Attractor and integrator networks in the brain — part 3/10

Cleaned-text code points [30560, 46007). Bibliography omitted here; source.md includes it.
affords ongoing de-noising is important for preventing spatial dispersion and temporal dissipation
of the activity packet during sequence generation.
As for stationary attractor manifolds, the small components of noise along the limit cycle attrac-
tors are not correctable and lead to a gradual accumulation of drift, which for sequence generation
is manifest as timing variability: the standard-deviation in the time of reaching the T th state in
the sequence is predicted to grow as
p
T for unbiased random drift along the attractor [53].
Evidence of attractors in the brain
Criteria for establishing attractor dynamics
The fundamental predictions of attractor models center on the state-space dynamics of the circuit,
as ﬁrst explicitly discussed and tested in a few papers [75, 9, 16, 76 ]: 1) That the system’s states
should be found localized at or around a much lower-dimensional set of states corresponding to
the attractors in the state space. 2) That perturbations of the system should ﬂow quickly back to
the low-dimensional states. 3) That the set of attractor states – quantiﬁed by either direct char-
acterization of the full state space or by the relationships between cells – should be invariant,
persisting over time and after removal of tuned input, across conditions, across behavioral states,
and even when there are induced variations in the mapping from internal states to external inputs
[75, 16, 76 ]. 4) Integrator networks should further exhibit the property of isometry , in which
lengths of coding space along a dimension are allocated to equal displacements along a dimen-
sion of the external variable. 5) Additional predictions of attractor dynamics models, that are not
as fundamental in the sense that they are not theoretically necessary or sufﬁcient but are nev-
ertheless of high signiﬁcance because they are highly supportive of the mechanisms of attractor
dynamics, are anatomical and structural correlates: the existence of low-dimensional structures
and symmetries in connectivity between cells.
Because attractor systems are characterized by their internally generated or autonomous dy-
namics5, putative attractor networks are best tested in conditions that minimize external cues that
are time-varying or tuned to provide localized inputs along the putative attractor .
Innovations in recording methods that made it possible to record multiple neurons simultane-
ously in animals performing naturalistic behaviors [77, 78, 79, 80], have enabled essential tests of
these state-space predictions of attractor models. The newest methods provide activity data from
⇠ 1000’s of neurons within a circuit [81, 82, 83 ], making it possible to directly characterize the
low-dimensional state-space dynamics of whole circuits [84, 85, 19, 86, 87 ].
When the attractor manifolds are < 3-dimensional, one can directly visualize them by project-
ing or embedding the high-dimensional state-spaces into  3 dimensions (using e.g. PCA, multi-
dimensional scaling, tensor factorization, and other linear methods for projection, or Isomap, LLE,
tSNE, VAEs, LF ADS, nonlinear tensor factorization, and so on, for nonlinear embedding [88, 89,
90, 91]). These methods can also be useful when manifolds have dimension  3 but are topolog-
ically simple [86, 92 ]. For topologically non-trivial structures (e.g. rings, torii), especially those
of dimension  3, topological data analysis methods become important[93, 94, 95, 19, 96, 97, 87].
5Attractor networks dynamics need not be used by the brain in an autonomous setting: inputs that drive attractor
networks can be an important part of their function, for instance in integration and evidence accumulation. However ,
for the purposes of identifying mechanism, it is important that their dynamics are probed in an (effectively) autonomous
setting
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T esting predictions 1)-3) requires examination of the state-space structure of the population
response, rather than the more conventional characterization of relationships (tuning curves) be-
tween cell activity and input or output variables. The most direct way to examine the state-
space structure is to record enough cells simultaneously that it is possible to characterize the
full state-space manifold [19, 96, 20 ]. However , the existence, stability , and invariance of lower-
dimensional state-space structures (predictions 1)-3)) can be inferred from smaller samples of
simultaneously recorded cells, by characterizing the invariant structure in pairwise cell-cell rela-
tionships, as has been successfully done in a number of studies [75, 98, 99, 76, 100, 101 ].
Predictions 1) and 2) are necessary but not sufﬁcient for identiﬁcation of recurrent attractor
dynamics in a target network. First, if the behaviors and inputs are themselves low-dimensional,
then any observed low dimensionality of the circuit states may be ascribed to the inputs and reveals
little about intrinsic constraints imposed by the circuit. Second, even if inputs and behaviors are
high-dimensional, a low-dimensional feedforward projection into the target network can generate
low-dimensional target states and rapid erasure of high-dimensional perturbations. The sina qua
non of attractor dynamics is prediction 3), which is that, because the states are internally generated
and stabilized by strong recurrent connectivity , the population states and cell-cell relationships
should be invariant when probed across time and across a wide and rich variety of input conditions
including the removal of tuned input and across waking and sleep. In simple terms, the states
observed in 1)-2) should be invariant across a broad range of conditions [16, 76].
