Neural Mass Research Articles

An adaptive dynamic surface control strategy for suppressing pathological oscillations in a neural mass model

Pathological oscillations that have a frequency in the [Formula: see text] band are widely recognized to be involved in Parkinson’s disease. Now, we present an adaptive dynamic surface control strategy for suppressing pathological oscillations that exist in the Parkinsonian state. First, the interactions between the subthalamic nucleus and external globus pallidus are fully considered and establish a neural mass model of Parkinson’s disease. Next, by reasonable state transformation, suppressing pathological oscillations can be converted into a tracking control study of a pure-feedback nonlinear system. Moreover, the dynamic surface control technique adopted reduces the dimensionality of the neural network input and effectively eliminates the complexity explosion problem commonly associated with existing methods. By Lyapunov stability analysis, it can be obtained that all the signals of the resulting closed-loop system are bounded, and the tracking error converges to a small neighborhood of zero. Finally, the simulation results demonstrate the effectiveness of the designed control strategy. This work may provide an effective approach to closed-loop deep brain stimulation optimization for the alleviation of the Parkinsonian state.

A whole-brain model of the aging brain during slow wave sleep.

Age-related brain changes affect sleep and are reflected in properties of sleep slow-waves, however the precise mechanisms behind these changes are still not completely understood. Here, we adapt a previously established whole-brain model relating structural connectivity changes to resting state dynamics, and extend it to a slow-wave sleep brain state. In particular, starting from a representative connectome at the beginning of the aging trajectory, we have gradually reduced the inter-hemispheric connections, and simulated sleep-like slow-wave activity. We show that the main empirically observed trends, namely a decrease in duration and increase in variability of the slow waves are captured by the model. Furthermore, comparing the simulated EEG activity to the source signals, we suggest that the empirically observed decrease in amplitude of the slow waves is caused by the decrease in synchrony between brain regions.Significance Statement Aging is characterized by changes in slow wave (SW) sleep features, yet the precise mechanisms driving these alterations remain elusive. Employing a connectome-based model, we implement the established age- related reductions in inter-hemispheric connectivity, successfully replicating the SW changes in the simulated activity. Our simulation of EEG activity also suggests that observed decreases in SW amplitude stems from diminished synchrony between brain regions. Our results support the notion that alterations in SW characteristics result from reductions in cortical excitatory drive-here facilitated by the inter-hemispheric connections. Our model serves as a robust foundation for extensions to population studies and interventional work in animal models of aging aimed at disentangling the contributions of network alterations, changes to local neural mass properties, and neuromodulation.

Bayesian inference of frequency-specific functional connectivity in MEG imaging using a spectral graph model

Abstract Understanding the relationship between structural connectivity (SC) and functional connectivity (FC) of the human brain is an important goal of neuroscience. Highly detailed mathematical models of neural masses exist that can simulate the interactions between functional activity and structural wiring. These models are often complex and require intensive computation. Most importantly, they do not provide a direct or intuitive interpretation of this structure–function relationship. In this study, we employ the emerging concepts of spectral graph theory to obtain this mapping in terms of graph harmonics, which are eigenvectors of the structural graph’s Laplacian matrix. In order to imbue these harmonics with biophysical underpinnings, we leverage recent advances in parsimonious spectral graph modeling (SGM) of brain activity. Here, we show that such a model can indeed be cast in terms of graph harmonics, and can provide a closed-form prediction of FC in an arbitrary frequency band. The model requires only three global, spatially invariant parameters, yet is capable of generating rich FC patterns in different frequency bands. Only a few harmonics are sufficient to reproduce realistic FC patterns. We applied the method to predict FC obtained from pairwise magnitude coherence of source-reconstructed resting-state magnetoencephalography (MEG) recordings of 36 healthy subjects. To enable efficient model inference, we adopted a deep neural network-based Bayesian procedure called simulation-based inference. Using this tool, we were able to speedily infer not only the single most likely model parameters, but also their full posterior distributions. We also implemented several other benchmark methods relating SC to FC, including graph diffusion and coupled neural mass models. The present method was shown to give the best performance overall. Notably, we discovered that a single biophysical parameterization is capable of fitting FCs from all relevant frequency bands simultaneously, an aspect that did not receive adequate attention in prior computational studies.

