Attractor and integrator networks in the brain — part 5/10

Cleaned-text code points [61044, 73105). Bibliography omitted here; source.md includes it.

Exploiting integration for rapid representation: reuse of continuous attractors
Strikingly , all established stationary continuous attractor networks in the brain are also integrators.
This seems surprising given that not all continuous variables represented in the brain need be
accumulation of evidence or navigation-like variables. Here we discuss how , even for just the
problem of representation, the functionality of integration could serve a vital role by enabling
rapid construction of new representations.
Building a representation (mapping values of an external variable to the internal states on a
continuous attractor) as in Fig. 2a can proceed by painstaking construction of a large set of asso-
ciative feedforward correspondences: Visit each external state and associate it with an attractor
state. Building this lookup table requires full exploration of the space. By contrast, if the attrac-
tor is an integrator , only two feedforward correspondences must be built: identify one value of
the external variable with one internal state – an anchoring process – then associate the velocity
signal with the shift mechanism in the integrator through a learned feedforward projection that
is independent of location on the attractor or in the external space, Fig. 2f [229]. The circuit
can now automatically generate appropriate and consistent representational states for future and
previously unvisited values of the variable based on displacements. This mapping is rapid and
does not require exhaustive exploration of the space or reconfiguration of the recurrent attractor
circuit, a form of generalizable and rapid learning [230, 231, 229 ] and zero-shot memory state
construction [229]. Moreover , an integrator network can correctly infer the current state upon
returning to it along a previously untraversed path (novel trajectory), a form of zero-shot inference
[231, 229, 232 ].
This rapid integration-based mapping process permits another use: A single attractor can be
easily reused to represent mutiple variables, Fig. 2f: By simply adding another anchor point for
another variable Z and driving the shift mechanism with velocities related to changes in Z, the
network can switch from representing variable X to de-novo representing Z, or can alternate
between representing X and Z without any reconfiguration of the attractor itself [229]. Consistent
with this idea, it appears that the brain (re)uses grid and place cells when navigating through both
the spatial environment and through non-spatial cognitive domains [233, 234, 235 ].
Multiple modular attractors for high-capacity representation
In general, a fully connected symmetric attractor network of N neurons permits the construction
of ⇠ N chosen attractor states (where the notation ⇠ refers to the functional scaling with N
with potential prefactors that do not depend on N ), a result established by a large body of work
from statistical physics [236, 237, 238 ] and information theory [239]. The result is independent
of learning rules [239, 236 ] and not qualitatively altered by adding hidden neurons [60]. It
also applies to (near-)continuous attractors, for which the total number of distinguishable states
(resolution) on an attractor manifold also scales in the same way .
A D-dimensional attractor with resolution ⇠ P per dimension would thus require a number of
neurons that grows exponentially with dimension, N ⇠ P D (this is one aspect of the curse of dimen-
sionality). For 2-dimensional space at a resolution of 10 cm per dimension, the full population
of ⇠ 106 rodent hippocampal cells – which have been hypothesized to function as a Hopfield-
like associative memory – could at most represent a combined 100m 2 area [163]. Similarly , the
number of cells in hippocampus is vastly smaller than the combinatorially large number of sparse
coding states of cortical neurons in rodents and humans, which might set the scale for the number
of items to be stored in memory . These arguments point to the need for much higher-capacity
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representations and memory than possible with fully-connected Hopfield-like networks.
Modular attractor networks permit efficient construction of a large number of representational
and memory states in neural networks [163, 129, 240, 241, 242, 243, 60, 214 ]. If M modular sub-
networks have ⇠ N discrete attractor states each, and these can update independently (uncoupled
modules), the combined system expresses a set of ⇠ N M states, exponential in the number of mod-
ules. Though these states are not attractors, it is possible to couple together these subnetworks to
generate exponentially many attractor states so they each have reasonable-sized basins and are
thus robust [244, 241, 242, 60, 57 ], Fig. 2. (By contrast, randomly connected Hopfield networks
also typically have exponentially many fixed points, but the basins are often small.) Similar ideas
have been applied in the temporal domain to show how networks might support exponentially
long activity sequences [245].
If each module expresses a continous attractor of dimension K and the subnetworks are inde-
pendent, their combined states define an MK dimensional manifold (with no error correction
between states on the manifold), solving the curse of dimensionality for representing higher-
dimensional variables while maintaining a structured representation for the variable that goes well
beyond random combinatorial codes [163, 129, 246]. These subnetworks can be coupled together
through their shift mechanisms, forcing a certain fixed relationship between modules in how their
relative states update [243], in which case the coupled system exhibits one K-dimensional attrac-
tor but with large capacity , containing ⇠ eM states per dimension.
