Participation is only available for CRM membres and affiliated researchers
CRM Maria de Maeztu Interdisciplinary Workshop: Mathematical Biology and Neuroscience
Sign into June 09, 2022
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INFORMATION
Morning sessions: CRM A1 Room
Afternoon sessions: CRM Polivalent Rooms 1 & 2
OBJECTIVES
One of the goals of the Maria de Maeztu programme at the CRM is to promote multidisciplinary and applied mathematical research to relevant problems in different areas such as Biology or Neuroscience. In this sense, the use or combination of different mathematical approaches can provide novel knowledge to problems related to health, ecology, phylogenetics, development, brain dynamics, cognition, or memory, among others. These interdisciplinary studies allow us to develop solid theories and to deepen our understanding of such complex systems which are highly nonlinear and tipically display novel emergent, collective phenomena.
The first CRMMaria de Maeztu Interdisciplinary Workshop seeks to boost collaborations between the researchers at CRM working in the areas of Mathematical Biology and Neuroscience with the rest of the researchers at the center working in other mathematical disciplines such as Dynamical Systems, Analysis and PDEs, Algebra, Geometry, Number theory, Topology, Combinatorics, Logics, and Algorithmics. This workshop will be organised in morning lectures, followed by thematic working sessions in the afternoon.
REGISTRATION
Registration is free but mandatory. If you wish to participate in the Workshop, please note that it will be necessary to register by clicking on SIGN IN (at the top of the page) and to fill in the following questionnaire for participants.
This workshop is only offered to CRM members and affiliated researchers and includes coffee breaks.
SCHEDULE
8 June 2022: Mathematical Biology Session  
9:50 10:00  Workshop Welcome 
10:00 10:45  Josep Sardanyés (CRM) 
10:45 11:30  Àngel Calsina (UABCRM) 
11:30 12:00  COFFEE BREAK 
12:00 12:45  Kevin Martínez (CRM) 
12:45 13:30  Jesús Fernández (UPCCRM) 
13:30 15:00  LUNCH 
15:00 18:00  Working Groups Activities 
9 June 2022: Neuroscience Session  
10:00 10:45  Antoni Guillamon (UPCCRM) 
10:45 11:30  Alex Roxin (CRM) 
11:30 12:00  COFFEE BREAK 
12:00 12:45  Gemma Huguet (UPCCRM) 
12:45 13:30  Alexandre Hyafil (CRM) 
13:30 15:00  LUNCH 
15:00 18:00  Working Groups Activities 
SPEAKERS
Àngel Calsina  Universitat Autònoma de Barcelona / Centre de Recerca Matemàtica  Abstract
Jesús Fernández  Universitat Politècnica de Catalunya / Centre de Recerca Matemàtica  Abstract
Antoni Guillamon  Universitat Politècnica de Catalunya / Centre de Recerca Matemàtica  Abstract
Gemma Huguet  Universitat Politècnica de Catalunya / Centre de Recerca Matemàtica  Abstract
Alexandre Hyafil  Centre de Recerca Matemàtica  Abstract
Kevin Martínez  Centre de Recerca Matemàtica  Abstract
Alex Roxin  Centre de Recerca Matemàtica  Abstract
Josep Sardanyés  Centre de Recerca Matemàtica  Abstract
WORK GROUPS
Mathematical Biology Work Groups (June, 8th)
WGBioMat1. Population dynamics, complexification, and game theory in Ecology and Virology
Organizers: Àngel Calsina, Sílvia Cuadrado, Josep Sardanyés, Blai Vidiella, Tomás Lázaro, Ernest Fontich
Topic: Population dynamics: qualitative theory for PDEs, global manifolds and bifurcations, complexification, and Game Theory applied to Ecology and Virology
Description: Differential equations (ODEs or PDEs) are used to investigate population dynamics with respect to their state/s (spatial position, age, abundances, phenotype, etc.). However, in the infinitedimensional (PDE) case there are no theorems allowing a qualitative theory based on linearization. The delay formulation, where the state variables are birth rates and, in the nonlinear case, interacting variables, based on Volterra integral equations, allows a semilinear formulation even when the corresponding PDEs are only quasilinear and consequently a consistent and rigorous qualitative theory. One of the goals of the WGBiomat1 will consist in studying the limits of the delay formulation through several study cases and the possible version for discretely structured populations. In the WGBiomat1 we will also introduce a system of ODEs models for preypredator dynamics with habitat loss. Our interest is to extend a published work by adding functional responses (saturation in predation) to the dynamics and provide local, and, especially, global information into the dynamics. These include how a described heteroclinic bifurcation changes with such functional responses. We will also address how chaotic intermittency can be studied using complexification techniques and holomorphic dynamics. Finally, we will address novel approaches to investigate virus dynamics using game theory.
