Stabilizing fractional dynamical networks suppresses epileptic seizures

Summary

We applied fractional-order dynamical systems to model brain network dynamics across seizure progression in 10 patients with drug-resistant epilepsy, examining how stability (eigenvalues) and multi-scale (fractional-order exponents) properties differ across interictal, pre-ictal, ictal, and post-ictal epileptic brain states. Discrete-time fractional-order dynamical systems provide a robust framework for modeling volatile iEEG data during seizures. Discrete-time fractional-order dynamical systems can be easily estimated from data using an expectation maximization algorithm 34. Furthermore, working in discrete time allows the controller design to remain analytically tractable and straightforward to implement. Our key contributions included achieving good model fit across all epileptic brain states with interpretable properties that revealed distinct patterns and a stabilizing control framework that satisfied the eigenvalue stability criterion in 77% of unstable ictal networks and reduced seizure signal amplitude by an average of 49%. These findings provide evidence that multi-scale and stability dynamics evolve distinctly across seizure progression, with inter-patient variability potentially reflecting diverse epilepsy etiologies. The integration of fractional-order modeling with control theory provides a foundation for developing personalized, model-based neurostimulation strategies targeting seizure suppression.

Fractional-order dynamical network models are suitable to represent epileptic brain states

Fractional-order dynamical networks achieve \(R^2\ge 0.5\) for 60% of all electrodes in 126 of 140 segments (90%), demonstrating that fractional-order systems effectively capture iEEG dynamics across diverse seizure states in most patients. The 14 segments where a good fit could not be obtained likely reflect limitations in our current estimation framework when confronted with extreme signal characteristics, rather than fundamental flaws in modeling the data with fractional-order dynamics. Our model validation revealed that the majority of rejected data occurred during the post-ictal segments, which had significantly smaller signal amplitudes (139.1±177.0 \(\upmu\)V) compared to successfully modeled post-ictal segments (457.8±928.7 \(\upmu\)V). This pattern may reflect post-ictal suppression, where the brain enters a recovery state with markedly lower electrical activity 35. This finding suggests that fractional-order dynamical network modeling is most informative when applied to periods with active neural dynamics rather than suppressed brain states. Patient HUP78 presented the opposite pattern, as all five of its interictal segments were excluded due to abnormally high amplitudes (238.5±176.3 \(\upmu\)V). HUP78 developed epilepsy from a traumatic brain injury, highlighting how individual patient characteristics and epilepsy etiology may influence the applicability of network-based modeling approaches.

Consistent multi-scale and stability properties across patients suggest reproducible brain network activity

Our analysis of fractional-order exponents and eigenvalues across seizure states reveals that epileptic brain networks exhibit consistent, multi-scale memory dynamics and structured stability properties. Fractional-order exponents quantify history-dependence in neural dynamics, a property that integer-order models cannot capture. These fractional-order exponents reveal how strongly current brain states depend on past neural activity, with smaller fractional-order exponents indicating stronger memory dependence and larger fractional-order exponents indicating weaker memory dependence, and values approaching 1 reducing to the integer-order case. In other words, smaller fractional-order exponents signify that the brain’s current activity is dependent on neural activity far in the past, while larger fractional-order exponents closer to 1 imply the current brain’s activity is dependent on very recent neural activity. From a neurophysiological perspective, \(\alpha\) represents the degree in which neural populations retain and integrate past activity through recurrent connectivity and synaptic mechanisms. Smaller \(\alpha\) values would reflect enhanced reverberatory dynamics and stronger long-term dependencies. Interictal fractional-order exponents indicate low memory dependence and a decrease in multi-scale properties. Larger fractional-order exponents during pre-ictal correspond to a higher memory dependence compared to fractional-order exponents during ictal, which is consistent with the recent work on critical slowing and seizure susceptibility that demonstrates that autocorrelation and signal variance increase before a seizure and reduce afterwards 36. Fractional-order exponents during the ictal period indicate potentially chaotic behavior and inconsistent multi-scale properties. On the other hand, post-ictal fractional-order exponents indicate a shift in memory dependence and multi-scale properties, without returning to interictal levels. Similarly, eigenvalues during interictal indicate unstable or potentially marginally stable behavior. Pre-ictal eigenvalue distributions generally reflect stabilized behavior, while the ictal eigenvalues suggest that the brain is moving closer towards instability. Finally, eigenvalues during post-ictal indicate instability, suggesting disrupted or incomplete recovery following seizure termination.

