Inferring brain plasticity rule under long-term stimulation with structured recurrent dynamics
Authors: Zhichao Liang
Jingzhe Lin
Xinyi Li
Guanyi Zhao
Quanying Liu
Venue: ICLR 2026 Poster (2026)
Keywords: brain plasticity
long-term stimulation
recurrent dynamics
Primary area: applications to neuroscience & cognitive science

Abstract: Understanding how long-term stimulation reshapes neural circuits requires uncovering the rules of brain plasticity. While short-term synaptic modifications have been extensively characterized, the principles that drive circuit-level reorganization across hours to weeks remain unknown. Here, we formalize these principles as a latent dynamical law that governs how recurrent connectivity evolves under repeated interventions. To capture this law, we introduce the Stimulus-Evoked Evolution Recurrent dynamics (STEER) framework, a dual-timescale model that disentangles fast neural activity from slow plastic changes. STEER represents plasticity as low-dimensional latent coefficients evolving under a learnable recurrence, enabling testable inference of plasticity rules rather than absorbing them into black-box parameters. 
We validate STEER with four benchmarks: synthetic Lorenz systems with controlled parameter shifts, BCM-based networks with biologically grounded plasticity, a task learning setting with adaptively optimized external stimulation and longitudinal recordings from Parkinsonian rats receiving closed-loop DBS. Our results demonstrate that STEER recovers interpretable update equations, predicts network adaptation under unseen stimulation schedules, and supports the design of improved intervention protocols. By elevating long-term plasticity from a hidden confound to an identifiable dynamical object, STEER provides a data-driven foundation for both mechanistic insight and principled optimization of brain stimulation. The source code of this study is available at https://github.com/ncclab-sustech/STEER.git.

Published as a conference paper at ICLR 2026
INFERRING BRAIN PLASTICITY RULE UNDER LONG- TERM STIMULATION WITH STRUCTURED RECURRENT DYNAMICS
Zhichao Liang!.2*, Jingzhe Lin' Xinyi Li Guanyi Zhao', Quanying Liu
1. Department of Biomedical Engineering, Southern University of Science and Technology
Center for Neurocognition and Social Behavior, Institute of Artificial Intelligence, SUAT
3. Shenzhen Loop Area Institute These authors contributed equally.
f Corresponding author: liuqy Sely@sssstechd.u..
ABSTRACT
Understanding how Long-term stimulation reshapes neural circuits requires uncov- ering the rules of brain plasticity. While short-term synaptic modifications have been extensively characterized, the principles that drive circuit-level reorganiza- tion across hours weeks remain unknown. Here, formalize these principles as latent dynamical law that governs how recurrent connectivity evolves under repeated interventions. To capture this law, we introduce the Stimulus- Siimulus-Evoked Evoked Evolution Recurrent dynamics (STEER) framework, a dual-timescale model that disentangles fast neural activity from slow plastic changes. STEER represents plasticity as low-dimensional latent coefficients evolving under a learnable recur- rence, enabling testable inference of plasticity rules rather than absorbing them into black-box parameters. We validate STEER with four benchmarks: synthetic Lorenz systems with controlled parameter shifts, BCM-based networks with bi- ologically grounded plasticity, task learning setting with adaptively optimized external stimulation and longitudinal recordings from Parkinsonian rats receiv- ing closed-loop DBS. Our results demonstrate that STEER recovers interpretable update equations, predicts network adaptation under unseen stimulation sched- ules, and supports the design of improved intervention protocols. By elevating long-term plasticity from a hidden confound to an identifiable dynamical object, STEER provides data-driven foundation for both mechanistic insight and princi- pled optimization of brain stimulation. The source code of this study is available / com/ ncc wwwtsssssssseeee STEER git
INTRODUCTION
Understanding long-horizon plasticity, the principles by which neural circuits reorganize over days weeks, is central problem in neuroscience and broader challenge for learning systems (Turri- giano 2012 Abbraham et al. 2019 Appelbaum et al.) (023). These rules determine how the brain adapts during learning and memory (Speranza al. (022 and how interventions reshape network function over repeated sessions (Huang et al. 20055] Suppa et al., 2016; SSandoval-Pistorius et al.)
2023). Yet despite decades of study, we still lack predictive rule that connects repeated stimula- tion to slow circuit reconfiguration. Without such a rule, we cannot anticipate how interventions accumulate their effects, nor can we design principled strategies for therapies such as deep brain stimulation (DBS) or transcranial magnetic stimulation (TMS). A data-driven, predictive, and testable description plasticity rules at these timescales therefore urgently needed, for both mechanistic understanding and clinical optimization.
Existing models fall into two extremes. Classical biophysical rules, such as Hebbian learning (Hebb) 1949) and spike-timing-deeendeer plasticity (STDP) (Caporale & Dan 2008 Dan & Poo 2004) describe local synaptic changes on millisecond -inddte scales. While interpretable, they are too nar-
row capture circuit-level reorganization across long horizons (Zenke & Gerstner, 2017 Benna & & Bontempi (2005). Conversely, machine learning approaches allow connec- tivity to vary across sessions (Pandarinath al. 2018 |Pellegrino al. 2023) Vermani al. 2025
Fusi

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Long-term stimulation induced network plasticity
Modeling unknown plasticity unit
network
Stimulation
unit
Wi, AWy
aW
D
Day
Planticity culer
dt
fs(W)
Figure 1: Motivation. Repeated stimulation induces slow evolution of network plasticity, with the underlying plasticity rules remaining unknown. The challenge of distinguishing slow plasticity from fast neural dynamics remains underdeveloped, with dynamical systems theory offering potential framework model these processes: dW So(W).
eu al. 2025), but typically absorb slow change into high-dimensional parameters, entangling fast activity with slow adaptation. This entanglement impedes the identification of the underlying rule and limits generalization beyond the observed protocols. What is missing is general framework that uncovers slow plasticity rules and separates them from fast neural dynamics, as well as that reliably extrapolate to unseen stimulation protocols.
We propose to treat long-term plasticity not as unstructured cession-to-sessiee variability but as latent dynamical law driven by external protocols. Concretely, we introduce stimulus-evoked evolution (SE-Evolution) as plasticity rule: repeated stimulation drives a low-dimensional latent state 2₁ that encodes plasticity embeddings. We formalize the SE-Evolution
2k+1 g0(2 up))
where Ug denotes the stimulation protocol and ge is the plasticity rule to be inferred. Once ge is identified, can predict how connectivity and activity will evolve under novel protocols, enabling counterfactual design.
STEER: dual-timescale, identifiable formulation. We develop Stimulus-Evoked Evolution Recurrent dynamics (STEER), a structured model that separates fast within-session activity from slow across-session adaptation. At the fast timescale, neural responses are generated by session- specific dynamics with structured recurrent connectivity capturing millisecond activity. At the slow timescale, latent coefficients Σą evolve via learnable recurrence ge that is the plasticity rule. STEER is trained to jointly optimize (i) within-session reconstruction, (ii) cross-session consistency, and (iii) structural regularizers that enforce motif interpretability. This yields identifiable separation between fast and slow processes and turns long-term teggtterm plasticity from confound into explicit, testable object of inference. The main contributions of STEER are summarized as follows:
Conceptually novel: We introduce the concept of SE-Evolution, framing long-term plas- ticity low-dimensional latent dynamical law inferred directly from longitudinal data.
Methodological advancement: We develop STEER, structured dual-timescale frame- work with priors that enable identifiable disentanglement of fast responses and slow net- work reconfiguration, yielding plasticity embeddings and motif-level readouts.
Predictive and out-of-distributtio: We show that the learned rule extrapolates to wn- seen stimulation protocols, predicting connectivity across (i) synthetic Lorenz systems, (ii) BCM-based neural models, (iii) task-learning through stimulation-inducer plasticity and (iv) longitudinal Parkinson's 's DBS data, with accurate neural activity forecasts.
RELATED WORK
2.1
NEURAL SYSTEM IDENTIFICATION: FLEXIBLE SURROGATES, MISSING SLOW LAWS
Neural system identification grounded in dynamical systems theory (DST) provides powerful surro- gates for uncovering latent flows in neural data (Durstewitz et al.) (023). -Ns-b surrogates can reconstruct nonlinear interactions and temporal dependencies from time series (Durstewitz et al.
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Luo al. 2025), and low-rank designs improve sample efficiency and interpretability by con- straining activity to low- liwdemenniooal manifolds (Mastrogiuseppe & Ostojic, 2018 Langdon et al, 2023
2023
Valente et al. 2022 Pals al. 2024). Methods such as SINDy further promote interpretable
discovery of governing equations and primitives (attractors, limit cycles) (Brunton et al. 2016) Driscoll et al. 2024), while anchitetturl/hoophyyiaal priors (e.g., neuromodulatory hypernetworks,
E/I constraints) increase biological plausibility and identifiability (Song et al., 2016 Zhang et al.
