Back to the Continuous Attractor
Authors: Ábel Ságodi
Guillermo Martín-Sánchez
Piotr A Sokol
Il Memming Park
Venue: NeurIPS 2024 poster (2024)
Keywords: continuous attractors
robustness
fast-slow decomposition
generalization
Primary area: neuroscience_and_cognitive_science
TL;DR: Even though continuous attractors are not structurally stable, they are functionally robust remain useful as a universal analogy for understanding analog memory.

Abstract: Continuous attractors offer a unique class of solutions for storing continuous-valued variables in recurrent system states for indefinitely long time intervals.
Unfortunately, continuous attractors suffer from severe structural instability in general---they are destroyed by most infinitesimal changes of the dynamical law that defines them.
This fragility limits their utility especially in biological systems as their recurrent dynamics are subject to constant perturbations.
We observe that the bifurcations from continuous attractors in theoretical neuroscience models display various structurally stable forms.
Although their asymptotic behaviors to maintain memory are categorically distinct, their finite-time behaviors are similar.
We build on the persistent manifold theory to explain the commonalities between bifurcations from and approximations of continuous attractors.
Fast-slow decomposition analysis uncovers the existence of a persistent slow manifold that survives the seemingly destructive bifurcation, relating the flow within the manifold to the size of the perturbation. Moreover, this allows the bounding of the memory error of these approximations of continuous attractors.
Finally, we train recurrent neural networks on analog memory tasks to support the appearance of these systems as solutions and their generalization capabilities.
Therefore, we conclude that continuous attractors are functionally robust and remain useful as a universal analogy for understanding analog memory.

Back to the Continuous Attractor
Ábel Ságodi, Guillermo Martín-Sánchez, Piotr Sokót, Il Memming Park
Champalimaud Centre for the Unknown
Champalimaud Foundation, Lishon, Portugal
(abel. sagodi, guillermo martin, memming. parrtin,emmming. Baaaannggaarrrrrrrreee park))researdde ch www.cch...happpe fchampal imaud. org piotr. weadr.ssok@@@rottonaaa com
Abstract
Continuous attractors offer a unique class of solutions for storing continuous- valued variables in recurrent system states for indefinitely long time intervals. Unfortunately, continuous attractors suffer from severe structural instability thneral-they are destroyed by most infinitesimal changes of the dynamical law that defines them. This fragility limits their utility especially in biological systems their recurrent dynamics are subject to constant perturbations. We observe that the bifurcations from continuous attractors theoretical neuroscience models display various structurally stable forms. Although their asymptotic behaviors maintain memory are categorically distinct, their finite-time behaviors are similar. We build on the persistent manifold theory to explain the commonalities between bifurcations from and approximations of continuous attractors. Fast-slow decom- position analysis uncovers the existence of persistent slow manifold that survives the seemingly destructive bifurcation, relating the flow within the manifold to the size of the perturbation. Moreover, this allows the bounding of the memory error these approximations of continuous attractors. Finally, we train recurrent neural networks analog memory tasks to support the appearance of these systems solutions and their generalization capabilities. Therefore, we conclude that contin- uous attractors are functionally robust and remain useful universal analogy for understanding analog memory.
1 Introduction
Biological systems exhibit robust behaviors that require neural information processing of analog variables such as intensity, direction, and distance. Virtually all neural models of working memory for continuous-valued information rely on persistent internal representations through recurrent dynamics. The continuous attractor structure in their recurrent dynamics has been pivotal theoretical tool due to their ability to maintain activity patterns indefinitely through neural population states' They are hypothesized to the neural mechanism for the maintenance of eye positions, heading direction, self-location, target location, sensory evidence, working memory, and decision variables, to name few 5-7 Observations of persistent neural activity across many brain areas, organisms, and tasks have corroborated the existence of continuous attractors 7-15
Despite their widespread adoption as models of analog memory, continuous attractors brittle mathematical objects, casting significant doubts on their ontological value and hence suitability in accurately representing biological functions. Even the smallest arbitrary change in recurrent dynamics can be problematic, destroying the continuum of fixed points essential for continuous-vale valuued working memory. In neuroscience, this vulnerability is well-known and often referred as the ffne-tuning problem" 5,16 There are two primary sources of perturbations in the recurrent network dynamics: (I) the stochastic nature of online learning signals that via synaptic plasticity, and (2) spontaneous fluctuations in synaptic weights 21,2 Thus, additional mechanisms are necessary to compensate for the degradation in particular implementations, by bringing the short-term behavior closer to that
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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of a continuous attractor this matters in practice, i.c. what are the effects of different levels of degradation on memory. This is fundamental to justify relying on the brittle concept of continuous attractors for understanding biological analog working memory.
16,23-29, However, we lack the theoretical basis to understand how much
In this study, we explore perturbations and approximations of continuous attractors in the space of dynamical models. We first report on the differences and similarities between the various structurally stable dynamics in the vicinity of continuous attractors in the space of dynamical systems models. Our analysis reveals the presence of "ghost" continuous attractor (a.k.a. slow manifold) in all of them (Sec. 2). By assuming normal hyperbolicity we separate the time scales to obtain a decomposition of the dynamics by separating out the fast flow normal to and the slow flow within the slow manifold. We derive theoretical results that ensure the existence of a slow manifold and determine its closeness to a continuous attractor (Sec. 3). We explore fask-trained recurrent neural networks (RNNs) to show that these systems appear naturally as solutions to the task (Sec. 4) and that their generalization capabiliitess can easily be studied as the distance to the continuous attractor (Sec. 5). The proposed decomposition applied to theoretical models and task-trained RNNs reveals a "universal motif"" of analog memory mechanism with various potential topologies. This leads to the connection of different systems with different topologies approximate continuous attractors (Sec. 6). Our theory guarantees that systems close to a continuous attractor (in the space of vector fields) will have similar behavior to it, implying that the concept of continuous attractors remains crucial framework for understanding the neural computation underlying analog memory (Sec. 3.4).
2 A critique of pure continuous attractors
We will first lay out a number of observations about the dynamics of bifurcations and approximations of continuous attractors used in theoretical neuroscience. Ordinary differential equations (ODEs) are commonly used to describe the dynamical laws governing the temporal evolution of firing rates or latent population states', In this framework, neural systems are viewed as implementing the continuous time evolution of neural states to perform computations. We will consider a continuous attractor as mechanism that implements analog memory computation: carrying particular memory representation over time. To define it formally, let x(t) E R denote the neural state, and x = f(x) represent its dynamies. Let M C RRd be a manifold. We say M a continuous attractor, (1) every state on the manifold is a fixed point, Vx fM, stable tangent to the manifold and stable normal to the manifold. In other words, the continuous attractor is continuum of equilibrium points such that the neural state near the manifold is attracted to it, and on the manifold, the state does not move. Marginal stability implies that continuous systems are structurally unstable, meaning that small perturbations parameters lead to significant changes in the system's 's behavior or stability 30-33, We will now study some examples of continuous attractors and how perturbations change their dynamics.
O, and (2) the fixed points are marginally
variations in the system's
2.1 Motivating example: bounded line attractor
As an illustrative examplc, we can construct a line attractor (a continuous attractor with line manifold) as follows:
x = - + Wx + b]+
(1)
where W = [0, -1; 11 and b- [1; 1J. and I+ max(0,-) is the threshold nonlinearity per unit We get x = 0 on the x1 = T2 + line segment in the first quadrant as the manifold (Fig. 1A, left; black line). Lincarization of the fixed points on the manifold exhibits two eigenvalues, 0 and -2; the 0 eigenvalue allows the continuum of fixed points, while -2 makes the flow normal to the manifold attractive (Fig. 1A, left; flow field).
In general, continuous attractors are not only structurally un they bifurcate almost certainly for arbitrary perturbation of f. In this example, small changes to the parameters (b, W) perturb the eigenvalues and any to the 0 eigenvalue destroys the continuous attractor: it bifurcates to either single stable fixed point (Fig. 1A. top) or two stable fixed points separated with saddle-node in between (Fig. IA, bottom). However, interestingly, after bifurcation, continuous attractors seemingly tend to leave "ghost" manifold topologically equivalent to the original continuous attractor (note the slow speed). Furthermore, the flow after the bifurcation is contained in the ghost manifold, i.e., it is invariant manifold. This phenomenon, wherein a continuous attractor is approximated by manifold within the neural space along which the drift occurs at a very slow pace, has previously been discussed35-3)
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Boundled Line AtAractor
Niturattnn (converging!
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Figure 1: The critical weakness of continuous attractors their inherent brittleness as they are rare in the parameter space, i.e., infinitesimal changes in parameters destroy the continuous attractor implemented in RNNs Some of the structure seems to remain: there is an invariant manifold that is topologically equivalent to the original continuous attractor. (A) Phase portraits for the bounded line attractor (Eq. (I)). Under perturbation of parameters, it bifurcates to systems without the continuous attractor. (B) The low-rank ring attractor approximation (Sec. (S3.4)). Different topologies exist for different realizations of low-rank attractor: different numbers of fixed points (4, 12), or a limit cycle (right bottom). Yet, they all share the existence of ring invariant set.
2.2 Theoretical models of ring attractors
For circular variables such as the goal-direction (e.g. for navigation 38,39 and working memory for communication in bees 40) or head direction, the temporal integration, and working memory functions are naturally solved by a ring attractor (continuous attractor with ring topology) '1? Other examples include integration of evidence for continuous perceptual judgments, e.g. hunting predator that needs to compute the net direction of motion of large group of prey In this section, we investigate the bifurcations of various implementations of continuous attractors. Continuous attractor network models of the head direction are based on the interactions of neurons that (anatomically) form ring-like overlapping neighbor connectivity 8,51-57, Similarly to the line attractor, the ring attractor bifurcates with almost any perturbation to the network dynamics. However, the resulting dynamics continue to follow a familiar pattern: they remain confined to ghost manifold that closely approximates the original continuous attractor.
Piecewise-linear ring attractor model of the central complex: Firstly, we discuss perturbations of a continuous ring attractor recently proposed as a model for the head direction representation in fruit flies This model is composed of N heading-tuned neurons with preferred headings 05 E (2mt/N)i==.... radians (Sec. S3.2). For sufficiently strong local excitation (given by the parameter JØ) and broad inhibition (Jz), this network will generate a stable bump of activity corresponding to the head direction. This continuum of fixed points forms N N-sided polygon.
We evaluate the effect of parametric perturbations the form W + W+V with V., iid A(O0, 1/100) on network of size N = 6 (forming hexagon, see also Sec. S3). We found that the continuum of fixed points can collapse to between and 12 isolated fixed points (Fig. 2A). As far as we know, this bifurcation from a ring of equilibria to saddle and node has not been described previously in the literature. The probability of cach type of bifurcation was numerically estimated (Sec. S3.2). Surprisingly, the number of fixed points is maintained throughout range of perturbation sizes and hence depends only on the direction of the perturbation (Fig. S8).
Bump attractor model: A well-established approach to form ring attractor in the limit of large network is with a connection matrix W with entries that follow circular Gaussian function of i j 58,00-662 (Sec. S3.3). This type of ring attractor network can support a stable "activity bump" that can move around the ring of nonlinear neurons in correspondence with changes in head direction 63 For finite-sized networks, the dynamics are constrained to an attractive invariant ring, covered with N stable fixed points for a network of size N (Fig. 2B). For such networks the number of fixed points can change with the size of the perturbation (Fig. S8).

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Figure 2: Perturbations of different implementations and approximations of ring attractors lead to bifurcations that all leave the ring invariant manifold intact. For each model, the network dynamics constrained a ring manifold with stable fixed points (green) and saddle nodes (red). (A) Perturbations to Noorman et al. 8 The ring attractor can be perturbed in systems with even number of fixed points (FPs) up to 2/V (stable and saddle points are paired). (B) Perturbations to a tanh approximation of ring attractor Seeholzer et al. (C) Different Embedding Manifolds with Population-leve Jacobians (EMPJ) approximations of ring attractor 59
Low-rank ring attractor model: Low-rank networks can be used to approximate ring attractors 64,65 In the limit of infinite-size networks, one can construct a ring attractor through rank 2 network by constraining the overlap of the right- and left-connectivity vectors (see Sec. S3,4), However, in simulations of finite-size networks with this constraint, the dynamics instead always converge to small number of stable fixed points arranged on ring (Fig. IB).
Embedding Manifolds with Population-level Jacobians: Approximate ring attractors be con- structed by constraining the connectivity so that the networks Jacobian satisfies certain requirements for a ring attractor exist 59 (see Sec. S3.5). The models constructed with this method also can contain an invariant ring manifold on which the dynamics contain stable and saddle fixed points (Fig. 2C). It has been observed that approximate continuous attractor emerge in networks trained on sampled points with other methods as well (6
Similarity between all bifurcations and approximations of continuous attractors: In all dis- cussed models ring attractors, we verify that they suffer from the fine-tuning problem. However, importantly, we also observe in all the systems the existence of ghosts of the continuous attractor (either through bifurcation or from finite-size effects) in the form an attractive invariant manifold. Therefore, while they not strictly continuous attractor in the mathematical sense, they approximate ring attractors in the sense that the fixed points and connecting orbits still form circle.
Is this lawful degradation universal phenomenon? And if so, how does it relate to the size of the perturbation? And what are the implications for the memory performance these approximations? (Sec. 3). Do these approximations appear natural solutions to the memory storage problem? (Sec. 4). And if so, how well do they generalize to longer time requirements? (Sec. 5). Finally, continuous attractors in practice still useful idealized model of how animals represent continuous variables? (Sec. 6).
3 Theory of Approximate Continuous Attractors
In this section, we theoretically answer in an implemeenatiionaannssii manner the degradation questions posed from the exploration. To do so, apply invariant manifold theory to continuous attractor models and translate the results for the neuroscience audience (see also Sec. S1).
3.1 Persistent Manifold Theorem
First, argue that the lawful degradation into system with slow manifold is universally guaranteed (as long as the perturbation is small, and the continuous attractor was normally hyperbolic). Let / be the intrinsic dimension of the manifold of equilibria that defines the continuous attractor. Given perturbation p(x) to the ODE that induces a bifurcation,
* f(x) + cp(x)

