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Planning as Dynamics Relaxation: Hippocampal Recurrent Network Realizes Optimal Goal-Directed Navigation

Yuhang He, Junfeng Zuo, Tianhao Chu, Si Wu

arXiv:2609.13219v1q-bio.NCcs.AI

TL;DR

How hippocampal circuits turn a stored spatial map into obstacle-avoiding routes remains unclear. The paper models recurrent transitions learned during exploration and uses goal-evoked relaxation to generate an LMDP-equivalent field. Simulations show bug-trap navigation and compact rerouting after local edits.

  • Problem

    Although hippocampal cognitive maps are well established, how neural circuits use them to produce obstacle-avoiding routes remains poorly understood.

    Source: §1 · p. 1

  • Method

    The paper models BTSP-written recurrent transitions and shows that goal-evoked network relaxation produces an LMDP desirability field whose log-gradient guides a place-tuned readout.

    Source: §1 · p. 2, §3.1 · p. 6

  • Results

    In simulations, every bug-trap run reached the goal with median path length 1.16× the shortest legal route, and a rank-2 correction captured 99.5% of field-change energy across 34 goals.

    Source: §4.2 · p. 8

  • Takeaways & Limitations

    The framework links cognitive-map storage and goal-directed action within one recurrent circuit and supports rerouting toward every recalled goal after a local change.

    Source: §1 · p. 2, §4.2 · p. 8

    The transition gate is idealized; real circuits may use self-motion signals and internally generated sequences together to restrict BTSP to accessible transitions.

    Source: §5 · p. 9

Abstract

from arXiv · show

Neural correlates of spatial cognitive map are well documented, yet exactly how neural circuits perform spatial navigation in complex environments - e.g., reaching a goal while avoiding obstacles - remains largely unclear. Here, we show that a hippocampal network with appropriate recurrent connections can naturally achieve optimal goal-directed navigation via its relaxation dynamics. Specifically, we consider that the recurrent weights between the neurons represent the transition probabilities between spatial locations encoded by neurons; obstacles such as walls and blocked corridors are therefore reflected by the vanishing of connection weights. This connection pattern can be learned in the hippocampus via behavioral-timescale synaptic plasticity (BTSP) while the animal is exploring the environment. When a goal signal is presented, the network dynamics will relax into an activity field representing the goal location. We prove that this field is mathematically equivalent to the desirability field of a Linearly-solvable Markov Decision Process (LMDP), and the local log-gradient of the field indicates the navigation direction. Both theoretical analyses and simulations demonstrate that this recurrent network dynamics-mediated navigation is efficient and robust in environments with complex obstacle layouts. Moreover, only low-rank updates of the network's connection pattern are needed when the environment has local changes. We hope this study offers insight into a general circuit principle for planning in abstract rational maps in the brain beyond spatial navigation.

1 Introduction

The paper addresses how a stored hippocampal map could generate obstacle-avoiding routes, proposing that recurrent-network relaxation itself performs goal-directed planning. The authors connect this mechanism to LMDP desirability, BTSP learning, and simulations of maze navigation.

1 Introduction

  • A hippocampal recurrent network can plan obstacle-avoiding routes through its own relaxation dynamics.Recurrent weights encode transitions, inaccessible transitions have zero weight, and goal recall produces a field whose log-gradient points toward the goal.Source: §1 · p. 2
  • The goal-locked field coincides with LMDP desirability, while a place-tuned readout converts it into a direction toward the goal.The paper also relates weak-cost relaxation to a Green’s function and a circuit-level basis for the spectral navigation account of Zuo et al. (2026).Source: §1 · p. 2
  • BTSP writes the required recurrent connectivity during ordinary, goal-free exploration.Local environmental changes require updates only to affected synapses; a low-rank correction then captures rerouting toward every recalled goal.Source: §1 · p. 2
  • Simulations on grid mazes, including a bug trap requiring a non-trivial detour, confirm the framework’s predictions.The framework recasts planning as recurrent relaxation, unifying cognitive-map storage and goal-directed action within one circuit.Source: §1 · p. 2

2 The model

The model stores spatial transitions in a recurrent hippocampal network and uses goal-evoked relaxation to produce a planning field. Its place-tuned readout extracts a local direction, and the same dynamics connect finite-cost planning to a weak-cost Green’s function limit.

