Unfolding the Black Box of Recurrent Neural Networks for Path Integration
Authors: Tianhao Chu
Yuling Wu
Neil Burgess
Zilong Ji
Si Wu
Venue: NeurIPS 2025 poster ()
Keywords: Path integration; Recurrent neural networks; Band cells; Grid cells
Primary area: neuroscience_and_cognitive_science

Abstract: Path integration is essential for spatial navigation. Experimental studies have identified neural correlates for path integration, but exactly how the neural system accomplishes this computation remains unresolved.
Here, we adopt recurrent neural networks (RNNs) trained to perform a path integration task to explore this issue. After training, we borrow neuroscience prior knowledge and methods to unfold the black box of the trained model, including: clarifying neuron types based on their receptive fields, 
dissecting information flows between neuron groups by pruning their connections, and
analyzing internal dynamics of neuron groups using the attractor framework. Intriguingly, we uncover a hierarchical information processing pathway embedded in the RNN model, along which velocity information of an agent is first forwarded to band cells, band and grid cells then coordinate to carry out path integration, and finally grid cells output the agent location. Inspired by the RNN-based study, we construct a neural circuit model, in which band cells form one-dimensional (1D) continuous attractor neural networks (CANNs) and serve as upstream neurons to support downstream grid cells to carry out path integration in the 2D space. Our study challenges the conventional view of considering grid cells as the principal velocity integrator, and supports a neural circuit model with the hierarchy of band and grid cells.

Unfolding the Black Box of Recurrent Neural Networks for Path Integration
Tianhao Chule, Yuling Wu Neil Burgess Zilong 33 Si Wu1..."
School of Psychological and Cognitive Sciences, Key Laboratory of Machine Perception (Ministry of Education), Peking University, China. 2 Peking-Tsinghun Center for Life Sciences, Academy for Advanced Interdisciplinary Studies, Peking University, China.
Institute of Cognitive Neuroscience, Department of Neuroscience, Physiology and Pharmacology, University College London, UK
*UCL Queen Square Institute of Neurology. University College London, UK BBijiing Key Laboratory of Behavior and Mental Health, IDG/McGovern Institute for Brain Research, Center of Quantitative Biology, Peking University, China.
'These authors contributed equally this work. Correspondence:
siwu @ pku.edu.cn
Abstract
Path integration is essential for spatial navigation. Experimental studies have identified neural correlates for path integration, but exactly how the neural system accomplishes this computation remains unresolved. Here, we adopt recurrent neural networks (RNNs) trained to perform a path integration task to explore this issue. After training, we borrow neuroscience prior knowledge and methods to unfold the black box of the trained model, including: clarifying neuron types based on their receptive fields, dissecting information flows between neuron groups by pruning their connections, and analyzing internal dynamics of neuron groups using the attractor framework. Intriguingly, we uncover a hierarchical information processing pathway embedded in the RNN model, along which velocity information of an agent is first forwarded to band cells, band and grid cells then coordinate to carry out path integration, and finally grid cells output the agent location. Inspired by the RNN-based study, we construct a neural circuit model, which band cells form one-dimensional (1D) continuous attractor neural networks (CANNs) and serve upstream neurons to support downstream grid cells to carry out path integration the 2D space. Our study challenges the conventional view of considering grid cells as the principal velocity integrator, and supports a neural circuit model with the hierarchy of band and grid cells.
Introduction
Path integration refers to the process of computing the position of an agent by continuously integrating self-motion information of the agent over time [0] It fundamental for spatial navigation in both humans and rodents, and also plays an important role in robot control
Experimental studies have shown that grid cells in the medial entorhinal cortex (MEC) are deeply involved in path integration B]. These cells exhibit periodic hexagonal firing patterns in the two- dimensional space, providing a metric representation for implementing path integration [1 4 Disruption of grid cells through medial septal inactivation or entorhinal lesions impairs spatial memory and navigation based path integration
5]. Despite these experimental evidence,
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

---

exactly how the neural system performs path integration remains largely unknown, and there are unresolved issues. First, in addition to grid cells, experimental data has shown that there exists a rich diversity of spatially modulated cell types in the MEC 6
7 g 9 In particular, band cells, found in
parasubiculum (an upstream region of MEC), exhibit periodic firing patterns that can be modeled as a superposition of bands, and they are hypothesized to perform path integration in a ID space [9] 10]. However, the functional relationship between grid and band cells remains to be clarified. Second, neural circuit models have been proposed to implement path integration. The conventional models consider only grid cells, and in order to implement 2D path integration, these models either consider a large amount of direction-cone conjunctive grid cells [1 [3] assume that the recurrent connections between grid cells are tuned instantly by the velocity signal [14]. A different model considering the hierarchy of band and grid cells for implementing path integration was proposed [T5] [6]. The
question of which model is biologically more plausible remains debated.
Related RNN-based navigated studies. Recently, Al approaches have been used to study the association between path integration and grid cells. Specifically, recurrent neural networks (RNNs) were trained to perform spatial navigation to ascertain the involvement of grid cells in path integration, and inconclusive results have been reported so far. In one line of works, it was found that training an RNN based solely motion velocity in a supervised way (i.e., reconstructing the place cell activity at the given location from pure velocity input) can lead to the emergence of grid-cell-like activity pattern in the RNN, especially under the biological constraints such non-negative firing rate [8]. This result implies that grid cells are associated with path integration. Building on this work. subsequent studies have extended the supervised training paradigm more unsupervised -gudded ones, using objective functions derived from group theory, information efficiency, and actionability, emphasizing the isometric structure, high coding capacity, and integrative functionality of grid cells [19 14 20)
These studies have systematically explored the conditions under which grid-like representations emerge in RNNs, examining how the network structure, the training objective, and biological priors jointly shape grid-like representations. In the other line of works, the necessity of grid cells for path integration is challenged. For instance, (2] demonstrated that grid-like firing patterns can arise as the principal components of place cell activities under the non-negativity constraint, without path integration involved. More recently, [22] argued that grid cells may emerge from a principle of efficient pattern formation for reconstructing place fields, rather than from the demand of path integration. Supporting this view, ablation studies of trained RNNs have shown that removing velocity inputs to high grid-score units has negligible impact on the path integration performance 233 24 Overall, current RNN-based studies have primarily focused on exploring the conditions under which grid cells emerge and on the necessity of grid cells for path integration. However, they have not addressed the question of how a trained RNN implements path integration, nor examined the The internal mechanisms of the trained RNN
functional role of band cells in this process 18 25 model thus remain a black box.
In this work, we take the same approach of training RNNs to perform path integration, but our goal different, that is, we aim to unveil the detailed mechanism concealed in the trained RNNs, and from which, to ascertain the neural mechanism of path integration. To achieve this goal, we will borrow neuroscience prior knowledge and methods to unfold the black box of the trained RNNs in much more detail than in previous studies. These include: clarifying neuron types based on their receptive fields, dissecting information flows between neuron groups by pruning their connections, and analyzing internal dynamics of neuron groups in the attractor framework. It turns out that there exists a hierarchical information processing pathway embedded in the RNN model, along which velocity information is first forwarded to band cells, band and grid cells then coordinate to carry out path integration, and finally grid cells output the location of the agent. Inspired by the RNN- based study and previous mechanistic models [16 26), we construct hierarchical neural circuit model for path integration in the brain, in which band and grid cells form ID and 2D continuous attractor neural networks (CANNs), respectively, and they coordinate to accomplish path integration. Contemporary models often assume that grid cells are primarily responsible for path integration study challenges this view and suggests that the hierarchy of band and grid cells is necessary. Our model also makes predictions testable in neuroscience experiments.
[2 13 Our
2

---

MEC
RNN
r (a.)
Band
Pr (xg)
Cell
Grid
Cell
Vo:T1
X1:T
M
W
Undefinned
Cell
Figure 1: (a) Illustration of the path integration task. (b) The structure of the RNN model. (c) Examples of band cell, grid cell and undefined cell found MEC (middle panel, adapted from 9 and in the trained RNN (right panel).
2 Neuroscience for AI: unfolding the black box
2.1 An RNN model for path integration
We adopt well-established paradigm train RNNs for path integration [17 22 obtaining the key results as reported here, which are further validated by consistent results from other modeling frameworks (see Supplement Information E) As illustrated Fig. Ta an agent is moving in square environment with velocity v. E R Mimicking place cells' position encoding in the neural system, the agent's location x; is represented by bump vector Pt E NR the 2D space centered at Xr, with N, the number of place cells. The task of path integration is formulated as: given the initial location Xg the agent, the RNN model infers subsequent positions XI.7 by using only self- motion information of the agent, i.e., the sequence of velocity vectors Vo:T no other position cue is available. Within the conceptual framework of the MEC-HPC loop, the model's information flow can be interpreted as follows: the initial state of the RNN (representing nascent MEC population) is set by a projection from the place cell population (representing hippocampus, HPC), i.c., ro = Mopo- where Po corresponds to the initial location X6 and the matrix M,- E IR N with N₂ the number of neurons the RNN (Fig. [p). This initialization embodies critical HPC > MEC projection signal that anchors the path-integration process to environmental cue.
At each time step, the RNN receives the agent's instantaneous velocity signal v, and updates its internal state following the recurrent dynamics, which is written as,
It+1 = (Jr, + Mv.),
(I)
where J E NR is the recurrent connection matrix between neurons in the RNN, the matrix M E R NRx2 and o()) is the ReLU function, enforcing non-negative firing rates of neurons.
At each time step, the agent's location is read out by,
p Wr.,
(2)
where the read-out matrix W E NRNNR
The free parameters of a RNN, including J, M, W, and Mo, are optimized using BPTT through minimizing the cross-entropy loss between the network outputs p. and the ground truth p. for all ,
time steps f = 1.....7. T and for all randomly generated motion trajectories. The details the model training and fixed parameters given in AI The trained RNN model effectively implements path integration, with quantitative results (SI indicating that it achieved competent performance on the task.
2.2 Interpreting the trained RNN model with neuroscience
straightforward to train the RNN model well achieve good path integration performance, but unveiling the underlying computational mechanism is challenging, which is known as the black box problem of Al-trained models. Without prior knowledge, we have no clue interpret the data. Here, borrow neuroscience prior knowledge and methods to unfold the black box.