Next, to the question of circuit localization: If a circuit exhibits the key signatures of attractor
dynamics, does it originate these dynamics or is it a readout of some other region? Localiza-
tion need not be a primary goal of establishing attractor dynamics: an important ﬁrst step is to
simply characterize whether the brain solves certain problems through attractor dynamics, re-
gardless which circuits create these dynamics. Nevertheless, the persistence of activity states in
attractors can lend a helping hand to localization efforts. If a region originates or is upstream of
the attractor dynamics, but not downstream of it, then perturbations that succeed in altering its
low-dimensional state should persist after the perturbing drive is removed [102].
As we illustrate next, theoretically-motivated analyses of population data have now ﬁrmly
established that low-dimensional attractor dynamics are ubiquitous in the brain, across levels in
the brain’s hierarchy and across species.
Discrete attractors
Up and down states
The simplest example of nontrivial discrete attractor dynamics (i.e., beyond a single point attrac-
tor) is bistability . Bistable dynamics are a feature of cortical activity in the form of up and down
states [103, 104, 105, 106, 4, 107, 108], in which the subthreshold membrane potential of neurons
switches between a hyperpolarized state and a relatively depolarized one, with long persistence
(100’s of milliseconds to seconds) per state, Fig. 3a. The two states are relatively invariant over
time, as seen in the relatively sharply peaked histograms (Fig. 3a; predictions 1), 3)), and despite
presumed internal noise in the system the peaks are well separated, suggesting a relatively rapid
corrective dynamics towards the two states (prediction 2)). There is little evidence of a critical con-
tribution from cellular bistability in supporting these states, suggesting that it is a network-driven
phenomenon involving self-excitation and global inhibition [109, 110, 111, 104, 105, 106, 4, 107].
Transitions are believed to be driven through adaptation (from up to down) and stochastic as well
as external coordinating events (from down to up) [108]. Though these states and switches can
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occur in cortex without input from thalamus and striatum, they tend to be synchronous across
cortex and striatum. Thus, the origin of up and down states may be highly distributed.
Perceptual bistability
Visual and auditory percepts including binocular rivalry , the Necker cube, and some auditory illu-
sions [112, 113, 114, 115, 116, 117, 118 ] offer clear examples of bistability in neural processing.
In these illusions, the brain (at the level of perceptual reports) selects one possible interpretation
of an ambiguous input, often switching between possibilities. Though the phenomenon has long
been known and studied, no localized circuit has been identiﬁed as the basis of perceptual bista-
bility . Indeed, some percepts may involve top-down activation and modulation of activity across
many brain areas [116], suggesting once again a widely distributed circuit for bistability .
Bistability in a premotor area
Recent studies identify and localize discrete attractor dynamics in a mouse premotor area, the
anterior lateral motor cortex (ALM) [119, 120, 121, 122]. In a cued 2-alternative delayed response
task ALM neurons exhibit persistent activity over a ⇠ 1s delay period. During the post-cue delay
period, activity evolves toward one of two states that guide the response, Fig. 3b (prediction
1)). The delay-period terminal states are similar for cues from different sensory modalities [123]
(partial test of prediction 3)). ALM perturbations during the delay are either erased (corrected)
by the circuit (Fig. 3b, top) or drive a jump to the opposite state (Fig. 3b, bottom), which results
in the animal making the wrong action, suggesting a bistable switching dynamics similar to the
mechanisms in either Fig.1b or Fig.2e (prediction 2)).
Given the long training time required for the task and the resulting tailoring of the ALM dy-
namics to the speciﬁc task structure – bistability for a two-choice task – it is likely that this system
acquires its dynamics through slow plasticity and thus that the network’s recurrent structure is
malleable in adult animals. New results showing the existence of small ( ⇠ 100µm scale) locally
recurrent clusters of neurons ALM that can maintain persistent responses to microstimulation
[124] may provide experimental evidence of the theoretically posited mixed modular networks
(below) hypothesized to support robust and high-capacity memory states [60].
