Discovered in 2005, grid cells are neurons located in the entorhinal cortex of mammals, responsible for processing information such as direction, position, and speed of the animal without the need for visual inputs. They play a central role in the neural representation of physical space in so-called cognitive maps. Even in complete darkness, the animal's brain can represent its physical movement in the environment, allowing it to return to the starting point of its path without visual information. This implies that the ability to retain previous information for subsequent decision-making is not necessarily tied to visual inputs. This capacity is referred to as Path Integration, and it is understood that the medial entorhinal cortex, along with the hippocampus, may be responsible for this integration based on grid cells, place cells, and head direction.

In the context of computational architectures for task resolution, it is known that neural networks with Transformer architecture have a great ability to solve problems that can be presented or described in a sequential format, such as text translation, mathematical demonstrations, and logical problems. The essence of the Transformer architecture is the mechanism called Self-Attention. Since the core mechanism of a Transformer can be related to models developed in the context of neuroscience, the aim is to understand whether this architecture can perform path integration as occurs in hippocampal formation.

The main objective of the proposed project was to model a Transformer architecture network to perform the task of integrating trajectories simulated using the Monte Carlo method, based on the Python library: RatInABox. However, this main objective can be broken down into four specific objectives:

1. Assess the network's ability to solve the problem and test its efficiency.
2. Analyze whether the synaptic weights of the network exhibit the hexagonal patterns of grid cells.
3. Relate the Self-Attention mechanism to the biological mechanisms of grid cells.
4. Describe the parameters necessary for the emergence of hexagonal patterns in the synaptic weights of the network.

Studying the relationship between the network addressed in this project and biological mechanisms is necessary for understanding the functioning of grid cells from a more abstract perspective, given that the proposed network was not developed specifically for the task of path integration. It is also understood that, with a significant capacity for generalization, the Transformer network can provide insights into how this generalization occurs in the formation of the hippocampus, in both grid cells and place and head direction cells.

To achieve the proposed objectives in this project, the methodology can be divided into two parts: data generation and architecture application. In the first part, we emphasize that the data was obtained using Monte Carlo algorithms to generate simulated trajectories of an animal in a confined environment using the RatInABox library, producing the necessary information about direction and speed for the integration task by the architecture. Once these generated data were validated, the process of adapting and organizing the Transformer architecture for application began. The training process involved training the network on complete trajectories so that its synaptic weights would converge to the solution of the integration problem. Subsequently, the network's functionality was tested on unknown trajectories, where the objective was to calculate the position of the next step presented in the simulation.

Results were generated after the training and testing stages for a simulation of 12,000 Monte Carlo steps (which can be assumed as 20 minutes of a rat's exploration in a square environment of 1 meter per side), with 9,600 steps used for training and the remaining 2,400 for testing. In distance metrics, the root mean square error (RMSE) of the network during training averaged 0.0466 with a standard deviation of 0.0074, while during testing the result averaged 0.0512 with a standard deviation of 0.0021. Although the proposed network was able to learn the trajectories in the sense of determining the next step, no hexagonal patterns were observed in the weights related to the attention mechanism.

A simple application of the Transformer architecture, trained on trajectories simulating the navigation statistics of a rat in a simple environment, allowing for the creation of a model that predicts the next step given a segment of the trajectory. However, this model does not exhibit the typical hexagonal weight structure of grid cells, indicating that a more complex learning paradigm may be necessary.