We note y the signal containing the high-frequency activity , and x the signal with slow frequency oscillations , also called the exogenous driver ., When a signal x results from a band-pass filtering step , we note the central frequency of the filter fx and the bandwidth Δfx ., The value of the signal x at time t is denoted x ( t ) ., To estimate PAC , the typical pipeline reported in the literature consists in four main processing steps: The Modulation Index ( MI ) described in the pioneering work of 6 is the mean over time of the composite signal z = a y e ϕ x ., The stronger the coupling between ϕx and ay , the more the MI deviates from zero ., This index has been further improved by Ozkurt et al . with a simple normalization 26 ., Another approach 23 , 44 has been to partition 0 , 2π into smaller intervals to get the time points t when ϕx ( t ) is within each interval , and to compute the mean of ay ( t ) on these time points ., PAC was then quantified by looking at how much the distribution of ay differs from uniformity with respect to ϕx ., For instance , a simple height ratio 44 , or a Kullback-Leibler divergence as proposed by Tort et al . 23 , can be computed between the estimated distribution and the uniform distribution ., Alternatively , it was proposed in 11 to use direct correlation between x and ay ., As this method yielded artificially weaker coupling values when the maximum amplitude ay was not exactly on the peaks or troughs of x , this method was later extended to generalized linear models ( GLM ) using both cos ( ϕx ) and sin ( ϕx ) by Penny et al . 22 ., Other approaches employed a measure of coherence 45 or the phase-locking value 46 ., All these last three approaches offer metrics which are independent of the phase at which the maximum amplitude occurs ., The methods of Tort et al . 23 , Ozkurt et al . 26 , and Penny et al . 22 will be considered for comparison in our experiments ., As one can see , there is a long list of methods to quantify CFC in neural time series ., Yet , a number of limitations which can significantly affect the outcomes and interpretations of neuroscientific findings exist with these approaches ., For example , in typical PAC analysis , a systematic bias rises where one constructs the so-called comodulogram ., A comodulogram is obtained by evaluating the chosen metric over a grid of frequency fx and fy ., This bias emerges from the choice of the bandpass filter , which involves the critical choice of the bandwidth Δfy ., It has been reported several times that to observe any amplitude modulation , the bandwidth of the fast oscillation Δfy has to be at least twice as high as the frequency of the slow oscillations fx: Δfy > 2fx 27 , 47 ., As a comodulogram uses different values for fy , many studies have used a variable bandwidth , by taking a fixed number of cycles in the filters ., The bandwidth is thus proportional to the center frequency: Δfy ∝ fy ., This choice leads to a systematic bias , as it hides any possible coupling below the diagonal fy = 2fx/α , where α = Δfy/fy is the proportionality factor ., Other studies have used a constant bandwidth Δfy; yet this also biases the results towards the low driver frequency fx , considering that it hides any coupling with fx > Δfy/2 ., A proper way to build a comodulogram would be to take a variable bandwidth Δfy ∝ fx , with Δfy > 2fx ., However , this is not common practice as it is computationally very demanding , because it implies to bandpass filter y again for each value of fx ., Another common issue arises with the use of the Hilbert transform to estimate the amplitude and the phase of real-valued signals ., Such estimations rely on the hypothesis that the signals x and y are narrow-band , i . e . almost sinusoidal ., However , numerous studies have used this technique on very wide-band signals such as the entire gamma band ( 80-150 Hz ) 6 ( see other examples in 28 ) ., The narrow-band assumption is debatable for high frequency activity and , consequently , using the Hilbert transform may yield non-meaningful amplitude estimations , and potentially poor estimations of PAC 27 , 28 ., Note also that , in this context , wavelet-based filtering is equivalent to the Hilbert transform 48 , 49 , and therefore does not provide a more valid alternative option ., Besides these issues of filtering and inappropriate use of Hilbert transforms , Hyafil 29 also warned that certain choices of bandwidth Δfy might mistake phase-frequency coupling for PAC , or create spurious amplitude-amplitude coupling; see also the more recent work in 24 for discussion and more practical recommendations for PAC analysis ., Here we advocate that the DAR models detailed in the next sections address a number of the limitations just mentioned ., They do not use bandpass filter or Hilbert transform on the high frequencies y ., They introduce a measure of goodness of fit , through the use of a probabilistic signal model whose quality can be assessed by evaluating the likelihood of the data under the model ., In practice , the likelihood quantifies how much variance of the signal can be explained by the model , and is similar to the R2 coefficient in generalized linear models ( GLM ) ., To the best of our knowledge , the only related model-based approach to measure PAC used GLM 22 ., With GLM , however , the modeling part is done independently on each signal yf , which is the band-pass filtered version of y around frequency f ., For each of these frequencies f a different model is fitted ., By doing so , a GLM approach cannot model the wide-band signal y as it is limited to multiple estimations frequency by frequency bin ., This largely limits the use of the likelihood to compare models or parameters ., On the contrary , we propose to model y globally , without filtering it in different frequency bands ., To conclude this section , and to position this work in the broader context of modeling approaches for