---

a
b
d
Band
Band
Band
Band g Grid
Grid
Grid
Grid
Undefined
Undelined
Undeffned
Undefined
0.05
to Band
00125
Band Output
Bond Coll
M
0.20
Band to Ca to Bland
to Grid
Grid Output
Gric Call
0.15
V to Undefined
Undefined Output
Undeffined Cell
0.075
0.30
0.050
0.1
0.025
200
500
200
2:0
400 500
Connections Pruned
Results of pruning the RNN model. Four pruning operations are carried out step-by- step, with each step building upon the previous one. Upper panels: (a) Pruning velocity inputs to each cell group one-by one-by-one; (b) Pruning read-out connections from each cell group one-by-one; (c) Pruning each cell type one-by-one; (d) Pruning the connections from band to grid and the connections from grid to band cells, respectively. Lower panels: Path integration error vs. the number of
Figure 2:
connections/cells pruned in each step. The path integration error measured by the standard division of location prediction errors divided by the arena size. For calculation details, see SI B
2.2.1 Clarifying neuron types
Our first step is to clarify neuron types in the trained RNN based on their receptive fields. Following the common practice in neuroscience, we first compute the spatial tuning maps of neurons in the trained RNN. Specifically, we consider that the agent performs random walks in the environment, covering the whole arena, and the corresponding velocity signals are used as inputs to stimulate neuronal responses in the RNN. We record the activities of each neuron at all locations, which gives a 2D heatmap for each neuron, reflecting their spatial tuning characteristics.
Inspired by the spatial tuning properties of grid and band cells observed in the neural system, we search for analogous unit types in the trained RNN. To classify neuron types quantitatively, we adopt established criteria from neuroscience (see SIB We have identified three mutually exclusive types based neuronal receptive fields (Fig. DT. right panel), which are: 1) Grid cell, which exhibits the hexagonally patterned firing field; 2) Band cell. whose receptive field consists of multiple equally spaced parallel lines (the so-called bands) (see more details below); and 3) Undefined cell, which displays neither grid-like nor band-like spatial tuning, All three cell types emerge robustly in the trained RNNs (see EE. For example, one trained RNN with NR = 4096 units, we identify 764 grid cells, 764 band cells, and 2568 undefined cells).
2.2.2 Dissecting information flows between neuron groups
After identifying neuron groups, we further investigate how information is propagated between them by applying pruning studies. We carry out four step-by-step pruning operations (see SI[),, with each of them building upon the previous one and aiming to probe one aspect of path integration. See SI for reproducibility results.
Step Pruning velocity inputs. To identify which cell types are primarily responsible for receiving and processing velocity inputs, selectively prune the afferent velocity inputs each cell group one-by-one (Fig. p upper panel). We find that pruning velocity inputs to band cells leads to a significant increase in the path integration error, while pruning velocity inputs to other groups has little effect (Fig. 2 lower panel). This indicates that the velocity information is predominantly processed by band cells, and we remove velocity inputs to other cell types in the followed pruning.
Step 2: Pruning read-out connections. To identify which cell types are primarily responsible for outputting the path integration result, we selectively prune the read-out connections from each cell group one-by-one (Fig. Pb, upper panel), and find that pruning the read-out connections from grid cells have the largest effect increasing the path integration error (Fig. 2p. lower panel). This

---

a
Band Cell
Grid Cell
102
1e2
Band Count
Grid Count
X
RNNI
RNNZ
RNNI
RNNZ
RNN1
RNN2
RNN
X
Figure 3: (a) The hierarchical path integration pathway embedded in trained RNNs. (b) A new RNN model taking the hierarchical pathway as a prior. (c) Neuron types found in the newly trained RNN. (d) An RNN model for location reconstruction task. For simulation details, see SI
indicates that grid cells are predominately responsible for outputting the path integration result, and we remove the read-out connections from other cell types in the followed pruning.
Step 3: Pruning cell types. To evaluate the relative contributions of different cell types on path integration, we compare the model performances after pruning one of them in equal numbers (Fig. P upper panel). We find that pruning band or grid cells causes substantial drop in the model performance, whereas pruning undefined cells has much less effect (Fig. (2.. lower panel). This suggests that band and grid cells play much more important roles than undefined cells in path integration, and we remove them in the followed pruning.
Step 4: Pruning connections between band and grid cells. To probe the functional relationship between band and grid cells in path integration, we separately prune the connections from band grid cells and the connections from grid to band cells (Fig. 2 upper panel). We find that pruning either type of connections has big impact on the model performance (Fig. 21. lower panel), indicating that the bidirectional, cooperative interactions between two cell types are critical for path integration.
2.3 A hierarchical path integration pathway embedded in the RNN
Combining pruning results while discarding less relevant components, we identify a hierarchical path integration pathway embedded in the trained RNN (Fig. (Fig.3p), along which velocity information first received and processed by band cells; band and grid cells then interact via reciprocal connections to integrate movement information; finally grid cells output the agent location in the 2D space.
It is surprising that such well-organized pathway naturally emerges in training RNN for path integration task. To further validate that this hierarchical pathway constitutes the functional core structure, we conduct two additional experiments. In one experiment, we construct new RNN model which takes the hierarchical pathway as a prior (Fig. (Fig.Bb), that is, the first module receives velocity inputs, intending play the role of band cells; the second module outputs the agent location, intending to play the role of grid cells; the two modules are reciprocally connected, intending to mimic the interactions between band and grid cells. We train this new model with the same path integration task and have two key observations as expected, which are: 1) the new model achieves a performance comparable to the original model; 2) band and grid cells emerge naturally as the dominant neuron types in the first and second modules, respectively (Fig. Bbb. In the other experiment, we change the task from path integration to pure location reconstruction (Fig. BD), i.c., the RNN receives the agent's 'ss location inputs and reconstruct them; no velocity information is available. After training the RNN with the new task, observe only grid cells without band cells (see SI(C). indicating the necessity of path integration for the emergence of band cells. Together, these two additional experiments further reinforce our finding that the hierarchical pathway, characterized by the hierarchy of band and grid cells, and their reciprocal interactions, is the core structure for path integration.
2.3.1 Continuous attractor dynamics of band cells
We now delve deeper into the internal structure and dynamics of band cells in the trained RNN.
5

---

b
d
90*
100
180
135
Percentage
120
.180
0
60
-135
45"
0
0.50
0.75
-90"
A (m)
Direction Score
9
80
0 =1880
0 =120
Average Weight
Count
Count
30
20
PC3
15
Time step
20
-20
0
-60
10
20
180
Prefer Direction
0
180
180
Prefer Direction
180
-20
PC1
20
-20
OpC2
Phase Difference
Figure 4: Properties of band cells in the trained RNN. (a) Illustration of the receptive field of an idealized band cells. Left panel: the receptive field consists of parallel bands, which are equally spaced with spacing À and have the same orientation A. Right panel: the positional separation between two band cells having the same spacing and orientation (blue and yellow) is quantified by phase 6, whose value is given by ø (A/A) x 2T, with A the offset between the adjacent bands of two band cells. (b) Band cells in the trained RNN are grouped into four clusters according to their spacing À and orientation 0. (c) An example band cell with direction-tuning æ. (d) Distribution of directional scores (a measurement of the direction-tuning level) of band cells in an example cluster. (c) Histograms of preferred directions of band cells in two example clusters, which have a bimodal shape peaked at either the cluster orientation 0 or its opposite direction 0 + .. (f) A 3D isomap embedding of the population activities of band cells in an example group with the same spacing and direction tuning. Coloured by time steps 5-20 of 200 trajectories. profile of band cells as a function of their phase difference. For detail calculations, see SI
(g) The averaged connection
Fig. 4h displays the receptive field of an idealized band cell, which consists of multiple equally spaced parallel lines (bands), on which the cell's response invariant. band cell can be characterized by three parameters: 1) spacing X, which is the distance between adjacent bands; 2) orientation 8, which the orientation bands band cells having the same spacing and orientation. Denote A E (-A/2, A/2) the offset between the adjacent bands of two band cells, their phase difference given by Φ (A/A) X π E (- T)
2D space; 3) phase which defines the positional separation between
First, we inspect the clustering property of band cells in the trained RNN. Using Fourier analysis, we extract the dominant spatial frequency of the receptive field of each band cell and obtain the cell's spacing and orientation (see SI for details). We find that band cells can be grouped into four well-separated clusters based on their spacing and orientation (Fig. 4p). For each cluster, we further differentiate cells" direction- direction-tuning by calculating how the averaged response of band cell varies with motion direction of the agent. Interestingly, we find that each cluster, the majority of band cells exhibit clear direction-tuninn (Fig. Fd- and they constitute two sub-groups with preferred directions along either the cluster orientation or opposite 0 + π (Fig. (4).
Second, inspired by recent successes in modeling spatially tuned neurons in the hippocampal-
set out to investigate whether the low-dimensional manifold of neural activity and its underlying connectivity conform CANN principles. Second, to uncover the latent structure and dynamics of band cells in each group having the same spacing and direction-tum ng. we performed dimensionality reduction dimensionality reduction analysis (Isomap, details shown in SI B on the activity trajectories of the group during path integration. Strikingly, we observed that the oowdiieesioona embedding of neural trajectories in each group forms a continuous ring (Fig. 4., consistent with the dynamics of a 1D CANN with periodicity. We also inspected the recurrent connectivity pattern between band cells, looking for the signature of the CANN structure. For each band cell group having the same spacing and direction-tuning calculated the averaged connection weights between cell pairs
entorhinal system using Continuous Attractor Neural Networks (CANNs) (27 28 29

