# mathematics of behavior and intelligence

## projects

### collective behavior

We study how animals, including humans, interact in groups. We have found that group decision-making experiments can be quantitatively explained using the hypothesis that each agent uses the behaviors of the others to estimate where to go in the world [PLOS Comp 2011, PNAS 2012]. Each agent interacts with very few neighbours [PNAS 2017], especially those close and moving fast [PLOS Comp 2019, Phil Trans Roy Soc B].

For mathematical modelling, we have used Bayesian and other frameworks for hypothesis-driven approaches [PLOS Comp 2011, PNAS 2012, PNAS 2017, Proceedings B 2017, Ther Ecol 2018, Interface 2023]. For data-driven approaches, we have used modular neural networks, with each module being a low-dimensional function that gives insights into the interactions [PLOS Comp 2017, Roy Soc 2023]. Modular networks give a nice balance of insight and predictability. We have also used the formalism of Reinforcement Learning to explain how local interactions give rise to global patterns of group behaviour like balls, mills and tornadoes [Fundamentals and Applications of AI, Frontiers Physics].

We have built the open-source systems IdTracker (MatLab) and idtracker.ai (Python; Torch) as the first multi-animal tracking system based on identification. These tracking systems are agnostic about species and idtracker. ai can track up to 100 animals. You can find all the data we used here. idtracker.ai is continuously updated to make it faster, more general and friendlier. See notebooks (Keras_version) and notebooks (Torch_version) for some pedagogical notebooks on Machine Learning including a notebook on how animals are identified (notebook 5). We have also built idmatcher.ai and ReactNet as tools to match identities in different videos and to find when animals are reacting to a stimulus, respectively.

In humans, we have studied how to make a good group decision even if the majority is wrong [PLOS Comp 2015, Front Robot AI 2017].

We are currently collaborating with Mike Orger to study zebrafish neuronal circuits in the presence of other animals.

### algebraic machine learning

We propose a new kind of learning method based only on an algebraic representation that guarantees generalization, without the need for the usual notions in Machine Learning of optimization, objectives or search. We gave a first description of the system here. Since that first description, we have studied its mathematical properties, how to make the computations more efficient and extended it to the continuous case.

We have shown how to express a task into an algebra here. Then we have described here how we use Birkhoff's result that an algebra can be decomposed into a set of indecomposable elements, which are going to be the algebraic atoms of our data and formal knowledge, and the set of atoms is our model. We have extended here these results to the infinite case to study continuous problems.

### modelling and AI in biology

We have studied a variety of biological problems using both hypothesis-driven theories as well as using AI approaches from which to distil biological knowledge. We have studied neuronal coding problems [J Theor Biol 2002, J Neurophys 2003, J Neurophys 2003b, Phys Rev E 2004, J Neuroscience 2005, J Neurophys 2006, Neural Comp 2006, Nat Neuro 2007, eLIFE 2017]. We have also studied wiring economy in the brain [PNAS 2007, Current Biology 2011, Current Biology 2014] and the structure of deviations from optimality in several biological systems [PNAS 2009].

Other problems we have modelled are body ownership in humans [Conc & Cog 2019], bursty behaviour [PLOS Comp 2011], aggression [Roc Soc Open 2018], and species diversity [Proc Roy Soc B].

We are currently working on applying machine learning to multiomics data [Bioinformatics 2023], genetics/epigenetics, and dolphin communication with German Sumbre.