Andrea K. Barreiro

Time scales of spike-train correlation for neural oscillators with common drive

We examine the effect of the phase-resetting curve on the transfer of correlated input signals into correlated output spikes in a class of neural models receiving noisy superthreshold stimulation. We use linear-response theory to approximate the spike correlation coefficient in terms of moments of the associated exit time problem and contrast the results for type I vs type II models and across the different time scales over which spike correlations can be assessed. We find that, on long time scales, type I oscillators transfer correlations much more efficiently than type II oscillators. On short time scales this trend reverses, with the relative efficiency switching at a time scale that depends on the mean and standard deviation of input currents. This switch occurs over time scales that could be exploited by downstream circuits.

Information theoretic approaches to understanding circuit function.

The analysis of stimulus/response patterns using information theoretic approaches requires the full probability distribution of stimuli and response. Recent progress in using information-based tools to understand circuit function has advanced understanding of neural coding at the single cell and population level. In advances over traditional reverse correlation approaches, the determination of receptive fields using information as a metric has allowed novel insights into stimulus representation and transformation. The application of maximum entropy methods to population codes has opened a rich exploration of the internal structure of these codes, revealing stimulus-driven functional connectivity. We speculate about the prospects and limitations of information as a general tool for dissecting neural circuits and relating their structure and function.

Neural integrators for decision making: a favorable tradeoff between robustness and sensitivity

A key step in many perceptual decision tasks is the integration of sensory inputs over time, but a fundamental questions remain about how this is accomplished in neural circuits. One possibility is to balance decay modes of membranes and synapses with recurrent excitation. To allow integration over long timescales, however, this balance must be exceedingly precise. The need for fine tuning can be overcome via a “robust integrator” mechanism in which momentary inputs must be above a preset limit to be registered by the circuit. The degree of this limiting embodies a tradeoff between sensitivity to the input stream and robustness against parameter mistuning. Here, we analyze the consequences of this tradeoff for decision-making performance. For concreteness, we focus on the well-studied random dot motion discrimination task and constrain stimulus parameters by experimental data. We show that mistuning feedback in an integrator circuit decreases decision performance but that the robust integrator mechanism can limit this loss. Intriguingly, even for perfectly tuned circuits with no immediate need for a robustness mechanism, including one often does not impose a substantial penalty for decision-making performance. The implication is that robust integrators may be well suited to subserve the basic function of evidence integration in many cognitive tasks. We develop these ideas using simulations of coupled neural units and the mathematics of sequential analysis.

When do microcircuits produce beyond-pairwise correlations?

Describing the collective activity of neural populations is a daunting task. Recent empirical studies in retina, however, suggest a vast simplification in how multi-neuron spiking occurs: the activity patterns of retinal ganglion cell (RGC) populations under some conditions are nearly completely captured by pairwise interactions among neurons. In other circumstances, higher-order statistics are required and appear to be shaped by input statistics and intrinsic circuit mechanisms. Here, we study the emergence of higher-order interactions in a model of the RGC circuit in which correlations are generated by common input. We quantify the impact of higher-order interactions by comparing the responses of mechanistic circuit models vs. “null” descriptions in which all higher-than-pairwise correlations have been accounted for by lower order statistics; these are known as pairwise maximum entropy (PME) models. We find that over a broad range of stimuli, output spiking patterns are surprisingly well captured by the pairwise model. To understand this finding, we study an analytically tractable simplification of the RGC model. We find that in the simplified model, bimodal input signals produce larger deviations from pairwise predictions than unimodal inputs. The characteristic light filtering properties of the upstream RGC circuitry suppress bimodality in light stimuli, thus removing a powerful source of higher-order interactions. This provides a novel explanation for the surprising empirical success of pairwise models.

Bifurcation theory explains waveform variability in a congenital eye movement disorder

In dynamical systems, configurations that permit flexible control are also prone to undesirable behavior. We study a bilateral model of the oculomotor pre-motor network that conforms with the neuroanatomical constraint that brainstem neurons project to cerebellar Purkinje cells on both sides, but Purkinje cells project back to brainstem neurons on the same side only. Bifurcation analysis reveals that this network asymmetry enables flexible control by the cerebellum of brainstem network dynamics, but small changes in connection pattern or strength lead to behavior that is unstable, oscillatory, or both. The model produces the full range of waveform types associated with the hereditary eye movement disorder know as congenital nystagmus, and is consistent with findings linking the disorder with abnormal connectivity or limited plasticity in the cerebellum.

