Testing non-linearity and directe...
Journal of Neuroscience Methods 94 (1999) 105���119 Testing non-linearity and directedness of interactions between neural groups in the macaque inferotemporal cortex Winrich A. Freiwald a,c,*, Pedro Valdes b, Jorge Bosch b, Rolando Biscay b, Juan Carlos Jimenez b, Luis Manuel Rodriguez b, Valia Rodriguez b, Andreas K. Kreiter a, Wolf Singer c a Institute for Brain Research, Uni6ersity of Bremen, FB2, P.O. Box 330440, D-28334 Bremen, Germany b Cuban Neuroscience Center, A6e 25 No. 5202 esquina 158 Cubanaca ��n, P.O. Box 6880, 6990 Ciudad Habana, Cuba c Max-Planck-Institute for Brain Research, Deutschordenstr. 46, D-60528 Frankfurt/Main, Germany Received 2 July 1999 accepted 9 August 1999 Abstract Information processing in the visual cortex depends on complex and context sensitive patterns of interactions between neuronal groups in many different cortical areas. Methods used to date for disentangling this functional connectivity presuppose either linearity or instantaneous interactions, assumptions that are not necessarily valid. In this paper a general framework that encompasses both linear and non-linear modelling of neurophysiological time series data by means of Local Linear Non-linear Autoregressive models (LLNAR) is described. Within this framework a new test for non-linearity of time series and for non-linearity of directedness of neural interactions based on LLNAR is presented. These tests assess the relative goodness of fit of linear versus non-linear models via the bootstrap technique. Additionally, a generalised definition of Granger causality is presented based on LLNAR that is valid for both linear and non-linear systems. Finally, the use of LLNAR for measuring non-linearity and directional influences is illustrated using artificial data, reference data as well as local field potentials (LFPs) from macaque area TE. LFP data is well described by the linear variant of LLNAR. Models of this sort, including lagged values of the preceding 25 to 60 ms, revealed the existence of both uni- and bi-directional influences between recording sites. �� 1999 Elsevier Science B.V. All rights reserved. Keywords: Non-linear dynamics Granger causality Multivariate non-linear autoregression Bootstrap test for non-linear time series Local field potential Inferotemporal cortex www.elsevier.com/locate/jneumeth 1. Introduction Visual information processing in the mammalian brain is based on a multitude of cortical and subcortical structures. Within the macaque cortex more than thirty visual areas have been described (Felleman and van Essen, 1991), a number likely to be paralleled in other higher mammals, including humans. Neuroanatomical, and electrophysiological evidence suggests, that these cortical areas are further subdivided into anatomical compartments composed of neurons with distinct phys- iological properties (Kaas and Krubitzer, 1991). Thus, multiple neuronal populations in different areas process different aspects of a visual stimulus. Since receptive fields of cortical cells usually behave like broadly tuned filters in a high dimensional feature space (Martin, 1994 van Essen et al., 1992), a given stimulus, which has different features like spatial position in the visual field, velocity, disparity, colour and form cues, will activate large neural populations within the same and in different cortical areas. These distributed responses have to be integrated into a coherent representation. The establishment of this representation requires exten- sive interactions between different neuronal populations within the same and in different cortical areas, since there is no final integration area in the brain onto which all processing pathways would converge. * Corresponding author. Tel.: +49-421-2189481 fax: +49-421- 2189004. E-mail address: firstname.lastname@example.org (W.A. Freiwald) 0165-0270/99/$ - see front matter �� 1999 Elsevier Science B.V. All rights reserved. PII: S0165-0270(99)00129-6
W.A. Freiwald et al. / Journal of Neuroscience Methods 94 (1999) 105���119 106 The structural properties of cortical networks sup- port such extensive interactions. Connections between cortical neurons are generally characterised by a high degree of divergence and convergence. Each cortical area is sending output connections to and is receiving input connections from several other cortical areas. These connections are so numerous that about one third of all possible connections between visual areas have been discovered and roughly one half of them are expected to exist (Felleman and van Essen, 1991). Based on these connectivity patterns between cortical areas, their strength, the spatial arrangement of areas and the relatedness of their functional properties, differ- ent schemes for their arrangement into processing path- ways have been proposed (Ungerleider and Mishkin, 1982 Felleman and van Essen, 1991 