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It was found that the rhythmicity of noise-induced events (spikes in neural systems, for example) depends significantly on the noise intensity. There is an optimal noise level at which the regularity (closeness to a periodic process) is maximal. Such a non-linear effect is known as coherence resonance and will be considered in detail below. In many ways, systems with noise-induced oscillations behave like noisy limit cycle oscillators. In the following sections we will study different types of synchronization that can arise between such systems using the concept of the coherence resonance oscillator.

In this paper, the periodic solution for a class of Cohen-Grossberg neural networks with variable coefficients is discussed. By using inequality analysis technique and matrix theory, some new sufficient conditions are obtained to ensure the existence, uniqueness, global attractivity and exponential stability of the periodic solution. An example is given to show the effectiveness of the obtained results.

In the previous chapter we have considered nonstationary VAR models with time invariant parameters. Nonstationarity, that is, time varying first and/or second moments of a process, can also be modeled in the framework of time varying parameter processes. Suppose, for instance, that the time series show a seasonal pattern. In that case a VAR(p) process with different intercept terms for each season may be a reasonable model: $$ y_t = v + A_1 y_{t - 1} + \cdots + A_p y_{t - p} + u_t $$ (12.1.1) Here v i is a (K × 1) intercept vector associated with the i-th season, that is, in (12.1.1) the time index t is assumed to be associated with the i-th season of the year. It is easy to see that such a process has a potentially different mean for each season of the year.

It will be useful to begin with a brief passage from The Critique of Pure Reason. At the commencement of his discussion of ‘time’ Kant says: “Time is not an empirical concept that has been derived from any experience. For neither coexistence nor succession would ever come within our perception if the representation of time were not presupposed as underlying them a priori. Only on the presupposition of time can we represent to ourselves a number of things as existing at one and the same time (simultaneously) or at different times (successively).”… “… the concept of alteration, and with it the concept of motion, as alteration of place, is possible only through and in the representation of time;…”

A short review of hypotheses on human time perception emphasises the distinction between endogenous and exogenous concepts. Methods are described, and the related problem of stationarity is discussed. A hypothesis is formulated which assumes oscillatory processes as the basis for the perception of short temporal intervals. Several experiments are described which test this hypothesis. The results of the first experiment suggest a temporal constant in the range of the conscious present. Periodic components in subjective random series which are demonstrated in another experiment, indicate a sequence of temporal units. The influence of physiological factors on time perception is illustrated; diurnal variations are discussed as an example. However, it is shown that the observed variations in time perception are not dependent on body temperature alone. The importance of informational cues is noted and an experiment is described which indicates the role of individual differences.