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Cross-frequency coupling (CFC) has been proposed to coordinate neural dynamics across spatial and temporal scales. Despite its potential relevance for understanding healthy and pathological brain function, the standard CFC analysis and physiological interpretation come with fundamental problems. For example, apparent CFC can appear because of spectral correlations due to common non-stationarities that may arise in the total absence of interactions between neural frequency components. To provide a road map towards an improved mechanistic understanding of CFC, we organize the available and potential novel statistical/modeling approaches according to their biophysical interpretability. While we do not provide solutions for all the problems described, we provide a list of practical recommendations to avoid common errors and to enhance the interpretability of CFC analysis.
Poster presentation from Twentieth Annual Computational Neuroscience Meeting: CNS*2011 Stockholm, Sweden. 23-28 July 2011. One of the central questions in neuroscience is how neural activity is organized across different spatial and temporal scales. As larger populations oscillate and synchronize at lower frequencies and smaller ensembles are active at higher frequencies, a cross-frequency coupling would facilitate flexible coordination of neural activity simultaneously in time and space. Although various experiments have revealed amplitude-to-amplitude and phase-to-phase coupling, the most common and most celebrated result is that the phase of the lower frequency component modulates the amplitude of the higher frequency component. Over the recent 5 years the amount of experimental works finding such phase-amplitude coupling in LFP, ECoG, EEG and MEG has been tremendous (summarized in [1]). We suggest that although the mechanism of cross-frequency-coupling (CFC) is theoretically very tempting, the current analysis methods might overestimate any physiological CFC actually evident in the signals of LFP, ECoG, EEG and MEG. In particular, we point out three conceptual problems in assessing the components and their correlations of a time series. Although we focus on phase-amplitude coupling, most of our argument is relevant for any type of coupling. 1) The first conceptual problem is related to isolating physiological frequency components of the recorded signal. The key point is to notice that there are many different mathematical representations for a time series but the physical interpretation we make out of them is dependent on the choice of the components to be analyzed. In particular, when one isolates the components by Fourier-representation based filtering, it is the width of the filtering bands what defines what we consider as our components and how their power or group phase change in time. We will discuss clear cut examples where the interpretation of the existence of CFC depends on the width of the filtering process. 2) A second problem deals with the origin of spectral correlations as detected by current cross-frequency analysis. It is known that non-stationarities are associated with spectral correlations in the Fourier space. Therefore, there are two possibilities regarding the interpretation of any observed CFC. One scenario is that basic neuronal mechanisms indeed generate an interaction across different time scales (or frequencies) resulting in processes with non-stationary features. The other and problematic possibility is that unspecific non-stationarities can also be associated with spectral correlations which in turn will be detected by cross frequency measures even if physiologically there is no causal interaction between the frequencies. 3) We discuss on the role of non-linearities as generators of cross frequency interactions. As an example we performed a phase-amplitude coupling analysis of two nonlinearly related signals: atmospheric noise and the square of it (Figure 1) observing an enhancement of phase-amplitude coupling in the second signal while no pattern is observed in the first. Finally, we discuss some minimal conditions need to be tested to solve some of the ambiguities here noted. In summary, we simply want to point out that finding a significant cross frequency pattern does not always have to imply that there indeed is physiological cross frequency interaction in the brain.
Background: Transfer entropy (TE) is a measure for the detection of directed interactions. Transfer entropy is an information theoretic implementation of Wiener's principle of observational causality. It offers an approach to the detection of neuronal interactions that is free of an explicit model of the interactions. Hence, it offers the power to analyze linear and nonlinear interactions alike. This allows for example the comprehensive analysis of directed interactions in neural networks at various levels of description. Here we present the open-source MATLAB toolbox TRENTOOL that allows the user to handle the considerable complexity of this measure and to validate the obtained results using non-parametrical statistical testing. We demonstrate the use of the toolbox and the performance of the algorithm on simulated data with nonlinear (quadratic) coupling and on local field potentials (LFP) recorded from the retina and the optic tectum of the turtle (Pseudemys scripta elegans) where a neuronal one-way connection is likely present.
Results: In simulated data TE detected information flow in the simulated direction reliably with false positives not exceeding the rates expected under the null hypothesis. In the LFP data we found directed interactions from the retina to the tectum, despite the complicated signal transformations between these stages. No false positive interactions in the reverse directions were detected.
Conclusions: TRENTOOL is an implementation of transfer entropy and mutual information analysis that aims to support the user in the application of this information theoretic measure. TRENTOOL is implemented as a MATLAB toolbox and available under an open source license (GPL v3). For the use with neural data TRENTOOL seamlessly integrates with the popular FieldTrip toolbox.
