34 research outputs found
Construction of some missing eigenvectors of the XYZ spin chain at the discrete coupling constants and the exponentially large spectral degeneracy of the transfer matrix
We discuss an algebraic method for constructing eigenvectors of the transfer
matrix of the eight vertex model at the discrete coupling parameters. We
consider the algebraic Bethe ansatz of the elliptic quantum group for the case where the parameter satisfies for arbitrary integers , and . When or
is odd, the eigenvectors thus obtained have not been discussed previously.
Furthermore, we construct a family of degenerate eigenvectors of the XYZ spin
chain, some of which are shown to be related to the loop algebra
symmetry of the XXZ spin chain. We show that the dimension of some degenerate
eigenspace of the XYZ spin chain on sites is given by , if
is an even integer. The construction of eigenvectors of the transfer matrices
of some related IRF models is also discussed.Comment: 19 pages, no figure (revisd version with three appendices
Irreducible representations of Upq[gl(2/2)]
The two-parametric quantum superalgebra and its
representations are considered. All finite-dimensional irreducible
representations of this quantum superalgebra can be constructed and classified
into typical and nontypical ones according to a proposition proved in the
present paper. This proposition is a nontrivial deformation from the one for
the classical superalgebra gl(2/2), unlike the case of one-parametric
deformations.Comment: Latex, 8 pages. A reference added in v.
Algebraic Bethe ansatz method for the exact calculation of energy spectra and form factors: applications to models of Bose-Einstein condensates and metallic nanograins
In this review we demonstrate how the algebraic Bethe ansatz is used for the
calculation of the energy spectra and form factors (operator matrix elements in
the basis of Hamiltonian eigenstates) in exactly solvable quantum systems. As
examples we apply the theory to several models of current interest in the study
of Bose-Einstein condensates, which have been successfully created using
ultracold dilute atomic gases. The first model we introduce describes Josephson
tunneling between two coupled Bose-Einstein condensates. It can be used not
only for the study of tunneling between condensates of atomic gases, but for
solid state Josephson junctions and coupled Cooper pair boxes. The theory is
also applicable to models of atomic-molecular Bose-Einstein condensates, with
two examples given and analysed. Additionally, these same two models are
relevant to studies in quantum optics. Finally, we discuss the model of
Bardeen, Cooper and Schrieffer in this framework, which is appropriate for
systems of ultracold fermionic atomic gases, as well as being applicable for
the description of superconducting correlations in metallic grains with
nanoscale dimensions. In applying all of the above models to physical
situations, the need for an exact analysis of small scale systems is
established due to large quantum fluctuations which render mean-field
approaches inaccurate.Comment: 49 pages, 1 figure, invited review for J. Phys. A., published version
available at http://stacks.iop.org/JPhysA/36/R6
Menelaus' theorem, Clifford configurations and inversive geometry of the Schwarzian KP hierarchy
It is shown that the integrable discrete Schwarzian KP (dSKP) equation which
constitutes an algebraic superposition formula associated with, for instance,
the Schwarzian KP hierarchy, the classical Darboux transformation and
quasi-conformal mappings encapsulates nothing but a fundamental theorem of
ancient Greek geometry. Thus, it is demonstrated that the connection with
Menelaus' theorem and, more generally, Clifford configurations renders the dSKP
equation a natural object of inversive geometry on the plane. The geometric and
algebraic integrability of dSKP lattices and their reductions to lattices of
Menelaus-Darboux, Schwarzian KdV, Schwarzian Boussinesq and Schramm type is
discussed. The dSKP and discrete Schwarzian Boussinesq equations are shown to
represent discretizations of families of quasi-conformal mappings.Comment: 26 pages, 9 figure
Brief wide-field photostimuli evoke and modulate oscillatory reverberating activity in cortical networks
Cell assemblies manipulation by optogenetics is pivotal to advance neuroscience and neuroengineering. In in vivo applications, photostimulation often broadly addresses a population of cells simultaneously, leading to feed-forward and to reverberating responses in recurrent microcircuits. The former arise from direct activation of targets downstream, and are straightforward to interpret. The latter are consequence of feedback connectivity and may reflect a variety of time-scales and complex dynamical properties. We investigated wide-field photostimulation in cortical networks in vitro, employing substrate-integrated microelectrode arrays and long-term cultured neuronal networks. We characterized the effect of brief light pulses, while restricting the expression of channelrhodopsin to principal neurons. We evoked robust reverberating responses, oscillating in the physiological gamma frequency range, and found that such a frequency could be reliably manipulated varying the light pulse duration, not its intensity. By pharmacology, mathematical modelling, and intracellular recordings, we conclude that gamma oscillations likely emerge as in vivo from the excitatory-inhibitory interplay and that, unexpectedly, the light stimuli transiently facilitate excitatory synaptic transmission. Of relevance for in vitro models of (dys)functional cortical microcircuitry and in vivo manipulations of cell assemblies, we give for the first time evidence of network-level consequences of the alteration of synaptic physiology by optogenetics
Innate Synchronous Oscillations in Freely-Organized Small Neuronal Circuits
BACKGROUND: Information processing in neuronal networks relies on the network's ability to generate temporal patterns of action potentials. Although the nature of neuronal network activity has been intensively investigated in the past several decades at the individual neuron level, the underlying principles of the collective network activity, such as the synchronization and coordination between neurons, are largely unknown. Here we focus on isolated neuronal clusters in culture and address the following simple, yet fundamental questions: What is the minimal number of cells needed to exhibit collective dynamics? What are the internal temporal characteristics of such dynamics and how do the temporal features of network activity alternate upon crossover from minimal networks to large networks? METHODOLOGY/PRINCIPAL FINDINGS: We used network engineering techniques to induce self-organization of cultured networks into neuronal clusters of different sizes. We found that small clusters made of as few as 40 cells already exhibit spontaneous collective events characterized by innate synchronous network oscillations in the range of 25 to 100 Hz. The oscillation frequency of each network appeared to be independent of cluster size. The duration and rate of the network events scale with cluster size but converge to that of large uniform networks. Finally, the investigation of two coupled clusters revealed clear activity propagation with master/slave asymmetry. CONCLUSIONS/SIGNIFICANCE: The nature of the activity patterns observed in small networks, namely the consistent emergence of similar activity across networks of different size and morphology, suggests that neuronal clusters self-regulate their activity to sustain network bursts with internal oscillatory features. We therefore suggest that clusters of as few as tens of cells can serve as a minimal but sufficient functional network, capable of sustaining oscillatory activity. Interestingly, the frequencies of these oscillations are similar those observed in vivo