2.1 Recurrent network model

  • Neurons encode locations, recurrent weights encode their transitions, and a goal-specific input clamps the goal neuron while the remaining population relaxes.The linearized network is stable when effective recurrent gain α lies between zero and one; α controls how far activity spreads along recurrent paths.Source: §2.1 · pp. 2–3

2.2 Goal-clamped relaxation produces the desirability field

  • Under reversibility, the recurrent steady state matches LMDP desirability up to a known factor when recurrent gain α equals planning discount γ.The goal-clamped field sums legal recurrent paths to the goal, with longer paths damped by α^L and blocked transitions contributing no paths.Source: §2.2 · pp. 3–4

2.3 A place-tuned readout of the goal-locked field

  • A downstream place-tuned population multiplies the goal field by local place inputs, yielding a population-vector direction along the field’s local log-gradient.The local approximation holds when the place field is narrower than the spatial scale over which the goal field varies.Source: §2.3 · pp. 4–5

2.4 Robustness to recall amplitude

  • Uniform rescaling of the goal field leaves the readout direction unchanged, so strong recall saturation mainly distorts field shape near the goal.The residual distortion decays exponentially with distance, making the shell radius grow only logarithmically with recall amplitude.Source: §2.3 · p. 5, §2.4 · p. 5

2.5 The weak-cost limit recovers Green’s function navigation

  • As step cost vanishes, the same relaxation approaches the Green’s function of the recurrent Laplacian, matching the navigation potential described by Zuo et al. (2026).In that limit, desirability flattens and its leading spatial correction is related to expected spontaneous steps to the goal.Source: §2.5 · pp. 5–6

3 Learning the recurrent connectivity

The paper proposes that goal-free BTSP writes a transition structure suitable for planning, and that local environmental edits support compact rerouting across goals.

3.1 Transition-gated symmetric plasticity

  • Transition-gated symmetric BTSP during goal-free exploration yields recurrent weights that vanish on inaccessible transitions and encode a reversible transition kernel.Finite-sample fluctuations and goal-directed behavioral biases can leave a small asymmetric residue.Source: §3.1 · p. 6

3.2 Local environmental change as a low-rank update

  • After local changes, a handful of traversals can update affected connectivity, and the next relaxation reroutes plans toward any recalled goal.Editing k synapses changes only those entries of S; changes in goal-locked fields decompose into a rank-O(k) propagation pattern plus a per-goal scaling that cancels in the readout.Source: §3.2 · p. 7

4 Experiments

Maze simulations test whether relaxation recovers the LMDP field, remains robust to recall amplitude, and supports navigation after BTSP learning and local edits.

4.1 The relaxation reproduces the LMDP field, and the readout is amplitude-robust

  • At α = 0.985, the corrected network field matched LMDP desirability to relative L2 error 9.5 × 10^-16, while the uncorrected field differed by 3.2%.The agreement held across tested α values.Source: §4.1 · p. 7
  • At strongest recall, readout direction outside the saturation shell matched the linear prediction to 0.15° on average and 0.60° at the 99th percentile.The shell contained about eight neurons at A = 5 and its radius grew as log A; at α = 0.985, outside-shell error stayed below 1° across the recall range.Source: §4.1 · p. 7, Figure 1 · p. 8

4.2 Goal-free BTSP solves the bug trap, and a local edit reroutes plans globally

  • After 3 transitions per neuron, BTSP recovered 68.9% of accessible connections with precision 1.000, and every run reached the bug-trap goal at median path length 1.16× the shortest legal route.The start readout pointed 93.8° away from the Euclidean direction toward the only legal entrance, while a Euclidean-guided agent stalled at the wall after 16 steps.Source: §4.2 · p. 8
  • After a near door was closed in a two-door maze, a rank-2 correction captured 99.5% of the energy in field changes across 34 recalled goals.The next relaxation rerouted plans through the far door without further learning.Source: §4.2 · p. 8

