Abstract:
Inflammation-induced sensitization of small diameter cutaneous neurons was associated with an increase in action potential duration and rate of decay of the afterhyperpolarization. However, no changes in voltage-gated K+ currents were detected. In contrast, Ca2+ modulated iberiotoxin sensitive and paxilline sensitive K+ (BKCa) currents were significantly smaller in small diameter IB4+ neurons. This decrease in current was not associated with a detectable change in total protein levels of the BKCa channel α or β subunits. Single cell PCR analysis revealed a significant change in the pattern of expression of α subunit splice variants and β subunits that were consistent, at least in part, with inflammation-induced changes in the biophysical properties of BKCa currents in cutaneous neurons.Results of this study provide additional support for the conclusion that it may be possible, if not necessary to selectively treat pain arising from specific body regions. Because a decrease in BKCa current appears to contribute to the inflammation-induced sensitization of cutaneous afferents, BKCa channel openers may be effective for the treatment of inflammatory pain.Peripheral inflammation is associated with pain and hyperalgesia that reflects, at least in part, the sensitization of primary afferents innervating the site of inflammation [1]. This increase in excitability reflects both acute (i.e., phosphorylation) and persistent (i.e., transcription) changes in a variety of ion channels [1] that control afferent excitability. Results from a series of studies on afferents innervating glabrous skin of the rat suggest that the impact of inflammation on the underlying mechanisms of sensitization is complex. Analysis of afferents in vivo indicate that the inflammation-induced increase in excitability is associated with changes in axon conduction velocity, [2] as well as changes in the action potential waveform invading the cell soma in a subpopulation of afferents [3]. Evidence from a r

Abstract:
Using Drosophila cultured cells we show that the full length Delta promotes accumulation of Daughterless protein, fringe RNA, and pangolin RNA in the absence of Scabrous or Notch. Scabrous binds Delta and suppresses this activity even though it increases the level of the Delta intracellular domain. We also show that Scabrous can promote Notch receptor activity, in the absence of Delta.Delta has activity that is independent of its activity as a ligand of Notch. Scabrous suppresses this Delta activity. Scabrous also promotes Notch activity that is dependent on Delta's ligand activity. Thus, Notch, Delta, and Scabrous might function in complex combinatorial or mutually exclusive interactions during development. The data reported here will be of significant help in understanding these interactions in vivo.Notch (N) and Delta (Dl) are cell surface proteins that are required for differentiation of almost all tissues in the fruit fly Drosophila melanogaster. They are evolutionarily conserved, functioning similarly in animals from worms to humans [1,2]. The best-known instance of their function is the process of lateral inhibition that initiates differentiation of the neuronal and epidermal tissues from proneural cells that are predisposed to making the neuronal tissue. Proneural cells express high levels of the neuronal transcription co-factors from the Achaete Scute Complex (ASC) or related genes [3,4]. These factors require their partner Daughterless (Da) to activate transcription of the neurogenesis genes [5-7]. Da is expressed at low levels in all Drosophila cells [8] and up regulated in proneural cells specified to differentiate the neurons [5]. Whether or not the up regulation of Da expression is part of lateral inhibition is not clear in Drosophila. In Caenorhabditis elegans, however, the differential accumulation of the Da homolog HLH-2 is the earliest detectable difference between the cells taking up alternate fates during lateral inhibition [9]. As N and Dl are k

Abstract:
Let $\mathfrak{D}$ be the space consists of pairs $(f,g)$, where $f$ is a univalent function on the unit disc with $f(0)=0$, $g$ is a univalent function on the exterior of the unit disc with $g(\infty)=\infty$ and $f'(0)g'(\infty)=1$. In this article, we define the time variables $t_n, n\in \Z$, on $\mathfrak{D}$ which are holomorphic with respect to the natural complex structure on $\mathfrak{D}$ and can serve as local complex coordinates for $\mathfrak{D}$. We show that the evolutions of the pair $(f,g)$ with respect to these time coordinates are governed by the dispersionless Toda hierarchy flows. An explicit tau function is constructed for the dispersionless Toda hierarchy. By restricting $\mathfrak{D}$ to the subspace $\Sigma$ consists of pairs where $f(w)=1/\bar{g(1/\bar{w})}$, we obtain the integrable hierarchy of conformal mappings considered by Wiegmann and Zabrodin \cite{WZ}. Since every $C^1$ homeomorphism $\gamma$ of the unit circle corresponds uniquely to an element $(f,g)$ of $\mathfrak{D}$ under the conformal welding $\gamma=g^{-1}\circ f$, the space $\text{Homeo}_{C}(S^1)$ can be naturally identified as a subspace of $\mathfrak{D}$ characterized by $f(S^1)=g(S^1)$. We show that we can naturally define complexified vector fields $\pa_n, n\in \Z$ on $\text{Homeo}_{C}(S^1)$ so that the evolutions of $(f,g)$ on $\text{Homeo}_{C}(S^1)$ with respect to $\pa_n$ satisfy the dispersionless Toda hierarchy. Finally, we show that there is a similar integrable structure for the Riemann mappings $(f^{-1}, g^{-1})$. Moreover, in the latter case, the time variables are Fourier coefficients of $\gamma$ and $1/\gamma^{-1}$.

