Protention and retention in biological systems

This paper proposes an abstract mathematical frame for describing some features of cognitive and biological time. We focus here on the so called "extended present" as a result of protentional and retentional activities (memory and anticipation). Memory, as retention, is treated in some physical theories (relaxation phenomena, which will inspire our approach), while protention (or anticipation) seems outside the scope of physics. We then suggest a simple functional representation of biological protention. This allows us to introduce the abstract notion of "biological inertia".Comment: This paper was made possible only as part of an extended collaboration with Francis Bailly (see references), a dear friend and "ma\^itre \'a penser", who contributed to the key ideas. Francis passed away in february 2008: we continue here our inspiring discussions and joint wor

Exact Diagonalization of SU(N)\mathrm{SU}(N) Fermi-Hubbard Models

We show how to perform exact diagonalizations of $\mathrm{SU}(N)$ Fermi-Hubbard models on $L$-site clusters separately in each irreducible representation ({irrep}) of $\mathrm{SU}(N)$. Using the representation theory of the unitary group $\mathrm{U}(L)$, we demonstrate that a convenient orthonormal basis, on which matrix elements of the Hamiltonian are very simple, is given by the set of {\it semistandard Young tableaux} (or, equivalently the Gelfand-Tsetlin patterns) corresponding to the targeted irrep. As an application of this color factorization, we study the robustness of some $\mathrm{SU}(N)$ phases predicted in the Heisenberg limit upon decreasing the on-site interaction $U$ on various lattices of size $L \leq 12$ and for $2 \leq N \leq 6$. In particular, we show that a long-range color ordered phase emerges for intermediate $U$ for $N=4$ at filling $1/4$ on the triangular lattice.Comment: 11 pages, 10 figure

A 2-dimensional Geometry for Biological Time

This paper proposes an abstract mathematical frame for describing some features of biological time. The key point is that usual physical (linear) representation of time is insufficient, in our view, for the understanding key phenomena of life, such as rhythms, both physical (circadian, seasonal ...) and properly biological (heart beating, respiration, metabolic ...). In particular, the role of biological rhythms do not seem to have any counterpart in mathematical formalization of physical clocks, which are based on frequencies along the usual (possibly thermodynamical, thus oriented) time. We then suggest a functional representation of biological time by a 2-dimensional manifold as a mathematical frame for accommodating autonomous biological rhythms. The "visual" representation of rhythms so obtained, in particular heart beatings, will provide, by a few examples, hints towards possible applications of our approach to the understanding of interspecific differences or intraspecific pathologies. The 3-dimensional embedding space, needed for purely mathematical reasons, allows to introduce a suitable extra-dimension for "representation time", with a cognitive significance.Comment: Presented in an invited Lecture, conference "Biologie e selezioni naturali", Florence, December 4-8, 200

Neural Substrates of Semantic Prospection – Evidence from the Dementias

The ability to envisage personally relevant events at a future time point represents an incredibly sophisticated cognitive endeavor and one that appears to be intimately linked to episodic memory integrity. Far less is known regarding the neurocognitive mechanisms underpinning the capacity to envisage non-personal future occurrences, known as semantic future thinking. Moreover the degree of overlap between the neural substrates supporting episodic and semantic forms of prospection remains unclear. To this end, we sought to investigate the capacity for episodic and semantic future thinking in Alzheimer’s disease (n = 15) and disease-matched behavioral-variant frontotemporal dementia (n = 15), neurodegenerative disorders characterized by significant medial temporal lobe (MTL) and frontal pathology. Participants completed an assessment of past and future thinking across personal (episodic) and non-personal (semantic) domains, as part of a larger neuropsychological battery investigating episodic and semantic processing, and their performance was contrasted with 20 age- and education-matched healthy older Controls. Participants underwent whole-brain T1-weighted structural imaging and voxel-based morphometry analysis was conducted to determine the relationship between gray matter integrity and episodic and semantic future thinking. Relative to Controls, both patient groups displayed marked future thinking impairments, extending across episodic and semantic domains. Analyses of covariance revealed that while episodic future thinking deficits could be explained solely in terms of episodic memory proficiency, semantic prospection deficits reflected the interplay between episodic and semantic processing. Distinct neural correlates emerged for each form of future simulation with differential involvement of prefrontal, lateral temporal, and medial temporal regions. Notably, the hippocampus was implicated irrespective of future thinking domain, with the suggestion of lateralization effects depending on the type of information being simulated. Whereas episodic future thinking related to right hippocampal integrity, semantic future thinking was found to relate to left hippocampal integrity. Our findings support previous observations of significant MTL involvement for semantic forms of prospection and point to distinct neurocognitive mechanisms which must be functional to support future-oriented forms of thought across personal and non-personal contexts