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essential) can find easy-to-use algorithmic help in Cambridge University Press’s Numerical Recipes series (Press et al, 1991).

The conceptual and communication gaps between applied mathematicians and physicists and the bench practitioners of the neurosciences, that inevitably lead one or the other, most often both, to surrender their deepest intuitions to jointly shared images that are inevitably more simplistic, are no longer inevitable. With her own hands on both the quantitative conjectural and experimental machinery, the motivated practicing neuroscientist can honor her own insights, read about and construct symbolic representations from her intuitions and do her own quantitative theory. Computerized numerical techniques have become so powerful and accessible that, even in academic settings, there is debate about whether fundamental analytic tools, such as series expansions, should be taught in undergraduate courses about differential equations.

The practice of “try it and see what happens", with the current name of experimental, computational mathematics, is accessible to all. In addition to the powerful general mathematical programs noted above, there exist several sets of more specifically targeted software with the capacity to generate, portray and quantify the behavior of nonlinear continuous and discrete abstract and real dynamical systems. These often also include algorithmic modules that are useful in tailoring new models and measures (see for examples, Parker and Chua, 1989; Baker and Gollub, 1991; Nusse and Yorke, 1991; Sprott and Rowlands, 1991; Sprott, 1993; Korsch and Jodl, 1994; Enns and McGuire, 1997). Learning from and using this software, along with only a little programming in the high level languages and computer algebra programs listed above, permit the non-mathematician neuroscientist, willing to read in the literature such as that described below, to do independent, cutting edge research in applied dynamical systems.

Described below will be the computational discoveries in abstract systems and real neuroscientific data that have led to multiple contexts of quantitative description. These include those that are: (1) Geometric and conserve metric distances; (2) Topological and conserve relative positions but not distances; (3) Single or multiple global quantitative descriptors such as scaling numbers or scaling

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