Extended Data Fig. 12: Left-right alternation is a stable regime for efficient spatial coverage.
From: Left–right-alternating theta sweeps in entorhinal–hippocampal maps of space
a, The sum of sweeps across grid modules has a characteristic beam-like spatial footprint which expands and decays with distance. Left: simulated spatial firing patterns (top row) and footprints of sweep terminal positions (bottom row) for three simulated grid-cell modules with Gaussian fields. Note the proportionality between the grid field size, grid inter-field spacing, sweep footprint size and sweep distance. Centre: when the sweeps of the three modules are overlaid, the beam-like shape of the sweep becomes evident. Right: sum of sweep footprints across the three modules, weighted such that each module makes the same total contribution. b, Creating a simplified, scale-free model of the multi-module sweep footprint. For a given point in the agent’s environment (red dot), the footprint intensity is calculated by multiplying two spatial functions. First, a beam-shaped angular intensity profile (left) is created by calculating the Von-Mises probability density function of the point’s angular offset relative to the sweep direction (blue arrow). Next, a decaying radial profile (center) is created by calculating the inverse squared distance from the agent. Multiplying these two intensity profiles yields a decaying beam shape, similar to the multi-module sum shown in a. c, Optimality and bimodality of the agent’s choices in 1,000 runs of the simulation in Fig. 6b. Left: average value of the largest (‘worst’) and smallest (‘best’) sweep overlap values at each time step, considering all possible sweep directions. The overlap value for a sweep at a given position and direction is defined as the product of the preexisting coverage trace with the footprint of the current sweep. Error bars indicate the 5th and 95th percentiles. Right: circular histogram of optimal sweep directions (smallest possible overlap with previous sweeps), relative to the direction of movement (‘front’). d, Alternation is stable within a range of sweep widths. Left: the angular concentration of the sweep is set by the Von Mises distribution parameter κ. Top right: alternation score (mean and 5th−95th percentiles) for 1,000 repeated simulations across a range of sweep widths. Bottom right: colour-coded histograms (columns) of optimal sweep directions for different sweep widths. Note the bimodality of sweep directions for the intermediate range of sweep widths that also yielded high alternation scores. e, Rapid forgetting yields consistent alternation in the open field. Left: temporal decay of the cumulative coverage trace is achieved by introducing an exponential forgetting factor τ. When τ < 1 (bottom) the penalty trace fades over time, meaning that the agent is less influenced by earlier sweeps (transparent blue arrows). The direction of the final sweep (opaque blue arrow) changes depending on how much the penalty from the first sweep has decayed. Right panels: exploration of κ and τ parameter space in simulations where the agent was tasked with choosing sweep directions based on the recorded behavioral trajectories of each rat. Top right: average alternation score of the agent’s chosen sweep directions, for each combination of κ and τ, based on the running trajectory of an animal during a recording in the open field (same session as Fig. 6e,f). Bottom right: circular correlation coefficient (‘ID correlation’) between decoded internal direction (ID) and the simulated sweep directions when the agent was run in ‘empirically driven’ mode (as illustrated in Fig. 6d). Blue crosses indicate the position of the maximum. Note that rapid forgetting and intermediate sweep widths results in robust alternation (τ: 0.01 ± 0.00, κ: 2.5 ± 0.67, mean ± s.e.m. across 13 animals) and high correlation with empirical sweep directions (τ: 0.04 ± 0.01, κ: 2.88 ± 0.11, mean ± s.e.m. across 13 animals). f, A single-sweep memory trace is sufficient to produce alternating sweeps. As in Fig. 6b, the agent moves along a linear path, generating a sweep at each time step. However, in this case the sweep memory trace retains only the most recent sweep. As in Fig. 6b, sweep directions (blue arrows) reliably alternate aside the direction of movement. g, Alternation of sweep directions increases with path straightness. Path straightness was computed for each time step of the recording by dividing net travel distance by cumulative travelled distance within a 2 s moving window (high values correspond to straight paths). Left: average alternation score of the directions chosen by the agent during an open field session, binned at different levels of path straightness. Each set of connected dots (with same color) shows data from a different animal (n = 13). Right: same, but for empirical decoded directions from the same animals. h, The agent accurately predicts empirical sweep directions. Heat map of decoded direction and predicted sweep directions (both in egocentric coordinates, relative to head axis) showing a high degree of correspondence between empirical and predicted sweep directions. i, Simulations based on multi-module sweep footprints. Sweeps were modelled as three Gaussian functions (corresponding to three grid modules, as illustrated in a) whose width and sweep length increased with a geometric ratio of 1.5. Each row shows a simulation in which the agent moves along a linear path, with sweeps (colored circles) generated by each module at every time step. In each simulation, the selection of sweep directions is governed by a different rule. Top (‘common’): all modules are governed by a single common sweep direction, which minimizes the overlap of the summed multi-module footprint with the summed multi-module trace. Middle (‘parallel’): same as top row, except that sweep directions are chosen individually for each module. On a given time step, the modules’ respective sweep directions are chosen in parallel (i.e. modules are agnostic of each other’s sweep directions within same time step). Note the preservation of coordinated alternation across modules, like in empirical data. Bottom (‘serial’): same as middle row, except that module sweep directions are now chosen sequentially within each time step (from the smallest module to the largest module). In this serial version, note (1) the absence of strong alternation in any module, and (2) the closer packing of each module’s footprints. These differences collectively suggest that alternating sweep directions in vivo are optimal for parallel processing in grid modules, rather than serial processing. Right of each row: mean alternation score for each sweep in the simulation, calculated over 1000 repetitions of the simulation with random initial conditions. Credit: rat, scidraw.io/Gil Costa; robot, openclipart.org/annares.
