Tau Function Research Articles

Rankin–Cohen brackets and Serre derivatives as Poincaré series

We give expressions for the Serre derivatives of Eisenstein and Poincar\'e series as well as their Rankin-Cohen brackets with arbitrary modular forms in terms of the Poincar\'e averaging construction, and derive several identities for the Ramanujan tau function as applications.

Tau functions, Hodge classes, and discriminant loci on moduli spaces of Hitchin’s spectral covers

We define two tau functions, τ and τ^, on moduli spaces of spectral covers of GL(n) Hitchin systems. Analyzing the properties of τ, we express the rational divisor class of the universal Hitchin’s discriminant in terms of standard generators of the rational Picard group of the moduli spaces of spectral covers with variable base. The function τ^ is used to compute the divisor of canonical 1-forms with multiple zeros.

Irregular conformal blocks and connection formulae for Painlevé V functions

We prove a Fredholm determinant and short-distance series representation of the Painlevé V tau function τt associated with generic monodromy data. Using a relation of τt to two different types of irregular c = 1 Virasoro conformal blocks and the confluence from Painlevé VI equation, connection formulas between the parameters of asymptotic expansions at 0 and i∞ are conjectured. Explicit evaluations of the connection constants relating the tau function asymptotics as t → 0, +∞, i∞ are obtained. We also show that irregular conformal blocks of rank 1, for arbitrary central charge, are obtained as confluent limits of the regular conformal blocks.

Tau Functions as Widom Constants

We define a tau function for a generic Riemann-Hilbert problem posed on a union of non-intersecting smooth closed curves with jump matrices analytic in their neighborhood. The tau function depends on parameters of the jumps and is expressed as the Fredholm determinant of an integral operator with block integrable kernel constructed in terms of elementary parametrices. Its logarithmic derivatives with respect to parameters are given by contour integrals involving these parametrices and the solution of the Riemann-Hilbert problem. In the case of one circle, the tau function coincides with Widom's determinant arising in the asymptotics of block Toeplitz matrices. Our construction gives the Jimbo-Miwa-Ueno tau function for Riemann-Hilbert problems of isomonodromic origin (Painlev\'e VI, V, III, Garnier system, etc) and the Sato-Segal-Wilson tau function for integrable hierarchies such as Gelfand-Dickey and Drinfeld-Sokolov.

Fredholm Determinant and Nekrasov Sum Representations of Isomonodromic Tau Functions

We derive Fredholm determinant representation for isomonodromic tau functions of Fuchsian systems with $n$ regular singular points on the Riemann sphere and generic monodromy in $\mathrm{GL}(N,\mathbb C)$. The corresponding operator acts in the direct sum of $N(n-3)$ copies of $L^2(S^1)$. Its kernel has a block integrable form and is expressed in terms of fundamental solutions of $n-2$ elementary 3-point Fuchsian systems whose monodromy is determined by monodromy of the relevant $n$-point system via a decomposition of the punctured sphere into pairs of pants. For $N=2$ these building blocks have hypergeometric representations, the kernel becomes completely explicit and has Cauchy type. In this case Fredholm determinant expansion yields multivariate series representation for the tau function of the Garnier system, obtained earlier via its identification with Fourier transform of Liouville conformal block (or a dual Nekrasov-Okounkov partition function). Further specialization to $n=4$ gives a series representation of the general solution to Painlev\'e VI equation.

On the modified KP hierarchy: Tau functions, squared eigenfunction symmetries and additional symmetries

In this paper, the modified KP hierarchy in the Kupershmidt–Kiso version is investigated starting from the tau functions. There are two tau functions for the modified KP hierarchy. We firstly give some properties about the tau functions, including Hirota bilinear identities and Fay identities. And it is showed that these two functions can be viewed as two different tau functions of the KP hierarchy. Then after constructing the spectral representations of the eigenfunction and the adjoint eigenfunctions, the expressions of all the squared eigenfunction potentials in the modified KP hierarchy are obtained. Next, the study of the squared eigenfunction symmetries and the additional symmetries is considered, whose actions on the two tau functions are derived. At last, the Virasoro additional symmetries in the constrained cases are given, also with the actions on the tau functions.

Structure, Function and Interactions of Tau: Particular Focus on Potential Drug Targets for the Treatment of Tauopathies.

