Using extracranial electrodes, EEG measures the extracellular ionic current produced by a graded postsynaptic potential of vertically oriented pyramidal neurons in the III, V, and VI cortical layers (Nunez and Srinivasan, 2006). The electrical dipole field created by the soma and apical dendrites of pyramidal neurons is propagated through layers of the cortex, cerebrospinal fluid (CSF), skull, and scalp via volume conduction (Schaul, 1998) and recordable at the scalp site (Mule et al., 2021). EEG technology has diverse applications, including characterizing brain dynamics in the early stages of Parkinson’s disease (PD) (Han et al., 2013), epileptic seizure detection (Zhou et al., 2018), motor imagery and movement classification in brain-computer interfaces (Thomas et al., 2009), emotion classification (Alhagry et al., 2017) etc.

Oscillations are a key feature of sleep EEG and are crucial in various physiological and cognitive processes (Lambert and Peter-Derex, 2023). For example, the presence of theta waves and reduction of an alpha wave during stage 1 sleep, Stage 2 sleep is characterized by bursts of oscillatory activity and increased cortical synchrony; delta wave is a marker of deep sleep also affected by diseases like insomnia (Bakker et al., 2023), theta oscillation in the hippocampus during REM stage, etc. EEG-based Coherence (Watanabe et al., 2023) and phase synchronizations (Ojha and Panda, 2024) are the quantitative measures of functional connectivity during sleep and temporal coordination of neuronal activity. Achermann and Borbély (1997) shows the significance of neuronal synchrony during deep sleep and the increase in delta (0.1–4 Hz) and theta (4–7 Hz) band power following sleep onset, particularly in the fronto-central region. Studies also report the earlier emergence of alpha (8–12 Hz) and frontal theta oscillations after sleep deprivation (Gorgoni et al., 2019), the effect of transcranial oscillatory stimulation on memory consolidation during NREM sleep (Marshall et al., 2006), and the role of oscillations as biomarkers of sleep homeostasis such as theta activity during wakefulness and slow-wave activity (delta) during sleep (Finelli et al., 2000). Sleep also plays a major role in synaptic homeostasis and memory consolidation, emphasizing the importance of oscillatory activity in brain function (Esser et al., 2007; Pankka et al., 2024; Bahador et al., 2021; Paul, 2020; Sano et al., 2018; Michielli et al., 2019; Zhao et al., 2024; Burke and de Paor, 2004; Weigenand et al., 2014; Ospeck, 2019; Ghosh et al., 2025; Krishnamurthy et al., 2022; Buzsáki, 2006; Achermann et al., 1993). The EEG-based sleep transition model (Ising model) describes the synchronous dynamics of the neuronal population (Acconito et al., 2023). The Kuramoto model describes phase synchronization among different brain regions during sleep (Ingendoh et al., 2023).

In the past decade, several efforts have been made to model the brain as a nonlinear dynamical system and describe brain dynamics using complex nonlinear dynamical networks (Başar, 1983). A phenomenological model comprising van der Pol–Duffing double oscillator networks was used to model EEG signals from healthy controls as well as Alzheimer’s disease patients (Ghorbanian et al., 2015b). The model results compare favorably with experimental results in terms of time series, power spectrum, and Shannon entropy (Ghorbanian et al., 2015a; Ghorbanian et al., 2015b; Ghorbanian et al., 2014). Another model used coupled Duffing-van der pol oscillators to generate EEG ictal patterns from the temporal lobe (Szuflitowska and Orlowski, 2021). By analyzing EEG time series, the presence of low dimensional chaos in NREM N1 and REM sleep stages was described by Babloyantz et al. (1985). Studies have been made using stochastic limit cycle oscillators to model EEG data from healthy subjects (Rankine et al., 2006; Burke and de Paor, 2004).

Hopf oscillator networks offer a robust framework for modeling complex brain dynamics across different states (Burke and de Paor, 2004). Neural mass models have been used to explain slow-wave activity and K-complexes (Weigenand et al., 2014). Additionally, Hopf oscillators provide a powerful approach to modeling the oscillatory neural dynamics underlying memory consolidation and sleep spindles (Ospeck, 2019). Their properties closely align with observed sleep and memory research phenomena, offering insights into how the brain processes, stores, and consolidates information through oscillatory activity patterns. These models connect the neuronal-level dynamics and population-level behavior observed in EEG recordings.

