January 22, 2021

# Mathematics to understand the brain | Science

The functioning of the brain remains one of the great mysteries facing science. The questions are almost infinite, and the answers are barely incipient. We know that all brain processes (which allow vision, speech, movement) are based on the transmission of nerve impulses between different types of neurons, but how do they coordinate to build these complex responses? Explaining neuronal synchronization is one of the great challenges of computational neuroscience, in which mathematics is key.

In the decade of 1950 the first models were proposed, that described of realistic form the operation of an isolated neuron. They tried to reproduce, with differential systems, what would be observed by looking at a working neuron with the microscope. The most popular model, by Alan Lloyd Hodgkin and Andrew Huxley, showed quite accurately how the action potential of the neuron starts and transmits over time, from the amounts of sodium, potassium, etc. in the ion channels. In current neuroscience this model, which was recognized with the Nobel Prize, is considered a good description of the functioning of different types of neurons.

However, in order to understand the neural processes, the study of a single neuron is not interesting, but the collective behavior of large sets of them. For example, how do the 10 ^ 7 cells of the cortex (some capable of distinguishing colors, other positions) that are used for vision or movement act together to recognize patterns and be able to interpret the image that is Are you watching?

The models of neural networks show an average of their activity, using as a rate the number of electric discharges per unit of time of an entire network (called firing rate), or of certain regions of it. Mathematically, the difficulty lies in the change of scale, that is, in establishing the firing rate of all the neurons in the network in a rigorous way, including the description of the microscopic models.

For this, the call is used kinetic theory, that allows to show with equations the behavior and macroscopic properties from a statistical description of the microscopic molecular processes. These ideas serve as a bridge between the micro and macro models, and are currently under development. José Antonio Carrillo de la Plata, researcher of the Imperial College London, has been working on these issues for decades. Together with Benoit Perthame (Paris VI) and María Cáceres (Granada University), they analyzed various models of macroscopic phenomena. Beyond developing new mathematics, they concluded that models could produce solutions that describe biological phenomena never observed by experimentalists. Therefore, they did not correspond completely with the observed reality, and it was convenient to rethink them. In this way, their work contributed to improve some of the models used by neuroscientists.

Currently, Carrillo has a great interest in those who describe the operation of the network cells (in English grid cells), neurons that allow humans and other animals to understand their position in space. The existence of these cells, which constitute the positioning system in the brain, was proposed by the English scientist John O'Keefe, and the Norwegians May-Britt Moser and Edvard I. Moser, which earned them the Nobel Prize in Medicine of 2014.

### Virtual mesh

The researchers observed that these neurons function as a virtual mesh that stores the information of the movement, so that for example, a rat can travel in the dark a path already known. In the mathematical models proposed by O'Keefe, Moser and Moser, the firing rate of all the neural networks that intervene in the process, and identify a coordination of that rate in the areas that determine the mesh. Specifically, a traveling wave is observed, so that the mesh advances with the movement of the animal.

Now Carrillo, along with other authors, is analyzing whether these proposed models can be obtained in a rigorously mathematical way. The point is, again, to verify that these macroscopic models are coherent with the information provided by classical microscopic models, using partial differential equations and their numerical simulation. With this analysis, one could determine how neurons are connected in this structure, and how the observed waves are created. Their advances could be key to improving current models of grid cells. In general, understanding the synchronization of neural networks in a mathematical way allows us to advance in the detailed study of the brain and in diseases that arise from synchronicity failures, such as epilepsy.

Agate A. Timón is responsible for Communication and Disclosure in the ICMAT.

Coffee and Theorems is a section dedicated to mathematics and the environment in which they are created, coordinated by the Institute of Mathematical Sciences (ICMAT), in which researchers and members of the center describe the latest advances in this discipline, share points of contact between mathematics and other social and cultural expressions, and remind those who marked their development and knew how to transform coffee into theorems. The name evokes the definition of the Hungarian mathematician Alfred Rényi: "A mathematician is a machine that transforms coffee into theorems".