concepts central to the interdisciplinary field of synergetics and nonlinear oscillator theory, is developed, which reproduces (among other features) the dramatic change in coordinative pattern observed between the hands.