Inferring Multi-Timescale Neural Dynamics with Switching Linear Dynamical Systems

Lulu Gong; Yongxu Zhang; Shreya Saxena

cs.LGOct 1, 2026

Inferring Multi-Timescale Neural Dynamics with Switching Linear Dynamical Systems

Authors: Lulu Gong, Yongxu Zhang, Shreya Saxena

Organizations: Center for Neurocomputation and Machine Intelligence, Wu Tsai Institute, Department of Biomedical Engineering, Yale University, New Haven, CT 06510, USA

Abstract

Neural activity often exhibits multiple timescales that can vary with behavioral states and task conditions. Identifying these timescales from neural recordings is important for better understanding neural computation and function. However, traditional approaches based on autocorrelation fitting are difficult to scale to high-dimensional population recordings and can become unreliable when neural dynamics change with behavior. State-space models have been a powerful framework for modeling high-dimensional neural population activity through latent dynamical systems, but standard formulations and inference methods do not explicitly account for multiple timescales and therefore do not guarantee accurate recovery of the underlying temporal structure. Motivated by these questions, we introduce the Multi-Timescale Switching Linear Dynamical System (MTS-SLDS), a framework for identifying regime-specific latent timescales from continuous or spiking neural observations. MTS-SLDS combines a multi-lag moment initialization, which captures temporal structure across multiple observation lags, with \textit{regime-conditioned} Laplace-EM inference, which reduces mixing of dynamical statistics across uncertain regimes. Characteristic timescales can then be extracted directly from the eigenvalues of the learned latent transition matrices. In synthetic and neural experiments with Gaussian and Poisson spike observations, MTS-SLDS accurately recovers timescales and switching structure over multiple datasets.

Figures & tables

Figure 1: Multiple neural timescales and the MTS-SLDS framework. A Neural activity can switch between regimes with different characteristic timescales (here $\tau_{1},\tau_{2}$ and $\tau_{3},\tau_{4}$ ). The pooled autocorrelation of such a signal is not well described by a single exponential, and a multi-exponential fit summarizes the decay without attributing components to regimes. B MTS-SLDS models Poisson spike counts or Gaussian signals as observations of a latent state $x_{t}$ whose linear dynamics $A_{z_{t}}$ are selected by a discrete regime $z_{t}$ . Multi-lag moment initialization and regime-conditioned Laplace EM (RC-L-EM) learn the regime sequence, latent

Figure 2: Timescale recovery in stationary and switching systems with Gaussian and Poisson observations. A Estimated timescales under stationary ( $K=1$ , $N_{\tau}=2$ ), where hollow markers represent different runs, and filled markers show medians. aABC shows the posterior median and $50\%$ / $95\%$ credible intervals. B Estimated timescales under switching ( $K=2$ , $N_{\tau}=3$ per regime) dynamics, where circles denote latent modes of timescales, and dots dentoe medians. C Example held-out latent continuous states and regime inference in switching dynamics. D, E Timescale MAPE versus Gaussian SNR and mean Poisson counts in switching dynamics, respectively, where shading denotes interquartile ranges and dashed lines denotes the reference $10\%$ error level.

Figure 3: Multiple timescale structure of V4 activity during fixation. A Fixation task and data analysis schematic. B Predictive gains over a constant-rate Poisson baseline: cross-validation (left, mean $\pm$ SE) and paired test trials at $H=8$ (right). C Example trial spike train, one-step rate predictions, and cumulative log-likelihood difference between MTS-SLDS and SLDS. D Observed and model-generated pooled autocorrelations and residuals over $100$ ms fitting window, where values in legend denote RMSE of each method. E Timescale estimates of different methods. For state-space models, small markers show 5-fold estimates and diamonds show medians. ACF-exp and aABC fits show estimates from pooled population autocorrelation, and aABC displays posterior medians and $50\%$ / $95\%$ credible intervals.

Figure 4: Task-related timescales in somatosensory Area 2. A Reach trajectories and onset-aligned movement speed during active reaching (purple) and passive perturbations (orange). B Held-out predictive log-likelihood gain versus latent dimension, where gray shading marks the selected

H=6

. C Inferred movement-regime probabilities with MTS-SLDS and SLDS. D Latent timescale spectra for pre-movement and movement regimes in the 20-300-ms analysis range. E Slowest effective timescales across regimes demonstrate different patterns between MTS-SLDS and SLDS. F Geometric mean of timescales for real and time-shuffled recordings. Lines in E,F connect paired estimates. G Behavioral decay times (individual estimates and means) compared with median slowest movement-regime timescales. MTS-SLDS inferred timescales are in close agreement with behavior decay times.

