Storing Infinite Dynamical Attractors in Nonreciprocal Associative Neural Networks

Abstract: We develop a dynamical mean-field theory for nonreciprocal associative networks that store an extensive number of dynamical attractors, from limit cycles to strange attractors. Using a path integral calculation under quenched disorder, we derive self-consistent dynamical mean-field equations for pattern overlaps, autocorrelations and response functions. Memory retrieval capacity is governed by the spectral structure of the coupling matrices encoding stored patterns. When their eigenvalues are coherently aligned, retarded self-interactions and quenched noise feed back destructively: at zero eigenphase (fixed point attractors) the classical equilibrium capacity bound is recovered, while for limit cycles retrieval collapses far below it. In contrast, for uniformly distributed eigenphases, retarded self-interactions and much of the quenched noise cancels, reducing the dynamics to an effective single-spin process and amplifying capacity substantially. We validate the theory against microscopic simulations for limit-cycle and chaotic attractors, identifying eigenvalue decoherence as the mechanism enabling enhanced storage of dynamical memories.

• The paper introduces a dynamical mean-field theory for recurrent neural networks capable of storing extensive temporal attractors, demonstrating that storage capacity is primarily determined by the spectral organization of encoding matrices over attractor topology.
• Coherent eigenphases across stored attractors result in destructive feedback and reduced storage capacity, while diverse eigenphases suppress these effects, enhancing retrieval quality.
• This approach significantly increases the zero-temperature capacity of fixed-point memories, achieving approximately 0.27 compared to the classical Hopfield capacity of 0.138, through uniform spectral distributions

Problem formulation and principal claim

“Storing Infinite Dynamical Attractors in Nonreciprocal Associative Neural Networks” develops a dynamical mean-field theory (DMFT) for recurrent networks that store an extensive number of temporal attractors, including fixed points, limit cycles, quasiperiodic trajectories, and chaotic attractors (2609.07341). The central claim is that storage capacity is determined less by the topology of an attractor than by the spectral organization of the matrices used to encode different attractors. In particular, coherent eigenphases across stored attractors generate destructive retarded self-interactions and quenched-noise correlations, whereas sufficiently diverse eigenphases suppress these terms and can substantially increase capacity.

The model consists of $N$ binary spins updated stochastically by Glauber dynamics. Each attractor block $\upsilon$ contains $M_\upsilon$ random binary patterns and an interaction matrix $A^\upsilon$,

$$J_{ij}=\frac{1}{N}\sum_{\upsilon=0}^{L}\sum_{a,b=1}^{M_\upsilon} A^\upsilon_{ab}\xi^a_{i,\upsilon}\xi^b_{j,\upsilon}, \qquad i\neq j.$$

The target attractor is assigned to block $\upsilon=0$, while the remaining blocks form quenched interference with load $\alpha=P/N$, where $P$ is the total number of non-target patterns. The update probability $\Delta$ interpolates between parallel and asynchronous dynamics; all numerical experiments use $\Delta=0.1$.

This construction generalizes classical Hopfield storage in two directions. First, the encoding matrices are nonreciprocal, so the target overlap dynamics need not converge to a fixed point. Second, the number of stored attractor blocks grows extensively with system size. The paper therefore addresses a regime not covered by analyses restricted to one stored sequence or to static, symmetric memories.

Dynamical mean-field reduction

The analytical treatment uses a generating-functional calculation over spin trajectories and averages over the quenched random patterns. The resulting effective process is characterized by three classes of macroscopic quantities: target overlaps $\upsilon$0, spin autocorrelations $\upsilon$1, and response functions $\upsilon$2. After the disorder average and saddle-point reduction, each microscopic spin experiences an effective field of the form

$\upsilon$3

where $\upsilon$4 labels one of the $\upsilon$5 target-pattern configurations, $\upsilon$6 is a retarded self-interaction kernel, and $\upsilon$7 is a temporally correlated Gaussian process with covariance $\upsilon$8.

The kernels are determined self-consistently by the response and correlation functions. For orthogonal disorder matrices with eigenvalues $\upsilon$9, they admit the expansions

$M_\upsilon$0

and

$M_\upsilon$1

These expressions identify the spectral mechanism underlying the numerical results. The kernel $M_\upsilon$2 retains memory of prior spin states and represents delayed feedback generated by asymmetric crosstalk. The kernel $M_\upsilon$3 describes correlated effective noise. Both depend on averages over powers of the eigenvalues, so their magnitude and temporal structure are controlled by the eigenphase distribution of the stored attractors.

