Attractor and integrator networks in the brain — part 2/10

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homeostatic plasticity [34, 35].
The attractor states in Hopﬁeld-like networks typically have highly mixed and overlapping neu-
ral memberships, even when they are well-separated in the state space, Fig. 1a (middle column).
In a special case of Hopﬁeld networks, neurons are partitioned into largely disjoint groups with
self-excitation within groups and inhibition between groups. In these winner-take-all networks,
Fig. 1b, the attractor states consist of non-overlapping active groups of cells, Fig. 1a,b (middle
columns).
Continuous attractors. How can one construct networks with a continuum of stationary
attractor states? Weights that are invariant to reﬂection about their diagonal lead to the formation
of discrete attractors, as we have seen. If the weights instead exhibit a continuous symmetry – for
instance, if the weights are invariant to continuous shifts in neural locations – then the set of
formed attractors will be related by the same symmetry and could thus form a continuous set.
The general principle for the formation of stationary continuous attractors is pattern forma-
tion [36, 37, 38, 39, 40, 41, 42 ]. Simple and spatially local competitive interactions lead to the
emergence of rich stable spatial activity patterns – neurons with excitatory coupling between them
become co-active, and suppress the rest of their neighbors through inhibition – a linear (T urning)
instability [36].
The following three elements provide a solution to the inverse problem for forming stationary
continuous attractors: 1) Nonlinear neurons with saturating responses or inhibition-dominated
recurrent interactions with a uniform excitatory drive [7, 16, 43, 44 ] to keep network activity
bounded. 2) Sufﬁciently strong recurrent weights with competitive dynamics in the form of local
excitation or disinhibition with broader inhibition to drive spontaneous pattern formation via the
T uring instability [36, 37, 38, 39, 40, 41, 42, 16 ]; these patterns become the attractor states. 3)
Some continuous symmetry in the weights (a continuous weight symmetry is one where, as some
variable is varied continuously , the weights remain invariant), such as translational or rotational
invariance, Fig. 1c-d, to ensure a continuum of attractor states. These conditions are generally
sufﬁcient, but not strictly necessary , for the construction of continuous attractors (see Box on “Cor-
respondences between attractor dynamics and anatomical layout”). If the continuous symmetry
of weights is sufﬁciently corrupted, the continuous attractor will fragment into a discretized set of
attractor states. Thus, the existence of stationary continuous attractors is fragile in the sense that
it depends on the maintenance of continuous symmetries.
A special set of networks, which do not involve pattern formation to generate continuous
attractor dynamics, are those with linear , planar , or hyperplanar attractors generated by neurons
with linear or near-linear response functions. In circuits of linear neurons, the network feedback
is a linear function of activity ( Wr where W is the weight matrix and r are the neural activities), as
is the activity decay (given by r). Such networks can stablize non-zero activity states simply by
tuning the strength of the feedback so that positive feedback cancels decay . The feedback matrixW
can direct feedback into the different dimensions of the state space; if feedback is directed largely
along one dimension, the network can support a line attractor , Fig. 1e. If it is directed equally
along two or more dimensions, it can support a plane or hyperplane attractor . T o create long-lived
attractors requires feedback to precisely cancel decay , thus the strength of network feedback must
be ﬁnely tuned [9, 45], in contrast with pattern-forming continuous attractor systems.
Non-stationary continuous attractors : Large non-symmetric (and nonlinear) networks with
strong connectivity generically exhibit limit cycle attractors or chaotic dynamics [46, 47]. Just as
point attractors emerge generically in large networks with strong symmetric weights and bounded
state spaces, chaotic attractors emerge generically in large recurrent networks with strong asym-
metric weights. Adequate asymmetries are easily achieved if excitatory and inhibtory synapses
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emerge from distinct sets of neurons neurons [47], as biologically necessitated by Dale’s law .
Despite the complexity of chaotic dynamics, these attractors are also highly structured in that
they are typically much lower-dimensional than the number of neurons in the network [48]. Non-
symmetric networks dominated by inhibition exhibit a single attractor at zero activity , though the
ﬂow towards the attractor in responose to perturbations can involve large transients in neural
activation that temporarily move the state further away from the attractor [49, 50].