Despite these efforts, the link between the mesoscopic EEG activity and the dynamics underlying neuronal circuits still needs to be fully unraveled. A weighted mean potential of a weakly-coupled, local cluster of Hindmarsh-Rose (HR) neurons collectively shows near-synchronization behavior and can optimally reconstruct epileptic EEG time series (Ren et al., 2017). A similar line of work was also proposed by Phuong and colleagues, in which networks of HR neurons and Kuramato oscillators were used to reconstruct EEG data in healthy and epileptic conditions (Nguyen et al., 2020). Ibáñez-Molina and Iglesias-Parro (2016) show a mean field of the Kuramoto oscillator to explore the dynamics of electroencephalographic (EEG) complexity during mind-wandering episodes. This work contributes to understanding how neural mechanisms underpin spontaneous thought processes and their representation in EEG signals. In another study (Das and Puthankattil, 2022), the authors used a phenomenological computational model of the Kuramoto oscillator to investigate functional connectivity and EEG complexity in mild cognitive impairment (MCI), a precursor to Alzheimer’s disease (AD). The study revealed that the brain’s dynamic repertoire results from the interplay between network topology and oscillatory dynamics by combining empirical structural and functional connectivity data with computational models (Kuramoto oscillatory model) of coupled oscillators (Cabral et al., 2022). This research highlights the role of synchronization mechanisms in shaping large-scale brain dynamics and offers a framework for understanding how the connectome supports diverse neural functions. Similar approaches have also been taken to model Functional magnetic resonance imaging (fMRI) (Logothetis et al., 2001) signals using the non-linear oscillators (Kuramoto, Hopf) model (Cabral et al., 2023; Breakspear, 2017; Deco et al., 2017). Several Hopf oscillatory models have been developed that explain several physiological phenomena such as cognitive behavior, sleep–wake cycle, Schizophrenia, and Alzheimer’s disease (Deco et al., 2015; Deco et al., 2017; Deco et al., 2021; Luppi et al., 2022; López-González et al., 2021; Deco and Kringelbach, 2014).

There is a growing interest in modeling large-scale brain activity using networks of nonlinear oscillators. A notable example of this kind is The Virtual Brain (TVB) framework, which uses large oscillatory networks to model various manifestations of functional brain dynamics like EEG, functional Magnetic Resonance Imaging (fMRI), and Magnetoencephalogram (MEG) (Sanz Leon et al., 2013). Al-Hossenat et al. (2019) proposed modeling of slow wave activity (delta and theta) of EEG from eight different regions using Jansen and Rit’s (JR) neural mass model and anatomical connectivity using the TVB framework. In another modeling study, using a similar kind of neural mass model, alpha wave activity was reproduced from four different brain regions (Al-Hossenat et al., 2017).

Several deep learning models have been proposed for EEG time series forecasting, including WaveNet (Pankka et al., 2024), correlation-based approaches (Bahador et al., 2021), and LSTM-based networks (Paul, 2020). LSTM-RNN networks have been applied to detect sleep stages from EEG signals (Sano et al., 2018; Michielli et al., 2019) and for sleep EEG reconstruction (Zhao et al., 2024), leveraging their ability to capture temporal dependencies in sleep EEG data. However, these networks primarily work in time series, not capturing the frequency and phase information. These approaches overlook key biological oscillatory features such as phase, frequency, and amplitude.

Feedforward spiking neural network models (Singanamalla and Lin, 2021; Zenke and Ganguli, 2018) generate synthetic EEG signals for Motor imagery and SSVEP EEG data. Yang et al. (2024) introduced a biologically inspired unsupervised learning framework for spiking neural networks (SNNs), enhancing neuromorphic vision systems with robust, efficient, and energy-adaptive visual perception for embodied applications in robotics and AI. The same group advanced SNN capabilities with a surrogate gradient learning framework (Yang and Chen, 2023a), demonstrating superior temporal precision and efficiency for neuromorphic computing and AI. Similar authors proposed SNIB (Yang and Chen, 2023b; Yang and Chen, 2024), a groundbreaking framework applying the nonlinear information bottleneck (NIB) principle to optimize the trade-off between information compression and retention, enabling more efficient, robust, and adaptive spike-based learning. There is a model of linearly coupled Hopf oscillatory model that can explain complex dynamics, and information processing of the human brain (Deco et al., 2017), where model dynamics depend on coupling coefficients among oscillators and oscillator amplitude (μ) (Deco et al., 2017). However, these kinds of models cannot accurately predict the neuroimaging data like fMRI (Hahn et al., 2021; Iravani et al., 2021).

