Attractor and integrator networks in the brain — part 1/10

Cleaned-text code points [0, 14954). Bibliography omitted here; source.md includes it.

Attractor and integrator networks in the brain
Mikail Khona1,2 and Ila R. Fiete ⇤1
1Department of Brain and Cognitive Sciences & McGovern Institute, MIT
2Department of Physics, MIT
Abstract
In this review , we describe the singular success of attractor neural network models in describing
how the brain maintains persistent activity states for working memory , error-corrects, and inte-
grates noisy cues. We consider the mechanisms by which simple and forgetful units can organize
to collectively generate dynamics on the long time-scales required for such computations. We
discuss the myriad potential uses of attractor dynamics for computation in the brain, and show-
case notable examples of brain systems in which inherently low-dimensional continuous attractor
dynamics have been concretely and rigorously identiﬁed. Thus, it is now possible to conclusively
state that the brain constructs and uses such systems for computation. Finally , we look ahead
by highlighting recent theoretical advances in understanding how the fundamental tradeoffs be-
tween robustness and capacity and between structure and ﬂexibility can be overcome by reusing
and recombining the same set of modular attractors for multiple functions, so they together pro-
duce representations that are structurally constrained and robust but exhibit high capacity and
are ﬂexible.
Introduction
One of Biology’s grand challenges is to explain how order and complex function spring from inan-
imate physical systems composed of much simpler parts. The brain creates order in its repre-
sentations of the world and performs complex functions through the collective interactions of
simpler elements. In this review , we will describe and evaluate the hypothesis that attractor dy-
namics in widespread regions of the central nervous system play a key role in constructing some
of these representations, generating long time-scales to support integration and memory func-
tions, and endowing all these functions with robustness. We will review the speciﬁc predictions of
attractor-based models and the now-extensive body of work testing these predictions. Thus, we
will illustrate that the theory and validation of computation with attractor dynamics in the brain
is one of the biggest success stories in systems neuroscience.
Some of the ﬁrst formal circuit-level models of brain function focused on the problem of as-
sociative memory and how neural circuits might generate spatially distributed, stable patterns
of activity that could function as such a memory [1, 2, 3, 4 ]. Hopﬁeld networks, with multi-
ple stable states learned from distributed input patterns, were proposed over four decades ago
⇤Corresponding author
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[5, 3, 6 ]. Network models possessing a continuous set of stable states, that could be used to rep-
resent continuous variables, were also ﬁrst proposed around the same period [7]. Subsequently ,
many canonical brain circuits for motor control, sensory ampliﬁcation and memory , motion inte-
gration, evidence integration, decision making, and spatial navigation have been modeled using
the same general principle – that a set of states stabilized through collective positive feedback can
be used for robust representation, memory , and to perform computations that involve memory
[8, 9, 10, 11, 12, 13, 14, 15, 16, 17 ].
Because these are circuit-level models, but were typically inspired by experimental charac-
terization of neurons recorded singly or a few at a time, the patterns of connectivity and cell
activity correlations in the models automatically became novel and relatively speciﬁc predictions
about the population dynamics and architecture of such circuits. As we will discuss below , the
combination of these prediction-rich (and often conceptually simple) models, modern experimen-
tal breakthroughs in the acquisition of cellular-resolution population activity data, and novel and
rigorous analyses of such data based on the model predictions has led to the accumulation of a pre-
ponderance of evidence that the brain constructs and exploits attractor networks for performing
several essential computations.
We will begin by deﬁning attractors, then describe proposed mechanisms for the construction
of attractor network models in neuroscience. We will provide an overview of why attractor net-
works can be important for computation in the brain and give criteria for determining whether a
system has non-trivial attractor dynamics. After this groundwork we will discuss several examples
of brain circuits with non-trivial attractor dynamics. Finally , we will end with new directions in
our understanding of how these simple circuits could contribute to ﬂexible computation through
reuse in multiple contexts.
What are attractors?