2022
Achterberg et al. 2023 Costacurta al. 2024).
Limitation relative to our goal. These works predominantly target fast neural dynamics within sessions. Even when session-to-sessionn variability allowed, the slow process typically treated as unstructured parameter drift inside a nigh-dimensional model. As a result, (i) fast and slow processes are entangled, obscuring the plasticity law; (ii) generalization unseen stimulation protocols not design objective; and (iii) there is no mechanism to make the slow rule identifialle rather than absorbed into flexible surrogates.
Our motivation. We elevate the slow process first-class target: a stimulas- low- dimensional dynamical law that governs across-session reconfiguration. This requires (a) an explicit dual-timescale factorization to avoid entanglement, (b) stracture/priors that restrict the hypothesis class of slow evolution, and (c) evaluation focused on unseen protocols.
2.2 OBTAINING PLASTICITY RULES: LOCAL/SHORT-TERM UPDATES VS. LONG-TERM RECONFIGURATION
A parallel line of work focuses on plasticity rules inference and derivation. Bredenberg et al. (2020) derived a task-dependent Hebbian plasticity rule to enable the learning of efficient, task speciffc representations under resource and noise constraints. Kepple et al. (2022) explore how curriculum learning can be used to uncover the learning principles underlying both artificial neural networks and biological brains. -radient-based parameterizations fit local synaptic update functions (Mehta) ict al. 2024). Meta-learning discovers self-organizing rules that stabilize sequence generation under synaptic turnover (Bell et al. 2024). These approaches illuminate short-term local adjustments, often emphasizing immediate pre/post activity or reward signals.
Limitation relative to long-term plasticity. Long-term reconfiguration unfolds over days-weeks and recruits homeostatic control, synaptic scaling, and neuromodulatory pathways (Turrigiano & Nelson 2004 Moulin et al. 2022; Marder 2012 Liu al. 2021). Prior methods typically: (i) focus on microscopic updates rather than the mesoscopic evolution of circuit motifs; (ii) do not ex- plicitly condition the rule on external stimulation protocols, limiting counterfactual design; (iii) treat cumulative effects as aggregated parameter drift, which makes the governing slow rule unidentifiable and hampers extrapolation.
Our motivation. We recast the objective from "recover local updates" to "learn slow dynamical law that maps stimulation histories to circuit-level coefficients." Doing so (i) aligns the modeling target with the experimental reality of longitudinal interventions, (ii) provides natural handle for stimulation design, and (iii) opens the window to falsifiable rule-level predictions.
2.3 CLASSICAL BIOPHYSICAL RULES: INTERPRETABILITY WITHOUT HORIZON
Biophysical rules such Hebbian learning and STDP (Hebb 1949 Caporale & Dan 2008 Dan &
Poo 2004) are mechanistically interpretable and operate on millisecond-minut scales. However, they struggle to capture circuit-level remodeling and memory consolidation across long horizons (Zenke & Gerstner 2017 Benna & Fusi| 2016 Frankland & Bontempil 2005), precisely where homeostasis and global neuromodulatory control become essential.
Limitation relative to our goal. Their temporal scope and locality make it difficult to predict multi-session consequences of repeated interventions, or to extrapolate across protocols that differ in schedule, intensity, or targeting.
How STEER addresses these gaps: (i) From parameter learning to plasticity law. We model across- sersssn change as dimeniinss,,nal, stimulus-evoked dynamical system, øk+1 go (zt, 24), rather than unstructured parameter drift. This centers the plasticity rule as the primary object of inference. (ii) Identifiable dual timescales. A structured decomposition separates fast
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Long-term neural recordings
Plasticity predictor
Seeslons
Plastialy Enpoder
Weights Inference
Plasticty entedding
Shared Low-rank state space
State
Reconstruction
recurrett connections
scaling factor
State Decoder
mm
Latent Dynamics
Figure 2: STEER Framework. A dual- -ime--imeseaee, identifiable formulation: within- dynam- ies (fast) are generated by structured recurrent connectivity; across-session evolution (slow) follows a stimulus-conditioned latent law that we infer the plasticity rule.
within-session responses from slow across-session evolution, augmented with causal/structural reg- ularizers to promote identifiability of fe. (iii) Morif-level interpretability. Reading plasticity embeddings as recurrent motif scales provides mesoscopic, circuit-level summaries consistent with biological priors (e.g., E/I structure, neuromodulation)
METHOD
Inspired by neuroscience domain knowledge that high-dimensional neural recordings often admit low-dimensional latent structure (Valente et al. 2022 yet these structures themselves drift slowly
across sessions during learning (Pellegrino et al. 2023 or long-term stimulation (Borra et al.) 2024).
we introduce STEER, a hierarchical model that parameterizes the fast recurrent dynamics within
each session, the slowly varying, indsxed connectivity governed by learnable slow law. To this end, given D ordered sessions with length-7 recordings (yf.r) the goal STEER is to use fast within-session dynamics to identify the slow, stimulus-evoked evolution that governs plasticity.
3.1 TIME- VARYING CONNECTIVITY VIA STRUCTURED LOW-RANK MOTIFS
We model N neural recording units across D sessions. Here, session denotes single contiguous block of population recording from the same preparation, over which we assume the recurrent con- nectivity is stationary, while is allowed to change across sessions. For session & E {l,..., D), W E R N X is the recurrent connectivity used by the fast dynamics. Stacking [W*] along the session axis yields tensor W W which we represent with rank-R CP decomposition,
R
W
Σ 1=1
a
D, C.E.D
(1)
Let A |a,, |ai,.... ...,,... an). bi..... bRj. and = (ci, Ci....., .......,c cnl DxR with session coefficients c = =C (ct,
Then W Σn ct b,, l
Identifiability structure. To reduce CP scaling/sign indeterminacy, we (i) constrain || ||
| ll2 and absorb scales into cf: (ii) penalize orthogonality with A AAA 1 +
B B
i and (iii) optionally impose Dale's principle (E/I structure) using fixed sign mask S E
{-1,0, NxN, W =S@ E ct a, which preserves prescribed signs.

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3.2 FAST TIMESCALE: LOW-RANK RNN FOR WITHIN- SESSION DYNAMICS
To uncover the low-dimensional dynamics underlying the complex long-term neural recordings, we introduce low-rank RNN the core model to capture the shared latent state space. We encode the session-wise neural recordings y into the shared low-rank latent state space h", where the latent state evolves as follows:
d (t) = h +
(E a. (th + Win (t).
(2)
Here, т is time constant, (b()) is element-wise nonlinearity, u (* is the time-varying input for session k, and Win in is the input matrix. A linear projector is used to decode the latent state back the observed space. During the implementation, we discretize equation 2 with Euler method.
3.3 SLOW TIMESCALE: STIMULUS EVOKEE EVOLUTION A LATENT LAW
Session-level plasticity embedding. We summarize each session by plasticity embedding z R²,, obtained from data (and optionally inputs) via
PlasticityEncoder (y uf.r),
(3)
where the encoder can be permutation-awe -ware across trials.
Stimulus-condittoone slow law. Across sessions, z follows discrete time residual evolution conditioned on a protocol summary üt = agg( (uf.r):
z k+1 go (z" ut) = (Wo o(z²) - zz B, a² + +b.)
(4)
From embedding to motif scales. readout
Plasticity modulates motif strengths via a linear (or sparse)
ch Mz b
W Σ a,b
(5)
which is analogous neuromodulatory gain control. Sparsity on M promotes interpretability (few embeddings per motif).
3.4 TRAINING OBJECTIVE AND OPTIMIZATION
We jointly learn fast and slow processes with a composite objective:
D
D-1
lyy
Aslow
Σ
b+1 go(z" u
A=
Ciast (within-session multi-step prediction)
Calow (across-session law consistency)
+ Asmooth
|
c
(6)
Lamooth (motif-scale smoothness)
where is the right-hand side of equation For Crsst we use H-step rollouts (teacher forcing at the first step, then fully autoregressive rollouts thereafter) to evaluate multi-step predictivity rather than one-step error, which better reflects dynamical fidelity over long horizons (Kramer et al. 2022] Brenner et al.) 2022 2024). The consistency term Lslow ensures the model captures the evolving plasticity embedding between sessions. The smoothness term Lamooth enforces gradual, continuous changes of motif scales between sessions. Since our framework has two timescales, we must choose how many steps into the future the model self- predicts the plasticity embedding before re-anchoring to data. This choice explicitly sets the amount of song-timescale teacher forcing and needed as model forecasts across sessions.