neuroscience data , we would like to stress that our proposed method can be considered as an encoding model for CFC , as opposed to a decoding model 50–52 ., Indeed , our model reports how much empirical data can be explained and by doing so enables us to test neuroscience hypotheses in a principled manner 51 ., The literature on the use of non-linear auto-regressive ( AR ) models is quite large and covers fields such as audio signal processing and econometrics ., For instance , AR models with conditional heteroskedasticity ( ARCH 53 , GARCH 54 ) are extremely popular in econometrics where they are used to model signals whose overall amplitude varies as a function of time ., Here , however , in the context of CFC and PAC , one would like to model variations in the spectrum itself , such as shifts in peak frequencies ( a . k . a . frequency modulations ) or changes in amplitude only within certain frequency bands ( a . k . a . amplitude modulations ) ., To achieve this , one idea is to define a linear AR model , whose coefficients are a function of time and change slowly depending on a non-linear function of the signal ., The first models based on this idea are SETAR models 41 , which switch between several AR models depending on the amplitude of the signal with respect to some thresholds ., To get a smoother transition between regimes , SETAR models have inspired other models like EXPAR 55 or STAR 42 , in which the AR coefficients change continuously depending on a non-linear function of the past of the signal ., These models share the same underlying motivation as the DAR models described below but , crucially , DAR models can be designed and parametrized to capture PAC phenomena independently of the phase in the driving signal at which the high frequency content is the strongest ., In other words , DAR models can work equivalently well if the high frequency peaks are in the troughs , the rising phase , the decreasing phase or the peaks of the low frequency driving signal ., Moreover , as DAR models do not require to infer the driving behavior from the signal itself and rather rely on the prior knowledge of the slow oscillation , the inference is significantly faster and more robust ., In this section , we present the outcome of using the model selection procedure to estimate the best filtering frequency fx and bandwidth Δfx to extract the driver x ., We first describe the outcome on simulated signals ( ground truth ) and then on empirical datasets ., Given that DAR models are parametric with a limited number of parameters to estimate , less time samples may be needed to estimate PAC as compared to non-parametric methods ., We tested this assumption using simulated signals of varying duration ., We computed their comodulograms ( as in Fig 7 ) and selected the frequencies of maximum coupling ., For each duration , we simulated 200 signals , and plotted the 2D histogram showing the fraction of time each frequency pairs corresponded to a maximum ., We then compared the same four methods: DAR models with ( p , m ) = ( 10 , 1 ) , the GLM-based model 22 , and two non-parametric methods 23 , 26 ., Results shown in Fig 9 show that parametric approaches provided a more robust estimation of PAC frequencies with short signals ( T = 2 sec ) than non-parametric methods ., The robustness to small sample size is a key feature of parametric models , as it significantly improves PAC analysis during shorter experiments ., When undertaking a PAC analysis across time using a sliding time window , parametric models should therefore provide more robust PAC estimates ., Note that the specific time values in these simulations should not be taken as general guidelines as they depend on the simulation parameters such as the signal-to-noise ratio ., However , across all tests , parametric methods consistently provided more accurate results than non-parametric ones ., One can note that in DAR models , the driver contains not only the phase of the slow oscillation , but also its amplitude ., As the driver is not a perfect sinusoid , its amplitude fluctuates with time ., On the contrary , most PAC metrics discard the amplitude fluctuations of the slow oscillation and only consider its phase ., To evaluate these two options , we compared two drivers using DAR models: the original ( complex ) driver x ( t ) , and the normalized driver x ˜ ( t ) = x ( t ) / | x ( t ) | ., This normalized driver only contains the phase information , as in most traditional PAC metrics ., Using cross-validation , we compared the log-likelihood of four fitted models , and found a difference always in favor of the non-normalized driver x ( t ) , as it can be visualized in S3 Fig . This result shows that the coupling phenomenon is associated with amplitude fluctuations , a kind of phase/amplitude-amplitude coupling , as it was previously observed in 25 ., Indeed , the GLM parametric method 22 was improved when taking into account the amplitude of the slow oscillation ., Here , we use our generative model framework to provide an easy comparison tool through the likelihood , to validate this neuroscientific insight from the signals ., In this section , we report the results of the directionality estimates using both simulations and neurophysiological signals ., It is noteworthy that in DAR models , we arbitrarily call driver the slow oscillation although the model makes no assumption on the directionality of the coupling ., Cross-frequency coupling ( CFC ) and phase-amplitude coupling ( PAC ) more specifically have been proposed to play a fundamental role in neural processes ranging from the encoding , maintenance and retrieval of information 3–5 , 8 , 17 , 81 , 82 , to large-scale communication across neural ensembles 7 , 19 , 83 , 84 ., While a steady increase in observations of PAC in neural data has been seen , how to best detect and quantify such phenomena remains difficult to settle ., We argue that a method using DAR models , as