Continuous-Time LTM: Foundations & Applications

• Continuous-Time LTM is defined by history-dependent, non-Markovian processes using fractional derivatives and nonlocal memory kernels.
• It constructs multi-slide memory architectures, modeling independent decaying traces that yield power-law or stretched-exponential decay.
• Statistical inference and computational consolidation methods validate its scalability in capturing long-context dynamics in both natural and artificial systems.

Continuous-Time Long-Term Memory (LTM) refers to history-dependent, non-Markovian mechanisms governing the retention, recall, and decay of information across diverse domains, including human cognition, statistical process modeling, complex materials, and artificial neural representations. The essential mathematical structure underlying LTM in continuous time is the use of nonlocal-memory kernels, often expressed via fractional derivatives and subordinated stochastic processes, resulting in characteristic power-law or stretched-exponential decays of system observables. These frameworks generalize both classical discrete-step and finite-state memory models to the functional-analytic setting needed for unbounded temporal domains and scale-free behavior.

1. Mathematical Foundations: Fractional Calculus and Subordination

The continuous-time modeling of LTM generically replaces local-in-time derivatives with fractional-order operators, capturing persistent effects that depend on the entire history of the system. In the human memory domain, Lubashevsky and Datsko (Lubashevsky et al., 2014) introduced a fractional Caputo derivative to model the chunk strength $F(t)$ of retrievable information:

$$\tau_m^{1-d}\,{}^C D_t^{1-d} F(t) = (\epsilon + F(t))^{\alpha}(1 - F(t))^{\beta} W(F(t))$$

Here, the fractional derivative of order %%%%1%%%% encodes slow, power-law forgetting ($d \in (0,1)$), while the right-hand side models attention, learning rate, and saturation effects.

In physical processes, Stanislavsky and Weron (Stanislavsky et al., 2011) employed two-time scale subordination, in which the evolution of observables is driven by the sum of two independent “operational” random times, each corresponding to heavy-tailed waiting-time distributions:

$Y_{\alpha, \beta}(t) = X(V_\alpha(t) + V_\beta(t))$

$$\Phi_{\alpha, \beta}(k, t) = \exp\left(-C_\alpha t^\alpha - C_\beta t^\beta\right)$$

This yields relaxation laws combining exponential and stretched-exponential decay, and fractional state equations of Bagley–Torvik type, further generalizing to multi-fractional settings via Caputo derivatives.

2. Multi-Slide (Multiple-Trace) Models and Memory Architecture

Rather than conceptualizing LTM as a monolithic fading trace, the continuous-time multi-slide architecture constructs memory strength as a superposition of individual, independently decaying traces laid down at different times. In the human memory framework (Lubashevsky et al., 2014), each learning episode establishes a new “slide,” whose capacity matches the unrecalled portion of the pattern:

$C(t) = 1 - F(t)$

$$f(t, t') = \left[1 + \frac{t - t'}{\tau_0}\right]^{-d}$$

$F(t) = \sum_{t'<t} C(t') f(t, t')$

Continuous-time limit and power-law kernel approximation ($(t - t')^{-d}$) yield nonlocal integral equations, which equivalently translate into fractional derivatives. The biological and computational justification rests on distributed ensemble models (multiple-trace theory), independent long-term decay rates, and competitive trace reactivation, aligning with hippocampal pattern completion/separation mechanisms.

3. Continuous-Time Stochastic Process Models and Inference

Formal statistical modeling of continuous-time LTM focuses on processes whose autocovariance or spectral density has slow, scale-free decay. Dedecker et al. (Haye et al., 2019) precisely define a stationary process with long memory:

$$\gamma_X(\tau) \sim C \,\ell(\tau) \, \tau^{2d-1} \quad (\tau \to \infty)$$

$$f_X(\lambda) \sim C' |\lambda|^{-2d} \ell(1/|\lambda|) \quad (\lambda \to 0)$$

Crucially, if sampled at random times (e.g., via renewal sampling), the resulting series preserves its memory exponent $d$ under broad conditions, even as joint Gaussianity is lost. For Poisson sampling, the spectral density transforms via an explicit integral kernel:

$$f_Y(\omega) = \frac{1}{2\pi} \int_{-\infty}^{\infty} p(\omega, \phi_A(\lambda)) f_X(\lambda) d \lambda$$

Consistent estimation of the memory parameter $d$ is achievable via local Whittle and periodogram techniques, subject to sampling-interval distribution constraints.

