View-tolerant face recognition and Hebbian learning imply mirror-symmetric neural tuning to head orientation

The primate brain contains a hierarchy of visual areas, dubbed the ventral stream, which rapidly computes object representations that are both specific for object identity and relatively robust against identity-preserving transformations like depth-rotations. Current computational models of object recognition, including recent deep learning networks, generate these properties through a hierarchy of alternating selectivity-increasing filtering and tolerance-increasing pooling operations, similar to simple-complex cells operations. While simulations of these models recapitulate the ventral stream's progression from early view-specific to late view-tolerant representations, they fail to generate the most salient property of the intermediate representation for faces found in the brain: mirror-symmetric tuning of the neural population to head orientation. Here we prove that a class of hierarchical architectures and a broad set of biologically plausible learning rules can provide approximate invariance at the top level of the network. While most of the learning rules do not yield mirror-symmetry in the mid-level representations, we characterize a specific biologically-plausible Hebb-type learning rule that is guaranteed to generate mirror-symmetric tuning to faces tuning at intermediate levels of the architecture

An OSpOSp extension of Canonical Tensor Model

Tensor models are generalizations of matrix models, and are studied as discrete models of quantum gravity for arbitrary dimensions. Among them, the canonical tensor model (CTM for short) is a rank-three tensor model formulated as a totally constrained system with a number of first-class constraints, which have a similar algebraic structure as the constraints of the ADM formalism of general relativity. In this paper, we formulate a super-extension of CTM as an attempt to incorporate fermionic degrees of freedom. The kinematical symmetry group is extended from $O(N)$ to $OSp(N,\tilde N)$, and the constraints are constructed so that they form a first-class constraint super-Poisson algebra. This is a straightforward super-extension, and the constraints and their algebraic structure are formally unchanged from the purely bosonic case, except for the additional signs associated to the order of the fermionic indices and dynamical variables. However, this extension of CTM leads to the existence of negative norm states in the quantized case, and requires some future improvements as quantum gravity with fermions. On the other hand, since this is a straightforward super-extension, various results obtained so far for the purely bosonic case are expected to have parallels also in the super-extended case, such as the exact physical wave functions and the connection to the dual statistical systems, i.e. randomly connected tensor networks.Comment: 27pages, 27 figure

Quantum Indistinguishability in Chemical Reactions

Quantum indistinguishability plays a crucial role in many low-energy physical phenomena, from quantum fluids to molecular spectroscopy. It is, however, typically ignored in most high temperature processes, particularly for ionic coordinates, implicitly assumed to be distinguishable, incoherent and thus well-approximated classically. We explore chemical reactions involving small symmetric molecules, and argue that in many situations a full quantum treatment of collective nuclear degrees of freedom is essential. Supported by several physical arguments, we conjecture a "Quantum Dynamical Selection" (QDS) rule for small symmetric molecules that precludes chemical processes that involve direct transitions from orbitally non-symmetric molecular states. As we propose and discuss, the implications of the Quantum Dynamical Selection rule include: (i) a differential chemical reactivity of para- and ortho-hydrogen, (ii) a mechanism for inducing inter-molecular quantum entanglement of nuclear spins, (iii) a new isotope fractionation mechanism, (iv) a novel explanation of the enhanced chemical activity of "Reactive Oxygen Species", (v) illuminating the importance of ortho-water molecules in modulating the quantum dynamics of liquid water, (vi) providing the critical quantum-to-biochemical linkage in the nuclear spin model of the (putative) quantum brain, among others.Comment: 12 pages, 5 figures. Clarified presentation and figure

Relaxation, closing probabilities and transition from oscillatory to chaotic attractors in asymmetric neural networks

Attractors in asymmetric neural networks with deterministic parallel dynamics were shown to present a "chaotic" regime at symmetry eta < 0.5, where the average length of the cycles increases exponentially with system size, and an oscillatory regime at high symmetry, where the typical length of the cycles is 2. We show, both with analytic arguments and numerically, that there is a sharp transition, at a critical symmetry \e_c=0.33, between a phase where the typical cycles have length 2 and basins of attraction of vanishing weight and a phase where the typical cycles are exponentially long with system size, and the weights of their attraction basins are distributed as in a Random Map with reversal symmetry. The time-scale after which cycles are reached grows exponentially with system size $N$, and the exponent vanishes in the symmetric limit, where $T\propto N^{2/3}$. The transition can be related to the dynamics of the infinite system (where cycles are never reached), using the closing probabilities as a tool. We also study the relaxation of the function $E(t)=-1/N\sum_i |h_i(t)|$, where $h_i$ is the local field experienced by the neuron $i$. In the symmetric system, it plays the role of a Ljapunov function which drives the system towards its minima through steepest descent. This interpretation survives, even if only on the average, also for small asymmetry. This acts like an effective temperature: the larger is the asymmetry, the faster is the relaxation of $E$, and the higher is the asymptotic value reached. $E$ reachs very deep minima in the fixed points of the dynamics, which are reached with vanishing probability, and attains a larger value on the typical attractors, which are cycles of length 2.Comment: 24 pages, 9 figures, accepted on Journal of Physics A: Math. Ge

Exactly solvable models of nuclei

In this paper a review is given of a class of sub-models of both approaches, characterized by the fact that they can be solved exactly, highlighting in the process a number of generic results related to both the nature of pair-correlated systems as well as collective modes of motion in the atomic nucleus.Comment: 34 pages, 8 figures accepted for publication in Scholarpedi