---

1D CANN
k1
band
20 manitiid
colls
neuron
k2
C
d
Low
Ground-rruth
High
Decoded
Figure 5: A neural circuit model for path integration. (a) The structure of a band cell module, which consists of three sub- solppooulation:: pure band cells, direetion-coojuunctie band cells ut, and direction-conjuuctive band cells v- A band cell module performs ID path integration in the form of continuous attractor dynamics, along either the module orientation
or the opposite direction V = 0 + (b) Two band cells modules with different orientations support the grid cell module of the 2D CANN structure to accomplish path integration in the 2D space. (c) Examples of band and grid cells in the neural circuit model. (d) Inferred trajectory of the neural circuit model closely matches the ground-trute ttunh trajectory. demonstrating the path integration capability of the model.
function of their phase differences. The result shows that the connectivity profile exhibits approximately symmetric shape with weight decreasing with phase difference (Fig. 4g) reminiscent of the center-surround connection pattern ID CANN 30).
In summary, the above study reveals that band cells in the trained RNN constitute modules different spacing and dtrection-tuning and each module displays the 1D CANN-like structure and dynamics.
3 AI for neuroscience: the neural mechanism for path integration
The above RNN-basee navigation study also gives us insight into the path integration mechanism in the neural system. Band cells were reported in a neuroscience experiment [9] and an early computational model [6] (in which band cells were called was also proposed, which considered that band cells are responsible for ID path integration, and multiple modules of band cells of different orientations jointly support grid cells to execute path integration in the 2D space. Our RNN-based study tends to support this view, and challenges the conventional models of relying only on grid cells for path integration.
(VCO))
3.1 A neural circuit model for path integration
Inspired by the findings in the trained RNN, we construct neural circuit model for path integration. It has the similar idea as that in [S] 16), but have different mathematical formulations. In the below, we introduce the main structure of the model, with details presented in SI
3.1.1 The modules of band cells
Motivated by the properties band cells found in the trained RNN and the 1D head-direction system in Drosophila [2). we propose a theoretical model for band cells. Specifically, we consider that band cells are clustered into multiple modules with varying spacing À and orientation Each module consists of three sub-populations (Fig. 5p), which are: 1) pure band cells, 2) direction-conjunctive band cells v and 3) direction-conjunettve band cells w- and they are aligned on a ID manifold according to their preferred phases (-T, m]. Each module of band cells implements path

---

integration in ID direction, along either the module orientation ex = 0 or the opposite direction +
The pure band cell population forms a ID CANN, whose dynamics is given by.
tul (o((,1)+1 [ Jold, rald' d tp Øt
T0 Σ (m, 23) rm ,1) de' ( (0,)) 1),
(3)
where us, t) and ro( t) denote the synaptic input and firing rate of pure band cells at 6, respec- tively. Jo(0, d') denotes the recurrent connections between band cells, and Wm. with m = =. denotes the connections from direction-conjunctive cells TtH to pure band cells. Ig denotes the input from grid cells (to be defined below). The firing rate of pure band cells given by (b( of((, 1 k S of t) dø') with KG regulating the global inhibitory strength to ensure stable bump activity of band cells.
Conjunctive band cells x receive excitatory inputs from pure band cells and are tuned by the moving direction of the agent, whose dynamics are written as,
(o,t) Ot = (o, + wy ro(0, t).
(4)
(+ = Lg0 + (v(() k)u (6, 0)+
(5)
where u- (o, t) and (d, denote the synaptic input and firing rate of conjunctive cells at phase 6, respectively. wy denotes the connection from pure band to conjunctive cells at the same phase. k (sin 8 //,, cos 0os0/N) 0/2) denotes module-speciffe vector, and the projection of the agent velocity v() on it, v(I) k, defines the moving speed of the agent along the orientation 0. go is a baseline constant. [+]+ denotes rectification. 5q reflects that the firing rates of conjunctive cells contain the projected ID speed information the agent along the module orientation (31].
Importantly, set the recurrent connections between pure band cells to be symmetric and translation-invariinn, i.e., Jo (') / (v (2TT)))) exp [-1ø 2' / [2o]]] while the connec- tions from conjunctive cells to pure band cells are offset with constant value 8, i.e., W=(o, = ( exp | ø' F 513 (2o])] Together, three populations of band cells in a module achieve ID path integration along the module orientation 31
3.1.2 The module of grid cells
single band cell module supports path integration in two directions 0 and 0 n. To realize 2D path integration, grid cells integrate the outputs of band cell modules of different orientations. With loss of generality, we consider only two band cell modules in the present study (Fig. 5p). The orientations of two modules are denoted as 02 and 02. respectively, and they are 60 apart. Grid cells are aligned on 2D toroidal manifold with a phase vectord (Φ1, Ø1). They form 2D CANN with dynamics given by,
Tg Ou,(C (0, (0, /f Jo( Φ') (r do + (1(1,, t) I2(
(6)
Ōt
where ( 1) and rg( I) the synaptic input and firing rate of grid cells phase Φb The recurrent connections between grid cells are symmetric and translation-invariant, i.e., Jy(Ø, d') Jg / (2x00) exp [-1lØ- Φ'1 0'113/ (2x2)) with || -1-11 llg denotes circular distance the torus and 0g controlling the bump activity size. The firing rate of grid cells also follows divisive normalization, i.e., rg (1, !)= u (1b !) [1 + kg ff "g 1) dep I, (#1, and 12(02, represent the inputs from two band cell modules having orientations 0, and 02, respectively (to be defined below).
3.1.3 Interactions between band and grid cells
We set the reciprocal connections between band and grid cells to be symmetric, which are written as,
Wo
W Wge (ot, (²)- =
exp
k= 1,2,
g
20
8

---

where k = 1, indexes the band cell module.
Thus, the inputs from grid cells to each band cell module given by,
Lo t) /f Woy(dh x²') rgl00 t) dø"
k 1,2,
(8)
and the inputs from each band module to grid cells are given by
t) = -f" Wor(0' of rrllhh t) doh h: == 1,2.
(9)
The forward connections from band to grid cells enable the grid cell module to integrate ID motion trajectories into 2D, while the feedback connections from grid to band cells enable the circuit to amend errors. They jointly support accurate path integration in the 2D space.
3.2 Simulation results
We carry out simulations to test the performance of the neural circuit model (for details, see SI We observe that band and grid cells of varying spacings and orientations appear in the hand-designed model (Fig. pp. The moving trajectory inferred by the model agrees well with ground truth, demonstrating the path integration capacity of the model (Fig. 5
should be noted that the path integration capability demonstrated here builds upon well-established computational frameworks 31) and thus we do not perform additional tests on its robustness in this regard. Furthermore, the absence of accumulated error in this simulation is due to the idealized setting where neither input signals nor the model contain any random noise. In more realistic scenarios where noise is present, small errors would accumulate over time, leading to gradual drift in the estimated position; corresponding results under noisy conditions are provided in SIF
4 Discussion
In the present study, by using neuroscience prior knowledge and methods, we unfold the black box of RNNs trained to perform path integration task, and identify a hierarchical path integration pathway embedded in trained RNNs. Using a combination of analyses (neuron type classification, connection pruning, attractor dynamics inspection et al.), we uncover that along this hierarchical pathway, velocity information of an agent is first conveyed to band cells, band and grid cells then coordinate to carry path integration, and finally grid cells output the agent location. Furthermore, we reveal that band cells form multiple modules of varying spacing and orientation, and each module constitutes a ID CANN-like structure. Inspired by the RNN-based study and other neuroscience evidence, we formulate a neural circuit model for path integration. In this model, band cells form multiple functional modules in the form of ID CANNS, and each of them is responsible for path integration along the module orientation; grid cells form a 2D CANN, which integrates the ID results of band cell modules to earry out 2D path integration and meanwhile provides feedback band cells to correct errors. We demonstrate that this computational model works well.
Related neural circuit models. Previous neural circuit models for path integration based only on grid cells require a large amount of direction-conjunciii grid cells, whereby these conjunctive cells receive velocity inputs and shift the activity of the downstream pure grid cells along the same direction to perform location updating [T]. Compared to this model, which requires N pure grid cells and least 4N2 conjunctive grid cells represent four allocentric directions (north, south, east, and west), the hierarchical model we propose requires only 2NV pure band cells, 4N conjunctive band cells (2.V per each module), and 2 grid cells. Thus, the total number of neurons required to perform path integration is significantly reduced from 5.22 to 6N + N without sacrificing the performance. This computational efficiency offers new insight for the advantage of employing the combination of band and grid cells. To reduce the neuron number, another neural circuit model based only on grid cells considers that the recurrent connection weights between grid cells are velocity-dependent [4 This requires that neuronal connections are instantly tuned by the agent' velocity and is hence unlikely biologically feasible. Our neural circuit model has the similar structure as that in [55 16) but has different mathematical formulations.

---

Model predictions. This body of work also generates predictions testable in experiments, including: 1) band cells should be organized into clusters based on their orientation and spacing in the brain, analogous to the way grid cells are organized along the dorsal-ventral axis in MEC [33 2) band cells in the same module should further be organized into attractor networks, similar to head-direction cells in the thalamic nuclei and postsubiculum [4); 3) disruption of band cells should lead to deficits in path integration in downstream grid cells, potentially offering insights into the mechanisms underlying navigation impairments observed in the early stages of Alzheimer's s disease [35]. Investigating these computationally derived predictions using experiments will not only prompt re-evaluation of the funetional role of grid cells in providing spatial metric for the environment, but also advance our understanding of the computational contribution of band cells in neural circuits.
Limitation and future works. Several important aspects remain to be explored in future studies. For example, it not fully clear whether undefined cells in the trained RNN hold certain spatial tuning properties, such as those seen in head direction cells [36] or boundary vector cells [37], and whether they play role in path integration; and if so, which point along the hierarchical pathway they contribute. Additionally, the nature of interactions between band and grid cells in the trained RNN remains not fully solved, and needs to be inspected more thoroughly whether these interactions correspond directly to the connectivity profile in our proposed mechanistic neural model. Finally, the current training paradigm is restricted to path integration in physical space. It will be valuable to explore whether this framework is generalizable to path integration in abstract spaces (e.g., social network, value space in decision making) and whether the same hierarchical pathway is preserved 38
Cross-talk between AI and neuroscience. Al approaches are powerful tool to train network models for executing tasks, but they also face the challenge that the trained networks are often hard to interpret. Without prior knowledge, one has no cue to open up the black box. Here, we demonstrate that by using neuroscience prior knowledge and methods, we can uncover the detailed hierarchical pathway embedded in the trained RNNs for path integration. On the other hand, the RNN-based findings can advance understanding of the neural circuit mechanism underlying path integration in the brain. This study presents an example of using cross cross-field knowledge to facilitate Al interpretation and neuroscience research.
Acknowledgments and Disclosure of Funding
This work was supported by the National Natural Science Foundation of China (no. T2421004 to S.W.), the National Key Research and Development Program of China (204YFF12065500 the Science and Technology Innovation 2030-Brain Science and Brain-inspired Intelligence Project (no. 202 2021200200244 S. W.).), and the Wellcome Principal Research Fellowship (222457/Z/21/Z2, N.B.).
10

