Timo Bröhl

Reservoir computing has established itself as a computationally efficient alternative to fully trained recurrent neural networks using a fixed high-dimensional nonlinear substrate to map temporal input into a feature space for a simple readout. Here, we argue that reservoir computing can be realized entirely within a coarse-grained, symbolic state space, and that constrained random walks on ordinal-pattern transition networks provide a natural instance of this realization, replacing the reservoir’s continuous state space with a discrete and directly data-derived one. We develop a formal correspondence between the two frameworks, show that the echo-state property has a natural discrete analog in the ergodicity and spectral gap of the walk’s underlying Markov chain, and illustrate the correspondence with a minimal example and an empirical one-step-ahead prediction task evaluated across five paradigmatic model systems. These systems exhibit dynamics that correspond to strictly periodic, quasiperiodic, chaotic, or stochastic signals. We argue that this symbolic reservoir perspective allows a redefinition of the reservoir substrate itself, replacing the requirement of a continuous, randomly generated dynamical system with any sufficiently ergodic, data-derived symbolic process. This opens a route toward interpretable, data-adaptive reservoirs whose internal structure is inferred rather than randomly assigned, with implications for surrogate-data methods and reservoir design more broadly.

Spatiotemporal signal propagation through the human brain during epileptic seizures remains poorly understood. Using intracranial EEG from subjects with medically refractory epilepsy, we introduce an edge-centric framework that tracks how influential functional connections in the time-evolving epileptic brain network form and transition as seizures unfold. Brain regions are grouped into three functional modules – seizure onset area, nearby ipsilateral regions, and other areas – with the most important edges followed over time. A probabilistic entropy approach reveals that path-based centralities exhibit a conserved circular motif of seizure propagation across subjects. In contrast, strength-based centralities separate seizures into two distinct classes: one where cross-module edges serve as dynamic bottlenecks guiding ictal spread, and another where these edges remain stable. These findings demonstrate that an edge-focused perspective may reveal vital information about ictal communication pathways, and may have potential application for improved prediction and classification in epilepsy.

We study the impact of the coupling topology on the ability of various networked dynamical systems to generate extreme events. By determining the coupling strength that is necessary to generate an extreme event in the collective dynamics of a given system, we observe a power-law-like relationship between this coupling threshold and both topological (edge density) and spectral (algebraic connectivity) properties of various coupling topologies. Interestingly, this relationship appears to be largely independent of both the investigated system and the underlying mechanism to generate extreme events. This may indicate that the observed relationship is primarily mediated by aspects of the coupling topology.

We introduce circulance, a scalar measure for classifying time series of dynamical systems. Circulance captures the extent of temporal regularity or irregularity that is encoded in the topology of a directed ordinal pattern transition network derived from a time series. We demonstrate numerically that circulance sensitively and robustly positions time series of canonical model systems, representative of preset dynamical regimes, along a continuous spectrum from regularity to randomness. Analyzing empirical data from long-term observations of high-dimensional, complex systems—human brain and the Sun—reveals that circulance aids in elucidating different dynamical regimes.

Background: The brain is never truly at rest. Even in the absence of external tasks, it remains active, continuously organizing itself into large-scale resting-state networks involved in shaping our internal thoughts and experiences. Understanding the networks’ structure and dynamics is key to uncovering how the brain functions as a whole. While previous studies have mapped resting-state networks and explored the roles of individual brain regions (network vertices), the relevance of the time-dependent functional interactions (network edges) between them remains largely unexplored. Methods: Here, we assess this relevance by elucidating the time-evolving importance of both brain regions and their interactions, associated with the networks’ constituents, using the fundamental concept of centrality. We investigate long-term electrophysiological recordings of brain dynamics from more than 100 participants and reveal new insights into how resting-state networks are organized over longer times. Results: Our findings reveal that the functional architecture of brain networks in a resting state is critically shaped by the dynamic interplay between brain regions. We identified functional importance backbones–core sets of dynamically central vertices and edges–whose configuration varies significantly between subgroups and further varies with different brain states, including wakefulness and sleep. Notably, regions associated with the default mode network exhibited adaptable patterns of centrality, challenging the notion of static network cores. Conclusions: By considering the temporal evolution of both vertices and edges, we provide a more comprehensive understanding of intrinsic brain activity and its functional relevance. This dynamic perspective reveals how the brain’s intrinsic activity is coordinated across space and time, highlighting the existence of functional importance backbones that adapt to different brain states.

