Research Summary

Research Statement (January 2017): [PDF]

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Research areas

Symmetries in Random Networks

Transfer of noisy, correlated signals in neural networks

Neural integrators: robustness and non-normality

Old stuff: Fluid dynamics

• Novel methods to find relative equilibria of point vortices

• Wave-driven vortex dynamics on barred beaches

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Symmetries in Random Networks

Publications

• Symmetries constrain dynamics in a family of balanced neural networks. A.K. Barreiro, J.N. Kutz, and E. Shlizerman (2016). Submitted (arXiv:1602.05092v2) [PDF]

Abstracts/Talks

• Symmetries constrain the transition to heterogeneous chaos in balanced networks: poster at CNS*2015 [PDF] Abstract

Transfer of noisy, correlated signals in neural networks

Multi-neuron spiking activity is known to be highly correlated. The resulting implications for stimulus coding are not obvious; correlations in a population code can enhance or diminish accuracy of stimulus representation depending on correlation structure. In addition, the development of this structure and its transfer between populations is highly nonlinear and nonintuitive, depending on intrinsic cell dynamics and stimulus statistics: even the (seemingly) simple question of how well a pair of neurons transmit a correlated signal arising from common synaptic input is challenging to answer.

More recently, I've been trying to understand how correlated signals propagate in recurrent networks, and how to use correlations to constrain neural networks.

Publications

• Timescales of spike-train correlation for neural oscillators with common drive. A.K. Barreiro, E.T. Shea-Brown and E.L. Thilo (2010). Physical Review E, Volume 81 (1), No. 011916 [PDF]
• A-current and type I/type II transition determine collective spiking from common input. A.K. Barreiro, E.L. Thilo and E.T. Shea-Brown (2012). Journal of Neurophysiology, Volume 108, p. 1631-1645 [PDF]
• When do microcircuits produce beyond-pairwise correlations? A.K. Barreiro, J. Gjorgieva, F. Rieke and E.T. Shea Brown (2014). Frontiers in Computational Neuroscience, Volume 8 (10) [PDF]
• When do correlations increase with firing rates in recurrent networks? A.K. Barreiro and C. Ly (2017). PLoS Computational Biology, Vol. 13, No. 4. DOI: 10.1371/journal.pcbi.1005506. [PDF]
• A practical approximation method for firing rate models of coupled neural networks with correlated inputs. A.K. Barreiro and C. Ly (2017). Submitted (arXiv:1702.03474) [PDF]
• A Theoretical Framework for Analyzing Coupled Neuronal Networks: Application to the Olfactory System. A.K. Barreiro, S.H. Gautam, W.L. Shew and C. Ly (2017). Submitted (arXiv:1702.03543) [PDF]

Abstracts/Talks

• When are microcircuits well-modeled by pairwise maximum entropy methods? Talk given at workshop on Linking neural dynamics and coding: correlations, synchrony, and information, BIRS, October 2010.
• Impact of single-neuron dynamics on transfer of correlations from common input, Talk in Workshop on Stochastic Neural Dynamics, CNS*2015, July 2015.
• Impact of single-neuron dynamics on transfer of correlations from common input, Modeling of Synchronous and Correlated Behavior in Neuronal Networks, SIAM Annual meeting, July 2016. (same name, but some different stuff)

Neural integrators: robustness and non-normality

The operation of maintaining persistent activity beyond the actual presence of a stimulus -- for example, in working memory -- can be conceptualized as mathematical integration. While integration is very easy to perform in calculus, there is significant debate about how it is performed by neural circuits. We have considered two issues that arise in recurrent feedback integrators: how to make circuits robust to noise and mistuning, and the effects of linear non-normality on function.

Most integrator models in the literature are normal; that is, if $A$ is a matrix governing the linear dynamics of the model, then $A^T A = A A^T$. Many realistic aspects of integrator function can be simulated by expanding potential circuits to include non-normal operators; improved signal-to-noise ratio, time-varying patterns of input, and the ability to operate without fine tuning of eigenvalues. In a recent paper we demonstrate two other beneficial aspects of non-normality --- enhanced plasticity and the ability to generate oscillations --- in a model of the oculomotor neural integrator. In the process, we introduce a novel method of spectral analysis for rank-perturbed operators that may be applicable to other problems in applied math.

