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J Neurosci. Author manuscript; available in PMC 2010 December 23.
Published in final edited form as:
PMCID: PMC2923068

Nonlinear coupling in the human motor system


The synchronous discharge of neuronal assemblies is thought to facilitate communication between areas within distributed networks in the human brain. This oscillatory activity is especially interesting, given the pathological modulation of specific frequencies in diseases affecting the motor system. Many studies investigating oscillatory activity have focussed on same frequency, or linear, coupling between areas of a network. In this study, our aim was to establish a functional architecture in the human motor system responsible for induced responses as measured in normal subjects with magnetoencephalography. Specifically, we looked for evidence for additional nonlinear (between-frequency) coupling among neuronal sources and, in particular, whether nonlinearities were found predominantly in connections within areas (intrinsic), between areas (extrinsic) or both. We modelled the event-related modulation of spectral responses during a simple hand-grip using dynamic casual modelling. We compared models with and without nonlinear connections under conditions of symmetric and asymmetric interhemispheric connectivity. Bayesian model comparison suggested that the task-dependent motor network was asymmetric during right hand movements. Furthermore, it revealed very strong evidence for nonlinear coupling between sources in this distributed network, but interactions among frequencies within a source appeared linear in nature. Our results provide empirical evidence for nonlinear coupling among distributed neuronal sources in the motor system and that these play an important role in modulating spectral responses under normal conditions.

Keywords: Nonlinear coupling, neuronal dynamics, induced responses, Dynamic Causal Modelling, spectral responses, motor network


Oscillatory activity is a ubiquitous feature of neuronal dynamics. It has been suggested that the synchronous discharge of neuronal assemblies might provide a mechanism to couple distributed processing over different neuronal populations (Singer and Gray, 1995; Kahana et al., 1999; Rodriguez et al., 1999). The central idea is that oscillations facilitate integration both within functionally segregated brain areas and between areas engaged by the same task. Many studies investigating oscillatory activity have focussed on linear coupling between nodes of a network at the same frequency (Andrew and Pfurtscheller, 1996; Gerloff et al., 1998; Gross et al., 2001; Serrien et al., 2005). More recently, evidence has emerged that suggests nonlinear coupling among different frequencies may play an equally important role in inter-areal communication (Breakspear, 2002; Chen et al., 2009; Jensen and Colgin, 2007; Tallon-Baudry and Bertrand, 1999; Varela et al., 2001; von Stein and Sarnthein, 2000). In this work, we use magnetoencephalography (MEG) and a recently validated method (dynamic causal modelling for induced responses; DCM for IR) (Chen et al. 2008) to characterise the modulation of frequency-dependent coupling among neuronal sources during action execution.

Hand movements have been shown to modulate oscillatory power at different frequencies, such as alpha event-related desynchronisation (ERD) and beta event-related synchronisation (ERS) in brain areas engaged by action execution; e.g., primary motor cortex (M1), supplementary motor area (SMA) and premotor areas (PM) (Pfurtscheller and Andrew, 1999). Movement-related modulations in power have been seen consistently across different spatial scales, from single unit recordings, through local field potential recordings, to macroscopic measures such as electroencephalogram (EEG) or magnetoencephalogram (MEG) (Crone et al., 1998a; Crone et al., 1998b; Kilner et al., 2003; Leocani et al., 1997). Given that action execution induces changes in different frequencies within connected brain regions, we wanted to model how these oscillations are orchestrated during motor control. The nature of this frequency-specific coupling is important, given that excessive synchronization at distinct frequencies is seen in some pathological conditions. For example, in Parkinson’s patients, synchronization at 4~6 Hz between the contralateral primary motor cortex and forearm muscles is thought to contribute to resting tremors, while excessive synchrony at 10- to 35 Hz in basal ganglia / subthalamic nucleus is associated with bradykinesia (for review, see Brown, 2007). The motivation for the work reported here was to establish a reference point for future clinical studies, using normal subjects and a simple established paradigm. This normative reference might help pinpoint where abnormal modulations of specific frequencies arise.