---

References
[1] Bruce McNaughton, Francesco P Battaglia, Ole Jensen, Edvard I Moser, and May-Britt Moser. Path integration and the neural basis of the' cognitive map'. Nature Reviews Neuroscience, 7(8):663-678 2006.
[2] Hugh Durrant- Durrant-Whyte Whyte and Tim Bailey. Simultancous localization and mapping: part i. IEEE robotics & automation magazine, 13(2):99-110, 2006.
[3] Mariana Gil, Mihai Ancau, Magdalene I Schlesiger, Angela Neitz, Kevin Allen, Rodrigo De Marco, and Hannah Monyer. Impaired path integration in mice with disrupted grid cell firing. Nature neuroscience, 21(1):81-911 2018.
[4] Torkel Hafting, Marianne Fyhn, Sturla Molden, May-Britt Moser, and Edvard I Moser. Mi- crostructure of a spatial map in the entorhinal cortex. Nature, 436(7052):801-866 2005.
[5] Johnson Ying, Alexandra T Keinath, Raphael Lavoie, Erika Vigneault, Salah El Mestikawy, and Mark P Brandon. Disruption of the grid cell network in mouse model of early alzheimer's disease. Nature Communications, 13(1):886, 2022.
[6] Emilio Kropff, James E Carmichael, May-Britt Moser, and Edvard Moser. Speed cells in the medial entorhinal cortex. Nature, 523(7561):419-444 2015.
[7] Francesca Sargolini, Marianne Fyhn, Torkel Hafting, Bruce L McNaughton, Menno P Witter, May-Britt Moser, and Edvard Moser. Conjunctive representation of position, direction, and velocity entorhinal cortex. Science, 312(5774):758-722 2006.
[8] Øyvind Arne Høydal, Emilie Ranheim Skytøen, Sebastian Ola Andersson, May-Britt Moser, and Edvard I Moser. Object-vector coding in the medial entorhinal cortex. Nature, 568(7752):4000- 404, 2019.
[9] Julija Krupic, Neil Burgess, and John Keeef.. Neural representations of location composed of spatially periodic bands. Science, (376660)) 2012.
[10] Neil Burgess, Caswell Barry, and John O'keefe. An oscillatory interference model of grid cell firing. Hippocampus, 1709):00-811, 2007.
[11] Yoram Burak and Ila R Fiete. Accurate path integration in continuous attractor network models of grid cells. PLOS computational biology, 5(2)=e10002)) 2009.
[12] Mark C Fuhs and David S Touretzky. A spin glass model of path integration in rat medial entorhinal cortex. Journal of Neuroscience, 26(16):4266 4276, 2006.
[13] Alexis Guanella, Daniel Kiper, and Paul Verschure. A model of grid cells based on twisted torus topology. International journal of neural systems, 17(04):231-244) 2007.
[14] Rylan Schaeffer, Mikail Khona, Tzuhsuan Ma, Cristobal Eyzaguirre, Sanmi Koyejo, and Ila Fiete. Self-supervised learning of representations for space generates multi-modular grid cells. Advances in Neural Information Processing Systems, 36:23 10-23140- 2023.
[15] Christopher P Burgess and Neil Burgess. Controlling phase noise in oscillatory interference models of grid cell firing. Journal of Neuroscience, 34(18))6224-6232, 2014.
[16] Daniel Bush and Neil Burgess. hybrid oscillatory interferenne/continuous attractor network model of grid cell firing. Journal of Neuroscience, 34(14):5065-507) 2014,
[17] Andrea Banino, Caswell Barry, Benigno Uria, Charles Blundell, Timothy Lillicrap, Piotr Mirowski, Alexander Pritzel, Martin Chadwick, Thomas Degris, Joseph Modayil, et al. Vector- based navigation using grid-like representations in artificial agents. Nature, 457 2018.
[18] Christopher Cueva and Xue-Xin Wei. Emergence of grid-like representations by training recurrent neural networks perform spatial localization. arXiv preprint 1003.v1810007700 2018.

---

[19] Dehong Xu, Ruiqi Gao, Wen-Hao Zhang, Xue-Xin Wei, and Ying Nian Wu. Conformal isometry of lie group representation in recurrent network of grid cells. arXiv preprint arXiv:2210.02684, 2022.
[20] William Dorell, Peter E Latham, Timothy EJ Behrens, and James CR Whittington. Actionable neural representations: Grid cells from minimal constraints. arXiv preprint arXiv:2209. 15563, 2022.
[21] Yedidyah Dordek, Daniel Soudry, Ron Meir, and Dori Derdikman. Extracting grid cell charac- teristics from place cell inputs using non-negative principal component analysis. Elife, 10040 2016.
[22] Ben Sorscher, Gabriel C Mel, Samuel A Ocko, Lisa M Giocomo, and Surya Ganguli. A unified theory for the computational and mechanistic origins of grid cells. Neuron, 111(1):211-337 2023.
[23] Vemund Schøyen, Markus Borud Pettersen, Konstantin Holzhausen, Marianne Fyhn, Anders MareheeS and Mikkel Elle Lepperød. Coherently remapping toroidal cells but not grid cells are responsible for path integration in virtual agents. Iscience, 26(11). 2023.
[24] Markus Pettersen, Vemund Sigmundson Schøyen, Mattis Dalsætra Østby, Anders Malthe- Sørenssen, and Mikkel Elle Lepperød. Self-supervised grid cells without path integration. bioRxiv, pages 2024-05, 2024.
[25] William Redman, Francisco Acosta, Santiago Acosta and Nina Miolane. Not so griddy: Internal representations of mns path integrating more than one agent. Advances in Neural Information Processing Systems, 37:22657-22688 2024.
[26]
Changmin Yu, Timothy EJ Behrens, and Neil Burgess. Prediction and generalisation directed actions by grid cells. arXiv preprint arXiv:2006.03355 2020.
[27] Tianhao Chu, Zilong Ji, Junfeng Zuo, Yuanyuan Mi, Wen-hao Zhang, Tiejun Huang, Daniel Bush, Neil Burgess, and Wu. Firing rate adaptation affords place cell theta sweeps, phase precession, and procession. Elife, 12:RP87055, 2024.
[28] Zilong Ji, Tianhao Chu, Si Wu, and Neil Burgess, A systems model of alternating theta sweeps via firing rate adaptation. Current Biology, 35(4):709-722, 2025.
[29] Zilong Ji, Tianhao Chu, Xingsi Dong, Changmin Yu, Daniel Bush, Neil Burgess, and Si Wu. Dynamical modulation of hippocampal replay sequences through firing rate adaptation. BioRxiv, pages 2024 09, 2024.
[30] Si Wu, Kosuke Hamaguchi, and Shun-ichi Amari. Dynamics and computation of continuous attractors. Neural computation, 20(4):994-1028 2008
[31] Wenhao Zhang, Ying Nian Wu, and Si Wu. Translation-equivarinn representation in recurrent networks with a continuous manifold of attractors. Advances in Neural Information Processing Systems, 35:15770-15783, 2022.
[32] Junfeng Zuo, Ying N Wu, Si Wu, and Wen-Hao Zhang. The motion planning neural circuit in goal-directed navigation as lie group operator search. Advances in Neural Information Processing Systems, 17:110999-110387, 2024.
[33] Hanne Stensola, Tor Stensola, Trygve Solstad, Kristian Frøland, May-Britt Moser, and Edvard I Moser. The entorhinal grid map discretized. Nature, 492(7427):72-78,, 2012.
[34] Jeffrey S Taube. The head direction signal: origins and sensory -otor integration. Annu. Rev. Neurosci,, 30(1):181-20) 2007.
[35] Coco Newton, Marianna Pope, Catarina Rua, Richard Henson, Zilong Ji, Neil Burgess, Christo- pher T Rodgers, Matthias Stangl, Maria-Fleni Dounavi, Andrea Castegnaro, et al. Entorhinal- based path integration selectively predicts midlife risk of alzheimer's disease, Alzheimer's & Dementia, 20(4):2779-279) 2024.
12

---

[36] Jeffrey S Taube, Robert U Muller, and James B Ranck. -irddiire cells recorded from the postsubieulum freely moving rats. description and quantitative analysis. Journal of Neuroscience, 10(2):420-4355 1990.
[37] Colin Lever, Stephen Burton, Ali Jeewajee, John O' Oeeefe, and Neil Burgess. Boundary vector cells the subiculum of the hippocampal formation. Journal of Neuroscience, 29(31):9771
9777, 2009.
[38] James CR Whittington, Timothy H. Muller, Shirley Mark, Guifen Chen, Caswell Barry, Neil Burgess, and Timothy EJ Behrens. The tolman-eichenbaun machine: unifying space and relational memory through generalization in the hippocampal formation. Cell, 18395):1249- 1263, 2020.
13

---

A Training details of RNN
To train RNNs on path integration tasks, we implemente an established computational frame- work adapted from the open-source repository developed by Ganguli Lab (https://github Code used for this paper is publicly available bandud/an___aa griirrdd hhierarcyy Vanilla RNN architectures are
at htttps://github com/ trained using the hyperparameters specified in Table [SI] Our weight matrices are initialized randomly using Py PoTch''ss default scheme and there are band-crafted localized initialization tricks. The trajectory used to train each iteration or test is sampled randomly from a distribution. The training loss is presented in Fig. ST
Table S1: Parameters used to train Vanilla RNNs
Parameter
Value
Arena size
(2.2m x 2.2m)
Average agent speed
0.1 m/sec
Place cells (Np)
512
0.12
T2
0.24 m
Recurrent units (Nn)
4096
Path length Epochs
20
100
Batch size
200
Batches per epoch
1000
Learning rate
10-1
12 regularization Optimizer
10 4
RMSProp
6.35
Value
6.30
6.25
6.20
W.A.M.A.
20
Epoch
Figure S1: Training loss for the vanilla RNN.
Place cell activity: The receptive field centers of place cells are randomly distributed across the square environment. Each place cell's 's firing activity is modeled using difference-off offfrrcaa--- tuning curve, analogous the Saarrenceeoo--uussiine function:
-1x- 1///0/1//20
e ||x-o;1//20
(t( DoG cf)
(10)
N e 0x
1x-2,11//22
where x the current location of the agent, c; denotes the center of the ith place cell's receptive field, and d and d2 parameterize the width of the center and surround, respectively. The network received velocity inputs derived from the simulated trajectory and was trained to produce the simulated place cell activities output.
Path integration error: Locations of the agent are decoded from the place cell activities by considering three maximally active place cells and averaging the x and y coordinates of their place
14

