Claudia Lainscsek

When a d-dimensional system is investigated through one of its variables v, a natural space for characterizing its dynamics is provided by the v-induced differential embedding spanned by that variable and its (d − 1) successive derivatives. The corresponding governing equations are called the canonical form of that system. Canonical forms—often called jerk equations when, in addition, the system is three-dimensional—can be used for classifying chaotic systems not only by comparing their algebraic structures, as done in previous works, but also by considering their parameter values, helping in identifying system’s parameter values for producing equivalent dynamics, that is, for producing a given type of attractor. Two types of equivalence can, therefore, be distinguished: (i) structural, when only the algebraic structures of canonical forms match, and (ii) dynamical, when the canonical parameter values coincide. Despite their potential, canonical forms have been only partially exploited for systematically relating and identifying chaotic systems. In this work, we address this gap by using the x-induced canonical form of the Lorenz system to investigate its relationships with other Lorenz-like systems. In particular, we show how canonical forms can be used to determine parameter values that reproduce a Lorenz attractor, even when the underlying algebraic structures differ. This provides a novel perspective in which the algebraic structure of the governing equations is explicitly leveraged to generate prescribed chaotic attractors, opening new ways for system identification and design.

Transcranial magnetic stimulation (TMS) has been widely used to study the mechanisms that underlie motor output. Yet, the extent to which TMS acts upon the cortical neurons implicated in volitional motor commands and the focal limitations of TMS remain subject to debate. Previous research links TMS to improved subject performance in behavioral tasks, including a bias in phoneme discrimination. Our study replicates this result, which implies a causal relationship between electro-magnetic stimulation and psychomotor activity, and tests whether TMS-facilitated psychomotor activity recorded via electroencephalography (EEG) may thus serve as a superior input for neural decoding. First, we illustrate that site-specific TMS elicits a double dissociation in discrimination ability for two phoneme categories. Next, we perform a classification analysis on the EEG signals recorded during TMS and find a dissociation between the stimulation site and decoding accuracy that parallels the behavioral results. We observe weak to moderate evidence for the alternative hypothesis in a Bayesian analysis of group means, with more robust results upon stimulation to a brain region governing multiple phoneme features. Overall, task accuracy was a significant predictor of decoding accuracy for phoneme categories (F(1,135) = 11.51, p < 0.0009) and individual phonemes (F(1,119) = 13.56, p < 0.0003), providing new evidence for a causal link between TMS, neural function, and behavior.

Delay Differential Analysis (DDA) is a nonlinear method for analyzing time series based on principles from nonlinear dynamical systems. DDA is extended here to incorporate network aspects to improve the dynamical characterization of complex systems. To demonstrate its effectiveness, DDA with network capabilities was first applied to the well-known Rössler system under different parameter regimes and noise conditions. Network-motif DDA, based on cortical regions, was then applied to invasive intracranial electroencephalographic data from drug-resistant epilepsy patients undergoing presurgical monitoring. The directional network motifs between brain areas that emerge from this analysis change dramatically before, during, and after seizures. Neural systems provide a rich source of complex data, arising from varying internal states generated by network interactions.

Differential equations serve as models for many physical systems. But, are these equations unique? We prove here that when a 3D system of ordinary differential equations for a dynamical system is transformed to the jerk or differential form, the jerk form is preserved in relation to a given variable and, therefore, the transformed system shares the time series of that given variable with the original untransformed system. Multiple algebraically different systems of ordinary differential equations can share the same jerk form. They may also share the same time series of the transformed variable depending on the parameters of the jerk form. Here, we studied 17 algebraically different Lorenz-like systems that share the same functional jerk form. There are groups of these systems that share the jerk parameters and, therefore, also have the same time series of the transformed variable.