Spectral gradient flow and equilibrium configurations 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. A generic configuration gives rise to a matrix with empty nullspace, and hence is not a relative equilibrium for any choice of vortex strengths. We formulate the problem as a gradient flow in the space of square covariance matrices MTM. The evolution equation for drives the configuration to one with a real nullspace, establishing the existence of an equilibrium for vortex strengths that are elements of the nullspace of the matrix. We formulate both the unconstrained gradient flow problem where the point vortex strengths are determined a posteriori by the nullspace of M and the constrained problem where the point vortex strengths are chosen a priori and one seeks configurations for which those strengths are elements of the nullspace.

When do correlations increase with firing rates in recurrent networks

A central question in neuroscience is to understand how noisy firing patterns are used to transmit information. Because neural spiking is noisy, spiking patterns are often quantified via pairwise correlations, or the probability that two cells will spike coincidentally, above and beyond their baseline firing rate. One observation frequently made in experiments, is that correlations can increase systematically with firing rate. Theoretical studies have determined that stimulus-dependent correlations that increase with firing rate can have beneficial effects on information coding; however, we still have an incomplete understanding of what circuit mechanisms do, or do not, produce this correlation-firing rate relationship. Here, we studied the relationship between pairwise correlations and firing rates in recurrently coupled excitatory-inhibitory spiking networks with conductance-based synapses. We found that with stronger excitatory coupling, a positive relationship emerged between pairwise correlations and firing rates. To explain these findings, we used linear response theory to predict the full correlation matrix and to decompose correlations in terms of graph motifs. We then used this decomposition to explain why covariation of correlations with firing rate—a relationship previously explained in feedforward networks driven by correlated input—emerges in some recurrent networks but not in others. Furthermore, when correlations covary with firing rate, this relationship is reflected in low-rank structure in the correlation matrix.

Data Assimilation in the Detection of Vortices

We develop new algorithms for target detection in multi-sensor environments. These methods are applied to study point vortex motion based on Lagrangian tracer information. First we solve analytically the nonlinear filtering problem for the special case of equal strength vortices. Recently developed methods, the particle filters that are based on importance sampling Monte Carlo simulations, are used for the detection of vortices in the the general case. Unlike the well-known extended Kalman filter, it is applicable to highly nonlinear systems with non-Gaussian uncertainties.

A geometric method for eigenvalue problems with low-rank perturbations

We consider the problem of finding the spectrum of an operator taking the form of a low-rank (rank one or two) non-normal perturbation of a well-understood operator, motivated by a number of problems of applied interest which take this form. We use the fact that the system is a low-rank perturbation of a solved problem, together with a simple idea of classical differential geometry (the envelope of a family of curves) to completely analyse the spectrum. We use these techniques to analyse three problems of this form: a model of the oculomotor integrator due to Anastasio & Gad (2007 J. Comput. Neurosci. 22, 239–254. (doi:10.1007/s10827-006-0010-x)), a continuum integrator model, and a non-local model of phase separation due to Rubinstein & Sternberg (1992 IMA J. Appl. Math. 48, 249–264. (doi:10.1093/imamat/48.3.249)).

A theoretical framework for analyzing coupled neuronal networks: Application to the olfactory system

Determining how synaptic coupling within and between regions is modulated during sensory processing is an important topic in neuroscience. Electrophysiological recordings provide detailed information about neural spiking but have traditionally been confined to a particular region or layer of cortex. Here we develop new theoretical methods to study interactions between and within two brain regions, based on experimental measurements of spiking activity simultaneously recorded from the two regions. By systematically comparing experimentally-obtained spiking statistics to (efficiently computed) model spike rate statistics, we identify regions in model parameter space that are consistent with the experimental data. We apply our new technique to dual micro-electrode array in vivo recordings from two distinct regions: olfactory bulb (OB) and anterior piriform cortex (PC). Our analysis predicts that: i) inhibition within the afferent region (OB) has to be weaker than the inhibition within PC, ii) excitation from PC to OB is generally stronger than excitation from OB to PC, iii) excitation from PC to OB and inhibition within PC have to both be relatively strong compared to presynaptic inputs from OB. These predictions are validated in a spiking neural network model of the OB–PC pathway that satisfies the many constraints from our experimental data. We find when the derived relationships are violated, the spiking statistics no longer satisfy the constraints from the data. In principle this modeling framework can be adapted to other systems and be used to investigate relationships between other neural attributes besides network connection strengths. Thus, this work can serve as a guide to further investigations into the relationships of various neural attributes within and across different regions during sensory processing.