T o deﬁne an attractor , we ﬁrst deﬁne a dynamical system and its states. A dynamical system is a
set of variables together with all the rules that determine their time-evolution. The instantaneous
value of these variables is called the state of the system at that moment. The state is a point
(vector) in the state space of the dynamical system. An attractor is the minimal set of states in a
state space, to which all nearby states eventually ﬂow [18]. One simple example of an attractor is a
stable ﬁxed point: all neighboring states ﬂow to it. Porting these crisp mathematical deﬁnitions to
the brain involves challenges and simpliﬁcations, which revolve around identifying a sufﬁciently
self-contained system and the variables necessary to determine its dynamics.
Deﬁning the state of a neural system: Inherent in the deﬁnition of a dynamical system is
the assumption that there are no external dynamical inputs to the system (equivalently , the sys-
tem deﬁnition includes all such variables). The ﬁrst simpliﬁcation in characterizing the dynamics
of a neural circuit is to assume that at least on the time-scale of interest, the system evolves au-
tonomously. Given that subcircuits in the brain are interconnected with others, and that the brain
itself interacts with the world, it is impossible to completely isolate these circuits into autonomous
systems. However , we may deﬁne a notion of effectively autonomous dynamics over time-scales
where inputs are not temporally varying and are untuned in the sense that they do not provide
differential drive to subsets of the putative set of attractor states. The second simpliﬁcation is in
deﬁning the states of the system. The time-evolution of a circuit in the brain may depend on the
detailed pattern of all the spikes in all neurons, the levels of associated ions, neurotransmitters
and modulators, the states of the ion channels. The weights and connections between neurons
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may be considered as parameters rather than variables on short timescales, but as variables if con-
sidering a longer time-scale. One widely used simpliﬁcation in describing a neural circuit on the
timescale of seconds is to use just the spiking outputs of the neurons in the circuit as the states,
often further simpliﬁed as time-varying spike rates. If such a description is sufﬁcient to predict
the evolution of the system at the relevant time-scales, it can be viewed as a reasonable dynamical
system model of the circuit. Even though spike or spike rate descriptions ignore sub-cellular and
molecular variables to make the grossly simplifying assumption that the relevant circuit dynamics
are governed by spikes, the state space of a vertebrate microcircuit described in this way is never-
theless very high-dimensional, comprising the number of neurons in the circuit, or 10 2 107 cells.
As we will see below , such simpliﬁed models can nevertheless yield rich and accurate predictions
about neural circuits.
Attractors exist in various ﬂavors: an attractor may consist of a single state or a set of states
that trace out a complex shape, such as a curved manifold 1, Fig. 1 (rightmost column). States
on an attractor may be stationary , or might ﬂow along the attractor to trace out trajectories that
are periodic (limit cycles, Fig. 1f, rightmost column) or chaotic (dynamics that are inherently
unpredictable due to high sensitivity to small changes in the state [21]). V arious combinations of
such attractors, of different dimension, geometry , and topology , may coexist in different regions
of the state space of a single dynamical system. Typically , the set of attractors in a dynamical
system comprises a small subset of the state space, and attractor manifolds are usually much
lower-dimensional than the state space. In cases where a system has multiple attractor states, the
initial condition determines the attractor state to which the system ﬂows.
Deﬁning attractors in the presence of noise: Any real physical system unavoidably behaves
non-deterministically from the perspective of a model of the system. This is because one cannot
observe and describe all variables, and all uncharacterized variables together with true stochas-
tic sources of variation (e.g., synaptic signalling noise from stochastic vesicle release [22], ionic
number ﬂuctuations in processes like spike initiation [23], calcium signaling, ﬂuctuations in small
copy numbers of proteins [24]) serve as effective sources of noise in the model. Noise can buffet
states so they do not strictly localize to the attractor described in a noise-free version of the model,
and can drive the system to escape from an attractor over time. However , the general idea of at-
tracting states remains in the following sense: If the system is initialized near such a state, it tends
to ﬂow toward it and subsequently remains localized around it, for extended periods. In sum,
since attractor states are where systems tend to localize (when not externally driven), they should
be observable in the autonomous dynamics of real systems. This basic property is the basis for
the most fundamental and robust tests of attractor dynamics in neural systems, as we will discuss
below . In a nutshell, the critical signatures of attractors in real systems (discussed in more detail
in later sections of this review) can be summarized as: localization of the states of a system to a
lower-dimensional subset, ﬂow of the states towards the subset after perturbation, and long-time
and (effectively) autonomous stability of states in that subset.
