Attractor and integrator networks in the brain — part 4/10

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(prediction 4)), as evidenced by a close match between the isometrically parameterized internal
ring states and the measured head direction, Fig. 5b (inset, right).
Natural perturbations away from the ring ﬂowed back to it, Fig.5d [19] (prediction 2)), and the
ring manifold was invariant across waking and REM sleep, Fig.5e [19, 96] (prediction 3)). These
ﬁndings explicitly validate the most fundamental predictions (predictions 1)-3)) of ring attractor
models and continuous attractor-based integrators (predictions 1)-3) with 5)), providing arguably
the most direct and compelling evidence of continuous attractor dynamics in the brain.
In a striking example of convergent evolution [155, 151 ], Drosophila compute HD estimates
using apparently very similar dynamics [156, 157, 153, 152 ]. The ﬂy neural compass circuit is
topographically organized such that the neuropil forms a physical ring-shaped structure in the
ellipsoid body , with a local moving activity peak that tracks head direction as the ﬂy turns, Fig. 5f.
Another notable advantage of the ﬂy circuit in the effort to characterize its mechanisms is that the
number of neurons is small and their morphology and connectivity has been fully traced [158],
Fig.5g. This detailed view of the circuit permits quantitative, not just qualitative, comparisons with
ring attractor models. The combined activity and connectivity data reveal that the ﬂy HD system
implements the copy-and-offset double-ring network architecture proposed for velocity integration
[159, 14]. The actual dimensionality of the ﬂy HD circuit and its full state-space dynamics remain
to be characterized; even though the circuit is organized physically as a ring network, recent
evidence suggests that the insect HD circuit may be involved in performing 2-dimensional path
integration as well [160, 161], and thus unlike the ADn network in mammals, may not be conﬁned
to a 1D ring of attractor states that fully factorizes head direction in its representation of spatial
variables.
Finally , the HD system can be re-anchored and reset based on tuned external cues [152, 153],
which can change the orientiation tuning curves of cells and moment-by-moment ﬁring rates of
cells in a way that remains consistent with prediction 3) for attractor dynamics.
Grid cells
Grid cells encode spatial location though a regular triangular-lattice discharge pattern that tiles
explored 2D spaces [162], representing 2D position as a set of spatially periodic 2D phases [163,
129]. They update their states while moving in the light and the dark [162], presumably based on
motion cues. Continuous attractor models [15, 164, 16, 165 ] predict that the population states of
a module – a set of grid cells with a common period – should be conﬁned to merely 2 dimensions
regardless of environment and behavioral state [16], forming a manifold that is topologically a
torus, Fig.1d (rightmost column).
Indeed, grid cells from one module (“co-modular cells”) have identical periods and orientations
and all possible 2D phases, suggesting a 2D set of states [166, 76 ] (prediction 1)). The relative
ﬁring phase and relative grid parameters of pairs of co-modular cells is tightly conserved even
as the spatial tuning and spatial phase of single cells varies across time and across familiar and
novel environments (Fig. 6a) [166, 76 ], across the dimension of the spatial environment (6b)
[167], and despite environmental rescaling that leads to large deformations in the spatial tuning
of grid cells [76] (prediction 3)). Moreover , the detailed cell-cell relationships (whether a pair
of cells is co-active, active in quadrature, or active fully out of phase) that are seen in waking
exploration are conserved across overnight REM and non-REM sleep for grid cells but not place
cells, Fig. 6c [100, 101], establishing that the low-dimensional states are autonomously generated
(prediction 3)). These ﬁndings established that the structure of the grid cell response is very low-
dimensional on a population level, invariant across environments, time, and behavioral states,
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and internally stabilized and autonomously generated – validating the fundamental predictions
[16] made by continuous attractor grid models. Most recently , these ﬁndings were reproduced
by a direct visualization of the state-space manifold, Fig6e made possible by large population
recordings of grid cells across waking and sleep that conﬁrmed the toroidal state-space structure
of co-modular cells [20].
A corollary of these ﬁndings is that the grid cell response is not derived from upstream place
cell inputs since place cells remap across environments and during sleep while grid cells retain
their population structure (Fig. 6c); this corollary belies and is inconsistetnt with models in which
the place cell response is primary to grid cells [168, 169, 170 ], as shown in [100].
