Abstract. A method that enables measurement of the degree of coupling between two signals is presented. The method is based on the de®nition of an uncoupling function calculating, by means of entropy rates, the minimum amount of independent information (i.e. the information carried by one signal which cannot be derived from the other). An estimator of the uncoupling function able to deal with short segments of data (a few hundred samples) is proposed, thus enabling the method to be used for usual experimental recordings. A syn- chronisation index is derived from the estimate of the uncoupling function by means of a minimisation proce- dure. It quanti®es the maximum amount of information exchanged between the two signals. Simulations in which non-linear coordination schemes are produced and changes in the coupling strength are arti®cially induced are used to check the ability of the proposed index to measure the degree of synchronisation between signals. The synchronisation analysis is utilised to measure the coupling strength between the beat-to-beat variability of the sympathetic discharge and ventilation in decerebrate arti®cially ventilated cats and the degree of synchronisation between the beat-to-beat variability of the heart period and ventricular repolarisation interval in normal subjects and myocardial infarction patients. The sympathetic discharge and ventilation are strongly coupled and the coupling strength is not aected by manoeuvres capable of increasing or de- pressing sympathetic activity. The synchronisation is lost after spinalisation. The synchronisation analysis con®rms that the heart period and ventricular repolar- isation interval are well coordinated. In normal subjects, the synchronisation index is not modi®ed by experimen- tal conditions inducing changes in the sympathovagal balance. On the contrary, it strongly decreases after myocardial infarction, thus detecting and measuring the uncoupling between the heart period and ventricular repolarisation interval. 1 Introduction Synchronisation occurs when patterns involving two signals contemporaneously are repetitive. Synchronisa- tion to a periodic input can be observed when the activity of a self-sustained oscillator is perturbed by a periodic input. If the coupling strength is low, a quasiperiodic dynamics can be found. The forcing input and undergoing activity are virtually independent as it occurs, for example, when the electrical activity of the squid giant axon is perturbed by a low-amplitude periodic forcing current (Aihara et al. 1986). In contrast, when N cycles of the undergoing activity occur every M cycles of the forcing input (N:M periodic dynamics), the two signals are strongly coordinated. Examples of N:M coordination are found between the electrical activity of aggregates of embryonic chick ventricular heart cells and the perturbing periodic current (Guevara et al. 1988) and between the phrenic eerent activity of paralysed anaesthetised cats and mechanical ventilation (Petrillo and Glass 1984). A certain degree of coupling could be present in sliding dynamics (Koepchen et al. 1991) when the changes in the coupling ratio are not completely unpredictable and in irregular dynamics when some bursts of the forced activity are still linked to the perturbing input (Porta et al. 1996). However, not only synchronisation to a periodic process can be found; examples of synchronised chaotic dynamics are given by Pecora and Carroll (1990). Quanti®cation of the degree of synchronisation (i.e. the coupling strength) between two signals is considered an important goal, as dierent levels of synchronisation are found between the RR interval and respiration (Hoyer et al. 1998), between coupled oscillators de- scribing biped animal gaits (Collins and Stewart 1993), between cardiac-related discharges of sympathetic Biol. Cybern. 81, 119±129 (1999) Conditional entropy approach for the evaluation of the coupling strength A. Porta 1 , G. Baselli 2 , F. Lombardi 3 , N. Montano 4 , A. Malliani 4 , S. Cerutti 5 1 Dipartimento di Scienze Precliniche, Universita' degli Studi di Milano, Milan, Italy 2 Dipartimento di Elettronica per l' Automazione, Universita' di Brescia, Brescia, Italy 3 Ospedale San Paolo, Milan, Italy 4 Centro Ricerche cardiovascolar CNR, Medicina Interna II, Ospedale L. Sacco, Universita' di Milano, Italy 5 Dipartimento di Bioingegneria, Politecnico di Milano, Italy Received: 29 October 1998 / Received in revised form: 4 March 1999 Correspondence to: Dr. Alberto Porta, Universita' degli Studi di Milano, Dipartimento di Scienze Precliniche, Ospedale L. Sacco, via G.B. Grassi 74, I-20157 Milan, Italy (e-mail: porta@imiucca.csi.unimi.it, Tel.: +39-02-38210945, Fax: +39-02-38210533)