The aim of this study was to model modulations of frequency-specific oscillations in the motor network induced by an established handgrip task (Ward et al., 2008). We modelled these modulations in terms of coupling between electromagnetic sources, where power in one source causes changes in the power expressed in others. Critically, we distinguished between within-frequency linear coupling and between-frequency nonlinear coupling. Specifically, we asked whether there is a difference in the relative contribution of linear and nonlinear mechanisms between intrinsic and extrinsic coupling. Our results suggest that extrinsic connectivity is best characterised as nonlinear coupling, whereas intrinsic connections are best modelled with linear coupling.

Material and Methods

Nine healthy, right-handed subjects (20~32 years of age, 5 female) were recruited. Written consent was obtained from all subjects, in accordance with the Declaration of Helsinki. The study was approved by the Joint Ethics Committee of the Institute of Neurology, UCL and National Hospital for Neurology and Neurosurgery, UCL Hospitals NHS Foundation Trust, London.

Subjects were instructed to perform a visually cued ballistic isometric grip, using their dominant hand with an inter-trial interval of 7±2 secs. Prior to scanning, subjects were asked to grip the manipulandum to generate a maximum voluntary contraction (MVC). The target force was set at 45% of MVC. Subjects were trained to approximate the target force with visual feedback prior to scanning. However, no visual feedback was provided during scanning, in order to minimise activity in occipital and parietal sources. Force output was recorded using a MEG-compatible gripper and used to identify the movement onset (i.e. the reaction time, from the onset of the visual cue until the onset of the ballistic grip), the grip duration and force level.

MEG signals were measured continuously at 240 Hz during task performance using a whole-head CTF Omega 275 MEG system. At the beginning and end of each measurement, the positions of three anatomical landmarks (bilateral pre-auricular points and nasion) were recorded to exclude excessive head movement (thresholded at 1.5cm and the measured maximal translation across subjects < 1.3 cm; 2.68~12.68 mm).

The MEG data were pre-processed offline using SPM8 (SPM8, Wellcome Trust Centre for Neuroimaging, The data were epoched from −500 to +1000 ms, where time zero indicates movement onset. Poorly performed (reaction times of more than one sec) and artefact contaminated (MEG amplitude > 500 fT) trials were excluded from further analysis; resulting in 88-98 artefact-free epochs (88 98 90 98 94 96 90 93 95) with 642.66±54.92 ms mean reaction time and 639.45±54.48 ms grip duration. The mean force level was 45±25 % of subject-specific MVC. These artefact-free epochs were projected from channel space to the sources using the generalised inverse of the lead-field matrix for our chosen sources (see Model specification below). The spectral density from 4-48Hz at each source was computed over peri-stimulus time using a time-frequency Morlet wavelet transform (wavelet number: 7). The frequency ranges cover the theta (4-8 Hz), alpha (8-15 Hz), beta (15-30 Hz), and gamma (>30 Hz) bands. The absolute values of the resulting time-frequency responses were averaged over trials and baseline-corrected by subtracting the frequency-specific power of the first time-bin. For computational expediency, we reduced the dimensionality of spectra into four principal frequency components derived from a Singular Value Decomposition of the spectra so that the extracted frequency modes are subject-specific. This procedure accounts for the huge inter-individual variability of engaged frequencies in motor system (Omlor et al., 2007; Aoki et al., 1999; Kilner et al., 2000; Kristeva et al., 2007), particularly within the alpha band (Pfurtscheller and Lopes da Silva, 1999). Those singular components preserve over 93 % of the spectral variance in all subjects (range 93% ~ 97%) of which 68.97% (ranged 56.03 % ~ 78.70 %) of total energy was contributed by the low frequency components (≤ 20 Hz). The resulting spectral dynamics enter DCM as the observations that the model is trying to explain.

DCM specification (sources and coupling)

The source locations were taken from the group results of an fMRI study using the identical task, where five subjects performed 25 ballistic isometric hand grips to 45% of MVC. Imaging data were analysed using SPM8 as described elsewhere (Ward et al., 2008). The localisations were taken as the peak coordinate in Montreal Neurological Institute (MNI) space within each significant cluster (voxels significant at p < 0.05, corrected for multiple comparisons across the whole brain). Peak increases in activity were seen in left primary motor cortex, (M1), bilateral premotor cortices (PM) and left supplementary motor area (SMA). In addition, right M1 was included because of significant task-related deactivation during hand grip secondary to transcallosal interhemispheric inhibition (Ward et al., 2008). Using these five sources we then specified 12 different connectivity models as shown in Figure 1.