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where IIp(-) = 1 and > 0 is the bifurcation parameter, we can reparameterize the dynamics around the manifold with coordinates IR' and the remaining ambient space with E R To describe an arbitrary bifurcation of interest, we introduce a sufficiently smooth function g and h, such that the following system is equivalent to the original ODE:
y cg(y 2, 2)
(tangent)
2 = h c.
(normaa)
(4)
where e = 0 gives the condition for the continuous attractor ý = 0. We denote the corresponding manifold of ( dimensions as MM = { h(y, 0}.
We say the flow around the manifold normally hyperbolic, if the flow normal to the manifold is hyperbolic, i.e. the Jacobians V.h evaluated at any point on the .Mp has d - I eigenvalues with their real part uniformly bounded away from zero, and Vyg has eigenvalues with zero real part. More specifically, for continuous attractors, the real parts of the eigenvalues of V,h strictly negative, representing sufficiently strong attractive flow toward the manifold. Equivalently, for the ODE, x f(x). the variational system of constant rank and has exactly (d - eigenvalues with negative real parts uniformly away from zero and I eigenvalues with zero real parts everywhere along the continuous attractor.
b(4)
Figure 3: Persistent manifold theorem applied to compact continuous attractor guarantees the flow on the slow manifold M, is invariant and continues to be attractive. The dashed line is trajectory "trapped" in the slow manifold (which has the same topology as the continuous attractor).
Mu
bituretion
slow montoodd
y - o
og 40
For any parameterization g. induces bifurcation of the continuous attractor. What can we say about the fate of the perturbed system? The continuous dependence theorem says that the trajectories will change continuously as function of without a guarantee how quickly they change. However, the topological structure and the asymptotic behavior of trajectories change discontinuously due to the bifurcation. Yet, surprisingly, there is strong connection in the geometry due to Fenichel's theorem 68
We informally present a special case due to Jones 73
Theorem 1 (Persistent Manifold), Let Mo be connected, compact", normally hyperbolic manifold of equilibria originating from a sufficiently smooth ODE. For a sufficiently small perturbation >> there exists manifold M, diffeomorphic to Mo and invariant under the flow of Eq. (3)-(4).
The manifold M, is called the slow manifold which is no longer necessarily continuum of equilibria. However, the invariance implies that trajectories remain within the manifold except potentially at the boundary. Furthermore, the non-zero flow the slow manifold is slow and given in the 0 limit as dy g(c y,),,00) 0) where et is rescaled time and (-) parameterizes the I dimensional slow manifold. In addition, the stable manifold of Mo is similarly persistent implying that the manifold M, remains attractive. Finally, the persisting invariant manifold is very close in space to the original continuous attractor (see also Theorem 3).
dr
These conditions are met for the examples in Fig. (see Sec. S2.1 for the corresponding fast- slow reparametrizationn...o ..As the theory predicts, it bifurcates into 1-dimensional slow manifold (Fig. 1, dark-colored regions) that contains fixed points and connecting orbits and is overall still attractive. Furthermore, Fenichel's Persistent Manifold theorem explains the bifurcation structure of the theoretical models discussed in Sec. 2.2. Because continuous ring attractors are bounded, they persist as invariant manifold and remain attractive under small perturbations" 78
"The Persistent Manifold Theorem has been successfully applied previously in neuroscience 09.70 for example reduce the dimensionality of the Hodgkin-Huse
model" 7172
See Eldering" 74 for results of persistence of noncompact invariant manifolds.
*As a technical note, the theory to apply to continuous piecewise- -inear system, it required that the invariant manifold is globally attracting 75 which is also the case for the BLA (see also 76,77 for a discussion of geometric singular perturbation theory for piecewise linear dynamical systems). Therefore, we consider systems that are at least continuous, but some extra conditions apply if system is not differentiable.
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3.2 Fast-slow decomposition and the revival of continuous attractors
Consider behaviorally relevant timescale for working memory, for example, roughly up few tens of seconds. If the relevant dynamical system is orders of magnitude slower, for example, 1000 sec or longer, its effect is too slow to have practical impact the behavior. This clear gap in the fast and slow time scales can be recast as normal hyperbolicity of the slow manifold by relaxing the zero real part to a separation of time scales (reciprocal of eigenvalues or Lyapunov exponents). In other words, the attractive flow normal to the manifold needs to be uniformly faster than the flow on the slow manifold. By taking the limit of the slow flow on the manifold to arbitrarily long time constant (i.e.. to zero flow), we achieve the reversal of the persistent manifold theorem.
Proposition (Revival of continuous attractor). Let Me be a connected, compact, attractive, normally hyperbolic slow manifold (as parametrized by Eq. (3)-(4). Let the uniform norm of the flow tangent to the manifold be |lyl = T/. There exists a perturbation with uniform norm at most that induces a bifurcation a continuous attractor manifold.
An explicit perturbation is derived in Sec. S5.1. This makes the uniform norm of the vector field on a (slow) manifold a useful measure of the distance an approximation to a continuous attractor. Prop. I can be extended to the case where the invariant manifold has additional dynamics to which the output mapping is invariant (see Theorem 7). These systems can be perturbed onto a decomposable system where one of the subsystems has slow flow.
3.3 Relevance of dynamics on the memory performance of the slow manifold
Third, we relate the flow of the manifold (and, through Prop. 1, the size of the perturbation) to the memory error of the approximation in short-time scale. We also discuss the implications of the theoretical insights on the memory error in the asymptotic time scale.
In the short-time scale the memory performance is bounded by the uniform norm of the flow tangent to the manifold. Let Xg E M, and yp = p(-) i be the vector field restricted to the manifold (following the notation in Eq. 2). The average deviation from the initial memory xo over time is bounded linearly (derivation in Sec. S6):
1
(5) Note that this bound is the worst case and tighter for sufficiently small 2 O. Furthermore, for compact invariant manifolds the error is bounded by the diameter of the manifold and hence this bound becomes irrelevant for large.
|x((,x)) - xol dxo S & 1l
(error bound)
M MM
While the uniform norm gives insight on the short-time scale behavior of the perturbed ODE, we expect that working memory tasks generalize to longer durations 19, The long-time scale behavior on the slow manifold is dominated by the stability structure, i.e., the topology of the dynamics. Although we have seen numerous topologies in Sec. 2, the Persistent Manifold Theorem says that this variability is fundamentally limited, especially in low dimensions (see for more details Sec. S4), This is especially relevant as previous works have identified a low-dimensional organization of neural activity to explain the brain's 's ability to adapt behavioral responses to changing stimuli and environments 79-81 For a ring attractor, this implies that the stability structure of the invariant manifold is either (1) composed of an equal number of stable fixed points and saddle nodes, placed alternatingly and with connecting orbits, or (2) a limit cycle. These different stability structures have different generalization properties (see Sec. 5). In more complex scenarios, such as wwo-dimensional attractors, fixed points can coexist with limit cycles, creating a rich tapestry of possible attractors.
3.4 Implications on experimental neuroscience
Animal behavior exhibits strong resilience changes their neural dynamics, such as the continuous fluctuations in the synapses or slight variations in ncuromodulator levels or temperature. Hence, any theoretical model of neural or cognitive function that requires fine-tuning, such as the continuous attractor model for analog working memory, raises concerns, as they are seemingly biologically irrelevant. This challenge is further compounded by the structural constraints imposed by the connectome, which defines the network's 's architecture and limits the possible configurations of synaptic and circuit dynamics 82,83 Moreover, unbiased data-driven models of time series data and task-trained recurrent network models cannot recover such continuous attractor theories precisely. Our theory shows that this apparent fragility is not as devastating as previously thought: despite the "qualitative differences" in the phase portrait, the "effective behavior" of the system can be arbitrarily

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close. especially in the behaviorally relevant time scales. We show that as long as the attractive flow the memory representation manifold is fast and the flow on the manifold is sufficiently slow, it represents an approximate continuous attractors. Furthermore, our theory bounds the error in working memory incurred over time for such approximate continuous attractors. Therefore, the concept of continuous attractors remains crucial framework for understanding the neural computation underlying analog memory, even if the ideal continuous attractor is never observed in practice. Experimental observations that indicate the slowly changing population representations during the "delay periods" where working memory is presumably required, do not necessarily contradict the continuous attractor hypothesis. Perturbative experiments can further measure the attractive nature of the manifold and their causal role through manipulating the memory content.
4 Numerical Experiments on Task-optimized Recurrent Networks
While our theory describes the abundance of approximate continuous attractors in the vicinity of a continuous attractor, it does not imply that there attractors. In this section, we use task-optimized RNNs as a means to search for plausible solutions for analog memory for circular variable. We train diverse set of RNNs, and then identify the solution type of trained RNNs to gain insights into its performance, error patterns, generalization capabilities, and, ultimately, proximity to a continuous attractor.
approximate solutions away from continuous
Understanding the implemented computation in neural systems in terms of dynamical systems is well-established practice 1.5.86, Researchers have analyzed task-optimized RNNs through non- linear dynamical systems analysis 87? -95 and to compare those artificial networks to biological circuits $6,96.97 Previously, systematic analysis of the variability in network dynamics has been surveyed in vanilla RNNs, and variations in dynamical solutions over architecture and nonlinearity have been quantified 36,87,0091,98 Furthermore, working memory mechanisms in RNNs had tendencies to find sequential or persistent representations through training depending on the task specifica- tion 99, We therefore investigated to what extent training RNNs on a task uniquely determines the low-dimensional dynamics, independent of neural architectures. We see that all the solutions have a slow invariant manifold, making all of them an instantiation of approximate continuous attractors.
4.1 Model Architectures and Training Procedure
Building upon prior work, which has shown their capabilities on such tasks, we trained RNNs to either (1) estimate head direction through integration of angular velocity 92.95 (Fig. 4A1) or (2) perform a memory-guided saccade task for targets on a circle 100,101 (Fig. 4A2, details in Sec. S7.1 and see Sec. S7.3 for how RNNs relate to Eq. 2). We numerically minimized the mean squared error loss LASE between the network output and the target output. For each activation function and each network architecture (vanilla RNN with ReLU, tanh, and rectified tanh activation functions, LSTM, and GRU), we trained 10 networks per hidden size: 64, 128, and 256 with (hidden) state noise.
4.2 Numerical Fast-Slow Decomposition
For each trained network, we find the slow manifold by integrating the autonomous dynamics, then selecting the parts of the trajectories that have speed slower than a threshold (Sec. 17.9.1). We identify the points on the invariant manifold from the simulated trajectories that are projected closest to a set of points in the output space relevant to the task after convergence, i.c. on the target ring. We parametrize the one-dimensional invariant manifold by fitting a cubic spline with periodic boundary constraints to these points (black line in Fig. 4B & C). Normal hyperbolicity is measured by a gap in the timescales of the system (the eigenvalue spectrum of the linearization along points on the invariant manifold, Fig. 4E and F).
We find the fixed points on the invariant ring by identifying regions where the direction of the flow flips (Sec. S7.6.3). Stable fixed points are identified where the flow directions are both pointing towards this flip point, while saddle nodes are identified where they are pointing away (Fig. 4B C.)
4.3 Variations in the Topologies of the Slow Manifold Solutions
To understand what solutions the RNNs found to solve the task, we investigate their memory mechanism. For this, we dissect the dynamics of RNNs by segregating time scales to delineate the rapid flow normal to the slow manifold, and the flow the manifold (Sec. S7.6.3). All solutions
'This correspond type of "ideal pattern"' in the vocabulary of Chirimuuta 81 Our framework proposes a general approach to abstract away irrelevant details in models for analog memory 8s
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involve slow manifold with the same topology as the relevant variable in the task. The different solutions are different their asymptotic dynamics (Fig. 4). The most often found solution is of the type fixed fixe point ring manifold (Fig. 4B and C). These solutions are consistent with observations that persistent activity relies on discrete attractors 102,103 Less commonly found topologies includes the slow torus around repulsive ring invariant manifold (Fig. 4D). This solution in turn is consistent with both observations of the possibility of using non-constant dynamics for memory storage 19,104 and neuronal circuits underlying persistent representations despite time-varying activity 105 All stability structures (fixed points and limit cycles) are mapped close to the target output circle (Figs. S15, S19, S20). We also verify the normal hyperbolicity of the trained networks shown in Fig. 4B and C. The largest eigenvalue of the Jacobian fluctuates around zero (the invariant manifold not continuous attractor), but it is removed from the second largest (Fig. 4E & F).
A1
B
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PCI
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F
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oigervalue spectrum
lack moddes R.w mndde
opanderr
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2m
Figure 4: Slow manifold approximation of different trained networks the nemory-guided saccade and angular velocity integration tasks. (A1) Output of an example trajectory on the angular velocity integration task. (A2) Output of example trajectories on the memory-guided saccade task. (B) An example fixed-point type solution to the memory-guided saccade task. Circles indicate fixed points of the system (filled for stable, empty for saddle) and the decoded angular value on the output ring is indicated with the color according to A1. (C) An example of a found solution to the angular velocity integration task. (D) An example slow-torus type solution to the memory-
saccade task. The colored curves indicate stable limit cycles of the system. (E+F) The eigenvalue spectra for the trained networks in and C show a gap between the first two largest eigenvalues.
4.4 Universality amongst Good Solutions
The fixed point topologies show lot of variation across networks (Fig. 4B.C, C, Fig. 5 and Fig. S20), much like the systems next to continuous attractors (Fig. 1 and Fig. 2). Previously, it has been observed that fixed point analysis has a major limitation, namely, that the number of fixed points must be equal across compared networks "1, Our methodology effectively addresses and overcomes this limitation. The universal structure of continuous attractor approximations as slow invariant manifolds allows us to connect different topologies as approximate continuous attractors (Sec. 3.3). For results on LSTMs and GRUs and a higher dimensional task, see Sec. S7.7 and Sec. S7.9, respectively.
Generalization Analysis
In this section, we use task-trained RNNs study the relationship between dynamics and general- ization capabilities. When neuroscientists study neural computations in animals, tasks have finite durations, leaving it unclear whether animals learn intended computation or finite-time approxi- mation. The same issue applies to trained neural networks. We will explore whether the networks possess the necessary memory for perfect recall or only perform the task within the timescale of their training.

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The two possible approximations of ring attractor, a limit cycle or a fixed point ring manifold (Sec. 3.3), exhibit markedly distinct generalization characteristics. Approximating the system as a limit cycle results in a memory trace that gradually diminishes over time (c.f. Park et al. 1).. Conversely, the alternative approximation's memory states are contingent upon the quantity and posi- tioning of stable fixed points within the system. We describe in detail the generalization properties of the trained networks on the angular velocity integration task at two different time scales; asymptotic and finite time.
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Figure 5: Temporal generalization validates theoretical predictions regardless of implementation
detail. (A) Average accumulated angular eтror versus the maximum flow on the manifold (Eq. 5), shown for finite time (task duration that the networks were trained on, Ti: filled markers) and at asymptotic time (hollow markers). (B) Normalized validation loss of all trained networks. (C) Average error and theoretical upper bound over time for two selected networks (corresponding to
arrows in panel D). (D) Average asymptotic error is roughly inversely proportional to the number of fixed points. (E) Memory capacity is predictive the average error.
Finite time: Along with the angular velocity integration component of the task, the trained networks learn to store a memory of angular variable. We assess the performance of the network to store the memory of the angle over time. The networks typically perform well on the timescale on which they have been trained, Ti 256 time steps 5C). The memory error for Ti is, as theoretically predicted (Eq. 5, see Prop. 2), bounded by the uniform norm of the vector field on the invariant manifold, and therefore by the distance to a continuous attractor (Prop. 1, Fig. 5A, Sec. S7.6.3 & S6).
Asymptotic time: Looking beyond the finite timescale provides valuable insights into the network's ability store information. For the asymptotic time scale, we capture the asymptotic behavior of the system by identifying to what part of the state space the system evolves to in the limit + Do (see also Sec.S7.6.2) For a oneddimensional system, this will either be fixed points or a limit cycle. For the fixed-point type solution, the maximal error is given by the maximal distance to the next fixed point, while for limit cycle, this will always be We calculate the average fixed point distance by taking the average of the inter-fixed-poin interval for each neighboring pair of fixed points.
We can also quantify the loss of information. Assuming uniform distribution over the angles, define the memory capacity as the negative conditional entropy of the continuous memory given the asymptotic state, i.e. the stable fixed points (see Sec. S7.6.2 and Eq. 68).
'e tested all networks with validation set and took a cutoff for the normalized MSE for the networks we consider for the analysis -20 dB (Fig. 5B).

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Error Accumulation in Neural Networks: The mean accumulated error at the time at which the task was trained has an exponential relationship with the number of fixed points (Fig. 5A). Furthermore, this error is bounded by the mean distance between stable and unstable fixed points (red dots in Fig. SD). This another indication that the networks rely on a ring invariant manifold to implement the task. Networks with different numbers of fixed points might have the same performance on the finite time scale (bounded by Ti l/ but have vastly different generalization properties because they differ in the number of fixed points (Fig. 5C).
6 Approximate Slow Manifolds are near Continuous Attractors
In Sec. 2.2, we presented theory of approximate solutions in the neighborhood of continuous attractors. When are approximate solutions to the analog working memory problem near a continuous attractor? We posit that there are four conditions (see for more detail Sec. S5.3): (C1) sufficiently smooth approximate bijection between neural activity and memory content, (C2) the speed of drift of memory content is bounded, (C3) robustness against state (S-type) noise, and (C4) robustness against dynamical (D-type) noise" 19, The correspondence implied by (C1) translates to the existence of a manifold in the neural activity space with the same topology as the memory content.' Persistence (C2) requires that the flow on the manifold is slow and bounded. S-type robustness (C3) implies non- expansive flow, i.e., non-positive Lyapunov exponents. Along with D-type robustness (C4), implies the manifold is "attractive", and normally hyperbolic (see also Sec. 15.3.
If these four conditions hold, for example for task-trained RNNs, there exists smooth function with a uniform norm matching the slowness on the manifold such that when added, the slow manifold becomes a continuous attractor (Prop. 1 and Theorem 7, see also Sec. S5.4). For the RNN experiments, we added state-noise while training using stochastic gradient descent, satisfying (C3) and (C4). We have also verified that (C2) holds (Fig. 5A). Although the stochastic optimization cannot lead to the continuous attractor solution, it gets to the neighborhood where all approximate solutions share the same main feature: having subsystem that has slow flow.
7 Discussion
Continuous attractors are highly prone to bifurcation under arbitrary perturbations unless they exist in special parametric forms. This sensitivity to perturbations has traditionally made them seem unsuitable for modeling neural computation in noisy biological systems, according to conventional views on robustness. Nevertheless, we demonstrate that continuous attractors can exhibit functional robustness, making them crucial concept in explaining the neural computation underlying analog memory. We show that approximations of analog memory (i.c., theoretical models that satisfy conditions (C1)-(C4)) must possess slow manifold dynamics, placing them near continuous attractors within the space of dynamical systems. This implies that both biological systems and artificial neural networks only need to be near continuous attractor to effectively solve problems in a manner similar to the ideal theoretical model, on behaviorally relevant timescales.
Although we expressed our theory in non-parametric manner with an arbitrary perturbation p(-). we can easily extend to particular parametric forms such as biophysical models or an RNN using a sensitivity of the flow to the parameters (e.g. synaptic weight). Our theory be applied to latent dynamical systems estimated from neural recordings 107 As a framework, it can abstract out the details in imperfect dynamical implementations, however, it is open problem to directly recover the continuous attractor from neural recordings or extend it to other ideal computational motifs.
Limitations Although, we only explicitly describe the topology and dimensionality of the identified invariant manifolds for representative set, the results indicate that most solutions have ring invariant manifold with a slow flow. Our numerical analysis relies on identifying a time scale separation from simulated trajectories. If the separation of time scales is too small, it may inadvertently identify parts of the state space that are only forward invariant (i.e., transient). However, this did not pose problem in our analysis of the trained RNNs, which is unsurprising, as the separation guaranteed by state noise robustness (due to injected state noise during training).
The possible solutions that the networks can find are restricted by having linear output mapping. In Park et al. 19 an alternative dynamical solution using oscillators (or quasi-periodic toroidal attractor) was described, however, a nonlinear readout may be necessary.
NNte that effectively feedforward solutions 106 do not satisfy (CI).
10

