Delay estimation for cortico-peripheral relations.

In neurophysiology, time delays between concurrently measured time series are usually estimated from the slope of a straight line fitted to the phase spectrum. We point out that this estimate is valid only in the case in which, one signal is a mere time-delayed copy of the other one. We present a procedure for delay estimation that applies to a much wider class of systems with nontrivial phase spectrum like for example lowpass filters. The procedure is based on the Hilbert transform relation between the phase of a linear system and its log gain. The Hilbert transform relation is nonlocal in frequency space, a fact that limits its applicability to experimental data. We explore these limits, and demonstrate that the method is applicable to neurophysiological time series. We present the successful application of the Hilbert transform behavior method to concurrently recorded epicortical brain activity and peripheral tremor. We point out and explain physiologically unreasonable delay estimates given by the traditional method. Finally, we discuss the assumptions underlying the applicability of the Hilbert transform method in the neuroscience context.