Journal of Neuroscience Methods 111 (2001) 127 – 139 Delay estimation for cortico-peripheral relations M. Lindemann a,b, *, J. Raethjen b , J. Timmer c , G. Deuschl b , G. Pfister a a Institute of Experimental and Applied Physics, Uniersity of Kiel, Leibnizstrasse 11, 24098 Kiel, Germany b Department of Neurology, Uniersity of Kiel, Leibnizstrasse, Germany c Department of Physics, Uniersity of Freiburg, Germany Received 16 May 2001; received in revised form 3 July 2001; accepted 9 July 2001 Abstract 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. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Time delay estimation; Linear system theory; Hilbert transform relation; Minimum phase; Cross-spectral analysis; Phase spectra www.elsevier.com/locate/jneumeth 1. Introduction In recent years, there has been a growing interest in the study of functional coupling between different parts of the nervous system. Stimulated by new techniques for measuring signals of high quality both by MEG and EEG there have been numerous studies into the cou- pling within the brain itself and with peripheral muscle activity; for a recent review see Mima and Hallett (1999a). Cross-spectral analysis provides very powerful tools for the analysis of neurophysiological systems in the frequency domain (Timmer et al., 2000). In particular, the existence of a linear association can be reliably tested for by the coherency function. Once a correlation has been found, the nature of the relation can be examined, and specifically one might ask whether there is a time delay between the processes. This question is of great neurophysiological importance as it might shed light on the pathways by which the processes interact. Methods for the identification of time delays in bio- logical systems have received a lot of attention. Delay estimation between cortical signals and peripheral mus- cle activity allows to differentiate a transmission via oligosynaptic corticospinal pathways, the conduction times of which are well known from cortical stimulation studies in humans (Rothwell et al., 1991), from a medi- ation via polysynaptic extrapyramidal systems conduct- ing much slower than the corticospinal tract. This delay estimation is to date usually accomplished by estimat- ing the slope of the phase spectrum by a straight line fit which has led to heterogeneous results (Mima and Hallett, 1999b; Mima et al., 1999; Brown et al., 1998Brown et al., 1999Halliday et al., 1998). We point out that this method can only be applied in the rather special case of one signal being a mere time-shifted copy of the other one. On the contrary, we demonstrate that using a procedure called Hilbert transform method * Corresponding author. Tel.: +49-431-880-4075; fax: +49-431- 880-3943. E -mail address: m.lindemann@neurologie.uni-kiel.de (M. Linde- mann). 0165-0270/01/$ - see front matter © 2001 Elsevier Science B.V. All rights reserved. PII:S0165-0270(01)00436-8