Research

White Matter Computation

Much of the focus in bio-inspired computing has been on replicating the properties of individual neurons, while largely overlooking the critical role that communication delays between these neurons play in shaping network dynamics. These delays are in fact thought to be central to many of the brain's natural computational processes, painting a picture of the brain as a highly distributed, delayed, parallel processing system.

The insulating material myelin, which gives white matter its colour, is now known to be plastic, adapting in response to neuronal activity. This plasticity adjusts the speed of signal transmission along axons, adding control over communication timing between neurons. While synaptic plasticity has been a major focus in neuroscience, myelin plasticity’s role in network dynamics is a far newer area of study. By modulating axonal delays, myelin impacts the timing and efficiency of signals, making communication delays a crucial aspect of brain-like network computation.

Our project aims to develop a new mathematical framework for understanding how these axonal delays, and their plasticity, influence the dynamics of biologically-inspired neural networks. We leverage tools from dynamical systems theory to create a novel paradigm for white matter computation, centred around the concept of "computation with network attractors." This approach allows us to explore the complex role of timing in the brain's distributed processing, offering new insights into how communication delays contribute to adaptive learning and neural computation. This work will not only deepen our understanding of how the brain organises and processes information, but it will also lay the groundwork for developing new computational frameworks inspired by natural neural systems.

Relevant publications

Phase-Isostable Descriptions of Coupled Oscillator Networks

Phase-reduced models, where oscillator dynamics are reduced to the dynamics of their phase on limit cycle, have been extremely successful in describing dynamical behaviours of networks of coupled oscillators in the case where individual oscillators possess a strongly attracting limit cycle and coupling between oscillators in the network is weak. However, for many biological (and particularly neural) oscillator networks these modelling assumptions are not appropriate.

An alternative approach to analysing oscillator network dynamics in cases where decay to the limit cycle is slow in some direction (one Floquet exponent is close to zero) is to consider an additional amplitude (or isostable) coordinate for each oscillator which gives a notion of distance from the limit cycle. In the case of instantaneous network interactions we have demonstrated that the phase-isostable framework is more accurate than higher-order phase reductions in capturing bifurcations of phase-locked states in the mean-field complex Ginzburg-Landau equation. We have also shown that the phase-isostable framework can capture dynamics of globally linearly coupled networks of Morris-Lecar neuron models (both two and many nodes) which cannot be discerned by a first-order phase description.

We have also extended this framework to allow for the treatment of delay induced oscillators where the phase and interaction functions required must be computed as solutions of delay differential equations. We use a harmonic balance approach to obtain approximations to these functions. We are now further extending this work to a phase-isostable description of networks of coupled oscillators with delayed network connections, with the aim of later exploring the effects of delay plasticity on network dynamics. Here a phase-isostable description is particularly desirable as it is able to capture the multiple branches of phase-locked solutions with different collective frequiencies and amplitudes which we observe when delays are large.

Relevant publications

Understanding Sensory Induced Visual Hallucinations

Explorations of visual hallucinations, and in particular those of Billock and Tsou, show that annular rings with a background flicker can induce visual hallucinations in humans that take the form of radial fan shapes as shown in the figure below. The well-known retino-cortical map tells us that the corresponding patterns of neural activity in the primary visual cortex for rings and arms in the retina are orthogonal stripe patterns. The implication is that cortical forcing by spatially periodic input can excite orthogonal modes of neural activity. We have shown that a simple scalar neural field model of primary visual cortex with state-dependent spatial forcing is capable of modelling this phenomenon. Moreover, we see that this occurs most robustly when the spatial forcing has a 2:1 resonance with modes that would otherwise be excited by a Turing instability. By utilising a weakly nonlinear multiple-scales analysis we have determined the relevant amplitude equations for uncovering the parameter regimes which favour the excitation of patterns orthogonal to sensory drive. In combination with direct numerical simulations we can use this approach to shed further light on the original psychophysical observations of Billock and Tsou.

Relevant publications

Networks of Piecewise Linear Oscillators

Coupled oscillators have been instrumental in advancing our understanding of real-world networks in myriad fields, ranging from neuroscience and ecology to sociology and optics. In many models, the dynamics of a single node in a network is governed by a system of nonlinear ordinary differential equations, which necessitates the numerical determination of periodic orbits in high-dimensional systems. In turn, this significantly complicates the analysis of emergent network states. In the SIAM review article below, we reviewed a suite of results of how — and why — piecewise linear node models are ideally suited to unravel the rich dynamics seen in coupled oscillatory networks.

The review discusses augmentations of phase reductions, phase-amplitude reductions, and the master stability function for the analysis of piecewise-linear systems, for which one can readily construct periodic orbits. This yields useful insights into network behavior, but the cost is that one needs to study nonsmooth dynamical systems requiring the use of saltation operators to treat the propagation of perturbations through switching manifolds to study stability and reveal bifurcations at the network level.

The techniques are illustrated with applications to neural systems including spiking neural networks (including cluster states in networks with symmetry) as in R Nicks, L Chambon and S Coombes 2018 as well as cardiac systems, networks of electromechanical oscillators, and cooperation in cattle herds.

Additional work on nonsmooth oscillatory systems include the analysis of the two-process model of sleep-wake regulation. Traditionally studied using the discrete-time mapping between sleep and wake times we show that an equivalent description can be obtained from a direct analysis of the underlying nonsmooth flow. We further show how to construct the Lyapunov exponent of the nonsmooth flow and use this to uncover a more detailed picture of the Arnol’d tongue structure of the model. See M Sayli, A C Skeldon, R Thul, R Nicks and S Coombes 2023.

Relevant publications

Pattern formation and neural fields on spherical domains

In collaboration with Prof Stephen Coombes and Dr Sid Visser, I have considered the spatiotemporal patterns that arise due to dynamic instability in a neural field model of cortex with axonal delays on a sphere. Symmetric bifurcation theory, together with centre manifold reduction and direct simulations with a bespoke numerical scheme, allowed investigation of the patterned states which can exist in addition to the quasi-periodic behaviour observed in numerical simulations. The movie here shows chaotic neural activity on the sphere (video courtesy of Sid Visser).

This work builds on the research I carried out for my PhD where I used equivariant bifurcation theory to consider both the symmetries of the time-periodic patterns which can be created at a Hopf bifurcation with spherical symmetry and the symmetric spiral patterns which can exist on spheres, such as the one in the image on the left. Equivariant bifurcation theory uses Lie group theory to study the model–independent behaviours of symmetric dynamical systems (those which depend on the symmetries alone) without any reference to the details of a particular model.

Relevant publications

Modelling solid crystals with defects

During my time as a research fellow at the University of Nottingham, I worked with Gareth Parry on the EPSRC funded project "Modelling continuous and discrete defective crystals". The discrete structures associated with defective crystals can be described by a set of lattice vectors and a dislocation density tensor, S, which describes the local "texture" of the crystal. When S is constant the theory of Lie groups can be used to describe the symmetries of the discrete structure. My research focused on the discrete structures and their symmetries for certain forms of the dislocation density tensor where the associated Lie group is nilpotent or solvable.

Relevant publications