5 Discussion

The section frames planning as relaxation over a hippocampal transition map, describing BTSP-stored accessible transitions and goal-evoked desirability fields alongside predictions, limitations, extensions, assumptions, and mathematical conditions. It also derives the KL-control Bellman equation and linear desirability representation, states the scope of the optimal-policy formula, and defines a passive-chain hitting time.

5 Discussion

  • BTSP stores accessible transitions in recurrent weights, and goal recall relaxes the same circuit into an LMDP desirability field whose local log-gradient guides the next step.In the weak-cost limit, the relaxation returns the recurrent Laplacian’s Green’s function, offering a circuit-level basis for the Laplacian account of grid, band, and boundary cells.Source: §5 · pp. 8–9
  • The model leaves the supporting hippocampal subfield and biological transition gate unresolved, and proposes tests of planning-pause activity and BTSP mechanisms in vivo.It also proposes extending transition-based relaxation beyond spatial maps to sensory continua, social hierarchies, and abstract task structures, with a step defined by each domain’s transitions.Source: §5 · pp. 9–10

A Related work

  • The related-work discussion links cognitive maps, predictive representations, LMDP and Laplacian planning, and recurrent dynamics by proposing one circuit that stores transitions and computes goal-conditioned fields.It assigns replay and theta sequences a role in writing transition weights through BTSP, separating offline map construction from online goal-clamped relaxation for action.Source: A Related work · pp. 16–17

5 Discussion

  • A local environmental edit can reroute plans for many goals: closing a doorway changes local connections, while relaxation propagates the change through a few common patterns.The discussion predicts goal-dependent subthreshold components alongside stable place-field centers; Figure 2 illustrates rerouting through a far door after a local edit and reports that two modes carry 99.5% of field-difference energy.Source: Figure 2 · p. 9, §5 · p. 9, A Related work · p. 17

B Notation and standing assumptions

  • The formal setup assumes a finite, reversible passive transition kernel on the goal-connected component; disconnected components have zero Dirichlet response and zero desirability for that goal.Fields are background-subtracted goal-induced perturbations, rather than raw firing rates.Source: B Notation and standing assumptions · pp. 18–19

C Linearization of the recurrent population; C.1 Expansion around the goal-free fixed point

  • The population analysis expands goal-driven activity around the goal-free fixed point, with recall equations applying at fixed position or during a quasi-static planning pause.The Taylor estimates require a twice-differentiable nonlinearity on the relevant bounded activity interval, or intervals that avoid kinks and hard saturation boundaries.Source: §C.1 · p. 20

C.2 Homogeneous-gain scalar-discount reduction; C.3 Diagonal-gain operator

  • With homogeneous gain, the linearized recurrent dynamics are stable and the scalar-discount operator is invertible; with nonuniform gain, the effective operator is symmetrizable but not generally a stochastic reversible kernel.The scalar-discount equivalence in the nonuniform case is treated as a perturbation or a state-dependent gain operator.Source: §C.2 · p. 21, §C.3 · p. 21

D.1 Bellman equation with KL control cost

  • The LMDP setup lets the agent choose a distribution over accessible successors and charges a positive per-step cost plus a KL penalty relative to passive transitions.The cost is q0 + λ DKL(µ(· | i) ∥ P0(· | i)), with q0 > 0 and λ > 0.Source: §D.1 · p. 22
  • The KL-control Bellman derivation uses a Lagrange-multiplier calculation for the constraint on P.Source: §D.1 · p. 22