---

b
X=5.8
500
Count
300
200
10
15
Band scores 18.69 16.98
Band Score
Figure S2: (a) Band score distribution. (b) The top 128 band cells with the highest scores.
field centers. We quantified decoding accuracy using the mean Euclidean distance between the decoded positions and ground truth locations:
(x. 2,)² + (yi 9x)²
(11)
where (x, y1) denotes the true position and (f;, D.) represents the decoded position for the i-th sample.
B
Using neuroscience knowledge to analyse RNN
B.1 Rate map construction
Rate mapping-a fundamental technique in systems cerroocienne-qqunntiie how neuronal firing rates vary with spatial location, providing crucial insights into neural representations of space. When applied RNNs trained on path integration, this approach has revealed the spontaneous emergence of functionally specialized units analogous to biological grid cells, and band cells observed in biological navigation systems. To do this, we generate test batch (n 200 trajectories) as inputs the RNN and record the activation states of all recurrent layer units, the agent's 's current position and head direction at each step.
To construct spatial rate maps, we partition the environment into an m m grid, resulting in m? spatial bins, and compute the mean activation for each unit within each bin. Here we set m 20 to compute set of low resolution maps to use for evaluating grid score and set m 50 to compute set of high resolution maps to plot the resulting tuning curves To construct angular activation profiles, we partition the head direction ranging from -T to into 100 bins, and compute the mean activation for each unit within each bin.
visualize the spatial firing patterns.
B.2 Neuron type classification
Band cell: Ideal band cells exhibit defining electrophysiological signature characterized by spatially periodic firing fields that form bands across the environment (9. To quantify this, we developed spectral analysis method. We first compute the 2D Fourier transform of the spatial rate map, suppressing negative frequencies. The resulting power spectrum is fitted with 2D Gaussian model parameterized by amplitude (A), spatial frequency (k), orientation angle (0), and bandwidth (a) using constrained nonlinear optimization. The band score is calculated the normalized correlation between the actual spectral power distribution and the Gaussian model, weighted by a:
(F. G)
Band score
(12)
F G
where:
F is the spectral power matrix obtained from the 2D Fourier transform of the rate map:
F(u, v) [F(R(z, v)) shifted
(13)
15

---

with negative frequencies suppressed (F(u, v) = 0 for < 2
is the fitted 2D Gaussian model:
(x k cos Φ) + (y - l: sin ø)
Giz, v) Axxp
(14)
202
Parameters are optimized via:
k' x. 0²) argmin
G||
(15)
A.K,0.o
subject to &s E [0.2, 1], o E [0, x|. a E [0.05, 0.5]
The spatial frequency components are derived as:
ke cos o
k- sin 0
k
ky
(16)
2Ax
2Ar
where Ar is the spatial bin size.
Band score distribution show Fig. S2p and cells with a band score over 5.8 are classified as band cells. Example band cells are shown Fig. S2b.
Grid cell: Ideal grid cells exhibit defining electrophysiologich signature characterized by spatially periodic firing fields that form a hexagonal lattice tessellation across the environment [4] Grid score
was quantified through rotational analysis of the autocorrelogram derived from the spatial rate map. Specifically, the autocorrelogram underwent circular rotations in 30 increments, followed by
computation of Pearson correlations between each rotated version and the original map. The final grid score metric was calculated as:
Grid Score p(609) + p(1209) p(30") + p(909) ) + p(150%)
(17)
2
where p(0) represents the correlation coefficient after 9-degree rotation. This measure captures the hexagonal periodicity signature by comparing correlations at the theoretical peaks (60% and 120%) versus troughs (30%, 90", and 1 150") of an ideal grid pattern. Finally, cells with grid score of over 0.88 are classified as grid cells.
Undetined cell: Any cells that are neither band nor grid cells are classified as undefined cells.
B.3 Properties of band cells tuning
Spatial tuning: To quantitatively characterize band cell activity patterns, derive three key spatial parameters from the spatial rate maps:
Spacing X, the distance between adjacent bands, calculated by A 2TT / 12 + Agg where
kz and ky the same in Eq 16
Orientation 6 the orientation of bands in the 2D space, computed through the following
spectral analysis procedure:
0 =
angle
Σ 2(0))
+ n
mod π
(18)
2
where ((w) = lan Lan -Jy/ the orientation of frequency component w (constrained to (0, T). P(w) the spectral power at frequency w, the angle(-) handles the z-periodicity of band patterns, the (-) π r sequence performs the requested axis transformation, and the modulo π operation ensures angular continuity;
Phase é. the positional separation between band cells having the same spacing and orien- tation. Denote A E (-x/2, 2/2) the offset between the adjacent bands of two band cells, their phase difference is given by (A/A) म (-T, T).
As illustrated in Fig. S h the connection weight of band cells varies with their phase difference, reminiscent to the center-surround connection pattern.
16

---

a
1e-3
Average Weight:
Averoge Weight
Average Woight
90*
90*
2
0
0
180
-33
-5"
-30
0
-1
-1
927
Difference
Phase Differance
Phase Difference
C
d
80
=1800
= 50
30
Time step
Count
Count
Caunt
-0
40-
20
PC3
8
0
10
8
-60
-12
20
10
20
PC1
9c2
Prefor Diroction
Preter Direction
Prefer Direction
-2
Figure S3: (a) The averaged connection profile of band cells as a function of their phase difference. In figures and colors indicate distinct band cell clusters, and each cluster has the same spacing and orientation. (b) Example band cells with opposite preferred direction. (c) Histograms of preferred directions of band cells in three clusters, which have bimodal shape peaked at either the cluster orientation 0 or its opposite direction + æ. (d) 3D isomap embedding population activity from a band cell cluster.
Directional tuning: To quantitatively characterize band cell direction, calculate two parameters from the angular activation profiles:
The direction score quantifies directional tuning by fitting angular activation patterns y:())) with a circular Gaussian function:
Y Ymuneel (1b) = A cxp (- d(})
(19)
20
where d(b,, (4) ) min( - u4l, 2n b-b- sa)) is the circular distance between angle e and preferred direction μs, and determines tuning width. Parameters (A, 14, σ)) are estimated via nonlinear least-squares fitting initialized at [1, 0, 1]. The direction score is computed as the normalized projection between observed (ydata) and fitted (Ymodel) activation patterns:
Ydata Ymodel
Direction Score
(20)
| [mmodel]
The preferred direction & [-x, T) for each band cell i is derived from its angular activation profile (1())) through circular moment analysis, computed as:
No.
E, arg Σ ys (c
(21)
where 1/a = -- 2, (k O,.., No 1) defines No equally spaced angular bins, eit represents the unit vector complex space, and arg(.) extracts the phase of the resultant vector.
We find that, for each cluster with the same spacing and direction tuning, the majority of band cells form two subgroups with opposite preferred directions (Fig. S3b-c).
B.4 Population activity analysis
Neural population dynamics were analyzed using the following pipeline:
Data extraction: A test batch (n 200 trajectories) was generated as inputs to the RNN to generate the network's corresponding latent representations. Then neuron subset X was extracted (shape: T' x N, where I' is timepoints and N is neurons to be analyzed).
17

---

Preprocessing: Neural activity was standardized using z-score normalization:
X scaled scaler (Xaw))
(22)
Dimensionality reduction:
PCA projected data to principal components
Isomap further reduced to 3D (k 8 neighbors, geodesic distance metric)
Jowdim
(23)
The 3D neural manifold was plotted against animal trajectory (5-frame offset corrected) using time-indexed coloring (viridis colormap).
Visualization:
Fig. S3d displays alternative Isomap embedding for distinct band cell group not analyzed in the main text.
C Dissecting information flows in RNN
C.1 Pruning experiments
To systematically investigate the computational roles of recurrent layer units in path integration, we performed four distinet pruning experiments with rigorous controls:
For each pruning level n E {l...., 7003, we randomly selecte n units across cell types and nullify their velocity input weights:
Pruning velocity inputs:
M n1)
O,
i Sn
where S,, denotes size-n random subset of units, and M Nax2 encodes velocity-to- recurrent layer weights.
Pruning read-out connections: We prune output projections from randomly selected units by zeroing their readout weights:
w 0, jE Sn
with e NR representing the recurrent-to-on torrenttto-outuut weight matrix.
Pruning cell types: For comprehensive cell inactivation, we simultaneously nullified all incoming and outgoing connections of selected units:
J f O, J -, O, Vie Sn
where Nu the recurrent weight matrix.
Pruning connections between band and grid cells:
For the cell-type specific pruning analysis, we conducted bidirectional connection pruning between band (A) and grid (B) cell populations:
A- Pruning: Nullified all forward connections from random subset of n. band cells to every grid cell:
A. + O, Ak S., Vj E (1,..,,,N
B >A Pruning: Symmetrically ablated all feedback connections from random subset of n grid cells to every band cell:
B Ba,A, t O, vk E Sn, Vj E (1,..,, NA}
Error bars represent + standard deviation across n 30 independent repetitions, reflecting the variability in individual measurements.
18

