Neutral coding - A report based on an NRP work session

Neural coding by impulses and trains on single and multiple channels, and representation of information in nonimpulse carrier

Counting Black Holes: The Cosmic Stellar Remnant Population and Implications for LIGO

We present an empirical approach for interpreting gravitational wave signals of binary black hole mergers under the assumption that the underlying black hole population is sourced by remnants of stellar evolution. Using the observed relationship between galaxy mass and stellar metallicity, we predict the black hole count as a function of galaxy stellar mass. We show, for example, that a galaxy like the Milky Way should host millions of $\sim 30~M_\odot$ black holes and dwarf satellite galaxies like Draco should host $\sim 100$ such remnants, with weak dependence on the assumed IMF and stellar evolution model. Most low-mass black holes ($\sim10 M_\odot$) typically reside within massive galaxies ($M_\star \simeq 10^{11} M_\odot$) while massive black holes ($\sim 50~M_\odot$) typically reside within dwarf galaxies ($M_\odot \simeq 10^9 M_\odot$) today. If roughly $1\%$ of black holes are involved in a binary black hole merger, then the reported merger rate densities from Advanced LIGO can be accommodated for a range of merger timescales, and the detection of mergers with $> 50~M_\odot$ black holes should be expected within the next decade. Identifying the host galaxy population of the mergers provides a way to constrain both the binary neutron star or black hole formation efficiencies and the merger timescale distributions; these events would be primarily localized in dwarf galaxies if the merger timescale is short compared to the age of the universe and in massive galaxies otherwise. As more mergers are detected, the prospect of identifying the host galaxy population, either directly through the detection of electromagnetic counterparts of binary neutron star mergers or indirectly through the anisotropy of the events, will become a realistic possibility.Comment: 10 pages, 8 figures. Accepted by MNRA

Quantum Fourier transform revisited

The fast Fourier transform (FFT) is one of the most successful numerical algorithms of the 20th century and has found numerous applications in many branches of computational science and engineering. The FFT algorithm can be derived from a particular matrix decomposition of the discrete Fourier transform (DFT) matrix. In this paper, we show that the quantum Fourier transform (QFT) can be derived by further decomposing the diagonal factors of the FFT matrix decomposition into products of matrices with Kronecker product structure. We analyze the implication of this Kronecker product structure on the discrete Fourier transform of rank-1 tensors on a classical computer. We also explain why such a structure can take advantage of an important quantum computer feature that enables the QFT algorithm to attain an exponential speedup on a quantum computer over the FFT algorithm on a classical computer. Further, the connection between the matrix decomposition of the DFT matrix and a quantum circuit is made. We also discuss a natural extension of a radix-2 QFT decomposition to a radix-d QFT decomposition. No prior knowledge of quantum computing is required to understand what is presented in this paper. Yet, we believe this paper may help readers to gain some rudimentary understanding of the nature of quantum computing from a matrix computation point of view

Asymptotically Optimal Quantum Circuits for d-level Systems

As a qubit is a two-level quantum system whose state space is spanned by |0>, |1>, so a qudit is a d-level quantum system whose state space is spanned by |0>,...,|d-1>. Quantum computation has stimulated much recent interest in algorithms factoring unitary evolutions of an n-qubit state space into component two-particle unitary evolutions. In the absence of symmetry, Shende, Markov and Bullock use Sard's theorem to prove that at least C 4^n two-qubit unitary evolutions are required, while Vartiainen, Moettoenen, and Salomaa (VMS) use the QR matrix factorization and Gray codes in an optimal order construction involving two-particle evolutions. In this work, we note that Sard's theorem demands C d^{2n} two-qudit unitary evolutions to construct a generic (symmetry-less) n-qudit evolution. However, the VMS result applied to virtual-qubits only recovers optimal order in the case that d is a power of two. We further construct a QR decomposition for d-multi-level quantum logics, proving a sharp asymptotic of Theta(d^{2n}) two-qudit gates and thus closing the complexity question for all d-level systems (d finite.) Gray codes are not required, and the optimal Theta(d^{2n}) asymptotic also applies to gate libraries where two-qudit interactions are restricted by a choice of certain architectures.Comment: 18 pages, 5 figures (very detailed.) MatLab files for factoring qudit unitary into gates in MATLAB directory of source arxiv format. v2: minor change

Development in a biologically inspired spinal neural network for movement control

In two phases, we develop neural network models of spinal circuitry which self-organises into networks with opponent channels for the control of an antagonistic muscle pair. The self-organisation is enabled by spontaneous activity present in the spinal cord. We show that after the process of self-organisation, the networks have developed the possibility to independently control the length and tension of the innerated muscles. This allows the specification of joint angle independent from the specification of joint stiffness. The first network comprises only motorneurons and inhibitory interneurons through which the two channels interact. The inhibitory interneurons prevent saturation of the motorneuron pools, which is a necessary condition for independent control. In the second network, however, the neurons in the motorneuron pools obey the size-principle, which is a threat to the desired invariance of joint angle for varying joint stiffness, because of the different amplification of inputs in the case these inputs are not equal. To restore the desired invariance the second network ha.s been expanded with Renshaw cells. The manner in which they are included in the circuitry corrects the problem caused by the addition of the size-principle. The results obtained from the two models compare favourably with the FLETE-model for spinal circuitry (Bullock & Grossberg, 1991; Bullock et al., HJ93; Bullock & Contreras-Vidal, 1993) which has been successful in explaining several phenomena related to motor control.Fulbright Scholarship; Office of Naval Research (N00014-92-J-1309, N00014-95-1-0409

A Vector-Integration-to-Endpoint Model for Performance of Viapoint Movements

Viapoint (VP) movements are movements to a desired point that are constrained to pass through an intermediate point. Studies have shown that VP movements possess properties, such as smooth curvature around the VP, that are not explicable by treating VP movements as strict concatenations of simpler point-to-point (PTP) movements. Such properties have led some theorists to propose whole-trajectory optimization models, which imply that the entire trajectory is pre-computed before movement initiation. This paper reports new experiments conducted to systematically compare VP with PTP trajectories. Analyses revealed a statistically significant early directional deviation in VP movements but no associated curvature change. An explanation of this effect is offered by extending the Vector-Integration-To-Endpoint (VITE) model (Bullock and Grossberg, 1988), which postulates that voluntary movement trajectories emerge as internal gating signals control the integration of continuously computed vector commands based on the evolving, perceptible difference between desired and actual position variables. The model explains the observed trajectories of VP and PTP movements as emergent properties of a dynamical system that does not precompute entire trajectories before movement initiation. The new model includes a working memory and a stage sensitive to time-to-contact information. These cooperate to control serial performance. The structural and functional relationships proposed in the model are consistent with available data on forebrain physiology and anatomy.Office of Naval Research (N00014-92-J-1309, N00014-93-1-1364, N0014-95-1-0409

Neural model of dopaminergic control of arm movements in Parkinson’s disease bradykinesia

Patients suffering from Parkinson’s disease display a number of symptoms such a resting tremor, bradykinesia, etc. Bradykinesia is the hallmark and most disabling symptom of Parkinson’s disease (PD). Herein, a basal ganglia-cortico-spinal circuit for the control of voluntary arm movements in PD bradykinesia is extended by incorporating DAergic innervation of cells in the cortical and spinal components of the circuit. The resultant model simulates successfully several of the main reported effects of DA depletion on neuronal, electromyographic and movement parameters of PD bradykinesia