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Code Availability
The code for this project is publicly available at httppp///ggtthu..ccmm/caaaap/ppp tile_continmous_attraae
Acknowledgments and Disclosure of Funding
We would like to extend our heartfelt thanks to Sukbin Lim, Srdjan Ostojic and Daniel Durstewitz for their invaluable feedback and thoughtful suggestions which significantly refined the work. This work was supported by NIH RF1-DA056404 and the Portuguese Recovery and Resilience Plan (PPR), through project number 62, Center for Responsible AI, and the Portuguese national funds, through FCT Fundação para Cíência Tecnologia- in the context of the project UIDB/04443/2020.
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Supplemental Material
S1 Intuitive definitions of several key concepts used in our paper
Manifold: A part of the state-space that locally resembles flat, ordinary space (such as plane or three- dimensional space, but more generally -dimensional Euclidean space) but can have more complicated global shape (such as donut).
Invariant set: A property of a set of points in the state space where, if you start within the set, all future states remain within the set and all past states belong the set as well.
Normally Hyperbolic Invariant Manifold: behavior of dynamical system where flow the direction orthogonal to the manifold converges (or diverges) to the manifold significantly faster than the direction that remains on the manifold.
Diffeomorphism: A diffcomorphism stretchable map that he used transform shape into another without tearing or gluing. differentiable map with differentiable inverse.
C neighborhood of C function: values and their first derivatives.
set of functions that close the function of hoth their
Compact Set: A set where every sequence of points has subsequence that converges to point within the Intuitively, it means the set is closed and bounded, making "finite" in certain
trajectory a dynamical system that connects two different equilibrium points or periodic orbits. Specifically, heteroclinic orbit connects distinct equilibrium points.
Connecting Orbit:
17

---

S2
The bounded line attractor
order to demonstrate the implications of the theory of the persistence of continuous attractors, we rigorously test the predictions of the theory on the stability of the Bounded Line Attractor (BLA). Our objective is to assess the practical implications of the theoretical findings of bounded continuous attractors in a small and tractable system, and second, to contribute empirical evidence that can help refine and extend existing thcoretical frameworks.
The BLA has a parameter that determines step size along line attractor a. Analogously for UBLA, these parameters determine the capacity the network. The inpuls push the input along the line attractor in two opposite directions, below. The BLA needs to be initialized at ((1, 1) and & (1, 1), respectively, for correct decoding, i.e., output projection.
Win a (1'
2 (-:) b- b- w (-r )
(1) bos = 0.
(i),
W.n
(6)
2a
Parameter that determines step size along line attractor a. The size determines the maximum number of clicks the difference between the two channels. This pushes the input along the line "attractor" in two opposite directions, see helow.
The results from such low-dimensional system can he extended to higher-dimensional systems through reduction methods from center manifold theory. On the center manifold the singular perturhation problem (as the case for continuous attractors) restricts to a regular perturbation problem Furthermore, relying on the Reduction Principle 109 one can always reduce all systems (independent of dimension) to the same canonical form, given that they have the same continuous attractor.
S2.1 Fast-slow form
First of all, we will show how to transform the BLA network to the slow-fast form in Eq. 3-4 to explicitly demonstrate that the theory applies to it. To achieve this, we transform the state space so that the line attractor aligns with the y-axis. So, we apply the affine transformation Ra( 2 with the rotation matrix Rs
cos 0 =
where we have set 0 - 7 So we perform the transformation r x' 1
sin 0 cos 0
1
Ro } and so we have R2 x' + 1 with RT Ro Then we get that
Ro 1' = ReLU
W(R x' 3-11 1)
Ro r'' 1 2
(7)
For a perturbed connection matrix W-
get
-1
R 1'' = ReLU
22
1
-1
1
1
-1
x²
+
2
+1
1
v2
(8)
1
1
-6-1
1
1
z'
x'
+
1
(9)
1
1
2
1
2v2
-1
2
n' = 0 0
0
-1
E
(10)
2
-1
2
-c
S2.2 Bifurcation analysis
We will now identify all possible bifurcations from the BLA, to show that indeed all perturbations preserve the continuous attractor an invariant manifold.
We consider all parametrized perturbations of the form W <- W + V for random matrix V E 2x2x2 to the BLA. The BLA can bifurcate in the following systems, characterized by their invariant sets: a system with single stable fixed point, a system with three fixed points (one unstable and two stable) and a system with two fixed points (one stable and the other a half-stable node) and a system with a (rotated) line attractor. Only the first two bifurcations (Fig. A) can happen with nonzero chance for the type of random perturbations we consider. The perturbations that leave the line attractor intact lead to system with two fixed points have measure zero in the parameter space. The perturbation that results in one fixed point happen with probahility 1. while perturbations lead to system with three fixed points with probability a- see Sec. S2.2.2. The (local) invariant manifold manifold is indeed persistent for the BLA and homeomorphic to the original (the bounded line).
Stabilty of the fixed point with full support We investigate how perturbations to the bounded line affect the Lyapunov spectrum. We calculate the cigenspectrum of the Jacobian:
det(W' (1 + X)[] = (e11
A) 1 X) (E12 1)((21 + +1)
x² - (2 + En Cz2) ) 611 622 611622 612 -22 612621
18

---

= e11 + 622) and v -611 + 611622
There are only two types of invariant set for the perturbations of the line attractor. Both have as invariant set lixed point the origin. What distinguishes them is that type of perturbations leads this fixed point being stable while the other one makes it unstable.
Stability of the fixed points on the axes where Wr > 0. There, the Jacobian is
We perform the stability analysis for the part of the state space
J = (1
1
i
(11)
We apply the perturbation
W
+
(12)
with
61
612
(13)
The cigenvalues are computed
det(W" X][] = (e11 1- A)(e22 1 - (e12 1)((21 1)
= = + (2 611 622) A 611 6z 6116 612 + 621 612621
Let u 2 622 and -611 E11E22
12 621 612621
A + Vu² 4v
(14)
2
Case I: Re(Vu2 40) < u, then 21,2 <0
Case 2: Re(v - 4v) > - then A < 0 and X2 > 0
Case 3: v 0, then À = ( u), i.c., X = 0 and 22 = -1
f11 622 f2 612 + 621 F12
(15)
We give some examples of the different types of perturbations to the bounded line attractor. The first type is when the invariant set composed of single fixed point, for example for the perturbation:
1
(-2
(16)
10
-2)
The second type is when invariant composed of three fixed points:
-2
(17)
10
2
The third type is when the invariant set is composed of two fixed points, both with partial support.
1)
(18)
10
The fourth and final type is when the line attractor is maintained rotated:
1
1
(19)
20
10
1
S2.2.1 Bifurcation landscape
We will now state our previous observations as Thcorem that characterizes the possible bifurcations of the BLA.
Theorem 2. All perturbations of the bounded line attractor of the types listed above.
Proof. We enumerate possibilities for the dynamies of ReLU activation network with two units. First all, note that there can be no limit cycle or chaotic orbits.
Now, we look at the different possible systems with fixed points. There can be at most three fixed points 110 Corollary 5.3. There has to be at least one fixed point, because the bias is non-zero.
General form (example):
1
(22
(20)
10
-2)
19

---

One fixed point with full support:
In this case we can assume W to be full rank.
ReLU
611
612
21 x22
(
21 202
621
622
Note that > iff 21 = c11 + (612
1)22
>
0. Similarly for 22 >> 0.
So for a fixed point with full support, we have
21 22
=
A -
(=1)
(21)
with
f11 -1
A =
621
622 -1)
Note that is not possible that 31 T2-
Now define
B --
1 612
det A (1-6) 621 611 -
with
det A C11622
612621 + 621
Hence, we have that 21, 22 > 0f B11 + B12 > 0, B2 + Boz > 0 and det > 0 or Bn + B12 < 0. B21 + B22 < 0 and det A < O.
This can be satisfied in two ways, If det A > 0, this is satisfied if e22 > 61 and 61 > 621, while if det A < 0. this is satistied if
612 and
621. This gives condition 1.
Finally, investigate the condition that specify that there fixed points with partial support. If æ1 0 then (e22 1)22 1 = 0 and 21 0. From the equality, we get that 22 T-e22" From the inequality, we get (e12 1) + 2 ie.. 1-612 2 #2. Hence,
1
612
622
and thus
62 S 612-
(22)
Similarly to have fixed point xx such that x = 0, wC must have that
611 S 621
(23)
Equation 22 and together form condition 2.
Then, get the following conditions for the different types of bifurcations:
If condition violated, but condition 2 satisfied with exactly one strict inequality, there fixed points on the boundary of the admissible quadrant.
2. If condition 1 is violated, and only one of the subconditions of condition 2 is satisfied, there is single fixed point one of the axes.
3. If condition 2 is violated. there is a single fixed point with full support.
If both conditions are satisfied, there are three fixed points.
We now look the possibility of the line attractor being preserved. This the case if It is not possible to have line attractor with fixed point off of for as there cannot be disjoint fixed points that are lincarly dependent 110 Lemma 5.2
S2.2.2 Probability of bifurcation types
We will now calculate which proportion proportion of the bifurcation parameter space is results in the different bifurcation types. The conditions that result in three fixed points are
0 < 611622 611
222 612621
612
621
62 & C1
61 < 621-
20

---

Therefore, because
62 S 612
611 S 621-
we always have
0 < 611622 - 611 622 612621 - 612 622
This implies that this bifurcation happens with probability e-hall around the BLA neural integrator with 1. We conclude that the single stable fixed point type perturbation happens with probability
S2.3 Structure of the parameter space
We will present the structure of the bifureation space through slice in which we fix e11 and 612- First, summarize which conditions result in which bifurcation Table derive that the local bifurcation this slice has the structure as shown in Fig. S6.
Table 1: Summary of the conditions for the different bifurcations.
IFP (full)
IFP (partial)
3FPs
2FPs
LA
Cl
x
C2
x
only Eq22 or 23
v
x
E,
E
det <
6
0 = Hb,
3FPs
b=
det 0
6,
8 > 81
8 = 8
2 = ac 11
2
2 11
det A >
Figure S6: slice of the parameter space of the BLA for a fixed 611 and 612
S2.4 Smoother activation functions
is well-known that activation functions (ar in Eqs. 61 and 64), which can take many forms, play a critical role in propagating gradients effectively through the network and backwards in time 1f-113 Activation functions that are C" for > the ones which the Persistence Theorem applies. The Persistence Thcorem further specifies how the smoothness of the activation have implications the smoothness of the persistent invariant manifold. situations where smoothness
persistent invariant manifold is importance, smoother activation functions might be preferable, such as the Exponential Linear Unit (ELU) 114 or the Continuously Differentiable Exponential Linear Units "55
21

---

S3 Ring perturbations
To computationally investigate neighborhood recurrent dynamical systems that implement continuous attractors, we investigate RNNs that are known priori to form or dimensional continuous attractors.
We define a local perturbation (i.c., a change to ODE with compact support) through the bump function (x) exp (1= for ||æ]|| < and zero outside, by multiplying it with uniform, unidirectional vector field. All such perturbations leave at least part of the continuous attracorr intact and preserve the invariant manifold, i.c. the parts where the fixed points disappear slow flow appears.
The parametrized perturbations are characterized the addition of random matrix the ODE.
S3.1 Simple ring attractor
We further analyzed simple (non- -oo-bboooggica) ring attractor, defined by the following ODE: i r(1 r), 0 = 0. This system has as fixed points the origin and the ring with radius one centered around zero, i.c., (0,0) U {(1,6) 0) 0 E [0, 2x)). }. We investigate bifurcations caused by parametric and bump perturbations of the ring invariant manifold (see Sec. S3), which is bounded and boundaryless. All perturbations maintain the topological structure of the invariant manifold.
S3.2 Heading direction network
The networks proposed in are composed of N heading- tuned neurons whose preferred headings 0g uniformly tile heading space, with an angular separation AO = 2*/x radians. These neurons can be arranged topologically a ring according to their preferred headings, with neurons locally exciting and broadly inhibiting their neighbors. The total input activity h, of cach neuron is governed by:
Th = -h + N Σ (W + Unn Wasyym (a(h)) + cs1,s = I,...,N,
(24)
with
jk Jy JE cos 0c),
(25)
where Jr and Jy respectively control the strength of the tuned and untuned components of recurrent connectivity between neurons with preferred headings 0, and and Van an angular velocity input which the network receives through asymmetric, velocity- modulated weights
Wrasym = sin(Os 0z).
(26)
Fixed points In the absence of an input (Vin = 0) fixed points of system can be found analytically by considering all submatrices Waym for all subsets (a C [n]} with[n] = (L...., N). A fixed point x² needs to satisfy
-WW cfs
(27)
and
23 0 for i c.
(28)
We bruteforce check all possible supports to find all fixed points. We the eigenvalues of the Jacobian to identify the stability of the found fixed points. We evaluate the effect parametrie perturbations of network of size N with Jg = Jr 2.4 by identifying bifurcations (Fig. 2A).
Connecting orbits We approximate connecting orbits through numerical integration of the ODE intialized in close to the identified saddle points along the unstable manifold.
22

---

Measure zero co-dimension bifurcations attractor network fall into two types, Fig. S7.
Measure zero co-dimension bifurcations of the ring
Figure S7: Measure zero co-dimension 1 bifurcations of the ring attractor network" 8
Measure zero co-dimension bifurcation The limit cycle is the only bifurcation that we found that can be achieved only measure zero set of parameter values around the parameter for the continuous attractor.
Independence of norm perturbation on bifurcation we can see Fig. S8, topology of the system is maintained through range of bifurcation sizes when the bifurcation direction is fixed.
Perturbation size
1e-10
1e-05
0.1
0.5
1.1
1.5
4
Number of fixed points
6
8
10
12
Figure S8: Rows show the bifurcations resulting from perturbations from the matrices with the same direction in Fig. 2A but with different norms (columns).
23

---

S3.3 Ring attractor approximation with tanh neurons
We investigated the bifurcations around the approximate ring attractor constructed with symmetric weight matrix for tanh network 5862 The functional form of W the sum of constant term plus Gaussian centered
8, 0:
(0. (0))
W( - 0,) J -J+
(29)
2o
with the dimensionless parameter J representing the strength of the weak crossdirectional connections, J+ the strength of the stronger isodirectional connections, and a the width of the connectivity footprint.
Such ring attractor approxinations are similar to the ones 0061,116.117 However, some have other nonlinearities. e.g., the sigmoid is used in 61 Another line of related models can be found in 118 119 andd" 120
Sensitivity of continuous attractors to perturbations We will show that there are differences how well approximations perform at different timescales. measure how performance of different models the representation of an angular variahle drop as a function of perturbation size Fig. $9 through the memory capacity metric (Sec.S7.6.2), For each perturbation size, we sample a low rank (rank 1,2 or 3) random matrix with norm equal to that perturbation size. We determine the location of the fixed points through the local flow direction criterion as described in Sec. 4.2 This invariant manifold close the the original invariant ring attractor. The initial ring had 2N fixed points (N stable, N saddle) on this invariant ring manifold. The memory capacity of this initial configuration is N log(N)) for the 2N uniformly spaced fixed points.
Loss of function:
found to be consistently
1.75
rank
1.50
memory capacity
1.25
1.00
0.75
0.50
0.25
0.00
1e-1
Degradation of performance across perturbation sizes. System behavior at the asymptotic time scales measured through memory capacity.
Figure S9:
S3.3.1 Mean field approaches
Another line of models 121-124 also relies on connection weights with translational invariance
A
(xc
((e-x')² x²)2
J(x,z')
exp
(30)
2/maa
2a
where J(r, x²) is the neural interaction from x' to and the ensemble of infinite neurons are lined up so that x E (-o0, 0o).
Weak random spatial fluctuations in the connection strength are be expected when learning the coupling function with Hebbian plasticity. the ID case, is well known that the presence of such synaptic heterogencity causes a drift of an input-inducet activity pattern to one of finite number of attractor positions which are randomly spread over representational space 6,20 Similarly, in 2D, spatial fluctuations in the connection strengths cause slight perturbation of the bump shape 125 Frozen stabilisation has been proposed as alternative method construct neural networks to self-organise to a state exhibiting (high-dimensional) memory manifolds with arbitrarily large time constants 126
127 analyzes an Ising network perturbed with specially structured noise at the thermodynamic limit. Although their analysis elegantly shows that the population activity of the perturbed system does not destroy the Fisher information about the input, they do not consider scenario where the ring attractor used as a working memory mechanism, it is rather used to encode instantaneous representation. In contrast, our analysis involves understanding how the working memory content degrades over time due to the dynamics. We are not aware of any mean field analysis that covers this aspect.
24