described here , is rich enough to capture the time-varying statistics of brain signals in addition to provide efficient inference algorithms ., These non-linear statistical models are probabilistic , allowing the estimation of their goodness of fit to the data , and allowing for an easy and fully controlled comparison across models and parameters ., In other words , they offer a unique principled data-driven model selection approach , an estimation strategy of phase/amplitude-amplitude coupling based on the approximation of the actual signals , a better temporal resolution of dynamic PAC and the estimation of coupling directionality ., One of the main features of PAC estimation through our method is the ability to compare models or parameters on non-synthetic data ., On the contrary , traditional PAC metrics cannot be compared on non-synthetic data , and two different choices of parameters can lead to different interpretations ., There is no legitimate way to decide which parameter shall be used with empirical data using traditional metrics ., The likelihood of the DAR model that can be estimated on left-out data offers a rigorous solution to this problem ., We presented here results on both simulated signals and empirical neurophysiological signals ., The simulations gave us an illustration of the phenomenon we want to model , and helped us understand how to visualize a fitted DAR model ., They also served a validation purpose for the bandwidth selection approach that we performed on real data ., Using the data-driven parameter selection on non-synthetic signals , we showed how to choose sensible parameters for the filtering of the slow oscillation ., All empirical signals are different , and it was for example reported in the neuroscience literature that peak frequencies vary between individuals 85 and that this should not be overlooked in the analysis of the data ., The parameter selection based on fitted DAR models makes it possible to fit parameters on individual datasets ., Our results also shed light on the asymmetrical and wide-band properties of the slow oscillation , which could denote crucial features involved in cognition 32 ., The second novelty of our method stands in considering the amplitude fluctuations of the slow oscillation in the PAC measure and not only its phase ., Using the rodent and human data , we showed that the instantaneous amplitude of the slow oscillation influences the coupling in PAC , as it was previously suggested in 25 ., The amplitude information should therefore not be discarded as it is done by existing PAC metrics ., For instance , the measure of alpha/gamma coupling reported during rest 86 , 87 should incorporate alpha fluctuations when studied in the context of visual tasks 88 , as an increase of alpha power is often concomitant with a decrease of gamma power 89 ., The comparison between DAR models considering or not these low-frequency power fluctuations would inform on the nature of the coupling: purely phase-amplitude , or rather phase/amplitude-amplitude ., In Tort et al . 14 , both theta power changes and modulation of theta/gamma PAC were reported in rats having to make a left or right decision to find a reward in a maze ., The use of our method could decipher whether the changes in coupling were related to the changes in power , informing on the underlying mechanisms of decision-making ., Moreover , as our method models the entire spectrum simultaneously , a phase-frequency coupling could potentially be captured in our models ., Therefore , our method is not limited to purely phase-amplitude coupling , and extends the traditional CFC analysis ., Furthermore , in those types of experiments , changes in PAC can be very fast depending on the cognitive state of the subject ., Therefore , the need for dynamic PAC estimates is growing 14 ., We showed with simulations that DAR models are more robust than non-parametric methods when estimating PAC on small time samples ., This robustness is critical for time-limited experiments and also when analyzing PAC across time in a fine manner , typically when dynamic processes are at play ., Last but not least , likelihood comparison can also be used to estimate the delay between the coupled components , which would give new insights on highly debated questions on the role of oscillations in neuronal communication 90 , 91 ., For example , a delay close to zero could suggest that the low and high frequency components of the coupling might be generated in the same area , whereas a large delay would suggest they might come from different areas ., As an alternative interpretation , the two components may come from the same area , but the coupling mechanism itself might be lagged ., In this case , a negative delay would suggest that the low frequency oscillation is driven by the high frequency oscillations , whereas a positive delay would suggest that the low frequency oscillation drives the high frequency amplitude modulation ., In any case , this type of analysis will provide valuable information to guide further experimental questions ., A recent concern in PAC analysis is that all PAC metrics may detect a coupling even though the signal is not composed of two cross-frequency coupled oscillators 30 , 92–95 ., It may happen for instance with sharp slow oscillations , described in humans intracranial recordings 68 ., Sharp edges are known not to be well described by a Fourier analysis , which decomposes the signal in a linear combination of sinusoids ., Indeed , such sharp slow oscillations create artificial high frequency activity at each sharp edge , and these high frequencies are thus artificially coupled with the slow oscillations ., This false positive detection is commonly referred to as “spurious” coupling 96 ., Fig 12 shows a comodulogram computed on a simulated spurious PAC dataset , using a spike train at 10 Hz , as described in 94 ., The figure shows that all four methods , including