Unexpected and often irreversible shifts in the state or the dynamics of a complex system often accumulate in extreme events with likely disastrous impact on the system and its environment. Detection, understanding, and possible prediction of such critical transitions are thus of paramount importance across a variety of scientific fields. The rather modest improvement achieved so far may be due previous research mostly concentrating on either particular subsystems, considered to be of vital importance for the generating mechanism of a critical transition, or on the system as a whole. These approaches only rarely take into account the intricate, time-dependent interrelatedness of subsystems that can essentially determine emerging behaviors underlying critical transitions. We uncover subsystems, network vertices, and the interrelatedness of certain subsystems, network edges, as tipping elements in a networked dynamical system, forming a time-evolving tipping subnetwork. We demonstrate the existence of tipping subnetworks in excitable complex networks and in human epileptic brains. These systems can repeatedly undergo critical transitions that result in extreme events. Our findings reveal that tipping subnetworks encapsulate key properties of mechanisms involved in critical transitions.

Background: Psychogenic non-epileptic seizures (PNES) are seizure-like episodes that resemble behavioral aspects observed for epileptic seizures but are without the abnormal electrical activity typically seen in epilepsy. The lack of an etiologic model for PNES as well as limitations of available diagnostic methods largely hinders a clear-cut distinction from epilepsy and from a normal functioning brain. Methods: In this study, we investigate the brain dynamics of people with PNES and people with epilepsy during phases far-off seizures and seizure-like events as well as the brain dynamics of a control group. Probing for differences between these groups, we utilise the network ansatz and explore local and global characteristics of time-evolving functional brain networks. We observe subject-specific differences in local network characteristics across the groups, highlighting the physiological functioning of specific brain regions. Furthermore, we observe significant differences in global network characteristics—relating to communication, robustness, and stability aspects of the brain. Conclusions: Our findings may provide new insights into the mechanisms underlying PNES and offer a promising diagnostic approach to differentiate them from epilepsy.

We investigate topological and spectral properties of models of European and US-American power grids and of paradigmatic network models as well as their implications for the synchronization dynamics of phase oscillators with heterogeneous natural frequencies. We employ the complex-valued order parameter—a widely used indicator for phase ordering—to assess the synchronization dynamics and observe the order parameter to exhibit either constant or periodic or non-periodic, possibly chaotic temporal evolutions for a given coupling strength but depending on initial conditions and the systems’ disorder. Interestingly, both topological and spectral characteristics of the power grids point to a diminished capability of these networks to support a temporarily stable synchronization dynamics. We find non-trivial commonalities between the synchronization dynamics of oscillators on seemingly opposing topologies.

Epilepsy is now considered a network disease that affects the brain across multiple levels of spatial and temporal scales. The paradigm shift from an epileptic focus—a discrete cortical area from which seizures originate—to a widespread epileptic network—spanning lobes and hemispheres—considerably advanced our understanding of epilepsy and continues to influence both research and clinical treatment of this multi-faceted high-impact neurological disorder. The epileptic network, however, is not static but evolves in time which requires novel approaches for an in-depth characterization. In this review, we discuss conceptual basics of network theory and critically examine state-of-the-art recording techniques and analysis tools used to assess and characterize a time-evolving human epileptic brain network. We give an account on current shortcomings and highlight potential developments towards an improved clinical management of epilepsy.