Department of Mathematical Sciences

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Chaos and Robustness in a Single Family of Genetic Oscillatory Networks
(2014-03) Fu, Daniel; Tan, Patrick; Kuznetsov, Alexey; Molkov, Yaroslav I
Genetic oscillatory networks can be mathematically modeled with delay differential equations (DDEs). Interpreting genetic networks with DDEs gives a more intuitive understanding from a biological standpoint. However, it presents a problem mathematically, for DDEs are by construction infinitely-dimensional and thus cannot be analyzed using methods common for systems of ordinary differential equations (ODEs). In our study, we address this problem by developing a method for reducing infinitely-dimensional DDEs to two- and three-dimensional systems of ODEs. We find that the three-dimensional reductions provide qualitative improvements over the two-dimensional reductions. We find that the reducibility of a DDE corresponds to its robustness. For non-robust DDEs that exhibit high-dimensional dynamics, we calculate analytic dimension lines to predict the dependence of the DDEs’ correlation dimension on parameters. From these lines, we deduce that the correlation dimension of non-robust DDEs grows linearly with the delay. On the other hand, for robust DDEs, we find that the period of oscillation grows linearly with delay. We find that DDEs with exclusively negative feedback are robust, whereas DDEs with feedback that changes its sign are not robust. We find that non-saturable degradation damps oscillations and narrows the range of parameter values for which oscillations exist. Finally, we deduce that natural genetic oscillators with highly-regular periods likely have solely negative feedback.
A Note on Gluing Dirac Type Operators on a Mirror Quantum Two-Sphere
(2014-03) Klimek, Slawomir; McBride, Matt
The goal of this paper is to introduce a class of operators, which we call quantum Dirac type operators on a noncommutative sphere, by a gluing construction from copies of noncommutative disks, subject to an appropriate local boundary condition. We show that the resulting operators have compact resolvents, and so they are elliptic operators.
A Closed-Loop Model of the Respiratory System: Focus on Hypercapnia and Active Expiration
(2014-10) Molkov, Yaroslav I; Shevtsova, Natalia A; Park, Choongseok; Ben-Tal, Alona; Smith, Jeffrey C; Rubin, Jonathan E; Rybak, Ilya A
Breathing is a vital process providing the exchange of gases between the lungs and atmosphere. During quiet breathing, pumping air from the lungs is mostly performed by contraction of the diaphragm during inspiration, and muscle contraction during expiration does not play a significant role in ventilation. In contrast, during intense exercise or severe hypercapnia forced or active expiration occurs in which the abdominal “expiratory” muscles become actively involved in breathing. The mechanisms of this transition remain unknown. To study these mechanisms, we developed a computational model of the closed-loop respiratory system that describes the brainstem respiratory network controlling the pulmonary subsystem representing lung biomechanics and gas (O2 and CO2) exchange and transport. The lung subsystem provides two types of feedback to the neural subsystem: a mechanical one from pulmonary stretch receptors and a chemical one from central chemoreceptors. The neural component of the model simulates the respiratory network that includes several interacting respiratory neuron types within the Bötzinger and pre-Bötzinger complexes, as well as the retrotrapezoid nucleus/parafacial respiratory group (RTN/pFRG) representing the central chemoreception module targeted by chemical feedback. The RTN/pFRG compartment contains an independent neural generator that is activated at an increased CO2 level and controls the abdominal motor output. The lung volume is controlled by two pumps, a major one driven by the diaphragm and an additional one activated by abdominal muscles and involved in active expiration. The model represents the first attempt to model the transition from quiet breathing to breathing with active expiration. The model suggests that the closed-loop respiratory control system switches to active expiration via a quantal acceleration of expiratory activity, when increases in breathing rate and phrenic amplitude no longer provide sufficient ventilation. The model can be used for simulation of closed-loop control of breathing under different conditions including respiratory disorders.
Superstable manifolds of invariant circles and codimension-one Böttcher functions
(2015-02) Kaschner, Scott R; Roeder, Roland K W
Let f : X ⇢ X be a dominant meromorphic self-map, where X is a compact, connected complex manifold of dimension n>1. Suppose that there is an embedded copy of P1 that is invariant under f, with f holomorphic and transversally superattracting with degree a in some neighborhood. Suppose that f restricted to this line is given by z↦zb, with resulting invariant circle S. We prove that if a≥b, then the local stable manifold Wsloc(S) is real analytic. In fact, we state and prove a suitable localized version that can be useful in wider contexts. We then show that the condition a≥b cannot be relaxed without adding additional hypotheses by presenting two examples with a
Mechanisms of Left-Right Coordination in Mammalian Locomotor Pattern Generation Circuits: A Mathematical Modeling View
(PLoS, 2015-05) Molkov, Yaroslav I.; Bacak, Bartholomew J.; Talpalar, Adolfo E.; Rybak, Ilya A.; Department of Mathematical Sciences, School of Science
The locomotor gait in limbed animals is defined by the left-right leg coordination and locomotor speed. Coordination between left and right neural activities in the spinal cord controlling left and right legs is provided by commissural interneurons (CINs). Several CIN types have been genetically identified, including the excitatory V3 and excitatory and inhibitory V0 types. Recent studies demonstrated that genetic elimination of all V0 CINs caused switching from a normal left-right alternating activity to a left-right synchronized “hopping” pattern. Furthermore, ablation of only the inhibitory V0 CINs (V0D subtype) resulted in a lack of left-right alternation at low locomotor frequencies and retaining this alternation at high frequencies, whereas selective ablation of the excitatory V0 neurons (V0V subtype) maintained the left–right alternation at low frequencies and switched to a hopping pattern at high frequencies. To analyze these findings, we developed a simplified mathematical model of neural circuits consisting of four pacemaker neurons representing left and right, flexor and extensor rhythm-generating centers interacting via commissural pathways representing V3, V0D, and V0V CINs. The locomotor frequency was controlled by a parameter defining the excitation of neurons and commissural pathways mimicking the effects of N-methyl-D-aspartate on locomotor frequency in isolated rodent spinal cord preparations. The model demonstrated a typical left-right alternating pattern under control conditions, switching to a hopping activity at any frequency after removing both V0 connections, a synchronized pattern at low frequencies with alternation at high frequencies after removing only V0D connections, and an alternating pattern at low frequencies with hopping at high frequencies after removing only V0V connections. We used bifurcation theory and fast-slow decomposition methods to analyze network behavior in the above regimes and transitions between them. The model reproduced, and suggested explanation for, a series of experimental phenomena and generated predictions available for experimental testing.