WGBioMat2. Phylogenetics and likelihood functions
Organizers: Marta Casanellas, Jesús Fernández
Topic: Computing the exact number of real and positive maxima for likelihood functions
Description: For small phylogenetic trees evolving on simple evolutionary models, there are conjectures stating that there is a unique real and positive local maximum of the likelihood function for generic data. In order to address these conjectures, tools from computational algebra and real algebraic geometry are needed.
WGBioMat3. Phylogenetics and transition matrices
Organizers: Marta Casanellas, Jesús Fernández
Topic: Evaluating the embeddability of transition matrices on real data
Description: The assumption that molecular substitution processes are time homogeneous is very restrictive mathematically speaking, as less than 1% of transition matrices are embeddable. However, this is usually assumed in phylogenetics. We want to analyze real data and test whether the estimated transition matrices can be assumed to be embeddable or not. This requires a good knowledge of statistics methods that can address this type of problem.
WGBioMat4. Wavelet theory applied to reactiondiffusion Systems
Organizers: Isaac Salazar Ciudad, Kevin Martínez
Topic: Harmonic analysis and wavelet theory applied to pattern formation
Description: Our results suggest that the classical Fourier approach to pattern formation in reactiondiffusion systems is not enough to account for some local features of the highly irregular patterns produced by some nonTuring networks and we wonder whether a different approach using different tools from Harmonic analysis or wavelet theory can be a fruitful attempt.
Neuroscience Work Groups (June, 9th)
WGNeuro1. Neuroscience and machine learning
Organizers: Àlex Roxin, Alex Hyafil, Klaus Wimmer
Topic: The relationship between biologically plausible plasticity rules and machinelearning rules
Description: In neuroscience we have models of several types of rules governing how synaptic weights change in response to pre and postsynaptic activity, inferred from experimental work. These rules are local and unsupervised, while it seems most rules used to train neural networks to perform some task are highly supervised. Is there some role for biologically realistic plasticity rules in machine learning? Another hot topic in neuroscience is whether perception relies on probabilistic codes and if so: What type of approximate inference algorithm (e.g. meanfield, expectationpropagation, MCMC, etc.) does the brain deploy to compute probabilities over the presence of feature in the environment. We would need a partner with expertise in machine learning for these to make sense.
WGNeuro2. Oscillations and topological data analysis
Organizers: Antoni Guillamon and Gemma Huguet
Topic: Mathematical Neuroscience, what are the challenges for mathematics?
Description: Many mathematical and quantitative tools have been applied successfully to Neuroscience in order to provide answers to the comprehension of the mechanisms underlying brain function. After a general overview in the morning session, in this WG we will discuss specific challenging problems that the field of Neuroscience is currently posing to mathematicians. In particular, we will focus on problems related with the study of brain oscillations with dynamical systems tools and classification of activity regimes in neuronal populations using topological data analysis.
LIST OF PARTICIPANTS
Name  Institution 

For inquiries about this event please contact the research programs coordinator Ms. Núria Hernández at nhernandez@crm.cat

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 coinfecting 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 nonlocal 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 longrange activatory and shortrange 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 overTuring 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: timehomogeneity 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 timehomogeneity) 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 timescales 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 singlecell properties, including longterm 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. meanfield 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.