Most integrators based on recurrent feedback are also extremely vulnerable to mistuning of parameters. If the weights are mistuned even slightly, then the system cannot do its job within biological constraints. Several authors have proposed integrators that overcome this problem through the inclusion of bistable, hysteretic components (Koulakov et al., Nat Neuro, 2002, Goldman et al., Cerebral Cortex, 2003). These models, however, are insensitive to small inputs that are too small to push the bistable system from one state to another. This introduces an inherent trade-off between robustness and accuracy; an integrator can survive mistuning only at the cost of ignoring potentially important stimuli.

Publications

• Bifurcation theory explains waveform variability in a congenital eye movement disorder. A.K. Barreiro, J.C. Bronski and T.J. Anastasio. Journal of Computational Neuroscience, 26, pp. 321-329 (2009). [PDF]
• Mechanisms of neural integration: recent results and relevance to nystagmus modeling. A.K. Barreiro. In: The Challenge of Nystagmus (Proceedings of the Second International Research Workshop on Nystagmus 2009). Eds. C.M. Harris, I. Gottlob, J. Sanders. ISBN: 978-0-9558940-2-2 (2012). [PDF]
• Neural integrators for decision making: A favorable tradeoff between robustness and sensitivity N.H. Cain, A.K. Barreiro, M.N. Shadlen and E.T. Shea-Brown. Journal of Neurophysiology, 109, pp. 2542–2559 [PDF]
• Self-adjoint eigenvalue problems with low rank perturbations. Anastasio, T.J., A.K. Barreiro, and J.C. Bronski (2017). Submitted (arXiv:nohere) [PDF]

Abstracts/Talks

• The effect of robust integrator dynamics on decision-making performance CNS 2010 Abstract (presented by Nick Cain)

Novel methods to find relative equilibria of point vortices

We formulate the problem of finding equilibrium configurations of N-point vortices in the plane in terms of a gradient flow on the smallest singular value of a skew-symmetric matrix M whose nullspace structure determines the (real) strengths, rotational frequency, and translational velocity of the configuration. This is to our knowledge the first method for finding large (N ~ 50), asymmetric configurations of point vortices. We also consider the constrained problem of quantized point vortex strengths.

Publications

• Spectral gradient flow and equilibrium configurations of point vortices. A.K. Barreiro, J.C. Bronski and P.K. Newton (2010). Proceedings of the Royal Society of London A, Vol. 466, Issue 2118, pp. 1687-1702. [PDF]

Abstracts/Talks

• Spectral Gradient Flow and Equilibrium Configurations of Point Vortices SIAM Conference on Applications of Dynamical Systems, May 2009.

Wave-driven vortex dynamics on barred beaches

It has long been known that obliquely incident waves breaking on the beach will cause an alongshore current running in the direction of the waves, due to momentum transfer from the breaking waves. Typical prediction models for such currents assume long-shore homogeneity in water depth and wave forcing, and a quasi-steady state in time. These assumptions lead to a one-dimensional momentum balance between the dissipation from wave breaking, friction, turbulent mixing and other processes. The resulting prediction includes a strong current at the location of strongest wave-breaking. The record of such models is not very good on barred beaches, on which an offshore minimum in water depth occurs (as above a sandbar). Typical observations show a strong current in the bar trough where little wave breaking is taking place. Buhler and Jacobson (JFM, 2001) suggest a mechanism based on vortex dynamics, which operates on bar-type topography where there is some alongshore inhomogeneity in either topography or wave forcing, to explain the discrepancy between observations and previous models. We develop a numerical model to study this mechanism and find that it offers a working mechanism for current dislocation. We also show, using recent results on the scaling of shallow water turbulence with quadratic drag, that the physical scales of the beach do not permit two-dimensional turbulence.

Publications

• Longshore current dislocation on barred beaches. A.K. Barreiro and O. Buhler.Journal of Geophysical Research, 113, C12004, doi:10.1029/2007JC004661. [PDF]
• Wave-driven vortex dynamics in the surf zone. A.K. Barreiro. PhD Thesis, New York University, 2006. [PDF] [GZIP (recommended)]

Abstracts/Talks/Videos

• Wave-driven vortex dynamics in the near-shore region Talk given at Physical Oceanography Dissertation Symposium 2006. (Warning: I imported this from and old version of Powerpoint: some of the equations have some "divide by" symbols in them which you can ignore!)
• (Some shallow-water turbulence videos to be posted here)