D.2 Linear desirability equation

  • The transformation z(i) = exp[−V(i)/λ] and γ = exp(−q0/λ) leads, by exponentiating Eq. (30), to the linear desirability equation.The equation is split into non-goal and goal terms, with z(g) = 1.Source: §D.2 · p. 22
  • The optimal policy is expressed in desirability coordinates, and its finite-value formula applies only on the goal-reachable component.There z(i) > 0 and the policy has the same support as P0(· | i); outside that component z = 0.Source: §D.2 · p. 22
  • The analysis defines τg as the first hitting time of g under the passive chain.Source: §D.2 · p. 22

D.3 Exact linear network–LMDP theorem

The theorem establishes network–LMDP equivalence under finite, reversible transition-kernel assumptions and a goal clamp. The accompanying sections describe symmetric coordinates, transition-gated learning, and reversible kernel constructions.

D.3 Exact linear network–LMDP theorem

  • Under the stated finite, irreducible, reversible-kernel assumptions and a Dirichlet goal clamp, the network field equals the LMDP desirability field.The proof establishes a unique solution on the goal-connected component.Source: §D.3 · p. 23

E.1 Symmetric network coordinate

  • The passage identifies S as the recurrent coordinate used in Theorem 2 and states that it is symmetric, with entries satisfying P0(j | i) = Sij.Source: §E.1 · p. 24

E.2 Transition-gated symmetric accumulation

  • Transition-gated accumulation yields symmetric expected weights under stationary bidirectional sampling and a time-reversal-invariant gate; blocked transitions are excluded from the learned support.Directed or policy-biased sampling appears as deviations from this symmetrized transition memory.Source: §E.2 · p. 24

E.3 Lazy normalization

  • Lazy normalization makes every row sum to one and, for symmetric accumulated weights, produces a symmetric reversible kernel with uniform stationary measure.Keeping ε fixed across local edits avoids rescaling all off-diagonal weights.Source: §E.3 · p. 25

E.4 Standard random-walk normalization

  • For isolated states under standard random-walk normalization, the state can be removed from the navigable component or treated as a closed singleton with P0(i | i) = 1.Source: §E.4 · p. 25

F.1 Reversible projection of empirical transition counts

  • Symmetrizing finite empirical transition counts before normalization yields an exactly reversible passive kernel, so the exact theorem applies to the projected kernel.Lazy normalization gives uniform stationary measure; random-walk normalization gives πi ∝ bdi.Source: §F.1 · p. 25

F.2 Perturbation by irreversible sampling error

The appendix bounds how sampling asymmetry and nonlinear goal recall affect the recurrent field and its readout. It also relates the place-tuned readout to the local LMDP policy under stated conditions.

F.2 Perturbation by irreversible sampling error

  • Small irreversible sampling errors produce controlled recurrent-field corrections when their operator norm is small relative to the stability margin; the reversible projection retains exact network–LMDP equivalence.The resolvent bound also applies to Dirichlet restrictions.Source: §F.2 · p. 26

F.3 Perturbation of a reversible stationary measure

  • Sampling or normalization changes to a reversible kernel’s stationary measure are controlled by the spectral gap.The first-order change in π is given by δπ = π̄∆P(I − P̄)# + O(|∆P|²).Source: §F.3 · pp. 26–27

F.4 Readout bias from nonuniform stationary measure

  • With nonuniform stationary measure, the readout’s edge log-gradient includes half the edgewise log-stationary-measure difference.An exact corrected readout has log-gradient equal to ∇log z_g.Source: §F.4 · p. 27

G.1 Continuous local moment expansion

  • To first order, the centered population-vector displacement equals the place-kernel covariance times the local field’s log-gradient.When the kernel mean is zero, raw and centered readouts agree to leading order.Source: §G.1 · p. 28

G.2 Centered-kernel correction

  • Local reflection symmetry zeros the third kernel moment, making the first shape correction fourth order in kernel width.The stated scales are the local kernel width σ_G and the field-variation length ℓ_f.Source: §G.2 · p. 28

G.3 Accessible directions

  • The place-kernel covariance projects the local log-gradient onto represented directions; a nonzero kernel mean adds resting drift that centering removes.Near walls, blocked-direction variance is suppressed, while corridor covariance is elongated along the corridor axis.Source: §G.3 · p. 29

