Introduction to a Theoretical Mechanics of Neural Networks

The goal of this book is to define the foundations for the theoretical approach to the electrical activity of the biological neural networks of the nervous system and brain. The book deals progressively with the physical basis of neuroelectric signalling; comparative input-output firing levels in neurons; coordinated dynamic firing patterns and memory representation in local neural networks; and structural-functional principles of organization at neuronal junctions and composite networks, including tentative theoretical models of the composite networks of the hippocampus and cerebral cortex.

Primary Orientation of This Book

The book strives to provide the conceptual and mathematical foundations of a theoretical mechanics for neuroelectric signalling over these several essential levels of organization.

The primary qualities offered by this work are

1. its foundations in explicit physical processes,

2. its formulations in terms of operational engineering design, including the focus on the few central causal driving agents and hierarchical relationing of these agents,

3. broad theoretical syntheses, sometimes over wide areas and sometimes clearly speculative

Each of these qualities speaks to essential needs within the scope of the current neural and brain sciences.

Perhaps the single most characteristic quality of the structure and operation of the brain is its tremendous richness and complexity. Virtually any of its operations, ranging from molecular through psychological levels is endowed with overwhelming richness of highly ordered detail, variability, and tantalizing combinations of autonomy and dependence on other levels of functioning. Any theory or model for any neural process is immediately guilty of the omission of some subset of ingredients that influence the system in some contexts. Sorely needed are clearer perspectives for placing the various neurobiological agents and their rich worlds of detail within carefully structured comprehensive, yet succinct, theoretical formulations, so that the operational significance of the particular neurobiological ingredients and their parameters can be seen in their proper places within the scheme of things. This concept of engineering operational design, with its focus on the few central causal physical agents and their hierarchical relationships, carefully housing the important neurobiological parameters, permeates all sections of this book.

For example, a serious central weakness across much of current brain research is the absence of common foundations among different contributing disciplines. In the field of neural networks for example, there is a very wide chasm concerning notions of foundational starting grounds between experimental neuroscientists and computational neural network theorists and simulators. This book claims that this is precisely because the central operational physical mechanisms that drive the neuroelectric processes studied experimentally have not been adequately incorporated in the computational neural net field in explicit relationship to the physiological processes they represent. The necessary knowledge has been present for some years, but the significance of the physical foundations for providing a common meeting ground and linking computational and theoretical studies to the physiological experimental world has simply not been recognized.

The rich complexity and high diversity of the various individual fields and disciplines sometimes obscures our vision of the essential operative principles, and their implications. This book attempts to strike a critical essential balance between the world of overwhelmingly rich biological detail and diversity, on the one hand, and obscuring operational, computational, and mathematical complexity, on the other, by resting on the critical fulcrum of succinct and simple, but accurate description of the essential physical processes.

We live in a time when perhaps 90% or more of all the scientists who ever lived are currently active. This has produced a tremendous degree of specialization. Many bureaucratic and some scientific pressures reinforce this specialization. Broad theoretical syntheses across disciplines are rare, particularly in the neural and brain sciences, where any such attempt is certain to contain some errors of detail and likely to contain some substantive conceptual misjudgments. Further, the neural and brain sciences are virtually 100% experimental. Current research, textbooks, thought processes, conferences, entire fields are conceived within the milieu, momentum, and constraints of experimental facts, experimental equipment, experimental paradigms. In the conventional scientific wisdom of "theory guides, experiment decides," there is a minimum of broad speculative theory in the neural sciences. The situation seems somewhat analogous to the state of the science of motion and mechanics before Newton. The deepest ambition of this book is to provide the initial formulations of a Newtonian-like mechanics for neuroelectric signalling.

At even broader levels in the neural and brain sciences, we should look forward to the time when psychologically directed theories, in, say, the various streams of psychiatry and psychoanalysis, can be linked more directly to operations of neuroelectric signalling in the brain. For such broad theories to be truly well grounded, their descriptions of brain operations should be rooted fundamentally in operative physical agents within the physiology, cast in the appropriate hierarchical structure.

Even short of these admittedly grand objectives, good, broad speculative theory can excite the imagination to visualize more broadly, and penetrate more deeply. The view of this book is that scientific knowledge within a given area is best contained within and in relation to succinct theoretical formulations of the essential operative processes in the area. The view here is further in concordance with the Popperian view that in a developing field it is the responsibility of theories to push conjectures over places of uncertainty so as to encourage the subsequent sharpening or perhaps replacement of the original speculations with new ones more in accordance with observations prompted by the original speculative views.

Structure and Themes of this Book

The first part of the book develops the conceptual and mathematical structure of the mechanics of the electrical signals of single neurons in terms of their underlying physical causal agents. The goal here is to provide a Newtonian-like formulation of this most fundamental level of the dynamics of neural networks.

Chapter 2 discusses the conceptual hierarchical engineering design envisioned for this formulation and its parallel to the Newtonian formulation of mechanics. The chapter also provides a brief, qualitative overview of principles of neural signalling. Subsequent chapters of the section deal with the biophysical principles of ionic fluxes and balances and the membrane equation for neuroelectric signals. This equation is seen here as the central governing equation of neuroelectric signalling, and the analog in this domain to Newton's second law of motion. Chapter 5 shows how to produce systematic theoretical formulations of neuroelectric signalling in entire neurons with arbitrary dendritic morphologies. This effectively integrates the membrane equation over space. These early chapters show that spatially and temporally localized selective modulations in membrane permeabilities are the central ingredient used by the nervous system in controlling its neuroelectric signalling. Chapter 6 shows several ways to bring descriptions of these permeability modulations into the theoretical formulation. This includes a particular and special but useful representation of the mechanism of repetitive firing in neurons based on principles of accommodation and postfiring refractoriness (adaptation). Chapter 7 shows how the theoretical mechanics developed in this section can be formulated in succinct, computer-efficient simulation programs.

The second part develops a theory of dynamic similarity for the comparative input-output firing behavior of neurons, expressed in terms of a few universal nondimensional characteristic parameters. This theory is prompted in part by the recognition that, although the nervous system utilizes great elegance in specifying particular classes of cell types to various regions, differing in size, shape, physiology, and distributions of input, we generally have very little idea as to the comparative principles of design at work here. Why are neurons involved in a given function in one particular region of the brain different from those in another function in another region? The dynamic similarity theory approach allows one to characterize and compare neurons in terms of universal nondimensional numbers, which represent the effective mechanical combination of their constituent physiological and morphological parameters.

Part III presents a model for the coordinated firing patterns used to represent meaning in neural populations. The current general view of neuroelectric signalling remains unduly limited to single-neuron concepts. The model presented here is intended to explicitly represent the language by which neuronal populations speak to each other as organic wholes. It is called the sequential configuration theory. Part III applies this model to describe the anatomical representation and dynamic recall of memory traces in local...