Elsevier

Journal of Neuroscience Methods

Volume 94, Issue 1, 15 December 1999, Pages 105-119
Journal of Neuroscience Methods

Testing non-linearity and directedness of interactions between neural groups in the macaque inferotemporal cortex

https://doi.org/10.1016/S0165-0270(99)00129-6Get rights and content

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.

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 physiological 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 extensive 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.

The structural properties of cortical networks support 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, different schemes for their arrangement into processing pathways 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 (Hupé 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 neuroanatomical 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 coherent 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 reviews). 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 internal behavioural state of the animal. Much previous work related to these concepts has focused on the strength of neural interactions as indicated by correlation 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 perception. 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 situations top-down processing should be more prominent than in other instances, e.g. in the case of rapid processing, in which the system might essentially operate in a feed-forward manner (Thorpe et al., 1996). Accordingly, 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 of observations from both sites at time t as zt=xtyt. 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 Iyx which will quantify the influence of time series yt on time series xt.

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 instantaneous, that is they depend only on the current state of the system.

Both of these assumptions may be summarised by the following equation:xt=ayttyt=bxttwhere the coefficients a and b express the linear instantaneous relationships between series y and series x. These may be written in matrix notation as:zt=Azt+εtwhere we have used the following notation:zt=xtyt A=0ab0 εt=ςtζt

The time series εt 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.

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 depend not only on the immediate past as is evidenced by the rich temporal structure of neural time series.

Therefore, more realistic signal models substitute Assumption 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,…, ytk,…, ytp], Xt=[xt, xt−1,…, xtk,…xtp], and Zt=XtYt, which contain all the information of both time series p points into the past.

The most frequently used linear model is the Multivariate Linear Autoregressive model:zt=k=1pAk · zt−k+εt

Based on the Multivariate Linear Autoregressive model a second generation of measures of influence has been proposed (Gersch, 1970, Gersch, 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 statistically assess causality as introduced by Granger and co-workers (Granger, 1963, Granger, 1969, Granger, 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 principle was originally formulated in a very general way, encompassing both linear and non-linear systems. However, Granger pointed out the difficulty of using non-linear models (Granger and Newbold, 1977) by stating: ‘Thus for purely pragmatic reasons, the ‘optimal prediction’ … should be replaced by ‘optimal linear prediction’’ (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=k=1pakxt−kt

In this case the innovation series ζt will have a variance σ2x|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=k=1pakxt−k+q=1rbkyt−qt

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 φt which will have a variance σ2x|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 (linear) influence measure (Granger, 1969):ILINyx=lnσxX2σxZ2

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 σ2x|Z will be equal to σ2x|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, Geweke, 1984).

Bernasconi and co-workers (Bernasconi et al., 1998, Bernasconi and König, 1999) have applied this type of influence measure to electrophysiological recordings. These authors also carried out a spectral decomposition of the causality measures and provided empirical confidence intervals for this spectrum by means of the bootstrap. The results obtained indicate that influence measures are indeed a useful tool for studying neural effective connectivity.

The above work presupposes that neural systems are linear. Neither the Hodgkin and Huxley equations of single cell neurophysiology, nor the modelling of synaptic interactions result in linear equations. Whether the ensemble behaviour of neural masses scales to a linear approximation is a matter of great importance to be determined empirically. The recent trend in signal modelling in neuroscience has been quite in the opposite direction to linear modelling. Results obtained with analytical methods derived from ‘chaos theory’ (see Elbert et al. (1994) for a review) have provided evidence for the essentially non-linear nature of large-scale EEG-, ECoG- and MEG-signals, even though the existence of underlying chaotic dynamics may not be demonstrable (Valdes et al., 1999). If the time series are non-linear then methods based on linear regression as those described above may be misleading.

There have been several previous attempts to generalise the first generation influence measures to the non-linear case as exemplified in the work of Lopes da Silva and Mars (1987) by using both information theory concepts (Pijn et al., 1990) and correlation concepts based on non-linear regression. Both works demonstrated that in specific instances the assumption of linear interactions was misleading and that rather than the relationship expressed in Eq. (2) the following expression should be used:zt=Fzt+εtwhere F is a non-linear relationship. This model and the aforementioned measures based on it suffer from the shortcomings of all first generation methods enumerated in the previous section.