Given the preserved internal structure and autonomous dynamics of grid cells across states,
time, and environments, it follows that various deviations in the spatial tuning curves of grid cells
from equilateral grid-like responses in 2- and 3-dimensional spaces [171, 172, 173, 174, 175, 176]
likely result from variability in how the invariant internal states are driven by and mapped to
external cues and states: for instance through altered velocity estimation [16, 76] or feedforward
inputs that shift the phase of the grid cell network [177, 178, 179 ]. This has been veriﬁed in the
case of the expansion of grid cells in novel environments [76] and is almost certain to hold – given
the preponderance of evidence of internal grid stability [166, 76, 180, 100, 101, 20] – when tested
in various other conditions that report grid deformations as well [172, 173, 174, 175, 176, 181 ].
In sum, the HD and grid cell systems conﬁrm that the same pattern formation principle – based
on local excitation or disinhibition, with broader inhibition – that is pivotal for morphogenesis in
plants and animals [39] is also fundamental to the genesis of stationary continuous attractor states
for computation and representation in the brain.
Graded working memory networks
In monkeys trained to make saccades to previously cued targets (selected from a set arranged in a
circle), neurons in PFC and PPC exhibit persistent activity tuned to the direction of the initial cue,
across the delay period after cue removal (predictions 1), 3)) [182, 183]. Analysis of delay-period
PFC activity [85] in a population of simultaneously recorded shows that the delay period activity
bump moves apparently randomly along a 1-dimensional manifold, with the characteristics of
a diffusion process, so that the variance in the location of the bump grows linearly with time,
as predicted by continuous attractor models [52, 16, 19 ], but the bump proﬁle remains largely
invariant over the duration of the delay (predictions 1) and 2), assuming that the diffusive process
is indicative of natural noise-driven perturbations of the system). The bump movement predicts
subsequent behavioral errors [85], suggesting that these states are repositories or readouts of the
memory .
The need for extensive training on the task and the observed tailoring of the attractor states
to this speciﬁc multi-cue task suggests that this attractor formed through learning in a ﬂexible
system rather by (re)using a genetically pre-speciﬁed circuit. Given the apparent malleability of
this attractor network, we might therefore also expect a loss of the neural correlation structure if
the animal is subsequently trained on other tasks, unlike with the grid and HD cell networks.
Attracting limit cycles and trajectories
The central and peripheral nervous systems contain numerous instances of periodic dynamics,
ranging from the spiking of single neurons [184, 185 ] to circadian rhythms and sleep cycle gen-
eration [186], to rhythmic activity in motor circuits. While linear oscillators have amplitudes set
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by the initial condition, attractive limit cycle oscillators have an intrinsic and invariant amplitude.
Thus, not all systems with oscillatory behavior are limit cycle attractors: oscillations that decay or
grow over time or whose long-term amplitude or frequency changes after a transient perturbation
are not limit cycles. Driven (non-autonomous) systems may exhibit limit cycles because of their
inputs rather than intrinsic attractor dynamics [187].
Many of the oscillations noted above maintain a ﬁxed amplitude, and because of their strong
functional imperatives for robustness to perturbation are almost certainly generated through at-
tractor dynamics. Particularly well-characterized examples are central pattern generators (CPGs)
in motor circuits of the peripheral nervous system, that drive swimming, crawling, walking, breath-
ing, and digestion, and differ in speciﬁcs across species but have common principles of mechanism
and operation, including high robustness [188, 189 ]. CPG circuits typically integrate external
feedback but have been shown to be able to operate in isolation without external drive [190].
Given the sizeable literature on these topics, we refer the reader to some excellent papers and
reviews [191, 192, 193, 194, 195, 186 ].
Departures from low-dimensional continuous attractor dynamics
Not all circuits hypothesized to exhibit low-dimensional attractor dynamics appear under further
experimentation to do so, or currently lack sufﬁcient evidence to establish such dynamics within
the circuit. We discuss three potential examples.