The diagonal constraint $M_\upsilon$4 is analytically consequential. It produces an Onsager contribution that exactly cancels the equal-time term in $M_\upsilon$5, leaving a strictly retarded self-interaction. Consequently, the effective process is causal despite the non-Markovian dependence on the full trajectory.

A necessary stability condition is that the spectral radius of $M_\upsilon$6 remain below one. When this condition fails, the resolvent expansions defining $M_\upsilon$7 and $M_\upsilon$8 diverge, implying unbounded amplification of quenched fluctuations. The paper thus distinguishes ordinary retrieval onset from noise stability: an attractor may exist at zero load but become impossible to retrieve for any positive $M_\upsilon$9 if its response dynamics crosses this instability.

Coherent eigenphases and the collapse of cyclic-memory capacity

The first main case considers identical two-dimensional rotation matrices,

$A^\upsilon$0

with eigenvalues $A^\upsilon$1. The angle $A^\upsilon$2 controls the temporal structure of the target attractor. At $A^\upsilon$3, the model reduces to fixed-point retrieval. For nonzero $A^\upsilon$4, it can support genuine limit cycles, although sufficiently large inverse temperature produces a heteroclinic transition in which the cycle slows and collapses onto fixed points.

When all stored attractors have the same eigenphase, the kernels become

$A^\upsilon$5

and

$A^\upsilon$6

The phase coherence prevents cancellation between different spectral contributions. In the limit-cycle regime, the retarded feedback and cross-correlations in $A^\upsilon$7 act destructively on retrieval, causing a sharp reduction in critical capacity. This is a strong and somewhat counterintuitive result: introducing a temporal cycle into a network that can store fixed points does not merely alter the attractor geometry; under coherent spectral encoding, it can make extensive storage substantially less reliable.

At $A^\upsilon$8, the authors recover the classical equilibrium capacity behavior for fixed-point memories. As $A^\upsilon$9 increases into the oscillatory regime, capacity decreases sharply. The reduction is associated with coherent feedback rather than with the existence of temporal structure per se.

Figure 1: Critical capacity as a function of temperature for coherently aligned versus uniformly distributed eigenphases; coherent limit-cycle encoding produces a pronounced capacity reduction, whereas spectral diversity largely removes it.

The phase structure at zero load clarifies which retrieval states are relevant. Three boundaries divide the $$J_{ij}=\frac{1}{N}\sum_{\upsilon=0}^{L}\sum_{a,b=1}^{M_\upsilon} A^\upsilon_{ab}\xi^a_{i,\upsilon}\xi^b_{j,\upsilon}, \qquad i\neq j.$$0 plane: the retrieval onset $$J_{ij}=\frac{1}{N}\sum_{\upsilon=0}^{L}\sum_{a,b=1}^{M_\upsilon} A^\upsilon_{ab}\xi^a_{i,\upsilon}\xi^b_{j,\upsilon}, \qquad i\neq j.$$1, the cycle-to-fixed-point boundary $$J_{ij}=\frac{1}{N}\sum_{\upsilon=0}^{L}\sum_{a,b=1}^{M_\upsilon} A^\upsilon_{ab}\xi^a_{i,\upsilon}\xi^b_{j,\upsilon}, \qquad i\neq j.$$2, and the noise-stability threshold $$J_{ij}=\frac{1}{N}\sum_{\upsilon=0}^{L}\sum_{a,b=1}^{M_\upsilon} A^\upsilon_{ab}\xi^a_{i,\upsilon}\xi^b_{j,\upsilon}, \qquad i\neq j.$$3. Between onset and noise stability, the target cycle exists but its response matrix causes the effective noise covariance to diverge at nonzero load. Only the region with bounded response supports robust extensive-memory retrieval.

Figure 2: Phase diagram distinguishing paramagnetic, unstable cyclic-retrieval, stable cyclic-retrieval, and fixed-point-retrieval regimes.

The paper therefore makes an important distinction between dynamical existence and storage feasibility. A limit cycle can be present for $$J_{ij}=\frac{1}{N}\sum_{\upsilon=0}^{L}\sum_{a,b=1}^{M_\upsilon} A^\upsilon_{ab}\xi^a_{i,\upsilon}\xi^b_{j,\upsilon}, \qquad i\neq j.$$4 while failing as an associative memory for every $$J_{ij}=\frac{1}{N}\sum_{\upsilon=0}^{L}\sum_{a,b=1}^{M_\upsilon} A^\upsilon_{ab}\xi^a_{i,\upsilon}\xi^b_{j,\upsilon}, \qquad i\neq j.$$5. This follows from the instability of the response dynamics, not from the absence of a target orbit.