In Fig. 3A, the interictal segments exhibit the weakest memory dependence (highest \(\alpha\)), while pre-ictal and ictal periods show strong memory effects. Post-ictal fractional-order exponents indicate that these segments are much more chaotic and do not return to interictal levels. This trend is mirrored in Fig. 3B with the eigenvalues and stability. The consistency in fractional-order exponents (\(\alpha\)) (median 0.75, IQR: 0.66-0.84) and eigenvalues (median 0.80, IQR: 0.76-0.85) across interictal periods reveals that epileptic brain networks maintain consistent dynamics between seizures. These uniform fractional-order exponent distributions indicate that interictal epileptic dynamics are fundamentally non-Markovian, with neural activity substantially coupled to its temporal history 17,30. We consistently observed eigenvalues greater than 1 during interictal periods, indicating unstable dynamics. Our within-patient analysis revealed that eigenvalues during interictal have the strongest separation from other epileptic brain states, with large effect sizes (Cohen’s \(d>1\)) for all comparisons except for eigenvalues during interictal vs post-ictal. These patterns and strong effect sizes highlight the structured nature of seizure dynamics, aligning with prior studies that have used EEG data to track physiological changes 37.

Seizure onset and termination have been modeled as critical transitions in dynamical systems 36,38,39,40,41, and recent work has explored seizure dynamics across spatial scales 38,42. Other studies have identified self-similar properties in EEG seizure data using multivariate eigen-wavelet analysis 43, generalized Hurst exponents to distinguish healthy from epileptic activity 44, and fractal features for machine learning classification in a clinical context 45. Our work echoes the conclusions of these previous studies by explicitly modeling multi-scale and stability properties, offering a unified framework to characterize and track multiscale seizure dynamics across time.

Multi-scale and stability properties uncover insights into seizure onset, progression, and recovery

The shift from interictal to ictal periods showed a decrease in fractional-order exponents (\(\alpha\)) (0.75 to 0.63) and eigenvalues (0.80 to 0.72). Lower fractional-order exponents (\(\alpha\)) indicates stronger history-dependence of neural activity, which suggests that seizures are self-sustaining events, where the brain cannot escape from pathological activity patterns 46. Similarly, eigenvalues also decreased in pre-ictal and ictal periods, with somewhat tightened distributions, suggesting a modest shift toward a more robust, stable region during seizures. While this observation may initially appear to contradict recent literature that hypothesizes that seizures can be mathematically represented as an instability in the brain 41,47,48,49,50, we note that asymptotic stability (assessed in this work) may not be suitable for characterizing seizure onset since seizure events have finite time horizons. Notably, the stability of a fractional-order dynamical network depends on both its fractional-order exponents (\(\alpha\)) and its spatial matrix (A). In this work, we observe that fractional-order exponents evolve together with eigenvalues across epileptic brain states.

Importantly, both fractional-order exponents (\(\alpha\)) and eigenvalues show small differences between pre-ictal and ictal segments. The similarity between pre-ictal and ictal suggests that the pathological network states characteristic of seizures are already established before clinical seizure onset, consistent with previous findings that seizures do not arise instantaneously but emerge from gradual network-level transitions 51,52. Furthermore, the pre-ictal period may represent an already altered brain state that is primed for seizure generation rather than a simple precursor to an abrupt change 46,53.

Post-ictal fractional-order exponents (\(\alpha\)) and eigenvalues did not return to interictal baseline values, revealing substantial heterogeneity in recovery dynamics. Some patients exhibited both fractional-order exponents and eigenvalues exceeding interictal baselines, suggesting hyperexcitable rebound, while others showed sustained suppression. The variability within post-ictal periods (wider IQR and numerous outliers for both metrics) indicates unstable network dynamics during recovery, where the brain may cycle through multiple states before re-establishing a baseline state. This finding indicates that post-seizure recovery is not a uniform process but rather reflects patient-specific network reorganization mechanisms 35,54,55.