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Counterfactual protocols and OOD evaluation. Given a novel stimulation protocol sequence {ū*), we compute i", iterate the slow law z = -0 ü*), obtain čh via equation assemble W and and simulate equation 2 to predict neural outcomes. This separation explicitly tests whether the learned rule (not merely within-session responses) governs across-session reconfiguration and supports design of unseen protocols. To assess out-of-distribution behavior in a realistic setting, we adopt a forward-in-tin interd----ime evaluation protocol. Models are trained on the early portion of the long-ten recording (first 60% of sessions) and tested on the later portion (last 40%). This temporal holdout mimics deployment where inference is made on future data and naturally captures distribution shifts over long timescales (e.g., shifts in stimulation/neural activity statistics).
EXPERIMENTS
To validate the STEER framework, we conducted series of experiments on synthetic datasets and real neural dataset. We generated synthetic data with ground truth using three standard benchmarks:
the Lorenz dynamical system (Lorenz 1963). the BCM plasticity model (Bienenstock et al. 1982), and task-learning setting with adaptively optimized external stimulation (Borra et al. (024). For synthetic datasets, we sampled multivariate time series by varying the underlying ODE param- eters. For the real data, used longitudinal neural recordings from Parkinsonian rats receiving closed-loop DBS (Wang et al., 2024). We compared our method with other meta-learnin leatnnl based black-box dynamical models, hierarchical PLRNN (Brenner et al. 2025 and a meta-dynamical state space model (MD-SSM) (Vermani et al.) 2025) on synthetic datasets. We compared against a hierarchical PLRNN to test whether low-rank inference can match model that directly infers a full-rank weight matrix, because the original framework does not generalize to our setting, we use an adapted version (implementation details in Appendix A.3). We also compared against MD- SSM, which makes similar low-rank assumption, to assess whether enforcing a aassion-invarians latent motif is important for recovering the plasticity rule, and whether our model can achieve both competitive fast-timescale accuracy and accurate slow-timescal forecasts (Appendix(AA A.3
4.1 LORENZ SYSTEMS WITH PARAMETER EVOLUTION
As classical chaotic system, the Lorenz system (equation 7 exhibits complex nonlinear dynamics and is highly sensitive to initial conditions (butterily effect), making it an ideal model for studying complex system dynamics. In this experiment, we aimed to model plausible dynamic process that mirrors what is observed in biological neural networks, where synaptic connections can be strengthened weakened. Therefore, predefined plasticity rules that simulate the stimulus- evoked evolution of system parameters (Fig, Blb, which are expressed as equation g
ayy x),
Fm
0 maax
-Ommax max Dinit) emm440 m/40 In(m 10)
(8)
x( 2) y,
Pm
Pinit
m
m +
zy Bz.
3m
Aiit +
40
20
where Pinit. = 10, o max = 15, Pinit = 48, Biit 3.5. The neural trajectories of the Lorenz systems were generated with fixed initial state tributional shifts are solely attributable to the induced parameter changes, rather than initialization effects or added noise. The parameter ranges were selected cover various dynamic regimes, each characterized by distinct attractor structures such limit cycles and chaotic behaviors. For each set of parameters, time series were generated with a maximum length of Tmaz = 10000. In total, we generated Lorenz dynamics for 100 parameter settings (ams Pms Bm) sampled along the schedule (with parameters held fixed within each system).
[1, 0, 20), ensuring that observed dis-
To optimize model performance, we first searched for the optimal parameters to determine the rank of the low-rank architecture. Since the three parameters showed no obvious correlation, we required rank-3 low-rank model for accurate approximation (Fig. 3p). This choice captures cross-system variations more effectively and improves predictive performance. The inferred motif scales are latent and therefore lack direct one-to-one mapping the ground-truth plasticity rule (Fig. (D). We

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a
Predefined plasticity rule
b
Predictive performance
e Predicting trajectory at two systems
training
inference
1.99
system 61
40
Metricss
5.0
EV-0.90 961
system
Rank
Similarity analysis
DSA+0.169
0.19
system 100
STEER
-42
DSA (L)
0.13
FTESH
C3
48
49
EV-0.974
25
0.4
0.6
0.8
1.0
system
EV(T)
Figure 3: STEER uncovers the parameter evolution in Lorenz system. (a) Plasticity rules for system parameter evolution. (b) Rank selection for a low-rank model, with rank-3 providing optimal approximation. (c) Predicted implicit scaling factor and similarity to the true plasticity rule (DSA = O. 169). (d) Similarity analysis among STEER and other baseline models (MD-SSM and hierarchical- PLRNNN,, with STEER having the lowest DSA and best EV ("w." and "w.o." denote with and without forecasting, respectively). (e) Prediction results for unseen data (systems 61 and 100) with high explained variance (EV 0.961 and EV 0.974).
quantified their correspondence using Dynamical Similarity Analysis (DSA) (Ostrow et al. 20233 yielding score of 16...9...666 Meanwhile, we compare our framework with two baselines (MD-SSM and Hicrarchical PLRNN) on both predictive performance (larger EV means better predictive capability) and dynamical similarity (lower DSA means better alignment). Fig. 31 indicates that STEER has better EV and lower DSA session- wise prediction. To further validate the model's performance, we visualized the prediction results on unseen data (Fig.3p). Both systems 61 and 100 exhibited high explained variance scores (EV 0.961 and EV 0.974, respectively), demonstrating the model's robust ability to predict unseen dynamics.
4.2 STIMULUS-EVOKER PLASTICITY IN BCM
To evaluate the performance of plasticity rule inference, we also simulated the dual-timescale dy- namies via BCM rule as controlled benchmark (Fig.44-) Appendix Fig. []. In this setting, the plasticity mechanism and stimulus-evoked modular organization are known, enabling direct assess- ment of the inferred network reconfiguration across sessions. We modeled the stimulus-evoked modifications of the recurrent weights across sessions through BCM plasticity rule (Fig. 4p). For session
W k++ Proje/1 ( hh 7/ (h+ + (h -0)) (k*)T),
(9)
where @ denotes element- multiplication. h is neuronal activities and 6 is a neuron -pprof- sliding threshold that sets the poteniiation-depression boundary, and itself adapts slowly across sessions (Fig. 4p) We preserved the E/I partition in Wet by fixing outgoing-weige weight signs (E > 0, S 0). For session k; the neuronal dynamics were governed by a leaky-integrator recurrent network with fixed W hihh and 8h (details in Appendix A.
We first compared STEER with MD-SSM variants. All models achieved similarly high within- session predictive accuracy across sessions (EV > 0.9; Fig. fi), with significant short-term differences once capacity is matched (t=0.4221, p>>0.05). In contrast, STEER more accurately re- covered stimulus-evoked cross-sessiee changes in recurrent connectivity: the inferred || |W] [|2 closely followed the ground truth, whereas MD-SSM systematically deviated (Fig. 4b. This suggests that treating each session effectively independent, only meta-learning trial-level variability, discards
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Ground truth dynamics of recurrent connections W &
Ground-truth dynamics of aeross-session thresholds
Proley (Mh. (# (e ()) (R))
ghri (1- sy (Tey
Stann Thrsshoke
esssion
Session 300
Session 1000
4002
250
302
150
Sess ion
Presyrapti
Traning
E Evllution of the inferred W L2-norm Mean across seeds
Predictive performance
Correlation between PCs and inferred motif scales
Std. across seeds
0:7-
14
18
[Mr algrmeen
2.2
0.8
FLAA
G
0.81
0.-
Session
320 600
500
-03
60
1.4
0.0e
Ground truth
STEER w. foracasting sessions
Fallvank ND.S51
PC1
STEER w.o. lotecasting
Low-ank MD-SSM
Figure 4: STEER captures stimulus-evoked plasticity in BCM. (a & b) We generate data via BCM-driven across-session simulation, subsequently fitting models with a 600/400 train-test split. (c) Inferred |2 evolution over sessions alignment with the ground truth. Left: Normalized L2- norm of inferred vs. true W over sessions (mean std. across random seeds). Right: [[ ||2 alignment (correlation between the session-by-session trajectories of the inferred and ground-truth [[[l2), comparing STEER with and without forecasting to adapted hierarchical-PL.RY. predictive performance (EV) on all sessions (mean + std. across random seeds), showing that STEER achieves competitive fast-timescale prediction accuracy. (e) STEER inferred motif scales significantly correlates with the ground-truth sliding thresholds 9 principal components.
full- and
low-rank MD-SSM baselines. (d)
inter-session structure that is essential for identifying slow network reconfiguration. By explicitly modeling coupled fast and slow timescales through shared latent motif, STEER preserves this information and yields a more faithful reconstruction of plasticity. Consistently, correlations be- tween e and the leading PCs of the ground-truth thresholds (Fig, 4 show that captures the same low-dimensional homeostatic manifold. Thus, с serves as an order parameter for stimulus- plasticity, supporting timescale disentanglement.