---

C
Loss Value
8.4
8.2
80
80
b
Epoch
P
6.35
750
Loss Value
Band Count
500
250
620-
Baseline Reconstruction
Epoch
Figure S4: (a) Training loss for the RNN with predefined hierarchical structure. (b) Training loss for RNN for location reconstruction. (c) Example grid cells found the newly trained RNN for location reconstruction. (d) The number of band cells found in the Vanilla RNN (Baseline) and newly trained RNN (Reconstruction).
C.2 RNN with predefined hierarchical structure
We implement& hierarchical RNN architecture that takes the hierarchical pathway as prior (Fig. 4b). The model consists of two reciprocally connected modules:
First module (RNN1): Processes velocity inputs (v:) combined with feedback from the second module, designed emulate the properties of band cells:
T (Jir(") + J2r, (2) + +Mv.)
(24)
where (222 represents the feedback from RNN2, J1 and J2 represent the recurrent weights of RNNI and RNN2 respectively.
Second module (RNN2): Receives inputs from RNNI and generates spatial representations, mimicking grid cell functionality:
(1) 2+1 (Jr J2r,
(25))
We train this hierarchical RNN model on the standard path integration task using the same hyperpa- rameters specified in Table ST The training loss presented in Fig. S4 a.
C.3 RNN trained for location reconstruction
We modify the task from path integration pure location reconstruction by replacing velocity inputs of the vanilla RNN with place cell activations at each timestep, expressed as:
T++1 (Jr, Mp.)
(26)
where P. E R denotes the ground truth place cell activation pattern corresponding to current
position. All hyperparameters are the same as in Table ST The training loss is presented in Fig. S4 b.
After training the RNN on this modified task, we observe the emergence of grid-like activity patterns while notably lacking band cell representations (Fig. S4 c-d).
19

---

D Quantitative Performance Analysis
D.I Error Distribution Analysis
We quantified path integration performance across 30 independent trials for each of 10 different RNN implementations (Seeds 0-9), The results demonstrate consistent performance, with errors (normalized by arena size) reported as mean standard deviation (units: %):
Seed 0 2.09 + 0.06
Seed 5 2.07 + 0.04
Seed 1 : 2.09 + 0.05
Seed 6 2.08 + 0.05
Seed 2 2.08 + 0.05
Seed 7 2.09 + 0.06
Seed 3 2.07 + 0.05
Seed 8: 2.13 + 0.06
Seed 4 2.12 + 0.07
Seed 9 2.11 + 0.07
D.2
Computational Complexity Scaling
We analyzed the relationship between hidden layer size (NR) and path integration error, with errors (normalized by arena size) reported as mean standard deviation (units: %):
Nn 1024 2.57 0.14
NR 2048 2.30 + 0.08
NR 4096 2.09 0.06
NR 8192 2.06 + 0.05
This analysis demonstrates that our model achieves stable performance (< 2.19% error) with N 2 4096 neurons, suggesting our approach provides computationally efficient spatial representation.
D.3
Pruning Baseline
For each pruning condition, we prune connections selected from random groups as baseline to assess the relative importance of our specific pruning choices. The key results are presented in Table [S2 which demonstrates that pruning specific functional connections (input-to-band or grid-read-out) produces significantly larger errors compared to random pruning, suggesting specialized functional roles for these specific cell groups.
Table S2: Path integration error increase (mean SD across 30 trials when 750 connections are pruned). Error is measured by the standard deviation of location prediction emoos divided by the arena size (unit: %).
Pruning Type
Band cell
Grid cell
Undefined cell
Random cell
Pruning velocity inputs
2.32 + 0.43
0.11 + 0.07
0.14 + 0.09
0.46 0.07
Pruning read-out connections
2.67 = 0.50
9.15 + 0.71
0.30 = 0.11
0.81 + 0.22
E
Reproducibility
E.1 Experiments Across Different Seeds
we repeat the experiments across 10 random seeds for Vanilla RNN. Our results robustly demonstrate that band cells consistently serve as the primary recipients of velocity inputs and grid cells reliably encode spatial location information. The additional results are:
20

---

Neuron Counts and Pruned Connections:
Seed
Band Cells
Grid Cells
Undefined Cells
Pruned Connections
0
868
443
2785
443
818
526
2752
526
2
914
267
2915
267
3
823
146
3127
146
4
782
457
2857
457
5
854
442
2800
442
6
972
228
2896
228
7
535
324
3237
324
8
553
597
2946
553
9
810
394
2892
394
Pruning Performance (mean = SD):
Velocity Input Pruning:
Seed
Band
Grid
Undefined
0
1.8282 + 0.1026
0.0003 + 0.1069
0.0106 + 0.0990
2.5350 + 0.5888
+ 0.0401 0.0616
0.0475 + 0.0720
2
0.7015 + 0.0786
0.0247 0.0687
0.0248 + 0.0736
3
0.3086 + 0.0679
0.0022 1: 0.0744
4
2.2449 + 0.1073
0.0133
5
2.0076 = 0.0884
0.0217 = 0.0480
0.0172
6
0.6119 + 0.0963
0.0123 + 0.0683
0.0003
7
1.2673 + 0.0806
0.0242 4 0.0697
0.0603 + 0.0803
8
2.7330 + 0.1191
0.0411 + 0.0625
9
1.3750 + 0.1083
0.0124 + 0.0815
0.0225 + 0.0795
Read-out Connection Pruning:
Seed
Band
Grid
Undefined
0
0.9446 + 0.2144
4.2009 4 0.6104
0.1558 + 1297
1.7184 + 0.2905
7.4244 0.7040
0.4433 + 0.1778
2
0.4024 + 0.1543
0.7765
+ 0.1850
0.1569 + 0.1119
3
0.1272 0.0737
0.2362
0.1024
0.1058 + 0.0894
4
0.5466 + 0.1479
1.6829
+ 0.2364
0.2440 0.09
5
0.7177 + 0.2362
2.3296 0.3476
0.1877 + 1093
6
0.2365 + 0.1038
0.7850 + 0.2799
0.0657 + 0.0990
0.3730 + 0.1556
1.0181 + 0.2656
0.0985 0.0967
8
2.6436 + 0.3403
8.3318 + 0.6056
0.2729 + 0.1016
9
0.6141 + 0.1355
2.8032 + 0.5064
0.1566 0.1104
E.2
Different Training Paragram
To check the flexibility of our results, we trained three additional
RNN variants:
Vanilla RNN -+ I.STM transition (different net structure).
Xu et al. (2022) [19 training paradigm (different net structure (RNN and LSTM), place field (Gaussian), and loss function (conformal isometry and path integration loss)).
Petterson et al. (2024) (24) training paradigm (different in initial state (MLP), loss function (distance preservation and capacity loss), and decoder (none)).
The quantitative analysis of path integration performance across the three RNN variants are presented below, which demonstrates that velocity inputs are preferentially processed by band cells across all three RNN variants, as evidenced by their greater sensitivity velocity pruning. However, the absence of decoder modules in some variants prevented systematic pruning read-out experiments, limiting direct comparisons of read-out performance.
21

---

E.2.1
Vanilla RNN LSTM transition
The pruning experiments show that models degrade severely when velocity inputs band cells are pruned, but remain relatively stable when inputs to grid cells or undefined cells are pruned under the same conditions:
Neuron Counts and Pruned Connections:
Seed
Band Cells
Grid Cells
Undefined Cells
Pruned Connections
0
346
534
3216
346
258
276
3562
258
2
524
554
3018
524
394
305
3397
305
4
500
186
3410
186
5
313
136
3647
136
6
294
57
3745
57
7
324
225
3547
225
8
258
298
3540
258
9
363
124
3609
124
Velocity Input Pruning Performance (mean + SD):
Seed
Band Cells
Grid Cells
Undefined Cells
0
1.3948 + 0.1030
0.0070 + 0.0667
0.0809 + 0.0785
1.3914 + 0.1392
0.0261 + 0.0729
0.0708 = 0.0718
2
1.8552 + 0.1261
0.0638 + 0.0974
0.1554 + 0.1085
3
1.1153 = 0.1696
0.0715 + 0.1594
0.0511 = 0.1202
4
0.2639 + 0.0781
0.0128 + 0.0605
0.0268 + 0.0613
0.3274 + 0.0774
0.0150 + 0.0647
0.0046 0.0776
6
0.0811 + 0.0649
0.0067 + 0.0548
0.0036 + 0.0597
7
0.8140 + 0.1018
0.0022 + 0.0689
0.0610 1 0.0805
8
0.7389 + 0.0691
0.0303 + 0.0568
0.0905 0.0544
0.2011 9
+ 0.1214
0.0223 + 0.1245
0.0191 + 0.1048
E.2.2 Xu et al. (2022)
The pruning experiments reveal that band cells exhibit significantly greater sensitivity input manip- ulation compared to grid or undefined cells, which is consistently observed across all experimental seeds except seed I:
Neuron Counts and Pruned Connections:
Seed
Band Cells
Grid Cells
Undefined Cells
Pruned Connections
0
347
403
1050
347
313
424
1063
313
2
303
363
1134
303
316
369
1115
316
4
312
384
1104
312
311
382
1107
311
6
318
427
1055
318
7
302
452
1046
302
8
318
418
1064
318
9
312
401
1087
312
22

---

Velocity Input Pruning Performance (mean + SD):
Seed
Band Cells
Grid Cells
Undefined Cells
0
8.3573 + 1.3764
0.3688 + 0.2071
2.9723 = 1.1574
1
9.6032 + 1.7588
0.0395 + 0.2031
10.0732 + 1.8683
2
13.5328 + 2.1336
0.0767 0.1197
4.0376 = 0.6832
3
14.8651 1.8007
0.8835 + 0.2180
4.9369 = 0.8729
4
15.2718 = 2.3267
0.0024
2.6171 = 0.4573
5
10.3343 2.0101
4.1531 = 0.7071
6
9.5627 + 1.3337
#.19000
5.5746 0.9934
7
9.3729 + 1.6383
0.2143 + 0.3917
5.8331 = 1.1967
8
11.2179 + 1.9553
3.3365 + 1.3821
4.1106 0.7282
5.1514 + 1.3389 9
-0.6553 + 0.8521
3.9936 = 1.6025
E.2.3
Petterson et al. (2024)
Our 10-seed replication of Petterson al. (2024) demonstrates robust preferential processing of velocity inputs by band cells. The additional results are:
Neuron Counts and Pruned Connections:
Seed
Band
Grid
Undefined
Pruned Connections
0
37
196
23
23
1
26
200
30
26
2
26
209
21
21
3
38
194
24
24
26
194
36
26
5
38
203
15
15
6
24
202
30
24
7
35
201
20
20
8
34
205
17
17
9
36
202
18
18
Pruning Performance (mean + SD):
Seed
Band
Grid
Undefined
2.5472 + 0.3644
0.0368 + 0.0278
0.6565 + 0.0061
1
5.2265 + 0.0205
0.0198 + 0.0121
0.7424 + 0.1176
2
3.4828 = 0.2735
0.0297 0.0187
1.2256 + 0.0086
3
3.1603 + 0.3524
0.0147
0.6654 +:44:000 0.0072
3.3173 + 0.0183
0.0315 1 0.0250
1.5762 1:2::00333 0.3514
5
1.8896 + 0.2988
0.0122 + 0.0071
0.1567
6
3.6600 = 0.0223
0.0223 + 0.0114
1.5868 0.2246
7
2.8943 = 0.3955
0.0117 0.0074
0.4566 0,0030
8
2.5261 + 0.3254
0.0147 + 0.0099
0.3661 + 0.0035
2.3297 + 0.3849 9
0.0205 + 0.0176
0.4389 + 0.0034
Neural circuit model details
F1 Overview of the model architecture
We provide mathematical overview of the neural circuit model for 2D path integration, consisting of band cell modules and a grid cell module. Each component performs specific computations described below.
1. Pure band cell dynamics (1D CANN)
Each band cell module maintains bump of activity along ID phase space (-1,, T], via recurrent dynamics:
Oup(s,)) -us _up(),))pp/ Jo( 1) +p Σ / Øt
de' (g( f):
(27)
23