Abstract:
We define and study dispersionless coupled modified KP hierarchy, which incorporates two different versions of dispersionless modified KP hierarchies.

Abstract:
By conformal welding, there is a pair of univalent functions $(f,g)$ associated to every point of the complex K\"ahler manifold $\Mob(S^1)\bk\Diff_+(S^1)$. For every integer $n\geq 1$, we generalize the definition of Faber polynomials to define some canonical bases of holomorphic $1-n$ and $n$ differentials associated to the pair $(f,g)$. Using these bases, we generalize the definition of Grunsky matrices to define matrices whose columns are the coefficients of the differentials with respect to standard bases of differentials on the unit disc and the exterior unit disc. We derive some identities among these matrices which are reminiscent of the Grunsky equality. By using these identities, we showed that we can define the Fredholm determinants of the period matrices of holomorphic $n$ differentials $N_n$, which are the Gram matrices of the canonical bases of holomorphic $n$-differentials with respect to the inner product given by the hyperbolic metric. Finally we proved that $\det N_n =(\det N_1)^{6n^2-6n+1}$ and $\pa\bar{\pa}\log\det N_n$ is $-(6n^2-6n+1)/(6\pi i)$ of the Weil-Petersson symplectic form.

Abstract:
In this article, we classify the solutions of the dispersionless Toda hierarchy into degenerate and non-degenerate cases. We show that every non-degenerate solution is determined by a function $\mathcal{H}(z_1,z_2)$ of two variables. We interpret these non-degenerate solutions as defining evolutions on the space $\mathfrak{D}$ of pairs of conformal mappings $(g,f)$, where $g$ is a univalent function on the exterior of the unit disc, $f$ is a univalent function on the unit disc, normalized such that $g(\infty)=\infty$, $f(0)=0$ and $f'(0)g'(\infty)=1$. For each solution, we show how to define the natural time variables $t_n, n\in\Z$, as complex coordinates on the space $\mathfrak{D}$. We also find explicit formulas for the tau function of the dispersionless Toda hierarchy in terms of $\mathcal{H}(z_1, z_2)$. Imposing some conditions on the function $\mathcal{H}(z_1, z_2)$, we show that the dispersionless Toda flows can be naturally restricted to the subspace $\Sigma$ of $\mathfrak{D}$ defined by $f(w)=1/\overline{g(1/\bar{w})}$. This recovers the result of Zabrodin.

Abstract:
In this paper, we derive the Fay-like identities of tau function for the Toda lattice hierarchy from the bilinear identity. We prove that the Fay-like identities are equivalent to the hierarchy. We also show that the dispersionless limit of the Fay-like identities are the dispersionless Hirota equations of the dispersionless Toda hierarchy.

Abstract:
Given a $C^1$ homeomorphism of the unit circle $\gamma$, let $f$ and $g$ be respectively the normalized conformal maps from the unit disc and its exterior so that $\gamma= g^{-1}\circ f$ on the unit circle. In this article, we show that by suitably defined time variables, the evolutions of the pairs $(g, f)$ and $(g^{-1}, f^{-1})$ can be described by an infinite set of nonlinear partial differential equations known as dispersionless Toda hierarchy. Relations to the integrable structure of conformal maps first studied by Wiegmann and Zabrodin \cite{WZ} are discussed. An extension of the hierarchy which contains both our solution and the solution of \cite{WZ} is defined.

Abstract:
We prove the dispersionless Hirota equations for the dispersionless Toda, dispersionless coupled modified KP and dispersionless KP hierarchies using an idea from classical complex analysis. We also prove that the Hirota equations characterize the tau functions for each of these hierarchies. As a result, we establish the links between the hierarchies.

Abstract:
In this article, we show that four sets of differential Fay identities of an $N$-component KP hierarchy derived from the bilinear relation satisfied by the tau function of the hierarchy are sufficient to derive the auxiliary linear equations for the wave functions. From this, we derive the Lax representation for the $N$-component KP hierarchy, which are equations satisfied by some pseudodifferential operators with matrix coefficients. Besides the Lax equations with respect to the time variables proposed in \cite{2}, we also obtain a set of equations relating different charge sectors, which can be considered as a generalization of the modified KP hierarchy proposed in \cite{3}.