To further test slow-timescale identifiability, we evaluated a STEER variant that forecasts slow parameters sessions ahead, alongside its non-forecast counterpart and the full-rank hicrarchical PLRNN baselines. Even in this harder setting, STEER preserved strong fast-timescale accuracy and closely tracked session-to-seseee
changes in W. Moreover, the non-forecast STEER variant did not differ significantly in either slow- fast-timescal performance from the full-rank hierarchi- cal model. However, the inferred recurrent weights (Appendix Fig. 2 showed that the hicrarchical PLRNN baseline yielded much weaker modular structure, whereas the low-rank models (STEER and MD-SSM) recovered modules that closely match the ground- -outh organization. This agreement highlights the advantage of a low-rank structure for recovering dimuulus-epeendee circuit motifs shaping population dynamics.
4.3 TASK LEARNING THROUGH STIMULLATION PLASTICITY
We construct a synthetic benchmark (Borra et al. 2024) to test whether STEER can recover stimulation-evokel reconfiguration of recurrent connectivity from population dynamics when the external drive is itself optimized in closed loop (see Appendix [..]). This setting contrasts with our BCM dataset, where synaptic updates follow a fixed plasticity rule under prespecified inputs: here, changes in W(t) arise from adaptive control, and the resulting neural trajectories are shaped by learned stimulus policy rather than by hand-designed input ensemble. A recurrent rate net- work with time-varying connectivity W(t) is driven by control inputs u(t), which are adaptively optimized across learning cycles k 0,..., K to steer W toward a ring-attracton target Wtarget
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Cycle-wise W cyde 856
Predictiee performance
cycle 250
cycle 650
eyde 1100
cydee 1400
0.95
Ground buth
0.50
01 0.65
0.75
0.70
foncanding
Similarit of eydle-wise W
Similanty
0.2
0.0
(one
Low-rant ND-S3V
foncced
Fullcanc MD S0-55MM
STEERW W firecssingg
Figure 5: STEER captures stimulus-evoked plasticity in cycle-wise task learning. (a) Cycle- wise weight dynamics during task learning and inference. (b) Predictive performance across various methods. (c) Similarity between cycle-wise weights in model inference and ground truth. STEER with forecasting demonstrates exceptional performance in prediction accuracy and weight similarity.
We first searched for the optimal rank of the low-rank structure of STEER. A rank-7 STEER model was necessary to achieve an accurate approximation and high similarity (Appendix Fig. 11. The STEER model effectively captures stimulus-evoked plasticity throughout cycle-wise task learning. As shown in Fig. 5h. the cycle-wise weight dynamics during task learning and inference highlight that STEER capable of inferring module-like weights. In Fig. 5pp and c, the predictive performance in fast-timescale neural dynamics and slow-timescake similarity between cycle-wise weights in the model inference and the ground truth emphasize STEER's exceptional performance in prediction accuracy and replicating the expected weight dynamics.
4.4 PD-DBS LONGITUDINAL NEURAL DATASET
We further validated the proposed framework in the publicly available longitudinal electrophysiol- ogy and behavior dataset of Parkinson's 's disease (PD) rats undergoing closed-loop deep brain stim- ulation (Wang et al. 2024). The original dataset study validated the efficacy of DBS treatment in alleviating the symptoms of PD. This dataset is well-suited our study of inferring brain plasticity rules based on long-ters term neural activity by quantifying dynamical and plastic changes across differ- cohorts. The dataset comprises three cohorts (Sham, PD without treatment, and PD with DBS) and was recorded from week 2 to week (four weekly time points, Fig. (6)) Signals were collected from layer V of primary motor cortex, providing raw wideband along with derived spikes and local field potentials (LFPs), accompanied by open-field behavior videos and quantitative measures (see Appendix A.I. The PD-DBS cohort received closed-loop STN-DBS on days 29 33 with simul- taneous neural recording, enabling comparisons within subjects before, during and after stimulation (Fig. (4).
After modeling, we revealed that a strong alignment between the inferred motif scales and func- tional connectivity (FC), indicating high level of consistency between plasticity factors and FC (Fig. G.. see Appendix A.4). Further, we assessed the cosine similarity between Ac and A FC, showing that all groups had similar cosine similarity between Ac and A FC across weeks, support- ing that Ac is closely aligned with the intrinsic functional connectivity (Fig. (0), see Appendix A.4). We also present the changes in the magnitude of Ac (ACmug) across different groups (PD, PD-DBS, Sham) over the DBS intervention interval (wa, - Ws), revealing that the PD-DBS group exhibited significantly higher Acmay values compared to the PD and Sham groups (Fig. (o).. This suggests that DBS stimulation effectively promotes the improvement of neural activity in PD rats. Overall, these

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Experiment pipeline
Long Long-lerm stimulation induces plasticity to alleviate PD
f eleckical stimuation recomtng
Siaan
stote
Healthy state
Wook11 (Day
Veeeid (Dey 14)
Veeia (Day 211
Weokd (Day 28)
Wess5 (Dey 34)
Alignment between inferred and FC
Similarity between åc and &FC
Amaag at difffrenn groups
algned
1.0
PDJDBS
sham
PC2
PC1
0.0
PD
PD-DBS
belween weeks
Figure 6: STEER interprets long-term DBS effect in PD rats. (a) Longitudinal experimental pipeline: repeated recordings weeks 2-5; PD-DBS receives closed-loop STN-DBS days 29-33. (b) Long-term stimulation induces functional connectivity plasticity to alleviate the symp- of PD. (c) Alignment between FC trajectories (blue, PCA space) and STEER motif scales e (red) across weeks. (d) Week-to-week cosine similarity between Ac and AFC. (c) Magnitude of plasticity increment (Acmug || DBS shows larger changes, consistent with stimulation- evoked slow update.
findings suggest that DBS is associated with a distinct plasticity-related neural signature, consistent with modulation of neural network plasticity.
5 DISCUSSION AND CONCLUSION
In this study, we presented the STEER framework to learn plasticity rules from neural recordings under long-term stimulation. STEER formalizes long-term plasticity as stimulus-conditioned la- tent dynamical system, 2k+1 = ge(2,, üx), and enforces an identifiable separation between fast within-session responses and slow network reconfiguration.
Methodological impact. By leveraging the dual-timescale nature of neural dynamics, STEER effectively separates fast, within-session neural responses from slower, session-wise plasticity changes. Across synthetic Lorenz dynamics, BCM-driven networks, task-learning benchmark and longitudinal PD-DBS recordings, this formulation consistently turns "uusstuctured variability" into protocol-
mnotocol-indexed indexed trajectories in low-dimensional plasticity space, yielding interpretable motif- level readouts and out-of-distribution predictions under unseen stimulation schedules.
Limitations and opportunities. First, our presented behavioral analyses are limited. Linking c-space trajectories to subject-level outcomes (e.g., motor scores) via mixed-effects models and prospective perturbations will be crucial. Second, richer biological structure (e.g., cell-type-speciee gains, homeostatic set points, neuromodulatoon neuromodulltorrdeeeeddee gating) could be embedded directly into ge and the c-> readout. Third, safety-aware design requires constraints on reachable sets of z and (e.g.. contractive regions and Lyapunov margins). Finally, broader validation across modalities (optogenetics, TES, TMS), species, and disease states will test the portability of the learned rules.
By making the slow rule explicit, STEER turns long-term plasticity from hidden con- found into a falsifiable object. We anticipate two immediate payoffs: (i) mechanistic inference, where lotif-level embeddings act as mesoscopic biomarkers of neuromodulatory control; and (ii) closed-loop design, where protocol parameters are optimized in the space of laws rather than tra- jectories. We believe this shift, from parameter drift to learned dynamical law, offers principled foundation for next-generation neurostimulation that is both interpretable and adaptive.
Outlook.
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ACKNOWLEDGMENTS
This work was supported by the National Natural Science Foundation of China (62472206), National Key R&D Program of China (2055YF34410000) Shenzhen Science and Technology Innovation TTX2003122109040555
111120200813111000000 GuangDong Basic and Applied Basic Research Foundation (2025A151501 1645), Guangdong Provincial Key Laboratory of Advanced Biomaterials (2022B12120100))
Committee
SYY2002001810021133
and the open research fund of the Guangdong Provincial Key Laboratory of Mathematical and Neural Dynamical Sys- tems, the Center for Computational Science and Engineering at Southern University of Science and Technology.
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A APPENDIX
A.1 DATASET DETAILS
Lorenz system. The famous 3D Lorenz attractor (Lorenz 1963) is widely used benchmark in data-driven system research. Its governing equations are
á = (y - 2).
ý = x(p 2) y: : = xy Bz.