4. Continuous-Time LTM in Artificial Systems: Consolidation and Attention

In long-context video-LLMs, the ∞-Video framework (Santos et al., 31 Jan 2025) operationalizes continuous-time LTM via dynamic consolidation. A global memory signal $\mathbf{x}(t)$ is maintained, reconstructed at each input chunk using contracted past and newly observed embeddings, all expressed over a continuous basis (e.g., boxcar functions):

$\mathbf{x}(t) = \mathbf{B}^\top \bm{\psi}(t)$

Continuous-time cross-attention deploys a Gibbs-weighted expectation over all time, with adaptive sampling ("sticky" attention) used for high-granularity memory at salient intervals. Consolidation is performed using ridge regression, and memory is contracted by a factor $\tau < 1$ at each step to enact gradual forgetting.

Comparison against standard short-term architectures reveals that the continuous-time LTM mechanism maintains scalable complexity (fixed-size state) and significantly improves performance in long-context question answering tasks, particularly when sticky sampling is used to allocate fine memory resolution adaptively to the most relevant video segments.

5. Power-Law Dynamics, Spacing Effects, and Combined Relaxation Laws

Characteristic continuous-time LTM models yield power-law or combined exponential–stretched exponential decay in the retention of information or physical observables. In human memory dynamics (Lubashevsky et al., 2014), after learning ceases ($W=0$), forgetting follows:

$F(t) \propto (t - T_L)^{-d}$

Conversely, ongoing learning produces power-law growth with a distinct exponent $d_L$, indicating that individual differences in learning and forgetting rates must be independently characterized. In physical systems, two-time scale subordinators generate observational decay laws of the form:

$$C(t) \propto \exp(-\lambda_1 t) \times \exp[-(\lambda_2 t)^\alpha]$$

as empirically observed in trapping-reaction kinetics (Stanislavsky et al., 2011). Spacing effects emerge naturally: in human learning, distributed practice with longer gaps yields longer retention intervals, with the trade-off being linearly proportional.

6. Limitations, Generalizations, and Open Questions

Limitations of these frameworks include dependence on kernel parameter tuning, possible under-representation of rapid, discontinuous events in basis-based embedding representations, and the challenge of extending from single-chunk to large networks of interacting memory units or multi-modal settings.

Open problems involve joint estimation of sampling-interval distribution and memory parameter in irregular data (Haye et al., 2019), end-to-end fine-tuning of consolidation modules in artificial LTM, adaptive selection of attention contraction factors, incorporation of complex kernels beyond simple boxcar bases, and interpretation of attention density peaks with respect to semantics or neural correlates.

Generalizations comprise multi-fractional and multi-parameter memory models, extension to systems consolidation architectures, mapping of memory exponents to neural replay and plasticity time-scales, and further exploration of fractional state equations in complex media (e.g., Bagley-Torvik models for viscoelasticity) (Stanislavsky et al., 2011).

7. Broader Implications and Applications

Continuous-time LTM frameworks unify phenomena across cognitive psychology, statistical physics, stochastic process theory, and artificial intelligence, providing principled, scalable models for history-dependence, retention, and gradual decay. Key predictions include scale-free retention curves, optimality criteria for spacing schedules, and experimentally testable trade-offs between consolidation effort and enduring memory.

Potential application domains comprise human learning, anomalous transport and relaxation in materials, fate and concentration dynamics in chemical and biological systems, irregularly sampled time-series in finance and biomedicine, and scalable context representations in neural network architectures for long-sequence modeling.