CRM- Maria de Maeztu Interdisciplinary Workshop: Mathematical Biology and Neuroscience

About global dynamics, complexification, and game theory in Biological systems (Josep Sardanyés)

Biological dynamics at all scales involve nonlinear interactions giving place to complex dynamics such as oscillations, transient chaos, or chaos. The investigation of global dynamics for such systems, including virus dynamics, cancer, or ecology, becomes crucial to understand global manifolds governing dynamics and involved in bifurcations. In this talk, I will focus on three topics: (i) some problems in Biology demanding for global dynamical analysis; (ii) a complexification approach to study chaotic intermittency; and (iii) game theory applied to viruses co-infecting with RNA satellites.

Integral formulation of structured population dynamics (Àngel Calsina)

The usual tool for formulating dynamics models for populations structured by physiological variables (age, size, phenotype,…) are hyperbolic PDEs with non-local terms. An alternative is to formulate the models using the integral renewal equations obtained from the integration along the characteristic lines. An advantage of the second option is that the resulting solution semigroup is differentiable and, as a result, rigorous linearization results (stability and instability) are obtained which in turn are only formal in the case of the corresponding quasilinear PDEs. We will see an example of hierarchical competition in which the analysis allows a fairly complete description of the dynamics and one of cell population growth in which the distribution is with respect to age and size.

Characterizing the space of gene networks capable of pattern formation (Kevin Martínez)

Traditionally, only two main classes of gene network topologies with cell signalling have been reported in the literature. On the one hand, Turing networks, based on the long-range activatory and short-range inhibitory signaling, produce extensive periodical spatial patterns that arise from a homogeneous steady state with little influence of the actual initial conditions. On the other hand, hierarchic networks present a strict hierarchy where no feedback from one element in the network to a previous element is allowed. These networks produce simple patterns where spatial information is concentrated at a small portion of the actual domain of the problem and whose final shape is strongly dependent on the initial conditions provided to the system. Here, we present a third class of gene networks, that we have called over-Turing gene networks, which constitute, to our knowledge, a whole new mechanism for pattern formation and, furthermore, we suggest that these three different architectures exhaust all the effective possibilities for pattern formation to occur in Developmental Biology.

Mathematical framework on phylogenetics: time-homogeneity and estimation of parameters (Jesús Fernández)

In this talk we will review the theoretical / mathematical framework for developing tools in phylogenetics (the study of the evolutionary relationships among species). We will put special emphasis on the problems related to the design of nucleotide substitution models (in particular, the problem with the assumption on time-homogeneity) and how to cope with them from an algebraic perspective. A second focus of interest is the estimation of the parameters of these models by using likelihood functions.

A mathematical walk through some neuroscience problems (Antoni Guillamon)

The purpose of this talk is to show neuroscience problems that intersect with research in several areas of mathematics, from dynamical systems theory to topology together with control theory, stochastic processes, among others.

Network mechanisms underlying representational drift in area CA1 of hippocampus (Alex Roxin)

Chronic imaging experiments in mice have revealed that the hippocampal code drifts over long time scales. Specifically, the subset of cells which are active on any given session in a familiar environment changes over the course of days and weeks. While some cells transition into or out of the code after a few sessions, others are stable over the entire experiment. Similar representational drift has also been observed in other cortical areas, raising the possibility of a common underlying mechanism, which, however, remains unknown. Here we show, through quantitative fitting of a network model to experimental data, that the statistics of representational drift in CA1 pyramidal cells are consistent with ongoing synaptic turnover in the main excitatory inputs to a neuronal circuit operating in the balanced regime. We find two distinct time-scales of drift: a fast shift in overall excitability operating over hours, and a slower drift in spatially modulated input which would lead to complete remapping of place fields after about a month. The observed heterogeneity in single-cell properties, including long-term stability, are explained by variability arising from random changes in the number of active inputs to cells from one session to the next. We furthermore show that these changes are, in turn, consistent with an ongoing process of learning via a Hebbian plasticity rule. We conclude that representational drift is the hallmark of a memory system which continually encodes new information.

Mathematical frameworks for neural oscillatory dynamics (Gemma Huguet)

In this talk I will focus on the study of neuronal oscillations, both regular and irregular. I will introduce some neuronal models and I will show how tools from dynamical systems theory such as the parameterization method for invariant manifolds or the separatrix map can be used to provide an analysis of the dynamics. We will show how the conclusions obtained may have implications in the study of the functional role of these oscillations.

Neural implementation of approximate inference in the human brain (Alexandre Hyafil)

How the human brain extracts information from raw sensory input to compose a representation of its immediate environment is still mysterious. A popular hypothesis in neuroscience (sometimes called the Bayesian brain) is that the brain encodes perceptual features in a probabilistic format, and represents known regularities in the world as a probabilistic graphical model. The jury is still very much out as to how neural activity represents probability distributions and propagates information to perform approximate inference. One hypothesis (called neural sampling) is that each neuron spike represents a sample from an underlying distribution, implementing some sort of sampling algorithm (e.g. Markov Chain Monte Carlo). Alternatively, neural firing rates could represent directly distribution parameters while the overall circuit dynamics would implement some form of variational inference (e.g. mean-field or Loopy Belief Algorithm). I will briefly review the state of the field and draw connections with statistical physics and machine learning – where the question of approximate inference in complex graphs is still an open question, despite great advances following the deep learning revolution. I will also present a short overview of a research programme I will deploy to try to test empirically which class of algorithms the human brain follows. Such a programme would greatly benefit from collaborations with specialists in approximate inference in machine learning.