BCM plasticity rule. The BCM rule employs sliding threshold mechanism dynamically adjust the plasticity gate. It achieves homeostatic regulation based on past activity levels. The update of the threshold 0 specified by:
k++ = (1 - (1)--))00 k* + 178 (h
(2)
where 02 denotes element-wise squaring, f1 E (0,1) 1) is the smoothing rate and h is the mean activity vector in session k:
h *
Σ
h
(3)
T t=0
where T denotes the session length. The neuron activity in each session k: is defined by
h h
5
(- h W tanh(h,) b W in u
(4)
Th
where utt) is two-channel, piecewise-constant one-hot stimulus alternating across channels (Ap- pendix Fig. [ We discretized the dynamics with forward- -uler step of size (set to 5 = 1 in practice) and update the state at each step. This threshold implements a homeostatic control mecha- nism (Abbott & Dayan 2001). The dynamics can be interpreted as follows: when the mean activity h is persistently elevated, 8 increases, thereby raising the plasticity threshold and counteracting runaway potentiation. Conversely, reduced activity drives 0 downward, permitting synaptic strength- ening. This negative feedback loop ensures stability while maintaining sensitivity to changes in input statistics.
Together with the weight update rule in equation 9 the sliding threshold allows the model to capture both activity-dependent potentiation/deeprssionn and homeostatic regulation across sessions. In simulations, this mechanism prevents weights from saturating and enables the emergence of stable modular connectivity patterns under repeated stimulation. Because the BCM rule Hebbian in nature. coactivation of units (Appendix Fig. T systematically strengthens their mutual connections (Appendix Fig. 2}. leading to a gradual increase the norm of the recurrent weight matrix over sessions.
Task Learning through Stimulation-Indueee Plasticity. To construct a dataset in which recur- rent connectivity evolves under biologically motivated plasticity, we adopt the rate-based network and plasticity model of (Borra al. 2024). The network contains N neurons with firing rates r(t) (f)) and a time-dependent recurrent connectivity matrix W(t) = (W) Neu- ral activity obeys the standard rate dynamics
(
dr: Tn dt
(t)= + Σ (w + +u(
(5)
where T, is the membrane time constant, (t(t) ) an externally applied control input on neuron i, and ((.) is a sigmoidal onput-output transfer function saturating a maximal firing rate Tmax- Similarly, the control signals (t) are chosen in the same form as in the original work, and are re-optimized across learning cycles to steer connectivity towards a predefined target structure.
Synaptic plasticity is modeled as a Hebbian covariance rule with homeostatic feedback and soft bounds on synaptic amplitudes. The dynamics of each synaptic weight (i)(() is given by
dW,, si Tg dt
(t) = m((s) (r)(t) ((a;)) (t) 3, (+,(())((((t))) (o(a,)²)
(6)
- (gssg((W,, (1) W),
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M(O)
Stm.
0.68
Neuron Activiy h(O)
0.41
Stm.
Exc.
0.34
-0.13
0.41
Inh.
.011
Time
Appendix Fig. 1: Stimulus selectivity in simulated neuronal populations. Simulated neuronal populations show clear stimulus selectivity, with neurons responding preferentially to specific stim- uli and exhibiting structured co-activation patterns.
Recument connections
Chonge recurren connections between sessions
(W;)
seo)
(Wasso))
Wase W'
Wioze
0.999
Ground truth
0.50
STEER
0.55
Hierarchical PLRNN
0.156
000
Low-ann MD-S5M
40022
6.01
Full-rank MD-SSM
6.00
Appendix Fig. Recurrent connection inference. (a) Recurrent weight matrices W three time points (sessions 1, 500, and 1000) for the ground truth, STEER, full-rank MD-SSM, and low-rank MD-SSM. (b) Corresponding changes in connectivity between sessions ())).
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where aj [E,1) denotes the type (excitator/inhibitorr) of presynaptic neuron j. 77(a)) is the learning rate for synapses from that neuron type, and f(a;) and fo(ay) are activity thresholds that stabilize the postsynaptic firing rate. The function
x 2 0 x2
h(x)
0, x < 0,
implements soft clipping of synaptic strengths at magnitude W, while 31 and B2 control the strengths of the two homeostatic terms. We work in the slow-plasticity regime Ts > S0 that firing rates are (quasi-)stationss
on the timescale over which ((1) changes.
Following (Borra et al. 2024), we consider structural target in which the excitatory and inhibitory neurons are arranged on concentric rings and the target connectivity Wilget supports a continuous ring attractor. Training aims reshape the initial random connectivity W towards Wilggt by optimizing the control inputs (u²(()) over a sequence of learning cycles k 0, 1,...,,. K. In each cycle, the cost
Lusk (W)+ = Σ War.ay (ii W Wr
(8)
penalizes deviations from the ring-attres iingaatrrcttor connectivity, with type-dependent weights Wa,.a, that balance contributions from all four connection classes (E-EE E->I, I-E,, I->I). Given an estimate
of the current connectivity W(A), we numerically optimize the control stimulation (1) over fixed control period E Tanl to approximately minimize Lusk (W((11)) under the plasticity
dynamics in equation 6 Applying u*t generates a trajectory of firing rates p())()) (t) and an updated
connectivity matrix W b+1)
Following (Borra et al., 2024), we model the effect of a fixed control stimulation u(t) applied for
a duration Tent starting from connectivity W(k as plasticity-inducee update AW(u, Tan, W())), obtained by integrating the rate and plasticity dynamics in equations and 6 The optimal control for learning cycle l then defined as
u & arg min Linkk (W()) + AW(u, Ton, W()))
(9)
which we solve numerically by gradient descent in the space of control signals, while keeping the Hebbian plasticity rule fixed.
To generate dataset, we simulate the coupled fast-slow dynamics starting from an initially ran- dom connectivity ( while iteratively optimizing the stimulation controls across learning cycles. For each cycle k, we record the evoked activity trajectory {t) under the optimized proto-
col = together with the pre- and post-cycle connectivity matrices W(A) and ((++1) 1) as ground
truths across sessions, which progressively approach the ring-attre attraccoo structure (Fig. 5 our experiments, we use (t)) and the optimized u as input to STEER. directly testing whether stimulation-drive connectivity reorganization can be inferred from neural activity alone.
PD-DBS Rat Dataset. We use the public dataset "A longitudinal electrophyysoological and behav- ior dataset for PD rat in response to deep brain stimulation" " (Wang al., 2024), released on the GIN (G-Node) repository at fitttps:: / doi. crq/ 10. 122122755 12w11121111 q node 1zvob5 The experi- mental protocol spanned five weeks and included three groups of animals: sham, PD, and PD-DBB.
Sham group. Rats underwent the same surgical implantation of stimulation and recording electrodes as other groups, but instead of the neurotoxin, they received a vehicle injection (ascorbic acid in saline) into the medial forebrain bundle (MFB; AP ~ -.,4 mm, ML ~ 1.2 mm, DV -7.8 mm). Recording electrodes were implanted only in the left primary motor cortex M1 (AP +2.5 mm, ML +3.0 mm, DV -1.6 mm), and stimulation electrodes in left STN (AP -3.6 mm, ML -2.6 mm. DV -7.6 mm). Sham rats never developed hemi- parkinsonism and did not receive DBS, serving as surgical and handling controls.
PD group. Rats received unilateral 6-hydroxydopamine (6-OHDA; 4 pl, 5 jug/ μd) injection into the left MFB to induce hemi-Parte Homi-Parkinsonism. confirmed on Day 6 with apomorphine- rotation screening (net >21 contralateral turns/30 min). Both stimulation and bilateral
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MI recording electrodes were implanted in the DBS group, but no DBS applied. These rats allowed assessment of electrophysiological and behavioral alterations due to PD pathology without stimulation.
PD-DBS group. Rats received the same unilateral 6-OHDA lesion and electrode implan- tation as the PD group. In addition, during Days 29-33 (Weeks 4-5), they underwent closed-loop STN-DBS (130 Hz. 90 ps, current individualized by motor threshold) with si- multaneous recording. The closed-loop controller used spectral features of MI LFPs (2- 50 Hz) to trigger stimulation. This group provided the longitudinal neural and behavioral data under chronic adaptive DBS.
Across all groups, electrodes were secured with dental acrylic, and placement was histologically verified post-mortem. Longitudinal data were organized into three daily recording episodes (morn- ing/afternoon/nighh)) with synchronized open-field behavior. In our study, for consistent temporal comparison of neural dynamics and plasticity across cohorts, we selected only the morning episodes (episode 1) from Weeks 2-5. For the PD-DBS group, all electrophysiology during the DBS inter- vention window (Days 29-33) was excluded. Within each selected episode, recordings were further segmented by behavioral state (sleep/wake/walk)) and we restricted our analyses to the walk seg- ments, which most clearly differentiate PD and non-PD animals. Otherwise, we adhered to the dataset's ss native channel mapping and indexing conventions.
