A universal instability of many-dimensional oscillator systems

Recent “connectionist” models provide a new explanatory alternative to the digital computer as a model for brain function. Evidence from our EEG research on the olfactory bulb suggests that the brain may indeed use computational mechanisms like those found in connectionist models. In the present paper we discuss our data and develop a model to describe the neural dynamics responsible for odor recognition and discrimination. The results indicate the existence of sensory- and motor-specific information in the spatial dimension of EEG activity and call for new physiological metaphors and techniques of analysis. Special emphasis is placed in our model on chaotic neural activity. We hypothesize that chaotic behavior serves as the essential ground state for the neural perceptual apparatus, and we propose a mechanism for acquiring new forms of patterned activity corresponding to new learned odors. Finally, some of the implications of our neural model for behavioral theories are briefly discussed. Our research, in concert with the connectionist work, encourages a reevaluation of explanatory models that are based only on the digital computer metaphor.

How brains make chaos in order to make sense of the world

How brains make chaos in order to make sense of the world. BBS 10:161-195. Response

The modification of mammalian membrane polyunsaturated fatty acid composition in relation to membrane fluidity and function

The term soliton has recently been coined to describe a pulselike nonlinear wave (solitary wave) which emerges from a collision with a similar pulse having unchanged shape and speed. To date at least seven distinct wave systems, representing a wide range of applications in applied science, have been found to exhibit such solutions. This review paper covers the current status of soliton research, paying particular attention to the very important "inverse method" whereby the initial value problem for a nonlinear wave system can be solved exactly through a succession of linear calculations.

The soliton: A new concept in applied science

Alternating periodic and chaotic regimes in a chemical reaction — Experiment and theory

This paper presents a variety of computer g.enerated evidence indicating that the Toda lattice behaves remarkably like· an integrable, nonlinear system, where here integrability means that the system Hamiltonian can be brought to an obviously integrable form. In particular, we investigate Toda lattices having three and six particles, using periodic boundary conditions. While computer calculations cannot rigorously prove integrability,. the evidence presented here is sufficiently strong to provide incentive for seeking the general, closed form, analytic so­ lution or, at least, some approximation to it. § I. Introduction The. T oda lattice 1J is of considerable interest to physical scientists because it serves as an example of a nonlinear lattice system which may rigorously be shown to propagafe certain wave forms without change of shape. Moreover the propagation of such unchanging wave forms has been observed experimentally 2> in various physical systems. Additionally in the continuum limit, the T thereby connecting the theory1> o{ the Toda lattice with the KdV soliton theory8> developed by Kruskal, Zabusky, and others. For the KdV equation the unchanging wave form, called a soliton, has been shown8> both theoretically and empirically (on a computer) to be remarkably stable. · Indeed, a recent paper by Zakharov and Faddeev4> definitively establishes the' validity and the source of this stability. In particular these authors' prove that· the KdV equation is a completely integrable Hamiltonian system, where integrability 6> here means that the system Hamiltonian can be reduced to an obvi­ ously integrable (or solvable) form6> by a canonical transformation analytic in the position and momentum variables. Because of the intimate connection7> between the Toda lattice and the KdV equation, the Zakharov-Faddeev result suggests that the Toda lattice may also be integrable and its associated solutions also highly stable; however, extreme caution is needed here. For the class of nonlinear Hamiltonian oscillator systems to which the Toda lattice belongs, Siegel 8) has shown that non-integrability is overwhelmingly the general case. Moreover, Saito9> and coworkers10> have presented computer evi

On the Integrability of the Toda Lattice

The first observation of quasiperiodic oscillations in a model of the Belousov–Zhabotinskii reaction is presented and the evolution along two parameter paths of the associated torus is described. Comparison is made with experiment and the hysteresis–Hopf normal form. The model dynamics is shown to be more complex than that of the normal form. The torus is also shown to be related to two other features of the BZ reaction: (1) the transition from small amplitude to mixed‐mode oscillations and (2) alternating periodic–chaotic sequences.

Observations of a torus in a model of the Belousov–Zhabotinskii reaction

In experiments on a stirred flow chemical reactor we have observed a sequence of periodic and chaotic regimes that alternate as a function of the flow rate. The periodic regimes are characterized by power spectra consisting of a single fundamental frequency component and harmonics and by limit cycle attractors, while the chaotic regimes are characterized by broadband power spectra and strange attractors. One of the chaotic regimes has been studied in detail and has been found to correspond to a smooth one-dimensional map that has a single maximum and a positive Lyapunov exponent.

Jack S. Turner

Papers

Alternating periodic and chaotic regimes in a chemical reaction — Experiment and theory

On the Integrability of the Toda Lattice

Observations of a torus in a model of the Belousov–Zhabotinskii reaction

Experimental observations of complex dynamics in a chemical reaction

Discontinuous thermotropic response of Tetrahymena membrane lipids correlated with specific lipid compositional changes.