The potential utility of attractors for computation in the brain
Networks with low-dimensional attractor dynamics exhibit myriad properties that can be vital for
computation in the brain 4. These include robust representation, memory , sequence generation,
integration, and robust classiﬁcation and decision making, ideas that have been extensively ex-
plored in the literature. In a later section, we will describe how , though attractor dynamics may
be rigid and invariant as needed for the roles listed above, recent theoretical and experimental
ﬁndings are beginning to reveal how these rigid constructions may also be exploited to perform
ﬂexible computation through reuse and re-combination across tasks.
Representation and memory
A representation of a set of inputs means the assignment of inputs to representational states (a
representation need not be injective), with the ability to reproducibly retrieve those states (’la-
bels’) when cued. Attractor networks provide a stable internal set of states that can be used for
reproducible representation of discrete or analog variables, by mapping states in the world to the
attractor states. One way to achieve this mapping is through a feedforward learning process that
associates each external state with an internal attractor state, Fig. 2a.
An attractor network can exhibit two kinds of memory: The ﬁrst is in the structure of the
weights, which specify the set of all attractors. If these weights are speciﬁed through an input-
driven learning process, this is a form of long-term memory about the inputs. The second is
the ability to maintain persistent activity in a stationary attractor state: if a system with multiple
stationary attractor states is initialized in one of them, it will tend to remain at or near the same
state for some time. In other words, the activation levels of the neurons contributing to that state
persist while the system remains in the state. This persistent activity response is thus a form of
short-term memory of the input that initialized the circuit. If these persistent memory states can
be activated without an explicit address, using just the content (or partial content) of the memory ,
they are content-addressable.
The short-term memory function of attractors depends on the prior formation of stable states
through long-term plasticity: For instance, in Hopﬁeld-like networks, states cannot persist if they
were not ﬁrst trained to be attractor states. Even models of short-term memory that are based on
synaptic facilitation rather than persistent activity rely implicitly on prior long-term plasticity to
construct recurrently stabilized neural ensembles that can be reinstated by random inputs [51]. In
4A system could theoretically be perfectly tuned such that every point in state space is a neutrally stable attractor ,
and thus the system has maximally high-dimensional attractor dynamics. However , because the robustness of attractor
networks is related to the low-dimensionality of the attractor states as quantiﬁed in the subsections below , the system
would lose most of its interesting computational properties: error correction /noise tolerance, nearest-neighbor com-
putation, pattern completion and content-addressable memory . It could perform integration but with no robustness to
noise.
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other words, these models cannot explain short-term memory for entirely novel inputs; however ,
combinations of attractors could enable more ﬂexible short-term memory , as we discuss later .
Denoising for ﬁdelity of representation and memory
If the representational states are attractors, then the representations are robust in the sense that
they perform denoising: If the input cues or initial conditions reﬂect noisy or corrupted versions
of an attractor state, the dynamics drive the state onto a point on the representational attractor ,
Fig. 2b (inset). When the attractors form a continuous manifold of dimension K ⌧ N , where
N is the number of neurons in the circuit, all noise in N K dimensions is erased. A noise ball
of unit radius in N dimensions (corresponding to random independent noise per neuron) has a
projection of size only ⇠
p
K/N ⌧ 1 along K dimensions. If K is low-dimensional, as is often the
case, and N ranges from 10 2 107 as estimated before for common microcircuits, this constitutes
a massive reduction in the sensitivity of the state to internal or input noise, Fig. 2b. Thus, most
noise is rendered impotent.
Denoising due to attractor dynamics is especially important for memory maintenance, as oth-
erwise noise-induced deviations would accumulate and grow over time. Discrete attractors con-
tinually erase all noise by mapping perturbed states back to the point attractor , resulting in zero
drift. With continuous attractors as memory states, all noise orthogonal to the manifold is cor-
rected, thus there is a net reduction of the effects of noise by the factor ⇠
p
K/N ⌧ 1 [52, 53 ].
However , all states on the attractor manifold are neutrally stable so movements along the attrac-
tor are allowed. Thus, components of noise along the K attractor dimensions are not internally
corrected and cause an accumulating drift away from the inital state, with variance proportional
to KT /N , where T is the elapsed time [52, 16, 53, 54 ]. Thus, even continuous memory states can
be well-stablized in sufﬁciently large attractor networks.