A.2 SCALING EFFECT FOR NETWORK CONNECTIONS IN STEER
To simulate the phenomenon of neural plasticity induced by long-term stimulation, we infer the motif scales c from neural signals, akin to neuromodulatory signals, which then influences the network's connectivity W R ding act as adaptive regulators of synaptic strength, where they either strengthen or weaken specific recurrent connections within the network. This dynamic modulation allows for the systematic rein- forcement or weakening of neural pathways, which in turn drives the plasticity effects in the neural system. Through this mechanism, the connectivity weights evolve in response to the slow, long-term changes in the neural embedding, reflecting the neural system's 's capacity for learning, adaptation, and memory formation. The evolving plasticity factors thus enable a flexible and sontext-sensitive adap- tation of the network's structure over time, facilitating the emergence of complex neural behaviors in response to environmental or internal stimuli.
-r=1 caa,b," The motif scales inferred from plasticity embed-
A.3 BASELINE IMPLEMENTATION DETAILS
The original hierarchical PLRNN learns session-specific parameters di- rectly rather than inferring session embeddings from the observed signals, and therefore does not generalize to unseen sessions in setting. To make it comparable to STEER, we adapted its frame- work so that session embeddings are inferred from the observations in the same way as in our model, and these embeddings are then used to generate session-specifie full-rank weights by projection.
Hierarchical PLRNN.
MD-SSM. Structurally, MD-SSM augments a session-specific low-rank transition with a shared full-rank recurrent matrix Win E N*N.
((a)
(+)) Whh-
(10)
Fall-rank MD-SSM
resulting in at least 22 additional parameters compared to STEER, which uses session embeddings only to modulate coefficients of a shared low-rank tensor. To control for learaalle parameter count, we also evaluated purely low-rank variant obtained by removing W Wh. while keeping all other modeling choices identical.
W (e)
(()
(11)
Low rank MD-SSM
This variant isolates the effect of the shared full-rank recurrent dynamics from the meta-dynamical, session-specific components.
A.4 PD-RAT ANALYSIS
Alignment between model plasticity and FC. After modeling, we observed a strong directional alignment between the inferred plasticity embedding e and empirical functional connectivity (FC)
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across weeks (Fig. (c). Concretely, we embedded week-wise FC (Fisher-2 upper triangles) and (subject-wise z-scored across weeks) into 3D using PCA to obtain trajectories {f,] and {ct)?= for each subject. Because the PCA for FC and was performed separately, the two 3D coordinate systems are only defined up to an arbitrary rotation/reflection and global scale, making their trajecto- ries not directly comparable. To remove these nuisance degrees of freedom and focus trajectory shape, we aligned the plasticity trajectory to the FC trajectory using a full Procrustes similarity transform.
Let F E IR and C E RR'x denote the stacked PCA coordinates with rows F. = fl] and C c, For each subject we solved the standard Procrustes problem
(15 R2 8) arg PI.AEERS, R R=1, 550
min
F -11n3 &c [
(12)
My and Pe are the row means of F and C, respectively; R is obtained from the SVD of (C ) - 1uf): and = || } |F/||C - 1u ll.. The aligned plasticity coordinates are then defined as:
è s R(ce -Me) + ps..
(13)
Directional consistency was quantified by the absolute cosine similarity between consecutive dis- placement vectors of the FC and aligned plasticity trajectories,
(Af.) (Aē)
E 10.
AG. - fet1 - fr Ac, = č2+1 - čt-
(14)
| 2 A
We take the absolute value because Procrustes alignment is defined up to reflection, which can flip the sign of displacement vectors without changing trajectory shape.
Change magnitudes and their relation. To relate plasticity changes to FC changes, we computed week-to-week change vectors Af, f++1 f, and Ac, Cc+1 čt, and summarized their mag- nitudes by & norms, AFC (1) = || la and ACmu (1) = |Ar Across weeks, all groups showed similar trends in the relation between ACmag and A FC (Fig. 6d), indicating that the learned plasticity closely tracks intrinsic FC dynamics.
Group comparison (DBS intervention interval). We compared the week-to-week plasticity change magnitude ACmag over the DBS intervention interval (w4 + Ws), corresponding to the period during which DBS was applied. The PD-DBS group showed significantly larger Acmag than both the PD and Sham groups (two-sample I-tests, p < 0.05; Fig. (e). consistent with stronger reorganization of the latent plasticity state during DBS.
A.5 SHUFFLE CONTROLS: TESTING RELIABILITY AND IDENTIFIABILITY PLASTICITY INFERENCE.
We introduce two complementary shuffle experiments because there are two distinct failure modes when claiming plasticity inference. One is reliablityynneerprtbblltt issue: a high plasticity score could arise from static session-to-session heterogeneity even there no coherent slow drift. The other is an identifiability issue: the model class might impose smooth trends through its in- ductive bias, making it unclear whether the recovered slow trajectory is truly learned from the data.
A.5.1 EVALUATION STAGE SHUFFLE (NULL. DISTRIBUTION FOR THE PLASTICITY METRICS).
We keep the model trained on the true session order fixed, then generate many session-order-shuufile data and recompute exactly the same plasticity metrics. Because shuffling breaks coherent slow evolution while preserving within-session statistics, this provides direct chance baseline for our plasticity readouts.
Lorenz systems with parameter evolution. Keeping the same trained model but evaluating it on data with a randomly shuffled session order (Appendix Fig. 33) destroys the smooth and monotonic drift structure, yielding highly variable motif scales with no apparent trend, with DSA 0.302,
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which is larger than the smooth structure. Since baseline models lack predictive capability. evaluate their performance using inferred embeddings rather than predicted ones. The resulting dynamical similarity analysis shows that baseline models exhibit higher DSA value compared to STEER (Appendix Fig. 4 and c).
Predefined plasticity rule training inference.
b Predicted embeddings with STEER
Inferred embeddings with STEER (shuffied data)
12.5
48.25
10.0
48
-42
DSA~0.180
-42.50
-42.25
DSA=0.302
-40
B
48.25
system
25
20
200
Appendix Fig. 3: Dynamical Similarity Analysis (shuffled data). (a) The predefined plasticity rule. (b) Predicted embeddings with STEER (no shuffle). (c) Inferred embeddings with STEER (random shuffled data).
a Predefined plasticity rule training
Inferred embeddings with Hierarchical PLRNN
Inferred embeddings MDSSM
inference
0.2232
12.5
27.0
0.2230
10.0
P
46
62
33.2
33.0
DSA=0.15 192
C2
0.13775
0.11800
DSA-0.461
Te-G- 901-D6e->
5.0
7.5
24.0
24.5
C3
25
50 system
300
25
Appendix Fig. Dynamical Similarity Analysis (baseline models). (a) The predefined plasticity rule. (b) Inferred embeddings with Hierarchical PLRNN. (c) Inferred embeddings with MD-SSM.
Stimulas-evoked plasticity via BCM. In this evaluation stage shuffle experiment, the trajectory of the motif scales collapses under this evaluation-time shuffle (as in Appendix Fig. Also, the plasticity-related metrics of original order evaluation significantly exceed metrics in shuffled null distribution (Appendix Fig. 5 The results indicate that the model's predictions rely on the real temporal organization rather than session-wisg wise independent statistics.
PD-DBS Longitudinal Neural Dataset. To test whether the observed alignment between week- to-week changes in plasticity and FC could be explained by trivial factors, we performed within- subject shuffle control. For each rat, we kept the trained model parameters fixed and generated surrogate datasets by randomly permuting the four weeks of data. For each shuffle, the plasticity trajectory c, projected it to 3D via PCA, and mapped it into the FC space using the Pro- crustes transform fitted the original (unshuffled) data. Directional similarity was then quantified using the same cosine-based metric as in the main text. Appendix Fig. 6 summarizes this analysis across groups (PD, PD-DBS, sham). For all three groups, the real data (no shuffle) show substan- tially higher Ac-AFC similarity than the week-shuffled baseline, indicating that the alignment cannot be explained by arbitrary rearrangements of weeks alone.
re-computed
A.5.2 TRAINING STAGE SHUFFLE (RULING OUT "THE MODEL HALLUCINATES DRIFT").
We also train the same architecture on session-order -oae---rr-r---ffeddddded---- data and ask whether it can still repro- duce the ordered slow-trend patterns. If a similar trend emerged despite the absence of chronologi-
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Appendix Fig. 5: Motif scales dynamics depend on the temporal ordering of sessions. (a) Predicted motif scales e (magenta) closely track the directly inferred motif scales (blue) for two example motifs when the model is trained and evaluated on data in the original chronological session order, yielding smooth gradual changes across sessions. (b) Using the same model but evaluating data with shuffled session order disrupts this structure, producing highly variable motif scales across ses- sions. (c) Original-order plasticity metrics significantly exceed the session-shuffled null distribution (t=-134, le for [[]] |l alignment and (=-888 < le 5 for max correlation).
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cal structure, it would indicate that the trend is largely artifact of model bias rather than inferred from temporal continuity. Instead, training on shuffled order removes the ordered slow-timescale trend, supporting that our slow plasticity dynamics are learned from coherent across across-session struc- ture present only in the correctly ordered data. Results in Appendix Tab. [ and Appendix Fig. 7 support the claim that temporal ordering provides essential information (and inductive bias) for capturing genuine cross-session slow dynamics rather than session-wise independent fitting or short- explanations.
