Cristóbal Quiñinao

• Professor
• Assistant professor at Pontifical Catholic University of Chile

In the modeling of a variety of models of large-scale systems of interacting agents, mathematical models consider that the coupling terms representing the interactions between agents are scaled according to the network size to prevent divergence of the coupling terms. In a previous paper of ours and follow-ups from colleagues, we have explored the...

We study the large-time behavior of an ensemble of entities obeying replicator-like stochastic dynamics with mean-field interactions as a model for a primordial ecology. We prove the propagation-of-chaos property and establish conditions for the strong persistence of the N-replicator system and the existence of invariant distributions for a class o...

We study the large-time behavior of an ensemble of entities obeying replicator-like stochastic dynamics with mean-field interactions as a model for a primordial ecology. We prove the propagation-of-chaos property and establish conditions for the strong persistence of the N-replicator system and the existence of invariant distributions for a class o...

In this paper, we consider a (control) optimization problem, which involves a stochastic dynamic. The model proposes selecting the best control function that keeps bounded a stochastic process over an interval of time with a high probability level. Here, the stochastic process is governed by a stochastic differential equation affected by a stochast...

We study the role of small-scale perturbations in the onset of avalanches in a rotating drum in the stick-slip regime. By vibrating the system along the axis of rotation with an amplitude orders of magnitude smaller than the particles’ diameter, we found that the order parameter that properly describes the system is the kinetic energy. We also show...

We propose a stochastic model for interacting species in a metacommunity in order to study the factors affecting the intensity of the competition/colonization trade-off as a coexistence mechanism in metacommunities. We particularly focus on the role of the number of local communities and the number of refuges for the inferior competitor. The stocha...

For large fully connected neuron networks, we study the dynamics of homogenous assemblies of interacting neurons described by time elapsed models. Under general assumptions on the firing rate which include the ones made in previous works [7, 8, 6], we establish accurate estimate on the long time behavior of the solutions in the weak and the strong...

We investigate the dynamics of a limit of interacting FitzHugh-Nagumo neurons in the regime of large interaction coefficients. We consider the dynamics described by a mean-field model given by a nonlinear evolution partial differential equation representing the probability distribution of one given neuron in a large network. The case of weak connec...

For large fully connected neuron networks, we study the dynamics of homogenous assemblies of interacting neurons described by time elapsed models. Under general assumptions on the firing rate which include the ones made in previous works [7, 8, 6], we establish accurate estimate on the long time behavior of the solutions in the weak and the strong...

Biodiversity is sustained by and is essential to the services that ecosystems provide. Different species would use these services in different ways, or adaptive strategies, which are sustained in time by continuous innovations. Using this framework, we postulate a model for a biological species (Homo sapiens) in a finite world where innovations, ai...

We introduce a microscopic spiking network consistent with the age-structured/renewal equation proposed by Pakdaman, Perthame and Salort. It is a jump process interacting through a global activity variable with random delays. We show the well-posedness of the particle system and the mean-field equation. Moreover we show the propagation of chaos pro...

We investigate existence and uniqueness of solutions of a McKean-Vlasov evolution PDE representing the macroscopic behaviour of interacting Fitzhugh-Nagumo neurons. This equation is hypoelliptic, nonlocal and has unbounded coefficients. We prove existence of a solution to the evolution equation and non trivial stationary solutions. Moreover, we dem...

One of the main challenges in cancer modelling is to improve the knowledge of tumor progression in areas related to tumor growth, tumor-induced angiogenesis and targeted therapies efficacy. For this purpose, incorporate the expertise from applied mathematicians, biologists and physicians is highly desirable. Despite the existence of a very wide ran...

This work is devoted to the study of mathematical questions arising from the modeling of biological systems combining analytic and probabilistic tools. In the first part, we are interested in the derivation of the mean-field equations related to some neuronal networks, and in the study of the convergence to the equilibria of the solutions to the li...

Boundary formation in the developing neuroepithelium decides on the position and size of compartments in the adult nervous system. In this study, we start from the French Flag model proposed by Lewis Wolpert, in which boundaries are formed through the combination of morphogen diffusion and of thresholds in cell responses. In contemporary terms, a r...

We analyze an inhomogeneous system of coupled reaction-diffusion equations representing the dynamics of gene expression during differentiation of nerve cells. The outcome of this developmental phase is the formation of distinct functional areas separated by sharp and smooth boundaries. It proceeds through the competition between the expression of t...

Networks of the brain are composed of a very large number of neurons connected through a random graph and interacting after random delays that both depend on the anatomical distance between cells. In order to comprehend the role of these random architectures on the dynamics of such networks, we analyze the mesoscopic and macroscopic limits of netwo...

This paper deals with a sub-critical Keller-Segel equation. Starting from the stochastic particle system associated with it, we show well-posedness results and the propagation of chaos property. More precisely, we show that the empirical measure of the system tends towards the unique solution of the limit equation as the number of particles goes to...

Modelamiento matemático de crecimiento de tumores, acoplado con angiogénesis tumoral y su uso en la investigación de nuevas estrategias de tratamiento del cáncer Resumen Considerando la importancia del cáncer a nivel mundial, es necesario planear estrategias para estudiar y controlar esta enfermedad. Sin embargo, es difícil tener un pronóstico y tr...