Attractor and integrator networks in the brain — part 6/10

Cleaned-text code points [73105, 86851). Bibliography omitted here; source.md includes it.
Correspondences between attractor dynamics and anatomical layout, and
weight symmetries in models and biology
Anatomical topography , in which functionally similar neurons are near one another , is
neither a necessary nor sufﬁcient condition for the existence of an attractor: Any low-
dimensional attractor network is mathematically unchanged if all weights are preserved but
neuron locations are scrambled. However , if the network is merely a spatially scrambled
version of the idealized model, then the symmetries of the weight matrix can be revealed af-
ter an appropriate reordering of the neurons. An advantage of anatomical topography from
a biological perspective is that it can reduce the complexity of development, in that wiring
decisions can be guided by spatial proximity rather than depending entirely on activity or
other target cell signalling mechanisms; for instance, the locally competitive interactions
of grid and HD circuit models could be largely constructed through local arborization. It
also reduces overall wiring length in the mature circuit[275]. However , a circuit with3-
dimensional dynamics that are represented in an unfactorizable form cannot be embedded
topographically in a 2-dimensional cell layout, which limits the utility of topographic lay-
outs for circuits representing higher-dimensional manifolds.
Second, the posited weight symmetries in simple models of attractors need not exist in
a biological instance of the circuit with the same dynamics: If low-dimensional attractor
dynamics is only needed downstream of the size-Nrecurrent network that generates the
dynamics, in a set ofM<Nneurons, then the symmetries required for continuous attractor
dynamics can be spread across the recurrent and readout weights and the recurrent weights
alone without taking into account the readout weights will not reﬂect the relevant symme-
tries[253]. These considerations suggest a hypothesis for circuits with 2-D continuous
attractors: Evolutionarily conserved circuits that do not require extensive early experience
[276, 277]should be topographically organized. We might thus predict that the circuit that
originates HD signals in mammals should be topographically organized. By contrast, if the
low-dimensional dynamics only emerges on the basis of activity-dependent plasticity with
repetitive training, we may not expect the circuit to be topographically organized (or even
localized to single brain regions).
Remarkably , despite these caveats and in a beautiful example of the predictive power of
simple theories in neuroscience, the recent empirical evidence from the anatomy of the
zebraﬁsh oculomotor integrator and the ﬂy HD circuit show that nature has used precisely
the hypothesized constructions proposed in simple circuit models to build some integrator
networks.
Figure 1(preceding page):Mechanisms of attractor formation.In all plots, open gray neurons represent neurons, connections
between them are excitatory (black lines ending in bars) or inhibitory (black lines ending in circles). Left column: layout of neurons
and connections; connectivity matries shown as inset, with blue to yellow colors indicating strongly inhibitory to excitatory inter-
actions. Middle column: examples of stable population activity patterns. Right column: state-space views of population states and
dynamics. Red circles with gray-green ring indicate the activity states shown in middle column. (a) A network with dense symmetric
connections determined by associative Hebbian learning on a set of input patterns (middle) stores them as stable attractor states.
This deﬁnes a Hopﬁeld network. (b) Disjoint groups of neurons interacting through within-group excitation and across-group mutual
inhibition leads to group winner-take-all dynamics. Stable states are any patterns with only one winning group (the state-space plot
collapses all activities of neurons in group gialong the axisrgi). (c) Neurons arranged in a ring with global inhibition and either
local excitation or a lack of local inhibition combined with uniform excitatory input to all neurons produces localized activity bumps
(middle) as the stable states. Bumps may be centered anywhere on the neural ring, deﬁning a near-continuum of attractor states
that form a ring in state-space (right). (d) Neurons arranged on a two-dimensional sheet, interacting through local inhibition and
either center-excitation or a lack of inhibition near the center together with uniform excitatory input to all neurons results in a pattern
of multiple, periodically spaced activity bumps (middle). Any two-dimensional phase shift of the periodic pattern upto the lattice
periodicity are distinct but equivalent stable states, then the states repeat; thus these are predicted to form a torus in the state-space.
(e) Two neuron groups with in-group excitation and across-group inhibition, precisely tuned interaction strengths, and quasi-linear
neural fI responses can counteract activity decay in the network and produce persistent activity over a continuum of activity levels
in the two populations, deﬁning ramp-like population activity states and a line of attractor states. (f) Neurons arranged on a ring
with asymmetric connections that bias neural activity to ﬂow in a particular direction (middle) The network forms localized activity
bumps that sequentially move around the ring in that direction (right) The state space contains a limit cycle. (g) The copy-and-offset
mechanism for constructing integrators, illustrated for the ring (left) and grid (right) attractor circuits. Each network copy receives
velocity inputs tuned to the corresponding shift direction.