---

$3.4 Ring attractor approximation with a low-rank network
The networks consisted of N firing rate units with a sigmoid input-output transfer function" "
(.( - --(0) + E Jijo(1, + J,
(31)
where (4) is the total input current unit i, Jay 9Xa, Py is the connectivity matrix, 6(x) tanl(()) the current-to-rate transfer function, and I, the external, feedforward input to unit i. The random component 9X considered unknown exeept for its statistics (mean O, variance 9² //N.. general structured component of rank r < N can be written as superposition of r independent unit-rank terms
(1) m
m n
Pg
(32)
N and is in principle characterized by 2r vectors m (1) and n (8)
N
To approximate ring attractor consider structured matrices where the two connectivity pairs m and (2 (2) and n (2) share two different overlap directions, defined by vectors yn and y2. We set:
m
/ 522
rf 211 riyh,
(33)
m
(2)
- -r3x2 r282,
(34)
n 552 - 19 x3 riy,
(35)
n (2)
E2 r2 x44 r2V2.
(36)
where 5 is the variance of the connectivity vectors and rî and r3 quantify the overlaps along the directions /1 and ya-
We keep the following parameters for analysis: E = 2, p = 1.9 and g = 0.1.
0.8
0.8
0.0
0.0
0.0
-0.8
-0.8
-0.8
0.8
0.0 mi
0.8
0.8
0.0
0.6
0.8
0.0
0.8
mi3
mil:
Figure S10: Some examples of networks dynamics for sizes N 10, 100, 1000.
Our theory explains the phenomenon of the existence of ring invariant manifold in these low-rank networks as follows. We can think the finite size realization as a small perturbation to infinite size network on the reduced dynamics in the m m plane (independent of the parameter for the random part of the matrix) (Fig. 2B). For very small networks the ring structure is destroyed and only the plane persists as slow manifold.
S3.5 Embedding Manifolds with Population-level Jacobians
We fit three networks with the Embedding Manifolds with Population-level Jacobians (EMPJ) method 59
The networks are RNNs of N = 10 neurons with a tanh activation function and r = 0.05 time constant. To use EMPJ we need to specify our desired & fixed points and m eigenvectors and eigenvalues per fixed point.
To do this we choose k = 3 points (( ) 1 in the 2D ring with radius 1 that equally spaced from 0.1 to 2n radians. Then we specify the vectors orthogonal to the ring in those points (V. = {(x,, y. ).) same the points) and tangent (V, [(- #x))
associate the negative eigenvalue 1// to the orthogonal eigenvectors and different eigenvalues to the tangent eigenvectors. We use -1, 0 or depending on whether we want the points to be stable, center or unstable in the direction of the slow ring manifold.
Finally
Since these determined dynamics are only in a 2D plane, we use a random linear mapping D: H² + R to map the fixed points and the eigenvectors to 10- dimensional space. First, sample a random from orthogonal matrix A from the O(V) Haar distribution and then take the first two columns of this matrix to be D = [A]12.
25

---

EMPJ then returns network parameters that satisfy these constraints, Furthermore, due to the particular solver used, that regularizes the magnitude of the parameters, get that all the other eigenvalues specified also set 1/r (for details see" ))
Finding fixed points As we remark in Sec. S3.5.1, EMPJ networks are not robust to S-type noise, therefore cannot apply analysis of identifying the invariant set through the convergence criterion of numerieally integrated trajectories. We therefore find fixed points through the Newton-Raphson method. We iteratively solve
J(x, dxx X,
(37)
where (x,) is the Jacobian of the system at x, and X = x, dx, The iteration stops when |dx] < for a tolerance threshold 6. We initialize Xa on the invariant ring uniformly. The maximum number of iterations was set to 10,000 and the tolerance level to 10
S3.5.1 Lack of S-type robustness
We remark that the resulting invariant manifold is robust to S-type perturbations. Although the fixed points that are constrained
fitting procedure are attractive in directions, some the points along the ring might not be. For on nn-manifold perturbations (in the plane in which the ring is embedded), S-type perturbations do lead to flow towards the invariant ring (Fig. S11 A). However, for (small) off-manifold perturbations, the trajectories typically diverge away from the invariant ring (Fig. S1 IB). This indicates that the basin of attration very small and hence this approximation is robust to S-type noise.
All the perturbations were sampled as x00) =00) 17 with z(0) span(D). For the on-manifold perturbations vy E span( (D) and || 10 2, For off-manifold perturbations v E R and ||v]| 10-5
A
B
Figure S11: Trajectories of the third network in Fig. 2C. Starting point in red, end of trajectory in blue. (A) On-manifold S-type perturbations from the ring. (B) An example of an off-manifold S-type perturbation from the ring.
S3.5.2 Higher dimensional manifolds
We furthermore fit torus and sphere continuous attractor with the EMPJJ. The networks used have N = 100 neurons. For finding fixed points with the Newton-Raut Raphhonn method, used 1.000 the maximum number of iterations and tolerance level
Torus Figure $12 illustrates the stability structures of the approximate torus attractor fitted with EMPJ. The ratio of the radii of the two rings is adjusted for visualization purposes.
Sphere Fig. S13 illustrates the stability structures of the approximate sphere attractor fitted with EMPJ. Similar to the torus, points initialized off the sphere converge onto the sphere attractor (Fig. S13).
S3.5.3 Other fixed point fitting methods
Storing multiple continuous attractors has been worked out in 128 with a similar approximation as our theory suggests. The different stored patterns form discrete approximation of the continuous map (resulting in quasicontinuous maps, which corrspond to the approximate contintuous attractors in our thcory).
26