Within-patient analysis revealed substantially larger KS test statistics and effect sizes compared to the population-pooled statistics. The strong within-patient separation between interictal and seizure-related states suggests that fractional-order exponents could serve as patient-specific biomarkers for predicting seizures, despite being limited on the population level. Additionally, since fractional-order exponents during pre-ictal and ictal states showed minimal differences, detecting the transition from interictal to pre-ictal may provide sufficient warning for impending seizures without requiring the exact time of seizure onset. This is further supported by the observation that interictal to ictal, and interictal to pre-ictal comparisons exhibited similarly strong separability, as shown in Fig. 5.

The pattern of effect sizes reveals an asymmetry in seizure dynamics. The sharp contrast between ictal and post-ictal states (\(d = 1.00\) for fractional-order exponents and 0.79 for eigenvalues) compared to the minimal difference between pre-ictal and ictal states (\(d = 0.28\) for fractional-order exponents and 0.25 for eigenvalues) may indicate that seizure termination triggers more dramatic network reorganization than seizure initiation. This finding aligns with recent modeling work showing that seizure onset and spread can emerge as critical transitions in brain networks driven by changes in excitability and connectivity 40. Furthermore, our finding that fractional-order exponents and eigenvalues separate interictal and pre-ictal states is supported by prior work highlighting the importance of identifying network-level changes as patient-specific biomarkers for seizure prediction 56.

Stabilizing control shows promise to suppress epileptic activity

To our knowledge, our work represents one of the most comprehensive demonstrations of fractional-order control for suppressing seizure dynamics applied to real patient iEEG data. Our stabilizing control framework achieved seizure suppression in 34/35 seizures, successfully stabilizing 77% of initially unstable seizures and reducing seizure amplitude by approximately 49% across all electrodes. Notably, our control strategy produced a similar average amplitude reduction in both initially stable and unstable seizures, showcasing effective suppression of ictal activity regardless of mathematical stability status. These findings suggest that fractional-order control strategies can generalize across diverse patient pools.

In contrast, previous work has focused on developing state-feedback stabilization strategies for linear time-invariant models of physiological networks 57. Another example is a generalized pole placement algorithm for linear time-invariant dynamics 11. The main issue with linear-time invariant models is that they do not capture the measurement dependence inherent in physiological networks 26.

Recent work has focused on designing linear networked models 58,59,60. For example, the work in 58 derives graph-theoretic conditions for structural vibrational stabilizability under which linear networked models can always be stabilized, and the work in 59 provides a method to design the vibrational inputs to stabilize the linear network. The work in 60 studies interconnected excitatory-inhibitory pairs with linear threshold dynamics, and presents strategies to design networks with desired robustness properties. These methods may offer theoretical guarantees under ideal conditions, but are not easily estimated from data and may not accommodate the heterogeneity observed across patients.

Open-loop vibrational control strategies have been proposed to stabilize networks of nonlinear oscillators 61,62,63. For example, recent work shows that vibrational control synchronizes clusters of nonlinear oscillators in a network by providing sufficient conditions for uniform exponential stability of a fixed point 61. Furthermore, the work in 62 focuses on stabilizing a nonlinear network of oscillators by providing the patterns of the pairwise relationships between the oscillators’ phases. In addition, the work in 63 provides sufficient conditions for vibrational inputs to stabilize cluster synchronization of a nonlinear network of oscillators and offers a tractable approach for designing vibrational control. Finally, for a bistable dynamical system, the authors provide conditions on external perturbations to ensure input-to-state stability 48. The main limitations of considering a network of nonlinear oscillators or bistable dynamical systems is the inability to estimate the parameters of the model from data and the issue of scalability.

Fractional-order dynamical networks can be efficiently estimated from data 34,64,65 and do not suffer from issues of scalability. In our prior work, we derived conditions for global asymptotic stability of time-invariant fractional-order systems in discrete-time 66,67. Furthermore, we derived two stabilizing feedback strategies by altering the parameters of fractional-order systems and provided the polynomial-time computational complexity to compute these feedback strategies 67.