Although content-addressable long-term memory and error reduction can be instantiated through
few-step feedforward computations [55, 56, 57] in place of attractor dynamics, recurrent attractor
dynamics are indispensable for the generation of persistent activity states (and thus for short-term
memory through persistent activity [58, 59]) and integration, as we discuss below .
Robust classiﬁcation
When the attractors form a set with a discrete component (e.g. a set of point attractors or a set
of continuous attractors), inputs that are not initially on one of the attractors will ﬂow to one of
the attractors and thus we may view the identity of the speciﬁc attractor to which the input ﬂows
as a classiﬁcation of the input as a class represented by that attractor . The process can perform a
pattern-completing nearest-neighbor computation, if the dynamics correctly drives non-attractor
states to the nearest attractor states. In other words, the dynamical basins of attraction must
align with the Voronoi regions of the attractor states, which is approximately the case for attractor
networks operating well below capacity . This property deteriorates when attractor networks are
pushed toward their capacity [60].
Integration
Single neurons integrate their inputs, but usually can only do this over the time-scales associated
with their membrane capacitances, typically 10-100 milliseconds. Continuous attractor dynamics
can allow neural circuits to integrate over much longer time-scales ( 1-100 seconds).
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A non-linear continuous attractor network requires an additional mechanism to gain the func-
tionality of an integrator: A way to shift the internal state along the attractor in response to an
input that encodes changes in the external variable, Fig. 2d (left). Conceptually the simplest way
to build a shift mechanism is by a copy-and-offset construction: construct multiple copies of the
attractor network, each with slightly offset (asymmetric) weights in the sense that active neurons
center their excitation or point of maximal disinhibition slightly offset from themselves on the
neural sheet (e.g Fig. 1g (left) represents a slightly asymmetric version of Fig. 1c). The states in
each such network will then form a limit cycle attractor , with patterns ﬂowing in the direction of
the asymmetry . If opposing copies are coupled together , the pattern is stabilized through a push-
pull balance. A velocity input whose components project differentially to the copies will break
the push-pull balance, allowing the more-active population of the moment to drive the pattern
along its ﬂow direction (cf Fig. 1g). Thus, the total direction and magnitude of the shift of the
pattern, corresponding to movement along the attractor manifold, represents the time-integral of
the velocity input to the network. This common principle uniﬁes the mechanisms across diverse
integrator models [61, 13, 14, 16, 62 ].
Decision making
If, instead of a velocity signal, the input to an integrator network consisted of temporally varying
positive and negative evidence in support of two options [63], Fig. 2d (right) (or in the case of
multiple options, evidence vectors instead of velocity vectors [64]), the network would integrate
those inputs and thus perform evidence accumulation.
Decision-making can be viewed as a selection process applied to the integrator , based on a
readout that detects when the integrator state has accumulated enough evidence and moved past a
decision threshold [65, 54]. The selection process can be external to the integrator in the form of a
readout circuit that detects such threshold crossings and outputs the decision; or it can be built into
the dynamics of the integrator itself, in the form of a more-complex attractor landscape: the states
evolve along a continuous attractor , but at some point the continuous attractor gives way to a pair
of discrete attractors toward which the states ﬂow , Fig. 2e. Neural winner-take-all (WT A) models
implement such a hybrid analog-discrete computation [63, 17, 66, 67, 68, 64 ]. The parameters in
WT A networks determine the balance between integration dynamics and competitive dynamics,
and thus how well the network integrates later evidence (when the network is tuned to be a perfect
integrator , its response to inputs is gradual and small amounts of evidence cause (reversible) ﬂow
along the continuous attractor manifold. In the case when competition dominates, the response
to evidence is a fast ﬂow toward one of the discrete attractors; beyond a point the ﬂow is nearly
irreversible, leading to rapid decision making and discounting of later evidence [69]).
Neural winner-take-all networks can accurately and rapidly (in ⇠ log(N ) time) make the best
decision among N alternatives, even if the presented data are noisy (ﬂuctuating over time around
their means) [68, 64] and the number of options varies over orders of magnitude [64].
Sequence generation
Attractor dynamics can be important for stabilizing another long time-scale behavior: the genera-
tion of sequences. Robust sequences can be constructed as low-dimensional limit cycle attractors,
in which high-dimensional perturbations are corrected, while along the attractor there is a sys-
tematic, periodic, or quasiperiodic ﬂow of states [70, 71, 72, 73, 74 ]. The attractor property that
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