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A
B
C
Figure $12: The approximate torus attractor. (A) Points initialized grid off of the torus (blue) converge onto the torus attractor (red). (B) The found fixed points on the approximate torus (green: stable, red: saddle). (C) The found fixed points projected onto the two 2D subspaces that defined the two rings of the torus.
A
B
2
Y
Figure $13: The approximate sphere attractor. (A) Points initialized on a grid off of the sphere (blue) converge onto the torus attractor (red). (B) The found fixed points on the approximate sphere (green: stable, red: saddle). The two subfigures show rotated versions of the location of the fixed points on the sphere.
S4 Persistence Theorem
Understanding the long-term behavior of dynamical systems is a fundamental question various fields, including mathematical biology, ccological modeling, and neuroscience. Fenichel's Persistence Theorem provides critical insights into the behavior of such systems, particularly in relation the stability and persistence of invariant manifolds under perturbations.
Fenichel's 's Persistence Theorem extends the classical theory of invariant manifolds, offering conditions under which normally hyperbolic invariant manifolds persist despite small perturbations. This theorem particularly powerful in analyzing systems where perturbations are inevitable, providing a framework for understanding how qualitative features of the system's 's dynamics are maintained. In this section, we delve into the specifics of Fenichel's Persistence Theorem, outlining its key components, assumptions, and implications.
Invariance One of the main concepts in the Persistence Theorem is the notion of an invariant manifold. Intuitively, this just means that trajectories stay inside the manifold for all time. Local invariance is bit more involved and allows for the possibility leaving manifold, but only through it's boundary.
Definition 1. A set M is locally invariant under the flow from Eq. 3-4 if it has neighborhood V that no trajectory can leave M without also leaving
other words, it is locally invariant for æ M, p(,, [0.4] C V implies that p(æ, [0, C M, similarly with [0, I] replaced by [4,0] when L <
Definition A set S said be forward invariant under a flow ye: for every point a and all L the image of æ under the flow at time i, denoted (z)),,, remains in S. This can be written as:
gi (x) S for x E S and 2 0.
Small perturbation In the context of Fenichel's Persistence Theorem, a "sufficiently small C" perturbation the vector field f" refers to perturbations that are small the C norm. The C" norm measures the size of a function and its derivatives up to order
Formally, let R _2" be the original smooth vector field and let f be perturbed vector field. The perturbation f is sufficiently small C" perturbation if the difference f f has small C" norm. Mathematically, this can be expressed as:
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| or
<c.
where é is small positive number, and the C" norm is defined as:
| /())]
max oSkSr ZER"
sup
or
Which gives a constraint on how much each of the k-th derivatives D* of the perturbed vector field can differ from the original.
In summary, perturbation f considered sufficiently small in the C" sense if the difference between f and f. along with their derivatives up to order r. is uniformly small across the domain. For C perturbations, this defines a Cd neighborhood of functions, within which the persistence of the manifold is ensured by Fenichel's theorem. For r = 0, we get the uniform norm (defined as [[] = (If)), see also Sec.S6).
Closeness Finally, to formalize what is meant by that the persistent invariant manifold very close to the original continuous attractor, we need notion of distance between the manifolds.
Definition 3. Let (M, d) be metric space. For each pair of non-empty subsets X C M and Y C M, the Hausdorff distance between X and Y is defined as
du (X, Y) == max sup d(x, Y), sup d(x,v)) v0)
wYY
where sup represents the supremum operator, inf the infimum operator, and where
d(a, B)
inf d(a.b) b)
bEB
quantifies the distance from a point a E to the subset B C X.
In conclusion, the Hausdorff distance provides a rigorous mathematical framework to quantify the "closeness" "similarity" between two sets.
S4.1 Fenichel's Persistent Manifold Theorem
This section will introduce the original Fenichel's Persistent Manifold Theorem, laying the groundwork for understanding how normally hyperbolic invariant manifolds persist under perturbations systems with distinct time scales. By examining this foundational theorem, we can build a deeper understanding of the stability and behavior of complex dynamical systems.
In the study of dynamical systems, particularly those involving multiple time scales, understanding the behavior of solutions near invariant manifolds is crucial. These manifolds often determine the long-tenn dynamics of the system and their stability properties. The Fenichel's Persistent Manifold Theorem provides a powerful framework for analyzing such systems by demonstrating the persistence of normally hyperbolic invariant manifolds under small perturbations. This theorem is particularly relevant in systems where variables evolve on different time scales- typically referred to as "slow" and "fast" dynamics. By reformulating the system with change of time-scale, we can explore how the dynamics on these manifolds behave, especially when perturbed. Consider the system given by Equations 3-4, which can be rewritten using a rescaled time variable, т, distinguish between the fast and slow dynamics as follows:
{
ea' = =(xx g)
(38)
v/ ((x.y..)
where = d and T = 1/.. The time scale given by is said to be fast whereas that for t is slow, as long as di 0 the two systems are equivalent.
The functions g and kin Eq. 3-4 are both assumed to be C" (for > 0 on set U x I where U C R is open, and / open interval, containing O.
Suppose that the set MD is subset of the (h(æ, v. 0} and is compact manifold, possibly with boundary, and normally hyperbolic relative to 38.
Theorem 3 (Thcorem in in")). If e >> O, but sufficiently small, there exists a manifold M. that lies within Oce) of Mo and is diffeumorphic to Mo. Moreover it is locully invariant under the flow of Eq. 3-4, and C", including for any 0 < oc. Finally, M. has O(e) Hausdroff distance to Mo and has the same smoothness as g and h.
If the invariant manifold Mo is attractive, then the only way trajectories can escape the invariant set M, after perturbation, is in negative time through the boundaries. This guarantees that the persistent manifold M, is still attractive.
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S4.2
for ring attractor). As you change the bifurcation parameter, Fundamental limitations of the topology of bifurcated continuous attractors
S4.2.1
Flow line
Theorem 4 (One Dimensional Equivalence). Two flows and in are topologically equivalent if and only if their equilibria, ordered the line, can be put into one-to-ome correspondence and have the same topological type (sink, source or semistable).
S4.2.2 Flow on ring
The flow ring can be described by the differential equation 129;
dl
dt = f(0):
(39)
with 0 [0,2m]
The simplest type of flow ring is the uniform circular flow, where each point with constant angular for all 9 there a circular flow with variable speed.
velocity f(0) w. [0)]
There is fixed point for each unique 0 for which f(0) O. The nature of the flow around these fixed points can classified into:
Stable fixed points: Points where the flow tends to as time progresses (f(0) > 0 0 [0,0] and f(0) 0 for all 0 [0, 2x] n V for some open ball V around 6).
Unstable fixed points: Points from which the flow diverges (f(0) < 0 for [0, n V and f(0) > 0 for all 0 E [0, 2m] O V for some open ball V around 0).
For detailed discussion of how bifurcations of a continuous attractors depend on the symmetry of the continuous attractor, see for example The Equivariant Branching Lemma (Lemma 1.31 (0)) the solutions are symmetric under rotations (a circular symmetry you find that new solutions appear that also respect this rotational symmetry. The lemma tells you that these new solutions will align with specific symmetries (like different rotation angles), which are described by the irreducible representations of the symmetry group.
S4.3 Consequences to system identification
Fenichel's Persistence Theorem has several significant implications modelling and system identification dynamical systems. Because the theorem provides a guarantee that small perturbations in the system do not lead to significant changes in the qualitative behavior of the system, we can be (slightly) wrong for example about the exact nonlinearity of neuron's transform function. For example, if neurons are only approximately ReLU the theory developed in 131 still holds (at the hehaviorally relevant timescales). More generally, when reconstructing computational system dynamics to understand how cognitive functions are implemented the brain, our theory shows that small deviations in the identified system can still lead to behaviorally equivalent models for neural computation".
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S5 Near Perfect Analog Memory Systems are close to Continuous Attractors
We will give some clarifications and proof' the claims on systems near perfect analog memory systems. S5.1 Revival of the continuous attractor from a slow manifold
We provide a proof of Prop. which is dependent on the c-Neighborhood Theorem"'. which we state here. Theorem (c-Neighborhood Theorem (22), enough c > 0, each w E Y, == {w E R" By E Y y wil < <) has unique closest point in Y. denoted T(w). Moreover, क Y + Y is submersion with xly = id.
1º smooth, compact manifold. Then for small
We now prove Prop.
Proof. is sufficient show that there is perturbation p that has flow off of M. but for which f f- on Me for full system as defined in Eq. Define
p(x)= Soa. -(xx (---y)FE F)
with the Dirac delta function 6(x) = 6(x, ((())))))))....)..)))))))))))))) .(xa). then easy to check that f(x) + p(x)
for all x E M. and f(x) + p(x) = f(x) for all x & M.. Hence, we have a continuous attractor at M...
smoothness important, we can construct following perturbation. From the e-Neighborhood Theorem
we get that there exists a smooth positive function 8: Me - IR+ such that if we let Ng be 6-neighborhood of Me.
M = ty E R ly x < 6(z) for some x E Me),
then each y E Na possesses unique closest point (y) in M. with the map wa: Ms M, being submersion.
can then define bump function
(v)) = exp (-1-p>)-)--))) e nz(1), Tz(1))
(40)
0
otherwise
Then the perturbation
p(y) = ++y)=(((((((()))
is smooth and creates a continuous attractor at M
S5.2 Output mapping
This paper focuses linear output mapping for simplicity, Errors can be minimized with nonlinear mapping. if there is an output that is mapped off of the output manifold, correct this, if space of output mappings is general enough. However, this can only be applied for errors off of the output manifold. If there is memory degradation along the output manifold, it not possible to choose another mapping that corrects for this errr. Therefore, we choose a linear output mapping for our analysis. In some cases, linear output mappings are found to support neural computation, for example for motion
always adjust the output mapping
S5.3 Approximate solutions to an analog working memory problem
We will now discuss the conditions for when approximate solutions to an analog working memory problem are near a continuous attractor. We consider approximate solutions to an analog working memory problem to be systems of the form 61 or 64 (in both cases following linear decnder), which have small memory error over time in output space.
$5.3.1 Robustness
Noise, practically defined unpredictable components of the system's behavior, comes from many sources. The concept of S- and D-type noise is based on
S-type robustness S-type noise encapsulates reversible changes in the neural state such that the deterministic part of the dynamics itself remains unchanged. Neural dynamics must be robust to perturbations and stimuli that push neuronal activity away from the continuous attractor
"oot be confused with the perturbation parameter
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D-type robustness D-type noise, in the space of recurrent network dynamics parameterized by the synaptic weights. This corresponds to slight changes the ODE, i.e, perturbations. Previously. it robustness D-type noise was considered to correspond structurally stability" 19
Here we shortly discuss what we consider to be a necessary and sufficient condition for D-type robustness. We can use the concept of Lipschitz persistence to define D-type robustness. Mañe showed that if an invariant manifold is Lipschitz persistent then it must be normally hyperbolic 136, To understand the concept of Lipschitz persistence, we need to define the Lipschitz section and Lipschitz constant.
Definition 4. Let
a C boundaryless manifold and V C C³ compact boundaryless submanifold. Assume that M is a submanifold Rf". Let NV be a C' subbundle of T'M satisfying TV @ NV = TM If 1) is section of NV. define the Lipschitz constant of by
[I((e) r(viil
Tip(n) = sup
22 E V V
| vll
We say that is Lipschitz section if Lip(n) < t00. Let ('c(NV) be the space of Lipschitz sections of N endowed with the norm
|mllcc = sup { [[((=)] E V) Lip(n)-
Let Diff' (M) (if be the space of Ca diffcomorphisms with the topology of the C* convergence on compact subsets. Definition 5. Let f E Diff' (M). We say that V is a Lipschitz persistent invariant manifold of f if there exists neighborhood U of V such that for all 8 > 0 there exists a neighhorhood Us of f such that if g E Us there exists 17 (NNVV) with |! < 8 satisfying Va = graph(n), where graph(n) (e(p) x E V).
V/ = (UU).
Observe that this definition implies Vy V. hence S(V) V. Moreover, the Lipschitz persistence independent of the bundle NV.
For a flow &r (coming from the solutions of an ODE) we can fix f = r E Ryu so that we get a homcomorphism Pr. This allows us to apply this result to apply our case.
In the where V is a point, then it is hyperbolic fixed point and the persistence follows trivially from the implicit function theorem.
Remark 1. It is sufficient take persistent manifold that is uniformly locally maximal. If V is persistent and uniformly locally maximal, then V has be normally hyperbolic
Definition 6 (Uniformly locally maximal). There exist neighborhoods U V in M and U of f in the space Diff' (M) of d'ddiffcomoppiisms of M, such that for any E U, N₂ = Nsez 9 (UU) is -uusumannifold close V. with N2 N. The latter property implies the uniqueness of the invariant submanifold.
S5.4
Near Perfect Analog Memory Systems are close to Decomposable Systems with a Continuous Attractor
We now prove more general statement about the kind of systems that are close to perfect analog memory systems. The theory guarantees that a system that satisfies conditions (CI)-(C4) will have continuous attractor in the following sense. For such a system there exists a decomposition such that the system can be effectively decomposed into a continuous attractor (attractive invariant manifold with zero flow) and a component on which a (possibly "fast" flow) can exist buy which get quenched by the out put projection. These additional dynamics orthogonal to decoding have been observed for motor movement preparation 157,138
For this part of the theory, we need to consider the output manifold Maruu the manifold on which we determine the error over time as in Sec. S6. For this section, we will consider the output mapping to be smooth (possibly nonlinear) mapping g: X -> Y between the neural state space X = p and the output space Y = Rdy For a circular variable this will be the ring $1
The construction of a perturbation relies on finding the necessary minimal structure in the invariant manifold for which can guarantee closeness continuous attractor. Therefore, first of all, need this closeness terms of the geometry of the manifold, which we guarantee through the notion a fibration the output mapping. Second, we need to guarantee that the flow is bounded in a sense so that our perturbation is also bounded by this amount. We will characterize this by the vector field normal to the fibers of the fiber hundle. A fiber bundle is mathematical structure that allows us to study spaces that are locally like product space but globally may have different structure. So first state the definition of fiber bundle and trivial fibration.
Definition 7. A fiber bundle is a structure (E, B, n. F') where:
E is the total space,
is the base space,
π É - BB is continuous surjection called the projection map, and
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I' is topological space called the fiber.
This structure must satisfy the local triviality condition: for each b B, there exists an open neighborhood U of b such that there is homeomorphism
9 (U.) + Ub x Fi
that commutes with the projection onto U. meaning that the following diagram commutes:
(U)) (U,)
Up Fb
pr.
U,
U
where pr, Ub Pb. is the projection onto the first factor.
So, around every point in the hase space, you can "zoom in" and see that the bundle looks like straightforward product of the base space and the fiber and is providing this trivialization.
Definition We say that a projection map E + B is locally trivial if each point b B is contained in an open set U having the property that Eg == π (U) trivial over U.
We will rely on the concept of a submersion to characterize how the output space needs to relate to our invariant manifold.
and N differentiable manifolds and f: M - N be a differentiable map between them. The map f is a submersion at point E M if its differential Df.: T,M - TTpp N is a surjective linear map.
Definition 9 (Submersion).
This allows to characterize what kind of structure the invariant manifold needs have, namely it be as direct product of output manifold and some other manifold which described over what part of the invariant manifold the output mapping is invariant.
Theorem (Ehresmann's 's lemma 1,9). smooth mapping f: M - N, where M and N smooth manifolds,
a surjective submersion, and
2. a proper map
then it a locally trivial fibration.
Remark 2. If a manifold M is compact, then the above smooth map is proper map. If we do not have submersion that the output mapping transversal to the output manifold, we have situation in which the flow on invariant manifold can arbitrarily fast (even though memory degrading slowly). This happens when the flow is in a singularity of the output mapping.
For statement we to the output manifold and another compact manifold: M, = MMm X Mmul with Matow = My. We can relax this to allow for the possibility of some torsion along the invariant manifold. For this to hold, it is sufficient that the output mapping must be a submersion because this makes it a locally trivial fibration. So we get that g: X Y detines a locally trivial fiber mapping.
that the invariant manifold can be decomposed into a space that is diffeomorphic
The second assumption need is have bound for the speed of trajectories along the fibers, which will correspond to speed along output manifold, resulting memory degradation. We need to assume that there a slow flow (in the direction of Matow). We characterize relevant maximal size of the vector field that needs to he perturbed as the supremum over the uniform norms of the vector field normal to the fibers of the fiber bundle.
Theorem Let Me be a connected, compuct. normally hyperbolic slow manifold (as parametrized by Eg. (3)- (4)) with the real part of the eigenvalues of V.h all negative. Further, assume that this manifold can be decomposed M. = Mamm x Mmutt with Matow = -My and that the uniform norm of the flow tangent to M, restricted to Mato be ([y))aulls = T3. Then, there exists a perturbation with uniform norm at most nj that induces biffrcation system that is decomposable into continuous attractor and a system with a non-zero flow.
other words, after applying this perturbation, the slow component of perturbed system satisties X K/lsww for the system X f(x) + pxx). Furthermore, the trajectories of the resulting system form a fiber bundle where the output projection serves as the bundle projection. Each fiber consists of trajectories that are mapped to the same value in Morpu, meaning that the fibers describe invariance under the output projection.
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Proef. Assume that the output mapping is submersion from the invariant manifold M, to the output manifold Mampa- Ehresmann's lemma implies that this implies that have locally trivial fibration (Me, MMM.Maa g. Mmu)).
construet the perturbation p as the part of the vector field that us normal to the fibers g (2) for each y Moupus By construction, this perturbation has uniform norm at most n, This perturbation makes the vector field normal to each fiber zero. That implies that cach fiber is an invariant submanifold of .M.. From the structure of the fiber bundle follows that there is continuum of such invariant submanifolds. Hence, the perturbed system is decomposable into system that is continuous attractor and system with non-zero flow.
Remark This perturbation can be made smoothness, along similar lines as above. We can take ac- Neighborhood around the invariant manifold on which we extend the above vector field with bump functions to get smooth vector field that still results in attractive invariant manifold.
Example An example of an approximate continuous attractor solution of the form with a decomposable system of which one of the subsystems is close to continuous attractor is the torus solution in Fig. 4D. In this case, the system can he perturbed slightly such that there exists continuum of limit cycles laid out over ring.
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S6 Upper bound for the memory performance on a short time scale
We will now formalize statement about an upper bound dependent on the uniform norm of vector field on the slow manifold in Scc.3.2 and provide a proof.
Proposition 2.
Let M be normally hyperbolic slow manifold in Prop. 3.2. Let xo M. and ip [ be the flow restricted to manifold. The average deviation from the initial memory Xy over time bounded linearly
1 vol M
/
[x((,xx)= - xol dxx S 11001
(41)
Proof. Numerical integration of the ODE gives
ætt, = /. p(u(T))dr + xo
/ 1111loodt +x0
1l1ypl1o0 +
From this, we get
Mol / xx(1,x0) xal dxo S vol //v 111/1110
= =11141l,0
We formulate here a theory of continuous attractor approximations in terms of memory loss over time. It can be used uniform norm of vector field the manifold to bound the memory performance the short-time scale. Xa E M, and و Pl. be the flow restricted the manifold. We will show that the average deviation from the initial memory Xo over time is bounded linearly as Eq. 5.
S6.1 Ring attractor
For ring attractor we can give more tight bounds on the accumulated error for the angular memory. Suppose we have dynamical system E R with autonomous dynamics á (x) and solutions x(t, xo) uniquely defined for each Xo. Let define the (i.e. the average deviation from the initial memory) for attractor
/
C(T) vol M
[x(1, xo) xp dxo
(42)
further assume that the memory is simply the state of the network, we need take into consideration decoder memory. Suppose that there is an invertible decoder mapping -> R a ring variable, we can take this to be the projection onto the plane Wou and then applying the arctan:
g(x) arctan W o..)
(43)
case the ring attractor, memory we would like encode is 21 [0. 2m). emror defined as Ixr - Vlo = or (|æ - vl) where:
i
On (n(x) = 2T
(44)
if x
we call (a t) s(pa(f(a)) we get the expression of the memory loss for this kind of memory
2m
((T)
(láo(as. () aol) daø
(45)
2x
General bounds Define the following functions:
f/d (láo(0o, (oool)
e (t) sup or (lão(as, t) - ool) 2 2x
on
1 2n
eT (t)
2T Jo Or (lão(as, 1) -xol) dep
(46)
2n
(t) inf (ao, t) -ool)=
/
(lāo(ao, ool)
27
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Then we get the bounds for the loss L(T) as:
8 T / (t)dt ((T) T '- (()dd s TJ, f c
(47)
Speed bounds
We can define the maximum, average and minimum memory speed as:
d'
U sup ag dt [ox ((āo(av, () -x
(or (lāo(au, t) - aul)) La 4m 1
(48)
dt
6-- inf
on dt (on (lāo 1) ocl))]c-o
Notice then that since:
ct (t) & min((e,+.m)
(49)
e (t) 2 tue-
then,
T ' e (t)dt S min
1
{" tu min (Tut
[T
(50)
T / 6 (ttddt 2
1 T
1 /
tve dt TU-
2
and we get:
Te. 1 2
S TT, //" E (t)dt s L(T) = 1 S/ eth (")(t S 7
1
(Teet
et (t)dt < min
x*)
(51)
Finally, if the error is uniform enough we can expect (²"(() ss tm and
C(T) 1 1 "" (")dt 1 1."0
1 (
1-" tu, di
Tvem
(52)
2
Within manifold Let's assume that we have managed the system Fo have slow manifold M R bijection with U, i.c. flm is not a mapping but bijective function and:
0f
(x))
(x))
Ve M á co(x)
do
(53)
Da 1
(x)))
Then we have slow manifold in the form of a ring attractor, we have f(0) f(2x) and:
ao /'" co s)ds
à t)
mod 2m
(54)
Then:
( f) Os ((( au + ( co(as, s)ds mod 2T av
0r (((a0 ao I'd co(os, s)ds mod 2n ox mod 2x
(55)
Os (1//0 e (aa, s)ds mod 2T
where we used og E [0, 2m) + ag = ap mod and that lo: mod y mod 2n| = [( - v) mad 2T).
The final equation of the loss in this case has the form:
1
on (1 soloa, s)ds mod 2n )
C(T)
(56)
2x
J
T
Slow manifold bounds
In this case, if we have N fixed points in the ring-like slow manifold, we know that:
2m
et (t) min
(57)
N
and therefore:
2m N π
C(T) S min
(58)
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S7
Slow manifold in trained RNNs
We will provide a detailed description the tasks, architectures, training methods, analysis techniques used our numerical experiments with trained RNNs.
S7.1 Tasks
Memory guided saccade task The total time length of trial is 512 steps. The time delay the output cue was sampled from
Taetay ~ L((0, 400).
(59)
We applied a mask m,, = 0 for 5 time steps (t = Tastoy + j for j = 0, after the go cue (Eq. 65).
Angular velocity integration task The time length of trial is 256 steps. The input is an angular velocity and the target output is the sine and cosine of the integrated angular velocity. Velocity every timestep sampled from as Gaussian Process (GP) for smooth movement trajectories, consistent with observed animal behavior in flies and rodents.
|| vll 2
k v) exp
(60)
with length scale f. The length scale of the kernel was fixed 1.
A
Network output
B
90
Head direction (sin, cos)
120
Angular velocity (".
-120-
180
0
-240-
-360-
0
2
4
10 Time (s)
6
8
12
270
Figure S14: Description of the angular velocity integration task. (A) The angular velocity integration task. (B) The output the angular velocity integration in the output space, color coded according to the integrated angle. An example of an input is shown with constant velocity and it is provided until one turn is completed.
Double angular velocity integration task The Double Angular Velocity Integration Task extension of the Angular Velocity Integration Task, where two independent instances of the task are performed simulta- neously. In this case, you have two separate angular velocities, each sampled from its own Gaussian Process, representing two distinct movement trajectories. For each of the two angular velocities, the integration over time performed separately, resulting four dimensional.
of outputs: for each angular velocity, making the output space
Grid cells are known to exhibit periodic firing patterns that form a hexagonal grid across an environment, and these patterns are often modeled as existing on a toroidal surface (a doughnut-shaper surface) 140l The reason for this is that the activity patterns of grid cells are continuous and wrap around scamlessly, meaning that if you move far enough in one direction, the grid pattern will repeat itself. This toroidal structure allows for the continuous representation of space without boundaries, which crucial for efficient path integration. In the context of the Double Angular Velocity Integration Task, where two independent angular velocities are integrated, the resulting four-dimensional output space can be considered as two 2D subspaces (one for cach angular velocity).
Regarding tasks with unbounded range, such navigation tasks, points bear mentioning. For planar attractors are diffeomorphic E note that they not conform the assumptions on normally hyperbolic invariant manifolds, since 2 is not compact. There are suitable generalizations of this theory to noncompact manifolds but we do not pursue them since they require more refined tools, which would only obscure the point that trying make. Tangentially, would also like point out that
Unbounded tasks
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assume that neural dynamics naturally bounded (e.g. by energy constraints) and hence sufficiently well described by compact invariant manifolds.
S7.2
Output projection of the invariant manifold
A
B
C
Figure S15: Output projection of slow manifold approximation of the trained networks of Fig. 4. All stability strutures are colored according the decoded angle shown in Fig. S14B, target output circle shown red. (A) An example fixed-point type solution to the memory-guided saccade task (Fig. 4B). (B) An example of found solution to the angular velocity integration task (Fig. 4C). (C) An example slow-torus type solution to the nemory-guided saccade task. The colored curves indicate stable limit cycles of the system (Fig. 4D).
The output projection of invariant manifolds and stability structures (fixed points and limit cycles) very close to the target output circle, as shown in Fig. S15.
$7.3 Discretization
Computational neuroscientists often train RNNs models of neural computation and interpret them as dynamical systems 3688,101 Our experiments connect to existing literature. We examined vanilla continuous-ting RNNs for this work. The time-discretized version of RNN network activity is given by
X + Wi + Wx b)) +
(61)
y: = W X; bou
where X. E Rd is the hidden state, yt the readout, I, E RK the input, Φ: R + R is an activation function that acts on each of the hidden dimensions, & vid N(0 = 1/100l) a state noise variable, and W, b, Wi, Wea, bos are parameters. We will shortly explain the discretization procedure, i.c., the steps for going from Eq. 64 Eq. 61. Let nAt.
The Muler-Maruyame method for stochastic differential equation (Eq. 64)
dx (-x ((T(() Wx b) odW;
given
Xn+1 Nn + (-xo + @WWn. + Wx. +)) AWWn, with AW, W(n1jjar - WnAi ~ N(0, At).
Now subsitute Af=
- xxx 6 Wx + b)) + +oWW,, =6(Wal, Wx, + b) oAW.
(62)
(63)
If we introduce the noise term <1 = Wc. which represents the discrete-time noise, we have derived the discrete-ting equation:
X x W_ Wx +) Çe-
So, assuming Maler-Manyyma integration with unit time step, the discrete-time RNN of (61) corresponds to the stochastic differential equation:
dt + T(t) + Wx + dt + dW:
(64)
where dW a Wiener process that models the intrinsic state noise in the brain. See for more detail on correspondences between discrete- and continuous time RNNs in 141 and 122 Our experiments connect to existing literature. In future studies, would be interesting perform experiments with Neural SDEs 143
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S7.4 Network architectures
In all network architectures a linear oulput is used. Furthermore, for the angular velocity integration tasks we used additional mapping from the output the hidden layer initialize the hidden state the initial position along the ring from which the network needed to integrate from.
Vanilla layer.
We used vanilla RNNs with different nonlincarities (ReLU, tanh and rectified tanh) for the recurrent
LSTM
The number of units the trained LSTMs was half of that of vanilla RNNs to match the number of paranters
144
GRU We also trained Gated Recurrent Units (GRU) 145 for which we used the number of hidden units the vanilla RNNs.
S7.5
Training methods
We trained RNNs with Poror 146 the three tasks Fig. S14. For the vanilla RNNs, we used a time step of
At = 0.1. The parameters were initialized were initialized using the Xavier normal distribution 147, For the recurrent weights we used W ~ N (0, 8/vx) with a high gain 1.5. The initial hidden state was initialized using the output to recurrent mapping matrix Wor R R N which was trained together with the other parameters.
Adam optimization with 3 0.9 and B2 = 0.999 was employed with a batch size of 64 and training was run for 5000 gradient updates. The batches were generated on-line, similar to how animals are trained with a new trial instead of iterating through dataset of trials.
The best learning rate 10 was chosen from set of values {10- 10 10 - 1005) by 5 initial runs for all nonlinearity and size pairings with the lowest average loss after 100 gradient steps. Training single network took around minutes on a CPU and occupied 10 percent of an 8GB RAM.
We numerically minimized the loss L which was mean squared error (MSE) between the network output (t) and the target output y(t):
LMSE = (mn,, 9r,e)))
(65)
with mask m,,1 with i the index of the output units and t the index for time. We implemented mask, m..., for modulating the loss with respect to certain time intervals
memory guided saccade task (see Sec. S7.1).
Although some of models did not learn the task, most networks converged a loss below 10-2 (Fig. S16).
10"
loss
10
10"
1000
2000
3000
1000
5000
gradient step
Figure $16: Training loss across gradient steps.
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S7.6
RNN analysis methods
S7.6.1
Evaluation Metric
After training, we report the normalized mean squared emror (NMSE) to asses the how good a found solution is:
[ 0)
NMSE
(66)
Elyl
where y the target and is the prediction.
S7.6.2 Asymptotic behavior and memory capacity
We determine the location of fixed points through the local flow direction criterion as described in Sec. 4.2 and determine the basin of attraction
Basin(x)) = { E lim ((t, {x }}-
(67)
E-000
through assesing the local flow direction for 1024 sample points the found invariant manifold.
We construct probability distribution of what part of space we end up in an infinite time through the calculation of the size of the basins of attraction stable fixed points as a proportion of the ring. We characterize the memory capacity of the network by calculating the entropy of this probability distribution of the network.
This follows from the following observation. If we assume that the angular variable that needs to be encoded X is uniformly distributed and the encoding Y is distributed according to the histogram given by the asymptotic behavior of the networks (i.e., the fixed points), then the memory capacity the negative conditional entropy of the continuous memory given the asymptotic state, i.e.
Σ p(y) VEY
p(zly) logp(zly)
vol vol Basin(y) log(vol Basin(y)). DEY
Σ
(68)
rEBasinty)
S7.6.3 Fast-slow decomposition
We simulated 1024 trajectories without noise with inputs from the task and let the networks evolve for 16 times the task definition lengths. We took the cutoff to identify the slow manifold to be 10 3 of the highest speed along each trajectory. We believe that this guarantees the identification of the slow manifold in system that has a fast-slow decomposition. We sampled 1024 points from these points fit a periodic, cubic spline (black line Fig. 4, S19, S20).
Finding fixed points We then find fixed points by identifying where the flow reverses by sampling the direction of the local flow for 1024 sample points along the found invariant manifold. We assess the direction through projection onto the output mapping and calculate the angular flow. If the flow pointing towards point where the flow reverses then we consider there to he stable fixed point. If the flow s pointing away from a reversal point then we consider the fixed point there to he a saddle. We find that long integrated trajectories of the network converge to the found stable fixed points through this independent method.
Eigenspectrum along the invariant manifold We use the eigenvaluc spectrum as evidence for normal hyperbolicity. Normal hyperbolicity an attractive ring invariant manifold implics that the eigenvalue spectrum has gap in its eigenvalue spectrum. To measure this, we lincarize at reference points on the invariant manifold (calculate the Jacobian) and calculate the eigenvalues. The largest eigenvalue (real part) for such a manifold needs be much closer to zero than the second largest. For LSTMs and GRUs the eigenspectrum was approximated by autodifferentiation the networks w.r.t. the states on the identified invariant manifold.
For a stable system, where the eigenvalues have negative real parts, the time constant T is given by the negative inverse of the eigenvalue's 's real part: 7 m(())* where R((A denotes the real part of the eigenvalue X. For the two example networks Fig.4, there a time scale separation between the dynamics on and off the invariant manifold because there is only one eigenvalue close to zero.
Vector field on invariant manifold We assess the vector field for the ODE (Eq. 64) without noise and input) on the found invariant manifold M by calculating it in the state space and then projecting it onto the output space:
à Woaa(())
(69)
for sampled points å E M. These points & E M on the manifold are associated with the points on the ring through the mapping a = W a.
This vector field in the output space captures in what direction and how quickly angular memory will decay. The vector field suggests that the system indeed has an invariant manifold (Fig. S17). Furthermore, the vector field and tixed points consistent with each other, the vector field flips direction around found fixed points.
39

