Does the entorhinal cortex use the Fourier transform?

Exploiting semantic information in a spiking neural SLAM system

To navigate in new environments, an animal must be able to keep track of its position while simultaneously creating and updating an internal map of features in the environment, a problem formulated as simultaneous localization and mapping (SLAM) in the field of robotics. This requires integrating information from different domains, including self-motion cues, sensory, and semantic information. Several specialized neuron classes have been identified in the mammalian brain as being involved in solving SLAM. While biology has inspired a whole class of SLAM algorithms, the use of semantic information has not been explored in such work. We present a novel, biologically plausible SLAM model called SSP-SLAM-a spiking neural network designed using tools for large scale cognitive modeling. Our model uses a vector representation of continuous spatial maps, which can be encoded via spiking neural activity and bound with other features (continuous and discrete) to create compressed structures containing semantic information from multiple domains (e.g., spatial, temporal, visual, conceptual). We demonstrate that the dynamics of these representations can be implemented with a hybrid oscillatory-interference and continuous attractor network of head direction cells. The estimated self-position from this network is used to learn an associative memory between semantically encoded landmarks and their positions, i.e., an environment map, which is used for loop closure. Our experiments demonstrate that environment maps can be learned accurately and their use greatly improves self-position estimation. Furthermore, grid cells, place cells, and object vector cells are observed by this model. We also run our path integrator network on the NengoLoihi neuromorphic emulator to demonstrate feasibility for a full neuromorphic implementation for energy efficient SLAM.

Why grid cells function as a metric for space

The brain is able to calculate the distance and direction to the desired position based on grid cells. Extensive neurophysiological studies of rodent navigation have postulated the grid cells function as a metric for space, and have inspired many computational studies to develop innovative navigation approaches. Furthermore, grid cells may provide a general encoding scheme for high-order nonspatial information. Built upon existing neuroscience and machine learning work, this paper provides theoretical clarity on that the grid cell population codes can be taken as a metric for space. The metric is generated by a shift-invariant positive definite kernel via kernel distance method and embeds isometrically in a Euclidean space, and the inner product of the grid cell population code exponentially converges to the kernel. We also provide a method to learn the distribution of grid cell population efficiently. Grid cells, as a scalable position encoding method, can encode the spatial relationships of places and enable grid cells to outperform place cells in navigation. Further, we extend the grid cell to images encoding and find that grid cells embed images into a mental map, where geometric relationships are conceptual relationships of images. The theoretical model and analysis would contribute to establishing the grid cell code as a generic coding scheme for both spatial and conceptual spaces, and is promising for a multitude of problems across spatial cognition, machine learning and semantic cognition.

A computational model for grid maps in neural populations

Grid cells in the entorhinal cortex, together with head direction, place, speed and border cells, are major contributors to the organization of spatial representations in the brain. In this work we introduce a novel theoretical and algorithmic framework able to explain the optimality of hexagonal grid-like response patterns. We show that this pattern is a result of minimal variance encoding of neurons together with maximal robustness to neurons' noise and minimal number of encoding neurons. The novelty lies in the formulation of the encoding problem considering neurons as an overcomplete basis (a frame) where the position information is encoded. Through the modern Frame Theory language, specifically that of tight and equiangular frames, we provide new insights about the optimality of hexagonal grid receptive fields. The proposed model is based on the well-accepted and tested hypothesis of Hebbian learning, providing a simplified cortical-based framework that does not require the presence of velocity-driven oscillations (oscillatory model) or translational symmetries in the synaptic connections (attractor model). We moreover demonstrate that the proposed encoding mechanism naturally explains axis alignment of neighbor grid cells and maps shifts, rotations and scaling of the stimuli onto the shape of grid cells' receptive fields, giving a straightforward explanation of the experimental evidence of grid cells remapping under transformations of environmental cues.

Gradients in the mammalian cerebellar cortex enable Fourier-like transformation and improve storing capacity

Cerebellar granule cells (GCs) make up the majority of all neurons in the vertebrate brain, but heterogeneities among GCs and potential functional consequences are poorly understood. Here, we identified unexpected gradients in the biophysical properties of GCs in mice. GCs closer to the white matter (inner-zone GCs) had higher firing thresholds and could sustain firing with larger current inputs than GCs closer to the Purkinje cell layer (outer-zone GCs). Dynamic Clamp experiments showed that inner- and outer-zone GCs preferentially respond to high- and low-frequency mossy fiber inputs, respectively, enabling dispersion of the mossy fiber input into its frequency components as performed by a Fourier transformation. Furthermore, inner-zone GCs have faster axonal conduction velocity and elicit faster synaptic potentials in Purkinje cells. Neuronal network modeling revealed that these gradients improve spike-timing precision of Purkinje cells and decrease the number of GCs required to learn spike-sequences. Thus, our study uncovers biophysical gradients in the cerebellar cortex enabling a Fourier-like transformation of mossy fiber inputs.