Orientation tuning in V1
The circuit of simple cells in V1 satisﬁes some key properties of ring attractor networks [11]: V1
and V2 cells exhibit strong orientation-tuned responses to real and illusory edges in the visual
world [196, 197, 198 ], and population-wide V1 spontaneous activity during sleep is correlated
with these tuned population coding states [199]. While these suggests that illusory edge re-
sponses may be driven by self-generated attractor dynamics, they tend to occur after a longer
latency than real edge responses, making them more likely to be driven by top-down inputs rather
than within-V1 dynamics. Next, moving an attractor state along a continuous attractor manifold
requires strong inputs and is slow [200, 201 ]. These features seem inconsistent with the imper-
atives of a perceptual system to respond rapidly to changes in input [202], and the dynamics of
state ﬂuctuations in sleep appear relatively rapid compared to attractor time-scales [199]. These
observations seem to lend more weight to models in which the circuit response is dominated by
feedforward drive [196, 203], possibly with recurrently generated but fast non-normal ampliﬁca-
tion processes [49, 204]. More quantitative characterizations of response speed will be important
to draw clear conclusions about competing models for V1 circuit dynamics.
Place cells
Place cells form stable and detailed representations of familiar 1-2 dimensional spatial environ-
ments [205], which can persist in the dark [206] and for short intervals after the animal has
fallen asleep [207, 208 ]. In any particular 2-dimensional environment, the population response
lies on a low-dimensional manifold in state-space [86]. Accordingly , the place cell circuit has
been modeled as a 2-dimensional continuous attractor network [209, 210] or as a superposition
(with overlapping neural membership) of a discrete number of such 2-dimensional continuous
attractors, each representing a different environment [211]. However , the capacity limitations of
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generating multiple high-resolution maps within one homogeneous attractor network are severe
[212, 213, 163, 214 ]. And, unlike grid cells, place cells do not, across long sleep bouts, exhibit
the spatial correlations measured in awake exploration Fig. 6c [100, 101, 207 ]. Even during
waking, cell-cell correlations are not preserved across environments because of the phenomenon
of remapping [215, 216]. Like V1 neurons, place cells might be better described as deriving their
tuning by forming conjunctions of multiple feedforward inputs, including from grid cells and cells
that encode external cues like borders, landmarks, and reward sites [217, 218, 129, 214 ]. At the
same time, place cells exhibit sequential activation of previous trajectories during activity replay
[208, 219, 220, 221], which is hypothesized to be generated by recurrent connections in CA3, sug-
gesting that recurrent and feedforward dynamics collaborate in the generation of place cell states;
recent models are beginning to capture this interplay [210]. Closing the book on the question of
autonomous low-dimensional dynamics in the far more complex response of place than grid cells
requires more detailed experimentation, analysis, and modeling.
Motor cortical trajectories
Finally , recordings of motor cortical activity during stereotyped primate arm movements reveal
the existence of stable low-dimensional trajectories [84, 222, 223, 224, 225 ], similar to the tra-
jectories in state space originally characterized in olfactory circuit responses to odors [226]. Limit
cycles and other low-dimensional attractors have been hypothesized to play a key role in corti-
cal movement generation [227, 228]. The behaviors typically performed during neural recording
are themselves restricted to be stereotyped and low-dimensional, thus it remains unclear whether
activity would remain equally low-dimensional across a richer set of behaviors (e.g. over the set
of all possible arm movements). Recent evidence from perturbation experiments [187] suggests
that neural trajectories in motor cortex during skilled movements are driven by input from the
thalamus, and thus that the circuits for motor pattern generation in the central nervous system
might be distributed across multiple brain regions. Characterizing the intrinsic dimensionality of
motor cortical activity , and determining whether the command to make more-complex motions
involves multiple upstream or distributed primitive attractors, remain important open questions
for both clinical brain-machine interfaces and neuroscience.
Flexibility despite rigidity: modern glimpses into the broader poten-
tial of attractor networks
Above, the key predictions and experimental validations of attractors in the brain hinged on their
invariance, or rigidity , across time and conditions. The identiﬁed attractor states were highly
structured and low-dimensional. The weight symmetries and asymmetries underlying these states
were precisely tailored to the speciﬁc tasks performed by the systems. These properties appear
to run counter to a key desideratum for representation, memory , and computation in the brain:
ﬂexibility .
Recent experimental and theoretical work are beginning to shed light on how the brain might
solve the perennial conundrum of stability versus ﬂexibility through attractor networks: the low-
dimensional and rigid attractor states might be reused and recombined to create versatile and
efﬁcient systems for novel situations.
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