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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.

 

Grid cells play a principal role in enabling cognitive representations of ambient environments. The key property of these cells—the regular arrangement of their firing fields—is commonly viewed as a means for establishing spatial scales or encoding specific locations. However, using grid cells’ spiking outputs for deducing geometric orderliness proves to be a strenuous task due to fairly irregular activation patterns triggered by the animal’s sporadic visits to the grid fields. This article addresses statistical mechanisms enabling emergent regularity of grid cell firing activity from the perspective of percolation theory. Using percolation phenomena for modeling the effect of the rat’s moves through the lattices of firing fields sheds new light on the mechanisms of spatial information processing, spatial learning, path integration, and establishing spatial metrics. It is also shown that physiological parameters required for spiking percolation match the experimental range, including the characteristic 2/3 ratio between the grid fields’ size and the grid spacing, pointing at a biological viability of the approach.

 

Our current understanding of brain rhythms is based on quantifying their instantaneous or time-averaged characteristics. What remains unexplored is the actual structure of the waves—their shapes and patterns over finite timescales. Here, we study brain wave patterning in different physiological contexts using two independent approaches: The first is based on quantifying stochasticity relative to the underlying mean behavior, and the second assesses “orderliness” of the waves’ features. The corresponding measures capture the waves’ characteristics and abnormal behaviors, such as atypical periodicity or excessive clustering, and demonstrate coupling between the patterns’ dynamics and the animal’s location, speed, and acceleration. Specifically, we studied patterns ofθ,γ, and ripple waves recorded in mice hippocampi and observed speed-modulated changes of the wave’s cadence, an antiphase relationship between orderliness and acceleration, as well as spatial selectiveness of patterns. Taken together, our results offer a complementary—mesoscale—perspective on brain wave structure, dynamics, and functionality.

 

Neurons in the brain are submerged into oscillating extracellular potential produced by synchronized synaptic currents. The dynamics of these oscillations is one of the principal characteristics of neurophysiological activity, broadly studied in basic neuroscience and used in applications. However, our interpretation of the brain waves' structure and hence our understanding of their functions depend on the mathematical and computational approaches used for data analysis. The oscillatory nature of the wave dynamics favors Fourier methods, which have dominated the field for several decades and currently constitute the only systematic approach to brain rhythms. In the following study, we outline an alternative framework for analyzing waves of local field potentials (LFPs) and discuss a set of new structures that it uncovers: a discrete set of frequency-modulated oscillatory processes—the brain wave oscillons and their transient spectral dynamics 

Seizure clusters are seizures that occur in rapid succession during periods of heightened seizure risk and are associated with substantial morbidity and sudden unexpected death in epilepsy. The objective of this feasibility study was to evaluate the performance of a novel seizure cluster forecasting algorithm. Chronic ambulatory electrocorticography recorded over an average of 38 months in 10 subjects with drug‐resistant epilepsies was analyzed pseudoprospectively by dividing data into training (first 85%) and validation periods. For each subject, the probability of seizure clustering, derived from the Kolmogorov–Smirnov statistic using a novel algorithm, was forecasted in the validation period using individualized autoregressive models that were optimized from training data. The primary outcome of this study was the mean absolute scaled error (MASE) of 1‐day horizon forecasts. From 10 subjects, 394 ± 142 (mean ± SD) electrocorticography‐based seizure events were extracted for analysis, representing a span of 38 ± 27 months of recording. MASE across all subjects was .74 ± .09, .78 ± .09, and .83 ± .07 at .5‐, 1‐, and 2‐day horizons. The feasibility study demonstrates that seizure clusters are quasiperiodic and can be forecasted to clinically meaningful horizons. Pending validation in larger cohorts, the forecasting approach described herein may herald chronotherapy during imminent heightened seizure vulnerability.

 

Topological data analyses are widely used for describing and conceptualizing large volumes of neurobiological data, e.g., for quantifying spiking outputs of large neuronal ensembles and thus understanding the functions of the corresponding networks. Below we discuss an approach in which convergent topological analyses produce insights into how information may be processed in mammalian hippocampus—a brain part that plays a key role in learning and memory. The resulting functional model provides a unifying framework for integrating spiking data at different timescales and following the course of spatial learning at different levels of spatiotemporal granularity. This approach allows accounting for contributions from various physiological phenomena into spatial cognition—the neuronal spiking statistics, the effects of spiking synchronization by different brain waves, the roles played by synaptic efficacies and so forth. In particular, it is possible to demonstrate that networks with plastic and transient synaptic architectures can encode stable cognitive maps, revealing the characteristic timescales of memory processing.

 

Various neurophysiological and cognitive functions are based on transferring information between spiking neurons via a complex system of synaptic connections. In particular, the capacity of presynaptic inputs to influence the postsynaptic outputs–the efficacy of the synapses–plays a principal role in all aspects of hippocampal neurophysiology. However, a direct link between the information processed at the level of individual synapses and the animal’s ability to form memories at the organismal level has not yet been fully understood. Here, we investigate the effect of synaptic transmission probabilities on the ability of the hippocampal place cell ensembles to produce a cognitive map of the environment. Using methods from algebraic topology, we find that weakening synaptic connections increase spatial learning times, produce topological defects in the large-scale representation of the ambient space and restrict the range of parameters for which place cell ensembles are capable of producing a map with correct topological structure. On the other hand, the results indicate a possibility of compensatory phenomena, namely that spatial learning deficiencies may be mitigated through enhancement of neuronal activity.