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A
B
10
2
sppae
Y 14
Y1
Y2
22
Figure S17: The projected vector field on the found invariant manifold for the system Fig 4C. (A) The found vector field aligns well with the ring in the projected output space. (B) The norm of the vector field is low around found fixed points as expected, but is higher for points that are just slow points.
There are some inconsistencies around saddle nodes, where the vector field seems to point off of the manifold. This is probably just inaccuracies coming from numerically calculating the vector field and the exact location of the invariant manifold. For the bound discussed Sec. 3.2, we calculate the uniform norm the found vector field
If = sup Woaf(a),
(70)
see also Sec. S6.
For LSTMs and GRUs the vector field was approximated by taking the difference the initial and the next state after initializing the network from states on the identified invariant manifold.
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S7.7 LSTM and GRU results
The trained LSTMs and GRUs share the same pattern observed in the trained vanilla RNNs: ring slow invariant manifold (Fig. S18C and D). The fixed point topologies in the LSTM and GRU networks show lot of variation the number of fixed points, paralleling the systems adjacent to continuous attractors from Fig. 1, as seen Figure 5D. These variations are similar to thosc discussed in Figures S18B, Additionally, the angular error and memory capacity measures across different time scales are comparable to those illustrated in Figure S18A, highlighting the generalization properties influenced by the topology of the solutions. These results underline the universality of our findings beyond vanilla RNNs.
A
B
RNN type
natwock 10x0
61
(mat)
1.5
206
2.0
T.Y
0.5
0.0
0.00
0.01
1102
RU05
0.04
0.05
mimber lood poits
Figure S18: The different measures for memory capacity reflect the generalization properties implied by the topology of the found solution. (A) The average accumulated angular error vs. the uniform norm the vector field shown. Angular error Ti trial length (filled markers) and lim Ti oo (hollow markers). Points are jittered to aid legibility. (B) The number of fixed points vs. average accumulated angular error, with the average distance between neighboring fixed points (magenta). (C,D) Invariant manifold (black) of trained LSTM (C) and GRU (D) with stable fixed points (green) and saddle nodes (red).
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S7.8 Identified invariant manifold output projections
The identified invariant manifolds in the trained networks (Fig. S19 and Fig. S20). However, not all solutions be meaningfully analyzed with the slow-fast decomposition method. For example, the solution the center of the tanh, .V 256 block, the found invariant manifold not correctly captured. This is true for the networks that have not Iearned the task correctly (networks with a NMSE higher than -20dB).
ReLU
tanh
rectified tanh
64
128
256
Figure S19: Representative identified invariant manifolds (projected onto the output space, black)
with the fixed points (cyan for stable, orange for saddle and black for unstable). The reference target ring is shown in grey.
42

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ReLU
tanh
rectified tanh
64
128
256
Figure S20: The identified invariant manifolds with the fixed points (cyan for stable, orange for saddle and black for unstable) for all inferred networks (except the ones in Fig S19).
43