Discriminating fingerprints of chronic neuropathic pain following spinal cord injury using artificial neural networks and mass spectrometry analysis of female mice serum

Spinal cord injury (SCI) often leads to central neuropathic pain, a condition associated with significant morbidity and is challenging in terms of the clinical management. Despite extensive efforts, identifying effective biomarkers for neuropathic pain remains elusive. Here we propose a novel approach combining matrix-assisted laser desorption/ionization time-of-flight mass spectrometry (MALDI-TOF MS) with artificial neural networks (ANNs) to discriminate between mass spectral profiles associated with chronic neuropathic pain induced by SCI in female mice. Functional evaluations revealed persistent chronic neuropathic pain following mild SCI as well as minor locomotor disruptions, confirming the value of collecting serum samples. Mass spectra analysis revealed distinct profiles between chronic SCI and sham controls. On applying ANNs, 100% success was achieved in distinguishing between the two groups through the intensities of m/z peaks. Additionally, the ANNs also successfully discriminated between chronic and acute SCI phases. When reflexive pain response data was integrated with mass spectra, there was no improvement in the classification. These findings offer insights into neuropathic pain pathophysiology and underscore the potential of MALDI-TOF MS coupled with ANNs as a diagnostic tool for chronic neuropathic pain, potentially guiding attempts to discover biomarkers and develop treatments.

Exploring nonlinear dynamics in brain functionality through phase portraits and fuzzy recurrence plots.

Much of the complexity and diversity found in nature is driven by nonlinear phenomena, and this holds true for the brain. Nonlinear dynamics theory has been successfully utilized in explaining brain functions from a biophysics standpoint, and the field of statistical physics continues to make substantial progress in understanding brain connectivity and function. This study delves into complex brain functional connectivity using biophysical nonlinear dynamics approaches. We aim to uncover hidden information in high-dimensional and nonlinear neural signals, with the hope of providing a useful tool for analyzing information transitions in functionally complex networks. By utilizing phase portraits and fuzzy recurrence plots, we investigated the latent information in the functional connectivity of complex brain networks. Our numerical experiments, which include synthetic linear dynamics neural time series and a biophysically realistic neural mass model, showed that phase portraits and fuzzy recurrence plots are highly sensitive to changes in neural dynamics and can also be used to predict functional connectivity based on structural connectivity. Furthermore, the results showed that phase trajectories of neuronal activity encode low-dimensional dynamics, and the geometric properties of the limit-cycle attractor formed by the phase portraits can be used to explain the neurodynamics. Additionally, our results showed that the phase portrait and fuzzy recurrence plots can be used as functional connectivity descriptors, and both metrics were able to capture and explain nonlinear dynamics behavior during specific cognitive tasks. In conclusion, our findings suggest that phase portraits and fuzzy recurrence plots could be highly effective as functional connectivity descriptors, providing valuable insights into nonlinear dynamics in the brain.

Exploring Nonlinear Dynamics In Brain Functionality Through Phase Portraits And Fuzzy Recurrence Plots.

Much of the complexity and diversity found in nature is driven by nonlinear phenomena, and this holds true for the brain. Nonlinear dynamics theory has been successfully utilized in explaining brain functions from a biophysics standpoint, and the field of statistical physics continues to make substantial progress in understanding brain connectivity and function. This study delves into complex brain functional connectivity using biophysical nonlinear dynamics approaches. We aim to uncover hidden information in high-dimensional and nonlinear neural signals, with the hope of providing a useful tool for analyzing information transitions in functionally complex networks. By utilizing phase portraits and fuzzy recurrence plots, we investigated the latent information in the functional connectivity of complex brain networks. Our numerical experiments, which include synthetic linear dynamics neural time series and a biophysically realistic neural mass model, showed that phase portraits and fuzzy recurrence plots are highly sensitive to changes in neural dynamics and can also be used to predict functional connectivity based on structural connectivity. Furthermore, the results showed that phase trajectories of neuronal activity encode low-dimensional dynamics, and the geometric properties of the limit-cycle attractor formed by the phase portraits can be used to explain the neurodynamics. Additionally, our results showed that the phase portrait and fuzzy recurrence plots can be used as functional connectivity descriptors, and both metrics were able to capture and explain nonlinear dynamics behavior during specific cognitive tasks. In conclusion, our findings suggest that phase portraits and fuzzy recurrence plots could be highly effective as functional connectivity descriptors, providing valuable insights into nonlinear dynamics in the brain.

