Origins: Brain and Self Organization

critical role in even this simplest of descriptions of neural dynamics. What should be amply clear from this study and our previous work is that there is no single model that describes the dynamics of neurons everywhere in the nervous system; even the very simple model of section III has its drawbacks, but it serves to provide a simple description of the role of noise and inhibition in the firing dynamics of a single cell. In fact, a generalization of this model, the so-called 'leaky' integrator characterized by the full dynamics (2) may well provide a good model of the bursting that is so ubiquitous a feature of axonal firing.

The Inter-Spike-Interval Histograms are not, by themselves, necessarily an indicator of the presence of stochastic resonance as an underlying cooperative effect in neurophysiology. For bistable models having the general form (1) they are, however, a product of correlated (between the noise and deterministic modulation) switching and, as we have indicated in section I, there is some evidence that such switching processes do occur in neurophysiology (although the form of the bistability may be different from the potential systems defined by dynamics such as (1)). Various features of these histograms can, however, lend themselves to explanations based on stochastic resonance. Perhaps the most important of these features is that the heights of successive peaks pass through a maximum as a function of the noise strength. This has been demonstrated in section III for the simple 'perfect' integrator model and is also known to occur in bistable systems of the form (1) [47,76]. So far, attempts to quantify this "resonance" as a matching of two characteristic rates have been inconclusive, for the bistable case, largely because of the difficulty of (numerically) producing good ISIHs with low noise. For the simple IF model of section III, however, this connection (or lack of it) might be possible to establish, due to the analytical tractability of the problem; this is an area currently being actively investigated. Since these models admit of a deterministic switching (or firing) mechanism, in contrast with the fundamental precept of no deterministic switching that underlies the bistable models, the mechanism for the observed "resonances" of figure 6 may well differ from the corresponding mechanism in the bistable case.

What should be clear from this paper is that effects that, qualitatively at least, appear to be similar can occur in systems that are quite different. A simple observation of the ISIH or the resonance in the peak heights as a function of noise is not sufficient to establish a model for the underlying dynamics. In different terms, it is probable that simple models of the type discussed in section III may provide a good fit to experimental ISIHs just as we have seen [24,44,45,48] in various forms of bistable descriptions. In fact, we have shown that, for the bistable case, a good fit to experimental data is obtained using dynamics of the form (1) with different flow functions f (x); the data shed no light on the precise form of the dynamics. This is a characteristic of our coarse-grained description of the dynamics in terms of probabilities and power spectra. In most theories of stochastic resonance in bistable systems, the quantity that critically mediates the dynamics is the ratio of the barrier height to the noise variance. This ratio, in fact, determines, as outlined in section II, the number and sizes of the peaks in the ISIH. So, one must end this review with the (by now) somewhat obvious question: "Which features of the ISIHs are due to underlying neurophysiological processes and which features are influenced by the statistics of the detection/measurement process or the statistics of the simple models such as (1) and (2) which have been invoked to explain the features of the ISIHS?"

It is a pleasure to acknowledge energizing conversations with Professor Karl Pribram ( Radford Univ.) and Drs. Andre Longtin ( Univ. of Ottawa), Frank Moss ( Univ. of Missouri, St. Louis), Dante Chialvo and A. Vania Apkarian ( Univ. of Syracuse), Stephen Schiff (Childrens' Hospital, Washington DC), Alianna Maren (Accurate Automation, Chattanooga, TN), Donatella Petracchi ( Univ. of Pisa) and Bill Jacobs, Gabor Schmera and Mario Inchiosa (NCCOSC, San Diego). Support from the Physics and Life Sciences

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