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$7.9 Double angular integration task
The amalysis of networks trained on the double angular velocity integration task indicates torus shaped invariant slow manifold predicted by theory (Fig.S21A and B). Furthermore, the angular memory error (measured the sum of the two separate angular errors) the trained networks show the same conformity the theoretical bound as defined by the uniform norm of the vector field on the identified invariant manifold. These results underline the universality of our findings beyond ID tasks.
A
D
25-
T,
T
PC2
PC
2.5-
B
4
0.-
86
0.000
0.025
0.000
ouons
100
0.125
0:150
0.175
0.000
100
3000
number of fixodd points
Figure S21: Networks trained on a double angular velocity integration task. (A) Initializations (blue) and fixed points (orange) of example network.
(B) Fixed points a 2D parametrization of the torus for the example network. (C) The sum of the total mean angular error (sum of the two seperate angular errors over the two rings) bounded by the uniform norm of the vector field. (D) Generalization for longer memory depends on the number of fixed points in the network.
Two dimensional attractors been proposed simultaneously represent heading direction and uncertainty it 82,148
$7.9.1 Methods
Fixed point search In study, we implemented method analyze the convergence and uniqueness of fixed points within our data. Specifically, we considered a convergence threshold of 10-4 meaning that iterative process was halted when the change in the solution between consecutive iterations fell below this value, indicating convergence. Additionally, to assess the uniqueness of the fixed points, we applied a broader threshold of 10-2 ensuring that any fixed points identified within this margin considered distinct.
Double Mean Angular Error The Mean Angular Error (MAE) was computed as the sum of the individual angular errors observed in the data. This measure provides an aggregate view of the angular errors for the two separate subtasks.
Uniform Norm of the (Projected) Vector Field evaluate uniform norm of (projected) vector field, we calculated the sum of the individual uniform norms the vector field components.
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S7.10
Comparison to other methods
Fixed point analysis and 149 are primarily concerned with pointwise definition of slowness, by compani- normal hyperbolicity requires uniform separation of timescales over the entire invariant manifold. In " was observed that structural perturbations (random gaussian noise deviation) still leads to the same approximate plane attractor dynamical structure is still in place, however no explanation is provided for these observations. Our theory can explain why perturbations to the trained RNN. The Persistence Manifold Theorem (Theorem 1) guarantees that for small perturbations (of size the persistent invariant manifold will be the approximate same place (it will he distance of order (()))
parameters zero mean and standard
Piecewise linear recurrent neural network identifies asymptotic behaviors dynamical systems, fixed point dynamics and more general cases cycles and chaos. We look beyond asymptotic behavior and characterize attractive invariant manifolds. thereby also identifying connecting orbits (or heteroclinic orbits) between fixed points. Although we developed new analysis methods for dynamical systems to find slow manifolds them, do not propose general framework for analysis of all dynamical systems. Finally, provides analysis tools for Piccewise-Lineat Dynamical Systems, while our methods are generally applicable to RNNs with any activation function.
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Reviews:
[{"review_id": "DKSvhun7t0", "reviewer": "Reviewer_xmvZ", "summary": "The study explores some bifurcations from continuous attractors in neuroscience models, revealing various structurally stable forms. Through fast-slow decomposition analysis, they uncover the persistent manifold surviving destructive bifurcations. Additionally, they analyze RNNs trained on analog memory tasks, showing approximate continuous attractors with predicted slow manifold structures. An important takeaway of their work is that continuous attractors demonstrate functional robustness and serve as a valuable universal analogy for understanding analog memory.", "questions": "How does the selection of a specific threshold value influence the identification and characterization of slow manifolds in neural networks with continuous attractors as discussed in the first lines of section 4.2? Could you elaborate on how different threshold settings impact the dynamics of network states and the emergence of persistent manifolds?", "limitations": "- The authors have discussed most limitations of the analysis in the discussion section, but I suggest making them more explicit. This could be done by either incorporating a dedicated (sub)section on limitations or adding a bold title \"**Limitations**\" at the beginning of the relevant paragraph within the discussion section. \n\n- As mentioned above, another important limitation is that it is difficult to determine how general their theoretical discussion is and whether it can be applied to investigate and obtain results for more general and high-dimensional cases.", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "- The main idea of the paper is interesting, and this work can have an important takeaway, as mentioned in \"Summary\". \n\n- The connection to Fenichel’s theorem is very nice. \n\n- The visualizations in Fig. 1 are very helpful to understand the overall message", "weaknesses": "1. They discussed some interesting theoretical techniques (e.g., Theorem 1, Proposition 1) in their study. However, their theoretical investigation and results are limited to a few very simple systems, low-dimensional systems either in Section S4 or low-dimensional RNNs with specific activation functions and restrictive settings, i.e., specific parameter values (e.g., equations (1) and (10)). The bifurcation analysis of the line attractors and fast-slow decomposition in Section S2 are also studied for very simple systems. Therefore, it is difficult to determine how general their theoretical discussion is and whether it can be applied to investigate and obtain results for more general and high-dimensional cases.\n\n2. In Sect. 3.1, the perturbation p(x) is not clear enough. Specifically, it is unclear:1) Under what conditions the perturbation function p(x) induces a bifurcation?  2) What types of (generic) bifurcations can arise from the perturbation p(x)? Likewise, the functions h and g are also not clear enough. It is unclear how one can obtain/choose the functions h and g such that the two systems defined by Eq. (2) and Eqs. (3) & (4) are equivalent.\n\n3. What does **sufficiently smooth** mean in Theorem 1? As mentioned by the authors after this theorem, it applies to continuous piecewise linear systems. However, it cannot be applied to all piecewise smooth (PWS) systems , such as Filippov systems. In particular, for these systems, bifurcations involving non-hyperbolic fixed points can be analyzed using similar slow (center) manifold approaches, but only for part of the phase space. However, discontinuity-induced bifurcations cannot be examined in the same way, as there is no slow manifold in these cases.\n\n4. It is unclear under what conditions RNN dynamics can be decomposed into slow-fast form to which we can apply Theorem 1.\n\n5. In Sect. 4.1, line 213,  it is vague how assuming an Euler integration with unit time step, the discrete-time RNN of (6) transforms to eq. (7). Is this transformation independent of the function f and matrix W in eq. (6)?\n\n6. In S4, the sentence \"All such perturbations leave at least a part of the continuous attractor intact and preserve the invariant manifold, i.e. the parts where the fixed points disappear a slow flow appears.\" needs more clarification. Could you explain the mathematical reasoning behind this assertion?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We would like to thank the reviewer for their valuable comments and suggestions.\n\n### Weaknesses:\n>their theoretical investigation and results are limited to a few very simple systems, low-dimensional systems [...] \n\nWe respectfully disagree with the stated limitation-- the role of the analysis, and numerical experiments is not to prove the generality of the theory but to illustrate it. Hence, we focus on visualizable low-dimensional systems and are more helpful in developing intuition.\nNevertheless, we include additional results with RNNs trained on a 2D task.\nIn the updated manuscript, we emphasize that the theory holds under broad, practically relevant conditions.\n1. We added statements that assure that our theory is applicable regardless of the dimensionality or the invariant manifold (see shared rebuttal).\n1. Furthermore, we will revise Theorem 1, to show that normal hyperbolicity is both **sufficient and necessary** for invariant manifolds to persist, see [1].\n\n> RNNs with specific activation functions and restrictive settings\n\nWe appreciate the reviewer's insightful comment. In response, we have conducted additional experiments with LSTMs and GRUs, for which the results are included in the supplementary document and discussed in the shared rebuttal.\n\n> Under what conditions the perturbation function p(x) induces a bifurcation?\n\nContinuous attractors satisfy the first-order conditions for a local bifurcation; that is, they are equilibria, and their Jacobian linearization possesses a non-trivial subspace with eigenvalues having zero real parts. Consequently, any generic perturbation $p(x)$ will induce a bifurcation of the system.\nFor a more comprehensive discussion on this topic, see [2].\n\n> What types of (generic) bifurcations can arise from the perturbation p(x)?\n\nWe are working to characterize codimension-1 bifurcations for a ring attractor and believe a simple polynomial normal form can be derived. \nCharacterizing bifurcations with codimension $n > 2$ is an open problem. \nPerturbations in the neighbourhood of a ring attractor (in $C^1$ topology) will result in no bifurcation, a limit cycle, or a ring slow manifold with fixed points. \n\n>the functions h and g are also not clear \n\nThe essence of Theorem 1 can be restated as follows: **if** the function $f$ has a normally hyperbolic invariant manifold (NHIM), **then** there exist vector fields $h$ and $g$ that satisfy the conditions for equivalence between the systems defined by Eq.2 and Eqs.3&4.\nThis means that the existence of these functions is guaranteed under the condition of having a NHIM, but their explicit forms is case specific.\n\n> What does sufficiently smooth mean in Theorem 1?\n\nWe mean that a system needs to be at least continuous, but some extra conditions apply if a system is not differentiable (discontinuous systems are not considered in our theory). Since all theoretical models and activation functions for RNNs are at least continuous and piecewise smooth, our theory is broadly applicable.\nCenter manifold are unique, and generally are local in **both** space and time and therefore invariance under the flow generally cannot be analyzed using them.\n\n> It is unclear under what conditions RNN dynamics can be decomposed into slow-fast form to which we can apply Theorem 1.\n\nTheorem 1 holds for all dynamical systems that have a normally hyperbolic continuous attractor. For example, RNNs with a ReLU activation functions can only have such continuous attractors. The continuous attractors and their approximations that we discuss are all normally hyperbolic. In fact, there is a huge benefit from having normal hyperbolicity as it can counteract state noise.\n\n> line 213 [...]\n\nThe transformation from the continuous-time to the discrete-time is independent on the function $f$ anf the matrix $W$.\nHowever, it is important to note that the discretization process can result in significantly different system behavior depending on the activation functions used. For instance, discretization can introduce dynamics that are not present in the continuous-time system.\n\n> In S4, the sentence \"All such [...]\n\nWe will reformulate it as \"all such perturbations leave the geometry of the continuous attractor intact as an attractive invariant slow manifold, i.e. the parts where the fixed points disappear a slow flow appears.\"\nThe persistence of the invariant manifold under perturbations is a direct consequence of the normal hyperbolicity condition in Theorem 1.\nTherefore, for a normally hyperbolic continuous attractor there will remain an attractive slow invariant manifold.\n\n**Questions:**\n>How does the selection of a specific threshold value...\n\nWe believe that the threshold value used to identify slow manifolds is robust- we used the same value for the newly added RNN models, without modification.\n\n>Could you elaborate [...] emergence of persistent manifolds?\n\nIt is unclear for us to which threshold the reviewer is referring. The emergence of persistent manifolds happens under the 4 conditions we discuss.\nWe demonstrate that all systems with a sufficiently good generalization property (in our case, defined as networks with NMSE lower than -20 dB) must have a NHIM. The persistence of these manifolds is a direct consequence of their normal hyperbolicity.\n\n**Limitations:**\n>The authors have discussed most limitations [...]\n\nWe appreciate the reviewer's suggestion to make the limitations of our analysis more explicit. In response, we have included a dedicated Limitations subsection in the discussion section of the manuscript. Please refer to the shared rebuttal for further details.\n\n[1] Mané, R. (1978). Persistent manifolds are normally hyperbolic. Transactions of the American Mathematical Society, 246, 261-283.\n\n[2] Kuznetsov, Y. A., Kuznetsov, I. A., & Kuznetsov, Y. (1998). Elements of applied bifurcation theory (Vol. 112, pp. xx+-591). New York: Springer.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "52Lfvt2BzJ", "reviewer": "Reviewer_CAVv", "summary": "This paper studies the fragility of continuous attractors, which have been used to explain various computations or functions in the brain related to memory and navigation, to perturbations. The authors mainly focus their analyses on ring attractors, which have been used to model continuous-valued memory. Under perturbation, the authors find that the bifurcations of continuous attractors exhibit structurally stable forms with different asymptotic behaviors but similar finite-time behaviors as the original attractor. For example, a stable limit cycle arising from a bifurcation of a ring attractor would be functionally similar to the ring attractor in finite time. Thus, the authors posit that signatures of the continuous attractor persist even under perturbation in the form of a persistent, attractive manifold which serves as an approximate continuous attractor. Experiments and analyses on recurrent neural networks performing analog memory tasks show that the networks learn approximate continuous attractors. Thus, the authors conclude from their theory and numerical experiments that approximate continuous attractors do remain a good model of analog memory.", "questions": "See the Weaknesses section. I have some additional questions:\n1. How do the authors identify the various kinds of approximations of the attractors? Can this be automated, perhaps by using to DSA to cluster the various types?\n2. At what level of performance are all trained networks compared? Are they all trained until the same loss value and how close is this MSE to 0?\n\n**Rebuttal update:** Score increased from 5 to 7.", "limitations": "There is no limitations section. Tthe authors have mentioned in the discussion that they do not explicitly characterize all possible invariant manifolds identified. However, there are other limitations as well such as the lack of diversity in the tasks/attractors explored. I would also encourage the others to unpack the limitations related to the numerical fast-slow analysis, such as how sensitive the results are to the threshold hyperparameter, cases or specific attractors where it does not work as well, etc.", "rating": 7, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "1. The authors' theoretically demonstrate the existence of a persistent attractive manifold in various bifurcations of a continuous attractor, and study the systems' finite time behaviors to show that they are functionally equivalent. To my knowledge, this is a novel contribution and an important result to bolster the continuous attractor hypothesis.\n2. Apart from the theory, the authors also carry out numerical experiments on a working memory task, characterized by a 1-D ring attractor. Through their analyses they validate their theory.\n3. The authors also study the generalization properties of the approximate attractors.", "weaknesses": "1. The experiments and associated analyses focus solely on networks that approximate 1D ring attractors. This is quite simplistic, and at least for the numerical expreiments, the authors could consider tasks like navigation where a planar 2D attractor is approximated by the networks.\n2. The authors have only qualitatively characterized the variations in the topologies of the networks. It is perhaps possible to quantitatively characterize this by using Dynamical Similarity Analysis [1] on various trained networks.\n3. For the generalization analysis, the authors could evaluate generalization performance by the nature/type of the approximate attractor as well. Furthermore, although I may have missed this, could the authors comment on what networks hyperparameters lead to which approximations?\n4. The figures and presentation could be improved:\n    1. On line 107 there is a comment that should be removed (\"add link to details\").\n    2. Fig. 4C, caption should indicate the nature of the solution found.\n    3. Fig. 5B, y-axis label is missing.\n    4. Fig. 5D, could also show the mean $\\pm$ std for classes of networks.\n    5. Fig. 5E, y-axis label is missing. Also, the authors could just use the normalized MSE on the axis could just follow the convention used in Fig. 5A instead of using dB.\n    6. Overall, the writing could be improved in several places to improve clarity. For example, the conclusions of the generalization analysis and their implications are not very clear, and how this connects to the various types of approximate attractors is not clear (related to W3).\n\n**References:**\n1. Ostrow et al. \"Beyond geometry: Comparing the temporal structure of computation in neural circuits with dynamical similarity analysis.\" Advances in Neural Information Processing Systems 36 (2024).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We would like to thank the reviewer for their valuable comments and suggestions.\n\n### Weaknesses\n> The experiments and associated analyses focus solely on networks that approximate 1D ring attractors.\n> This is quite simplistic, and at least for the numerical experiments, the authors could consider tasks like navigation where a planar 2D attractor is approximated by the networks.\n\nWe appreciate the remark, and have included an additional task where the approximate continuous attractor is of higher dimension, namely a double angular velocity integration task. Please see the shared reply to all reviewers on our new findings.\nThe networks develop corresponding approximate continuous attractors that have the same structure as the task requires (in this case a torus).\n\nThat being said, we would like to reiterate that the primary contribution is theoretical and the numerical experiments are meant to illustrate the theory.\nRegarding navigation tasks, two points bear mentioning.\n1. For planar attractors are diffeomorphic to $R^2$, note that they do not conform to the assumptions on normally hyperbolic invariant manifolds, since $R^2$ isn't compact.\nThere are suitable generalizations of this theory to noncompact manifolds [1], but we do not pursue them since they require more refined tools, which would only obscure the point that we are trying to make.\n2. Tangentially, we would also like to point out that we assume that neural dynamics are naturally bounded (e.g. by energy constraints) and hence sufficiently well described by compact invariant manifolds.\n\nIn the revised version of the manuscript, we will include the above limitations and provide reference to [2].\n\n> The authors have only qualitatively characterized ...\n\nWe thank the reviewer for pointing out the reference; we applied DSA to our numerical results.\nOur preliminary observations are that DSA reflects the fact that the geometry of the invariant manifold is preserved, but it cannot detect the emergence of fixed-points and saddles on the perturbed manifold.\nThe DSA values clustered around two points regardless of the number of fixed points.\nThis appears to be consistent with the results reported in the referenced paper, c.f. Figure 4 shows a gradual increase in DSA as $\\alpha \\to 1$ despite having a bifurcation at $\\alpha = 1$.\n\nLastly, we would like to note that the analysis using DSA cannot be trivially automated. As pointed out by the authors of DSA:\n1. The DSA 'score' is relative; one needs to compare different dynamics.\n1. DSA essentially requires 'learning' or fitting a separate model, which implicitly requires performing model selection with respect to the delay embedding, rank of the linear operator.\n\nFor these reasons, we would like to adhere to our initial analyses.\n\n\n> For the generalization analysis, the authors could evaluate generalization performance by the nature/type of the approximate attractor as well.\n\nWe looked at the generalization performance by the nature/type of the approximate attractor (Fig.5D MSE vs number of fixed points).\n\n>Furthermore, although I may have missed this, could the authors comment on what networks hyperparameters lead to which approximations?\n\nThe only networks hyperparameters that we varied were the nonlinearity and the size.\nIn all our figures we show which nonlinearity and size corresponds to which fixed point topology (which we characterize through the number of fixed points on the invariant ring).\n\n\n> The figures and presentation could be improved [...]\n\nWe appreciate the comments, and changed the manuscript accordingly.\n\n> Overall, the writing could be improved in several places to improve clarity.\n\nWe improved the writing, focusing on overall clarity. See the shared comments.\n\n\n>the conclusions of the generalization analysis and their implications are not very clear...\n\nIn the revised version, we will make a stronger point that connects the inherent slowness of the invariant manifold to the generalizability of the approximate solutions.\nWe also added a longer description of the implications of our numerical experiments to the main text.\n\n### Questions:\n\n> How do the authors identify the various kinds of approximations of the attractors? Can this be automated, perhaps by using to DSA to cluster the various types?\n\nWe identify approximations by their (1) attractive invariant manifold (as motivated by the theory) and (2) asymptotic behavior (as motivated by our analysis of perturbations and approximations of ring attractors).\nThe invariant manifold in our examples typically take the structure of a ring with fixed points and transient trajectories on it.\nIn the supplementary document (FigR.1) we show that the identified invariant manifold indeed reflects the fast-slow separation expected for a normally hyperbolic system.\nWe find the fixed points and their stabilities by identifying where the flow reverses by sampling the direction of the local flow for 1024 sample points along the found invariant manifold.\nThe only example we found that is of another type is the attractive torus (Fig.4D).\nFor this network, instead of finding the fixed points, we identifies stable limit cycles where there was a recurrence of the simulated trajectories, i.e., where the flow returned back to an initial chosen number of time steps (up to a distance of $10^{-4}$).\n\nFor the difficulties of using DSA, see above.\n\n\n> At what level of performance are all trained networks compared? Are they all trained until the same loss value and how close is this MSE to 0?\n\nAll networks are trained for 5000 gradient steps.\nWe exclude those networks from the analysis that are performing less than -20dB in terms of normalized mean squared error tested on a  version of the task that is 16 times as long as the task on which the networks were trained.\n\n[2] Eldering, J. (2013). Normally hyperbolic invariant manifolds: The noncompact case (Vol. 2). Atlantis Press.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "rHJcvdzpqf", "reviewer": "Reviewer_8M6u", "summary": "The manuscript investigates the stability and robustness of continuous attractors to small deformations in vanilla recurrent neural networks. Continuous attractors were once heavily studied in the context of working memory, but they are inherently fragile and susceptible to even small perturbations and lead to very different topologically distinct attractor landscapes. This study, on the other hand, focuses on finite time behavior around these distinct attractor landscapes. Specifically, the authors theoretically discuss and empirically show that continuous attractors exhibit slow manifold structures under perturbations, which persist despite their theoretical instability.", "questions": "I believe the work warrants a borderline accept as is, yet I would feel very supportive of its publication (up to a strong accept) if the authors performed the following changes:\n\n- I believe the slow manifold picture the authors are introducing here is not too different from [1], specifically Fig. 7. Can the authors please clarify the differences in the main text?\n\n- The fast-slow decomposition does not seem to be specific to vanilla RNNs. Can the authors please include experiments with LSTMs and GRUs, which would support their claims on generality.\n\n- Similarly, the experiments on Section 4 are centered around ring-like attractors. Can you please show an example with other structures? For example, you can consider the delayed addition/multiplication tasks discussed in [2]. Relatedly, similar slow dynamics arguments are made in [2], which I believe the authors should list as a close related work and explain the differences in their approach. \n\n- A more detailed discussion of Section 5 is desired. I was able to understand the main takeaways, but could not evaluate the validity of the claims. Perhaps the authors may want to explain in simpler terms the evidence presented in Fig. 5.\n\n- The title is not descriptive of the manuscript's content and feels like as if it belongs to a blog post. Could you please update the title to be representative of the paper's conclusions?\n\nCitations\n\n[1] Opening the Black Box: Low-Dimensional Dynamics in High-Dimensional Recurrent Neural Networks, David Sussillo and Omri Barak, Neural Computation, 2013\n\n[2] Schmidt, D., Koppe, G., Monfared, Z., Beutelspacher, M., & Durstewitz, D. (2019). Identifying nonlinear dynamical systems with multiple time scales and long-range dependencies. arXiv preprint arXiv:1910.03471.\n\n\n**Edit:** My main remaining concern is the presentation, which I find to be unnecessary complicated. That being said, this work is an important piece of contribution to neuroscience and I support its acceptance.", "limitations": "Yes.", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 2, "contribution": 4, "strengths": "- The study of continuous attractors in the finite time limit, opposite of what is usually studied with attractor landscapes, is a fresh look, novel, original, and interesting.\n- The authors provide a superb theoretical motivation, which convinced me of their results correctness.", "weaknesses": "- The presentation in section 5 becomes unclear, and perhaps too dense. The authors may want to expand on this section significantly.\n- Some more control experiments need to be added for proving the generality (See below).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We would like to thank the reviewer for their valuable comments and suggestions (and appreciate the recent edit). They have significantly enhanced the quality of our manuscript.\n\n### Weaknesses:\n\n> Some more control experiments need to be added for proving the generality (See below).\n\nWe address the specific questions below, but would like to emphasize that the main contribution of this submission is theoretical.\nThe numerical experiments are meant to illustrate our results, rather than prove them.\n\n### Questions:\n> I believe the slow manifold picture the authors are introducing here is not too different from [1], specifically Fig. 7. Can the authors please clarify the differences in the main text?\n\nWe appreciate the remark, and will revise the text to include the following:\n1. The two approaches are indeed similar, however there are subtle technical differences.\n1. In [1], Sussillo and Barak are primarily concerned with a pointwise definition of slowness, by comparison normal hyperbolicity requires a uniform separation of timescales over the entire invariant manifold. \n1. Our theory can explain why perturbations to the trained RNN (with random gaussian noise with zero mean and standard deviation) still lead to the same approximate plane attractor dynamical structure is still in place. The Persistence Theorem guarantees that for small perturbations (of size $\\epsilon$) the persistent invariant manifold will be at the approximate same place (it will be at a distance of order $\\mathcal{O}(\\epsilon))$. See Figure 9 for the experiments of the structural perturbations in [1]. They do not provide an explanation for their observations.\n\n\n> The fast-slow decomposition does not seem to be specific to vanilla RNNs. Can the authors please include experiments with LSTMs and GRUs, which would support their claims on generality.\n\nWe have now trained and analyzed both LSTMs and GRUs on the angular integration task. We include our preliminary results in a separate document uploaded to OpenReview.\n\n\n> Similarly, the experiments on Section 4 are centered around ring-like attractors...\n\nWe appreciate the comment, and have included an additional task where the approximate continuous attractor is of higher dimension, namely a double angular velocity integration task. Please see the shared reply to all reviewers on our new findings.\nSpecifically regarding addition or multiplication tasks, an idealized solution to either would require that the RNN represent $G = (\\mathbb{R},+)$ or $G = (\\mathbb{R}_{+},\\times)$, which are not compact.\nBecause this contradicts our technical assumptions, we opt to focus on tasks where the invariant manifolds are naturally compact.\n\n\n> similar slow dynamics arguments are made in [2]...\n\nWe thank the reviewer for pointing out this work, we will reference it accordingly.\nThis work identifies asymptotic behaviors in dynamical systems, fixed point dynamics and more general cases cycles and chaos. \nWe look beyond asymptotic behavior and characterize attractive invariant manifolds, thereby also identifying connecting orbits (or heteroclinic orbits) between fixed points.\nWe would like to reiterate that we believe that the main contribution of the paper is a new theory of approximations of continuous attractors. \nAlthough we developed new analysis methods for dynamical systems to find slow manifolds in them, we do not propose a new general framework for analysis of all dynamical systems.\nFinally, [2] provides analysis tools for Piecewise-Linear Dynamical Systems, while our methods are generally applicable to RNNs with any activation function.  \n\n> The presentation in section 5 becomes unclear, and perhaps too dense...\n\nWe agree with the reviewer, the updated manuscript will include a substantial revision of section 5. In it, we will focus on clarity.\nWe have split the section into generalization properties of trained RNNs (all relating to Fig.5 and in relation to our theoretical prediction on the error bound) and the four conditions that guarantee that a system that approximates an analog memory system will be near a continuous attractor (which was section 5.2).\nPlease, see the shared rebuttal for our clarification of Section 5.2.\n\n\n> the evidence presented in Fig. 5.\n\nThe main message of Fig.5 is to show the validity of our theoretical predictions about the bound to the memory based on the uniform norm of the flow.\nBesides, we can demonstrate that even though all networks learned a continuous attractor approximation, they are distinguished from one another by their fixed point topology, which determines their asymptotic behavior and hence generalization properties (Fig.5D and E).\nThese results indicate the distance to a continuous attractor as was discussed in Sec.3.2 as measured by the uniform norm of the invariant slow manifold. \n\n\n> The title is not descriptive of the manuscript's content and feels like as if it belongs to a blog post. Could you please update the title to be representative of the paper's conclusions?\n\nThe title references the reversal of the Persistence Manifold theorem, i.e., how to get back to a continuous attractor.\nFurthermore, it references that we can return to continuous attractors as a useful concept to describe neural computation because of the deep connection of all continuous attractor approximations. \nWe can however propose to revise the title to: \"A theory of analog memory approximation: Back to the continuous attractor.\"\n\n\n### References\n[1] Opening the Black Box: Low-Dimensional Dynamics in High-Dimensional Recurrent Neural Networks, David Sussillo and Omri Barak, Neural Computation, 2013\n\n[2] Schmidt, D., Koppe, G., Monfared, Z., Beutelspacher, M., & Durstewitz, D. (2019). Identifying nonlinear dynamical systems with multiple time scales and long-range dependencies. arXiv preprint arXiv:1910.03471.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "ujmjqlsZx9", "reviewer": "Reviewer_TfFn", "summary": "The authors study continuous attractor networks, and their famous instability under noise. They show that continuous attractors, despite being unstable, are functionally robust, and analyse some behaviours when noise is introduced.", "questions": "My, many, questions and confusions ended up being in the weaknesses section.", "limitations": "The limitations were somewhat discussed.", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "The main thrust of the paper was very interesting and very novel. I am a theoretical neuroscientist, and the studied noise-instability of CANs is a classic problem that we are taught in lectures. This paper appears to resolve this problem, using the very clever proof technique of 'knowing a relatively esoteric piece of maths (to me!) that solves the problem of interest as a corollary'. If I've understood this right, it should be applauded!\n\nThe rest of the paper was a smorgasbord of somewhat interesting characterisation of the behaviour of noise perturbed CANs, illustrating some interesting phenomenology.\n\nA lot of the exposition was very clear, especially on the micro-scale: individual ideas were explained well. Figures 1 and 3 were good.\n\nThe bound on error in figure 5A was cool (if true).", "weaknesses": "I got very confused by what most of the paper was trying to show. The main result, section 3, is, almost draw-droppingly, nice. (Though I'm partly trusting that the authors are applying the results correctly; they seem to be, because the step from the stated theorem to their claims is not large) The rest is, ..., a bit more meh, especially in how it was presented. I would have liked a lot more signposting: why are these sections the right ones to care about? What are you showing, and why?\n\nFor example, section 2 shows that noise degrades CANs - a nice literature review - and then shows a categorisation of perturbed fixed point dynamics. I guess the idea was to show that noise-instability was a problem for all CAN types in the literature? If so you could do with signposting that. If not, why did you include all of that? Why was it important that some patterns appear and not others in the perturbed dynamics?\n\nThen there were a lot of quite confusing things, especially in the figures: \n\nFig 1) A was a great figure, B was never mentioned in the text, despite being very pretty.\n\nFig 2) Why were there hexagons everywhere? I never could find any reason for there to be hexagonal figures, did you just make the ring a hexagon for fun? If so, tell the reader! Further, in B and C, are the blue lines numerical? How did you choose what to colour blue? Should I be using it as evidence for your big theoretical claim? Or is it just illustrative? \n\nFig 3) What am I looking at in figure 4A2: it says example trajectory? But there are many trajectories, no? The dots in B and C are presumably fixed points, why then is it called a limit cycle (line 246)? It doesn't look like that? Why is 4D described as around a repulsive ring invariant manifold? Are you describing the ring inside the torus? (rather than the presumably attractive limit cycles that are marked on the figure) What does the colouring of the torus (the greys) denote? I didn't get told what the task was for figure A1 so had to go to the appendix to see that this is apparently the output of the integration? Why are you plotting the output space, and not the recurrent space as in all the other figures? \n\nOn that last point, why include details of the (standard) discretisation scheme, MSE loss, and network details, when key steps to understand what I am looking at (e.g. figure A1 = output space) are missing?\n\nFigure 5A) Why did one of the finite time results go above the line? Shouldn't this be an exact theoretical result, yet it appears not to be true? \n\nSeemed obtuse to claim a linear relationship then show a log-linear plot, fig 5C? How should I see this?\n\nDid you define the memory capacity? \n\nDid you need to introduce omega-limit set, especially given the likely audience for this paper?\n\nFinall, some other points:\n\nYou should definitely cite and discuss the relationship to this paper: Information content in continuous attractor neural networks is preserved in the presence of moderate disordered background connectivity, Kuhn & Monasson, 2023.\n\nWhat was section 5.2 trying to show? First it claims that 2.2 presents a theory of approximate solutions in the neighbourhood of continuous attractors (news to me, as far as I could tell, that section showed me that all CAN models were unstable to noise and turn into a variety of different fixed points under noise, that doesn't sound like a theory at all? Section 3 seems to be the theory?) Then you list four conditions on what sounds like exactly the same problem? What is the difference between dynamical systems having robust manifolds, and the working memory task being solved, isn't the whole point of the model that these two are the same? (i.e. you can solve a working memory task with a CAN). Is this supposed to be a concluding section that says when working memory can be solved? Then why have you suddenly defined state and dynamical noise that haven't been used before, I thought we had a perfectly nice definition of perturbations on the network (equation 2)? This section seemed... strange, in my eyes the paper would be improved by removing it\n\nSmaller things:\n- line 107 - comment left in doc\n- Figure 2 caption line 1, missing 'of' and pluralisation of 'implementation'\n\nSo all in all, I think the exposition, despite, as I said, often being very clear when describing a single idea, is, on a macro scale, a mess. I had a very hard time following, and the figures were quite tough-going. I think this paper should probably get in, but I think it should be given a good clean up first, at least for a humble neuro-theorist, rather than a dynamical systems guru, to understand.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We would like to thank the reviewer for their valuable comments. They have significantly enhanced the quality of our manuscript.\n\n> signposting\n\nWe thank the reviewer for and agree with this feedback. We improved the text by:\n- Changing the last paragraph of the introduction, summarizing each section\n- Highlighting the main questions arising from the discussed examples\n- Adding sentences at the start of each (sub)section to point out the main message\n\n> Fig 1) A was a great figure...\n\nWe now added the needed reference to Fig.1B.\n\n> Fig 2) Why were there hexagons everywhere?\n\nThis is a quality of the ring attractor in [1]. The attractor implemented by 6 neurons is made up of 6 line attractors, fused at their ends. At these fusion points there is a lack of smoothness (similar to the function abs(x)). Therefore, this piecewise straight \"ring\" attractor, when projected onto two dimensions looks like a hexagon. We have added clarification to this on the main text.\n\n> Further, in B and C...\n\nWe find the connecting orbits between the fixed points by identifying the slow part of simulated trajectories. From these slow trajectories, we choose the trajectory that is closest to two fixed points. These trajectories go from a saddle node to a stable fixed point. We do this for every pair of neighbouring fixed points.\n\n> Should I be using it as evidence for your big theoretical claim?\n\nIn this section, we aim to motivate the relevance of our theory by demonstrating two key points: 1) that continuous attractors are inherently fragile to noise in the parameters (a well-known fact), and 2) that all bifurcations from and approximations of continuous attractors share a common feature: a slow invariant attractive manifold. \nIn the subsequent sections, we provide an explanation of these universal features. \n\n> What am I looking at in figure 4A2...\n\nWe appreciate the remarks and have revised the text accordingly.\n1. You are correct; this figure shows multiple example trajectories. We have corrected this in the text and caption.\n1. We have corrected the mistake of referring to Fig. 4B and C as a limit cycle. \n1. We did not include the slow repulsive points that we found as it decreased the interpretability of the figure and this structure is not relevant for how the network solves the task.\n1. The grey lines on the torus represent simulated trajectories, indicating the network dynamics after a saccade.\n1. Fig. 4A1 shows the output of an integrated angular velocity, illustrating the task in addition to the solution type in Fig. 4C.\nFig. 4A1 and 4A2 are different in that they show how the networks behave (in output space).\nThe other subfigures in Fig.4 illustrate the stability structures of the networks, i.e., to which part of state space they tend over time in the absence of input.\n\n> On that last point...\n\nWe included these details to support reproducibility.\n\n> Figure 5A) Why did one of the finite time results go above the line?\n\nThis is indeed an exact theoretical result; however, our numerical methods are not exact. Because we numerically approximate the invariant manifold of each trained RNN, on which we calculate the uniform norm of the vector field, we cannot guarantee the vector field simulated trajectories follow to be exact. Additionally, the network initialization along the invariant manifold is approximate due to our parametrization (using a cubic spline). Nevertheless, it is important to note that this method has only a single violation of our theory among approximately 250 networks tested.\n\n> Seemed obtuse...\n\nWe appreciate the reviewer pointing this out. We have corrected the mistake and now observe the log-linear relationship, which aligns with the theoretical expectation. The angular error should asymptotically approach zero as the number of fixed points increases, making the log-linear plot the appropriate representation for this relationship.\n\n> Did you define the memory capacity?\n\nWe determine the location of the fixed points through points of reversal of the flow direction (the system evolves along a 1D subspace). We calculate the probability of a point converging to a stable fixed point by assessing the local flow direction (which allows us to characterize the basin of attraction). The memory capacity is the entropy of this probability distribution. The definition is in S.4.3.1., referenced in the main text. We hope these definitions are clearer in the new version of Sec. 5.\n\n> Did you need to introduce omega-limit set...\n\nWe believe that this definition is supporting the definition of memory capacity.\nAs we explained above, the memory capacity is calculated from the omega-limit set of each of the points on the invariant ring.\nThis idea can be more generally applied to systems with other omega-limit sets, like limit cycles or chaotic orbits and therefore included this definition.\n\n> You should definitely cite...\n\n[2] analyzes an Ising network perturbed with a specially structured noise at the thermodynamic limit.\nAlthough their analysis elegantly shows that the population activity of the perturbed system does not destroy the Fisher information about the input to study instantaneous encoding, they do not consider a scenario where the ring attractor is used as a working memory mechanism. In contrast, our analysis involves understanding how the working memory content degrades over time due to the dynamics. We are not aware of any mean field analysis that covers this aspect.\nWe include this work to discuss continuous attractors in mean field approaches.\n\n> What was section 5.2 trying to show? \n\nSee the shared rebuttal for our clarification of Sec. 5.2.\n\n### References\n[1] Noorman, M. et al. (2022). Accurate angular integration with only a handful of neurons. bioRxiv, 2022-05.\n\n[2] Kühn, T., & Monasson, R. (2023). Information content in continuous attractor neural networks is preserved in the presence of moderate disordered background connectivity. Physical Review E, 108(6), 064301.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:
We are grateful and encouraged that the reviewers found our work novel and interesting. Reviewers remarked that it is "a novel contribution and an important result to bolster the continuous attractor hypothesis", "fresh look, novel, original, and interesting [and] superb theoretical motivation", that "the main thrust of the paper was very interesting and very novel" and "it should be applauded."
We feel many of your suggestions have led us to changes and additions that better position the paper. 

