Hypercomplex polynomials, vietoris’ rational numbers and a related integer numbers sequence

This paper aims to give new insights into homogeneous hypercomplex Appell polynomials through the study of some interesting arithmetical properties of their coefficients. Here Appell polynomials are introduced as constituting a hypercomplex generalized geometric series whose fundamental role sometimes seems to have been neglected. Surprisingly, in the simplest non-commutative case their rational coefficient sequence reduces to a coefficient sequence S used in a celebrated theorem on positive trigonometric sums by Vietoris (Sitzungsber Österr Akad Wiss 167:125–135, 1958). For S a generating function is obtained which allows to derive an interesting relation to a result deduced by Askey and Steinig (Trans AMS 187(1):295–307, 1974) about some trigonometric series. The further study of S is concerned with a sequence of integers leading to its irreducible representation and its relation to central binomial coefficients.The work of the first and third authors was supported by Portuguese funds through the CIDMA - Center for Research and Development in Mathematics and Applications, and the Portuguese Foundation for Science and Technology (“FCT-Fundação para a Ciência e Tecnologia”), within project PEstOE/MAT/UI4106/2013. The work of the second author was supported by Portuguese funds through the CMAT - Centre of Mathematics and FCT within the Project UID/MAT/00013/2013.info:eu-repo/semantics/publishedVersio

Enhanced Hippocampal Long-Term Potentiation and Fear Memory in Btbd9 Mutant Mice

Polymorphisms in BTBD9 have recently been associated with higher risk of restless legs syndrome (RLS), a neurological disorder characterized by uncomfortable sensations in the legs at rest that are relieved by movement. The BTBD9 protein contains a BTB/POZ domain and a BACK domain, but its function is unknown. To elucidate its function and potential role in the pathophysiology of RLS, we generated a line of mutant Btbd9 mice derived from a commercial gene-trap embryonic stem cell clone. Btbd9 is the mouse homolog of the human BTBD9. Proteins that contain a BTB/POZ domain have been reported to be associated with synaptic transmission and plasticity. We found that Btbd9 is naturally expressed in the hippocampus of our mutant mice, a region critical for learning and memory. As electrophysiological characteristics of CA3-CA1 synapses of the hippocampus are well characterized, we performed electrophysiological recordings in this region. The mutant mice showed normal input-output relationship, a significant impairment in pre-synaptic activity, and an enhanced long-term potentiation. We further performed an analysis of fear memory and found the mutant mice had an enhanced cued and contextual fear memory. To elucidate a possible molecular basis for these enhancements, we analyzed proteins that have been associated with synaptic plasticity. We found an elevated level of dynamin 1, an enzyme associated with endocytosis, in the mutant mice. These results suggest the first identified function of Btbd9 as being involved in regulating synaptic plasticity and memory. Recent studies have suggested that enhanced synaptic plasticity, analogous to what we have observed, in other regions of the brain could enhance sensory perception similar to what is seen in RLS patients. Further analyses of the mutant mice will help shine light on the function of BTBD9 and its role in RLS

Biophysical Basis for Three Distinct Dynamical Mechanisms of Action Potential Initiation

Transduction of graded synaptic input into trains of all-or-none action potentials (spikes) is a crucial step in neural coding. Hodgkin identified three classes of neurons with qualitatively different analog-to-digital transduction properties. Despite widespread use of this classification scheme, a generalizable explanation of its biophysical basis has not been described. We recorded from spinal sensory neurons representing each class and reproduced their transduction properties in a minimal model. With phase plane and bifurcation analysis, each class of excitability was shown to derive from distinct spike initiating dynamics. Excitability could be converted between all three classes by varying single parameters; moreover, several parameters, when varied one at a time, had functionally equivalent effects on excitability. From this, we conclude that the spike-initiating dynamics associated with each of Hodgkin's classes represent different outcomes in a nonlinear competition between oppositely directed, kinetically mismatched currents. Class 1 excitability occurs through a saddle node on invariant circle bifurcation when net current at perithreshold potentials is inward (depolarizing) at steady state. Class 2 excitability occurs through a Hopf bifurcation when, despite net current being outward (hyperpolarizing) at steady state, spike initiation occurs because inward current activates faster than outward current. Class 3 excitability occurs through a quasi-separatrix crossing when fast-activating inward current overpowers slow-activating outward current during a stimulus transient, although slow-activating outward current dominates during constant stimulation. Experiments confirmed that different classes of spinal lamina I neurons express the subthreshold currents predicted by our simulations and, further, that those currents are necessary for the excitability in each cell class. Thus, our results demonstrate that all three classes of excitability arise from a continuum in the direction and magnitude of subthreshold currents. Through detailed analysis of the spike-initiating process, we have explained a fundamental link between biophysical properties and qualitative differences in how neurons encode sensory input