Neurodegenrative diseases are among the most widespread lifethreatening disorders around the world in elderly ages. The common feature of a group of neurodegenerative disorders, called tauopathies, is an accumulation of microtubule associated protein tau inside the neurons. The exact mechanism underlying tauopathies is not well-understood but several factors such as traumatic brain injuries and genetics are considered as potential risk factors. Although tau protein is well-known for its key role in stabilizing and organization of axonal microtubule network, it bears a broad range of functions including DNA protection and participation in signaling pathways. Moreover, the flexible unfolded structure of tau facilitates modification of tau by a wide range of intracellular enzymes which in turn broadens tau function and interaction spectrum. The distinctive properties of tau protein concomitant with the crucial role of tau interaction partners in the progression of neurodegeneration suggest tau and its binding partners as potential drug targets for the treatment of neurodegenerative diseases. This review aims to give a detailed description of structure, functions and interactions of tau protein in order to provide insight into potential therapeutic targets for treatment of tauopathies.

Accessory parameters in conformal mapping: exploiting the isomonodromic tau function for Painlevé VI.

We present a novel method to solve the accessory parameter problem arising in constructing conformal maps from a canonical simply connected planar region to the interior of a circular arc quadrilateral. The Schwarz–Christoffel accessory parameter problem, relevant when all sides have zero curvature, is also captured within our approach. The method exploits the isomonodromic tau function associated with the Painlevé VI equation. Recently, these tau functions have been shown to be related to certain correlation functions in conformal field theory and asymptotic expansions have been given in terms of tuples of the Young diagrams. After showing how to extract the monodromy data associated with the target domain, we show how a numerical approach based on the known asymptotic expansions can be used to solve the conformal mapping accessory parameter problem. The viability of this new method is demonstrated by explicit examples and we discuss its extension to circular arc polygons with more than four sides.

Stochastic Representation of Tau Functions With an Application to the Korteweg-De Vries Equation

Stochastic Representation of Tau Functions With an Application to the Korteweg-De Vries Equation

Conformational heterogeneity of tau: Implication on intrinsic disorder, acid stability and fibrillation in Alzheimer's disease

The self-assembly of intrinsically disordered protein tau into paired helical filament forms one of the hallmarks of Alzheimer's disease. However, the facets of innately disordered structure of tau and its conversion to a β-sheet-rich fibril during several tauopathies are poorly understood. Here, we provide a direct insight into the ensemble of highly heterogeneous conformational families of tau at physiological pH, by nano-electrospray mass spectrometry coupled with ion mobility. The average collision cross section of the most unfolded conformer was higher by >2 fold than that of the most folded one. Acidic pH largely induced unfolding in tau, obliterating the compact conformers completely. The highly unfolded conformers were the key species bestowing the unusual solubility to tau at low pH, with limited contribution from intramolecular long-range interfaces giving rise to ordered conformers. Contrarily, alkaline pH shifted tau towards folded conformations due to charge neutralization, keeping the overall random coil architecture intact. Intriguingly, the heparin-induced in vitro aggregation propensity of the protein attenuated at both acidic and alkaline pH, illustrating the significance of altered conformations in pathological functions of tau. Our observations at low pH indicate that a reorganization of the intricate network of momentary long-range contacts in tau might have implication in its aggregation pathology. Disease-modifying therapies for Alzheimer's disease targeting either to disrupt the essential fibril-forming interaction at third microtubule-binding repeat of tau or to perturb specific binding interaction of tau with endogenous polyanionic species might be of high benefit.

On Some Hamiltonian Properties of the Isomonodromic Tau Functions

We discuss some new aspects of the theory of the Jimbo–Miwa–Ueno tau function which have come to light within the recent developments in the global asymptotic analysis of the tau functions related to the Painlevé equations. Specifically, we show that up to the total differentials the logarithmic derivatives of the Painlevé tau functions coincide with the corresponding classical action differential. This fact simplifies considerably the evaluation of the constant factors in the asymptotics of tau functions, which has been a long-standing problem of the asymptotic theory of Painlevé equations. Furthermore, we believe that this observation is yet another manifestation of L. D. Faddeev’s emphasis of the key role which the Hamiltonian aspects play in the theory of integrable system.

