Yoram Burak

Accurate path integration in continuous attractor network models of grid cells

Grid cells in the rat entorhinal cortex display strikingly regular firing responses to the animals position in 2-D space and have been hypothesized to form the neural substrate for dead-reckoning. However, errors accumulate rapidly when velocity inputs are integrated in existing models of grid cell activity. To produce grid-cell-like responses, these models would require frequent resets triggered by external sensory cues. Such inadequacies, shared by various models, cast doubt on the dead-reckoning potential of the grid cell system. Here we focus on the question of accurate path integration, specifically in continuous attractor models of grid cell activity. We show, in contrast to previous models, that continuous attractor models can generate regular triangular grid responses, based on inputs that encode only the rats velocity and heading direction. We consider the role of the network boundary in the integration performance of the network and show that both periodic and aperiodic networks are capable of accurate path integration, despite important differences in their attractor manifolds. We quantify the rate at which errors in the velocity integration accumulate as a function of network size and intrinsic noise within the network. With a plausible range of parameters and the inclusion of spike variability, our model networks can accurately integrate velocity inputs over a maximum of ∼10–100 meters and ∼1–10 minutes. These findings form a proof-of-concept that continuous attractor dynamics may underlie velocity integration in the dorsolateral medial entorhinal cortex. The simulations also generate pertinent upper bounds on the accuracy of integration that may be achieved by continuous attractor dynamics in the grid cell network. We suggest experiments to test the continuous attractor model and differentiate it from models in which single cells establish their responses independently of each other.

What Grid Cells Convey about Rat Location

We characterize the relationship between the simultaneously recorded quantities of rodent grid cell firing and the position of the rat. The formalization reveals various properties of grid cell activity when considered as a neural code for representing and updating estimates of the rats location. We show that, although the spatially periodic response of grid cells appears wasteful, the code is fully combinatorial in capacity. The resulting range for unambiguous position representation is vastly greater than the ≈1–10 m periods of individual lattices, allowing for unique high-resolution position specification over the behavioral foraging ranges of rats, with excess capacity that could be used for error correction. Next, we show that the merits of the grid cell code for position representation extend well beyond capacity and include arithmetic properties that facilitate position updating. We conclude by considering the numerous implications, for downstream readouts and experimental tests, of the properties of the grid cell code.

The Basal Ganglia Is Necessary for Learning Spectral, but Not Temporal, Features of Birdsong

Executing a motor skill requires the brain to control which muscles to activate at what times. How these aspects of control-motor implementation and timing-are acquired, and whether the learning processes underlying them differ, is not well understood. To address this, we used a reinforcement learning paradigm to independently manipulate both spectral and temporal features of birdsong, a complex learned motor sequence, while recording and perturbing activity in underlying circuits. Our results uncovered a striking dissociation in how neural circuits underlie learning in the two domains. The basal ganglia was required for modifying spectral, but not temporal, structure. This functional dissociation extended to the descending motor pathway, where recordings from a premotor cortex analog nucleus reflected changes to temporal, but not spectral, structure. Our results reveal a strategy in which the nervous system employs different and largely independent circuits to learn distinct aspects of a motor skill.

Onset of DNA Aggregation in Presence of Monovalent and Multivalent Counterions

We address theoretically aggregation of DNA segments by multivalent polyamines such as spermine and spermidine. In experiments, the aggregation occurs above a certain threshold concentration of multivalent ions. We demonstrate that the dependence of this threshold on the concentration of DNA has a simple form. When the DNA concentration c(DNA) is smaller than the monovalent salt concentration, the threshold multivalent ion concentration depends linearly on c(DNA), having the form alphac(DNA) + beta. The coefficients alpha and beta are related to the density profile of multivalent counterions around isolated DNA chains, at the onset of their aggregation. This analysis agrees extremely well with recent detailed measurements on DNA aggregation in the presence of spermine. From the fit to the experimental data, the number of condensed multivalent counterions per DNA chain can be deduced. A few other conclusions can then be reached: 1), the number of condensed spermine ions at the onset of aggregation decreases with the addition of monovalent salt; 2), the Poisson-Boltzmann theory overestimates the number of condensed multivalent ions at high monovalent salt concentrations; and 3), our analysis of the data indicates that the DNA charge is not overcompensated by spermine at the onset of aggregation.

Grid cells: The position code, neural network models of activity, and the problem of learning

We review progress on the modeling and theoretical fronts in the quest to unravel the computational properties of the grid cell code and to explain the mechanisms underlying grid cell dynamics. The goals of the review are to outline a coherent framework for understanding the dynamics of grid cells and their representation of space; to critically present and draw contrasts between recurrent network models of grid cells based on continuous attractor dynamics and independent‐neuron models based on temporal interference; and to suggest open questions for experiment and theory.

