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Predicting future system behaviour from past observed behaviour (time series) is fundamental to science and engineering. In computational neuroscience, the prediction of future epileptic seizures from brain activity measurements, using EEG data, remains largely unresolved despite much dedicated research effort. Based on a longitudinal and state-of-the-art data set using intercranial EEG measurements from people with epilepsy, we consider the automated discovery of predictive features (or biomarkers) to forecast seizures in a patient-specific way. To this end, we use the path signature, a recent development in the analysis of data streams, to map from measured time series to seizure prediction. The predictor is based on linear classification, here augmented with sparsity constraints, to discern time series with and without an impending seizure. This approach may be seen as a step towards a generic pattern recognition pipeline where the main advantages are simplicity and ease of customisation, while maintaining forecasting performance on par with modern machine learning. Nevertheless, it turns out that although the path signature method has some powerful theoretical guarantees, appropriate time series statistics can achieve essentially the same results in our context of seizure prediction. This suggests that, due to their inherent complexity and non-stationarity, the brain's dynamics are not identifiable from the available EEG measurement data, and, more concretely, epileptic episode prediction is not reliably achieved using EEG measurement data alone.

The empirical success of machine learning models with many more parameters than measurements has generated an interest in the theory of overparameterisation, i.e., underdetermined models. This paradigm has recently been studied in domains such as deep learning, where one is interested in good (local) minima of complex, nonlinear loss functions. Optimisers, like gradient descent, perform well and consistently reach good solutions. Similarly, nonlinear optimisation problems are encountered in the field of system identification. Examples of such high-dimensional problems are optimisation tasks ensuing from the reconstruction of model states and parameters of an assumed known dynamical system from observed time series. In this work, we identify explicit parallels in the benefits of overparameterisation between what has been analysed in the deep learning context and system identification. We test multiple chaotic time series models, analysing the optimisation process for unknown model states and parameters in batch mode. We find that gradient descent reaches better solutions if we assume more parameters to be unknown. We hypothesise that, indeed, overparameterisation leads us towards better minima, and that more degrees of freedom in the optimisation are beneficial so long as the system is, in principle, observable.

Autoregressive models are ubiquitous tools for the analysis of time series in many domains such as computational neuroscience and biomedical engineering. In these domains, data is, for example, collected from measurements of brain activity. Crucially, this data is subject to measurement errors as well as uncertainties in the underlying system model. As a result, standard signal processing using autoregressive model estimators may be biased. We present a framework for autoregressive modelling that incorporates these uncertainties explicitly via an overparameterised loss function. To optimise this loss, we derive an algorithm that alternates between state and parameter estimation. Our work shows that the procedure is able to successfully denoise time series and successfully reconstruct system parameters. This new paradigm can be used in a multitude of applications in neuroscience such as brain-computer interface data analysis and better understanding of brain dynamics in diseases such as epilepsy.

This paper examines the relationship between sparse random network architectures and neural network stability by examining the eigenvalue spectral distribution. Specifically, we generalise classical eigenspectral results to sparse connectivity matrices obeying Dale's law: neurons function as either excitatory (E) or inhibitory (I). By defining sparsity as the probability that a neutron is connected to another neutron, we give explicit formulae that shows how sparsity interacts with the E/I population statistics to scale key features of the eigenspectrum, in both the balanced and unbalanced cases. Our results show that the eigenspectral outlier is linearly scaled by sparsity, but the eigenspectral radius and density now depend on a nonlinear interaction between sparsity, the E/I population means and variances. Contrary to previous results, we demonstrate that a non-uniform eigenspectral density results if any of the E/I population statistics differ, not just the E/I population variances. We also find that 'local' eigenvalue-outliers are present for sparse random matrices obeying Dale's law, and demonstrate that these eigenvalues can be controlled by a modified zero row-sum constraint for the balanced case, however, they persist in the unbalanced case. We examine all levels of connection (sparsity), and distributed E/I population weights, to describe a general class of sparse connectivity structures which unifies all the previous results as special cases of our framework. Sparsity and Dale's law are both fundamental anatomical properties of biological neural networks. We generalise their combined effects on the eigenspectrum of random neural networks, thereby gaining insight into network stability, state transitions and the structure-function relationship., Comment: 17 pages, 6 figures

Mathematical modelling of the macroscopic electrical activity of the brain is highly non-trivial and requires a detailed understanding of not only the associated mathematical techniques, but also the underlying physiology and anatomy. Neural field theory is a population-level approach to modelling the non-linear dynamics of large populations of neurons, while maintaining a degree of mathematical tractability. This class of models provides a solid theoretical perspective on fundamental processes of neural tissue such as state transitions between different brain activities as observed during epilepsy or sleep. Various anatomical, physiological, and mathematical assumptions are essential for deriving a minimal set of equations that strike a balance between biophysical realism and mathematical tractability. However, these assumptions are not always made explicit throughout the literature. Even though neural field models (NFMs) first appeared in the literature in the early 1970's, the relationships between them have not been systematically addressed. This may partially be explained by the fact that the inter-dependencies between these models are often implicit and non-trivial. Herein we provide a review of key stages of the history and development of neural field theory and contemporary uses of this branch of mathematical neuroscience. First, the principles of the theory are summarised throughout a discussion of the pioneering models by Wilson and Cowan, Amari and Nunez. Upon thorough review of these models, we then present a unified mathematical framework in which all neural field models can be derived by applying different assumptions. We then use this framework to i) derive contemporary models by Robinson, Jansen and Rit, Wendling, Liley, and Steyn-Ross, and ii) make explicit the many significant inherited assumptions that exist in the current literature., Comment: 67 pages, 10 figures, 1 table

Changes in brain states, as found in many neurological diseases such as epilepsy, are often described as bifurcations in mesoscopic neural models. Nearly all of these models rely on a mathematically convenient, but biophysically inaccurate, description of the synaptic input to neurons called current-based synapses. We develop a novel analytical framework to analyze the effects of a more biophysically realistic description, known as conductance-based synapses. These are implemented in a mesoscopic neural model and compared to the standard approximation via a single parameter homotopic mapping. A bifurcation analysis using the homotopy parameter demonstrates that if a more realistic synaptic coupling mechanism is used in this class of models, then a bifurcation or transition to an abnormal brain state does not occur in the same parameter space. We show that the more realistic coupling has additional mathematical parameters that require a fundamentally different biophysical mechanism to undergo a state transition. These results demonstrate the importance of incorporating more realistic synapses in mesoscopic neural models and challenge the accuracy of previous models, especially those describing brain state transitions such as epilepsy., Comment: This is the submitted version