Research Summary
Research Statement (January 2017):
[PDF]
Research areas
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
Multineuron 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 spiketrain correlation for neural oscillators with common drive. A.K. Barreiro, E.T. SheaBrown and E.L. Thilo (2010). Physical Review E, Volume 81 (1), No. 011916 [PDF]

Acurrent and type I/type II transition determine collective spiking from common input. A.K. Barreiro, E.L. Thilo and E.T. SheaBrown (2012). Journal of Neurophysiology, Volume 108, p. 16311645 [PDF]

When do microcircuits produce beyondpairwise 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 wellmodeled by pairwise maximum entropy methods? Talk given at workshop on Linking neural dynamics and coding: correlations, synchrony, and information, BIRS, October 2010.
 Impact of singleneuron dynamics on transfer of correlations from common input, Talk in Workshop on Stochastic Neural Dynamics, CNS*2015, July 2015.
 Impact of singleneuron 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)
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 nonnormality 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 nonnormal operators;
improved signaltonoise ratio,
timevarying patterns of input,
and the ability to operate without fine tuning of eigenvalues.
In a recent paper we demonstrate two other beneficial aspects of nonnormality 
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 rankperturbed 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 tradeoff 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. 321329 (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: 9780955894022 (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. SheaBrown. Journal of Neurophysiology, 109, pp. 2542–2559
[PDF]

Selfadjoint eigenvalue problems with low rank perturbations.
Anastasio, T.J., A.K. Barreiro, and J.C. Bronski (2017).
Submitted (arXiv:nohere)
[PDF]
Abstracts/Talks
We formulate the problem of finding equilibrium
configurations of Npoint vortices in the plane in
terms of a gradient flow on the smallest singular
value of a skewsymmetric 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. 16871702.
[PDF]
Abstracts/Talks
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 longshore homogeneity in water depth and wave
forcing, and a quasisteady state in time. These
assumptions lead to a onedimensional 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 wavebreaking. 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 bartype
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
twodimensional 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]

Wavedriven vortex dynamics in the surf zone. A.K. Barreiro.
PhD Thesis, New York University, 2006. [PDF] [GZIP (recommended)]
Abstracts/Talks/Videos
 Wavedriven vortex dynamics in the nearshore 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 shallowwater turbulence videos to be posted here)