An impressive review of brain neurophysiology provides the basis for modelling the dynamics of transmission in neural circuits, using appropriate nonlinear mathematics. The coverage is unbalanced, however: the parallel dynamics at the level of behaviour and sensory-cognitive processes are sparsely addressed, so the final chapter fails to indicate the complexity and subtlety of relevant modern work.

A data-screening procedure for identifying the dynamic structure of psychological data series is presented, based on information theory measures. It can generate signatures of random, periodic, and edge-of-chaos series which then serve as a preliminary step to pursuing fuller analyses, using appropriate statistics that are already available in software. It is no longer necessary to assume that real data are either linear, noisy, or periodic in order to obtain measures of the matching of theory to data. A glossary of technical terms is appended.

The research on mental rotation is now extensive, and diverse in its approach and level. The phenomenon was originally described in terms of perceptual performance, and has been well replicated in that sense. That human observers can apparently turn images about axes of rotation or reflection, and correctly construct identifications of objects whose images have been transformed in a virtual mental space, is uncontested but in some ways unexplained. The phenomenon raises fundamental questions about the nature of mathematical assumptions within psychological theory, and suggests a need for the reconciliation of phenomenological, neuropsychological and mathematical descriptions within one compatible framework. It is concluded that what actually happens is not rotation in the geometrical sense but a serial operation that may be confused with rotation, and may sometimes yield the same terminal result. A bibliography of major sources on the experimental and mathematical analyses of the mental rotation of images of 2-d and 3-d figures has been compiled.

Contemporary statistical theory has moved away in its balance from the emphasis on significance levels and hypothesis testing that is built into many computer packages familiar to, and obviously created for, psychologists. Serious fundamental criticisms have been raised by statisticians about the tradition of inference associated with Neyman, which he himself had found convincing half a century ago, and which often is still the only theory found in textbooks of statistics-for-psychologists. A resurgence of interest in Bayesian inference, and the solution of associated computational problems, has made available methods and data interpretations that are still not widely known or taught in psychology departments. An elementary example to distinguish between "significance" levels, likelihoods, and informational content is offered.

Although one can endorse the complexity of the data and processes that Phillips & Singer (P&S) review, their mathematical suggestions can be compared critically with cases in nonlinear psychophysics, where the theoretician is faced with analogous problems. Owing to P&S's failure adequately to recognise both the intricate properties of nonlinear dynamics in networks and the constraints of metabolic demands on the temporal generation of patterns in biological nets their conclusions fail to meet the problems they properly address.

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A sequel study to that reported by Gregson, Britton, Campbell and Gates (1990), partially replicating and extending findings which relate estimates of the dimensionality of the EEG to the complexity of task load in a visual scanning task, is described. The correlation dimensionality D2 of the attractor was computed using a variant of the Grassberger-Procaccia algorithm, and was shown to change in the expected direction, increasing as the task became more complicated. The effects are slight but consistent, and may be attenuated by nonstationarity over time, and by idiosyncratic factors. The results are numerically and qualitatively compatible with other recent reported studies, and support an interpretation linking brain dynamics to implied cognitive processes.

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The problem of how a consistent sensory input can be given an invariant name is addressed within the framework of nonlinear psychophysical theory. In order to link sensory transduction processes to cognitive operations an extension of purely psychophysical modelling is used. A hybrid model of sensory switching, produced by injecting a Λ recursion into a non‐Lipschitzian dynamics evolved by Zak, has the capacity to differentiate inputs and encode inputs into classes, in a form that makes vector inputs to neural networks possible. This is a necessary precursor to learning new sensory‐verbal mappings. The distinction between using piecewise linear models and using continuous nonlinear dynamics in theory construction is emphasized.

The details of response distributions in the method of limits are known to violate both simple assumptions of monotone increasing stimulus–response functions, and the derived idea that change points on such functions, those points where sensory phenomenology changes abruptly, are strictly ordered correspondingly. The predictions of nonlinear psychophysics include mappings from responses back onto stimuli which are not single‐valued. Such predictions imply that outliers to change point distributions expressed in physical units will exist under some conditions. The detection of outliers is tractable from a Bayesian approach. An experiment using fractionation of perceived brightness by continuous response adjustment is used as an illustrative example both for theory and for statistical analyses.

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A comparative study of water supplies in three townships in New South Wales, Australia, employing a rating scale method and various levels of dilution of source water with deionised water, was conducted using untrained tasters drawn from the same townships as the waters. Each taster thus tasted, without knowledge of its source, 36 samples in a balanced design, permitting comparisons in terms of source, dilution, familiarity and order effects within the test. Clear differences between source waters were found; these are compared with some European results reported by Zoeteman.

It is possible to predict the topology of isointensity plots under conditions of extreme imbalance of the stimulus inputs, without making any assumptions specific to the circumstances in which Fechner's Paradox is sometimes observed. This is done by extending a nonlinear model for a sensory channel, by postulating a form of cross-coupling or interference between two channels which represents other phenomena in psychophysics. It is noted that the form in which data are usually reported is not an adequate basis for testing all the predictions of a nonlinear model in sensory psychophysics. The physiologist Panum (1958), and later Fechner (1860) reported that the apparent brightness of an object viewed binocularly could, under conditions where the input to one eye was diminished by filtering, be less than its brightness viewed monocularly by the unfiltered eye. To a first approximation, binocular brightness is more like an averaging of two monocular inputs than a summation of those same inputs. For over 120 years this phenomenon, which came to be called "Fechner's Paradox", though Panum should perhaps have had some credit, has been the subject of experimental investigation and associated mathematical modelling. If one consults a dictionary of psychological terms, for example Evans (1978), one may read something like Fechner's Paradox: The name give to the observation that something [which is] viewed binocularly seems to increase in brightness when one eye is closed. And yet we now know that this definition is misleading, because the same phenomenon in pooling two sensory inputs has its analogues in audition (Lehky 1983) and in olfaction (Gregson 1986). Gilchrist and McIver (1985) have now shown an analogue of the paradox exists in ocular contrast sensitivity. The definition also goes awry when the input luminance to one eye is zero, or when the luminance and ocular adaptation are closely matched for the two eyes. It is wiser, in the light of results reporting individual differences in the existence and extent of the paradox, and its sensitivity to stimulus conditions, to side with Blake and Fox (1973) when they observed that it is not unreasonable to suppose that various stimulus conditions might yield varying amounts of summation or even inhibition. Empirical reviews of relevant data in vision have been given by Roelofs and Zeeman (1914), Blake and Fox (1973), and Blake et al. (1981), but a theoretical model of interest as a starting point is that of Lehky (1983).