Compared with fMRI, a smaller number of brain modeling approaches have been explored with EEG. EEG is less expensive than fMRI, more easily available, portable, and has high temporal resolution. Therefore, we can use all the advantages of EEG to understand brain dynamics. With this motivation, we propose a network of Hopf oscillators described in the complex domain and show how the network can be trained to model high-dimensional EEG data in the waking and sleep stages and publicly available BONN epilepsy dataset (Andrzejak et al., 2001). Sleep is a complex, naturally recurring dynamic process that occurs periodically in most animals (Březinová, 1974). Three critical physiological mechanisms or rhythms regulate sleep, viz., circadian rhythm, homeostasis, and ultradian rhythm (Fisher et al., 2013; Achermann and Borbély, 2003). Polysomnogram (Wolpert, 1969), which jointly measures brain electrical activity (EEG), muscle activity (Electromyogram—EMG), eye movement (Electrooculogram—EOG), and heart rate (Electrocardiogram—ECG), is a standard method of recording sleep activity. Sleep stages can be broadly categorized into five stages: waking, non-rapid eye movement N1 (NREM N1), non-rapid eye movement N2 (NREM N2), non-rapid eye movement N3 (NREM N3), and rapid eye movement (REM) sleep (Šušmáková, 2004). Sleep facilitates important neural and physiological functions, including memory consolidation (Siegel, 2001), and emotion control (Goldstein and Walker, 2014).

In earlier work, we showed how to achieve a stable phase relationship between oscillators with arbitrarily different frequencies using a special form of coupling known as power coupling (Biswas et al., 2021). The difficulties that arise in a pair of coupled oscillators, depicted by Arnold tongues, seem to be overcome effectively with power coupling. It was shown how networks of such coupled oscillator systems can be trained to learn a small number of EEG channels. In the present study, we add a hidden layer of sigmoidal neurons and geometrically constrain the network to accurately learn high-dimensional, “whole brain” EEG signals under various sleep conditions.

In this work, we have developed a Deep Oscillatory Neural Network (DONN) to reconstruct and predict sleep EEG and epileptic EEG time series. This type of model combines oscillatory neurons and sigmoid neurons (Ghosh et al., 2025). RNNs (Krishnamurthy et al., 2022) with gating mechanisms are excellent at sequence processing capabilities but fail to show biological plausibility. In contrast, neural activity in the brain exhibits complex dynamics characterized by key frequency bands such as alpha, beta, gamma, and theta (Buzsáki, 2006). Our proposed oscillatory network consists of Hopf oscillators, where each unit exhibits both amplitude and phase dynamics. The oscillators presented in the network effectively carry out a Fourier decomposition of the teaching signal (Biswas et al., 2021). Additionally, we demonstrate that a single oscillatory neuron is computationally more efficient than single LSTM neurons in terms of processing time, moreover, the DONN network has significantly fewer trainable parameters compared to an LSTM (Ghosh et al., 2025).

Our study shows that the EEG signal can be reconstructed as well as predicted optimally compared to the Kuramoto and HR neuron-based model proposed (Nguyen et al., 2020). Our model can reconstruct and predict the next 5 s of EEG data (2,500 data points) from 5 sleep stages. Various signal features like power spectrum density, Hurst exponent (Supplementary section 11), and Higuchi fractal (Supplementary section 12) dimension show good agreement with model-predicted EEG data compared to empirical EEG Data. The key contributions of this work are summarized below (1–5):

The outline of the paper is as follows. This article begins with an account of sleep EEG recording followed by preprocessing methodology. The two stages of training of the proposed network are described in the ‘1st stage of training’ and ‘2nd stage of training’ sections, respectively. The section ‘Insertion of hidden layer’ describes the deep oscillator network, which is a combination of an oscillatory layer and a feedforward network. The section ‘Prediction of EEG Data’ describes the prediction of EEG data using a trained Hopf oscillator model. The section ‘spatial distribution of oscillator’ shows how oscillators are distributed on a rectangular grid and spherical shell geometry. Results from the reconstruction, EEG data prediction, and statistical analysis are described in the results section. A discussion of the work is presented in the last section.