Figure 1
The connectivity architectures for all the models considered. The models in the four columns differ according to whether the intrinsic or extrinsic connections are linear or nonlinear. The models in the different rows differ in the deployment of inter-hemispheric ...

This model space was constructed using three model attributes. First, whether intrinsic (I) connections are linear (L) or non-linear (N) and second, whether extrinsic (E) connections are linear or non-linear. This results in four sorts of models. The third attribute was the lateralization of cross-hemispheric coupling between PM and M1. We modelled three levels of this lateralisation: right PM to left M1 (r-), or left PM to right M1 (l-) or both (b-). This gave a total of 12 models, as shown in Figure 1. We use the designation r-ILEL to mean a right-lateralised architecture, where Intrinsic connections are Linear and Extrinsic connections are Linear.

Dynamic causal modelling for induced responses

Dynamic causal modelling (DCM) entails the use of dynamic or state-space models to explain observed time-series of data features (Friston et al., 2003; David et al.,2006). Recently, we described a dynamic causal model for spectral responses as measured by EEG or MEG (Chen et al., 2008) and have used it to assess nonlinear coupling among visual areas during face processing (Chen et al, 2009). In brief, DCM for IR enables the user to model event-related spectral responses, over a range of frequencies, as the response of a distributed set of coupled electromagnetic sources to a spectral perturbation. The model parameters encode the frequency response of each source to exogenous input and the coupling among sources and different frequencies. Bayesian inversion of these models allows one to compare different models or hypotheses and enables inferences about the parameters of the best model.

Inference on models: Bayesian Model Comparison (BMC)

At a single subject level, we compared the difference in log model evidence or marginal likelihoods between models, i.e. log Bayes factor (Penny et al. 2004) to identify the best among the models tested. To identify the model with the most evidence at the group level we added the log-evidences from each subject, under the assumption that each subject’s data are conditionally independent of each other. To ensure differences in log-evidence were consistent across subjects, the log-evidences for each model, over the nine subjects, were entered into a repeated-measures analysis of variance (ANOVA) with three within subject factors (distance with two levels, nonlinearity with two levels and laterality with three levels – corresponding to the columns and rows of Figure 1 respectively).

Model parameters: Visualization of coupling matrices

To quantify the coupling under the best model, for each intrinsic or extrinsic connection the (frequency to frequency) matrices of subject-specific estimates were smoothed (to account for inter-subject variability in frequency-to-frequency coupling using a Gaussian kernel with Full-width half-maximum of 8 Hz). These were then averaged by entering them into a conventional SPM analysis to identify reliable frequency-specific connectivity. We report the average coupling strengths in, and only in, non-zero frequency bins (at p<0.005 uncorrected).


Time-frequency responses at source level

The estimated event-related spectral responses at the source level of a representative subject are shown in Figure 2 (upper panel), where the alpha power decreases bilaterally in M1 from movement onset to 300 ms. This pattern was observed consistently across all subjects. Transient beta power increases were seen in bilateral M1, SMA and left PM (6 out of 9 subjects), and enhanced gamma was seen in SMA and PM bilaterally (8 out of 9). We also observed transient bilaterally M1 beta power decreases in some subjects (5 out of 9). Taken together, these time-frequency responses are in line with previous findings (Crone et al., 1998a; Crone et al., 1998b; Kilner et al., 2003; Leocani et al., 1997). The lower panel in Figure 2 shows the predicted spectral responses, at the source level, by the best (l-ILEN) model (see ‘Inference on models’ for details). Note that the spectra are normalized individually with respect to their maximum. It is clear that the model can explain the data very well. On average, the l-ILEN model explains 94.17% data variance (range 86.78~98.01%).

Figure 2
The observed (upper panel) and predicted (lower panel) spectral responses for a representative subject under the best mode (l-ILEN) from the Bayesian model comparison.