In short, modular subnetworks can work together to greatly expand representational capacity
in terms of number of attractor states while also maintaining a large denoising capability [244,
241, 242, 60, 57, 163, 129, 246, 243 ]. However , the vast set of attractor states made possible
by these coupled-module constructions are a rather structured set pre-defined as combinations of
the states in the individual modules, rather than being arbitrarily specified as the states are in a
standard Hopfield network. They do not directly encode user-defined patterns as the attractor
states, and thus do not violate the capacity limitations of neural networks [236, 237, 238, 239 ].
A critical question is how high-capacity and robust but structured sets of attractor states could be
leveraged for general memory and computation. Three recent works have begun to address this
question, showing that structured attractor states can be leveraged for robust labeling and action
selection [247], robust classification [248], and as a component of a heterogeneously structured
general associative memory that exhibits smooth degradation instead of catastrophic memory loss
as more patterns beyond capacity are added to the network sharma2022content.
Mixed modular codes for flexible representation
Finally ,M modular subnetworks that are each integrators in K dimensions can be (re)used with-
out any rewiring of recurrent weights to represent and store inputs of any input dimension  MK ,
Fig. 2h, using a mixed-modular coding scheme [229]. This scheme combines five concepts: Rapid
representation learning with the integration mechanism, the reuse of the same attractors for dif-
ferent variables, the capacity of modular attractors, the fact that a high-dimensional variable can
be represented unambiguously by multiple independent lower-dimensional projections, and the
fact that random projections tend to be independent.
In mixed modular coding, movement along each dimension in the external space is randomly
projected to the shift mechanisms of all modular integrators. Every module is thus involved in rep-
resenting every input dimension – a form of holographic representation [249, 250]. If the number
of input dimensions D is smaller than MK , all input dimensions are represented without informa-
tion loss, and excess module capacity ( MK  D excess dimensions) is automatically convertible
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into extending the coding range or resolution for each of the D input dimensions, instantly trading
off the number of represented dimensions and the dynamic range of each represented dimension
without recurrent plasticity .
Looking ahead
The theory of attractor dynamics in the brain has provided a powerful and unifying conceptual
framework for understanding integration, representation, memory , error-correction, and efficient
learning and inference in the brain. The experimental effort to study candidate attractor circuits
and test their predictions has been a fertile field of research, and population-wide physiology
techniques have led to breathtaking direct visualizations of attractor dynamics at work in the
brain.
The theory is also proving to be a powerful tool in interpreting how artifical neural networks
(ANNs) solve complex tasks. ANNs trained to robustly solve memory , integration, and decision-
making tasks in domains as diverse as spatial navigation, vision, and language, develop attractor
dynamics [251, 252, 253, 254, 255 ], suggesting that not only are attractor networks able to solve
such problems but might be necessary when the computing elements are memoryless neurons.
Further , equipping networks with preconfigured attractor networks can help to produce faster ,
more data-efficient and generalizable learning [230, 229]. Because ANNs can be trained on com-
plex tasks and then fully examined after learning, they will potentially more readily contribute
to the next chapter in our understanding of how continuous attractor networks can interact and
combine with other mechanisms to allow the brain to solve richer problems associated with intel-
ligence.
Notable mechanistic questions about attractor networks also remain open, including: Mov-
ing away from the high-firing-rate asynchronous spiking regime [256, 257] to better understand
whether low-firing-rate synchronous spiking networks might support attractor dynamics – and
thus permit a combination of fast time-scale dynamics like spike synchronization and oscillatory
phase dynamics [256, 258, 259 ]. For continuous attractors, understanding how the brain deals
with the problem of fine-tuning in linear networks or the imposition and maintenance of a con-
tinuous symmetry across neurons remains unknown and ripe for resolution [34, 260].
A few continuous attractor development models show how they could emerge simply through
unsupervised associative plasticity [10, 211, 178 ]; others are based on combining feedback of
known or plausible error signals with neural activity in relatively simple learning rules [261, 10,
262]; the rest train networks on a high-level goal with error backpropagation, combined with
other constraints on architecture or the form the solutions should take [263, 264, 252, 253, 230,
265, 254 ]. These models are incomplete for different reasons: the unsupervised models require
uniform exploration of the input variable space and suppression of recurrent weights during their
training; the backpropagation models do not offer an account of how the loss functions, learning,
and additional constraints might be generated in biological systems.
There is much left to do in the field and an exciting vista ahead. On the experimental side, tools
for high-resolution population-level neural recrdings and perturbation across multiple brain areas
[266, 82, 83] let us peer further and deeper than ever . On the theory side, future developments will
help us conceptualize how such circuits could help underwrite intellegent computation through
the formation, interaction, and reuse of multiple low-dimensional structures.
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