1Attractor manifolds: If the number of attractor states in a network is large and the points are close to one another ,
they can behave effectively as a continuous set. If this near-continuous set traces out a surface that is locally Euclidean,
it is called a manifold. Nonlinear continuous attractor manifolds can be curved and topologically complex (e.g. rings,
torii, etc., Fig. 1c-d, rightmost column [19, 20].
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Mechanisms: The construction of neural attractors
The general principle underlying the formation of non-trivial attractor states in neural circuits is
strong recurrent positive feedback 2. Positive feedback ﬁghts activity decay to stabilize certain
states, and has been conjectured by James, Hebb and others [25, 26, 27, 28 ] as the basis for the
stabilization of memory traces and persistent activity in the brain. Which states become stabilized
into attractors depends on how the network sculpts the positive feedback, which according to the
synaptic hypothesis is determined by the synaptic weights. [29, 30, 31 ]
In general, characterizing the relationship between structure and function in a large collection
of interacting elements is extremely difﬁcult, as described by Anderson in "More is different” [32].
For instance, a large collection of simple polar 3-atom molecules of hydrogen and oxygen give
rise to emergent phenomena like liquidness and wetness and freezing into a solid, which cannot
be predicted through intuition or drawing box-and-arrow diagrams. On the ﬂip side, there is also
emergent simplicity , in that the transitions and properties of the emergent states can be described
with very few key parameters and variables.
One way to characterize the relationship between synaptic weights and attractor dynamics is to
ask what attractor states a given set of weights produces (the “forward” problem). With the given
weights, one can simulate the circuit and explore the resulting dynamics to ﬁnd attractors of the
system. A more powerful method, the Lyapunov function approach, holds for symmetric weight
matrices ( Wij = Wji ) and rate-based neural dynamics. For this class of models, a generalized
energy function (the Lyapunov function), which is a function of the weights and neural activation
function [6, 5, 2 ], analytically speciﬁes the network’s dynamics. Stable (unstable) attractor states
are the energy minima (maxima) of the derived landscape, and the network’s state ﬂows downhill
towards the attractors Fig. 2e in the way a ball rolls down a gravitational potential.
Another way to characterize the relationship between attractors and network structure is to
consider the “inverse” problem: given a set of attractors, what network structure could generate
it? Neuroscientists want to solve the inverse problem to make predictions about underlying mech-
anism (and since neural activations are more readily observed than synaptic weights, the inverse
problem is more frequently encountered than the forward one), while evolution, the brain, and
artiﬁcially intelligent systems need to solve the inverse problem to be able to perform computa-
tions that require a given type of attractor dynamics (which we will discuss below). Theoretical
neuroscience has discovered some solutions to the inverse problem for different types of attractors,
as we describe next.
Discrete attractors. A well-known prescription for creating a discrete set of stable attrac-
tors at user-deﬁned points in state space is given by the Hopﬁeld network model [5], Fig. 1a:
an externally induced and distributed pattern of neural activity is encoded into the weights by a
Hebbian-like learning rule that causes co-activated neurons to excite each other and inhibit all
the rest. As a result, these patterns become stable attractor states. If a sufﬁciently small number
of patterns are inscribed into the weights, they can be retrieved from partial or corrupted ver-
sions of the stored states, thus the network is a content-addressable memory . More generally , the
attractors of simple rate-based networks without synaptic delays and with arbitrary symmetric
weight matrices3 consist entirely of ﬁxed points. Some non-symmetric networks can also support
point attractors [33], but this is not the generic case and can require additional mechanisms like
2Non-trivial attractor states refer to any state other than the null activity state. Positive feedback is not synonymous
with excitatory feedback: like mutual excitation, disinhibition or inhibition of one’s inhibitor is also a form of positive
feedback.
3A symmetric weight matrix W satisﬁes W T = W : it is invariant to reﬂection of its entries about its diagonal.
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