Inference on models: Nonlinear effect and motor networks

Figure 3 summarizes the results of our model comparison. It is immediately obvious that the models which fit the data best are those with nonlinear extrinsic coupling (Figure. 3A). Models with one or more nonlinear connection were consistently better than purely linear models in every subject (Figure. 3B) and an ILEN model was the best in six out of nine subjects. In addition, model comparison also clearly shows strong evidence for model ILEN, summed over the factors of lateralization and subjects (assuming all subjects expressed the same unknown model; see Stephan et al., 2009 and Figure. 3C). This is supplemental evidence that shows we can differentiate the nonlinear intrinsic coupling from linear coupling, when the extrinsic coupling is nonlinear. The l-ILEN model was the best model amongst those tested at the group level, with a Bayes Factor of 1857 (l-ILEN vs. r-INEN; i.e. the best v.s the next best). Bayes Factor measures the marginal likelihood ratio of two models using their model evidence and usually a Bayes Factor of about twenty is considered as “strong” evidence in favour of one model relative to another (Penny et al. 2004). It is important to note that the most complex model (b-INEN) is not necessarily the best (only one subject had this model as the best). This is because the log-evidence includes a penalty term for complexity (see Friston, 2002, Friston et al., 2003 and Penny et al., 2004 for details). The repeated-measure ANOVA confirms our BMS result that there is strong evidence for nonlinearity (F(1,7)=15.355, p=0.006), distance (F(1,7)=19.539, p=0.003) and their interaction (F(1,7)=15.615, p=0.006) in terms of consistency over subjects. This mirrors the BMS conclusion that extrinsic connectivity was nonlinear while intrinsic connectivity was essentially linear. There was no effect of laterality (levels: right, bilateral and left lateralized models; F(1.13,9.14)=0.951, p=0.381) nor any interaction with the other two factors.

Figure 3
Results of Bayesian Model Selection (BMS). (A) Pooled log-evidences of the twelve DCMs tested). It can be seen that the best model is l-ILEN (log-evidence = −24297) and the next best is b-ILEN (log-evidence = −25880). (B) Comparison of ...

In summary, in normal subjects we found very strong evidence for nonlinear coupling between areas and linear interactions between areas. Note that our conclusion about intrinsic linear coupling is based on the fact that there was no consistent evidence for intrinsic nonlinear coupling. This lack of evidence might suggest that our data were simply insufficient to reveal a small nonlinear contribution within area; however, the clear evidence for nonlinear effects in the extrinsic connections suggests that, at least quantitatively, nonlinearities predominate in extrinsic, relative to intrinsic interactions. In addition, there is evidence for an asymmetry of interhemispheric interactions in the right hand movement task we used; although this was less consistent over subjects.

Coupling parameters

Statistical tests are repeated for each element of the coupling parameter matrices from all subjects, under the l-ILEN model. Each of these matrices represents the frequency-to-frequency coupling associated with one connection. The corresponding SPMs of the T-statistic (thresholded at p<0.005 uncorrected) are shown for ‘excitatory’ (positive; Figure 4A) and ‘inhibitory’ (negative; Figure 4B) effects respectively. As seen in Figure 4, we found several instances of consistent nonlinear interactions. A summary of these T-test results is provided in Table 1. These seem to be more profound when the coupling is negative. For instance, several consistent regions of negative coupling are found far from the (within-frequency), leading diagonal compared to the positive coupling SPMs. When considering reciprocal connections, the frequencies entailed by nonlinear coupling appear asymmetric. For example, in Figure 4B (arrows), the negative coupling from LPM to RM1 involves alpha-gamma coupling, while the reciprocal RM1 to LPM connection shows significant gamma-theta coupling. In the same vein, theta oscillations in SMA suppress gamma oscillations in LM1 but, from LM1 to SMA, the negative coupling was between gamma and alpha. The more quantitative characterisations of the nonlinear coupling identified by Bayesian model comparison speak to the complicated nature of nonlinear interactions in the brain, even when modelled as simply as with DCM for induced responses. Note that the significance of the negative linear intrinsic coupling in Figure is due largely to prior constraints on the parameters. We use these priors to ensure the system is dissipative (see Friston et al., 2003 for discussion of this constraint). In other words, the system will decay to a stable fixed point (equilibrium point) after a period of time. Given that we know the rough time-constants of induced electromagnetic responses, we can use this information to place a prior expectation on the decay rates, following experimental perturbation. This decay is implemented by intrinsic (self) inhibition (c.f. Friston et al 2003).

