A NONLINEAR DYNAMICS PERSPECTIVE OF WOLFRAM'S NEW KIND OF SCIENCE PART V: FRACTALS EVERYWHERE

This fifth installment is devoted to an in-depth study of CA Characteristic Functions, a unified global representation for all 256 one-dimensional Cellular Automata local rules. Except for eight rather special local rules whose global dynamics are described by an affine (mod 1) function of only one binary cell state variable, all characteristic functions exhibit a fractal geometry where self-similar two-dimensional substructures manifest themselves, ad infinitum, as the number of cells (I + 1) → ∞. In addition to a complete gallery of time-1 characteristic functions for all 256 local rules, an accompanying table of explicit formulas is given for generating these characteristic functions directly from binary bit-strings, as in a digital-to-analog converter. To illustrate the potential applications of these fundamental formulas, we prove rigorously that the "right-copycat" local rule [Formula: see text] is equivalent globally to the classic "left-shift" Bernoulli map. Similarly, we prove the "left-copycat" local rule [Formula: see text] is equivalent globally to the "right-shift" inverse Bernoulli map. Various geometrical and analytical properties have been identified from each characteristic function and explained rigorously. In particular, two-level stratified subpatterns found in most characteristic functions are shown to emerge if, and only if, b1 ≠ 0, where b1 is the "synaptic coefficient" associated with the cell differential equation developed in Part I. Gardens of Eden are derived from the decimal range of the characteristic function of each local rule and tabulated. Each of these binary strings has no predecessors (pre-image) and has therefore no past, but only the present and the future. Even more fascinating, many local rules are endowed with binary configurations which not only have no predecessors, but are also fixed points of the characteristic functions. To dramatize that such points have no past, and no future, they are henceforth christened "Isles of Eden". They too have been identified and tabulated.

Synapse Models for Neural Networks: From Ion Channel Kinetics to Multiplicative Coefficient wij

This paper relates different levels at which the modeling of synaptic transmission can be grounded in neural networks: the level of ion channel kinetics, the level of synaptic conductance dynamics, and the level of a scalar synaptic coefficient. The important assumptions to reduce a synapse model from one level to the next are explicitly exhibited. This coherent progression provides control on what is discarded and what is retained in the modeling process, and is useful to appreciate the significance and limitations of the resulting neural networks. This methodic simplification terminates with a scalar synaptic efficacy as it is very often used in neural networks, but here its conditions of validity are explicitly displayed. This scalar synapse also comes with an expression that directly relates it to basic quantities of synaptic functioning, and it can be endowed with meaningful physical units and realistic numerical values. In addition, it is shown that the scalar synapse does not receive the same expression in neural networks operating with spikes or with firing rates. These coherent modeling elements can help to improve, adjust, and refine the investigation of neural systems and their remarkable collective properties for information processing.