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Figure 2: Utility of low-dimensional attractor networks. (a) Persistent and stable states generated by attractor networks (red) can be
used to represent and remember external variables (blue) by constructing an appropriate mapping between them (vertical lines). (b)
Noise-robustness: attractor networks error-correct by mapping noisy states to the nearest attractor state [267]. Main: N -dimensional
noise drawn from the unit sphere centered on a 1D attractor has a projection strength of only 1 /N along the attractor: in this counter-
intuitive high-dimensional geometry , a ball is more like a pancake with the attractor orthogonal to the large dimensions [19]. (c)
Flow to the nearest (continuous or discrete) attractor can perform a nearest-neighbor computation and thus perform classiﬁcation:
e.g. the two attractors may represent ”cat” and ”dog” perceptual manifolds, and the blue dot a speciﬁc input data point. (d) Left:
Continuous attractors can become integrators if velocities or movements in the external space are inputs to the network and induce
proportional shifts in the internal attractor state: The current state on the attractor is then the integral of past velocity inputs relative
to the starting state. Right: if the input to an integrating attractor consists of temporally varying evidence pulses (bottom, evidence
about one option in blue and evidence about the opposing option in khaki), these will move the state on the attractor (top) so its
current state reﬂects the integral of the total evidence.(e) The energy landscape of a combined integration and decision making
network: inputs push the state left or right, and as it integrates, the network state also moves toward one of two discrete attractors
(left and right; white arrows: two sample trajectories). Arrival to the neighborhood of one of the discrete attractors is a decision point
[63, 64]. (f) An integrator can be quickly re-purposed to represent multiple different and new external variables simply by yoking its
velocity shift mechanism to different external velocities cues by feedforward learning. It also does zero-shot learning and inference:
Given an initial state and an input velocity trajectory , it will generate a self-consistent representation for the current state even if the
trajectory if different and new each time [229, 231, 232 ]. (g) A set of (continuous or discrete) attractor subnetworks (red boxes at
bottom) can interact bidirectionally with a shared network to form a high-capacity attractor network [241, 242, 268, 60 ]. (h) Mixed
modular representations can enable representation of inputs of different dimensions by resuing the same attractors of ﬁxed dimension
each. V elocities (vi ) from external spaces of potentially different dimension are selected by a set of selection signals ( si ). The selected
velocity (green) is routed through random projections to a set of M modular integrator networks of dimension K each. This kind
of mixed modular circuit can interchangeably represent a variety of input spaces of dimension D  MK while smoothly trading off
resolution for dimension [229].
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filtered MU
filtered Vm
MU
Vm
a
time to go cue (s) prop. trials
activity along choice dirn. (AU)
unsuccessful perts end points
successful perts
b c
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odor
odor
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Figure 3: Evidence of discrete attractor dynamics in the brain. (a) Multi-unit (above) and single-unit (middle) activity during
cortical up- and down-states show signatures of bistability (clusters and histograms at bottom). Reproduced with permission from
[269]. (b) Delay-period dynamics in rodent premotor area ALM during a binary decision task before the animal can make a motor
report of its decision appears to converge to one of two discrete end points (blue and red curves and histograms, top). Perturbations
are either robustly ignored (top), or ﬂip the dynamics so that the end points are reversed (bottom). Reproduced with permission
from [123, 119]. (c) Evidence of all-to-all inhibition and competitive winner-take-all recurrent dynamics in the ﬂy olfactory system:
Kenyon cell (KC) responses to odors, with input from the globally projecting APL inhibitory neuron, are sparse (top left, Ca ﬂuorescence
response to odor IA) and decorrelated across odors (top right); blockage of KC drive to APL or APL inhibition to KCs results in dense
and correlated odor responses (middle, bottom). Reproduced with permission from [137].
4 s
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Figure 4: Linear attractor dynamics generated by nework feedback in the oculomotor integrator . (a) Stable horizontal gaze
during ﬁxation at different angular positions (top two traces) is supported by stable steps in ﬁring rate by oculomotor integrator
neurons (bottom two traces) that integrate transient ( ⇠ 100 ms) saccadic command bursts. Reproduced with permission from [140].
(b) Oculomotor neurons maintain eye position through linearly ramping tuning curves (bottom); responses are the same in the light
and dark (top) and thus do not depend on visual input for gaze stabilization on the time-scale of seconds. Reproduced with permission
from [270]. (c) Transient current injection into single oculomotor neurons reveals a transient, not persistent, decrease or elevation
(inset) in ﬁring rate, consistent with lack of a cellular origin for persistent intersaccadic ﬁring. Reproduced with permission from
[140]. (d) Injection of kainic acid into the oculomotor integrator produces leaky dynamics even in the light (inset, faster leak in the
dark), consistent with network models. Adapted with permission from [271]. (e) Visual feedback mimicking leaky or unstable eye
positions in goldﬁsh can mistune the oculomotor integrator , making it unstable or leaky , respectively . Adapted with permission from
[272, 145].
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measured
SPUD
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Figure 5: The head direction circuit: a ring attractor in the brain. (a) Activity of two cells in the rat HD circuit during free foraging
in a 2-dimensional circular arena with a globally orienting cue (top). When the cue is removed (bottom), the ﬁelds rotate but the
cells maintain their tuning shapes and relative tuning angles (pale curves: top plot, globally rotated. Adapted with permission from
[75, 98 ]. (b) A nonlinear 2-dimensional embedding of the population-level states of the thousands-of-neurons sized mammalian
thalamic area ADn recorded during 2-dimensional free-foraging: the states are conﬁned to a 1-dimensional ring (cf. Fig. 1c); here,
colors encode the measured head direction of the rodent. Inset: Non-linear embedding of states from a different rodent (left), with
coloring obtained by an isometric parametrization along the ring by SPUD [19]. Reproduced with permission from [19]. (c) A close
match between unsupervised isometric parametrization of the manifold from (b, inset) and the externally measured head direction
of the rodent. Reproduced with permission from [19]. (d-e) The same cells as in (b, inset), recorded during REM sleep (green): the
states during REM remain conﬁned to a 1-dimensional ring that precisely overlays the ring of waking states (blue, in (e)), and exhibit
large ﬂows back toward the ring (in (d)) Reproduced with permission from [19]. (f) Calcium imaging of activity in the physically
ring-shaped Drosophila ellipsoid body reveals a localized bump of excitation that follows the movement of a cue in the ﬂy’s visual ﬁeld.
Reproduced with permission from [156]. (g) Combination of electrophysiology and EM imaging of the central complex in bees (left)
and ﬂies (right) provides detailed layout and connectivity data for comparison with predicted connectivity in ring attractor models.
Reproduced with permission from [161, 273].
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