Neural dynamics and seizure correlations: Insights from neural mass models in a Tetanus Toxin rat model of epilepsy

This study focuses on the use of a neural mass model to investigate potential relationships between functional connectivity and seizure frequency in epilepsy. We fitted a three-layer neural mass model of a cortical column to intracranial EEG (iEEG) data from a Tetanus Toxin rat model of epilepsy, which also included responses to periodic electrical stimulation. Our results show that some of the connectivity weights between different neural populations correlate significantly with the number of seizures each day, offering valuable insights into the dynamics of neural circuits during epileptogenesis. We also simulated single-pulse electrical stimulation of the neuronal populations to observe their responses after the connectivity weights were optimized to fit background (non-seizure) EEG data. The recovery time, defined as the time from stimulation until the membrane potential returns to baseline, was measured as a representation of the critical slowing down phenomenon observed in nonlinear systems operating near a bifurcation boundary. The results revealed that recovery times in the responses of the computational model fitted to the EEG data were longer during 5 min periods preceding seizures compared to 1 hr before seizures in four out of six rats. Analysis of the iEEG recorded in response to electrical stimulation revealed results similar to the computational model in four out of six rats. This study supports the potential use of this computational model as a model-based biomarker for seizure prediction when direct electrical stimulation to the brain is not feasible.

On the influence of input triggering on the dynamics of the Jansen–Rit oscillators network

Our investigation delves into the intricate dynamical properties of a network of coupled neural oscillators, each comprising identical Jansen–Rit masses. The connections between these nodes follow the renowned Watts–Strogatz topology. Every node receives inputs from two primary sources: external and internal inputs derived from its neighboring nodes’ outputs. In this paper, we aim to explore the consequences of changing these two inputs and analyze the results generated. To begin with, we analyzed the model using the mean-field approximation, assuming identical outputs across all nodes due to identical internal and external inputs. Subsequently, we relaxed this assumption, and a more detailed analysis of both states is discussed. As a result of the mean-field approach, we observed that there was no phase transition, despite apparent changes in behavior. However, the relaxation of the mean-field assumption led to the occurrence of the first (discontinuous) and second (continuous) phase transitions. Moreover, our findings indicate that by relaxing the mean-field assumption in our analysis of the single Jansen–Rit neural mass model, theta waves can be generated. Our study provides a comprehensive examination of various observed behaviors by the model, including transitions between synchrony and asynchrony. Additionally, our work highlights the potential contribution of external and internal inputs in studying phase transition and synchronization of neural mass models. Overall, our investigation sheds light on the intricate dynamics of coupled neural oscillators and their sensitivity to varying inputs.

Neural mass modelling for the masses: Democratising access to whole-brain biophysical modelling with FastDMF

Abstract Different whole-brain computational models have been recently developed to investigate hypotheses related to brain mechanisms. Among these, the Dynamic Mean Field (DMF) model is particularly attractive, combining a biophysically realistic model that is scaled up via a mean-field approach and multimodal imaging data. However, an important barrier to widespread usage of the DMF model is that current implementations are computationally expensive, supporting only simulations on brain parcellations that consider less than 100 brain regions. Here, we introduce an efficient and accessible implementation of the DMF model: the FastDMF. By leveraging analytical and numerical advances – including a novel estimation of the feedback inhibition control parameter, and a Bayesian optimization algorithm – the FastDMF circumvents various computational bottlenecks of previous implementations, improving interpretability, performance and memory use. Furthermore, these advances allow the FastDMF to increase the number of simulated regions by one order of magnitude, as confirmed by the good fit to fMRI data parcellated at 90 and 1000 regions. These advances open the way to the widespread use of biophysically grounded whole-brain models for investigating the interplay between anatomy, function, and brain dynamics, and to identify mechanistic explanations of recent results obtained from fine-grained neuroimaging recordings.