In this work, we demonstrate the capability of our state-of-the-art stabilizing state feedback control scheme to effectively suppress epileptic activity in 34 out of 35 patients. Our approach admits a computationally tractable control solution that is straightforward to implement. Future work will focus on comparing our state feedback control method with traditional stabilizing state feedback methods for linear time-invariant dynamics.

The eight seizures that failed to stabilize exhibited severely ill-conditioned optimization problems, with the median condition numbers 50x larger than those of successful cases (1383 vs 28), which suggests that a feasible solution to the optimization problem presented in (12) may not exist. Physiologically, this may indicate that certain seizure dynamics involve such profound network reorganization or high-dimensional complexity that linear coupling modifications may be unable to restore stability 69. Importantly, most control failures occurred in seizures with good model fits, indicating that accurate system identification does not guarantee a feasible solution to the stabilizing control problem formulated as an optimization problem in (12). Future work will focus on developing feasibility conditions for solving the stabilizing control problem.

Limitations

A potential limitation of this study is the sample size (10 patients, 35 seizures), which may limit statistical power despite large effect sizes. Larger cohorts would enable the identification of patient subgroups with distinct seizure dynamics profiles and improve statistical power for detecting group-level patterns.

We observed that the goodness of fit of our model varied across segments, with post-ictal periods being particularly challenging. In our work, we considered a 3 second sliding window to estimate fractional-order dynamical networks. In future work, we will investigate the effects of window size selection on estimation. A systematic investigation of window size and stride parameters could optimize the temporal resolution for capturing seizure transitions. Additionally, more work is needed to assess the sensitivity of our estimation scheme to noise and extreme perturbations.

Our current control method may face real-world implementation challenges. In future work, we plan to address the issue of robustness and develop a scheme to overcome delays. Furthermore, our current strategy ignores stimulation artifacts and safety constraints, which we plan to address in a future study.

In this study, we focused on analyzing eigenvalues and fractional-order exponents (\(\alpha\)) of fractional-order dynamical networks, which show significant promise. In future work, we plan to analyze eigenvectors, which may help to identify the spatial patterns of network reorganization across the seizure states and reveal which electrode drives transitions between states, which has the potential to improve targeting for neurostimulation.

Future work will focus on developing conditions to guarantee a feasible control solution by verifying stabilizability with state-feedback. Certain network configurations that are well-characterized by fractional-order models may be fundamentally resistant to state-feedback interventions due to structural constraints, pathological states, or intrinsic nonlinearities that emerge during control attempts 69. The numerical ill-conditioning observed in failed cases may thus reflect genuine physiological properties of these seizures rather than modeling inadequacies. The heterogeneity in control outcomes also points toward the need for patient-specific or etiology-specific controller designs. Patients with structural lesions, certain epilepsy etiologies, or particular network architectures may require alternative control strategies, such as nonlinear control or targeting different network nodes. Patient HUP78, who developed epilepsy from traumatic injury, failed to achieve stabilization in all five of their seizures, including three of their seizures that were initially stable. This complete failure suggests that traumatic brain injury-induced structural damage, gliosis, and altered network connectivity 68 may require control strategies different from other epilepsy etiologies. Larger studies grouping patients by clinical characteristics may reveal which characteristics are more favorable to state-feedback control versus those requiring more sophisticated interventions 69,70.

Finally, we evaluated our control framework in silico simulations using estimated models from real-world iEEG data. More work is needed to ensure a clinical translation pathway for the proposed control strategy. Future work includes translating control inputs to safe neurostimulation paradigms, such as electrical stimulation. Another important consideration is to ensure real-time performance by overcoming any latency constraints through incorporating time delays into the theoretical framework. In the future, we plan to verify the effectiveness of our proposed strategy in animal models. Finally, we plan to test the effectiveness of this approach in a non-invasive stimulation paradigm, such as TMS or focused ultrasound, which would help us assess the real-world feasibility and efficacy of fractional-order model-based neurostimulation for seizure suppression. We recognize our approach will first have to pass regulations, pre-clinical safety studies, and review before any human trials.