Grid cells generate an analog error-correcting code for singularly precise neural computation

Multiscale codes in the nervous system: the problem of noise correlations and the ambiguity of periodic scales

Encoding information about continuous variables using noisy computational units is a challenge; nonetheless, asymptotic theory shows that combining multiple periodic scales for coding can be highly precise despite the corrupting influence of noise [Mathis, Herz, and Stemmler, Phys. Rev. Lett. 109, 018103 (2012)]. Indeed, the cortex seems to use periodic, multiscale grid codes to represent position accurately. Here we show how such codes can be read out without taking the long-term limit; even on short time scales, the precision of such codes scales exponentially in the number N of neurons. Does this finding also hold for neurons that are not firing in a statistically independent fashion? To assess the extent to which biological grid codes are subject to statistical dependences, we first analyze the noise correlations between pairs of grid code neurons in behaving rodents. We find that if the grids of two neurons align and have the same length scale, the noise correlations between the neurons can reach values as high as 0.8. For increasing mismatches between the grids of the two neurons, the noise correlations fall rapidly. Incorporating such correlations into a population coding model reveals that the correlations lessen the resolution, but the exponential scaling of resolution with N is unaffected.

Specific evidence of low-dimensional continuous attractor dynamics in grid cells

We examined simultaneously recorded spikes from multiple rat grid cells, to explain mechanisms underlying their activity. Among grid cells with similar spatial periods, the population activity was confined to lie close to a two-dimensional (2D) manifold: grid cells differed only along two dimensions of their responses and otherwise were nearly identical. Relationships between cell pairs were conserved despite extensive deformations of single-neuron responses. Results from novel environments suggest such structure is not inherited from hippocampal or external sensory inputs. Across conditions, cell-cell relationships are better conserved than responses of single cells. Finally, the system is continually subject to perturbations that, were the 2D manifold not attractive, would drive the system to inhabit a different region of state space than observed. These findings have strong implications for theories of grid-cell activity and substantiate the general hypothesis that the brain computes using low-dimensional continuous attractors.

Synthesizing cognition in neuromorphic electronic systems

The quest to implement intelligent processing in electronic neuromorphic systems lacks methods for achieving reliable behavioral dynamics on substrates of inherently imprecise and noisy neurons. Here we report a solution to this problem that involves first mapping an unreliable hardware layer of spiking silicon neurons into an abstract computational layer composed of generic reliable subnetworks of model neurons and then composing the target behavioral dynamics as a "soft state machine" running on these reliable subnets. In the first step, the neural networks of the abstract layer are realized on the hardware substrate by mapping the neuron circuit bias voltages to the model parameters. This mapping is obtained by an automatic method in which the electronic circuit biases are calibrated against the model parameters by a series of population activity measurements. The abstract computational layer is formed by configuring neural networks as generic soft winner-take-all subnetworks that provide reliable processing by virtue of their active gain, signal restoration, and multistability. The necessary states and transitions of the desired high-level behavior are then easily embedded in the computational layer by introducing only sparse connections between some neurons of the various subnets. We demonstrate this synthesis method for a neuromorphic sensory agent that performs real-time context-dependent classification of motion patterns observed by a silicon retina.

Spatial scale and place field stability in a grid-to-place cell model of the dorsoventral axis of the hippocampus

The rodent hippocampus and entorhinal cortex contain spatially modulated cells that serve as the basis for spatial coding. Both medial entorhinal grid cells and hippocampal place cells have been shown to encode spatial information across multiple spatial scales that increase along the dorsoventral axis of these structures. Place cells near the dorsal pole possess small, stable, and spatially selective firing fields, while ventral cells have larger, less stable, and less spatially selective firing fields. One possible explanation for these dorsoventral changes in place field properties is that they arise as a result of similar dorsoventral differences in the properties of the grid cell inputs to place cells. Here, we test the alternative hypothesis that dorsoventral place field differences are due to higher amounts of nonspatial inputs to ventral hippocampal cells. We use a computational model of the entorhinal-hippocampal network to assess the relative contributions of grid scale and nonspatial inputs in determining place field size and stability. In addition, we assess the consequences of grid node firing rate heterogeneity on place field stability. Our results suggest that dorsoventral differences in place cell properties can be better explained by changes in the amount of nonspatial inputs, rather than by changes in the scale of grid cell inputs, and that grid node heterogeneity may have important functional consequences. The observed gradient in field size may reflect a shift from processing primarily spatial information in the dorsal hippocampus to processing more nonspatial, contextual, and emotional information near the ventral hippocampus.

The entorhinal grid map is discretized

The medial entorhinal cortex (MEC) is part of the brain's circuit for dynamic representation of self-location. The metric of this representation is provided by grid cells, cells with spatial firing fields that tile environments in a periodic hexagonal pattern. Limited anatomical sampling has obscured whether the grid system operates as a unified system or a conglomerate of independent modules. Here we show with recordings from up to 186 grid cells in individual rats that grid cells cluster into a small number of layer-spanning anatomically overlapping modules with distinct scale, orientation, asymmetry and theta-frequency modulation. These modules can respond independently to changes in the geometry of the environment. The discrete topography of the grid-map, and the apparent autonomy of the modules, differ from the graded topography of maps for continuous variables in several sensory systems, raising the possibility that the modularity of the grid map is a product of local self-organizing network dynamics.

