Measurement of the Topological Dimension of Hippocampal Place Cell Activity

The fundamental property of topological dimension of neural activity has never been measured before. We measured the topological dimension of the activity of 89 rat hippocampal place cells recorded during foraging using the neighborhood boundary countdown method. Points were rate vectors forming a manifold in an 89 dimensional rate space Since a boundary has one less dimension than the neighborhood it encloses, reducing the dimension in sequential radial cuts finally arrives at boundary that is two points. The number of cuts required is the topological dimension of the set. Due to sparsity, we used a shell-like boundary with thickness. As expected, we found the large inductive dimension of this set of points to be two. To examine the robustness of the method, we used both real and modeled place cells and varied the number of neurons, duration of the time step and smoothing, radius and thickness for the boundary, decrement of the radius with each cut, requirements for the center point, and number of points in the final cluster. With the exception of shell thickness, the result was insensitive to these variations. Knowing the topological dimension allows application of rigorous topological principles to the activity of neuron populations. The method potentially can be applied to other neural classes and other behaviors, to enumerate firing variables when they are unknown, even count the factors influencing neural activity in sleep.