Time and Frequency Domain Analysis of Spike Train and Time Series Data

The concept of a spike triggered average will be familiar to many neurophysiologists. The first application in neurophysiology by Mendell and Henneman (1968, 1971) was used to examine the magnitude of monosynaptic excitatory postsynaptic potentials (EPSP) from muscle spindle Ia afferents onto homonymous motoneurons, which provided a major piece of evidence in the development of the size principle for motoneuron recruitment (see Henneman and Mendell, 1981). The technique has gained widespread acceptance, and been widely used to investigate the strength of synaptic connections in the mammalian central nervous system (e.g. Watt et al., 1976; Stauffer et al., 1976; Kirkwood and Sears, 1980; Cope et al., 1987), leading to new insights and an increased understanding of basic neurophysiological mechanisms. The basic principle in the above studies is that averaging of an intracellular recording from a motoneuron triggered by action potentials from a single intact afferent will reveal a waveform which is taken as an estimate of the postsynaptic potential (PSP) for that input. Averaging is required due to the presence of unrelated activity within the cell, which can be regarded as a noise component. Spike triggered averaging can detect weak effects (Cope et al., 1987), averages involving 105 or more triggers are commonly used. A similar procedure is often used to assess the coupling between two simultaneously recorded sequences of action potentials. In this case it is the timing of spikes in one spike train which is averaged with respect to the timing of spikes in a second spike train. This leads to a histogram based measure, frequently referred to as the cross-correlation histogram, which shows the relative timing of spikes in one spike train with respect to a second spike train. First used by Griffith and Horn (1963) to study functional coupling between cells in cat visual cortex, this method has subsequently been widely accepted and used in other areas of neurophysiology (e.g. Sears and Stagg, 1976; Kirkwood and Sears, 1978; Datta and Stephens, 1990). Spike triggered averaging and cross-correlation histogram analysis can be considered as the detection of a correlation between two signals. Both are affected by the presence of noise, particularly when studying weak interactions. In most cases the signals which are being studied contain noise, and can therefore be considered as stochastic processes. The study of stochastic signals, and the detection of correlated activity in the presence of noise are major research areas in engineering and statistics, an extensive literature exists on the related questions of characterizing stochastic signals, and the estimation of correlated activity in the presence of noise (e.g. Brillinger and Tukey, 1984). In this chapter we are concerned with the question: Given two stochastic signals (which