In this section, we describe the performance of the models described in the previous section on modelling high-dimensional, whole-brain EEG data (56 electrodes). In order to model a large number of electrodes, as well as the essential frequency band (0.1 to 20 Hz), a large number (N = 200) of oscillators are used. Although the model is trained on the original EEG time series. In order to depict the signal spectrum, instead of using normal FFT, we use average periodogram method known as the “Welch method” (Naderi and Mahdavi-Nasab, 2010). We use the Welch method with a Hamming window of size 1 s and 50% overlap throughout the paper.

In the present study, we model a 56-channel EEG signal with a network of oscillatory neurons. The proposed network is able to model (both reconstruction and prediction) whole-brain EEG data. The network is able to successfully predict test signals over a significant duration beyond the training duration (5 s), and is able to retain properties (Hurst component and Higuchi fractal dimension) of the actual EEG signals over the frequency band of interest.

In the present study, insertion of a hidden layer between the oscillator layer and the output layer is proven to improve reconstruction quality (Supplementary section 7) significantly. This is because when there was no hidden layer, the output signal was essentially approximated by a finite set of sinusoidal signals represented by the oscillators of the input layer. But once we applied the hidden layer in our 2nd network, those sinusoids pass through the nonlinear sigmoid functions of the hidden layer. Even when a single sinusoid passes through a nonlinear sigmoid, we can get all the infinite harmonics. Furthermore, when a mixture of sinusoids is passed through a sigmoid function, we get the harmonics not only of the original frequencies but the harmonics of all the mixtures (e.g., ± n i ω i ± n j ω j ± n k ω k ). Hence, the hidden layer is expanding the spectrum that is available at the output layer. Thus, the finite number of frequencies available when we did not apply a hidden layer suddenly explode to an infinite set of frequencies available when we apply a hidden layer.

In comparison with Nguyen et al. (2020), where 3,000 neurons used to simulate EEG data whereas in our model the network with only 250 Hopf oscillators was able to predict more accurately even without using any data reduction techniques like the PCA.

Our proposed method can predict EEG Data in the order of 5 s (2,500 Data points). The predicted signal has good agreement with actual EEG with respect to power spectrum, Hurst exponent (Supplementary material) and complexity measure (Higuchi fractal dimension) (Supplementary material) which are the key results of our study. A potential application of our network can be synthetic EEG generation for research and education.

Also, we have successfully demonstrated spatial localization architecture for oscillator reservoirs. The performance of spatially arranged oscillators with a hidden layer is good, as demonstrated by RMSE values. However, in our spatial geometry of oscillators, we only considered locally connected regions. Also, real structural data can be implemented on this proposed model to realize a large-scale TVB type of model (Sanz Leon et al., 2013; Al-Hossenat et al., 2019). In the future, we can easily expand our model to a real MRI-based surface where oscillators are placed according to structural-functional connectivity nodes. Thus, the problem regarding the unequal distribution of oscillators in the case of “spherical shell” geometry can be eliminated.

Each EEG channel was modeled by a single Hindmarsh Rose neuron in an ADHD study (Ansarinasab et al., 2023); since EEG represents the collective activity populations of neurons, in the present model, each EEG channel is modeled by a network of oscillators. Also, a graph-based brain topology was used in the same study (Ansarinasab et al., 2023), whereas in the proposed model, we distribute the oscillators spatially as per two different geometries: rectangular and spherical. It is very difficult to predict non-stationary signal like EEG beyond the training duration, however, deep learning generative model [GAN type model (Panwar et al., 2020; Aznan et al., 2019a)] requires a huge training sample, that problem also can be solved by our proposed oscillatory generative model.

While both works utilize, like a good number of existing models of brain dynamics (Ghorbanian et al., 2014; Ghorbanian et al., 2015a; Ghorbanian et al., 2015b; Szuflitowska and Orlowski, 2021; Babloyantz et al., 1985; Rankine et al., 2006; Burke and de Paor, 2004; Ren et al., 2017; Nguyen et al., 2020; Logothetis et al., 2001; Cabral et al., 2023; Breakspear, 2017; Deco et al., 2015; Deco et al., 2017; Deco et al., 2021; Luppi et al., 2022; López-González et al., 2021; Deco and Kringelbach, 2014; Ponce-Alvarez and Deco, 2024), our case, network of Hopf oscillators as the basic source of oscillations, the specific methods and the outcomes of the two studies are significantly different.

