Modelling Dendrites

Neurons receive information through thin, branched structures called dendrites. During development dendrites grow, retract, and branch. Crucially, retraction and branching make it necessary to track whole dendrites rather than only their endpoints. While this is handled implicitly in some existing simulations, we have developed an explicitly path-valued stochastic process for dendrite growth. 

The figure to the right shows a simulation of the model at four timepoints. Branch sections are marked as active (solid lines) or inactive (dashed lines) depending on whether their endpoints are free to move. This prevents retracting braches causing the neuron to become disconnected.

Our goal is to use this model alongside neuron imaging data to help understand dendrite development in different environments. 

Age Structure 

We incorporate age structure in a classical model of continuous opinion formation, studying both a stochastic differential equation (SDE) model and the corresponding mean-field partial differential equation (PDE). We prove the existence of stationary states of the PDE model, but also see non-uniqueness of stationary states and periodic behaviour, an example of which is shown on the left. 

You can find a video description of this paper here! 

Comparing Modelling Approaches

We derive ODEs for opinion dynamics from a stochatsic ABM with pairwise interactions by simultaneously rescaling the time-step and strength of interactions. The video shows realisations of the ABM in solid lines and the limiting ODE in dashed lines. This connection answers questions in both settings, such as the motivation of multiplicative noise terms in SDE models or the link between selection noise and mollification of interaction functions.

Early Warning Signals

Critical slowing down states that systems display increasing relaxation times prior to a critical transition, an effect that can be observed in timeseries statistics to give an early warning of the transition. However, in epidemiological models there is frequent disagreement with this general theory, and the alternative theory of critical speeding up predicts contradictory behaviour of early warning signals. We describe the behaviour of common early warning signals in terms of a system’s potential surface and noise around a quasi-steady state, then describe an equation-free method to obtain these key features from timeseries, using a version of the SIS model as a case study. The figure shows example reconstructed potential surfaces.