Eigenphase decoherence and enhanced storage

The central positive result arises when the eigenvalues of the disorder matrices are uniformly distributed on the unit circle. In this case, phase averaging eliminates all nonresonant terms. Specifically,

$$J_{ij}=\frac{1}{N}\sum_{\upsilon=0}^{L}\sum_{a,b=1}^{M_\upsilon} A^\upsilon_{ab}\xi^a_{i,\upsilon}\xi^b_{j,\upsilon}, \qquad i\neq j.$$6

so the retarded self-interaction disappears and $$J_{ij}=\frac{1}{N}\sum_{\upsilon=0}^{L}\sum_{a,b=1}^{M_\upsilon} A^\upsilon_{ab}\xi^a_{i,\upsilon}\xi^b_{j,\upsilon}, \qquad i\neq j.$$7 satisfies a discrete Lyapunov equation,

$$J_{ij}=\frac{1}{N}\sum_{\upsilon=0}^{L}\sum_{a,b=1}^{M_\upsilon} A^\upsilon_{ab}\xi^a_{i,\upsilon}\xi^b_{j,\upsilon}, \qquad i\neq j.$$8

The effective dynamics consequently reduces to a single-spin process driven by colored Gaussian noise but without delayed self-coupling. This cancellation is the paper’s principal mechanism for capacity enhancement. Spectral diversity does not simply distribute interference across frequencies; it removes the coherent components that otherwise reinforce retarded feedback.

For independently sampled rotation angles, the zero-temperature capacity at $$J_{ij}=\frac{1}{N}\sum_{\upsilon=0}^{L}\sum_{a,b=1}^{M_\upsilon} A^\upsilon_{ab}\xi^a_{i,\upsilon}\xi^b_{j,\upsilon}, \qquad i\neq j.$$9 reaches approximately

$\upsilon=0$0

compared with the classical Hopfield value of approximately $\upsilon=0$1. Thus, under the paper’s normalization and update convention, the capacity is roughly doubled. The result is especially notable because it occurs in a network storing dynamical attractor blocks rather than only static patterns.

The improvement is not universal across all target phases. For target angles beyond approximately $\upsilon=0$2, the authors find that attractors cannot be stored, even at arbitrarily small positive load. In this regime, the response matrix becomes unstable at sufficiently low temperature, causing $\upsilon=0$3 to diverge. Hence, eigenphase diversity cannot compensate for an intrinsically unstable target dynamics.

The simulations with partially dispersed eigenphases provide a more direct test. When phases are sampled from a distribution with central angle $\upsilon=0$4 and standard deviation $\upsilon=0$5, increasing $\upsilon=0$6 generally increases capacity. Except near the largest central angles, the reported enhancement ranges from a factor of approximately $\upsilon=0$7 to $\upsilon=0$8 relative to the coherent case after normalization. The monotonicity breaks down close to the response-instability boundary, where the target attractor itself is poorly conditioned.

Figure 3: Capacity increases with eigenphase dispersion for most central angles, although the enhancement disappears near the instability boundary.

This result supports a spectral, rather than purely dynamical, interpretation of interference. Two networks may encode attractors with the same local topology—such as two-dimensional cycles—but exhibit very different capacities depending on whether their eigenphases are aligned or dispersed.

Chaotic attractor retrieval

The framework extends beyond rotations in two dimensions. The authors construct a four-dimensional orthogonal encoding matrix with two conjugate eigenvalue pairs and use a genetic search to identify parameters producing a chaotic overlap trajectory while maintaining response stability. The selected target has leading Lyapunov exponents approximately

$\upsilon=0$9

The positive leading exponent indicates chaos, while the near-zero second exponent is consistent with weakly contracting or marginal structure in the projected overlap dynamics. The resulting attractor is therefore not a periodic sequence represented with a long period; it is generated directly by the nonreciprocal overlap map.

Figure 4: Retrieval of a chaotic overlap attractor and comparison between DMFT predictions and microscopic simulations under conjugate and uniformly distributed eigenvalue spectra.