H Relation between the readout and the LMDP optimal policy

  • The centered place-kernel readout matches the local LMDP policy mean to leading order when their local means and covariances approximately match.Otherwise, the readout is a geometrically preconditioned log-desirability readout, using the place-kernel covariance.Source: H Relation between the readout and the… · p. 29

I Nonlinear perturbation under strong goal recall

  • Exact network–LMDP equivalence holds for homogeneous linearized dynamics; nonlinear distortion under strong recall is controlled in the perturbative small-gain regime.The nonlinear solution exists uniquely within the stated small-gain ball, while its relative correction is generally not forced to decay by the bound alone.Source: I Nonlinear perturbation under strong… · p. 30, §I.3 · p. 31

I.4 Far-field amplitude projection

These sections analyze far-field effects, weak-cost Green functions, and the structure of local connectivity updates. The bounds and update dimensions depend on geometric or normalization conditions stated in the appendix.

I.4 Far-field amplitude projection

  • Under the stated far-field ratio condition, the affected shell radius depends logarithmically on effective recall amplitude; bounded saturating activation gives that radius a finite upper bound.The condition holds for some standard geometries but is an explicit assumption on arbitrary finite graphs.Source: §I.4 · pp. 31–32

J Weak-cost limit and Green functions

  • In the weak-cost limit, the recurrent Laplacian shares eigenvectors with the cited discrete Laplacian under lazy normalization with symmetric transition counts.The recurrent Laplacian coincides with that discrete Laplacian up to a positive scalar.Source: J Weak-cost limit and Green functions · p. 32

J.1 Screened Dirichlet operator; J.2 Boundary-driven weak-cost limit

  • The goal-clamped linear equation yields a screened Dirichlet Green operator; the boundary-driven weak-cost field is a hitting-probability field.Source: §J.1 · p. 33, §J.2 · pp. 33–34

J.3 Dirichlet source-response Green function

  • In the weak-cost limit, a localized interior source produces a Dirichlet Green-function response, distinct from the boundary-driven field.Source: §J.3 · p. 34

J.4 Full-graph mean-zero Green function

  • For the full-graph mean-zero Green function, arithmetic mean-centering matches the stated centering exactly when π is uniform.Source: §J.4 · p. 35

K.1 Rank of a local connectivity edit

  • Local environmental edits change O(k) recurrent-matrix entries under the stated normalizations and graph conditions.The passage notes bounded-degree graphs for the random-walk normalization case.Source: §K.1 · p. 35

K.2 Fixed-goal Dirichlet update

  • For a fixed goal, the field change lies in an O(k)-dimensional subspace, including when the boundary column changes locally under the stated normalizations.Source: §K.2 · p. 36

K.3 Shared full-resolvent update

  • With a fixed source matrix, a shared full-resolvent update has rank at most the edit rank; a local source change adds another low-rank term.Source: §K.3 · p. 37

K.4 Unit-clamped all-goal fields

The all-goal update separates a low-rank propagation term from source renormalization needed to preserve clamp amplitudes. Per-goal log-gradient readouts cancel scalar renormalization, while locality estimates weaken in the weak-cost limit.

K.4 Unit-clamped all-goal fields

  • The all-goal field update’s propagation term has rank at most O(k), but the source-renormalization term can have full algebraic rank.The latter keeps each goal clamped to the same amplitude and can change diagonal entries for many goals.Source: §K.4 · p. 37
  • The discarded-term locality estimate weakens as α approaches one because the decay rate may decrease and resolvent constants may grow.The exponential Frobenius-energy decay is stated for graphs with subexponential or sufficiently slow volume growth.Source: §K.4 · p. 38
  • Per-goal diagonal source renormalization cancels exactly in log-gradient readouts, although it changes raw all-goal field matrices.Source: §K.4 · p. 38

L Numerical experiment details

The numerical protocol tests field equivalence, strong-recall stability, BTSP maze connectivity and local remapping, and Green-function and sampling-asymmetry predictions. Simulations use a legal weighted grid graph and specified field, readout, and rollout conventions.

