Search

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.

Computational modeling helps neuroscientists to integrate and explain experimental data obtained through neurophysiological and anatomical studies, thus providing a mechanism by which we can better understand and predict the principles of neural computation. Computational modeling of the neuronal pathways of the visual cortex has been successful in developing theories of biological motion processing. This review describes a range of computational models that have been inspired by neurophysiological experiments. Theories of local motion integration and pattern motion processing are presented, together with suggested neurophysiological experiments designed to test those hypotheses.

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

The identification of states and parameters from noisy measurements of a dynamical system is of great practical significance and has received a lot of attention. Classically, this problem is expressed as optimization over a class of models. This work presents such a method, where we augment the system in such a way that there is no distinction between parameter and state reconstruction. We pose the resulting problem as a batch problem: given the model, reconstruct the state from a finite sequence of output measurements. In the case the model is linear, we derive an analytical expression for the state reconstruction given the model and the output measurements. Importantly, we estimate the state trajectory in its entirety and do not aim to estimate just an initial condition: that is, we use more degrees of freedom than strictly necessary in the optimization step. This particular approach can be reinterpreted as training of a neural network that estimates the state trajectory from available measurements. The technology associated with neural network optimization/training allows an easy extension to nonlinear models. The proposed framework is relatively easy to implement, does not depend on an informed initial guess, and provides an estimate for the state trajectory (which incorporates an estimate for the unknown parameters) over a given finite time horizon.

Previous work by Linsker revealed how simple cells can emerge in the absence of structured environmental input, via a self-organisation learning process. He empirically showed the development of spatial-opponent cells driven only by input noise, emerging as a result of structure in the initial synaptic connectivity distribution. To date, a complete set of radial eigenfunctions have not been provided for this multi-layer network. In this paper, the complete set of eigenfunctions and eigenvalues for a three-layered network is for the first time analytically derived. Initially a simplified learning equation is considered for which the homeostatic parameters are set to zero. To extend the eigenfunction analysis to the full learning equation, including non-zero homeostatic parameters, a perturbation analysis is used., Comment: 24 pages, 6 figures

The mechanisms underlying how activity in the visual pathway may give rise through neural plasticity to many of the features observed experimentally in the early stages of visual processing was provided by Linkser in a seminal, three-paper series. Owing to the complexity of multi-layer models, an implicit assumption in Linsker's and subsequent papers has been that propagation delay is homogeneous and plays little functional role in neural behaviour. We relax this assumption to examine the impact of distance-dependent axonal propagation delay on neural learning. We show that propagation delay induces low-pass filtering by dispersing the arrival times of spikes from presynaptic neurons, providing a natural correlation cancellation mechanism for distal connections. The cut-off frequency decreases as the radial propagation delay within a layer increases relative to propagation delay between the layers, introducing an upper limit on temporal resolution. Given that the PSP also acts as a low-pass filter, we show that the effective time constant of each should enable the processing of similar scales of temporal information. This result has implications for the visual system, in which receptive field size and, thus, radial propagation delay, increases with eccentricity. Furthermore, the network response is frequency dependent since higher frequencies require increased input amplitude to compensate for attenuation. This concords with frequency-dependent contrast sensitivity in the visual system, which changes with eccentricity and receptive field size. We further show that the proportion of inhibition relative to excitation is larger where radial propagation delay is long relative to inter-laminar propagation delay. We show that the addition of propagation delay reduces the range in the cell's on-center size, providing stability to variations in homeostatic parameters., Comment: 27 pages, 6 figures

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