---

where us and rb are synaptic imput and firing rate of pure band cells. The recurrent connectivity is symmetric and translation-invariin invariant:
Jg
Jo
exp
(28)
V2TOb
20
Their firing rates are normalized to stabilize the bump:
n (0, L)
ro((, + kbp J " 1) dap
(29)
2. Direction-conjunctiv band cells
These cells receive excitatory imput from pure band cells and encode the agent's projected velocity along the module's preferred orientation 0 and its opposite direction 0 + æ. Their dynamics are defined as:
(p, -uu (0, () + wh ro(d, (),
(30)
Ø1 (0, Lgo (v(t) k)u=(0,t)) t)]4
(31)
where u (), and ri (o, are the synaptic input and firing rate of the conjunctive cells. The velocity projection vector is k = (sin 0 X, cos 0/)), determining the speed component along the module's axis. The firing rates reflect the movement direction and strength of the agent.
Feedback to pure band cells. Direction-conjunctive cells modulate the activity bump in pure band cells via asymmetric projections. The connection weights from conjunctive to pure band cells are:
W
6 0' O
exp
(32)
V2TO,
2
These connections are offset by 5 along the phase axis and thus induce a translation of the activity bump, driving ID path integration along the module orientation. Specifically, W+ promotes bump motion in the forward direction (0), and W_ in the opposite direction (0 + x).
Grid cells
Grid cells are organized on a 2D toroidal manifold and integrate inputs from two band cell modules with orientations 0, and 02, which are 60" apart. They form 2D continuous attractor neural network (CANN) with the following dynamics:
Fug (b, 1) Tg Öt
(d, JI J 6') rg( as' (01, + I2(02, t).
(33)
where Ug ug(0, t) and rg( t) are the synaptic input and firing rate of grid cell at phase & (Φ), d2). The recurrent connectivity Jg is translation-invarinn and shaped to support hexagonal grid patterns:
J
||
1 4')
exp
(34)
2TO
20
with firing rates given by a divisive normalization:
(b) t)
(0 () 1 kg IJ 12 dō
(35)
The distance metric ||- llg between grid cell phases is defined to produce hexagonal pattern. Specifically, letting AΦ = Φb, we apply periodic wrapping and linear transformation:
A wrap( o().
(36)
A DΦ2 wrap( db));
(37)
8 AdI,
(38)
5, = 2
1
Adz A
(39)
v3
11 - Φ'11,
82 + 8y:
(40)
24

---

where wrap(-) maps values to (- π to account for periodic boundary conditions. The linear transformation aligns the phase coordinates with the axes of hexagonal lattice.
This design ensures that the bump activity on the grid cell manifold has hexagonal structure rather than square one, consistent with biological grid cell tuning.
4. Reciprocal connections between grid and band modules
Bidirectional connections allow information exchange between grid and band cells.
From grid cells to band cells:
// Was rt(oo,, k= ==1, 1,2
Ig(dk.t) t)
(41)
From band cells to grid cells:
/ W ro(d))))))))))))))))))))))) do'
I& (
1,2
(42)
The weights are symmetric and phase-alignet
W Wor Φ) -
exp
(43)
V2TTogb
20 gb
F.2 Parameters
The model is composed of multiple path integration modules, each responsible for encoding spatial location a specific scale. Fach module consists of two band cell modules and one grid cell module. The two band modules within path integration module differ only in their preferred orientation (offset by 60%), but share all other dynamic and connectivity parameters. The five modules operate at distinct spatial scales ( which influence their velocity projection vectors (()) and resulting grid periodicity.
Within each module, band cells perform velocity integration using directionally tuned input, coupled through asymmetric recurrent and cross-module connections. Their outputs drive the activity of grid cells, which integrate spatial phase across the two orientations generate hexagonal grid patterns. The grid cell dynamics are shaped by their own recurrent connectivity and normalization mechanisms.
All modules share common set of parameters for band and grid dynamics, except for the spatial scale À and the orientation 8, which define the Shocity-ttophaae projection. These design choices allow for consistent multi-scale encoding of space, which is later used for spatial decoding via grid-to-place projection.
The table below summarizes all parameters used in the model and their hierarchical roles.
F.3
Simulation details
We performed two simulations to evaluate the functionality of the proposed neural circuit model for path integration.
Simulation I: Activity mapping in full environment. We simulated a freely foraging rat in square box environment, generating trajectory of 100,000 time steps with a fixed time step size At = 0.05. The simulated trajectory densely covers the entire spatial environment. At each time step, we recorded the activity of different neural populations in the model, including band cells (from each module) and grid cells. We then computed the average firing rate at each spatial location to obtain the spatial tuning maps (i.c., heatmaps) for different cells.
Simulation 2: Decoding trajectory from neural activity. In this simulation, we generated continuous trajectory of 1,000 time steps, again with A/ = 0.05. Using the spatial tuning maps (heatmaps) of grid cells obtained from Simulation 1, we decoded the animal's position at each time step from the instantaneous population activity of grid cells. The decoding procedure was carried out in two stages:
25

---

Table S3: Parameters of the path integration model. Each path integration module consists of two band cell modules (offset by 60 in orientation) and one grid cell module, with shared dynamics and connectivity parameters.
Symbol Description
Value
Global Parameters
At
Simulation time step
ms
nA
Number of path integration modules (scales)
Spatial scale of module
2.5 + 0.3i
0 ((2)
Orientation of the band cell modules
0%, 60
Band Cell Modules (per module, shared across scales)
Time constant of pure band cell dynamics Time constant of conjunctive band cell dynamics Neuron number of pure band cells
100 ms ms 180
T
No
N+
Neuron number of u* band cells
180
N
Neuron number of o band cells
180
J
Strength of recurrent excitation in pure band cells
1.1
Ob
Width of Gaussian kernel for recurrent/fecdforaar connections Inhibition strength in divisive normalization Strength of pure-to-conjunctive band cell connections
g
2
kb
10-4
wb
Strength of conjunctive-tte teeeetto-urreee feedback connections
0.2
8
Offset in phase space from direction-specific input oz
0.265
90
Baseline activity level of conjunctive band cells
0.2
Grid Cell Module (per module, shared across scales)
Tg
Time constant of grid cell dynamics
10ms
J
Strength of recurrent connectivity among grid cells Width of grid cell recurrent kernel
dg
g
kg
Inhibition strength in grid cell divisive normalization Strength of connection between band and grid cells
* 10
W
0.1
gb
gb
Width of Gaussian kernel for band-to-grid connections
2 π
Grid-to-place cell transformation: The grid cell activity vector at each time step was linearly projected onto population of place cells. Each place cell was assigned spatial location (zz y) on a uniform 2D grid. To define the synaptic connection between a grid cell and place cell, we first projected the place cell's spatial location into the phase space (ø1, 02) of the two band cell modules used to construct grid cells. This projection was done by computing the inner product of the place field center (z, y) with the two module wave vectors k₃ and Kg, i.e.,
Φx = 2n mod (ks (2,,),, - k 1
This yields the wrapped phases (#h, (b2) corresponding to the grid cell's toroidal phase space. The connection strength between each grid cell (with preferred phase ΦO") and place cell (located at (ar, y) with corresponding dp (1), 22))) is then defined by Gaussian kernel on the torus:
Wipe
P -001 0
W Pe Pe
2TT
exp
20
pe
where the toroidal phase distance - l1g is computed using the same hexagonal metric in the grid cell CANN. Specifically, for d = &
Φ" (wrapped to [-T,, T))
5,
50 v3/2
2
|d||
82 +
with 5, = di and 8, = dz.
This construction ensures that the grid-to-place connection matrix respects the periodic and hexagonal structure of the underlying grid cell code.
26

---

2. Population vector decoding: After obtaining the activity of place cells from the grid cell imput, we decoded the position by computing a population vector (center-of-man -fassss) over place field centers, weighted by the activity of each place cell at each time step.
We then evaluated the model's path integration accuracy by comparing the decoded positions with the trajectory. Results obtained under idealized conditions- model were free of random noise are presented in Fig. [ In more realistic noisy scenarios, small errors accumulate over time, resulting gradual drift the estimated position, illustrated in Fig. S5
both the input signals and the
a
b
Decoding Time
0.0150
0.0125
Ground-truth Decoded
Ener (unts))
0.0100
0.0075
0.0050
0.0000
Time
Figure S5: (a) Inferred trajectory of the noisy neural circuit model compared with the ground-truth trajectory. (b) Decoding error of the neural circuit under noisy conditions.
27

