TL;DR
Qualitative analyses of training curves, loss landscapes, and latent embeddings support the interpretation that Hopfield retrieval smooths optimization and encourages more compact, linearly separable class manifolds.
The Hopfield network established that static memories can be stored as energy minima of a recurrent dynamical system, yet real intelligent agents must navigate \emph{sequences} of memories rather than isolated snapshots.Biological cortex addresses this through a separation of timescales: fast synaptic dynamics encode individual states while slow neuromodulatory processes govern transitions between them. Inspired by this multi-timescale organization, we propose a sequential memory architecture that is fully continuous, admits exponential storage capacity, and is learnable in the sense that the transition structure is carried by a separate routing matrix -- decoupled from the stored patterns, driven by the input context, and free to be chosen, optimized, or learned from data rather than hard-wired into the memory substrate. Specifically, we construct an autonomous three-timescale dynamical system with three coupled layers: a fast Kuramoto layer that stores phase patterns as exponentially stable phase-locked configurations, an intermediate hysteresis layer that enforces reliable dwell times, and a slow attention layer that routes sequential transitions. We provide a complete theoretical analysis of the stability and robustness of each layer, and we validate the full system through numerical simulations of sequential memory retrieval.
SpikeWorld, a 1.45M-parameter sparse spiking model jointly trained for heterogeneous sensory prediction, semantics, image-text binding and action-conditioned dynamics, is introduced, showing that the contribution is not superior linear identification, but its integration with a frozen multimodal spiking checkpoint.
This chapter reconstructs the Hopfield network as a physical theory of memory rather than merely an early neural-network algorithm. It begins with the problem as it stood before 1982-threshold logic, Hebbian association, correlation memories, and recurrent binary networks-and isolates what Hopfield's synthesis added: a dynamical definition of content-addressable memory, a symmetric recurrent architecture with a Lyapunov function, a Hebbian embedding of patterns in its couplings, and a physical account of basins, robustness, and graceful degradation. The binary and graded-response energy functions are derived in full, together with the signal-crosstalk decomposition governing pattern stability, the mean-field theory of retrieval at extensive load, and the zero-temperature retrieval spinodal at (alpha 0.138) established by Amit, Gutfreund, and Sompolinsky. The energy-based program is then followed through analog optimization networks, polynomial dense associative memories, exponential interactions, and modern continuous Hopfield updates, including the precise conditions under which the update becomes scaled dot-product attention. Throughout, capacity claims are tied to their disorder ensemble, scaling limit, and success criterion, showing why numerically different storage limits need not conflict. A closing assessment distinguishes established results from surviving principles, assumption-bound limitations, and open problems, treating the Hopfield network as an effective theory whose symmetry, locality, and point-neuron assumptions delimit its biological reach. Fixed-seed numerical experiments expose the mechanisms discussed but do not substitute for analytical results.
For an open quantum reservoir, how the system forgets is part of how it computes. Quantum reservoir computing processes input streams with fixed quantum dynamics and trains only a linear readout. Dissipation can make old inputs fade, but prior studies commonly fix the environmental process and tune only its strength. Here we show numerically that the coupling pattern, meaning whether transitions connect to separate or shared environmental channels, changes which parts of the input history remain accessible. Paired simulations of finite spin reservoirs keep the Hamiltonian, inputs, measurements, and readout fixed. The tested patterns produce distinct task profiles, with no universal winner. Shared relaxation preserves more recent input history than independent local loss, and the retained memory changes when the qubits contribute with different relative phases to the shared decay channel. This ordering recurs across system sizes, Hamiltonians, input protocols, and targeted controls. Environmental coupling is therefore more than a damping parameter: it is a design layer that shapes not only how quickly information fades, but which input history remains available for computation.
Echo-state networks enable efficient temporal learning by fixing the recurrent dynamics and training only a linear readout. However, conventional reservoirs typically accommodate signal mixing, memory retention, and stability within a single random recurrent matrix. Existing structured designs improve topology, norm preservation, leakage, or depth, but generally do not provide separate modal control of reversible mixing and irreversible forgetting together with a direct global stability guarantee. We introduce a classical Lindblad-inspired multi-timescale reservoir that bridges open-system dynamical principles with structured state-space modeling. The recurrent operator is assembled from exactly discretized damped rotational modes, so rotation and decay become independent design variables governing phase mixing and memory loss. Orthogonal mode mixing preserves normality, while the decay spectrum directly determines the echo-state stability margin without post-hoc spectral-radius rescaling. We evaluate the method over ten aligned seeds against standard, leaky, deep, orthogonal, cycle, and next-generation reservoirs, together with a compact trained gated recurrent unit, across linear memory, nonlinear recurrence, chaotic forecasting, delayed logic, and real sensor calibration. Across the benchmark suite, the proposed reservoir achieves the best fixed-reservoir performance on bounded NARMA-20 and the lowest mean error on Lorenz-63, matches the strongest linear-memory result, and remains broadly competitive across broad range of benchmarks. Ablation studies show that rotation increases state diversity, whereas dissipation provides controlled forgetting and improves predictive conditioning. The resulting framework offers an interpretable recurrent architecture in which mixing, memory, and stability are explicit and independently tunable design variables.
Training RNNs on delayed decision-making tasks with progressively increasing temporal demands shows that temporal and decision-related computations can emerge through multiple dynamical regimes, while maintaining structured low-dimensional representations and comparable behavioural performance, mirroring biological principles of degeneracy and functional redundancy.
We use cookies to run the site and, with your consent, for analytics and to show ads. See our Cookie Policy.
