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

Is there chaos in the brain? II. Experimental evidence and related models.

Using the concepts of chaotic dynamical systems, we present an interpretation of dynamic neural activity found in cortical and subcortical areas. The discovery of chaotic itinerancy in high-dimensional dynamical systems with and without a noise term has motivated a new interpretation of this dynamic neural activity, cast in terms of the high-dimensional transitory dynamics among "exotic" attractors. This interpretation is quite different from the conventional one, cast in terms of simple behavior on low-dimensional attractors. Skarda and Freeman (1987) presented evidence in support of the conclusion that animals cannot memorize odor without chaotic activity of neuron populations. Following their work, we study the role of chaotic dynamics in biological information processing, perception, and memory. We propose a new coding scheme of information in chaos-driven contracting systems we refer to as Cantor coding. Since these systems are found in the hippocampal formation and also in the olfactory system, the proposed coding scheme should be of biological significance. Based on these intensive studies, a hypothesis regarding the formation of episodic memory is given.

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Toward an interpretation of dynamic neural activity in terms of chaotic dynamical systems

Original Contribution: Dynamic link of memory-Chaotic memory map in nonequilibrium neural networks

1. Introduction.- 1.1 What This Book Is About.- 1.2 Statement of the Problem.- 1.3 Some Preliminary Definitions of Complexity and Organization.- 1.3.1 Complexity.- 1.3.2 Organization.- 2. Preliminaries from Nonlinear Dynamics and Statistical Physics.- 2.1 Symmetries and Conservation Principles.- 2.2 Instabilities at the Root of Broken Symmetries, Dissipation, and Irreversibility for Low-Dimensional (Not Statistical) Dynamical Systems.- 2.2.1 The Role of Gravitation.- 2.2.2 Comments on the Role of Coupling Among the Four Basic Interactions in Evolution.- 2.2.3 The Overdamped Nonlinear Oscillator: A Case of Spontaneously Breaking Symmetry.- 2.2.4 The Laser: A Case of Broken Symmetry.- 2.2.5 The Rotating Pendulum: A Case of Bifurcation Leading to Spontaneous Symmetry Breaking.- 2.2.6 Broken Symmetry Through a Hysteresis-Like Process.- 2.2.7 Essentials of Stability Theory.- a) General Criterion.- b) Specific Analyses.- 2.2.8 Behavior of a Two-Dimensional Dynamical System in the Vicinity of Singular Points (Steady States).- 2.2.9 First Encounter with Nontrivial Dissipative Systems: The Concept of the Attractor in Two Dimensions (Limit Cycle).- 2.3 Elements of Statistical Physics and Their Relevance to Evolutionary Phenomena.- 2.3.1 Some Characteristics of Stochastic Systems.- 2.3.2 Informational Entropy, Physical Entropy, Thermodynamic Entropy.- 2.3.3 Entropy of a Perfect Gas at Thermodynamic Equilibrium.- 2.3.4 Entropy of a Photon Gas at Thermodynamic Equilibrium.- 2.3.5 Elements of Newtonian Big Bang Cosmology.- 2.3.6 Expansion of a Mixture of Matter and Radiation. Differential Cooling and Entropy Production.- 2.3.7 The Concept of Complexity.- a) Structural Complexity and Its Relationship to the Stability of a System.- b) Algorithmic Complexity.- 2.4 Concluding Remarks.- 3. The Role of Spherical Electromagnetic Waves as Information Carriers.- 3.1 Radiation from Accelerated Charge in Vacuo. The Concept of "Self"-Force. Thermodynamics of Electromagnetic Radiation.- 3.1.1 Radiation in Vacuum.- 3.1.2 The Concept of Self-Force.- 3.1.3 Thermodynamics of Electromagnetic Radiation.- 3.2 Electromagnetic Wave Propagation in Dispersive Media and Lossy Media.- 3.3 Analysis of a Spherical Wave in Terms of Elemental "Rays". The Mode Theory of Wave Propagation. Excitable Modes (Degrees of Freedom) in a Closed Cavity.- 3.3.1 Spectral Decomposition of a Spherical Wave.- 3.3.2 The Wave-Guide Mode Theory of Wave Propagation.- 3.3.3 A Cavity Resonator.- 3.4 The Entropy of Electromagnetic Radiation. Information Received by an Electromagnetic Wave Impinging on a Finite Aperture. Ambiguity of Perception.- 4. Elements of Information and Coding Theory, with Applications.- 4.1 Information Transfer and the Concept of Channel Capacity for Discrete and Continuous Memoryless Signals.- 4.2 Some Ideas from Coding Theory Instrumental in Minimizing Reception Error.- 4.3 Some Efficient Coding Algorithms for Source-Channel Matching and Single-Error Detection and Correction.- 4.3.1 Coding for Source-Channel Matching.- 4.3.2 Coding for Error Detection and Correction.- 4.4 Information Sources with Memory. Markov Chains.- 4.5 Specific Examples of Some Useful Channels and Calculations of Their Capacities.- 4.5.1 Capacity of a Homogeneously Turbulent Channel.- 4.5.2 The Lossless Channel.- 4.5.3 The Deterministic Channel.