Criticality meets learning : criticality signatures in a self-organizing recurrent neural network
(2017)
Many experiments have suggested that the brain operates close to a critical state, based on signatures of criticality such as power-law distributed neuronal avalanches. In neural network models, criticality is a dynamical state that maximizes information processing capacities, e.g. sensitivity to input, dynamical range and storage capacity, which makes it a favorable candidate state for brain function. Although models that self-organize towards a critical state have been proposed, the relation between criticality signatures and learning is still unclear. Here, we investigate signatures of criticality in a self-organizing recurrent neural network (SORN). Investigating criticality in the SORN is of particular interest because it has not been developed to show criticality. Instead, the SORN has been shown to exhibit spatio-temporal pattern learning through a combination of neural plasticity mechanisms and it reproduces a number of biological findings on neural variability and the statistics and fluctuations of synaptic efficacies. We show that, after a transient, the SORN spontaneously self-organizes into a dynamical state that shows criticality signatures comparable to those found in experiments. The plasticity mechanisms are necessary to attain that dynamical state, but not to maintain it. Furthermore, onset of external input transiently changes the slope of the avalanche distributions – matching recent experimental findings. Interestingly, the membrane noise level necessary for the occurrence of the criticality signatures reduces the model’s performance in simple learning tasks. Overall, our work shows that the biologically inspired plasticity and homeostasis mechanisms responsible for the SORN’s spatio-temporal learning abilities can give rise to criticality signatures in its activity when driven by random input, but these break down under the structured input of short repeating sequences.
Background Many systems in nature are characterized by complex behaviour where large cascades of events, or avalanches, unpredictably alternate with periods of little activity. Snow avalanches are an example. Often the size distribution f(s) of a system's avalanches follows a power law, and the branching parameter sigma, the average number of events triggered by a single preceding event, is unity. A power law for f(s), and sigma=1, are hallmark features of self-organized critical (SOC) systems, and both have been found for neuronal activity in vitro. Therefore, and since SOC systems and neuronal activity both show large variability, long-term stability and memory capabilities, SOC has been proposed to govern neuronal dynamics in vivo. Testing this hypothesis is difficult because neuronal activity is spatially or temporally subsampled, while theories of SOC systems assume full sampling. To close this gap, we investigated how subsampling affects f(s) and sigma by imposing subsampling on three different SOC models. We then compared f(s) and sigma of the subsampled models with those of multielectrode local field potential (LFP) activity recorded in three macaque monkeys performing a short term memory task. Results Neither the LFP nor the subsampled SOC models showed a power law for f(s). Both, f(s) and sigma, depended sensitively on the subsampling geometry and the dynamics of the model. Only one of the SOC models, the Abelian Sandpile Model, exhibited f(s) and sigma similar to those calculated from LFP activity. Conclusions Since subsampling can prevent the observation of the characteristic power law and sigma in SOC systems, misclassifications of critical systems as sub- or supercritical are possible. Nevertheless, the system specific scaling of f(s) and sigma under subsampling conditions may prove useful to select physiologically motivated models of brain function. Models that better reproduce f(s) and sigma calculated from the physiological recordings may be selected over alternatives.
Neuronal dynamics differs between wakefulness and sleep stages, so does the cognitive state. In contrast, a single attractor state, called self-organized critical (SOC), has been proposed to govern human brain dynamics for its optimal information coding and processing capabilities. Here we address two open questions: First, does the human brain always operate in this computationally optimal state, even during deep sleep? Second, previous evidence for SOC was based on activity within single brain areas, however, the interaction between brain areas may be organized differently. Here we asked whether the interaction between brain areas is SOC. ...
Neuronal activity differs between wakefulness and sleep states. In contrast, an attractor state, called self-organized critical (SOC), was proposed to govern brain dynamics because it allows for optimal information coding. But is the human brain SOC for each vigilance state despite the variations in neuronal dynamics? We characterized neuronal avalanches – spatiotemporal waves of enhanced activity - from dense intracranial depth recordings in humans. We showed that avalanche distributions closely follow a power law – the hallmark feature of SOC - for each vigilance state. However, avalanches clearly differ with vigilance states: slow wave sleep (SWS) shows large avalanches, wakefulness intermediate, and rapid eye movement (REM) sleep small ones. Our SOC model, together with the data, suggested first that the differences are mediated by global but tiny changes in synaptic strength, and second, that the changes with vigilance states reflect small deviations from criticality to the subcritical regime, implying that the human brain does not operate at criticality proper but close to SOC. Independent of criticality, the analysis confirms that SWS shows increased correlations between cortical areas, and reveals that REM sleep shows more fragmented cortical dynamics.