L Numerical experiment details

  • The simulations test four claims: LMDP-field equivalence, strong-clamp readout stability, obstacle-maze planning with BTSP and local remapping, and Green-function and asymmetry predictions.Source: L Numerical experiment details · p. 38

L.1 Common graph, field, and readout conventions

  • The common memory graph uses open cells and legal eight-neighbor edges, removing diagonal corner cutting; axial, diagonal, and blocked transitions receive weights e^−1/2, e^−1, and zero.The matrix is symmetrized after edge construction.Source: §L.1 · p. 38
  • The linear recurrent field and LMDP desirability field are computed separately, and readout-angle errors compare vectors at the same state.The symmetric-coordinate recurrent matrix makes the stationary-measure factor visible near boundaries and obstacles.Source: §L.1 · pp. 38–39
  • The population-vector readout is projected onto legal neighboring moves for rollouts, which choose the neighbor with the largest visit-adjusted score.Source: §L.1 · p. 39

L.2 Equivalence and strong-recall robustness

The two-room diagnostic tests the network’s LMDP equivalence and whether strong recall preserves its action-relevant readout away from the goal.

L.2 Equivalence and strong-recall robustness

  • At α = 0.985, the corrected network field matched LMDP desirability with relative error 9.55 × 10−16 and correlation 1.000.Across the α grid, corrected relative error ranged from 6.29 × 10−16 to 9.59 × 10−16; the uncorrected field differed by 3.17% at α = 0.985.Source: §L.2 · pp. 39–40
  • The nonlinear test mapped the field to LMDP coordinates and aligned it to the linear field by a scalar, removing uniform amplitude changes that cannot affect the log-gradient readout.The test used tanh activity and recall amplitudes from 0.05 to 5.Source: §L.2 · p. 40
  • At A = 5, outside-shell readout error was 0.080° on average and 0.60° at the 99th percentile, with an eight-cell shell.The shell radius grew approximately as log A (R² = 0.94).Source: §L.2 · p. 40

L.3 Goal-free BTSP learning and local low-rank remapping

Goal-free transition sampling learns legal connectivity for navigation, and local edits reroute fields through a compact set of propagation patterns.

L.3 Goal-free BTSP learning and local low-rank remapping

  • Exploration sampled under-visited legal transitions without goal input or shortest-path information, and symmetric transition counts formed the learned BTSP memory.A self-only numerical floor prevented early sparse rows from making the linear system singular without adding untraversed geometry.Source: §L.3 · p. 40
  • After 3 transitions per neuron, learned connectivity had 68.9% edge recall, precision 1.000, and field correlation 0.988 ± 0.003; all five seeds reached the goal with median route ratio 1.16.The Euclidean greedy baseline stalled at the wall after 16 legal steps.Source: §L.3 · p. 41
  • With 2.0 transitions per assembly, the two-door diagnostic achieved 49.4% mean edge recall, precision 1.000, field correlation 0.973 ± 0.006, and median route ratio 1.27 across five successful seeds.The displayed open-door seed had route ratio 1.20.Source: §L.3 · p. 41
  • Closing the near door by removing 16 crossing edges rerouted the displayed goal through the far door, with route ratio 1.09 and no other weight edits or new exploration.The Woodbury update matched a direct solve with relative error 1.82 × 10−15.Source: §L.3 · p. 41
  • Across 34 recalled goals, the first two singular modes explained 99.46% of the squared energy in field differences after the local edit.The fields were corrected to LMDP coordinates before differencing.Source: §L.3 · p. 41

L.4 Weak-cost Green-function limit

The experiment tests how recurrent source responses relate to screened Green fields and how those fields approach the Laplacian Green shape as cost weakens.