- 4.5.4 The Uniform Channel.- 4.5.5 The Binary Symmetrical Channel.- 4.5.6 The Binary "Erasure" Channel.- 4.5.7 Capacity of an Optical Channel.- 4.5.8 Role of Quantum Noise in an Optical Channel.- 4.5.9 An Introduction to the "Genetic Channel" and the Genetic Code.- 4.5.10 The Phase-Locked Loop in the Absence and Presence of Noise.- 4.6 Modeling of Stochastic Time Series.- 4.7 Communication Between Two Hierarchical Systems Modeled by Controlled Markov Chains.- 4.7.1 Introduction: Elaboration of the Nature of Hierarchical Systems.- 4.7.2 Dynamics at the Base Levels Q, Q' and the Underlying Game.- 4.7.3 A Semi-Markov Chain Model for the Hierarchical Levels W and W'.- 4.7.4 The Control Problem.- a) Biological Rhythms Underlying the Games.- b) Description of the Communication and Control Processes.- c) Selection of Control Mechanisms.- 4.7.5 Computer Simulation.- 4.7.6 Biological Relevance of the Model.- 4.8 Emergence of New Hierarchical Levels in a Self-Organizing System.- 4.8.1 Formulation of the Problem.- 4.8.2 Creation of a New Hierarchical Level.- 4.8.3 A Comment on Typical Cases of "Psychosomatic Disturbances".- 5. Elements of Game Theory, with Applications.- 5.1 Constant-Sum Games.- 5.1.1 Both Players Have a Dominant Strategy.- 5.1.2 Only One Player Has a Dominant Strategy.- 5.1.3 Neither Player Has a Dominant Strategy.- 5.1.4 Mixed Strategies.- 5.2 Non-Constant-Sum Games.- 5.2.1 Non-Constant-Sum "Negotiable" Games.- 5.2.2 Non-Constant-Sum, Nonnegotiable "Paradoxical" Games.- 5.3 Competing Species.- 5.4 Survival and Extinction.- 5.5 Some Elementary Knowledge from Genetics: Selection and Fitness.- 5.6 Games Between Animals Adopting Specific Modes of Behavior (Roles). Concepts of Evolutionarily Stable Strategy.- 5.7 The Game of Competitive-Cooperative Production and Exchange. The Concept of "Parasite" at a Symbolic Level.- 5.8 Epidemiology of Rumors.- 6. Stochasticiky Due to Deterministic Dynamics in Three- or Higher-Dimensional Space: Chaos and Strange Attractors.- 6.1 A Reappraisal of Classical Statistical Mechanics. The Kolmogorov-Arnold-Moser Theorem.- 6.2 Dynamics in Three-Dimensional State Space (Three Degrees of Freedom). Steady States, Limit Cycles, Attracting Tori.- 6.3 Strange Attractors.- 6.3.1 One-Dimensional Maps on the Interval. The "Logistic" Model.- 6.3.2 Fractal Dimensionality. The Cantor Set.- 6.3.3 The Concept of the Lyapounov Exponents for the Period-Doubling and Chaotic Regimes.- 6.3.4 A Typical Three-Dimensional Strange Attractor. The Lorenz Model.- 6.3.5 The Rate of Information Production by the Lorenz Attractor.- 6.4 Parameters Characterizing the Average Behavior of Strange Attractors: Dimensions, Entropies, and Lyapounov Exponents.- 6.4.1 The Concept of Information Dimension.- 6.4.2 The Concept of Characteristic Lyapounov Exponents and Their Relation to Information Dimension.- 6.4.3 The Concept of Metric (Kolmogorov-Sinai) Entropy and Its Relation to Information Dimension.- 6.5 A Possible Role for Chaos in Reliable Information Processing.- 6.5.1 Theoretical Considerations and General Discussion.- 6.5.2 Application: The Electrical Activity of the Brain - Should It Be Chaotic?.- 6.5.3 Experimental Data from EEG Research.- 6.5.4 The Model.- 6.5.5 The Dual Role of Intermittency in Information Processing.- 6.5.6 The Origin of Conflict in Communicating Hierarchical Systems.- 6.6 Comments on the Effects of Internal Fluctuations and External Noise on the Stability Properties of Dynamical Systems.- 7. Epilogue: Relevance of Chaos to Biology and Related Fields.- 7.1 Computational Complexity.- 7.2 Towards a Dynamic Theory of Language.- 7.2.1 The Nature of the Problem.- 7.2.2 Structural and Functional Hierarchical Levels.- 7.2.3 An Evolutionary Linguistic Model: Digits and Patterns.- a) Total Entrainment.- b) Part Entrainment, Part "Jittery" Phase Locking.- c) Chaos.- 7.2.4 Unresolved Problems: Communication Between Two Hierarchical Systems.- 7.3 Concluding Remarks.- A. A View of the Role of External Noise at a Neuronal Hierarchical Level.- A.1 Introduction to the Problem.- A.2 Organization Through Weak Stationary-Amplitude Noise.- A.3 Relevance of the Model to Neuronal and Cognitive Organization.- B. On the Difficulty of Treating the Transaction Between Two Hierarchical Levels with Continuous Nonlinear Dynamics.- B.1 The Level Q of Partner I.- B.2 Homeostasis and Cross-Correlations.- B.3 The Level W of Partner I.- B.4 The Controller.- C. Noisy Entrainment of a Slightly Nonlinear Relaxation Oscillator by an External Harmonic Excitation.- C.1 General Description of the Model.- C.2 A Method for the Study of Entrainment.- C.2.1 Strict Entrainment.- C.2.2 Loose or "Jittery" Entrainment.- C.2.3 Pure "Free-Running" Oscillation.- C.2.4 Free-Running Oscillation.- C.3 Mathematical Treatment and Computer Simulation.- C.4 Behavior of the Oscillator Under an Applied Harmonic Excitation (Entrainment).- References.