Fundamental limits on persistent activity in networks of noisy neurons

Neural noise limits the fidelity of representations in the brain. This limitation has been extensively analyzed for sensory coding. However, in short-term memory and integrator networks, where noise accumulates and can play an even more prominent role, much less is known about how neural noise interacts with neural and network parameters to determine the accuracy of the computation. Here we analytically derive how the stored memory in continuous attractor networks of probabilistically spiking neurons will degrade over time through diffusion. By combining statistical and dynamical approaches, we establish a fundamental limit on the network’s ability to maintain a persistent state: The noise-induced drift of the memory state over time within the network is strictly lower-bounded by the accuracy of estimation of the network’s instantaneous memory state by an ideal external observer. This result takes the form of an information-diffusion inequality. We derive some unexpected consequences: Despite the persistence time of short-term memory networks, it does not pay to accumulate spikes for longer than the cellular time-constant to read out their contents. For certain neural transfer functions, the conditions for optimal sensory coding coincide with those for optimal storage, implying that short-term memory may be co-localized with sensory representation.

Order and Stochastic Dynamics in Drosophila Planar Cell Polarity

Cells in the wing blade of Drosophila melanogaster exhibit an in-plane polarization causing distal orientation of hairs. Establishment of the Planar Cell Polarity (PCP) involves intercellular interactions as well as a global orienting signal. Many of the genetic and molecular components underlying this process have been experimentally identified and a recently advanced system-level model has suggested that the observed mutant phenotypes can be understood in terms of intercellular interactions involving asymmetric localization of membrane bound proteins. Among key open questions in understanding the emergence of ordered polarization is the effect of stochasticity and the role of the global orienting signal. These issues relate closely to our understanding of ferromagnetism in physical systems. Here we pursue this analogy to understand the emergence of PCP order. To this end we develop a semi-phenomenological representation of the underlying molecular processes and define a “phase diagram” of the model which provides a global view of the dependence of the phenotype on parameters. We show that the dynamics of PCP has two regimes: rapid growth in the amplitude of local polarization followed by a slower process of alignment which progresses from small to large scales. We discuss the response of the tissue to various types of orienting signals and show that global PCP order can be achieved with a weak orienting signal provided that it acts during the early phase of the process. Finally we define and discuss some of the experimental predictions of the model.

Hydration interactions: Aqueous solvent effects in electric double layers

A model for ionic solutions with an attractive short-range pair interaction between the ions is presented. The short-range interaction is accounted for by adding a quadratic nonlocal term to the Poisson-Boltzmann free energy. The model is used to study solvent effects in a planar electric double layer. The counterion density is found to increase near the charged surface, as compared with the Poisson-Boltzmann theory, and to decrease at larger distances. The ion density profile is studied analytically in the case where the ion distribution near the plate is dominated only by counterions. Further away from the plate the density distribution can be described using a Poisson-Boltzmann theory, with an effective surface charge that is smaller than the actual one.

The Origins of Specificity in Polyketide Synthase Protein Interactions

Polyketides, a diverse group of heteropolymers with antibiotic and antitumor properties, are assembled in bacteria by multiprotein chains of modular polyketide synthase (PKS) proteins. Specific protein–protein interactions determine the order of proteins within a multiprotein chain, and thereby the order in which chemically distinct monomers are added to the growing polyketide product. Here we investigate the evolutionary and molecular origins of protein interaction specificity. We focus on the short, conserved N- and C-terminal docking domains that mediate interactions between modular PKS proteins. Our computational analysis, which combines protein sequence data with experimental protein interaction data, reveals a hierarchical interaction specificity code. PKS docking domains are descended from a single ancestral interacting pair, but have split into three phylogenetic classes that are mutually noninteracting. Specificity within one such compatibility class is determined by a few key residues, which can be used to define compatibility subclasses. We identify these residues using a novel, highly sensitive co-evolution detection algorithm called CRoSS (correlated residues of statistical significance). The residue pairs selected by CRoSS are involved in direct physical interactions in a docked-domain NMR structure. A single PKS system can use docking domain pairs from multiple classes, as well as domain pairs from multiple subclasses of any given class. The termini of individual proteins are frequently shuffled, but docking domain pairs straddling two interacting proteins are linked as an evolutionary module. The hierarchical and modular organization of the specificity code is intimately related to the processes by which bacteria generate new PKS pathways.

Weak- and strong-coupling electrostatic interactions between asymmetrically charged planar surfaces.

We compare weak- and strong-coupling theory of counterion-mediated electrostatic interactions between two asymmetrically charged plates with extensive Monte Carlo simulations. Analytical results in both weak- and strong-coupling limits compare excellently with simulations in their respective regimes of validity. The system shows a surprisingly rich structure in terms of interactions between the surfaces as well as fundamental qualitative differences in behavior in the weak- and the strong-coupling limits.