Nonlinear forms of coprimeness preserving extensions to the Somos-4 recurrence and the two-dimensional Toda lattice equation—investigation into their extended Laurent properties

Coprimeness property was introduced to study the singularity structure of discrete dynamical systems. In this paper we shall extend the coprimeness property and the Laurent property to further investigate discrete equations with complicated pattern of singularities. As examples we study extensions to the Somos-4 recurrence and the 2D discrete Toda equation. By considering their non-autonomous polynomial forms, we prove that their tau function analogues possess the extended Laurent property with respect to their initial variables and some extra factors related to the non-autonomous terms. Using this Laurent property, we prove that these equations satisfy the extended coprimeness property. This coprimeness property reflects the singularities that trivially arise from the equations.

Characterisation of tau in the human and rodent enteric nervous system under physiological conditions and in tauopathy

Tau is normally a highly soluble phosphoprotein found predominantly in neurons. Six different isoforms of tau are expressed in the adult human CNS. Under pathological conditions, phosphorylated tau aggregates are a defining feature of neurodegenerative disorders called tauopathies. Recent findings have suggested a potential role of the gut-brain axis in CNS homeostasis, and therefore we set out to examine the isoform profile and phosphorylation state of tau in the enteric nervous system (ENS) under physiological conditions and in tauopathies. Surgical specimens of human colon from controls, Parkinson’s disease (PD) and progressive supranuclear palsy (PSP) patients were analyzed by Western Blot and immunohistochemistry using a panel of anti-tau antibodies. We found that adult human ENS primarily expresses two tau isoforms, localized in the cell bodies and neuronal processes. We did not observe any difference in the enteric tau isoform profile and phosphorylation state between PSP, PD and control subjects. The htau mouse model of tauopathy also expressed two main isoforms of human tau in the ENS, and there were no apparent differences in ENS tau localization or phosphorylation between wild-type and htau mice. Tau in both human and mouse ENS was found to be phosphorylated but poorly susceptible to dephosphorylation with lambda phosphatase. To investigate ENS tau phosphorylation further, primary cultures from rat enteric neurons, which express four isoforms of tau, were pharmacologically manipulated to show that ENS tau phosphorylation state can be regulated, at least in vitro. Our study is the first to characterize tau in the rodent and human ENS. As a whole, our findings provide a basis to unravel the functions of tau in the ENS and to further investigate the possibility of pathological changes in enteric neuropathies and tauopathies.

Combinatorial interpretations of Ramanujan’s tau function

We use a q-series identity by Ramanujan to give a combinatorial interpretation of Ramanujan’s tau function which involves t-cores and a new class of partitions which we call (m,k)-capsids. The same method can be applied in conjunction with other related identities yielding alternative combinatorial interpretations of the tau function.

Do Cardiometabolic Risk Factors Influence Amyloid, Tau, and Neuronal Function in APOE4 Carriers and Non-Carriers in Alzheimer's Disease Trajectory?

Cardiovascular risk could be calculated using Qrisk2. It is suggested that cardiovascular risk factors influence the progression of Alzheimer's disease (AD). However, studies have not specifically evaluated the influence of cardiovascular risk using Qrisk2 on neuropathological progression and AD biomarkers. The aim of the study was to evaluate the influence of cardiovascular risk factors using Qrisk2 on CSF amyloid-β (Aβ) and tau, 18F-AV45-PET, 18F-FDG-PET, MRI, and cognitive measures in APOE4 negative cognitively normal and mild cognitive impairment (MCI) subjects. 614 cognitively normal, early, and late MCI subjects were selected from the ADNI cohort with a 2-year follow-up. CSF Aβ and tau, 18F-AV45-PET, 18F-FDG-PET, MRI, and cognitive measures along with modified Qrisk2 were evaluated. APOE4 non-carrier, high cardiovascular risk sub-group of early and late MCI and cognitively normal subjects, demonstrated worse biomarker and cognitive profile at baseline and during follow up compared to low cardiovascular risk group. Additionally, similar pattern was also observed in APOE4 carriers. We demonstrated that Qrisk2 and APOE4 were independent predictors of biomarker and clinical progression in AD trajectory. High cardiovascular risk is associated with biomarker changes in APOE4 non-carriers in prodromal AD, which may suggest that treatment of cardiovascular risk is an effective prevention strategy even in APOE4 negative subjects and may influence disease progression independent of amyloid pathology. Demonstration of accelerated neuropathological changes in both APOE4 carriers and non-carriers suggest that focusing on modifiable cardiovascular risk factors is an effective preventative strategy while we eagerly waiting for new treatments.