Preparatory activity of anterior insula predicts conflict errors: integrating convolutional neural networks and neural mass models

Preparatory brain activity is a cornerstone of proactive cognitive control, a top-down process optimizing attention, perception, and inhibition, fostering cognitive flexibility and adaptive attention control in the human brain. In this study, we proposed a neuroimaging-informed convolutional neural network model to predict cognitive control performance from the baseline pre-stimulus preparatory electrophysiological activity of core cognitive control regions. Particularly, combined with perturbation-based occlusion sensitivity analysis, we pinpointed regions with the most predictive preparatory activity for proactive cognitive control. We found that preparatory arrhythmic broadband neural dynamics in the right anterior insula, right precentral gyrus, and the right opercular part of inferior frontal gyrus (posterior ventrolateral prefrontal cortex), are highly predictive of prospective cognitive control performance. The pre-stimulus preparatory activity in these regions corresponds to readiness for conflict detection, inhibitory control, and overall elaborate attentional processing. We integrated the convolutional neural network with biologically inspired Jansen-Rit neural mass model to investigate neurostimulation effects on cognitive control. High-frequency stimulation (130 Hz) of the left anterior insula provides significant cognitive enhancement, especially in reducing conflict errors, despite the right anterior insula’s higher predictive value for prospective cognitive control performance. Thus, effective neurostimulation targets may differ from regions showing biomarker activity. Finally, we validated our theoretical finding by evaluating intrinsic neuromodulation through neurofeedback-guided volitional control in an independent dataset. We found that left anterior insula was intrinsically modulated in real-time by volitional control of emotional valence, but not arousal. Our findings further highlight central role of anterior insula in orchestrating proactive cognitive control processes, positioning it at the top of hierarchy for cognitive control.

Incorporating slow NMDA-type receptors with nonlinear voltage-dependent magnesium block in a next generation neural mass model: derivation and dynamics

We derive a next generation neural mass model of a population of quadratic-integrate-and-fire neurons, with slow adaptation, and conductance-based AMPAR, GABAR and nonlinear NMDAR synapses. We show that the Lorentzian ansatz assumption can be satisfied by introducing a piece-wise polynomial approximation of the nonlinear voltage-dependent magnesium block of NMDAR current. We study the dynamics of the resulting system for two example cases of excitatory cortical neurons and inhibitory striatal neurons. Bifurcation diagrams are presented comparing the different dynamical regimes as compared to the case of linear NMDAR currents, along with sample comparison simulation time series demonstrating different possible oscillatory solutions. The omission of the nonlinearity of NMDAR currents results in a shift in the range (and possible disappearance) of the constant high firing rate regime, along with a modulation in the amplitude and frequency power spectrum of oscillations. Moreover, nonlinear NMDAR action is seen to be state-dependent and can have opposite effects depending on the type of neurons involved and the level of input firing rate received. The presented model can serve as a computationally efficient building block in whole brain network models for investigating the differential modulation of different types of synapses under neuromodulatory influence or receptor specific malfunction.

Criticality and partial synchronization analysis in Wilson-Cowan and Jansen-Rit neural mass models.

Synchronization is a phenomenon observed in neuronal networks involved in diverse brain activities. Neural mass models such as Wilson-Cowan (WC) and Jansen-Rit (JR) manifest synchronized states. Despite extensive research on these models over the past several decades, their potential of manifesting second-order phase transitions (SOPT) and criticality has not been sufficiently acknowledged. In this study, two networks of coupled WC and JR nodes with small-world topologies were constructed and Kuramoto order parameter (KOP) was used to quantify the amount of synchronization. In addition, we investigated the presence of SOPT using the synchronization coefficient of variation. Both networks reached high synchrony by changing the coupling weight between their nodes. Moreover, they exhibited abrupt changes in the synchronization at certain values of the control parameter not necessarily related to a phase transition. While SOPT was observed only in JR model, neither WC nor JR model showed power-law behavior. Our study further investigated the global synchronization phenomenon that is known to exist in pathological brain states, such as seizure. JR model showed global synchronization, while WC model seemed to be more suitable in producing partially synchronized patterns.

Modeling the contribution of theta-gamma coupling to sequential memory, imagination, and dreaming.

