Andrey Babichev

Topological Schemas of Cognitive Maps and Spatial Learning

Spatial navigation in mammals is based on building a mental representation of their environment—a cognitive map. However, both the nature of this cognitive map and its underpinning in neural structures and activity remains vague. A key difficulty is that these maps are collective, emergent phenomena that cannot be reduced to a simple combination of inputs provided by individual neurons. In this paper we suggest computational frameworks for integrating the spiking signals of individual cells into a spatial map, which we call schemas. We provide examples of four schemas defined by different types of topological relations that may be neurophysiologically encoded in the brain and demonstrate that each schema provides its own large-scale characteristics of the environment—the schema integrals. Moreover, we find that, in all cases, these integrals are learned at a rate which is faster than the rate of complete training of neural networks. Thus, the proposed schema framework differentiates between the cognitive aspect of spatial learning and the physiological aspect at the neural network level.

Persistent Memories in Transient Networks

Spatial awareness in mammals is based on an internalized representation of the environment, encoded by large networks of spiking neurons. While such representations can last for a long time, the underlying neuronal network is transient : neuronal cells die every day, synaptic connections appear and disappear, the networks constantly change their architecture due to various forms of synaptic and structural plasticity. How can a network with a dynamic architecture encode a stable map of space? We address this question using a physiological model of a “flickering” neuronal network and demonstrate that it can maintain a robust topological representation of space.

Transient cell assembly networks encode stable spatial memories

One of the mysteries of memory is that it can last despite changes in the underlying synaptic architecture. How can we, for example, maintain an internal spatial map of an environment over months or years when the underlying network is full of transient connections? In the following, we propose a computational model for describing the emergence of the hippocampal cognitive map in a network of transient place cell assemblies and demonstrate, using methods of algebraic topology, how such a network can maintain spatial memory over time.

A topological approach to synaptic connectivity and spatial memory

In the hippocampus, a network of place cells generates a cognitive map of space, in which each cell is responsive to a particular area of the environment -- its place field. The peak response of each cell and the size of each place field have considerable variability. Experimental evidence suggests that place cells encode a topological map of space that serves as a basis of spatial memory and spatial awareness. Using a computational model based on Persistent Homology Theory we demonstrate that if the parameters of the place cells spiking activity fall inside of the physiological range, the network correctly encodes the topological features of the environment. We next introduce parameters of synaptic connectivity into the model and demonstrate that failures in synapses that detect coincident neuronal activity lead to spatial learning deficiencies similar to the ones that are observed in rodent models of neurodegenerative diseases. Moreover, we show that these learning deficiencies may be mitigated by increasing the number of active cells and/or by increasing their firing rate, suggesting the existence of a compensatory mechanism inherent to the cognitive map.

Robust spatial memory maps encoded by networks with transient connections

The spiking activity of principal cells in mammalian hippocampus encodes an internalized neuronal representation of the ambient space—a cognitive map. Once learned, such a map enables the animal to navigate a given environment for a long period. However, the neuronal substrate that produces this map is transient: the synaptic connections in the hippocampus and in the downstream neuronal networks never cease to form and to deteriorate at a rapid rate. How can the brain maintain a robust, reliable representation of space using a network that constantly changes its architecture? We address this question using a computational framework that allows evaluating the effect produced by the decaying connections between simulated hippocampal neurons on the properties of the cognitive map. Using novel Algebraic Topology techniques, we demonstrate that emergence of stable cognitive maps produced by networks with transient architectures is a generic phenomenon. The model also points out that deterioration of the cognitive map caused by weakening or lost connections between neurons may be compensated by simulating the neuronal activity. Lastly, the model explicates the importance of the complementary learning systems for processing spatial information at different levels of spatiotemporal granularity.