Appendix figures & tables8 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 5: Pooling can obscure latent temporal modes in a single recording. A Per-mode recovery fractions from three-exponential ACF fits in the setting. B Mean BIC-selected exponential order in both count settings. A and B reuse the same generated recording at every pooling level, with 30 channel draws per level; full-pooling repetitions vary only optimization. C Effective timescales of one- and two-exponential fits at full pooling in the setting. Marker size indicates relative ACF amplitude; the lowest-residual fit at each order is displayed. D Dotted lines indicate the true order or timescales.

Figure 6: Effects of initialization and fitting configuration. A Timescale MAPE for

K=2

under RC-Laplace-EM with moment or ssm initialization. The upper strip shows iterations to reach 15% MAPE across ten seeds (triangle, threshold not reached). B, C Minimum per-regime recall and timescale MAPE for

K=3

under the same initialization comparison. Curves in A-C show example runs; right-hand points in B, C show final outcomes across seeds. D Timescale MAPE before and after RC-Laplace-EM or Laplace-EM fitting from moment-based starts. Thick lines highlight one seed; light lines show the others.

Figure 7: Regime inference and sensitivity to transition priors. A True and inferred regimes for a trial in MTS-SLDS recovery among 15 eligible trials; numbers indicate balanced accuracy. The two ssm rows use identical fitted parameters with native or regime-conditioned inference. B Timescale MAPE across 6 transition-prior settings, including an oracle prior matched to the generating transition probability. Open circles show individual seeds (

n=5

), bars show medians, and shading indicates interquartile ranges. The blue line marks the MTS-SLDS median.

Figure 8: A,B Decay times and frequencies recovered by MTS-SLDS for oscillatory dynamics. C Recovery of three decay timescales by MTS-SLDS, SLDS, and ACF-exp. Black horizontal lines indicate ground truth, where red and blue markers denote MTS-SLDS and SLDS, respectively, and gray diamonds denote ACF-exp.

Figure 9: Held-out V4 prediction at $H=8$ . (A) Raw spike counts from the same trial 59. (B,C) Observed and predicted rates averaged across channels and for a selected channel example (channel 9). (D) Cumulative predictive log-likelihood difference between MTS-SLDS and SLDS. (E) Per-channel one-step predictive gains over the constant-rate Poisson baseline.

Figure 10: Effective timescales and exponential-fit diagnostics in V4. (a) Estimates from

H=2

state-space models and two-timescale aABC. Points show fold fits, diamonds show medians, and aABC intervals. (b) Components from one-, two-, and three-exponential ACF fits across fitting windows.

Figure 11: Held-out trial activity prediction and regime inference in Area 2. A Spike counts from an example held-out active trial. B,C Observed smoothed firing rates (black) and estimates from MTS-SLDS (red) and SLDS (blue dashed), pooled across channels B and for a single channel C. D Movement-regime posterior probabilities across held-out trials during active reaching (left) and passive perturbations (right), where each row represents one trial and vertical dashed lines mark movement onset.

Figure 12: Timescales estimated by sPCF-EM are shown for pre-movement and movement regimes during active (red) and passive (blue) conditions.

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Spiking neural networks (SNNs) promise low-power event-driven computation for temporally rich tasks, but commonly used neuron models often trade off gradient-based trainability, dynamical richness, and high activity sparsity. These limitations are acute in regression, where approximation error, noise and spike discretization can severely degrade continuous-valued outputs. Indeed, many state-of-the-art (SOTA) SNNs rely on simple phenomenological dynamics trained with surrogate gradients and offer limited control over spiking diversity and sparsity. To overcome such limitations, we introduce multi-timescale conductance spiking networks, a gradient-trainable framework in which neural dynamics emerge from shaping the current-voltage (I-V) curve by tuning fast, slow and ultra-slow conductances. This parametrization allows systematic control over excitability, can be implemented efficiently in analog circuits, and yields rich firing regimes including tonic, phasic and bursting responses within a single model. We derive a discrete-time formulation of these differentiable dynamics, enabling direct backpropagation through time without surrogate-gradient approximations. To probe both trainability and accuracy, we evaluate feedforward networks of these neurons at the predictability limit of Mackey-Glass time-series regression and compare them to baseline LIF and SOTA AdLIF networks. Our model outperforms LIF and AdLIF networks, while exhibiting substantially sparser activity from both communication and computational perspectives. These results highlight multi-timescale conductance spiking neurons as a promising building block for energy-aware temporal processing and neuromorphic implementation.

Abstract

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