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Abstract A common approach to interpreting spiking activity is based on identifying the firing fields—regions in physical or configuration spaces that elicit responses of neurons. Common examples include hippocampal place cells that fire at preferred locations in the navigated environment, head direction cells that fire at preferred orientations of the animal’s head, view cells that respond to preferred spots in the visual field, etc. In all these cases, firing fields were discovered empirically, by trial and error. We argue that the existence and a number of properties of the firing fields can be established theoretically, through topological analyses of the neuronal spiking activity. In particular, we use Leray criterion powered by persistent homology theory, Eckhoff conditions and Region Connection Calculus to verify consistency of neuronal responses with a single coherent representation of space.
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Grid Cell Percolation
Abstract Grid cells play a principal role in enabling cognitive representations of ambient environments. The key property of these cells—the regular arrangement of their firing fields—is commonly viewed as a means for establishing spatial scales or encoding specific locations. However, using grid cells’ spiking outputs for deducing geometric orderliness proves to be a strenuous task due to fairly irregular activation patterns triggered by the animal’s sporadic visits to the grid fields. This article addresses statistical mechanisms enabling emergent regularity of grid cell firing activity from the perspective of percolation theory. Using percolation phenomena for modeling the effect of the rat’s moves through the lattices of firing fields sheds new light on the mechanisms of spatial information processing, spatial learning, path integration, and establishing spatial metrics. It is also shown that physiological parameters required for spiking percolation match the experimental range, including the characteristic 2/3 ratio between the grid fields’ size and the grid spacing, pointing at a biological viability of the approach.
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- Award ID(s):
- 1901338
- PAR ID:
- 10554726
- Editor(s):
- Sharpee, T
- Publisher / Repository:
- Neural Computation
- Date Published:
- Journal Name:
- Neural Computation
- Volume:
- 35
- Issue:
- 10
- ISSN:
- 0899-7667
- Page Range / eLocation ID:
- 1609 to 1626
- Subject(s) / Keyword(s):
- grid cells, place cells, percolation, phase transition, spatial learning
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Abstract Place and grid cells in the hippocampal formation are commonly thought to support a unified and coherent cognitive map of space. This mapping mechanism faces a challenge when a navigator is placed in a familiar environment that has been deformed from its original shape. Under such circumstances, many transformations could plausibly serve to map a navigator's familiar cognitive map to the deformed space. Previous empirical results indicate that the firing fields of rodent place and grid cells stretch or compress in a manner that approximately matches the environmental deformation, and human spatial memory exhibits similar distortions. These effects have been interpreted as evidence that reshaping a familiar environment elicits an analogously reshaped cognitive map. However, recent work has suggested an alternative explanation, whereby deformation‐induced distortions of the grid code are attributable to a mechanism that dynamically anchors grid fields to the most recently experienced boundary, thus causing history‐dependent shifts in grid phase. This interpretation raises the possibility that human spatial memory will exhibit similar history‐dependent dynamics. To test this prediction, we taught participants the locations of objects in a virtual environment and then probed their memory for these locations in deformed versions of this environment. Across three experiments with variable access to visual and vestibular cues, we observed the predicted pattern, whereby the remembered locations of objects were shifted from trial to trial depending on the boundary of origin of the participant's movement trajectory. These results provide evidence for a dynamic anchoring mechanism that governs both neuronal firing and spatial memory.more » « less
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Spatial periodicity in grid cell firing has been interpreted as a neural metric for space providing animals with a coordinate system in navigating physical and mental spaces. However, the specific computational problem being solved by grid cells has remained elusive. Here, we provide mathematical proof that spatial periodicity in grid cell firing is the only possible solution to a neural sequence code of 2-D trajectories and that the hexagonal firing pattern of grid cells is the most parsimonious solution to such a sequence code. We thereby provide a likely teleological cause for the existence of grid cells and reveal the underlying nature of the global geometric organization in grid maps as a direct consequence of a simple local sequence code. A sequence code by grid cells provides intuitive explanations for many previously puzzling experimental observations and may transform our thinking about grid cells.more » « less
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null (Ed.)Grid cells in the brain fire in strikingly regular hexagonal patterns across space. There are currently two seemingly unrelated frameworks for understanding these patterns. Mechanistic models account for hexagonal firing fields as the result of pattern-forming dynamics in a recurrent neural network with hand-tuned center-surround connectivity. Normative models specify a neural architecture, a learning rule, and a navigational task, and observe that grid-like firing fields emerge due to the constraints of solving this task. Here we provide an analytic theory that unifies the two perspectives by casting the learning dynamics of neural networks trained on navigational tasks as a pattern forming dynamical system. This theory provides insight into the optimal solutions of diverse formulations of the normative task, and shows that symmetries in the representation of space correctly predict the structure of learned firing fields in trained neural networks. Further, our theory proves that a nonnegativity constraint on firing rates induces a symmetry-breaking mechanism which favors hexagonal firing fields. We extend this theory to the case of learning multiple grid maps and demonstrate that optimal solutions consist of a hierarchy of maps with increasing length scales. These results unify previous accounts of grid cell firing and provide a novel framework for predicting the learned representations of recurrent neural networks.more » « less
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Wu, Si (Ed.)We propose a mechanism enabling the appearance of border cells—neurons firing at the boundaries of the navigated enclosures. The approach is based on the recent discovery of discrete complex analysis on a triangular lattice, which allows constructing discrete epitomes of complex-analytic functions and making use of their inherent ability to attain maximal values at the boundaries of generic lattice domains. As it turns out, certain elements of the discrete-complex framework readily appear in the oscillatory models of grid cells. We demonstrate that these models can extend further, producing cells that increase their activity toward the frontiers of the navigated environments. We also construct a network model of neurons with border-bound firing that conforms with the oscillatory models.more » « less
- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1162/neco_a_01606