the proposed one , detect some significant PAC , even though there is no nested oscillations in the signal ., Even though our method does not use filtering in the high frequencies , it does not solve this issue and is affected in the same way as other traditional PAC metrics ., Indeed , our work shed light on the wide-band property of the slow oscillations , but DAR models cannot cope with full-band slow oscillations , which contain strong harmonic components in the high frequencies ., However , we consider that such “spurious” PAC can also be a relevant feature of a signal , as stated in 68 ., In their study , they show that abnormal beta oscillations ( 13-30 Hz ) in the basal ganglia and motor cortex underlie some “spurious” PAC , but are actually a strong feature associated with Parkinson’s disease ., A robust way to disentangle the different mechanisms that lead to similar PAC results remains to be developed ., The method we presented in this paper uses univariate signals obtained invasively in rodents or humans ., As a lot of neurophysiological research uses non-invasive MEG or EEG recordings containing multiple channels , a multivariate analysis could be of high interest ., One way to use data from multiple channels is to estimate a single signal using a spatial filter such as in 97 ., Such a method is therefore complementary to univariate PAC metrics like ours which can be applied to the output of the spatial filter ., The method from 97 builds spatial filters that maximize the difference between , say , high-frequency activity that appears during peaks of a low-frequency oscillation versus high-frequency activity that is unrelated to the low-frequency oscillation ., Again , from the signal obtained with the spatial filter , it is straightforward to adapt most PAC metrics such as our method ., Neurophysiological signals have all the statistical properties to make them a challenge from a signal processing perspective ., They contain non-linearities , non-stationarities , they are noisy and they can be long , hence posing important computational challenges ., Our method based on DAR models offer novel and more robust possibilities to analyse neurophysiological signals , paving the way for new insights on how our brain functions via spectral interactions using local or distant coupling mechanisms ., Inline with the open science philosophy of this journal , our method is fully available as an open source package that comes with documentation , tests , and examples: https://pactools . github . io .
We address the issue of reliably detecting and quantifying cross-frequency coupling ( CFC ) in neural time series ., Based on non-linear auto-regressive models , the proposed method provides a generative and parametric model of the time-varying spectral content of the signals ., As this method models the entire spectrum simultaneously , it avoids the pitfalls related to incorrect filtering or the use of the Hilbert transform on wide-band signals ., As the model is probabilistic , it also provides a score of the model “goodness of fit” via the likelihood , enabling easy and legitimate model selection and parameter comparison; this data-driven feature is unique to our model-based approach ., Using three datasets obtained with invasive neurophysiological recordings in humans and rodents , we demonstrate that these models are able to replicate previous results obtained with other metrics , but also reveal new insights such as the influence of the amplitude of the slow oscillation ., Using simulations , we demonstrate that our parametric method can reveal neural couplings with shorter signals than non-parametric methods ., We also show how the likelihood can be used to find optimal filtering parameters , suggesting new properties on the spectrum of the driving signal , but also to estimate the optimal delay between the coupled signals , enabling a directionality estimation in the coupling .
Neural oscillations synchronize information across brain areas at various anatomical and temporal scales ., Of particular relevance , slow fluctuations of brain activity have been shown to affect high frequency neural activity , by regulating the excitability level of neural populations ., Such cross-frequency-coupling can take several forms ., In the most frequently observed type , the power of high frequency activity is time-locked to a specific phase of slow frequency oscillations , yielding phase-amplitude-coupling ( PAC ) ., Even when readily observed in neural recordings , such non-linear coupling is particularly challenging to formally characterize ., Typically , neuroscientists use band-pass filtering and Hilbert transforms with ad-hoc correlations ., Here , we explicitly address current limitations and propose an alternative probabilistic signal modeling approach , for which statistical inference is fast and well-posed ., To statistically model PAC , we propose to use non-linear auto-regressive models which estimate the spectral modulation of a signal conditionally to a driving signal ., This conditional spectral analysis enables easy model selection and clear hypothesis-testing by using the likelihood of a given model ., We demonstrate the advantage of the model-based approach on three datasets acquired in rats and in humans ., We further provide novel neuroscientific insights on previously reported PAC phenomena , capturing two mechanisms in PAC: influence of amplitude and directionality estimation .
acoustics, medicine and health sciences, sine waves, engineering and technology, signal processing, vertebrates, electrophysiology, neuroscience, animals, mammals, signal filtering, bandwidth (signal processing), research and analysis methods, mathematical functions, mathematical and statistical techniques, physics, rodents, bandpass filters, eukaryota, speech signal processing, physiology, biology and life sciences, physical sciences, amniotes, neurophysiology, organisms, acoustic signals