Neural field models with transmission delays and diffusion

A neural field models the large scale behaviour of large groups of neurons. We extend previous results for these models by including a diffusion term into the neural field, which models direct, electrical connections. We extend known and prove new sun-star calculus results for delay equations to be able to include diffusion and explicitly characterise the essential spectrum. For a certain class of connectivity functions in the neural field model, we are able to compute its spectral properties and the first Lyapunov coefficient of a Hopf bifurcation. By examining a numerical example, we find that the addition of diffusion suppresses non-synchronised steady-states while favouring synchronised oscillatory modes.

Introducing divisive inhibition in the Wilson-Cowan model

Both subtractive and divisive inhibition has been recorded in cortical circuits and recent findings suggest that different interneuronal populations are responsible for the different types of inhibition. This calls for the formulation and description of these inhibitory mechanisms in computational models of cortical networks. Neural mass and neural field models typically only feature subtractive inhibition. Here, we introduce how divisive inhibition can be incorporated in such models, using the Wilson-Cowan modelling formalism as an example. In addition, we show how the subtractive and divisive modulations can be combined. Including divisive inhibition in neural mass models is a crucial step in understanding its role in shaping oscillatory phenomena in cortical networks.

Neural Field Models for Latent State Inference: Application to Large-Scale Neuronal Recordings

Large-scale neural recordings are becoming increasingly better at providing a window into functional neural networks in the living organism. Interpreting such rich data sets, however, poses fundamental statistical challenges. The neural field models of Wilson, Cowan and colleagues remain the mainstay of mathematical population modeling owing to their interpretable, mechanistic parameters and amenability to mathematical analysis. We developed a method based on moment closure to interpret neural field models as latent state-space point-process models, making mean field models amenable to statistical inference. We demonstrate that this approach can infer latent neural states, such as active and refractory neurons, in large populations. After validating this approach with synthetic data, we apply it to high-density recordings of spiking activity in the developing mouse retina. This confirms the essential role of a long lasting refractory state in shaping spatio-temporal properties of neonatal retinal waves. This conceptual and methodological advance opens up new theoretical connections between mathematical theory and point-process state-space models in neural data analysis.SignificanceDeveloping statistical tools to connect single-neuron activity to emergent collective dynamics is vital for building interpretable models of neural activity. Neural field models relate single-neuron activity to emergent collective dynamics in neural populations, but integrating them with data remains challenging. Recently, latent state-space models have emerged as a powerful tool for constructing phenomenological models of neural population activity. The advent of high-density multi-electrode array recordings now enables us to examine large-scale collective neural activity. We show that classical neural field approaches can yield latent statespace equations and demonstrate inference for a neural field model of excitatory spatiotemporal waves that emerge in the developing retina.