Do Cardiometabolic Risk Factors Influence Amyloid, Tau, and Neuronal Function in APOE4 Carriers and Non-Carriers in Alzheimer’s Disease Trajectory?

Do Cardiometabolic Risk Factors Influence Amyloid, Tau, and Neuronal Function in APOE4 Carriers and Non-Carriers in Alzheimer’s Disease Trajectory?

An isomorphism between the convolution product and the componentwise sum connected to the D’Arcais numbers and the Ramanujan tau function

Given a commutative ring R with identity, let $$H_R$$ be the set of sequences of elements in R. We investigate a novel isomorphism between $$(H_R, +)$$ and $$(\tilde{H}_R,*)$$ , where $$+$$ is the componentwise sum, $$*$$ is the convolution product (or Cauchy product) and $$\tilde{H}_R$$ the set of sequences starting with $$1_R$$ . We also define a recursive transform over $$H_R$$ that, together to the isomorphism, allows to highlight new relations among some well studied integer sequences. Moreover, these connections allow to introduce a family of polynomials connected to the D’Arcais numbers and the Ramanujan tau function. In this way, we also deduce relations involving the Bell polynomials, the divisor function and the Ramanujan tau function. Finally, we highlight a connection between Cauchy and Dirichlet products.

Dihedral Group, 4-Torsion on an Elliptic Curve, and a Peculiar Eigenform Modulo 4

We work out a non-trivial example of lifting a so-called weak eigenform to a true, characteristic 0 eigenform. The weak eigenform is closely related to Ramanujan's tau function whereas the characteristic 0 eigenform is attached to an elliptic curve defined over ${\mathbb Q}$. We produce the lift by showing that the coefficients of the initial, weak eigenform (almost all) occur as traces of Frobenii in the Galois representation on the 4-torsion of the elliptic curve. The example is remarkable as the initial form is known not to be liftable to any characteristic 0 eigenform of level 1. We use this example as illustrating certain questions that have arisen lately in the theory of modular forms modulo prime powers. We give a brief survey of those questions.

Alzheimer's Disease and Type 2 Diabetes: A Critical Assessment of the Shared Pathological Traits

Alzheimer's disease (AD) and Type 2 Diabetes Mellitus (T2DM) are two of the most prevalent diseases in the elderly population worldwide. A growing body of epidemiological studies suggest that people with T2DM are at a higher risk of developing AD. Likewise, AD brains are less capable of glucose uptake from the surroundings resembling a condition of brain insulin resistance. Pathologically AD is characterized by extracellular plaques of Aβ and intracellular neurofibrillary tangles of hyperphosphorylated tau. T2DM, on the other hand is a metabolic disorder characterized by hyperglycemia and insulin resistance. In this review we have discussed how Insulin resistance in T2DM directly exacerbates Aβ and tau pathologies and elucidated the pathophysiological traits of synaptic dysfunction, inflammation, and autophagic impairments that are common to both diseases and indirectly impact Aβ and tau functions in the neurons. Elucidation of the underlying pathways that connect these two diseases will be immensely valuable for designing novel drug targets for Alzheimer's disease.

The direct and indirect effects of α-synuclein on microtubule stability in the pathogenesis of Parkinson's disease.

Despite decades of research, the mechanism of Parkinson’s disease pathogenesis remains unclear. Studies have focused heavily on the protein α-synuclein, which is the primary component of Lewy bodies, the pathologic inclusions that are the hallmark of Parkinson’s on the cellular level. While the roles of α-synuclein in causing mitochondrial dysfunction and disruptions to the proteasomal system have been well documented, recently, its role in effecting microtubule dynamics has been investigated as a potential source of pathogenicity. Here, we evaluate the evidence for and against the role of α-synuclein in destabilizing microtubules, causing axonal transport deficits and eventually neurodegeneration. We present evidence for a model where α-synuclein has both a direct and indirect effect on microtubule stability. Directly, it may act as a microtubule-associated protein, binding to microtubules and directly effecting their dynamics. Indirectly, it may promote the hyperphosphorylation of the microtubule stabilizing protein, tau, leading to tau aggregation with other microtubule stabilizing proteins, hence indirectly causing microtubule destabilization. This model provides insights into the function of α-synuclein and tau in Parkinson’s disease pathogenesis and raises the possibility that this role that may also be conserved in Alzheimer’s disease.