Topological Data Analysis di neuroni tipo Grid-Cell emergenti in Reti Neurali Ricorrenti allenate nel compito di path integration ​

Grid cells are a class of neurons that has been observed within the medial enthorinal cortex (MEC) of many mammals: their firing activity is place-modulated, which means that it depends on the location of the animal in the space. The name refers to their unique pattern of activation: the firing locations of each grid cell define a regular 2-dimensional hexagonal grid covering the entire surface of the environment explored by the animal. For this reason, each grid cell generates a periodic, low- dimensional representation of space. Since their discovery, grid cells have been considered an essential component of the brain’s circuit involved in path integration, i.e. the ability of an animal to keep track of its spatial location, in absence of external sensory references, using only internal information about its own direction and velocity. Recently, studies have shown the emergence of units with grid-like activation in the hidden layers of Recurrent Neural Networks (RNN) trained to path integrate. In this Master thesis, we replicated these studies, implementing and training many RNNs, both vanilla and Long Short-Term Memory, to predict the animal position at discrete time steps, based on velocity inputs. Data were artificially produced, generating trajectories over surfaces of various kinds: square arena with closed border, square arena with periodic border corresponding to a torus, and non-orientable surfaces as the Klein bottle. We studied the effectiveness of the networks to resolve the task in different environments, and the robustness in predicting the correct trajectories with the increasing of the number of time steps. We then studied the representation of space produced within the hidden layers: we computed the ratemaps of the hidden units, that represent the mean activity as a function of the location on the arena. We observed units with pattern of activation corresponding to the hexagonal grid of the grid cells, and also units with other spatial correlates, that have been recorded in biological experiments. In particular, we found border-like units, that fire maximally near the border of the environment, and band-link units, that generate periodic patterns of activation composed of parallel bands. To classify and categorize units based on their pattern of activation, we used recently developed techniques of Topological Data Analysis. In particular, we compute the persistent homology of the super-level set filtration of the ratemaps, generating persistence diagrams that summarize the topological properties of each unit’s pattern of activation. Based on these topological features we applied a clustering algorithm to group similar ratemaps; the results of the classification have been validated, comparing them with classification methods already established in literature. Classification based on TDA presents significant advantages: first of all topological features are inherently invariant for translation and rotation, so they can be extracted directly from the ratemaps, without the preprocessing that is needed to apply previously used methods. Moreover, our method can be generalized and applied in dimension above 2, to study the representation of space in 3D, or to investigate the representations in even higher dimension. This can be useful because recently some studies have suggested that the brain could use the same coding scheme of grid cells to represent variables of conceptual spaces of arbitrary dimension. ​

I neuroni grid-cells sono stati osservati nella corteccia media entorinale (MEC) di diversi mammiferi. Il nome è dovuto al loro peculiare pattern di attivazione che forma dei reticoli 2D esagonali regolari mentre il soggetto esplora un ambiente. Fin dalla loro scoperta è stato ipotizzato che questi neuroni siano fondamentali per l'abilità di path integration, cioè la capacità di un animale di tenere traccia della propria posizione a partire dalle informazioni sulla propria velocità. Recentemente alcuni studi hanno mostrato l'emergere di unità nascoste con attivazione tipo grid-cells, in reti neurali ricorrenti addestrate nel compito di path integration. In questa tesi magistrale abbiamo addestrato reti neurali ricorrenti, semplici e di tipo LSTM, a tenere traccia della posizione utilizzando la velocità come input. Il training è stato effettuato generando delle traiettorie su superfici di diverso tipo: arena con i bordi non oltrepassabili, superficie periodica toroidale e infine su superfici non orientabili (bottiglia di Klein). I dati sono stati generati artificalmente. Abbiamo osservato l'emergere di unità nascoste sia tipo grid-cells, sia con altri correlati spaziali, come le border-cells e le band-cells, anch'esse osservate sperimentalmente. Per categorizzare e classificare i diversi tipi di unità, abbiamo utilizzato le recenti tecniche di Topological Data Analisys, che ben si adattano a questo scopo, data la loro intrinseca invarianza rotazionale e traslazionale. ​