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Appendix Tab. 1: Performance of model training and evaluating by original and shuffled session order data in BCM synthetic dataset.
A.6 JOINT TRAINING VS SEQUANTIAL TRAINING OF FAST AND SLOW DYNAMICS
In our model, the "fast" module learns set of cross-session motifs, and the "slow" module captures how these motifs are expressed over longer timescales. Training fast dynamics first and then fitting the slow dynamics in two sequential parameter blocks (fast, then slow), rather than jointly. If the motifs are not held fixed in the second stage, this procedure effectively reduces standard joint optimization and the distinction between "fast" and "already-learned" components becomes moot. A direct comparison of the two training procedures in the BCM experiment is shown Appendix Tab. and Appendix Fig. [8 Across sessions, both approaches produce very similar trajectories of the inferred motif scales, indicating that the learned slow dynamics are largely unaffected by whether fast and slow components optimized jointly or in sequence. Joint training yields slightly smaller discrepancies early and late sessions, but the overall across-session trends and 5-session- forecasts are essentially indistinguishable between the two schemes.
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Appendix Tab. 2: Model performance by different training procedures in BCM synthetic dataset.
A.7 HYPERPARAMETER EVALUATION
Lorenz systems with parameter evolution. We examined the sensitivity of the model to learning rate, tensor rank, hidden dimension, and the regularization parameter Xatow (Aymooth is the same value as )stow). Here, evaluate the predictive performance of model. For the fast time
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scale, we used EV to evaluate the predictive performance. The results in Appendix Fig. 9 indicate that, within a wide range, the choice of these hyperparameters has little impact short-timescale prediction.
Stimulus-evoked plasticity via BCM. We examined the sensitivity of the model learning rate, tensor rank and the regularization parameter Astow (we do not consider Asmooth in this experiment). Here, we evaluate our model on two timescales. For the fidelity of the inferred slow law, we used max correlation between PCs of 0 and inferred motif scales c, and the alignment of the inferred ||]|2 dynamics with the ground truth. For the fast time scale, we used EV to evaluate the pre- dictive performance. The results in Appendix Fig. 10 indicate that, within wide range, the choice of these hyperparameters has little impact on either short-timescale prediction or on the accuracy of inferred plasticity. In contrast, varying the number of forecasting sessions revealed dissoci- ation between short-term prediction and long-term plasticity inference. With longer forecasting horizons, fithin-session stayed high while both the max correlation and W dynamics alignment declined systematically. Thus, extending the forecasting window preserves short-term behavioral performance of the model but compromises its ability to faithfully reconstruct the latent plasticity dynamics.
Task Learning through Stimulation-Induen Plasticity. To assess robustness the task learn- ing dataset, we varied the rank of the shared low-rank latent state space, the learning rate and X. In Appendix Fig. [T] the rank-7 7 model the best one to approximate the task-learning dynamics. The results in Appendix Fig. ITp and e indicate that, within wide range of learning rates, À has little impact on weight similarity.
PD-DBS Longitudinal Neural Dataset. To assess robustness the PD-DBS dataset, we varied the rank, the hidden dimension of the shared low-rank latent state space, and the regularization parameter Astow (Aemooth is the same value \slow), over the ranges shown in Fig. 12 For each configuration, we evaluated within-session predictive performance (EV) and the similarity between inferred coupling changes Ac and observed functional connectivity changes AFC. Across this
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Appendix Fig. 9: Predictive performance across different hyperparameters. (a) Predictive perfor- mance across ranks under different hidden dimensions (rank=3 is stable to explain most of the vari- ance in most of the hidden dimension settings). (b) Predictive performance across under different hidden dimensions at rank=3. (c) Predictive performance across learning rates at rank=3 (A higher hidden dimension has stable predictive performance across learning rates).
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sweep. both metrics remain consistently high and change only modestly, indicating broad plateau of good performance. The hyperparameters used in our main PD-DBS experiments lie well within this stable region. Thus, our conclusions on the PD-DBS data do not rely on fine-tuning: STEER reliably reconstructs neural activity and recovers the relationship between slow coupling changes and functional connectivity across a wide range of settings.
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A.8 IMPLEMENTATION DETAILS
Lorenz dynamics.
Low-rank RNN: RNN (28), rank 3.
Plasticity Predictor: RNN (28)
State Encoder: MLP (28, 28)
State Decoder: Linear Projection (28 +
Plasticity Encoder: MLP (28, 28)
Weight Inference: Linear Projection (28 - 3)
Stimulus-evoked plasticity via BCM.
Low-rank RNN: RNN (50), rank 2.
Plasticity Predictor: RNN (10)
State Encoder: MLP (50,50)
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State Decoder: Linear Projection (50 - 50)
Plasticity Encoder: GRU (50)
Weight Inference: Linear Projection (10 - 2)
Task learning through stimulation-induced plasticity.
Low-rank RNN: RNN (50), rank 7.
Plasticity Predictor: RNN (50)
State Encoder: MLP (50, 50)
State Decoder: Identity Matrix
Plasticity Encoder: MLP (50,50)
Weight Inference: Linear Projection (50 - 7)
PD-DBS.
Low-rank RNN: RNN (200), rank 24.
Plasticity Predictor: RNN (200)
State Encoder: MLP (200, 200)
State Decoder: Linear Projection (200 channel)
Plasticity Encoder: MLP (200, 200)
Weight Inference: Linear Projection (200 - 24)
All networks are trained with the Adam optimizer, and the learning rate is tuned by selecting the best predictive performance. We initialize parameters using Xavier normal, Xavier uniform, and zero- mean Gaussian initializations; remaining nn.Linear weights follow PyTorch's default initialization (with optional orthogonality constraints via GeoTorch).
A.9 USE OF LARGE LANGUAGE MODELS (LLMS)
To enhance the overall quality and presentation of the manuscript, we made limited use large language model (LLM) during the writing process. LLM assisted us in refining wording, improving sentence clarity, and checking grammar and narrative fluency. In addition, provided support in organizing certain sections' logical structure and narrative flow, helping present our research in clearer, more coherent, and more persuasive manner.
It is important to note that the LLM's involvement was strictly confined to textual polishing and structural expression. The study's core ideas, research design, experimental procedures, data anal- ysis. and scientific conclusions were conceived and carried out entirely by the authors. All content of the manuscript, including any suggestions generated by the model, was thoroughly reviewed, edited, and verified by the authors, who take full responsibility for the scientific rigor, accuracy, and originality of the final work.