A diversity of interneurons and Hebbian plasticity facilitate rapid compressible learning in the hippocampus

The hippocampus is able to rapidly learn incoming information, even if that information is only observed once. Furthermore, this information can be replayed in a compressed format in either forward or reverse modes during sharp wave-ripples (SPW-Rs). We leveraged state-of-the-art techniques in training recurrent spiking networks to demonstrate how primarily interneuron networks can achieve the following: (1) generate internal theta sequences to bind externally elicited spikes in the presence of inhibition from the medial septum; (2) compress learned spike sequences in the form of a SPW-R when septal inhibition is removed; (3) generate and refine high-frequency assemblies during SPW-R-mediated compression; and (4) regulate the inter-SPW interval timing between SPW-Rs in ripple clusters. From the fast timescale of neurons to the slow timescale of behaviors, interneuron networks serve as the scaffolding for one-shot learning by replaying, reversing, refining, and regulating spike sequences.

Using Grid Cells for Navigation

Mammals are able to navigate to hidden goal locations by direct routes that may traverse previously unvisited terrain. Empirical evidence suggests that this “vector navigation” relies on an internal representation of space provided by the hippocampal formation. The periodic spatial firing patterns of grid cells in the hippocampal formation offer a compact combinatorial code for location within large-scale space. Here, we consider the computational problem of how to determine the vector between start and goal locations encoded by the firing of grid cells when this vector may be much longer than the largest grid scale. First, we present an algorithmic solution to the problem, inspired by the Fourier shift theorem. Second, we describe several potential neural network implementations of this solution that combine efficiency of search and biological plausibility. Finally, we discuss the empirical predictions of these implementations and their relationship to the anatomy and electrophysiology of the hippocampal formation.

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Grid cells (GCs) are believed to provide a path integration input to place cells

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However, GCs also provide a powerful context-independent metric for large-scale space

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Hence, we show how GCs can be used for vector navigation between arbitrary locations

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We simulate various neural implementations and make testable experimental predictions

Oscillator-interference models of path integration do not require theta oscillations

Navigation and path integration in rodents seems to involve place cells, grid cells, and theta oscillations (4-12 Hz) in the local field potential. Two main theories have been proposed to explain the neurological underpinnings of how these phenomena relate to navigation and to each other. Attractor network (AN) models revolve around the idea that local excitation and long-range inhibition connectivity can spontaneously generate grid-cell-like activity patterns. Oscillator interference (OI) models propose that spatial patterns of activity are caused by the interference patterns between neural oscillators. In rats, these oscillators have a frequency close to the theta frequency. Recent studies have shown that bats do not exhibit a theta cycle when they crawl, and yet they still have grid cells. This has been interpreted as a criticism of OI models. However, OI models do not require theta oscillations. We explain why the absence of theta oscillations does not contradict OI models and discuss how the two families of models might be distinguished experimentally.

Controlling phase noise in oscillatory interference models of grid cell firing

Oscillatory interference models account for the spatial firing properties of grid cells in terms of neuronal oscillators with frequencies modulated by the animal's movement velocity. The phase of such a "velocity-controlled oscillator" (VCO) relative to a baseline (theta-band) oscillation tracks displacement along a preferred direction. Input from multiple VCOs with appropriate preferred directions causes a grid cell's grid-like firing pattern. However, accumulating phase noise causes the firing pattern to drift and become corrupted. Here we show how multiple redundant VCOs can automatically compensate for phase noise. By entraining the baseline frequency to the mean VCO frequency, VCO phases remain consistent, ensuring a coherent grid pattern and reducing its spatial drift. We show how the spatial stability of grid firing depends on the variability in VCO phases, e.g., a phase SD of 3 ms per 125 ms cycle results in stable grids for 1 min. Finally, coupling N VCOs with similar preferred directions as a ring attractor, so that their relative phases remain constant, produces grid cells with consistently offset grids, and reduces VCO phase variability of the order square root of N. The results suggest a viable functional organization of the grid cell network, and highlight the benefit of integrating displacement along multiple redundant directions for the purpose of path integration.

A hybrid oscillatory interference/continuous attractor network model of grid cell firing

Grid cells in the rodent medial entorhinal cortex exhibit remarkably regular spatial firing patterns that tessellate all environments visited by the animal. Two theoretical mechanisms that could generate this spatially periodic activity pattern have been proposed: oscillatory interference and continuous attractor dynamics. Although a variety of evidence has been cited in support of each, some aspects of the two mechanisms are complementary, suggesting that a combined model may best account for experimental data. The oscillatory interference model proposes that the grid pattern is formed from linear interference patterns or "periodic bands" in which velocity-controlled oscillators integrate self-motion to code displacement along preferred directions. However, it also allows the use of symmetric recurrent connectivity between grid cells to provide relative stability and continuous attractor dynamics. Here, we present simulations of this type of hybrid model, demonstrate that it generates intracellular membrane potential profiles that closely match those observed in vivo, addresses several criticisms aimed at pure oscillatory interference and continuous attractor models, and provides testable predictions for future empirical studies.