Abstract. This paper introduces a new frequency- domain approach to describe the relationships direction of information ¯ow) between multivariate time series based on the decomposition of multivariate partial coherences computed from multivariate autoregressive models. We discuss its application and compare its performance to other approaches to the problem of determining neural structure relations from the simul- taneous measurement of neural electrophysiological signals. The new concept is shown to re¯ect a frequen- cy-domain representation of the concept of Granger causality. 1 Introduction For many years the monitoring multiple electric signals derived from neuronal depolarization has been used to infer functional aspects of both normal and patholog- ical brain processes. Among numerous techniques for displaying and analyzing neural signals of various types electroencephalograms, local ®eld potentials, and multi and single neural unit activity), methods based on the estimation of correlation/coherence functions between the activity of pairs of simultaneously analyzed struc- tures have been the most popular approaches. These include investigating issues of physiological interest such as the determination of the source of neural activity in epileptic seizures Duckrow and Spencer 1992) and in physiological oscillations, e.g., the alpha and theta rhythms Lopes da Silva et al. 1973; Kocsis et al. 1994), and the activation of brain centers related to speci®c behavioral tasks or cognitive processes Toyama et al. 1981; Melssen and Epping 1987; Egger- mont 1990; Bressler et al. 1993; Pawelzik 1994), or the studies of the correlation between EEG waveforms and brain behavioral state, as characterized by speci®c patterns like typical signal amplitude and frequency as, for example, in staging the sleep state Barlow 1979). This state of aairs has remained largely unchang- ed despite the practical and theoretical limitations of coherence analysis, which merely describes in- stances when pairs of structures are in synchronous activity. In fact little attention has been given to the evo- lution of the concept of coherence: the idea of directed coherence DC) between pairs of structures. Directed coherence, rather than merely describing mutual synchronicity, tells us whether and how two structures under study are functionally connected. While ordi- nary coherence focuses on the structures themselves and the mutual synchrony of their activity, DC stresses their relative structural relationships by decomposing their interactions into ``feedforward'' and ``feedback'' aspects. This shift is specially relevant as even today much of the actual structural and func- tional connectivity in the brain is still derived from the post mortem anatomical study of experimental animals, which is unrevealing as to whether links be- tween structures are ``active'' in a given scenario of brain processing that underlies the generation of some speci®c behavior. In this paper Sect. 2), we review the notion of DC as generalized to the simultaneous analysis of more than just pairs of neural structures, and place it in the per- spective of the more fundamental concept of Granger causality Granger 1969) in Sect. 3, where we further introduce a new approach of structural analysis in the frequency domain that we name partial directed coher- ence PDC). In the examples Sect. 4), we contrast PDC with DC to show how PDC provides direct structural information for multivariate autoregressive MAR) models that simultaneously model many time series. PDC is used next to reveal a reversal in the direction of information ¯ow between the cortex and the hippo- campus during a spindle episode within a record of slow- wave sleep. Correspondence to: L. A. Baccala e-mail: baccala@lcs.poli.usp.br) Biol. Cybern. 84, 463±474 2001) Partial directed coherence: a new concept in neural structure determination Luiz A. Baccala 1 , Koichi Sameshima 2 1 Telecommunications and Control Engineering, Escola PoliteÂcnica, University of S~ ao Paulo, Av. Prof. Luciano Gualberto, Trav. 3, 158, CEP 05508-900, Brazil 2 Discipline of Medical Informatics & Functional Neurosurgery Laboratory, School of Medicine, University of S~ ao Paulo, Brazil Received: 25 April 2000 / Accepted in revised form: 13 November 2000