Dynamics of Hierarchical Systems: An Evolutionary Approach

Chaotic dynamics of information processing: the "magic number seven plus-minus two" revisited.

The human mind possesses the unique capability of “mapping” the external (as well as part of the organism's internal) world ie it “compresses” long and complex strings of impinging environmental stimuli (“observations”) and then uses these “minimal length algorithms” in order to simulate physical phenomena‐thereby revealing the “laws of nature” In this paper we theorize that this process of “Self”‐organization and category formation is implimented via a set of coexisting (strange) attractors in the cognizant apparatus each one of which attracts (and therefore compresses) whole subsets of “initial conditions” the sum‐total of which constitute the set of external stimuli This set of the initial conditions forms the “Basin” of the attractors and the processes of partition and category formation in the mind involves the topology of the separatrixes amongst the individual subsets of the Basin We examine in particular how the information processing is mediated by the thalamocortical pacemaker of the brain and, therefore, what might be the role of EEG (which is measurable on a routine basis) in Cognition

Chaotic dynamics of information processing with relevance to cognitive brain functions

This paper considers a non-conservative pendulum-type non-linear oscillator which emulates some behavioural features of systems displaying adaptive response, or biological rhythms, under the influence of an environmental excitation (‘ Zeitgeber ’) of electromagnetic origin, for instance. Of special interest are : (a) The entraining or locking-in process of the oscillation under a sudden change Δω of the Zeitgeber frequency ; (b) the time interval within which locking-in is achieved, and (c) the evolution of the oscillation under constant excitation of varying amplitude It is shown that the range of entertainment (pull-in region) increases with the amplitude of excitation and depends on the sign of Δω, and the oscillation damps out for constant excitation beyond a certain critical amplitude The above results are deduced on a quantitative basis by introducing a method of following up the entrainment process in the plane ‘ phase-error—instantaneous frequency offset ’.

A frequency entrainment model with relevance to systems displaying adaptive behaviour

A problem of fundamental importance in the design of “self”-organizing systems is the “theoretical minimum” amount of hardware complexity (CH) necessary to drive a given functional repertoire (software complexity CS). In general it is assumed that the curve CH = f(CS) is a monotonically rising one with steepness depending on the specific “wiring mechanism” or the architecture of the given system. This conviction comes traditionally from the communication engineering practice where the act of information processing includes a sequence of an “expansion” and a “contraction” of the dimensionality of the state space, i.e., a proliferation and a subsequent compression of the degrees of freedom of the transmitted message.

John S. Nicolis

Papers

Dynamics of Hierarchical Systems: An Evolutionary Approach

Chaotic dynamics of information processing: the "magic number seven plus-minus two" revisited.

Chaotic dynamics of information processing with relevance to cognitive brain functions

A frequency entrainment model with relevance to systems displaying adaptive behaviour

The role of chaos in reliable information processing