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.

# Past Industrial and Applied Mathematics Seminar

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).

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).

Many systems in nature consist of stochastically interacting agents or particles. Stochastic processes have been widely used to model such systems, yet they are notoriously difficult to analyse. In this talk I will show how ideas from statistics can be used to tackle some challenging problems in the field of stochastic processes.

In the first part, I will consider the problem of inference from experimental data for stochastic reaction-diffusion processes. I will show that multi-time distributions of such processes can be approximated by spatio-temporal Cox processes, a well-studied class of models from computational statistics. The resulting approximation allows us to naturally define an approximate likelihood, which can be efficiently optimised with respect to the kinetic parameters of the model.

In the second part, we consider more general path properties of a certain class of stochastic processes. Specifically, we consider the problem of computing first-passage times for Markov jump processes, which are used to describe systems where the spatial locations of particles can be ignored. I will show that this important class of generally intractable problems can be exactly recast in terms of a Bayesian inference problem by introducing auxiliary observations. This leads us to derive an efficient approximation scheme to compute first-passage time distributions by solving a small, closed set of ordinary differential equations.

In Soft Matter, octupolar order is not just an exotic mathematical curio. Liquid crystals have already provided a noticeable case of soft ordered materials for which a (second-rank) quadrupolar order tensor may not suffice to capture the complexity of the condensed phases they can exhibit. This lecture will discuss the properties of a third-rank order tensor capable of describing these more complex phases. In particular, it will be shown that octupolar order tensors come in two separate, equally abundant variants. This fact, which will be given a simple geometric interpretation, anticipates the possible existence of two distinct octupolar sub-phases.

**Bubble Dynamics**

We shall discuss certain generalisations of the Rayleigh Plesset equation for bubble dynamics

**Self-assembly of a filament by curvature-inducing proteins**

We explore a simplified macroscopic model of membrane shaping by means of curvature-sensing proteins. Equations describing the interplay between the shape of a freely floating filament in a fluid and the adhesion kinetics of proteins are derived from mechanical principles. The constant curvature solutions that arise from this system are studied using weakly nonlinear analysis. We show that the stability of the filament’s shape is completely characterized by the parameters associated with protein recruitment and establish that in the bistable regime, proteins aggregate on the filament forming regions of high and low curvatures. This pattern formation is then followed by phase-coarsening that resolves on a time-scale dependent on protein diffusion and drift across the filament, which contend to smooth and maintain the pattern respectively. The model is generalized for multiple species of proteins and we show that the stability of the assembled shape is determined by a competition between proteins attaching on opposing sides.

In recent years, ideas from the world of partial differential equations (PDEs) have found their way into the arena of graph and network problems. In this talk I will discuss how techniques based on nonlinear PDE models, such as the Allen-Cahn equation and the Merriman-Bence-Osher threshold dynamics scheme can be used to (approximately) detect particular structures in graphs, such as densely connected subgraphs (clustering and classification, minimum cuts) and bipartite subgraphs (maximum cuts). Such techniques not only often lead to fast algorithms that can be applied to large networks, but also pose interesting theoretical questions about the relationships between the graph models and their continuum counterparts, and about connections between the different graph models.

The immune system finds very rare amounts of pathogens and responds against them in a timely and efficient manner. The time to find and respond against pathogens does not vary appreciably with the size of the host animal (scale invariant search and response). This is surprising since the search and response against pathogens is harder in larger animals.

The first part of the talk will focus on using techniques from computer science to solve problems in immunology, specifically how the immune system achieves scale invariant search and response. I use machine learning techniques, ordinary differential equation models and spatially explicit agent based models to understand the dynamics of the immune system. I will talk about Hierarchical Bayesian non-linear mixed effects models to simulate immune response in different species.

The second part of the talk will focus on taking inspiration from the immune system to solve problems in computer science. I will talk about a model that describes the optimal architecture of the immune system and then show how architectures and strategies inspired by the immune system can be used to create distributed systems with faster search and response characteristics.

I argue that techniques from computer science can be applied to the immune system and that the immune system can provide valuable inspiration for robust computing in human engineered distributed systems.

**What makes cities successful? A complex systems approach to modelling urban economies**

Urban centres draw a diverse range of people, attracted by opportunity, amenities, and the energy of crowds. Yet, while benefiting from density and proximity of people, cities also suffer from issues surrounding crime, congestion and density. Seeking to uncover the mechanisms behind the success of cities using novel tools from the mathematical and data sciences, this work uses network techniques to model the opportunity landscape of cities. Under the theory that cities move into new economic activities that share inputs with existing capabilities, path dependent industrial diversification can be described using a network of industries. Edges represent shared necessary capabilities, and are empirically estimated via flows of workers moving between industries. The position of a city in this network (i.e., the subnetwork of its current industries) will determine its future diversification potential. A city located in a central well-connected region has many options, but one with only few peripheral industries has limited opportunities.

We develop this framework to explain the large variation in labour formality rates across cities in the developing world, using data from Colombia. We show that, as cities become larger, they move into increasingly complex industries as firms combine complementary capabilities derived from a more diverse pool of workers. We further show that a level of agglomeration equivalent to between 45 and 75 minutes of commuting time maximizes the ability of cities to generate formal employment using the variety of skills available. Our results suggest that rather than discouraging the expansion of metropolitan areas, cities should invest in transportation to enable firms to take advantage of urban diversity.

This talk will be based on joint work with Eduardo Lora and Andres Gomez at Harvard University.

**Hamilton-Jacobi-Bellman equations for dynamic pricing**

I will discuss the Hamilton-Jacobi-Bellman (HJB) equation, which is a nonlinear, second-order, terminal value PDE problem. The equation arises in optimal control theory as an optimality condition.

Consider a dynamic pricing problem: over a given period, what is the best strategy to maximise revenues and minimise the cost of unsold items?

This is formulated as a stochastic control problem in continuous time, where we try to find a function that controls a stochastic differential equation based on the current state of the system.

The optimal control function can be found by solving the corresponding HJB equation.

I will present the solution of the HJB equation using a toy problem, for a risk-neutral and a risk-averse decision maker.

Cells adapt their metabolic state in response to changes in the environment. I will present a systematic framework for the construction of flux graphs to represent organism-wide metabolic networks. These graphs encode the directionality of metabolic fluxes via links that represent the flow of metabolites from source to target reactions. The weights of the links have a precise interpretation in terms of probabilities or metabolite flow per unit time. The methodology can be applied both in the absence of a specific biological context, or tailored to different environmental conditions by incorporating flux distributions computed from constraint-based modelling (e.g., Flux-Balance Analysis). I will illustrate the approach on the central carbon metabolism of Escherichia coli, revealing drastic changes in the topological and community structure of the metabolic graphs, which capture the re-routing of metabolic fluxes under each growth condition.

By integrating Flux Balance Analysis and tools from network science, our framework allows for the interrogation of environment-specific metabolic responses beyond fixed, standard pathway descriptions.