A Bartlett-type formula is proposed for the asymptotic distribution of the sample autocorrelations of nonlinear processes. The asymptotic covariances between sample autocorrelations are expressed as the sum of two terms. The first term corresponds to the standard Bartlett's formula for linear processes, involving only the autocorrelation function of the observed process. The second term, which is specific to nonlinear processes, involves the autocorrelation function of the observed process, the kurtosis of the linear innovation process and the autocorrelation function of its square. This formula is obtained under a symmetry assumption on the linear innovation process. It is illustrated on ARMA–GARCH models and compared to the standard formula. An empirical application on financial time series is proposed.

In the olfactory bulb, both the spatial distribution and the temporal structure of neuronal activity appear to be important for processing odor information, but it is currently impossible to measure both of these simultaneously with high resolution and in all layers of the bulb. We have developed a biologically realistic model of the mammalian olfactory bulb, incorporating the mitral and granule cells and the dendrodendritic synapses between them, which allows us to observe the network behavior in detail. The cell models were based on previously published work. The attributes of the synapses were obtained from the literature. The pattern of synaptic connections was based on the limited experimental data in the literature on the statistics of connections between neurons in the bulb. The results of simulation experiments with electrical stimulation agree closely in most details with published experimental data. This gives confidence that the model is capturing features of network interactions in the real olfactory bulb. The model predicts that the time course of dendrodendritic inhibition is dependent on the network connectivity as well as on the intrinsic parameters of the synapses. In response to simulated odor stimulation, strongly activated mitral cells tend to suppress neighboring cells, the mitral cells readily synchronize their firing, and increasing the stimulus intensity increases the degree of synchronization. Preliminary experiments suggest that slow temporal changes in the degree of synchronization are more useful in distinguishing between very similar odorants than is the spatial distribution of mean firing rate.