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

Abstract

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 = 5µ 3 /(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.

Cosine Directional Tuning of Theta Cell Burst Frequencies: Evidence for Spatial Coding by Oscillatory Interference

“…Burgess et al (2005Burgess et al ( , 2007 introduced an "oscillatory interference" theory, hypothesizing that theta oscillations are generated by velocity-controlled oscillators (VCOs), which perform path integration by modulating their frequencies in proportion with the speed and direction of a rat's translational movements. Supporting this idea, theta frequency is indeed modulated by a rat's movement speed (Rivas et al, 1996;Geisler et al, 2007), and oscillatory properties of spatial neurons are correlated with their spatial tuning parameters in accordance with predictions of oscillatory interference models (Burgess et al, 2007;Giocomo et al, 2007;Jeewajee et al, 2008a;Zilli et al, 2009). However, oscillatory interference models explicitly require that VCO frequencies vary as the cosine of an animal's movement direction, and such directional modulation of theta oscillations has never been observed.…”

Controlling Phase Noise in Oscillatory Interference Models of Grid Cell Firing

“…Extending this scheme to the in vivo situation, in which location-specific information (perhaps mediated by place cells) might reset VCO phases (Burgess et al, 2007;Monaco et al, 2011;Sreenivasan and Fiete, 2011;Bush and Burgess, 2014) and VCOs with constant phase offsets could be arranged as ring attractors (Blair et al, 2008), potentially provides unlimited additional stability. This situation is more consistent with the spatially stable theta-phase of firing observed in grid cells (Hafting et al, 2008;Reifenstein et al, 2012) and largescale coherence of the septohippocampal theta system (Buzsáki, 2002) than analyses of MPOs in vitro (Zilli et al, 2009;Remme et al, 2010). The provision of a local baseline oscillation might be a role for local circuits in MEC, while the septohippocampal theta system may contain the proposed coherent sets of VCOs (Welday et al, 2011).…”

“…Coupling VCOs with the same preferred direction Oscillatory MPOs measured in MEC slices have a phase SD of ϳ15 ms per cycle (Zilli et al, 2009), which would give stable grid firing for only ϳ3 s. However, the slice preparation may well disconnect grid cells from the ascending theta system that may contain coherent VCO networks (Welday et al, 2011). Furthermore, coupling neuronal oscillators to entrain one another to the same phase , or arranging them as a ring attractor such that they fire with sequentially offset phases (Blair et al, 2008;Welday et al, 2011), would achieve much lower variance and might correspond to the extended theta network operating in vivo.…”

“…However, a key question concerns whether the level of noise in neuronal oscillations is too high to allow the spatial stability observed in grid fields (Zilli et al, 2009;Remme et al, 2010). The observation of theta-phase precession in grid cells (Hafting et al, 2008;Reifenstein et al, 2012) implies that grid cell populations do maintain reliable spatially demarcated phase relationships in vivo.…”

“…For example, the baseline (local field potential) oscillation against which individual neuronal firing precesses may be the mean of the population activity (Burgess et al, 1993;Geisler et al, 2010), and oscillatory activity may be stabilized relative to the environment by location-specific phase-resetting inputs (Burgess et al, , 2007; O' Keefe and Burgess, 2005;Monaco et al, 2011;Sreenivasan and Fiete, 2011;Bush and Burgess, 2014). Zilli and Hasselmo (2010) showed how synaptically or electrically coupling VCOs with the same preferred direction reduces their overall frequency variability. Here we show how coupling multiple VCOs carrying information about displacement along different directions (via entrainment of the baseline oscillation), or with different spatial offsets (via ring-attractor dynamics; Blair et al, 2008), could produce the phase reliability and consistency needed to generate stable and coherent grid-like firing patterns.…”