Universally optimal designs for two interference models
(2015) Zheng, Wei; Department of Mathematical Sciences, School of Science
A systematic study is carried out regarding universally optimal designs under the interference model, previously investigated by Kunert and Martin [Ann. Statist. 28 (2000) 1728–1742] and Kunert and Mersmann [J. Statist. Plann. Inference 141 (2011) 1623–1632]. Parallel results are also provided for the undirectional interference model, where the left and right neighbor effects are equal. It is further shown that the efficiency of any design under the latter model is at least its efficiency under the former model. Designs universally optimal for both models are also identified. Most importantly, this paper provides Kushner’s type linear equations system as a necessary and sufficient condition for a design to be universally optimal. This result is novel for models with at least two sets of treatment-related nuisance parameters, which are left and right neighbor effects here. It sheds light on other models in deriving asymmetric optimal or efficient designs.
Continuous Statistical Models: With or Without Truncation Parameters?
(Springer, 2015) Vancak, V.; Goldberg, Y.; Bar-Lev, S. K.; Boukai, Benzion; Department of Mathematical Sciences, School of Science
Lifetime data are usually assumed to stem from a continuous distribution supported on [0, b) for some b ≤ ∞. The continuity assumption implies that the support of the distribution does not have atom points, particularly not at 0. Accordingly, it seems reasonable that with an accurate measurement tool all data observations will be positive. This suggests that the true support may be truncated from the left. In this work we investigate the effects of adding a left truncation parameter to a continuous lifetime data statistical model. We consider two main settings: right truncation parametric models with possible left truncation, and exponential family models with possible left truncation. We analyze the performance of some optimal estimators constructed under the assumption of no left truncation when left truncation is present, and vice versa. We investigate both asymptotic and finite-sample behavior of the estimators. We show that when left truncation is not assumed but is, in fact present, the estimators have a constant bias term, and therefore will result in inaccurate and inefficient estimation. We also show that assuming left truncation where actually there is none, typically does not result in substantial inefficiency, and some estimators in this case are asymptotically unbiased and efficient.
COMPUTING DYNAMICAL DEGREES OF RATIONAL MAPS ON MODULI SPACE
(Cambridge, 2015) Koch, Sarah; Roeder, Roland K.; Department of Mathematical Sciences, School of Science
The dynamical degrees of a rational map f:X⇢X are fundamental invariants describing the rate of growth of the action of iterates of f on the cohomology of X. When f has non-empty indeterminacy set, these quantities can be very difficult to determine. We study rational maps f:XN⇢XN, where XN is isomorphic to the Deligne–Mumford compactification M¯¯¯¯0,N+3. We exploit the stratified structure of XN to provide new examples of rational maps, in arbitrary dimension, for which the action on cohomology behaves functorially under iteration. From this, all dynamical degrees can be readily computed (given enough book-keeping and computing time). In this paper, we explicitly compute all of the dynamical degrees for all such maps f:XN⇢XN, where dim(XN)≤3 and the first dynamical degrees for the mappings where dim(XN)≤5. These examples naturally arise in the setting of Thurston’s topological characterization of rational maps.
Invariants of the vacuum module associated with the Lie superalgebra gl(1|1)
(IOP, 2015-07) Molev, Alexander; Mukhin, Evgeny E.; Department of Mathematical Sciences, School of Science
We describe the algebra of invariants of the vacuum module associated with an aﬃnization of the Lie superalgebra gl(1|1). We give a formula for its Hilbert–Poincare´ series in a fermionic (cancellation-free) form which turns out to coincide with the generating function of the plane partitions over the (1, 1)-hook. Our arguments are based on a super version of the Beilinson–Drinfeld–Raı¨s–Tauvel theorem which we prove by producing an explicit basis of invariants of the symmetric algebra of polynomial currents associated with gl(1|1). We identify the invariants with aﬃne supersymmetric polynomials via a version of the Chevalley theorem.
NUTTALL’S THEOREM WITH ANALYTIC WEIGHTS ON ALGEBRAIC S-CONTOURS
(Elsevier, 2015-02) Yattselev, Maxim L.; Department of Mathematical Sciences, School of Science
Given a function f holomorphic at infinity, the nth diagonal Padé approximant to f, denoted by [n/n]f, is a rational function of type (n,n) that has the highest order of contact with f at infinity. Nuttall’s theorem provides an asymptotic formula for the error of approximation f−[n/n]f in the case where f is the Cauchy integral of a smooth density with respect to the arcsine distribution on [−1,1]. In this note, Nuttall’s theorem is extended to Cauchy integrals of analytic densities on the so-called algebraic S-contours (in the sense of Nuttall and Stahl).

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