SummaryWhen and how infants begin to discriminate noxious from innocuous stimuli is a fundamental question in neuroscience [1]. However, little is known about the development of the necessary cortical somatosensory functional prerequisites in the intact human brain. Recent studies of developing brain networks have emphasized the importance of transient spontaneous and evoked neuronal bursting activity in the formation of functional circuits [2, 3]. These neuronal bursts are present during development and precede the onset of sensory functions [4, 5]. Their disappearance and the emergence of more adult-like activity are therefore thought to signal the maturation of functional brain circuitry [2, 4]. Here we show the changing patterns of neuronal activity that underlie the onset of nociception and touch discrimination in the preterm infant. We have conducted noninvasive electroencephalogram (EEG) recording of the brain neuronal activity in response to time-locked touches and clinically essential noxious lances of the heel in infants aged 28–45 weeks gestation. We show a transition in brain response following tactile and noxious stimulation from nonspecific, evenly dispersed neuronal bursts to modality-specific, localized, evoked potentials. The results suggest that specific neural circuits necessary for discrimination between touch and nociception emerge from 35–37 weeks gestation in the human brain.

The influence of higher-order wavelet properties on the analytic wavelet transform behavior is investigated, and wavelet functions offering advantageous performance are identified. This is accomplished through detailed investigation of the generalized Morse wavelets, a two-parameter family of exactly analytic continuous wavelets. The degree of time/frequency localization, the existence of a mapping between scale and frequency, and the bias involved in estimating properties of modulated oscillatory signals, are proposed as important considerations. Wavelet behavior is found to be strongly impacted by the degree of asymmetry of the wavelet in both the frequency and the time domain, as quantified by the third central moments. A particular subset of the generalized Morse wavelets, recognized as deriving from an inhomogeneous Airy function, emerge as having particularly desirable properties. These "Airy wavelets" substantially outperform the only approximately analytic Morlet wavelets for high time localization. Special cases of the generalized Morse wavelets are examined, revealing a broad range of behaviors which can be matched to the characteristics of a signal.Comment: 15 pages, 6 Postscript figure

The generalized Morse wavelets are shown to constitute a superfamily that essentially encompasses all other commonly used analytic wavelets, subsuming eight apparently distinct types of analysis filters into a single common form. This superfamily of analytic wavelets provides a framework for systematically investigating wavelet suitability for various applications. In addition to a parameter controlling the time-domain duration or Fourier-domain bandwidth, the wavelet {\em shape} with fixed bandwidth may be modified by varying a second parameter, called $\gamma$. For integer values of $\gamma$, the most symmetric, most nearly Gaussian, and generally most time-frequency concentrated member of the superfamily is found to occur for $\gamma=3$. These wavelets, known as "Airy wavelets," capture the essential idea of popular Morlet wavelet, while avoiding its deficiencies. They may be recommended as an ideal starting point for general purpose use

Abstract-This paper examines the class of generalized Morse wavelets, which are eigenfunction wavelets suitable for use in time-varying spectrum estimation via averaging of time-scale eigenscalograms. Generalized Morse wavelets of order (the corresponding eigenvalue order) depend on a doublet of parameters ( , ); we extend results derived for the special case = = 1 and include a proof of "the resolution of identity." The wavelets are easy to compute using the discrete Fourier transform (DFT) and, for ( ) = (2 2), can be computed exactly. A correction of a previously published eigenvalue formula is given. This shows that for 1, generalized Morse wavelets can outperform the Hermites in energy concentration, contrary to a conclusion based on the = 1 case.For complex signals, scalogram analyses must be carried out using both the analytic and anti-analytic complex wavelets or odd and even real wavelets, whereas for real signals, the analytic complex wavelet is sufficient.

An exact and general expression for the analytic wavelet transform of a real-valued signal is constructed, resolving the time-dependent effects of non-negligible amplitude and frequency modulation. The analytic signal is first locally represented as a modulated oscillation, demodulated by its own instantaneous frequency, and then Taylor-expanded at each point in time. The terms in this expansion, called the instantaneous modulation functions, are time-varying functions which quantify, at increasingly higher orders, the local departures of the signal from a uniform sinusoidal oscillation. Closed-form expressions for these functions are found in terms of Bell polynomials and derivatives of the signal's instantaneous frequency and bandwidth. The analytic wavelet transform is shown to depend upon the interaction between the signal's instantaneous modulation functions and frequency-domain derivatives of the wavelet, inducing a hierarchy of departures of the transform away from a perfect representation of the signal. The form of these deviation terms suggests a set of conditions for matching the wavelet properties to suit the variability of the signal, in which case our expressions simplify considerably. One may then quantify the time-varying bias associated with signal estimation via wavelet ridge analysis, and choose wavelets to minimize this bias