#### Recent preprints & publications

##

Context-dependent representation in recurrent neural networks

G. Wainrib,

**Preprint pdf**

In order to assess the short-term memory performance of non-linear random neural networks, we introduce a measure to quantify the dependence of a neural representation upon the past context. We study this measure both numerically and theoretically using the mean-field theory for random neural networks, showing the existence of an optimal level of synaptic weights heterogeneity. We further investigate the influence of the network topology, in particular the symmetry of reciprocal synaptic connections, on this measure of context dependence, revealing the importance of considering the interplay between non-linearities and connectivity structure.

Ulcerative
Colitis and smoking: an integrative network-based analysis of detoxification gene
expression data

Y-P. Ding et al., submitted.

Inflammatory bowel diseases (IBD) including Crohn’s disease (CD) and ulcerative colitis (UC) are severe chronic intestinal disorders common in developed countries. Besides the well-characterized genetic contribution to IBD predisposition, environmental factors affect the incidence and medical history of IBD, among which active smoking has been shown as the most robust risk factor. The effect of smoking seems to be ambivalent since active smoking improves UC while it worsens CD. Although this clinical relationship between IBD and tobacco is well established, only a few experimental works have investigated the effect of smoking on the colonic barrier homeostasis focusing on xenobiotic detoxification genes. We performed a comprehensive and integrated comparative analysis of the global xenobiotic detoxification capacity of the normal colonic mucosa of healthy smokers and non-smokers versus the non-affected colonic mucosa of UC patients to improve our understanding of the colon susceptibility to environmental aggression. Among the 244 detoxification genes investigated, 65 were significantly dysregulated in UC patients, which corresponds to a specific disease signature. We then developed a network-based data analysis approach for differentially assessing changes in genetic interactions allowing identifying unexpected regulatory detoxification genes which could play a major role in the pathogenesis of UC or in the beneficial effect of smoking on the colonic mucosa of UC patients. These observations could help clinicians to better understand the protective effect of cigarette smoking in UC and will be useful to develop new therapeutic avenues and automated diagnostic strategies.

## Branching random walks on binary strings and application to adaptive immunity

I. Balelli, V. Milisic and G. Wainrib, submitted [**arxiv**]

During the germinal center reaction, B lymphocytes proliferate, mutate and differentiate, while being submitted to a powerful selection, creating a micro-evolutionary mechanism at the heart of adaptive immunity. We introduce and analyze a simplified mathematical model of the division-mutation process, by considering random walks and branching random walks on graphs, whose structure reflects the associated mutation rules. In particular, we investigate how the combination of various division and mutation models influences the typical time-scales characterizing the efficiency of state space exploration for these processes, such as hitting times and cover times. Beyond the initial biological motivation, this framework is not limited to the modelling of B-cell learning process in germinal centers, as it may be relevant to model other evolutionary systems, but also information propagation in networks, gossip models or epidemic processes.

The real Ginibre ensemble with k=0(n) real eigenvalues

L.C. Garcia del Molino, K. Pakdaman, J.Touboul and G. Wainrib, submitted [**arxiv**]

We consider the ensemble of Real Ginibre matrices with a positive fraction α > 0 of real eigenvalues. We demonstrate a large deviations principle for the joint eigenvalue density of such matrices and we introduce a two phase log-gas whose stationary distribution coincides with the spectral measure of the ensemble. Using these tools we provide an asymptotic expansion for the probability pnαn that an n × n Ginibre matrix has k = αn real eigenvalues and we characterize the spectral measures of these matrices.

## Mathematical modeling of lymphocytes selection in the germinal center

V.Milisic and G.Wainrib, submitted [**arxiv**]

Lymphocyte selection is a fundamental operation of adaptive immunity. In order to produce B-lymphocytes with a desired antigenic profile, a process of mutation-selection occurs in the germinal center, which is part of the lymph nodes. We introduce in this article a simplified mathematical model of this process, taking into account the main mechanisms of division, mutation and selection. This model is written as a non-linear, non-local, inhomogeneous second order partial differential equation, for which we develop a mathematical analysis in the case of piecewise-constant coefficients. We assess, mathematically and numerically, the performance of the biological function by evaluating the duration of this production process as a function of several parameters such as the mutation rate or the selection profile, in various asymptotic regimes.
## Network modeling of Crohn's disease

Jean-Marc Victor, Gaëlle Debret,
Annick Lesne, Leigh Pascoe, Pascal Carrivain,
Gilles Wainrib, Jean-Pierre Hugot, submitted.

Crohn's Disease (CD) is a complex genetic disorder related to genetic and environmental risk factors. We modelled the disease as a modular network of patho-physiological functions, each summarizing gene-environment interactions. The disease resulted from one or few specific combinations of the modules' functional states. Network aging dynamics was able to reproduce age-specific CD incidence curves and their variations over the past century in Western countries. The model allowed translating the Odds Ratios associated to at-risk alleles in terms of disease propensities of the functional modules. Finally, this modelling was successfully applied to other complex genetic disorders including ulcerative colitis, ankylosing spondylarthritis, multiple sclerosis and schizophrenia.