Evaluation of the Oscillatory Interference Model of Grid Cell Firing through Analysis and Measured Period Variance of Some Biological Oscillators

Models of the hexagonally arrayed spatial activity pattern of grid cell firing in the literature generally fall into two main categories: continuous attractor models or oscillatory interference models. Burak and Fiete (2009, PLoS Comput Biol) recently examined noise in two continuous attractor models, but did not consider oscillatory interference models in detail. Here we analyze an oscillatory interference model to examine the effects of noise on its stability and spatial firing properties. We show analytically that the square of the drift in encoded position due to noise is proportional to time and inversely proportional to the number of oscillators. We also show there is a relatively fixed breakdown point, independent of many parameters of the model, past which noise overwhelms the spatial signal. Based on this result, we show that a pair of oscillators are expected to maintain a stable grid for approximately t = 5µ3/(4πσ)2 seconds where µ is the mean period of an oscillator in seconds and σ2 its variance in seconds2. We apply this criterion to recordings of individual persistent spiking neurons in postsubiculum (dorsal presubiculum) and layers III and V of entorhinal cortex, to subthreshold membrane potential oscillation recordings in layer II stellate cells of medial entorhinal cortex and to values from the literature regarding medial septum theta bursting cells. All oscillators examined have expected stability times far below those seen in experimental recordings of grid cells, suggesting the examined biological oscillators are unfit as a substrate for current implementations of oscillatory interference models. However, oscillatory interference models can tolerate small amounts of noise, suggesting the utility of circuit level effects which might reduce oscillator variability. Further implications for grid cell models are discussed.Author Summary: For many animals, including rats, accurate spatial memory over relatively large areas is important in order to find food and shelter. Just as unique points in time can be efficiently represented by combinations of repeating elements like hours, days, and months, points in space can be represented as combinations of elements that repeat at different spatial scales. Just such a code has been identified in the brains of rats and it shows an intriguing triangular spacing of encoded locations. Two different explanations have been developed as to what general mechanism in the brain might be able to generate this unusual code. However, to date there is not conclusive experimental evidence indicating whether either of the two explanations is correct. Here we show in detail that one of the explanations, called oscillatory interference, has specific requirements regarding the amount of variability in the system that implements it. We then report data experimentally examining candidate systems to evaluate their levels of noise. The large amount of noise that we find presents a challenge to the currently suggested biological implementations of oscillatory interference, but it does not provide support for the alternative explanation.