Our group recently developed a similar network using fMRI signals. In that work, Bandyopadhyay et al. (2023) demonstrated a strong alignment between predicted and empirical functional connectivity (FC), as validated through graph-theoretical analysis. Authors have shown that structural damage resulted in cascading disruptions in static and dynamic FC patterns. Computational interventions revealed that optimizing the coupling coefficient (μ) could restore functional integrity. This in silico perturbation study further highlighted how targeted parameter adjustments could compensate for structural degradation, providing insights into potential therapeutic applications. These findings underscore the model’s potential in understanding and addressing disruptions in brain network dynamics.

However, a key distinction between our proposed study and that of Bandyopadhyay et al. (2023) lies in the testing regime. While their work primarily focused on training, signal reconstruction, and the connection between structural and functional connectivity, they did not discuss the network’s behavior beyond training. In contrast, our proposed network can retain learned patterns beyond the training signal. Additionally, we show that the network-generated signals preserve key properties in both the time and frequency domains. Furthermore, we introduced a spherical shell model based on the 10–20 EEG electrode system, enabling EEG signal reconstruction while utilizing shared oscillators’ natural frequencies and phases. We also explored optimal network parameters to achieve the best fit, enhancing the model’s reliability and performance. Data augmentation for EEG has gained significant research attention in recent years (Torma and Szegletes, 2025; Aznan et al., 2019b; Kalaganis et al., 2020; Hartmann et al., 2018). Collecting EEG data has several challenges, primarily due to the strict requirements of the environment and the variability in subjects’ psychological and physiological conditions. Due to the limited accessibility of the highly dense EEG data, applying several deep learning models is challenging. To address this issue, data augmentation techniques have emerged as a viable solution. This deep oscillatory neural generative model offers strong potential for synthesizing realistic EEG data. Additionally, we have included a comparison table (Table 8) comparing our method with that of Bandyopadhyay et al. (2023).

Baseline models Ren et al. (2017), Ghorbanian et al. (2015a), and Ghorbanian et al. (2015b) utilized the same modeling and training methodologies. Our assessment of existing EEG modelling approaches highlights significant differences and enhancements in our proposed framework (Table 9). This structured comparison emphasizes our framework’s advancements in achieving a more biologically plausible network with improved computational traceability.

Additionally, we introduced a spherical shell model based on the 10–20 electrode geometry, enabling the reconstruction of EEG signals. In this model, the natural frequencies and phases of the shared oscillators were utilized effectively. Furthermore, we explored and identified optimal network parameters to achieve the best fit, ensuring superior signal reconstruction and validation performance.

Future efforts will be directed to developing the current model into a more realistic model of sleep dynamics. In the current modelling approach, separate networks are trained to produce EEG signals of various sleep stages and the waking stage. However, in a more authentic model of sleep, it is desirable to generate the various stages in a single model and explicitly demonstrate the transitions from one stage to the next. Ideally, such a model will show the sleep–wake cycle at a longer or diurnal time scale, and also the entire architecture of sleep substages within the 8-h long sleep stage. The model also will permit a minimal representation of the main neural substrates of sleep regulation such as the Suprachiasmatic Nucleus (SCN), hypothalamic and thalamic nuclei and the neuromodulatory systems involved in sleep regulation, and the Reticular Activating System (RAS). Our approach combines empirical EEG data with a mathematical model to create virtual representations of the brain’s oscillatory dynamics. It can help to explore the impact of neuronal excitability, synaptic plasticity, and network connectivity on sleep stage transitions (Tatti and Cacciola, 2023).

In the future sleep model that we envisage, the interactions among the various subcortical circuits and neural systems produce the subrhythms of sleep, which, acting on the cortex, generate the EEG activity patterns characteristic of the relevant sleep stage. In this paper, we primarily focus on sleep EEG. However, our model is a general-purpose, universal framework that can be applied to any EEG dataset. To validate its universality, we tested the model on the publicly available BONN dataset, demonstrating improved performance compared to the previously reported study (Nguyen et al., 2020).