A minimal mathematical formalization for grid cells using automata theory

Abstract

Do animals have an internal map of the world? The discovery of grid cells, a type of neuron found in the entorhinal part of animals’ brains, certainly supports the affrmative side of this debate. Grid cells are theorised to play a part in the navigation of some animals by fring in a spatially periodic manner. Tracking where the neuron fres in the environment over time during animal exploration reveals the emergence of a hexagonal grid pattern. Whether this grid serves as a map to the animal cannot be said with certainty, but this avenue of exploration has motivated many to try and recreate grid cells computationally to better understand them.

In this thesis, a computer-theoretic approach is taken, and a single grid cell is modeled as a strongly connected deterministic fnite automaton (DFA). By coupling this automaton with a discrete planar environment through a shared set of actions, it is shown through simulation that the hexagonal pattern produced by grid cells can emerge if an exploring agent uses the automaton as its control unit. Furthermore, it is shown that any repeating pattern can be created through this construction as long as there are certain constraints set upon the automaton. These constraints are investigated more in depth in a 1D environment, leading to the discovery that the suffcient property for such repeating patterns to emerge is for the automaton to be circular. By circular, it is meant that for every transition between states, there is an inverse transition. This corresponds to an agent having the possibility to move to a new position in the environment and then back to the position where it was previously in while doing the same in its automaton with regard to its states. To reach this conclusion, it was frst discovered that the fniteness of the automaton meant that there would be only a number of environmental positions which can be linked to an automaton state, before the same states start to be observed. Because of this cycling of states, it was possible to prove that all other positions in the environment must be copies of those original, unique positions. Therefore, if any of the original unique positions are such that the automaton is in a fring state (i.e., the grid cell is fring), then there will be many other positions in the environment with the same property. Intuitively, this shows that if a neuron recognizes a location in the agent’s environment, then it recognizes multiple other locations as well. The mathematical characterization proves that the only way for this to be possible is if the automaton is either circular or bisumulation equivalent to circular, meaning that the pattern created by the bisimulation equivalent automaton can also be created by a circular automaton. As a result, a minimal alternative to the current existing computational models of grid cells is created, which opens the door to new discoveries by giving a different perspective.