To respond to the reviewer's comments, we have performed the following analysis:

 * quantify the fast-slow time scale separation on the manifold found in task-trained RNNs (Fig R1)

 * trained LSTM and GRU networks (Fig R2)

 * trained RNNs on a 2D task where the continuous attractor manifold is a torus (Fig R3)

# Generality of the theory
While most bifurcation analyses in theoretical neuroscience and machine learning are based on a particular parameterization (e.g., pairwise weight matrix), our theory applies to any differentiable dynamical system and to continuous piecewise smooth systems (with a global continuous attractor). Hence, the robustness of most continuous attractors are covered. The only necessary condition is the *normal hyperbolicity* as demonstrated via separation of eigenvalues (Fig R1).
## Architecture
We tested our theory with LSTMs and GRUs to support our claim about the universality on trained RNNs.
These networks form the same normally hyperbolic invariant slow ring manifold just like Vanilla RNNs (Fig R2C,D) and on this manifold we find fixed points (Fig R2A,B). This consistency of structure across different RNN architectures provides further validation of our theoretical framework.
## Simple systems
The analysis of theoretical models and numerical experiments is intended to illustrate the theory's practical applicability rather than to prove its generality.
We focused on low-dimensional systems because they are easier to visualize and are a better guide to developing intuition.
We include results on RNNs trained on a 2D task (a double angular velocity integration task) to further demonstrate our theory's relevance. In the trained RNNs (Fig R3A,B), we find a slow, attractive, invariant manifold in the shape of a torus with a point topology. Additionally, we find evidence supporting the relevance of the error bound in these trained RNNs (Fig R3C,D).
## Broader impact
Approximate continuous attractor theory applies to dynamical systems inferred from data, from task-trained neuralODEs and RNNs, finite size effects on theoretical models, and due to lack of expressive power parametrized dynamics.
We believe our theory opens up a new way of grouping similar dynamical systems for understanding the essential computation.

# Clarity
We acknowledge the reviewers' concerns regarding clarity and add details on the main topics highlighted in the reviews. If accepted, our final manuscript will be updated.
## Section 5.2
Section 5.2 outlines the conditions under which approximations to an analog working memory problem are near a continuous attractor. This section is crucial for clarifying when a situation like Proposition 1 would occur. These conditions are met for RNNs:

 * C1: This translates to the existence of a manifold in the neural activity space with the same topology as the memory content. We formalize the dependence as the output mapping being a locally trivial fibration over the output manifold.
 * C2: Persistence, as per the reverse of the Persistent Manifold Theorem, requires the flow on the manifold to be slow and bounded.
 * C3+C4: Non-positive Lyapunov exponents correspond to negative eigenvalues of $\nabla_zh$. Along with dynamics robustness (corresponding to the persistence of the manifold), this implies normal hyperbolicity. We have expanded on this correspondence by building on [1].

## Parameter dependence for the analysis
The threshold parameter for identifying invariant slow manifolds was chosen to reflect the bimodal distributions of speeds along the integrated trajectories.
The supplementary document (Fig R1) shows that the identified invariant manifold accurately reflects the fast-slow separation expected for a normally hyperbolic system, thereby validating our method's legitimacy.
The number of fixed points (NFPS) identified depends on the number of points sampled for the angular flow on the invariant ring, but converges to the true NFPS as the grid of initial points is refined.
# Limitations
We will add a separate **Limitations** subsection:

Although our theory is general, since the magnitude of perturbation is measured in uniform norm, for specific parameterizations, further analysis is needed. If the parameters are not flexible enough, the theory may not apply, for example, RNNs with "built-in" continuous attractors such as LSTM without a forget gate cannot be destroyed. However, in biological systems, this is highly unlikely at least at the network level.

Our empirical analysis requires the fast-slow decomposition around the manifold. Not all dynamical system solutions to the tasks that require analog memory have this property (hence sec 5.2). Solutions such as the quasi-periodic toroidal attractors or constant spirals represent challenges to the current framework of analysis in understanding the general phenomena of analog memory without continuous attractors.

Our numerical analysis relies on identifying a time scale separation from simulated trajectories. If the separation of time scales is too small, it may inadvertently identify parts of the state space that are only forward invariant (i.e., transient). However, this did not pose a problem in our analysis of the trained RNNs, which is unsurprising, as the separation is guaranteed by state noise robustness (due to injected state noise during training).

[1] Mané, R. (1978). Persistent manifolds are normally hyperbolic. Transactions of the American Mathematical Society, 246, 261-283.

Meta-review: This work revives the neuroscience idea of using manifold attractors for storing continuously varying quantities in long memory of recurrent systems. The common critique with such systems is that they are structurally unstable, but anchored in solid and elegant theoretical work and topological considerations, the authors show this actually may not be so much of a problem on functionally relevant time scales. These are important observations that may give new impetus to a powerful concept in theoretical neuroscience and beyond. While some of the referees initially had some reservations about this work, all of them after a strong rebuttal unanimously support publication.

I would like to add, however, that the idea that the vicinity of a manifold attractor may still be good enough to support behaviorally relevant time scales is not entirely new [1,2]. This should be mentioned and discussed in the revision.

[1] https://www.frontiersin.org/journals/computational-neuroscience/articles/10.3389/fncom.2011.00040/full

[2] https://www.cell.com/cell-reports/fulltext/S2211-1247(22)00360-6
Decision: Accept (poster)