Grid cells, border cells, and discrete complex analysis

Published on October 10, 2023

Imagine you’re exploring a new city, and as you stroll through its streets, you can’t help but notice that your excitement builds as you approach its borders. Well, it turns out that the brain’s navigational cells, called grid cells, behave in a similar way. Scientists have discovered a mechanism called discrete complex analysis on a triangular lattice, which allows for the creation of imaginary functions that have maximum values at the boundaries of specific areas. By applying this concept to grid cell models, researchers were able to identify border cells—neurons that fire when the boundaries of an environment are reached. These border cells enhance their activity near the frontiers of the spaces being explored. To support their findings, the researchers even developed a network model of neurons that exhibit border-bound firing. This exciting research sheds light on the fascinating interplay between mathematics and neuroscience, showing how mathematical concepts can inform our understanding of brain function. Dive into the details of this study to uncover more about the connection between discrete complex analysis and navigational cells!

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.

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