Electroencephalography (EEG) holds promise for brain-computer interface (BCI) devices as a non-invasive measure of neural activity. With increased attention to EEG-based BCI systems, publicly available datasets that can represent the complex tasks required for naturalistic speech decoding are necessary to establish a common standard of performance within the BCI community. Effective solutions must overcome various kinds of noise in the EEG signal and remain reliable across sessions and subjects without overfitting to a specific dataset or task. We present two validated datasets (N=8 and N=16) for classification at the phoneme and word level and by the articulatory properties of phonemes. EEG signals were recorded from 64 channels while subjects listened to and repeated six consonants and five vowels. Individual phonemes were combined in different phonetic environments to produce coarticulated variation in forty consonant-vowel pairs, twenty real words, and twenty pseudowords. Phoneme pairs and words were presented during a control condition and during transcranial magnetic stimulation targeted to inhibit or augment the EEG signal associated with specific articulatory processes.

State-of-the-art technologies in neural speech decoding utilize data collected from microwires or microarrays implanted directly into the cerebral cortex. Yet as a tool accessible only to individuals with implanted electrodes, speech decoding from devices of this nature is severely limited in its implementation, and cannot be considered a viable solution for widespread application. Speech decoding from non-invasive EEG signals can achieve relatively high accuracy (70-80%), but only from very small classification tasks, with more complex tasks typically yielding a limited (20-50%) classification accuracy. We propose a novel combination of technologies in which transcranial magnetic stimulation (TMS) is first applied to augment the neural signals of interest, producing a greater signal-to-noise ratio in the EEG data. Next, delay differential analysis (DDA) – a cutting-edge computational method based on nonlinear dynamics – is implemented to capture the widest range of information available in the neural signal, by incorporating both linear and nonlinear dynamics.

Determining synchronization, causality, and dynamical similarity in highly complex nonlinear systems like brains is challenging. Although distinct, these measures are related by the unknown deterministic structure of the underlying dynamical system. For two systems that are not independent on each other, either because they result from a common process or they are already synchronized, causality measures typically fail. Here, we introduce dynamical ergodicity to assess dynamical similarity between time series and then combine this new measure with cross-dynamical delay differential analysis to estimate causal interactions between time series. We first tested this approach on simulated data from coupled Rössler systems where ground truth was known. We then applied it to intracranial electroencephalographic data from patients with epilepsy and found distinct dynamical states that were highly predictive of epileptic seizures.

In 1994, Sprott [Phys. Rev. E 50, 647–650 (1994)] proposed a set of 19 different simple dynamical systems producing chaotic attractors. Among them, 14 systems have a single nonlinear term. To the best of our knowledge, their diffeomorphical equivalence and the topological equivalence of their chaotic attractors were never systematically investigated. This is the aim of this paper. We here propose to check their diffeomorphical equivalence through the jerk functions, which are obtained when the system is rewritten in terms of one of the variables and its first two derivatives (two systems are thus diffeomorphically equivalent when they have exactly the same jerk function, that is, the same functional form and the same coefficients). The chaotic attractors produced by these systems—for parameter values close to the ones initially proposed by Sprott—are characterized by a branched manifold. Systems B and C produce chaotic attractors, which are observed in the Lorenz system and are also briefly discussed. Those systems are classified according to their diffeomorphical and topological equivalence.

Dynamic functional brain connectivity facilitates adaptive cognition and behavior. Abnormal alterations within such connectivity could result in disrupted functions observed across various neurological conditions. As one of the most common neurological disorders, epilepsy is defined by the seemingly random occurrence of spontaneous seizures. A central but unresolved question concerns the mechanisms by which extraordinarily diverse dynamics of seizures emerge. Here, we apply a graph-theoretical approach to assess dynamic reconfigurations in the functional brain connectivity before, during, and after seizures that display heterogeneous propagation patterns despite sharing similar origins. We demonstrate unique reconfigurations in globally-defined network properties preceding seizure onset that predict propagation patterns of impending seizures, and in locally-defined network properties that differentiate post-onset dynamics. These results characterize quantitative network features underlying the heterogeneity of seizure dynamics and the accompanying clinical manifestations. Decoding these network properties could improve personalized preventative treatment strategies for epilepsy as well as other neurological disorders.