Topographic maps are a brain structure connecting pre-synpatic and
post-synaptic brain regions. Topographic development is dependent on
Hebbian-based plasticity mechanisms working in conjunction with spontaneous
patterns of neural activity generated in the pre-synaptic regions. Studies
performed in mouse have shown that these spontaneous patterns can exhibit
complex spatial-temporal structures which existing models cannot incorporate.
Neural field theories are appropriate modelling paradigms for topographic
systems due to the dense nature of the connections between regions and can be
augmented with a plasticity rule general enough to capture complex time-varying
structures.
We propose a theoretical framework for studying the development of topography
in the context of complex spatial-temporal activity fed-forward from the
pre-synaptic to post-synaptic regions. Analysis of the model leads to an
analytic solution corroborating the conclusion that activity can drive the
refinement of topographic projections. The analysis also suggests that
biological noise is used in the development of topography to stabilise the
dynamics. MCMC simulations are used to analyse and understand the differences
in topographic refinement between wild-type and the $\beta2$ knock-out mutant
in mice. The time scale of the synaptic plasticity window is estimated as
$0.56$ seconds in this context with a model fit of $R^2 = 0.81$.
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 […]
We study a model of interacting neurons. The structure of this neural system
is composed of two layers of neurons such that the neurons of the first layer
send their spikes to the neurons of the second one: if $N$ is the number of
neurons of the first layer, at each spiking time of the first layer, every
neuron of both layers receives an amount of potential of the form $U/\sqrt{N},$
where $U$ is a centered random variable. This kind of structure of neurons can
model a part of the structure of the visual cortex: the first layer represents
the primary visual cortex V1 and the second one the visual area V2. The model
consists of two stochastic processes, one modelling the membrane potential of
the neurons of the first layer, and the other the membrane potential of the
neurons of the second one. We prove the convergence of these processes as the
number of neurons~$N$ goes to infinity and obtain a convergence speed. The
proofs rely on similar arguments as those used in [Erny, LĂ¶cherbach,
Loukianova (2022)]: the convergence speed of the semigroups of the processes is
obtained from the convergence speed of their infinitesimal generators using a
Trotter-Kato formula, and from the regularity of the limit semigroup.
Among other models aimed at understanding self-sustained oscillations in neural networks, the NNLIF model with synaptic delay was developed almost twenty years ago to model fast global oscillations in networks of weakly firing inhibitory neurons. Periodic solutions were numerically observed in this model, but despite its intensive study by researchers in PDEs and probability, there is up-to-date no analytical result on this topic. In this article, we propose to approximate formally these solutions by a Gaussian wave whose periodic movement is described by an associate delay differential equation (DDE). We prove the existence of a periodic solution for this DDE and we give a rigorous asymptotic result on these solutions when the connectivity parameter $b$ goes to $-\infty$. Lastly, we provide heuristic and numerical evidence of the validity of our approximation.