spatial scales aligned to the horizontal and vertical axis (see Fig. S8)
14
. In grid cells, the spatial scales tend
to be approximately constant in all directions unless the environment changes
55
. The principal components
of Gaussian place field activity are mathematically related to the SR eigenvectors, and naturally also have
grid fields that scale independently along the perpendicular boundaries of a rectangular room. However,
Dordek et al. found that when the components were constrained to have non-negative values and the
constraint that components be orthogonal was relaxed, the scaling became uniform in all directions and
the lattices became more hexagonal
23
. This suggests that the difference between SR eigenvectors and
recorded grid cells is not fundamental to the idea that grid cells are applying a spectral dimensionality
reduction. Rather, additional constraints such as non-negativity are required.
The SR can be viewed as occupying a middle ground between model-free and model-based learning.
Model-free learning requires storing a look-up table of cached values estimated from the reward history
1, 56
.
Should the reward structure of the environment change, the entire look-up table must be re-estimated.
By decomposing the value function into a predictive representation and a reward representation, the SR
allows an agent to flexibly recompute values when rewards change, without sacrificing the computational
efficiency of model-free methods
4
. Model-based learning is robust to changes in the reward structure, but
requires inefficient algorithms like tree search to compute values
1, 15
.
Certain behaviors often attributed to a model-based system can be explained by a model in which
predictions based on state dynamics and the reward function are learned separately. For instance, the
context preexposure facilitation effect refers to the finding that contextual fear conditioning is acquired
more rapidly if the animal has the chance to explore the environment for several minutes before the first
shock
57
. The facilitation effect is classically believed to arise from the development of a conjunctive
representation of the context in the hippocampus, though areas outside the hippocampus may also develop
a conjunctive representation in the absence of the hippocampus, albeit less efficiently
58
. The SR provides a
somewhat different interpretation: over the course of preexposure, the hippocampus develops a predictive
representation of the context, such that subsequent learning is rapidly propagated across space. Figure S5
shows a simulation of this process and how it accounts for the facilitation effect.
Recent work has elucidated connections between models of episodic memory and the SR. Specifically,
Gershman et al. demonstrated that the SR is closely related to the Temporal Context Model (TCM) of
episodic memory
16, 19
. The core idea of TCM is that items are bound to their temporal context (a running
average of recently experienced items), and the currently active temporal context is used to cue retrieval of
other items, which in turn cause their temporal context to be retrieved. The SR can be seen as encoding a
set of item-context associations. The connection to episodic memory is especially interesting given the
crucial mnemonic role played by the hippocampus and entorhinal cortex in episodic memory. Howard
and colleagues
59
have laid out a detailed mapping between TCM and the medial temporal lobe (including
entorhinal and hippocampal regions).
Spectral graph theory provides insight into the topological structure encoded by the SR. We showed
specifically that eigenvectors of the SR can be used to discover a hierarchical decomposition of the
environment for use in hierarchical RL. Mahadevan et al. demonstrated that the related Laplacian
eigenvectors are useful as a representational basis for approximating value functions, dubbing these
eigenvectors “protovalue functions”
60
. Spectral analysis has frequently been invoked as a computational
motivation for entorhinal grid cells (e.g., by Krupic and colleagues
61
). The fact that any function can
be reconstructed by sums of sinusoids suggests that the entorhinal cortex implements a kind of Fourier
transform of space. However, Fourier analysis is not the right mathematical tool when dealing with spatial
representations in a topologically structured environment, since we do not expect functions to be smooth
over boundaries in the environment. This is precisely the purpose of spectral graph theory: Instead of being
maximally smooth over Euclidean space, the eigenvectors of the graph Laplacian embed the smoothest
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The copyright holder for this preprint (which was not. http://dx.doi.org/10.1101/097170doi: bioRxiv preprint first posted online Dec. 28, 2016;