Goodale and Milner, 1992 Scannell et al., 1995 Hilgetag et al., 1996). These pathways are characterised by extensive feedback connections, lateral connections to areas at the same processing level and connections by-passing intermediate levels of the hierarchy (see, e.g. Rockland and van Hoesen (1994)). Recent physiological data show that feedback projections can exert substantial effects onto earlier processing stages (Hupe �� et al., 1998). Large temporal overlap of the response periods of neurons even in areas at very different levels of the processing hierarchy (Nowak and Bullier, 1997) further support mutual influences. Thus, current neuroanatom- ical and neurophysiological evidence suggests extensive mutual interactions between distributed groups of neurons. Despite these results, the mechanisms which serve to integrate the activities of different neurons into a coher- ent representation are still a much debated issue. A recent concept of information processing in the cortex, extending Hebb���s cell assembly concept (Hebb, 1949), stresses the importance of the relative timing of action potentials to express relatedness of responses (von der Malsburg, 1981 Singer et al., 1990). According to this temporal binding hypothesis, neurons belonging to the same assembly should synchronise their responses, while cells belonging to different assemblies should fire asynchronously. Indeed, many experimental findings in the visual cortex are in agreement with this theory, including the existence and stimulus dependency of inter- and intra-areal synchronisation (see, e.g. Singer and Gray (1995) Engel et al., (1997) for recent re- views). According to this conceptual framework as well as to similar ones (Johannesma et al., 1986 Gerstein et al., 1989 Abeles, 1991 Aertsen et al., 1991 Sporns et al., 1991 Ahissar et al., 1992 Prut et al., 1998), neural interactions change in relation to current processing requirements defined by external stimuli and the inter- nal behavioural state of the animal. Much previous work related to these concepts has focused on the strength of neural interactions as indicated by correla- tion measures. Less attention has been paid to the direction of these interactions as a further dynamic property of functional connectivity. However, in our opinion, directed influences (or causal relations) that individual neurons (Abeles, 1982 Gerstein and Aertsen, 1985) or larger neural groups exert on each other and their variation in time might be of prime importance for cortical information processing. This idea seems to follow naturally from considerations of visual percep- tion. Recent psychophysical research provided evidence that the perception even of the most elementary aspects of a visual scene may depend on factors like attention, past experience, or the segmentation of the visual scene into different objects (Braddick, 1996). In these situa- tions top-down processing should be more prominent than in other instances, e.g. in the case of rapid process- ing, in which the system might essentially operate in a feed-forward manner (Thorpe et al., 1996). Accord- ingly, the relative influence of a ���higher level��� neural group on a second, ���lower level��� one might be stronger in the former condition than in the latter. Even during the response to a stimulus, the pattern of these relative influences between individual neurons or ensembles of neurons might change over time. Thus, in analysing information processing in the visual system, there is a strong interest to study the interactions of neuronal groups, i.e. to infer the direction of these influences from simultaneous electrophysiological recordings. To define this problem formally, let us denote by xt,yt the values of electrical recordings at time t obtained from any of two sites. Let us also denote the vector ofn observations from both sites at time t as zt = xt yt . Henceforth we shall use lower boldface type to indicate vectors and upper boldface type to indicate matrices. With this notation in place, our problem can then be formalised as defining a measure I(y x) which will quantify the influence of time series yt on time series xt. 1.1. First generation influence measures: linear instantaneous influences A first generation of methods (Gerstein et al., 1978) assessed neural interactions by means of correlation methods, based on the use of linear regression. There have been many recent papers along these lines in the neuro-imaging literature, specific instances being path analysis (McIntosh and Gonzalez-Lima, 1994), partial least squares (McIntosh et al., 1996) and the general concept of ���functional connectivity��� (Friston, 1994). These methods are based upon two implicit assumptions: 1. Interactions between neural ensembles are linear. 2. Interactions between neural ensembles are instanta- neous, that is they depend only on the current state of the system.