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Reviews:
[{"review_id": "TqUBX4mE7o", "reviewer": "Reviewer_TgyK", "summary": "This paper introduces STEER, a method to infer the dynamical rules underlying long-term evolution of recurrent connectivity over sessions/hours/days from neural data. The framework learns low-dimensional latent coefficients for the neural dynamics underlying a session, which themselves evolve over slower timescales – thus allowing the inference of plasticity rules explicitly. The framework is trained by jointly optimising for reconstruction fidelity within a session, regularisation for consistency of inferred plasticity rule across sessions, and also regularisation for smoothness. The authors validate their framework on three tasks – two synthetic tasks involving learning a Lorenz system with varying coefficients and synthetic data generated with a specific, known plasticity rule; and one real dataset of neural recordings from rats undergoing DBS treatment for Parkinson's disease. The main claim is that STEER disentangles effectively the within-session dynamics from cross-session dynamics, allowing better extraction and interpretability of the rules/changes underlying cross-session variability.", "questions": "* Minor typo in Fig. 1a: \"Brian\" -> \"Brain\".\n* Why is the evolution of coefficients jagged and not smooth in the Lorenz system plots?\n* What is the difference between Fig. 4 and App. Fig. 1? Why are the results so different qualitatively?\n* Could the authors provide additional information/motivation on the consistency and smoothness terms? How are the coefficient $\\lambda$s tuned?", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "* The framework seems principled in its design and the approach overcomes disadvantages of certain prior meta-learning approaches by not assuming a specific functional form of the underlying learning rule (e.g., not restricted to just variants of Oja's rule).\n* The presentation and figures are mostly clear.\n* The experimental evaluations span both synthetic tasks and real data, which is important for such works.", "weaknesses": "* The use of DSA is good but you only report a single DSA value of 0.63 with no baseline or control/chance value. DSA is a relative metric, as emphasised in the original paper. Thus, it is not possible to know whether a DSA of 0.63 is good without a baseline comparison. Could the authors provide a baseline on shuffled values of the implicit factor, for example, and show that the actual inferred dynamics have a higher DSA score with the true dynamics?\n* In Fig. 4f I am not sure you can claim that you perform comparably to MD-SSM without a statistical significance test. The avg. score for STEER seems lower even when considering error bars. What are the error bars over – sessions or seeds? If the result of a stat. test is not significant then that validates the claim of comparable performance. I would also ask that the plot y axis range be restricted so it's clearer what the difference in performance is.\n* There is no quantification for Fig. 4d where it is claimed that STEER better recovers the $\\Delta \\mathbf{W}$ compared to MD-SSM. As far as I'm concerned, the block structure of both the MD-SSM nor STEER weight changes do not particularly resemble the ground truth weight changes. Could the authors comment on this? The value scales are also quite different and while I understand that exact values/scale may not matter, maybe they need to be normalised so the plots can be compared better?\n* There is no comparison against the meta-learning approaches mentioned in the introduction. While I can understand the difference between these methods, I think it is important to see clear evidence to back up the claim that they cannot model the effects of DBS, for example, and also that STEER can recover plasticity rules as those methods do in short-term settings.\n* Minor nit: in lines 130-131 you cite Bredenberg et al. and Kepple et al. as fitting synaptic plasticity rules through gradient-based optimisation on observational data. While Bredenberg et al. does do this, to my knowledge, they do not do it on real data. Meanwhile, Kepple et al. do not fit parameters for learning rules – they mainly evaluate different learning rules and curricula in an in silico setting on the basis of learning speeds (and classifying between these on the basis of such observations), again not working with real data to my knowledge.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "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": "lHYb39LUEV", "reviewer": "Reviewer_B3ZH", "summary": "This paper introduces STEER, a dual-timescale recurrent framework for inferring long-term neural plasticity rules from longitudinal stimulation data. STEER models fast neural activity within sessions using a low-rank recurrent network, while slow structural changes across sessions are captured by a learnable latent dynamical system driven by stimulation input. Experiments on synthetic benchmarks and Parkinson’s DBS data show that STEER allows disentangle short- and long-term dynamics, recovers biologically interpretable motifs, and generalizes to unseen stimulation protocols.", "questions": "1. Could the authors discuss how choices such as rank, learning rate, or regularization impact performance? What might explain the prediction gap between the ground truth in Fig. 4(a) and the inferred results in Fig. 4(b)?\n\n2. How does the method scale to larger networks or longer time series? Please comment on computational cost and potential limitations for larger-scale simulations.\n\n3. Since the model infers latent dynamics and connectivity from observed activity, how does it avoid learning spurious correlations due to partial observations and noises?\n\n4. More implementation details (e.g., initialization, optimizer settings, runtime) would be helpful for reproducibility.", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "1. **Motivation:** The paper addresses an underexplored area by extending short-term plasticity modeling toward the longer timescales of circuit reorganization. The proposed framework offers a structured, data-driven way to describe how stimulation may gradually reshape network connectivity.\n2. **References:** The related research is carefully reviewed, linking established neuroscience findings on Hebbian and homeostatic mechanisms with recent machine learning approaches for recurrent dynamics and meta-learning. This grounding strengthens the biological and methodological motivation.\n3. **Evaluation:** The method is tested on two synthetic datasets (Lorenz and BCM) and one real Parkinson’s DBS dataset. The experiments provide supporting evidence that the model can capture slow network adaptations and generate interpretable patterns, though further validation would be beneficial.", "weaknesses": "1. **Evaluation:** Results on the BCM and Parkinson’s DBS datasets appear modest, and there is a visible mismatch between the ground truth in Fig. 4(a) and the model output in Fig. 4(b), suggesting partial recovery of the connection change. \n\n2. **Baselines:** Only one baseline (MD-SSM) is evaluated on the BCM simulation. Including other relevant machine learning approaches discussed in the related work would strengthen the empirical comparison.\n\n3. **Benchmark:** The evaluation spans two synthetic and one real dataset. Additional simulations or real neural datasets would better demonstrate the method’s generalization and robustness across experimental settings.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "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": "SQde6wkXRX", "reviewer": "Reviewer_sTJk", "summary": "This paper developed a framework, named Stimulus-Evoked Evolution Recurrent dynamics (STEER), a structured model that separates fast within-session activity from slow across-session adaptation. The method explored how recurrent connectivity evolves under repeated interventions.", "questions": "1. If you learn fast dynamics at first, then learn slow dynamics of these fast ones, will the results be different? Learning jointly requires hyperparameter search such as lambda slow, are the results sensitive to these parameters? \n2. Could the author define delta W? (Figure 4D) Since there are many Ws in the paper.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The paper treats long-horizon plasticity as a latent dynamical law, rather than unstructured parameter drift.\n2. The model separates fast within-session dynamics and slow across-session evolution.", "weaknesses": "1. The dynamical systems have input weights and readouts, which can also encode some information of connectivity. Therefore, it is still unsure if the connectivity recovered by the model is true or believable. (This may have been claimed by the author in the limitation part, but it remains a substantive concern.)\n2. The authors stated that the proposed method enforces an identifiable separation between fast within-session responses and slow network reconfiguration. But there’s no theorem or ablation demonstrating uniqueness of learned dynamics.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "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": "ZCzPWqZ1EN", "reviewer": "Reviewer_YUkh", "summary": "The original idea and intuition behind this paper are excellent. \n\nSTEER represents recurrent connectivity as a low-rank CP tensor decomposition and learns a stimulus-conditioned dynamical law $z_{k+1}=g_{\\theta}(z_k,\\bar{u}_k)$  governing how low-dimensional motif coefficients evolve across sessions. Identifiability is promoted through unit-norm and orthogonality constraints and an optional sign mask enforcing Dale’s principle. The approach is validated on increasingly realistic tasks: synthetic Lorenz systems, a Bienenstock–Cooper–Munro (BCM) plasticity model, and a longitudinal Parkinson’s disease DBS dataset.\n\nSeveral essential details are relegated to the appendix and the presentation undermine the overall quality of the work. Some of those choices make me dubious on the validity of the model.\n\nI am giving the paper an overall 4 \"marginally below the acceptance threshold\" but will consider increasing that grade if the weaknesses are adresses.", "questions": "* Identifiability and factor uniqueness: The authors reduce CP scaling/sign indeterminacy by constraining motif factors to unit‑norm and penalising orthogonality.  Do these constraints guarantee a unique solution, or could different factor orderings or scalings still yield the same connectivity?  How do they interpret the motifs biologically when multiple equivalent factorizations exist?\n\n* Report performance on the inferred plasticity, not just on state prediction. The paper emphasises within‑session predictive accuracy ($R^2$) and explained variance on held‑out trajectories, but these metrics speak only to how well the model predicts neural activity.  For a method whose main goal is to infer the slow plasticity rule, one would also expect quantitative assessments of how accurately the latent plasticity dynamics $z_{k+1}=g_\\theta(z_k,\\bar{u}_k)$ and the motif coefficients $c_k$ recover the ground truth.  In the Lorenz and BCM benchmarks, the authors report a dynamical‑similarity score (DSA = 0.63) and correlations with BCM thresholds, but there is no systematic evaluation of plasticity inference across rank choices or hyperparameters.  Likewise, in the DBS experiment they assess alignment between $\\Delta c$ and functional connectivity rather than the fidelity of the inferred slow law.  Expanding these analyses to provide error metrics on the inferred plasticity would strengthen the claim that STEER recovers the underlying rule.", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 4, "strengths": "This work is well motivated and conceptually strong and addresses a fundamental challenge in neuroscience of understanding how long-term stimulation reshapes neural circuits. The proposed framework introduces a dual-timescale formulation that separates fast within-session dynamics from slow plasticity adaptation.\n\nThe paper is innovative. By employing a low-rank decomposition of the recurrent connectivity tensor, STEER enforces structure and identifiability, leading to interpretable motif-level representations of plasticity. \n\nThe experimental design is particularly strong. The authors carefully structure their validation in increasing order of biological realism: starting from the synthetic Lorenz system, progressing to a controlled Bienenstock–Cooper–Munro (BCM) plasticity model, and culminating with real longitudinal deep-brain stimulation (DBS) data in Parkinsonian rats.", "weaknesses": "The presentation needs a brush\n\nClarity:\n-\"session\" is never really well defined in the beginning and is such a vague term. Does itrefers to a single day, trial block, animal or experiment ? \n- section 4.1. Lorentz equation should be in main text, not hidden in the appendix\n- l299-204.  the are no motivation behind equation 10\n- Figure 4 is too small \n\nMotivation \n-  In the Lorentz experiment, it's not clear why this specific synthetic plasticity is chosen; they appear arbitrary.  A brief justification would be okay and would help readers evaluate whether the benchmark reflects plausible plasticity.  (you already justify parameters range, does the same justification work for the function form ?)\n\nConceptually:\n- you are presenting the performances on the states but the evaluation of plasticity (evaluation of the parameters) changes from benchmark to benchmark\n- similarly you don't provide systematic comparison with other methods", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "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:

Decision: Accept (Poster)