Momentum dynamics of one dimensional quantum walks

We derive the momentum space dynamic equations and state functions for one dimensional quantum walks by using linear systems and Lie group theory. The momentum space provides an analytic capability similar to that contributed by the z transform in discrete systems theory. The state functions at each time step are expressed as a simple sum of three Chebyshev polynomials. The functions provide an analytic expression for the development of the walks with time.Ian Fuss, Langord B. White, Peter J. Sherman, Sanjeev Naguleswara

Gold as an inflation hedge?

This paper attempts to reconcile an apparent contradiction between short-run and long-run movements in the price of gold. The theoretical model suggests a set of conditions under which the price of gold rises over time at the general rate of inflation and hence be an effective hedge against inflation. The model also demonstrates that short-run changes in the gold lease rate, the real interest rate, convenience yield, default risk, the covariance of gold returns with other assets and the dollar/world exchange rate can disturb this equilibrium relationship and generate short-run price volatility. Using monthly gold price data (1976-1999), and cointegration regression techniques, an empirical analysis confirms the central hypotheses of the theoretical model

Restoration of SMN in Schwann cells reverses myelination defects and improves neuromuscular function in spinal muscular atrophy

Spinal muscular atrophy (SMA) is a neuromuscular disease caused by low levels of SMN protein, primarily affecting lower motor neurons. Recent evidence from SMA and related conditions suggests that glial cells can influence disease severity. Here, we investigated the role of glial cells in the peripheral nervous system by creating SMA mice selectively overexpressing SMN in myelinating Schwann cells (Smn(−/−);SMN2(tg/0);SMN1(SC)). Restoration of SMN protein levels restricted solely to Schwann cells reversed myelination defects, significantly improved neuromuscular function and ameliorated neuromuscular junction pathology in SMA mice. However, restoration of SMN in Schwann cells had no impact on motor neuron soma loss from the spinal cord or ongoing systemic and peripheral pathology. This study provides evidence for a defined, intrinsic contribution of glial cells to SMA disease pathogenesis and suggests that therapies designed to include Schwann cells in their target tissues are likely to be required in order to rescue myelination defects and associated disease symptoms

Drp2 and Periaxin Form Cajal Bands with Dystroglycan But Have Distinct Roles in Schwann Cell Growth

Cajal bands are cytoplasmic channels flanked by appositions where the abaxonal surface of Schwann cell myelin apposes and adheres to the overlying plasma membrane. These appositions contain a dystroglycan complex that includes periaxin and dystrophin-related protein 2 (Drp2). Loss of periaxin disrupts appositions and Cajal bands in Schwann cells and causes a severe demyelinating neuropathy in mouse and man. Here we have investigated the role of mouse Drp2 in apposition assembly and Cajal band function and compared it to periaxin. We show that Periaxin and Drp2 are not only both required to form appositions, but they must also interact. Periaxin-Drp2 interaction is also required for Drp2 phosphorylation but phosphorylation is not required for the assembly of appositions. Drp2 loss causes corresponding increases in Dystrophin family members, utrophin and dystrophin Dp116 though dystroglycan remains unchanged. We also show that all dystroglycan complexes in Schwann cells utilise the uncleaved form of β-dystroglycan. Drp2-null Schwann cells have disrupted appositions and Cajal bands, and they undergoe focal hypermyelination and concomitant demyelination. Nevertheless, they do not have the short internodal lengths and associated reduced nerve conduction velocity seen in the absence of periaxin, showing that periaxin regulates Schwann cell elongation independent of its role in the dystroglycan complex. We conclude that the primary role of the dystroglycan complex in appositions is to stabilize and limit the radial growth of myelin

Astrocyte Ca2+-evoked ATP release regulates myelinated axon excitability and conduction speed*