Figure 4
Statistical results of coupling parameters. (A) SPM {F} of the positive coupling strengths (one-tailed), thresholded at p<0.005 uncorrected. This shows regions of frequency-frequency space, where the average coupling was great than zero. (B) SPM ...
Table 1
Summary of the SPM analysis of the coupling parameters


In this study, we provide empirical evidence for nonlinear coupling among distributed neuronal sources in the motor system and that these play an important role in modulating spectral responses under normal conditions. Interestingly, we found no evidence for nonlinear or between-frequency coupling intrinsic to each source, suggesting that linear or driving mechanisms may be a sufficient account of interactions among local neuronal populations. In addition, we found evidence for an asymmetric inter-hemispheric interaction involving right PM in our right-hand movement task. A quantitative examination of the extrinsic or long-range coupling parameters, showed some interesting asymmetries in frequency space and that this coupling was predominantly negative or suppressive.

Intrinsic (local) linear and extrinsic (global) nonlinear effects

A recent pharmacological study in rat brains has shown that co-application of kainic acid and carbachol to layer V in M1 can reliably induce synchronous oscillatory activity in the beta frequency band in layer II to VI (Yamawaki et al., 2008). These results imply that inter-laminar influences may be mediated by driving or linear effects because they induce distributed oscillations at the same frequency. Furthermore, it has been shown that in pathological conditions such as Parkinsonism, abnormal oscillatory synchronization of neuronal populations in cortex, subthalamic nucleus and basal ganglia can lead to movement impairment (Brown, 2007; Levy et al., 2002; Marceglia et al., 2006; Priori et al., 2004). Our findings suggest that the local interactions may be predominantly linear under normal conditions. Other studies have demonstrated nonlinear coupling in EEG/MEG signals in a variety of tasks, systems and pathological conditions (Breakspear, 2002;Antoniou et al., 2004; Breakspear, 2002; Chavez et al., 2003; Kotini and Anninos, 2002) leading to suggestions that nonlinear coupling is an important aspect of functional integration (Chen et al, 2009; Jensen and Colgin, 2007; Tallon-Baudry and Bertrand, 1999; Varela et al., 2001). We have shown that during the performance of a simple motor task both linear and nonlinear coupling is likely to be present. Specifically, our results demonstrate that local interactions can be explained by linear coupling, but that coupling between regions is nonlinear in nature.

In terms of cross-frequency coupling, bicoherence and nested oscillations are other important expressions of nonlinear coupling that may reflect key physiological mechanisms of functional integration and dynamic coordination in the brain. Bicoherence has been intensively used as an index of anaesthesia (Hagihira et al., 2002; Hayashi et al., 2008; Jeleazcov et al., 2005) while nested oscillations are observed largely during memory tasks, such as theta-gamma, theta-beta and theta-gamma, theta-beta/gamma coupling (Lisman and Idiart, 1995; Mormann et al. 2005; Schack et al. 2002) and during sleep (infra-slow oscillations ; ISOs: 0.02 −0.2 Hz – 1 Hz) (Vanhatalo et al., 2004) and even during the resting state :alpha-high gamma frequencies (Osipova et al. 2007). However, we are not aware of any applications within the central motor system using these methods. Our results suggest that nonlinearities in coupling are expressed predominantly in extrinsic connections. This may reflect the fact afferent inputs from other regions exert nonlinear effects on intrinsic dynamics. This nonlinearity may be at the level of post-synaptic receptors (e.g., NMDA receptors that are preferentially targeted by extrinsic backward connections in the brain); or at the level of lumped time-constants governing intrinsic population dynamics.

Asymmetry of inter-hemispheric connections

Asymmetry in the human brain architectures has been shown in many studies, both functionally or anatomically (Amunts et al., 1996; Friston, 2005; Rockland and Pandya, 1979; Zeki and Shipp, 1988). In the motor system, handedness and experience-dependent plasticity are thought to be the main factors subtending these asymmetric properties (Haaland et al., 2000; Karni et al., 1995; Kloppel et al., 2007). Moreover, functional lateralization of motor control can be altered by pathological or traumatic changes (Ward et al., 2004). In this study, we were able to quantify task-dependent frequency-specific causal influences mediating the observed spectral responses and characterise functional asymmetry in terms of long-range coupling. Bayesian model comparison suggested that the left lateralised ILEN model was superior to the symmetric homologue. Our left-lateralised model conforms to the left hemisphere dominance, expressed for example in the asymmetric engagement of premotor cortex during skilled movement in right-handed adults (Pollok et al., 2006).