For both conjugate and uniformly distributed eigenvalue ensembles, the DMFT predictions agree quantitatively with microscopic simulations of retrieval quality as a function of load. Uniform eigenvalue distributions again produce higher capacity than conjugate spectra. This observation extends the decoherence mechanism beyond limit cycles: the capacity advantage is present for a chaotic attractor and is therefore not tied to periodicity.

The chaotic example also establishes a methodological point. The theory does not require the target dynamics to be analytically solvable in closed form. Once the target overlap process and response functions are specified, the effective single-spin description can be evaluated numerically. The remaining limitation is that the chaotic encoding matrix is selected by an optimization procedure rather than sampled from a principled ensemble. The example demonstrates feasibility and consistency, but it does not characterize the capacity of generic four-dimensional chaotic encoders.

Relation between spectral structure and attractor topology

The paper’s strongest interpretive claim is that the diversity of eigenphases, rather than attractor topology, is the primary determinant of capacity. The evidence consists of three comparisons. First, coherent rotations sharply reduce capacity in the cyclic regime. Second, independent uniform eigenphases eliminate $\alpha=P/N$0 and improve capacity even when the target remains a cycle. Third, the same improvement appears for a chaotic target when the disorder spectrum is decorrelated.

The claim should be understood at the level of the derived mean-field kernels. Attractor topology determines the target overlap trajectory and its susceptibility, but eigenphase statistics determine whether different disorder paths add coherently in $\alpha=P/N$1 and $\alpha=P/N$2. Uniform phase averages suppress terms with unequal powers of the eigenvalues. In this sense, spectral diversity reduces structured crosstalk without necessarily reducing the strength of the individual stored interactions.

The mechanism has a formal analogy with other asymmetric or diluted models in which crosstalk terms disappear because interactions become sufficiently decorrelated. The present model achieves a related effect without dilution: cancellation is induced by the spectral organization of the encoding matrices. This distinction matters because the network remains densely connected while its effective disorder becomes less self-reinforcing.

Limitations and open questions

The analytical theory assumes i.i.d. binary patterns and focuses primarily on orthogonal encoding matrices. The simplifications leading to $\alpha=P/N$3 and the Lyapunov equation for $\alpha=P/N$4 depend specifically on uniform eigenphase distributions and do not apply to arbitrary nonnormal or nonorthogonal matrices. Since nonnormality can strongly amplify transient responses, the condition based on the spectral radius of $\alpha=P/N$5 may not fully characterize finite-time amplification in more general ensembles.

All numerical experiments use $\alpha=P/N$6. The dependence of the capacity enhancement on the update schedule is therefore not systematically established. The paper notes that the fixed-point result at $\alpha=P/N$7 has a correspondence with sequential-memory results at other update conventions, but this does not imply that the full spectral-decoherence mechanism is invariant under $\alpha=P/N$8.

Capacity is operationally defined through a threshold on the long-time RMS overlap, using one-half of the zero-load retrieval value. This is a reasonable numerical criterion, but different definitions based on basin volume, phase fidelity, Lyapunov-spectrum preservation, or finite-time prediction could yield different critical loads, particularly for chaotic attractors.

Finally, the chaotic example relies on a genetically optimized four-dimensional matrix and reports only one principal attractor construction. The results establish that chaotic retrieval is compatible with the DMFT and that eigenphase dispersion can improve it, but they leave open how capacity scales with attractor dimension, how it depends on the full Lyapunov spectrum, and whether generic high-dimensional chaotic encoders exhibit the same quantitative enhancement.

Conclusion

The paper derives a DMFT for densely connected nonreciprocal associative networks storing an extensive number of dynamical attractors. Its main result is that storage capacity is controlled by the spectral coherence of the attractor-encoding matrices. Coherent eigenphases generate retarded self-interactions and correlated quenched noise that can severely impair cyclic and chaotic retrieval. Uniform or partially dispersed eigenphases suppress these terms, remove the retarded kernel in the idealized uniform case, and can increase capacity by factors of $\alpha=P/N$9 to $P$0, with a reported zero-temperature capacity near $P$1 in the most favorable cycle-encoding setting.

The theory also identifies response instability as a separate obstruction to retrieval: an attractor may exist without disorder but become nonretrievable at any positive memory load. Together, these results provide a quantitative framework for analyzing extensive storage of nonstationary attractors and establish eigenphase organization as a central control parameter for nonreciprocal associative memory (2609.07341).