A theory of superconductivity is presented, based on the fact that the interaction between electrons resulting from virtual exchange of phonons is attractive when the energy difference between the electrons states involved is less than the phonon energy, Ace. It is favorable to form a superconducting phase when this attractive interaction dominates the repulsive screened Coulomb interaction. The normal phase is described by the Bloch individual-particle model. The ground state of a superconductor, formed from a linear combination of normal state configurations in which electrons are virtually excited in pairs of opposite spin and momentum, is lower in energy than the normal state by amount proportional to an average (Ace)', consistent with the isotope effect. A mutually orthogonal set of excited states in one-to-one correspondence with those of the normal phase is obtained by specifying occupation of certain Bloch states and by using the rest to form a linear combination of virtual pair configurations. The theory yields a second-order phase transition and a Meissner effect in the form suggested by Pippard. Calculated values of specihc heats and penetration depths and their temperature variation are in good agreement with experiment. There is an energy gap for individual-particle excitations which decreases from about 3.5kT, at T=O'K to zero at T,. Tables of matrix elements of single-particle operators between the excited-state superconducting wave functions, useful for perturbation expansions and calculations of transition probabilities, are given.

ED I TOR momentum operators by applying projection operators and we form an orthogonal set. For the uniform solution, St=i+ (LkAtg kt+CC). k»0 Noting that the method has a variational aspect, L» need not be taken from the small-oscillation analysis but may be freely chosen to describe average largeamplitude effects. We obtain improved ground-state energy and single-and multiple-excitation spectra.

NOVEM8 ER 15~1956 : etters to t.ae. 'Kcitor UBLICATIOII of brief reports of insportant discoveries in physics may be secured by addressing them to this department. The closing date for this departngnt is fsve weeks prior to the date of issue X. o proof will be sent to the authors The B. oard of Editors does not hold itself responsible for the opinions expressed by the corre spondents. Communications should not exceed 600 words in length and should be submitted in duplicate.

Synapses in the brain are bidirectionally modifiable, but the routes of induction are diverse. In various experimental paradigms, N-methyl-D-aspartate receptor-dependent long-term depression and long-term potentiation have been induced selectively by varying the membrane potential of the postsynaptic neurons during presynaptic stimulation of a constant frequency, the rate of presynaptic stimulation, and the timing of pre-and postsynaptic action potentials. In this paper, we present a mathematical embodiment of bidirectional synaptic plasticity that is able to explain diverse induction protocols with a fixed set of parameters. The key assumptions and consequences of the model can be tested experimentally; further, the model provides the foundation for a unified theory of N-methyl-D-aspartate receptordependent synaptic plasticity.S ynapses throughout the brain are bidirectionally modifiable.This property, postulated in almost every theoretical description of synaptic plasticity, has been most clearly demonstrated at the Schaffer collateral-CA1 synapse in the hippocampus. Here, it was shown that a low-frequency tetanus induces long-term depression (LTD), whereas high-frequency stimulation produces long-term potentiation (LTP) of the stimulated synapses, and that LTD and LTP are inversely related (1-4). Similar results have been obtained at excitatory synapses throughout the brain.A considerable body of evidence indicates that the important variable is actually the amount of integrated postsynaptic N-methyl-D-aspartate (NMDA) receptor (NMDAR) activation during conditioning (1, 2, 4-6). Modest NMDAR activation induces LTD, whereas strong activation produces LTP. Because of their voltage dependence, the contribution of NMDARs to synaptic transmission during conditioning stimulation varies with the level of postsynaptic depolarization. Thus, it is possible to induce LTD or LTP with a constant stimulation frequency by clamping the postsynaptic membrane potential at different values (approximately Ϫ50 mV for LTD and Ϫ20 mV for LTP).Recently, it has been demonstrated that synaptic modification also can depend on the precise timing of pre-and postsynaptic action potentials (7-11). If a presynaptic action potential occurs in a window of several tens of milliseconds before a back-propagating postsynaptic action potential, LTP is induced. In contrast, if a presynaptic action potential occurs after the postsynaptic spike, LTD is induced.Ideally, one would like to develop a unified description of bidirectional synaptic plasticity that can account for all routes of induction. One approach is to look beyond the various induction protocols to the critical role of calcium influx through NMDARs. One attractive idea is that modest increases in postsynaptic calcium trigger LTD, whereas large increases trigger . This hypothesis is consistent with the classical rate-based induction protocols if it is assumed that high-frequency stimulation triggers a larger rise in postsynaptic calcium than does low-frequency stimulation. Indeed, recent...

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Thirty years have passed since the publication of Elie Bienenstock, Leon Cooper and Paul Munro's 'Theory for the development of neuron selectivity: orientation specificity and binocular interaction in visual cortex', known as the BCM theory of synaptic plasticity. This theory has guided experimentalists to discover some fundamental properties of synaptic plasticity and has provided a mathematical structure that bridges molecular mechanisms and systems-level consequences of learning and memory storage.

The calculational aspects of solving certain classes of location-allocation problems are presented. Both exact extremal equations and a heuristic method are presented for solving these problems. Directions for further investigation are also indicated because of the need for methods to shorten the total amount of calculation involved.

Mesenchymal stem/stromal cells (MSCs) are promising cell sources for regenerative therapies due to their multipotency and ready availability, but their application can be complicated by patient-specific factors like age or illness. MSCs have been investigated for the treatment of many musculoskeletal disorders, including osteoarthritis and osteoporosis. Due to the prevalence of these diseases in older populations, researchers have studied how aging affects MSC properties and have found that proliferation and differentiation potential are impaired. However, these effects have never been compared among MSCs isolated from multiple tissue sources in the same, healthy donor. Revealing differences in how MSCs are affected by age could help identify an optimal cell source for musculoskeletal therapies targeting older patients. MSCs were isolated from young and old rabbit bone marrow, muscle, and adipose tissue. Cell yield and viability were quantified after isolation procedures, and expansion properties were assessed using assays for proliferation, senescence, and colony formation. Multipotency was also examined using lineage-specific stains and spectrophotometry of metabolites. Results were compared between age groups and among MSC sources. Results showed that MSCs are differentially influenced by aging, with bone marrow-derived stem cells having impaired proliferation, senescence, and chondrogenic response, whereas muscle-derived stem cells and adipose-derived stem cells exhibited no negative effects. While age reduced overall cell yield and adipogenic potential of all MSC populations, osteogenesis and clonogenicity remained unchanged. These findings indicate the importance of age as a factor when designing cell-based therapies for older patients.