Discrete multistability
Hopﬁeld networks and winner-take-all (WT A) networks are models of multistability beyond bista-
bility6
T o date, there are somewhat less direct data and exhaustive analyses to establish discrete mul-
tistability as a circuit-level brain process, in comparison to the evidence for continuous attractor
networks (described next). However , there are many likely candidates systems and brain regions
with dynamics suggestive of and consistent with discrete multistability , at least of the special case
of WT A attractor dynamics, including in mammalian hippocampus and auditory cortex, and the
ﬂy and mamalian olfactory system [133, 134, 135, 136, 137, 138 ]. In particular , many of these
circuits exhibit global inhibition that clearly narrows and reﬁnes activity in the circuit (predic-
tion 5)), Fig. 3c top versus middle, and also show evidence of selective recurrent excitation that
6WT A networks[125, 126, 127, 67, 128, 129, 130, 131, 132 ] may be viewed as a special case of Hopﬁeld networks,
and bistable switch networks are a special case of WT A networks. As noted earlier , both can express mulitple discrete
attractor states, but while the Hopﬁeld network attractors have highly mixed and overlapping neural membership, WT A
attractors consist of activity in largely disjoint groups of neurons.
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leads to multiple distinct and stably correlated input responses in distinct subpopulations of cells,
Fig. 3c, middle versus bottom [133, 134, 135, 136, 137, 138 ]. In our view , it is likely that these
circuits exhibit multiple discrete attractor states but quantitative testing of predictions 1)-3) and
direct demonstration of these states as stable and invariant remains an important future direction
for characterizing these circuits.
Continuous attractors
The oculomotor integrator
The oculomotor integrator , together with the HD circuit, was one of the ﬁrst systems in neuro-
science to be studied theoretically [8, 139, 9 ] and experimentally [140] as a continuous attractor
network – speciﬁcally as a line attractor , Fig. 1e. This network, presynaptic to the motor neurons
that control horizontal eye position, is highly conserved across vertebrates, from ﬁsh [141, 140 ]
to primates [142, 143]. It integrates pulse-like saccadic eye movement command signals to gener-
ate step-like stable muscle tension command signals (Fig.4a) that persist autonomously at various
graded activity levels after removal of the movement cue and even in the dark in the absence
of visual feedback (Fig.4b; prediction 3)) and thus enable stable gaze ﬁxation at various eccen-
tricities. Saccadic inputs knock the system slightly off the linear response states, but the neural
responses rapidly decay back towards the persistent ﬁring states (prediction 2)). Remarkably ,
the same system also integrates smooth head velocity signals to permit gaze stabilization during
head movement. Integration functionality is a network-level rather than single-cell process: single
neurons do not generate persistent responses to transient current injections, Fig. 4c (inset), while
decreasing network feedback through synaptic blockers reduces the time-constant of integration
and results in a leaky integrator [144], Fig.4c. It is possible to induce a reduction or increase
in network feedback through training with a virtual surround that generates an artiﬁcial retinal
slip percept, Fig.4d, showing that the system is capable of error-driven ﬁne-tuning to maintain a
high degree of persistence [145]. Finally , a recent EM reconstruction [146, 147 ] ﬁnds recurrent
synaptic interconnectivty between integrator neurons, with excitatory connections between ipsi-
lateral neurons and primarily inhibitory contralateral projections, in excellent agreement with line
attractor models of the oculomotor circuit [9], Fig.1e (prediction 5)).
Head direction cells
Some of the earliest experiments to suggest the existence of low-dimensional continuous attrac-
tor dynamics were done in the rodent head-direction (HD) circuit [98, 75, 148 ], Fig.5a,b. The
HD circuit in mammals maintains an updated internal compass estimate of heading direction
(relative to some arbitrary external reference) as animals move around. It does so by integrat-
ing internal rotational velocity estimates during navigation and incorporating information from
external cues [149, 150, 151, 152, 153 ]. The HD circuit is modeled by the ring attractor net-
work [10, 61, 11, 154, 13, 14 ], Fig.1c, g (left). Before large population recordings became avail-
able, cell-cell correlations established that the network states remained invariant on a very low-
dimensional manifold across environments [98, 75, 148], Fig.5a (predictions 1), 3)). Recently , the
complete set of states of the several thousand neuron-sized mammalian HD network was shown to
consist solely of a 1-dimensional ring, Fig.5b [19, 96] (prediction 1)), revealing that the brain has
completely factorized its navigational representations to dedicate a circuit only to head direction.
Further , intervals in the state-space ring manifold map isometrically to intervals of head direction
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