My research interests lie at the intersections of mathematics, computer science, and their applications to life sciences and neuroscience in particular.

### Machine learning in life sciences

My main focus currently is about the development of techniques of machine learning able to handle data from life sciences, and brain imaging in particular. I’m interested in both supervised and unsupervised approaches. For the latter, my collaborators and I have recently published our first results about the use of cluster analysis on electroencephalography (EEG) data, where we group together the signal segments of an EEG dataset in order to recover the experimental conditions and the similarities between the participants. I am currently continuing this work in order to treat correctly **feature extraction** and **artefact rejection**. I am also working on different distances to compare the signal segments in ways that account for the possible time shifts and delays in the response, such as the **Dynamic Time Warping** metric.

The next steps in this line of research are the study of interactions of classical data mining techniques such as **clustering** and mathematical techniques that aim to find the internal structure of a set, such as **Topological Data Analysis**.

I am also very interested in the **big data** issues related to this project, coming from the big amounts of data that are needed to properly understand their structure.

### Modelling of neural dynamics

I work on models and simulations of neural dynamics. In particular, since the power of such dynamics lies in the mutual interactions of a huge number of interconnected units, one of the central notions here is that of **network** or **graph**. I run in-silico simulations and study how the shape (or topology) of a network is related with its dynamics.

#### Applied algebraic topology

One of the mathematical tools I use is algebraic topology. One way to do so is the construction of a **simplicial complex** associated to the graph and the use of toolbox of homology to understand it. This produces nice **invariants of the network**, such as the **Euler characteristic** and the **Betti numbers**, to name a couple of them, which can be computed algorithmically and describe and encode the shape of the network. Computing these invariants as the network evolves allows us to see how the topology is influencing the evolution and the activation of the network.

The current work in this project is about finding substructures in a network that can reveal and localise the interesting **temporal activation patterns** in its dynamics.

#### Paradigms of neural computation

In a recent project, my co-authors and I have studied finite-state automata and boolean neural networks and we have made a construction that associates to an automaton a corresponding network, similar to the classical construction by Minsky, but in a more robust way that doesn’t need a continuous feed of input, and instead has a self-sustained activity. The key notion here is that of **synfire ring**, which is a multi-layered network in a ring shape. This structure is capable of maintaining its activation once it’s firing.

I’m also interested in the correspondence between artificial boolean recurrent neural networks and finite-state automata, to give a notion of classification of neural networks in terms of the (infinite) sequences of inputs that create interesting periodical *sequences of states* in the network.

#### Decision-making and neuroeconomics

I collaborate with my colleagues at the Neuroheuristic Research Group in Lausanne in experiments in decision-making and neuroeconomics. We perform recordings with brain imaging techniques such as EEG and NIRS. During these experiments, participants wear helmets with many electrodes/optodes that record the brain activity at fixed locations on the scalp.

### Past work in pure mathematics

I spent several years (i.e. most of the time during my education) thinking about problems in pure maths: I had the chance to learn some Algebraic Topology while I was in Copenhagen (of course in *homotopy-theoretic* flavour!), where I wrote my master’s thesis on Equivariant Homotopy Theory. During my years as a PhD student at Aarhus University I was part of the Centre for Quantum Geometry of Moduli Spaces (QGM) and my focus switched towards differential geometry and quantization, under the supervision of Prof. Jørgen Andersen. In a nutshell, I studied **star products**, which are a way to formalize **quantization** (in particular, deformation quantization). More on my PhD project here!

Chi cerca trova, chi ricerca ritrova. – *Ennio De Giorgi*