Integrate and Fire Neural Networks, Piecewise Contractive Maps and Limit Cycles

We study the global dynamics of integrate and fire neural networks composed of an arbitrary number of identical neurons interacting by inhibition and excitation. We prove that if the interactions are strong enough, then the support of the stable asymptotic dynamics consists of limit cycles. We also find sufficient conditions for the synchronization of networks containing excitatory neurons. The proofs are based on the analysis of the equivalent dynamics of a piecewise continuous Poincar\'e map associated to the system. We show that for strong interactions the Poincar\'e map is piecewise contractive. Using this contraction property, we prove that there exist a countable number of limit cycles attracting all the orbits dropping into the stable subset of the phase space. This result applies not only to the Poincar\'e map under study, but also to a wide class of general n-dimensional piecewise contractive maps.Comment: 46 pages. In this version we added many comments suggested by the referees all along the paper, we changed the introduction and the section containing the conclusions. The final version will appear in Journal of Mathematical Biology of SPRINGER and will be available at http://www.springerlink.com/content/0303-681

Inter-Cohort Validation of SuStaIn Model for Alzheimer's Disease

Alzheimer's disease (AD) is a neurodegenerative disorder which spans several years from preclinical manifestations to dementia. In recent years, interest in the application of machine learning (ML) algorithms to personalized medicine has grown considerably, and a major challenge that such models face is the transferability from the research settings to clinical practice. The objective of this work was to demonstrate the transferability of the Subtype and Stage Inference (SuStaIn) model from well-characterized research data set, employed as training set, to independent less-structured and heterogeneous test sets representative of the clinical setting. The training set was composed of MRI data of 1043 subjects from the Alzheimer's disease Neuroimaging Initiative (ADNI), and the test set was composed of data from 767 subjects from OASIS, Pharma-Cog, and ViTA clinical datasets. Both sets included subjects covering the entire spectrum of AD, and for both sets volumes of relevant brain regions were derived from T1-3D MRI scans processed with Freesurfer v5.3 cross-sectional stream. In order to assess the predictive value of the model, subpopulations of subjects with stable mild cognitive impairment (MCI) and MCIs that progressed to AD dementia (pMCI) were identified in both sets. SuStaIn identified three disease subtypes, of which the most prevalent corresponded to the typical atrophy pattern of AD. The other SuStaIn subtypes exhibited similarities with the previously defined hippocampal sparing and limbic predominant atrophy patterns of AD. Subject subtyping proved to be consistent in time for all cohorts and the staging provided by the model was correlated with cognitive performance. Classification of subjects on the basis of a combination of SuStaIn subtype and stage, mini mental state examination and amyloid-β1-42 cerebrospinal fluid concentration was proven to predict conversion from MCI to AD dementia on par with other novel statistical algorithms, with ROC curves that were not statistically different for the training and test sets and with area under curve respectively equal to 0.77 and 0.76. This study proves the transferability of a SuStaIn model for AD from research data to less-structured clinical cohorts, and indicates transferability to the clinical setting

Primate modularity and evolution: first anatomical network analysis of primate head and neck musculoskeletal system

Network theory is increasingly being used to study morphological modularity and integration. Anatomical network analysis (AnNA) is a framework for quantitatively characterizing the topological organization of anatomical structures and providing an operational way to compare structural integration and modularity. Here we apply AnNA for the first time to study the macroevolution of the musculoskeletal system of the head and neck in primates and their closest living relatives, paying special attention to the evolution of structures associated with facial and vocal communication. We show that well-defined left and right facial modules are plesiomorphic for primates, while anthropoids consistently have asymmetrical facial modules that include structures of both sides, a change likely related to the ability to display more complex, asymmetrical facial expressions. However, no clear trends in network organization were found regarding the evolution of structures related to speech. Remarkably, the increase in the number of head and neck muscles – and thus of musculoskeletal structures – in human evolution led to a decrease in network density and complexity in humans

Limitations of perturbative techniques in the analysis of rhythms and oscillations

Perturbation theory is an important tool in the analysis of oscillators and their response to external stimuli. It is predicated on the assumption that the perturbations in question are “sufficiently weak”, an assumption that is not always valid when perturbative methods are applied. In this paper, we identify a number of concrete dynamical scenarios in which a standard perturbative technique, based on the infinitesimal phase response curve (PRC), is shown to give different predictions than the full model. Shear-induced chaos, i.e., chaotic behavior that results from the amplification of small perturbations by underlying shear, is missed entirely by the PRC. We show also that the presence of “sticky” phase–space structures tend to cause perturbative techniques to overestimate the frequencies and regularity of the oscillations. The phenomena we describe can all be observed in a simple 2D neuron model, which we choose for illustration as the PRC is widely used in mathematical neuroscience