W.A. Freiwald et al. / Journal of Neuroscience Methods 94 (1999) 105���119 107 Both of these assumptions may be summarised by the following equation: xt =ayt +wt yt =bxt +zt (1) where the coefficients a and b express the linear instan- taneous relationships between series y and series x. These may be written in matrix notation as: zt =Azt +ot (2) where we have used following notation: zt = xt yt n A= 0 b a 0 nthe ot = w t z t n The time series ot is known as the ���innovation��� and can be viewed alternatively as a white noise process driving the system or as the error of prediction of one time series given the other. Note that all relations involve only the current time t this is what is meant by instantaneous interactions. First generation influence measures are defined as association coefficients that quantify how much of the total variation of the time series are explained by instantaneous linear relationships. 1.2. Second generation influence measures: Granger causality The second assumption of first generation influence measures, that of instantaneous neural interactions, is clearly not realistic since it ignores: The delay of transmission of information from one neural site to another. The fact that the evolution of the system may de- pend not only on the immediate past as is evidenced by the rich temporal structure of neural time series. Therefore, more realistic signal models substitute As- sumption 2 above by: 3. The evolution of the state of the system may be described as a function of a finite number of past states. On the basis of this assumption, Eq. (2) may be generalised by stating a dependence of zt not only on its own value, but also upon a set of p past vectors. (We will refer to p as ���the number of lagged values��� in the remainder of this paper, and accordingly use ���lagged values��� or ���lagged vectors���.) These can be stacked into ���delay matrices��� Yt =[yt, yt-1,���, yt-k,���, yt-p ], Xt = [xt, xt-1,���, xt-k,���xt-p ], and Zt = Xt Yt n , which con- tain all the information of both time series p points into the past. The most frequently used linear model is the Multi- variate Linear Autoregressive model: zt = % p k=1 Ak �� zt-k +ot (3) Based on the Multivariate Linear Autoregressive model a second generation of measures of influence has been proposed (Gersch, 1970, 1972 Akaike, 1974 Franaszczuk et al., 1985). These not only take into account the correlation structure within and between the observed time series, but they also allow use of the ���arrow of time��� to devise influence measures to statisti- cally assess causality as introduced by Granger and co-workers (Granger, 1963, 1969, 1980 Granger and Lin, 1995). Granger reasoned thus: if time series xt is influencing yt then adding past values of xt to the regression of yt will improve its prediction. This princi- ple was originally formulated in a very general way, encompassing both linear and non-linear systems. How- ever, Granger pointed out the difficulty of using non- linear models (Granger and Newbold, 1977) by stating: ���Thus for purely pragmatic reasons, the ���optimal predic- tion��� ��� should be replaced by ���optimal linear predic- tion������ (p. 226). Almost all specific measures of Granger causality have therefore been based on linear models. To be specific, consider the prediction of xt based only on its own past: xt = % p k=1 ak xt-k +wt (4) In this case the innovation series wt will have a variance s 2 x X where the suffix indicates that the error variance is that of series xt predicted only by its own delay matrix Xt. Now consider the following model, which adds the past values of yt as predictors of xt : xt = % p k=1 ak xt-k + % r q=1 bk yt-q +ft (5) Note that the number of delays of series yt used to predict series xt is r and thus does not have to be equal to p. In this case the prediction error is now ft which will have a variance s 2 x Z where the suffix indicates that series xt is now predicted by the complete delay matrix Zt which includes the past of both series. Based on these definitions, Granger introduced the following (lin- ear) influence measure (Granger, 1969): ILIN(y x)=ln s x X 2 s x Z 2 (6) Note that this measure of influence has the right properties. If the past of series yt does not improve the prediction of series xt then s 2 x Z will be equal to s 2 x X and the influence measure will be zero. Any improvement in prediction leads to a decrease in the denominator in Eq. (6) and therefore increases the value of the influence measure. A symmetrical definition of the influence of series xt on yt is possible. In fact, Geweke and others have generalised these definitions to multivariate time series and have defined influence measures between two sets of time series conditional on a third set of time series (Geweke, 1982, 1984).