L.4 Weak-cost Green-function limit

  • The identity in Eq. (141) stayed at floating-point precision, with maximum relative error 2.67 × 10−13 at α = 0.9995.Source: §L.4 · p. 42
  • As mα decreased from 0.111 to 5.0 × 10−4, screened Green shape error fell from 1.168 to 0.162, indicating convergence to the Laplacian Green shape.The comparison used mean-zero fields projected and normalized before shape comparison.Source: §L.4 · p. 42
  • Low-frequency Laplacian modes reconstruct the screened Green response, with the plotted shape error tracked as modes are added.The reconstruction used the 72 smallest eigenpairs of L.Source: §L.4 · p. 42
  • The recurrent source response and screened Green operator agreed with relative error 2.0 × 10−15 in the displayed experiment.The comparison normalized both fields by their goal values.Source: §L.4 · p. 42

L.5 Asymmetric BTSP perturbation and bilateral replay

The asymmetric-BTSP experiments compare field and navigation deviations using a recurrently amplified irreversibility scale, and test bilateral replay as a recovery mechanism.

L.5 Asymmetric BTSP perturbation and bilateral replay

  • The perturbation parameter η scales the directional residue while the symmetric support stays fixed; χ combines that asymmetry with recurrent amplification α/(1 − α).The experiment used legal support and did not change the maze.Source: §L.5 · pp. 43–44
  • For χ ≤ 0.3, raw asymmetric BTSP had median field error 0.0040, 75th-percentile field error 0.0070, median readout-angle error 0.13°, and median route ratio 1.03; all raw rollouts succeeded.For 1 ≤ χ < 2, median field error was 0.053, median readout-angle error 1.88°, and median route ratio 1.14; two raw rollouts failed near χ ≈ 1.7.Source: §L.5 · p. 44
  • Field error and median readout-angle error followed single increasing trends when plotted against χ across perturbation strengths and recurrent gains.The panels encode η by color, α by marker shape, and use log(1 + χ) on the horizontal axis.Source: §L.5 · pp. 44–45
  • The bilateral projection kept every rollout successful across the sweep, with route ratios between 1.02 and 1.08.It projects the asymmetric transition memory back to its reversible part, which is exactly P0 after normalization.Source: §L.5 · p. 44

L.6 reproduction

  • The simulations require no external data files; saved metrics and generated fields and rollouts produce the figures, with listed seeds determining the simulations.Source: §L.6 · pp. 45–46

NeurIPS Paper Checklist

The checklist states that the paper’s claims are supported by its theory and simulations and are scoped to the model assumptions and numerical maze settings. It also reports reproducibility provisions, statistical reporting, and limitations, while noting that asset licenses are not explicitly supplied.

Claims and scope

  • The paper’s claims are supported by the equivalence theorem, readout analysis, BTSP learning model, low-rank update analysis, and grid-maze experiments, within the stated model assumptions and numerical settings.The experiments use numerical maze settings detailed in App. L.Source: NeurIPS Paper Checklist · p. 47

Limitations

  • The model does not specify a hippocampal subfield, and its transition gate is idealized; identifying the cellular mechanism and in-vivo parameters remains future work.The authors suggest intracellular recordings during planning pauses as a way to test which subfield supports the relaxation.Source: NeurIPS Paper Checklist · p. 47

Reproducibility

  • The supplemental materials provide code, instructions, result files, figures, and reproduction commands for the reported experimental results.The checklist says App. L specifies experimental settings and protocols.Source: NeurIPS Paper Checklist · pp. 48–49

Statistical reporting

  • Stochastic BTSP results report means and SEM across five exploration seeds, while deterministic checks report errors, sweep values, and numerical tolerances instead of statistical error bars.The seed source of variability is stated.Source: NeurIPS Paper Checklist · p. 50

Asset documentation

  • The supplemental package documents the released code and synthetic results but does not explicitly include licenses, copyright notices, or terms of use for the new assets.The checklist also notes that software dependency licenses are not explicitly enumerated.Source: NeurIPS Paper Checklist · p. 52
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