A third generation of influence measures is obtained by the application of Granger's most general concept of causality (Granger and Newbold, 1977), in the context of a specific non-linear multivariate model:zt=FZt+εtin which F is not necessarily linear. The difficulty of this task has been the specification of a tractable framework for non-linear time series analysis. One such framework was proposed by Ozaki (1985) and later generalised by Tong (1990). This consists in specifying a linear Autoregressive model in which the coefficients Ak will now depend on the previous states of the system:zt=k=1pAkZt · zt−k+εt(This is a generalisation of Eq. (3) which has coefficients Ak independent of the delay matrix Zt.)

A number of recent papers implementing Granger causality measures are based on particular non-linear time series models (Teräsyirta, 1998, Warne et al., 1999). It must be stressed, however, that the model selected for implementing causality measures must be matched to the dynamic characteristics of the time series studied. Recent work (reviewed in Valdes et al. (1999)) has shown that not all non-linear models are capable of capturing the complex characteristics of neural signals. The class of models that offered a good trade-off between computational complexity and descriptive properties were the use of locally weighted polynomial non-parametric regression (Fan and Gijbels, 1995, Fan and Gijbels, 1996). Bell et al., 1996, Bell et al., 1998 have devised Granger causality measures for a specific class of additive local polynomial models. In a series of recent papers Valdes and co-workers (reviewed in Valdes et al. (1999)) have applied Local Linear polynomial regression to the analysis of neural signals. On the basis of this technique they have implemented a specific measure of Granger causality for the analysis of non-linear multivariate neurophysiological signals and have carried out the preliminary evaluation of these measures (Valdes et al., 1996). This family of models includes ordinary linear Autoregression as a special case.

The purpose of this paper is fivefold

  • 1.

    To describe a general framework that encompasses both linear and non-linear modelling of neurophysiological time series by means of Local Linear Non-linear Autoregressive models (LLNAR).

  • 2.

    Within this framework to describe new tests for a) non-linearity of time series and for b) non-linearity of neural interactions, both based on the LLNAR model.

  • 3.

    To introduce a specific measure of Granger Causality for directed influences based on the LLNAR model and a test of significance for this measure.

  • 4.

    To show the advantages of this measure of causality for non-linear data.

  • 5.

    To show examples of the use of LLNAR with non-linear reference data and local field potentials (LFPs).

Section snippets

The electrophysiological data

To test the new data analysis methods, recordings were taken from the visual cortex of awake macaque monkeys. Since numerous previous crosscorrelation studies have shown that the likelihood to find synchronous activity generally declines with cortical separation of recording sites (see, e.g. (Ts'o et al., 1986, Ts'o and Gilbert, 1988, Krüger and Aiple, 1988, Engel et al., 1990)), we chose to apply our methods to data obtained from recordings from within the same cortical area in order to

Results for test time series

The methods developed above were applied to two test data sets, and afterwards to LFP data. For the first test a set of 40 time series with a length of 600 data points were generated from a bivariate linear Autoregressive model with p=2. Half the time series were generated as interdependent by construction of the autoregression matrices. The other half was generated as a set of independent time series. In all 40 cases the type of LLNAR model selected corresponded to a bandwidth of infinity,

Discussion

The major points emphasised in this article are:

  • 1.

    The introduction of non-parametric non-linear Autoregressive methods, originally developed for econometrics, for the analysis of neural signals. A significance test for non-linearity of time series is presented.

  • 2.

    The introduction of particular methods for detecting the non-linear character of neural time series and the presence of non-linear interactions. It is now possible to explore these non-linear characteristics of neural data with methods

Acknowledgements

The authors are grateful to Johanna Klon-Lipock and Petra Janson for excellent technical assistance. Detlef Wegener's help with organising the data analysis is gratefully acknowledged as well as the assistance of Sabine Melchert and Sunita Mandon during the literature search. Special thanks are due to the two anonymous referees and to Corrado Bernasconi whose comments during the review process helped us to clarify and generally improve an earlier manuscript. The anatomical MRI scans have been

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