Why place cells fire in many fields across large spaces: a hierarchical predictive map ties field statistics to the scale of behavior

The predictive map no longer forecasts future position one primitive step at a time; it writes subgoal-directed subroutines, such as heading for a doorway or the end of a corridor, into the map as single compound transitions, so one representation both transfers across tasks and stays usable as the environment grows. The spatial scale of the standard successor representation is fixed by a single discount factor, so in large environments it either blurs local position and aliases distant locations with broad fields or cannot cover a large space with a fixed population using narrow fields; adding subroutines directly to the successor representation instead produced negative transfer, with learning slowing across tasks. Agents ran 20 navigation tasks from a shared start to different goals in topologically identical four-room environments of side 11, 17 and 25 states, with readout weights reset at each goal so only the state representation carried information between tasks; the hierarchical successor representation with eigenoptions beat the naive successor-plus-options combination at all three sizes, and the advantage held when the number of options changed, when representations were learned online and when options were constructed online.

The multiple fields of a place cell can be read as one task-shared predictive basis, with each cell a reusable part of that basis rather than a one-to-one stand-in for a state. The classical move identifies place cells with the columns of the predictive map, which caps the number of modelled neurons at the number of states and, because the successor and hierarchical successor representations have a dominant diagonal, would give every modelled cell at least one field, contradicting the finding that only a minority of CA1 pyramidal neurons express fields in a given environment. A non-negative, overcomplete decomposition of the predictive maps of all tasks yields one basis vector per model cell; on the 48 m rat track the model's field counts were heavy-tailed and a gamma-Poisson description beat the equal-Poisson description by a wide maximum-likelihood margin, matching rat CA1 recordings on the same track, while the factorised successor representation and raw hierarchical successor columns did not reproduce this structure.

Field statistics change approximately as power laws with environment length, because the range of subroutine scales expands as the track lengthens while the finest scale stays fixed. The earlier random-process account reproduced field sizes, spacing and shapes within a single environment, but its key parameters were fitted separately to each environment and it did not explain why those parameters should change with size. On 10, 24 and 48 m rat tracks and 15, 130, 180 and 200 m bat tracks, spatial resolution and all model parameters were fixed, so only the number of states and the resulting number of options changed; model and both recordings shared four trends, with mean fields per cell, the proportion of active cells and the within-cell largest-to-smallest field-width ratio rising with track length while the fraction of track covered by one cell's fields fell.

Population firing rates carry a spatially periodic modulation whose wavelength is set by the finest subroutine scale and therefore does not change with environment length. This was a new prediction of the model that had not been tested in experimental data, and populations with independently placed fields drawn from equal-Poisson or gamma-Poisson processes do not produce the modulation. On 10, 24 and 48 m tracks the model's summed population activity peaked in spatial autocorrelation at similar non-zero lags and had similar spectral centroids, both significantly different from shuffles that relocated each cell's fields at random; rat CA1 place cells showed qualitatively similar periodicity with no detectable trend in wavelength with track length, and bat cells on 130 to 200 m tracks showed a wavelength an order of magnitude longer with no detectable difference between lengths.

The comparison is confined to rat and bat CA1 recordings on linear tracks, with model parameters and spatial resolution fixed across all track lengths so that only the number of states and options changes; the authors note that natural navigation unfolds in large two-dimensional spaces and predict elevated population activity at bottlenecks where options terminate in two dimensions, which requires new recordings to test.

Quantitative deviations remain between model and data, and the authors suggest factors outside the normative theory, such as heterogeneous excitability, also shape field statistics; modelling place cells as a single task-agnostic basis gives each model cell identical fields in the two running directions, at odds with the directional firing observed on linear tracks; on the 15 m bat track the gamma-Poisson and equal-Poisson descriptions could not be distinguished, which the authors attribute possibly to few neurons recorded there.