What can maths tell us about this topic? Do mathematicians even have a seat at the table, and should we? What do we know about directed networks and dynamical systems that can contribute to this?

We consider the implications of the mathematical modelling and analysis of neurone-to-neurone dynamical complex networks. We explain how the dynamical behaviour of relatively small scale strongly connected networks lead naturally to non-binary information processing and thus to multiple hypothesis decision making, even at the very lowest level of the brain’s architecture. This all looks a like a a loose  coupled array of  k-dimensional clocks. There are lots of challenges for maths here. We build on these ideas to address the "hard problem" of consciousness - which other disciplines say is beyond any mathematical explanation for ever! 

We discuss how a proposed “dual hierarchy model”, made up from both externally perceived, physical, elements of increasing complexity, and internally experienced, mental elements (which we argue are equivalent to feelings), may support a leaning and evolving consciousness. We introduce the idea that a human brain ought to be able to re-conjure subjective mental feelings at will. An immediate consequence of this model  is that finite human brains must always be learning and forgetting and that any possible subjective internal feeling that might be fully idealised only with a countable infinity of facets, could never be learned completely a priori by zombies or automata: it may be experienced more and more fully by an evolving human brain (yet never in totality, not even in a lifetime). 

Results on the existence and stability of clustered spike patterns for biological reaction‐diffusion systems with two small diffusivities will be presented. In particular we consider a consumer chain model and the Gierer‐Meinhardt activator-inhibitor system with a precursor gradient. A clustered spike pattern consists of multiple spikes which all approach the same limiting point as the diffusivities tend to zero. We will present results on the asymptotic behaviour of the spikes including their shapes, positions and amplitudes. We will also compute the asymptotic behaviour of the eigenvalues of the system linearised around a clustered spike pattern. These systems and their solutions play an important role in biological modelling to account for the bridging of lengthscales, e.g. between genetic, nuclear, intra‐cellular, cellular and tissue levels, or for the time-hierarchy of biological processes, e.g. a large‐scale structure, which appears first, induces patterns on smaller scales. This is joint work with Juncheng Wei.
 

Considerable attention has been recently focused on the study of muscle tissues viewed as prototypes of new materials that can actively generate stresses. The intriguing mechanical properties of such systems can be linked to hierarchical internal architecture. To complicate matters further, they are driven internally by endogenous mechanisms supplying energy and maintaining non-equilibrium.  In this talk we review the principal mechanisms of force generation in muscles and discuss the adequacy of the available mathematical models.

Reinhard Farwig and Chenyin Qian

Consider the autonomous quasi-geostrophic equation with fractional dissipation in $\mathbb{R}^2$
  \begin{equation} \label{a}
 \theta_t+u\cdot\nabla\theta+(-\Delta)^{\alpha}\theta=f(x,\theta)
 \end{equation}
in the subcritical case $1/2<\alpha\leq1$, with initial condition $\theta(x, 0)= \theta^{0}$ and given external force $f(x,\theta)$. Here the real scalar function $\theta$ is the so-called potential temperature, and the incompressible velocity field $u=(u_1,u_2)=(-\mathcal {R}_2\theta,\mathcal {R}_1\theta)$ is determined from $\theta$ via Riesz operators.  Our aim is to prove the existence of the compact global attractor $\mathcal{A}$ in the Bessel potential space $H^s(\mathbb{R}^2)$ when $s>2(1-\alpha)$.

The  construction of the attractor is based on the existence of an absorbing set in $L^2(\mathbb{R}^2)$ and $H^s(\mathbb{R}^2)$ where $s>2(1-\alpha)$. A second major step is usually based on compact Sobolev embeddings which unfortunately do not hold for unbounded domains. To circumvent this problem we exploit compact Sobolev embeddings on  balls $B_R \subset \mathbb{R}^2$ and uniform smallness estimates of solutions on $\mathbb{R}^2 \setminus B_R$. In the literature the latter estimates are obtained by a damping term $\lambda\theta$, $\lambda<0$, as part of the right hand side $f$ to guarantee exponential decay estimates. In our approach we exploit a much weaker nonlocal damping term of convolution type $\rho*\theta$ where $\widehat \rho<0$. 

In everyday language, this talk studies the question about the optimal shape and location of a thermometer of a given volume to reconstruct the temperature distribution in an entire room. For random initial conditions, this problem was considered by Privat, Trelat and Zuazua (ARMA, 2015), and for short times we remove both the randomness and geometric assumptions in their article. Analytically, we obtain quantitative estimates for the well-posedness of an inverse problem, in which one determines the solution in the whole domain from its restriction to a subset of given volume. Using a new decomposition of $L^2(\Rd)$ into heat packets from microlocal analysis, we conclude that there exists a unique optimal such subset, that it is semi-analytic and can be approximated numerically by solving a sequence of finite-dimensional optimization problems. (joint with Alden Waters)
 

I will report some recent analytical results on microstructures in low-hysteresis shape memory alloys. The modelling assumption is that the width of the thermal hysteresis is closely related to the minimal energy that is necessary to build a martensitic nucleus in an austenitic matrix. This energy barrier is typically modeled by (singularly perturbed) nonconvex elasticity functionals. In this talk, I will discuss recent results on the resulting variational problems, including stress-free inclusions and microstructures in the case of almost compatible phases. This talk is partly based on joint works with S. Conti, J. Diermeier, M. Klar, and D. Melching.

Networks of interacting oscillators give rise to collective dynamics such as localized frequency synchrony. In networks of neuronal oscillators, for example, the location of frequency synchrony could encode information. We discuss some recent persistence results for certain dynamically invariant sets called weak chimeras, which show localized frequency synchrony of oscillators. We then explore how the network structure and interaction allows for dynamic switching of the spatial location of frequency synchrony: these dynamics are induced by stable heteroclinic connections between weak chimeras. Part of this work is joined with Peter Ashwin (Exeter).