Poster presentation: Self-organized critical (SOC) systems are complex dynamical systems that may express cascades of events, called avalanches [1]. The SOC state was proposed to govern brain function, because of its activity fluctuations over many orders of magnitude, its sensitivity to small input and its long term stability [2,3]. In addition, the critical state is optimal for information storage and processing [4]. Both hallmark features of SOC systems, a power law distribution f(s) for the avalanche size s and a branching parameter (bp) of unity, were found for neuronal avalanches recorded in vitro [5]. However, recordings in vivo yielded contradictory results [6]. Electrophysiological recordings in vivo only cover a small fraction of the brain, while criticality analysis assumes that the complete system is sampled. We hypothesized that spatial subsampling might influence the observed avalanche statistics. In addition, SOC models can have different connectivity, but always show a power law for f(s) and bp = 1 when fully sampled. This may not be the case under subsampling, however. Here, we wanted to know whether a state change from awake to asleep could be modeled by changing the connectivity of a SOC model without leaving the critical state. We simulated a SOC model [1] and calculated f(s) and bp obtained from sampling only the activity of a set of 4 × 4 sites, representing the electrode positions in the cortex. We compared these results with results obtained from multielectrode recordings of local field potentials (LFP) in the cortex of behaving monkeys. We calculated f(s) and bp for the LFP activity recorded while the monkey was either awake or asleep and compared these results to results obtained from two subsampled SOC model with different connectivity. f(s) and bp were very similar for both the experiments and the subsampled SOC model, but in contrast to the fully sampled model, f(s) did not show a power law and bp was smaller than unity. With increasing the distance between the sampling sites, f(s) changed from "apparently supercritical" to "apparently subcritical" distributions in both the model and the LFP data. f(s) and bp calculated from LFP recorded during awake and asleep differed. These changes could be explained by altering the connectivity in the SOC model. Our results show that subsampling can prevent the observation of the characteristic power law and bp in SOC systems, and misclassifications of critical systems as sub- or supercritical are possible. In addition, a change in f(s) and bp for different states (awake/asleep) does not necessarily imply a change from criticality to sub- or supercriticality, but can also be explained by a change in the effective connectivity of the network without leaving the critical state.
When studying real world complex networks, one rarely has full access to all their components. As an example, the central nervous system of the human consists of 1011 neurons which are each connected to thousands of other neurons. Of these 100 billion neurons, at most a few hundred can be recorded in parallel. Thus observations are hampered by immense subsampling. While subsampling does not affect the observables of single neuron activity, it can heavily distort observables which characterize interactions between pairs or groups of neurons. Without a precise understanding how subsampling affects these observables, inference on neural network dynamics from subsampled neural data remains limited.
We systematically studied subsampling effects in three self-organized critical (SOC) models, since this class of models can reproduce the spatio-temporal activity of spontaneous activity observed in vivo. The models differed in their topology and in their precise interaction rules. The first model consisted of locally connected integrate- and fire units, thereby resembling cortical activity propagation mechanisms. The second model had the same interaction rules but random connectivity. The third model had local connectivity but different activity propagation rules. As a measure of network dynamics, we characterized the spatio-temporal waves of activity, called avalanches. Avalanches are characteristic for SOC models and neural tissue. Avalanche measures A (e.g. size, duration, shape) were calculated for the fully sampled and the subsampled models. To mimic subsampling in the models, we considered the activity of a subset of units only, discarding the activity of all the other units.
Under subsampling the avalanche measures A depended on three main factors: First, A depended on the interaction rules of the model and its topology, thus each model showed its own characteristic subsampling effects on A. Second, A depended on the number of sampled sites n. With small and intermediate n, the true A¬ could not be recovered in any of the models. Third, A depended on the distance d between sampled sites. With small d, A was overestimated, while with large d, A was underestimated.
Since under subsampling, the observables depended on the model's topology and interaction mechanisms, we propose that systematic subsampling can be exploited to compare models with neural data: When changing the number and the distance between electrodes in neural tissue and sampled units in a model analogously, the observables in a correct model should behave the same as in the neural tissue. Thereby, incorrect models can easily be discarded. Thus, systematic subsampling offers a promising and unique approach to model selection, even if brain activity was far from being fully sampled.
In self-organized critical (SOC) systems avalanche size distributions follow power-laws. Power-laws have also been observed for neural activity, and so it has been proposed that SOC underlies brain organization as well. Surprisingly, for spiking activity in vivo, evidence for SOC is still lacking. Therefore, we analyzed highly parallel spike recordings from awake rats and monkeys, anesthetized cats, and also local field potentials from humans. We compared these to spiking activity from two established critical models: the Bak-Tang-Wiesenfeld model, and a stochastic branching model. We found fundamental differences between the neural and the model activity. These differences could be overcome for both models through a combination of three modifications: (1) subsampling, (2) increasing the input to the model (this way eliminating the separation of time scales, which is fundamental to SOC and its avalanche definition), and (3) making the model slightly sub-critical. The match between the neural activity and the modified models held not only for the classical avalanche size distributions and estimated branching parameters, but also for two novel measures (mean avalanche size, and frequency of single spikes), and for the dependence of all these measures on the temporal bin size. Our results suggest that neural activity in vivo shows a mélange of avalanches, and not temporally separated ones, and that their global activity propagation can be approximated by the principle that one spike on average triggers a little less than one spike in the next step. This implies that neural activity does not reflect a SOC state but a slightly sub-critical regime without a separation of time scales. Potential advantages of this regime may be faster information processing, and a safety margin from super-criticality, which has been linked to epilepsy.