Gamma oscillations nested in a theta rhythm are observed in the hippocampus, where are assumed to play a role in sequential episodic memory, i.e., memorization and retrieval of events that unfold in time. In this work, we present an original neurocomputational model based on neural masses, which simulates the encoding of sequences of events in the hippocampus and subsequent retrieval by exploiting the theta-gamma code. The model is based on a three-layer structure in which individual Units oscillate with a gamma rhythm and code for individual features of an episode. The first layer (working memory in the prefrontal cortex) maintains a cue in memory until a new signal is presented. The second layer (CA3 cells) implements an auto-associative memory, exploiting excitatory and inhibitory plastic synapses to recover an entire episode from a single feature. Units in this layer are disinhibited by a theta rhythm from an external source (septum or Papez circuit). The third layer (CA1 cells) implements a hetero-associative net with the previous layer, able to recover a sequence of episodes from the first one. During an encoding phase, simulating high-acetylcholine levels, the network is trained with Hebbian (synchronizing) and anti-Hebbian (desynchronizing) rules. During retrieval (low-acetylcholine), the network can correctly recover sequences from an initial cue using gamma oscillations nested inside the theta rhythm. Moreover, in high noise, the network isolated from the environment simulates a mind-wandering condition, randomly replicating previous sequences. Interestingly, in a state simulating sleep, with increased noise and reduced synapses, the network can "dream" by creatively combining sequences, exploiting features shared by different episodes. Finally, an irrational behavior (erroneous superimposition of features in various episodes, like "delusion") occurs after pathological-like reduction in fast inhibitory synapses. The model can represent a straightforward and innovative tool to help mechanistically understand the theta-gamma code in different mental states.

Coexistence of asynchronous and clustered dynamics in noisy inhibitory neural networks

A regime of coexistence of asynchronous and clustered dynamics is analysed for globally coupled homogeneous and heterogeneous inhibitory networks of quadratic integrate-and-fire (QIF) neurons subject to Gaussian noise. The analysis is based on accurate extensive simulations and complemented by a mean-field description in terms of low-dimensional next generation neural mass models for heterogeneously distributed synaptic couplings. The asynchronous regime is observable at low noise and becomes unstable via a sub-critical Hopf bifurcation at sufficiently large noise. This gives rise to a coexistence region between the asynchronous and the clustered regime. The clustered phase is characterised by population bursts in the γ-range (30–120 Hz), where neurons are split in two equally populated clusters firing in alternation. This clustering behaviour is quite peculiar: despite the global activity being essentially periodic, single neurons display switching between the two clusters due to heterogeneity and/or noise.

Simulated dynamical transitions in a heterogeneous marmoset pFC cluster.

Network analysis of the marmoset cortical connectivity data indicates a significant 3D cluster in and around the pre-frontal cortex. A multi-node, heterogeneous neural mass model of this six-node cluster was constructed. Its parameters were informed by available experimental and simulation data so that each neural mass oscillated in a characteristic frequency band. Nodes were connected with directed, weighted links derived from the marmoset structural connectivity data. Heterogeneity arose from the different link weights and model parameters for each node. Stimulation of the cluster with an incident pulse train modulated in the standard frequency bands induced a variety of dynamical state transitions that lasted in the range of 5-10 s, suggestive of timescales relevant to short-term memory. A short gamma burst rapidly reset the beta-induced transition. The theta-induced transition state showed a spontaneous, delayed reset to the resting state. An additional, continuous gamma wave stimulus induced a new beating oscillatory state. Longer or repeated gamma bursts were phase-aligned with the beta oscillation, delivering increasing energy input and causing shorter transition times. The relevance of these results to working memory is yet to be established, but they suggest interesting opportunities.

Some aspects of bidomain modeling with volume conductors

The concept of dynamic multiscaling has changed our approach to multi-neuronal cable theory. Previously, computational neuroscientists considered individual neurons as neural masses or compartmental models, but now, a distributed representation of single neurons as ionic cable structures is most likely to lead to a greater understanding of how the distribution of ionic channels and synaptic input along the dendrites of a few neurons can offset the collective behavior of a large ensemble of neurons and, therefore, provide a measure of the dynamical brain. This change in perspective forms the basis of volume conductor-bidomain modeling, a new method that captures multiscalar electrophysiology.