Asymmetry of hierarchical connections

In addition to hemispheric asymmetries, frequency-specific coupling was distinct in forward and backward connections, especially between the SMA and premotor sources. Furthermore, predominant positive and negative couplings are located in different frequency bands in most connections. For example, the gamma rhythm in left M1 inhibits the alpha activity in SMA but no consistent positive connection was found.

It is important to establish the normal pattern of the frequency-specific interactions in the motor system because several movement disorders show frequency-related abnormalities, such as resting tremors (4~6 Hz) and bradykinesia (10- to 35 Hz) (Brown, 2007). However, the details of the underlying mechanisms remain largely unknown. Our study provides a qualitative and quantitative characterisation of frequency-specific effects under normal conditions, which we hope will be useful when studying induced responses in patients. To our knowledge, this is the first study of frequency-specific coupling in the motor network under normal conditions. Given that, in the motor system, induced responses depend on the task and show substantial between-subject variability (Aoki et al., 1999; Kilner et al., 2000; Kristeva et al., 2007; Omlor et al., 2007), we do not anticipate out results will generalize to other movement-related networks. Rather, we consider our study as a reference point for similar studies in patients using the same paradigm.

The question of model specification

It is important to note that any DCM analysis is conditioned on the models considered. For example, our conclusions are specific to the particular sources that we chose to model the data. Specifically, we mapped the observed MEG data into source space by multiplying the generalized inverse lead field matrix given the source locations (see Chen et al, 2008 for details). This generalised inverse of the lead-field is one of many inversion schemes that can be used to project data from channel to source space (Darvas et al., 2004; Friston et al., 2008; Kiebel et al., 2008; Michel et al., 2004). The advantage of this projection is that there is a unique solution for the data features, given the prior specification of source locations. This approach leads to a source spectrum, which is a reasonable summary of the real neuronal sources generating the data. Therefore, if any sources are omitted or misplaced, there might be a better model of the data and possibly a different conclusion from model comparison. If one did not know where the spectral signals were generated, the beam-former method could be one useful strategy that allows one to localize the source positions and estimate spectral features empirically (Singh et al 2003). Once these locations have been established, the generalised inverse of the associated lead-field matrix furnishes a near-optimum Equivalent Current Dipoles (ECD) summary of activity that avoids suppression of local correlated activity. In this study, we adopted fMRI results as our prior on source locations. This is based on the fact that there is a congruity between fMRI activity and the corresponding MEG dipoles in sensorimotor cortex (Korvenoja et al., 2006; Roberts and Rowley, 1997; Stippich et al., 1998). The sensitivity of our conclusions to the particular sources chosen may not be very severe for two reasons. First, there is little localising information in electromagnetic signals. This means that if we had changed the source locations, the data features would not change very much (this is because we pool spectral estimates over all orientations and the orientation of the implicit dipole is essentially free). Second, including more sources would not affect the conclusions about effective connectivity; in the sense that effective connectivity can be polysynaptic and could be mediated by intervening (un-modelled) sources. The only worry here is that choosing the wrong number of sources leaves variance in the data-features unexplained (or wrongly attributed). This is unlikely, given the high proportion of variance explained by the sources we used.

In conclusion, we have established the prevalence of nonlinear or between-frequency coupling among distributed components of the motor system during a simple motor task. These extrinsic nonlinear interactions appear to unfold in the context of local or intrinsic linear coupling within each area. The associated task-dependent motor network has asymmetric features, as reflected in both the deployment of connections and the frequency specificity of reciprocal connections. In a future paper, we will show how this motor network is affected by healthy aging and by corticospinal system damage secondary to stroke.


The Wellcome Trust funded this work. CC Chen is funded by National Science Council of Taiwan (TMS-094-1-A-037). The authors thank Peter Aston and Eric Featherstone for the design and programming involved in creating the hand grip manipulandum and Dr. Vladimir Litvak for comments on the presentation of this work


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