Neural representations of location composed of spatially periodic bands

The mammalian hippocampal formation provides neuronal representations of environmental location, but the underlying mechanisms are poorly understood. Here, we report a class of cells whose spatially periodic firing patterns are composed of plane waves (or bands) drawn from a discrete set of orientations and wavelengths. The majority of cells recorded in parasubicular and medial entorhinal cortices of freely moving rats belonged to this class and included grid cells, an important subset that corresponds to three bands at 60° orientations and has the most stable firing pattern. Occasional changes between hexagonal and nonhexagonal patterns imply a common underlying mechanism. Our results indicate a Fourier-like spatial analysis underlying neuronal representations of location, and suggest that path integration is performed by integrating displacement along a restricted set of directions.

Resolution of nested neuronal representations can be exponential in the number of neurons

Collective computation is typically polynomial in the number of computational elements, such as transistors or neurons, whether one considers the storage capacity of a memory device or the number of floating-point operations per second of a CPU. However, we show here that the capacity of a computational network to resolve real-valued signals of arbitrary dimensions can be exponential in N, even if the individual elements are noisy and unreliable. Nested, modular codes that achieve such high resolutions mirror the properties of grid cells in vertebrates, which underlie spatial navigation.

From A to Z: a potential role for grid cells in spatial navigation

Since their discovery, the strikingly regular and spatially stable firing of entorhinal grid cells has attracted the attention of experimentalists and theoreticians alike. The bulk of this work has focused either on the assumption that the principal role of grid cells is to support path integration or the extent to which their multiple firing locations can drive the sparse activity of hippocampal place cells. Here, we propose that grid cells are best understood as part of a network that combines self-motion and environmental cues to accurately track an animal’s location in space. Furthermore, that grid cells - more so than place cells - efficiently encode self-location in allocentric coordinates. Finally, that the regular structure of grid firing fields represents information about the relative structure of space and, as such, may be used to guide goal directed navigation.

The abrupt development of adult-like grid cell firing in the medial entorhinal cortex

Understanding the development of the neural circuits subserving specific cognitive functions such as navigation remains a central problem in neuroscience. Here, we characterize the development of grid cells in the medial entorhinal cortex, which, by nature of their regularly spaced firing fields, are thought to provide a distance metric to the hippocampal neural representation of space. Grid cells emerge at the time of weaning in the rat, at around 3 weeks of age. We investigated whether grid cells in young rats are functionally equivalent to those observed in the adult as soon as they appear, or if instead they follow a gradual developmental trajectory. We find that, from the very youngest ages at which reproducible grid firing is observed (postnatal day 19): grid cells display adult-like firing fields that tessellate to form a coherent map of the local environment; that this map is universal, maintaining its internal structure across different environments; and that grid cells in young rats, as in adults, also encode a representation of direction and speed. To further investigate the developmental processes leading up to the appearance of grid cells, we present data from individual medial entorhinal cortex cells recorded across more than 1 day, spanning the period before and after the grid firing pattern emerged. We find that increasing spatial stability of firing was correlated with increasing gridness.

Linear look-ahead in conjunctive cells: an entorhinal mechanism for vector-based navigation

The crisp organization of the “firing bumps” of entorhinal grid cells and conjunctive cells leads to the notion that the entorhinal cortex may compute linear navigation routes. Specifically, we propose a process, termed “linear look-ahead,” by which a stationary animal could compute a series of locations in the direction it is facing. We speculate that this computation could be achieved through learned patterns of connection strengths among entorhinal neurons. This paper has three sections. First, we describe the minimal grid cell properties that will be built into our network. Specifically, the network relies on “rigid modules” of neurons, where all members have identical grid scale and orientation, but differ in spatial phase. Additionally, these neurons must be densely interconnected with synapses that are modifiable early in the animal’s life. Second, we investigate whether plasticity during short bouts of locomotion could induce patterns of connections amongst grid cells or conjunctive cells. Finally, we run a simulation to test whether the learned connection patterns can exhibit linear look-ahead. Our results are straightforward. A simulated 30-min walk produces weak strengthening of synapses between grid cells that do not support linear look-ahead. Similar training in a conjunctive cell module produces a small subset of very strong connections between cells. These strong pairs have three properties: the pre- and post-synaptic cells have similar heading direction. The cell pairs have neighboring grid bumps. Finally, the spatial offset of firing bumps of the cell pair is in the direction of the common heading preference. Such a module can produce strong and accurate linear look-ahead starting in any location and extending in any direction. We speculate that this process may: (1) compute linear paths to goals; (2) update grid cell firing during navigation; and (3) stabilize the rigid modules of grid cells and conjunctive cells.

Self-organization of multiple spatial and context memories in the hippocampus

One obstacle to understanding the exact processes unfolding inside the hippocampus is that it is still difficult to clearly define what the hippocampus actually does, at the system level. Associated for a long time with the formation of episodic and semantic memories, and with their temporary storage, the hippocampus is also regarded as a structure involved in spatial navigation. These two independent perspectives on the hippocampus are not necessarily exclusive: proposals have been put forward to make them fit into the same conceptual frame. We review both approaches and argue that three critical developments need consideration: (a) recordings of neuronal activity in rodents, revealing beautiful spatial codes expressed in entorhinal cortex, upstream of the hippocampus; (b) comparative behavioral results suggesting, in an evolutionary perspective, qualitative similarity of function across homologous structures with a distinct internal organization; (c) quantitative measures of information, shifting the focus from who does what to how much each neuronal population expresses each code. These developments take the hippocampus away from philosophical discussions of all-or-none cause-effect relations, and into the quantitative mainstream of modern neural science.