Applied Mathematics Seminar

Synopsis:

Spontaneous waves are ubiquitous across many brain regions during early development. Activity from waves occurring in the retina are propagated to downstream areas, such as the thalamus and primary visual cortex, and are hypothesized to drive the development of receptive fields (RFs). Different stages of retinal spontaneous waves coincide with the development of the retinotopic map, ON-OFF segregation, and orientation selectivity in the early visual pathway of mammals. However, the mechanisms underlying the influence of such retinal waves on RF refinement are not well understood. In this work, we build a biologically-constrained mathematical model describing the development of the feed-forward RF of neurons in the primary visual cortex. These feed-forward synapses are driven by retinal waves using a spike-timing-dependent triplet plasticity rule. Using this model, we propose a possible mechanism underlying a pruning process leading to different RF spatial structures. In particular, we quantify how key characteristics of the retinal wave, such as wave speed and width, affect the simulated pruning result and shape of the receptive field. Through the derivation and analysis of a reduced rate model, we uncover mechanisms for the formation of a range of spatially structured RFs, including a periodic RF, which may help to understand related periodic RF development in other brain areas such as grid cells in the entorhinal cortex.

Synopsis:

We study the emergence, stability and evolution of solitons and compactons in a class of Klein-Gordon equations

\(u_{tt} - u_{xx} + u = u^{1+n} - \kappa_{1+2n} u^{1+2n}, \quad n= 1, 2, \ldots\)

endowed with trivial and non-trivial stable equilibria, and demonstrate that similarly to the classical \(\kappa_{1+2n} = 0\) cases, solitons are linearly unstable, but their instability weakens as \(\kappa_{1+2n}\) increases, and vanishes at a critical \(\kappa_{1+2n}^{crit} = (1+n)/(2+n)^2\), where solitons disappear and kinks form.

As the growing amplitude of the unstable soliton approaches the non-trivial equilibrium, it morphs into ”meson”, a robust box shaped sharp pulse with a flat-top plateau which expands at a sonic speed. In the \(\kappa_{1+2n}^{crit}\) vicinity, where the instability is suppressed, and the internal modes hardly change, solitons persist for a very long time and rather than turn into meson, convert to breather.

Linear damping tempers the conversion and slows it. When \(-1/2 < n < 0\), compactons emerge and being unstable morph either to meson or to breather.

Joint work with Slava Krylov.

Synopsis:

Spreading of innovations and epidemics is a classical problem. Traditionally, they have been analyzed using compartmental models (Bass, SI, …), which implicitly assume that all the individuals are homogeneous and connected to each other. To relax these assumptions, research has gradually shifted to the more fundamental network models, which are particle models for the stochastic adoption/infection by each individual.

In this talk I will present an emerging mathematical theory for the Bass and SI models on networks. I will present analytic tools that enable us to obtain explicit expressions for the expected adoption/infection level on various networks (complete, circular, d-regular, Erdos-Renyi, …), without employing mean-field type approximations. The main focus of the talk will be on the effect of network structure. For example, which networks yield the slowest and fastest spreading? What is the effect of boundaries? Of heterogeneity among individuals? How does the network structure influence the optimal promotional strategy?