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The purpose of this work is a unified and general treatment of activity in neural networks from a mathematical pOint of view. Possible applications of the theory presented are indica ted throughout the text. However, they are not explored in de tail for two reasons : first, the universal character of n- ral activity in nearly all animals requires some type of a general approach~ secondly, the mathematical perspicuity would suffer if too many experimental details and empirical peculiarities were interspersed among the mathematical investigation. A guide to many applications is supplied by the references concerning a variety of specific issues. Of course the theory does not aim at covering all individual problems. Moreover there are other approaches to neural network theory (see e.g. Poggio-Torre, 1978) based on the different lev els at which the nervous system may be viewed. The theory is a deterministic one reflecting the average be havior of neurons or neuron pools. In this respect the essay is written in the spirit of the work of Cowan, Feldman, and Wilson (see sect. 2.2). The networks are described by systems of nonlinear integral equations. Therefore the paper can also be read as a course in nonlinear system theory. The interpretation of the elements as neurons is not a necessary one. However, for vividness the mathematical results are often expressed in neurophysiological terms, such as excitation, inhibition, membrane potentials, and impulse frequencies. The nonlinearities are essential constituents of the theory.
Table of Contents1. The general form of a neural network.- 1.1 Introduction.- 1.2 The transformation of impulse frequencies into generator potentials (intercellular transmission).- 1.3 The transformation of generator potentials into impulse frequencies (intracellular transmission).- 1.4 Structures of neural networks.- 2. On the relations between several models for neural networks.- 2.1 The retinal network of Limulus polyphemus; the Hartline-Ratliff equations.- 2.2 A statistical approach: activities in coupled neuron pools; models of Cowan, Feldman, Wilson.- 2.3 Discrete models.- a) the logical neurons of McCulloch and Pitts.- b) discrete time and continuous states.- 3. Existence and uniqueness of time dependent solutions.- 4. Steady states of finite-dimensional networks.- 4.1 Existence problem.- 4.2 The number of steady states.- a) single neurons.- b) pairs of neurons.- c) arbitrarily many neurons.- 4.3 Input-output behavior of stationary networks.- 4.4 An example of spatial hysteresis.- 5. Local stability analysis of nets with finitely many neurons.- 5.1 Introduction.- 5.2 The linearization principle.- 5.3 Some simple general criteria for asymptotic stability.- 5.4 Single neurons.- 5.5 Pairs of neurons.- 5.6 Closed chains of neurons.- 6. Oscillations in nets with finitely many neurons.- 6.1 Introduction.- 6.2 Oscillations in closed chains of neurons.- 6.3 Oscillations in systems with delays.- 7. Homogeneous tissues with lateral excitation or lateral inhibition.- 7.1 Introduction.- 7.2 The model.- 7.3 Stationary solutions and their stability.- 7.4 Thresholds in bistable tissues.- 7.5 Traveling fronts.- 7.6 Diverging pairs of fronts (spread of excitation or depression).- 8. Homogeneous tissues with lateral excitation and self-inhibition.- 8.1 The model.- 8.2 Bulk oscillations.- 8.3 Traveling pulses (solitary waves).- 8.4 Traveling wave trains.- 9. Homogeneous tissues with lateral inhibition and self- or local excitation.- 9.1 The model.- 9.2 Periodic spatial patterns.- 9.3 Stability.- 9.4 Miscellaneous topics.- References.- List of symbols.