public 01:09:19

Benoit Charbonneau : Hilbert series and K-polynomials

  -   Colloquium ( 232 Views )

public 01:34:43

Johan Brauer : The Stabilisation of Equilibria in Evolutionary Game Dynamics through Mutation

  -   Probability ( 194 Views )

The multi-population replicator dynamics (RD) can be considered a dynamic approach to the study of multi-player games, where it was shown to be related to Cross-learning, as well as of systems of co-evolving populations. However, not all of its equilibria are Nash equilibria (NE) of the underlying game, and neither convergence to an NE nor convergence in general are guaranteed. Although interior equilibria are guaranteed to be NE, no interior equilibrium can be asymptotically stable in the multi-population RD, resulting, e.g., in cyclic orbits around a single interior NE. We report on our investigation of a new notion of equilibria of RD, called mutation limits, which is based on the inclusion of a naturally arising, simple form of mutation, but is invariant under the specific choice of mutation parameters. We prove the existence of such mutation limits for a large range of games, and consider an interesting subclass, that of attracting mutation limits. Attracting mutation limits are approximated by asymptotically stable equilibria of the (mutation-)perturbed RD, and hence, offer an approximate dynamic solution of the underlying game, especially if the original dynamic has no asymptotically stable equilibria. Therefore, the presence of mutation will indeed stabilise the system in certain cases and make attracting mutation limits near-attainable. Furthermore, the relevance of attracting mutation limits as a game theoretic equilibrium concept is emphasised by the relation of (mutation-)perturbed RD to the Q-learning algorithm in the context of multi-agent reinforcement learning. However, in contrast to the guaranteed existence of mutation limits, attracting mutation limits do not exist in all games, raising the question of their characterization.

public 01:34:58

Alex Blumenthal : Chaotic regimes for random dynamical systems

  -   Probability ( 154 Views )

It is anticipated that chaotic regimes (e.g., strange attractors) arise in a wide variety of dynamical systems, including those arising from the study of ensembles of gas particles and fluid mechanics. However, in most cases the problem of rigorously verifying asymptotic chaotic regimes is notoriously difficult. For volume-preserving systems (e.g., incompressible fluid flow or Hamiltonian systems), these issues are exemplified by coexistence phenomena: even in quite simple models which should be chaotic, e.g. the Chirikov standard map, completely opposite dynamical regimes (elliptic islands vs. hyperbolic sets) can be tangled together in phase space in a convoluted way. Recent developments have indicated, however, that verifying chaos is tractable for systems subjected to a small amount of noise— from the perspective of modeling, this is not so unnatural, as the real world is inherently noisy. In this talk, I will discuss two recent results: (1) a large positive Lyapunov exponent for (extremely small) random perturbations of the Chirikov standard map, and (2) a positive Lyapunov exponent for the Lagrangian flow corresponding to various incompressible stochastic fluids models, including stochastic 2D Navier-Stokes and 3D hyperviscous Navier-Stokes on the periodic box. The work in this talk is joint with Jacob Bedrossian, Samuel Punshon-Smith, Jinxin Xue and Lai-Sang Young.

public 01:34:51

Christopher Cornwell : TBA

  -   Graduate/Faculty Seminar ( 129 Views )


public 01:14:48

Steven Baer : Multiscale Modeling of Neural Subcircuits and Feedback Mechanisms in the Outer Plexiform Layer of the Retina

  -   Mathematical Biology ( 127 Views )

Visual processing begins in the outer plexiform layer of the retina, where
bipolar, horizontal, and photoreceptor cells interact. In vertebrates, the
onset of dim backgrounds can enhance small spot flicker responses of
retinal horizontal cells. This flicker response is called background-
induced flicker enhancement. The underlying mechanism for the feedback
is unclear but competing hypotheses have been proposed. One is the GABA
hypothesis, which states that the inhibitory neurotransmitter GABA,
released from horizontal cells, mediates the feedback by blocking calcium
channels. Another is the ephaptic hypothesis, which contends that calcium
entry is regulated by changes in the electrical potential within the
intersynaptic space between cones and horizontal cells. In this study, a
continuum spine model of cone-horizontal cell synaptic circuitry is
formulated. The model captures two spatial scales - the scale of an
individual synapse and the scale of the receptive field involving hundreds
to thousands of synapses. We show that the ephaptic mechanism produces
reasonable qualitative agreement with the temporal dynamics exhibited by
flicker enhancement experiments. We find that although GABA produces
enhancement, this mechanism alone is insufficient to reproduce the
experimental results. We view this multiscale continuum approach as a
first step in formulating a multi-layer mathematical model of retinal
circuitry, which would include the other ‘brain nuclei’ within the retina:
the inner plexiform layer where bipolar, amacrine, interplexiform, and
ganglion cells interact.

public 01:34:46

Gregory Herschlag : Optimal reservoir conditions for material extraction across pumping and porous channels

  -   Mathematical Biology ( 113 Views )

In this talk, I will discuss a new result in fluid flows through channels with permeable membranes with simple pumping dynamics. Fluid will be exchanged and metabolized in a simple reservoir and I will demonstrate the existence of optimal reservoir properties that may either maximize or minimized the amount of fluid being extracted across the channel walls. The biological relevance of this work may be seen by noting that all living organisms of a sufficient size rely on complex systems of tubular networks to efficiently collect, transport and distribute nutrients or waste. These networks exchange material with the interstitium via embedded channels leading to effective permeabilities across the wall separating the channel interior from the interstitium. In many invertebrates, for example, respiratory systems are made of complex tracheal systems that branch out through the entire body allowing for passive exchange of oxygen and carbon dioxide. In many of these systems, certain animals utilize various pumping mechanisms that alter the flow of the air or fluid being transported. Although the net effect of pumping of the averaged rates of fluid flow through the channel is typically well understood, it is still a largely open problem to understand how, and in what circumstances, pumping enables and enhances the exchange of material across channel walls. It has been demonstrated experimentally, for example, that when certain insects flap their wings, compression of the trachea allow for more efficient oxygen extraction, however it is unclear if this pumping is optimized for flight, oxygen uptake or neither, and understanding this problem quantitatively will shed insight on this biological process. Many of these interesting scenarios occur at low Reynolds number and this regime will be the focus of the presentation.