Effects of the spatial resolution of the Virtual Epileptic Patient on the identification of epileptogenic networks

Abstract Digital twins play an increasing role in clinical decision making. This study evaluates a digital brain twin approach in presurgical evaluation, the Virtual Epileptic Patient (VEP), which estimates the epileptogenic zone in patients with drug-resistant epilepsy. We built the personalized digital brain twins of 14 patients and a series of synthetic dataset by considering different spatial configurations of the epileptogenic and/or propagation zone networks (EZN and PZN, respectively). Brain source signals were simulated with a high spatial resolution neural field model (NFM) composed of 81942 nodes, embedding both long-range (between brain regions) and short-range (within brain regions) coupling. Brain signals were then projected to stereotactic electroencephalographic (SEEG) contacts with an accurate forward solution. An inversion procedure based on a low spatial resolution neural mass model (NMM) composed of 162 nodes was applied to estimate the excitability of each region in each simulation. The ensuing estimated EZN/PZN was compared to the simulated ground truth by means of classification metrics. Overall, we observed correct but degraded performance when using an NMM to estimate the EZN from data simulated with an NFM, which was significant for the simplest spatial configurations. We quantified the reduced performance and demonstrated that the oversimplification of the forward problem is its principal cause. We showed that the absence of local coupling in the NMM affects the inversion process by an overestimation of the excitability, representing a significant clinical impact when using this procedure in the context of presurgical planning. In conclusion, this study highlighted the importance to shift from an NMM towards a full NFM modeling approach for the estimation of EZN, with a particularly relevant need when considering the most complex clinical cases.

Ketosis regulates K+ ion channels, strengthening brain-wide signaling disrupted by age

Abstract Aging is associated with impaired signaling between brain regions when measured using resting-state functional magnetic resonance imaging (fMRI). This age-related destabilization and desynchronization of brain networks reverses itself when the brain switches from metabolizing glucose to ketones. Here, we probe the mechanistic basis for these effects. First, we confirmed their robustness across measurement modalities using two datasets acquired from resting-state EEG (Lifespan: standard diet, 20–80 years, N = 201; Metabolic: individually weight-dosed and calorically-matched glucose and ketone ester challenge, μage = 26.9 ±11.2 years, N = 36). Then, using a multiscale conductance-based neural mass model, we identified the unique set of mechanistic parameters consistent with our clinical data. Together, our results implicate potassium (K+) gradient dysregulation as a mechanism for age-related neural desynchronization and its reversal with ketosis, the latter finding of which is consistent with direct measurement of ion channels. As such, the approach facilitates the connection between macroscopic brain activity and cellular-level mechanisms.

Personalized whole-brain neural mass models reveal combined Aβ and tau hyperexcitable influences in Alzheimer’s disease

Neuronal dysfunction and cognitive deterioration in Alzheimer’s disease (AD) are likely caused by multiple pathophysiological factors. However, mechanistic evidence in humans remains scarce, requiring improved non-invasive techniques and integrative models. We introduce personalized AD computational models built on whole-brain Wilson-Cowan oscillators and incorporating resting-state functional MRI, amyloid-β (Aβ) and tau-PET from 132 individuals in the AD spectrum to evaluate the direct impact of toxic protein deposition on neuronal activity. This subject-specific approach uncovers key patho-mechanistic interactions, including synergistic Aβ and tau effects on cognitive impairment and neuronal excitability increases with disease progression. The data-derived neuronal excitability values strongly predict clinically relevant AD plasma biomarker concentrations (p-tau217, p-tau231, p-tau181, GFAP) and grey matter atrophy obtained through voxel-based morphometry. Furthermore, reconstructed EEG proxy quantities show the hallmark AD electrophysiological alterations (theta band activity enhancement and alpha reductions) which occur with Aβ-positivity and after limbic tau involvement. Microglial activation influences on neuronal activity are less definitive, potentially due to neuroimaging limitations in mapping neuroprotective vs detrimental activation phenotypes. Mechanistic brain activity models can further clarify intricate neurodegenerative processes and accelerate preventive/treatment interventions.

Collective dynamics and shot-noise-induced switching in a two-population neural network.

Neural mass models are a powerful tool for modeling of neural populations. Such models are often used as building blocks for the simulation of large-scale neural networks and the whole brain. Here, we carry out systematic bifurcation analysis of a neural mass model for the basic motif of various neural circuits, a system of two populations, an excitatory, and an inhibitory ones. We describe the scenarios for the emergence of complex collective behavior, including chaotic oscillations and multistability. We also compare the dynamics of the neural mass model and the exact microscopic system and show that their agreement may be far from perfect. The discrepancy can be interpreted as the action of the so-called shot noise originating from finite-size effects. This shot noise can lead to the blurring of the neural mass dynamics or even turn its attractors into metastable states between which the system switches recurrently.