INTRODUCTION: Astrocytes support neuronal function throughout the central nervous system. In the gray matter, they regulate synapse number during development, remove synaptically released neurotransmitters to terminate their action and prevent excitotoxicity, control the extracellular potassium concentration to prevent hyperexcitability, regulate blood flow to ensure an adequate energy supply, provide lactate to neurons for energy, and respond to rises of intracellular calcium concentration ([Ca2+]i) by releasing adenosine triphosphate (ATP) and other gliotransmitters that act on neuronal receptors to modulate information processing. However, their role is unclear in the white matter, which transmits information rapidly between gray matter areas using axons wrapped with capacitance-reducing myelin (although they have been suggested to regulate myelination during development and during normal function). RATIONALE: Recently, it has been suggested that learning and memory may reflect not only changes in synaptic function in the gray matter, but also changes in white matter function. In particular, neural circuit function might be regulated by changes in the conduction speed of myelinated axons that result in an altered arrival time of action potentials at a distant neuron. These speed changes might be brought about by alterations of the properties of the passively conducting myelinated internodes or of the intervening excitable nodes of Ranvier, where the action potential is generated. We applied immunohistochemistry to assess how astrocytes interact with myelinated axons, neuronal stimulation and light-evoked calcium uncaging in astrocytes to evoke Ca2+-dependent release of gliotransmitters, and electrophysiology and pharmacology to characterize how astrocyte-released substances might affect the axon initial segment (AIS) and nodes of Ranvier of myelinated neurons. Measurements of conduction velocity and computer modeling allowed us to interpret the results. RESULTS: Astrocytes closely approach the axons of myelinated neurons in layer V of the cerebral cortex that enter the corpus callosum. Uncaging Ca2+ within astrocytes or stimulating spike trains in neurons evoked a rise of astrocyte [Ca2+]i that triggered the release of ATP-containing vesicles from these cells. This evoked an inward current in the AIS and nodes of Ranvier of the pyramidal neurons. Pharmacology showed that this was mediated by the activation of Gs-linked adenosine A2a receptors (A2aRs), implying that the released ATP was converted to adenosine by extracellular enzymes. The A2aRs raise the intracellular concentration of cyclic AMP, which activates hyperpolarization-activated cyclic nucleotide–gated (HCN) channels mediating the inward hyperpolarization-activated current (Ih) and thus depolarizes the cell. In the AIS, the activation of A2aRs alters excitability and hence action potential generation, whereas in the nodes of Ranvier, it decreases the conduction speed of the action potential along the axon. CONCLUSION: As in the gray matter, astrocyte [Ca2+]i regulates the release of ATP into the extracellular space in the white matter. After conversion to adenosine, this regulates the excitability and conduction speed of myelinated axons. The changes in excitability at the AIS will lead to changes in the relationship between the synaptic input and action potential output of the cell. The altered conduction speed of the myelinated axon may change neural circuit function by changing the action potential arrival time at the cell’s output synapses, thus altering the integration of signals in postsynaptic neurons. Variations in astrocyte-derived adenosine level can occur between wake and sleep states, and the extracellular adenosine concentration rises during energy deprivation conditions. These changes in adenosine level could thus control white matter information flow and neural circuit function

Increasing Internodal Distance in Myelinated Nerves Accelerates Nerve Conduction to a Flat Maximum

SummaryPredictions that conduction velocities are sensitive to the distance between nodes of Ranvier in myelinated axons have implications for nervous system function during growth and repair [1–3]. Internodal lengths defined by Schwann cells in hindlimb nerves, for example, can undergo a 4-fold increase during mouse development, and regenerated nerves have internodes that are uniformly short [4, 5]. Nevertheless, the influence of internodal length on conduction speed has limited experimental support. Here, we examined this problem in mice expressing a mutant version of periaxin, a protein required for Schwann cell elongation [4]. Importantly, elongation of mutant Schwann cells was retarded without significant derangements to myelination or axon caliber. In young mice with short mutant Schwann cells, nerve conduction velocity was reduced and motor function was impaired. This demonstrates a functional relationship between internodal distance and conduction speed. Moreover, as internodes lengthened during postnatal growth, conduction velocities recovered to normal values and mutant mice exhibited normal motor and sensory behavior. This restoration of function confirms a further prediction by Huxley and Stämpfli that conduction speeds should increase as internodal distances lengthen until a “flat maximum” is reached, beyond which no further gains in conduction velocity accrue [6]

FAK is required for axonal sorting by Schwann cells

Signaling by laminins and axonal neuregulin has been implicated in regulating axon sorting by myelin-forming Schwann cells. However, the signal transduction mechanisms are unknown. Focal adhesion kinase (FAK) has been linked to α6β1 integrin and ErbB receptor signaling, and we show that myelination by Schwann cells lacking FAK is severely impaired. Mutant Schwann cells could interdigitate between axon bundles, indicating that FAK signaling was not required for process extension. However, Schwann cell FAK was required to stimulate cell proliferation, suggesting that amyelination was caused by insufficient Schwann cells. ErbB2 receptor and AKT were robustly phosphorylated in mutant Schwann cells, indicating that neuregulin signaling from axons was unimpaired. These findings demonstrate the vital relationship between axon defasciculation and Schwann cell number and show the importance of FAK in regulating cell proliferation in the developing nervous system

Effect of Limb Lengthening on Internodal Length and Conduction Velocity of Peripheral Nerve

The influences of axon diameter, myelin thickness, and internodal length on the velocity of conduction of peripheral nerve action potentials are unclear. Previous studies have demonstrated a strong dependence of conduction velocity on internodal length. However, a theoretical analysis has suggested that this relationship may be lost above a nodal separation of ∼0.6 mm. Here we measured nerve conduction velocities in a rabbit model of limb lengthening that produced compensatory increases in peripheral nerve growth. Divided tibial bones in one hindlimb were gradually lengthened at 0.7 mm per day using an external frame attached to the bone. This was associated with a significant increase (33%) of internodal length (0.95–1.3 mm) in axons of the tibial nerve that varied in proportion to the mechanical strain in the nerve of the lengthened limb. Axonal diameter, myelin thickness, and g-ratios were not significantly altered by limb lengthening. Despite the substantial increase in internodal length, no significant change was detected in conduction velocity (∼43 m/s) measured either in vivo or in isolated tibial nerves. The results demonstrate that the internode remains plastic in the adult but that increases in internodal length of myelinated adult nerve axons do not result in either deficiency or proportionate increases in their conduction velocity and support the view that the internodal lengths of nerves reach a plateau beyond which their conduction velocities are no longer sensitive to increases in internodal length