---

NeurIPS Paper Checklist
1. Claims
Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
Answer: [Yes]
Justification: The main claims made in the abstract and introduction truthfully summarize the paper's 's contributions and scope.
Guidelines:
The answer NA means that the abstract and introduction do not include the claims made in the paper.
The abstract and/or introduction should clearly state the claims made, including the contributions made the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers.
The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings.
It is fine to include aspirational goals as motivation long is clear that these goals are not attained by the paper.
2. Limitations
Question:
Does the paper discuss the limitations of the work performed by the authors?
Answer: [Yes]
The discussion section explicitly examines the study's limitations, ensuring transparent acknowledgment of the work's boundaries.
Justification:
Guidelines:
The answer NA means that the paper has limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.
The authors are encouraged to create a separate "Limitations" section in their paper.
The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (c.g.. independence assumptions, noiseless settings, model sell-speciiiatiinn asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be.
The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on few datasets or with few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.
The authors should reflect on the factors that influence the performance of the approach. For example, facial recognition algorithm may perform poorly when image resolution low or images are taken in low lighting. Or speech-to-tex, system might not be used reliably to provide closed captions for online lectures because it fails handle technical jargon.
The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.
If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and faimesss.
While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't 't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an impor- tant role in developing norms that preserve the integrity of the community, Reviewers will be specifically instructed to not penalize honesty concerning limitations.
3. Theory assumptions and proofs
Question: For each theoretical result, does the paper provide the full set of assumptions and complete (and correct) proof?
28

---

Answer:
[NA]
Justification:
The paper does not include theoretical results.
Guidelines:
The answer NA means that the paper does not include theoretical results.
All the theorems, formulas, and proofs in the paper should be numbered and cross- referenced.
All assumptions should be clearly stated or referenced in the statement of any theorems.
The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide a short proof sketch provide intuition.
Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material.
Theorems and Lemmas that the proof relies upon should be properly referenced.
4. Experimental result reproducibility
Question: Does the paper fully disclose all information needed to reproduce the main ex- perimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?
Answer: [Yes]
The paper provides sufficient methodological details, hyperparameters, and experimental setups to allow reproduction of the key results supporting its claims. Details are presented in Sec. and SI.
Justification:
Guidelines:
The answer NA means that the paper does not include experiments.
If the paper includes experiments, a No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not.
If the contribution is dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable.
Depending on the contribution, reproducibility can be accomplished in various ways. For example, if the contribution is novel architecture, describing the architecture fully might suffice, or if the contribution a specific model and empirical evaluation, it may be necessary to either make it possible for others to replicate the model with the same dataset, or provide access to the model. In general. releasing code and data is often one good way accomplish this, but reproducibility instructions for how to replicate the results, access to a hosted model (e.g.. in the case of large language model), releasing of a model checkpoint, other means that appropriate to the research performed.
also be provided via detailed
While NeurIPS does not require releasing code, the conference does require all submis- sions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. For example
(a) If the contribution is primarily a new algorithm, the paper should make it clear how to reproduce that algorithm.
(b) the contribution is primarily new model architecture, the paper should describe the architecture clearly and fully.
(c) If the contribution is a new model (e.g., a large language model), then there should either way to access this model for reproducing the results or way to reproduce the model (e.g., with an open-source dataset or instructions for how to construct the dataset).
(d) We recognize that reprodueibility may be tricky in some cases, in which authors are welcome to describe the particular way they provide for reproducibility. In the case of closed-source models, it may be that access to the model is limited in some way (e.g., to registered users), but it should be possible for other researchers have some path reproducing or verifying the results.
5. Open access to data and code
29

---

Does the paper provide open access to the data and code, with sufficient instruc- tions faithfully reproduce the main experimental results, as described in supplemental material?
Question:
Answer: [Yes]
Justification:
We provide the link to our publicly available Github repository the SI.
Guidelines:
The answer NA means that paper does not include experiments requiring code.
Please see the NeurlPS code and data submission guidelines (https://nips.com cc/
for more details.
While we encourage the release of code and data, we understand that this might not be possible, "No" an acceptable answer. Papers cannot be rejected simply for including code, unless this is central to the contribution (e.g., for new open-source benchmark).
The instructions should contain the exact command and environment needed to run to reproduce the results. See the NeurlPS code and data submission guidelines (https
/ priips.cc/publiccguiieee CoodeSumeeieeeeee ssi iessonPolll mollicyy for more details.
The authors should provide instructions on data access and preparation, including how to access the raw data, preprocessed data, intermediate data, and generated data, etc.
The authors should provide scripts to reproduce all experimental results for the new proposed method and baselines. If only subset of experiments are reproducible, they should state which ones are omitted from the script and why.
At submission time, to preserve anonymity, the authors should release anonymized versions (if applicable).
Providing much information possible in supplemental material (appended to the paper) recommended, but including URLs to data and code permitted.
6. Experimental setting/details
Question: Does the paper specify all the training and test details (e.g., data splits, hyper- parameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?
Answer: [Yes]
Justification: The paper thoroughly documents all the training and test details required to comprehend and replicate the reported findings in SI.
Guidelines:
The answer NA means that the paper does not include experiments.
The experimental setting should be presented in the core of the paper to a level of detail that is necessary appreciate the results and make sense of them.
The full details can provided either with the code, in appendix, or as supplemental material.
7. Experiment statistical significance
Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
Answer: [Yes]
Justification: The paper reports error bars and provides appropriate information about the statistical significance of the experiments, as evidenced in the presented figures and SI.
Guidelines:
The answer NA means that the paper does not include experiments.
The authors should answer "Yes" if the results are accompanied by error bars, confi- dence intervals, or statistical significance tests, least for the experiments that support the main claims of the paper.
The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall with given experimental conditions).
30

---

The method for calculating the error bars should be explained (closed form formula, call a library function, bootstrap, etc.)
The assumptions made should be given (e.g., Normally distributed errors).
It should be clear whether the error bar is the standard deviation or the standard error of the mean.
OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have 96% CI, if the hypothesis of Normality errors is not verified.
For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that out of range (e.g. negative error rates).
If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.
8. Experiments compute resources
For each experiment, does the paper provide sufficient information on the com- puter resources (type of compute workers, memory, time execution) needed to reproduce the experiments?
Question:
Answer: [No]
The manuscript does not explicitly report computational resource requirements, the lightweight architecture of the proposed model renders such specifications essential for reproducibility.
Justification:
Guidelines:
The answer NA means that the paper does include experiments.
The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage.
The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute.
The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make it into the paper).
9. Code of ethics
Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https: ftctps://neurips: wttpiipps.cpppuaaae cc/ppublic/thhiccsu.d........
Answer: [Yes]
Justification: The research fully complies with the NeurIPS Code of Ethics in all aspects by ensuring transparency, reproducibility, and fairness in reporting experimental results.
Guidelines:
The answer NA means that the authors have not reviewed the NeurIPS Code of Ethics.
If the authors answer No, they should explain the special circumstances that require deviation from the Code of Ethics.
The authors should make sure to preserve anonymity (e.g., there special consid- eration due to laws or regulations in their jurisdiction).
10. Broader impacts
Question: Does the paper discuss both potential positive socictal impacts and negative socictal impacts of the work performed?
Answer: [NA]
Justification: There is no societal impact of the work performed.
Guidelines:
The answer NA means that there no societal impact of the work performed.
If the authors answer NA or No, they should explain why their work has no societal impact why the paper does not address societal impact.
31

---

Examples of negative societal impacts include potential malicious unintended uses (e.g., disinformation, generating fake profiles, surveillance), faireess considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.
The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster.
The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional unintentional) misuse of the technology.
If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML).
11. Safeguards
Does the paper describe safeguards that have been put in place for responsible release of data models that have high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?
Question:
Answer: [NA]
Justification:
The paper poses such risks.
Guidelines:
The answer NA means that the paper poses no such risks.
Released models that have high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere usage guidelines restrictions to access the model or implementing safety filters.
Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images.
We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make best faith effort.
12. Licenses for existing assets
Are the creators or original owners of assets (e.g.. code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?
Question:
Answer: [Yes]
Justification: We appropriately acknowledge all third-party code and data sources through proper citations of original works and the code base is also linked in SI.
Guidelines:
The answer NA means that the paper does use existing assets.
The authors should cite the original paper that produced the code package or dataset.
The authors should state which version of the asset is used and, if possible, include URL.
The name of the license (e.g., CC-BY 4.0) should be included for each asset.
For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided.
32

---

If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswi thcode. com/dataseta
latasets.papersstt
has curated licenses for some datasets. Their licensing guide can help determine the license of dataset.
For existing datasets that are re-packaged, both the original license and the license of the derived asset (if has changed) should be provided.
this information is not available online, the authors are encouraged to reach out to the asset's s creators.
13. New assets
Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
Question:
Answer: [Yes]
The completeness of documentation for newly introduced assets will be verified upon release the data and code on GitHub, ensuring reproducibility requirements are met.
Justification:
Guidelines:
The answer NA means that the paper does release assets.
Researchers should communicate the details the dataset/code/mooel as part their submissions via structured templates. This includes details about training, license, limitations, etc.
The paper should discuss whether and how consent was obtained from people whose asset used.
At submission time, remember to anonymize your assets (if applicable). You can either create anonymized URL or include anonymized zip file.
14. Crowdsourcing and research with human subjects
Question: For crowdsoureing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well details about compensation (if any)?
Answer: [NA]
Justification:
The paper does not involve crowdsourcing nor research with human subjects.
Guidelines:
The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.
Including this information the supplemental material is fine, but if the main contribu- tion of the paper involves human subjects, then as much detail as possible should be included the main paper.
According to the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector.
15. Institutional review board (IRB) approvals or equivalent for research with human subjects
Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?
Question:
Answer: [NA]
Justification:
The paper does not involve crowdsoureing nor research with human subjects.
Guidelines:
The answer NA means that the paper does not involve crowdsoureing nor research with human subjects.
33

---

Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.
We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution.
For initial submissions, do not include any information that would break anonymity (if' applicable), such as the institution conducting the review.
16. Declaration of LLM usage
Does the paper describe the usage of LLMs if is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, originality of the research, declaration is not required.
Question:
Answer: [NA]
Justification: The core method development in this research does not involve LLMS as any important, original, non-standard components.
Guidelines:
The answer NA means that the core method development in this research does not involve LLMs as any important, original, non-standard components.
Please refer to our LLM policy hatt:p///earrrppcc///ooeeeeeeee/ //t/////ooeeeeeccc////////..1. for what should or should not be described.
34


Reviews:
[{"review_id": "zhB4MRmj38", "reviewer": "Reviewer_nMie", "summary": "An approach based on recurrent neural networks is taken to realise path integration task. From the trained model, difference types of neurons are extracted and connectivity is probed based on the attractor dynamics of the system, for which porpuse various pruning processes are employed, As later specifically defined connections are reintroduced it is not clear whether a hierarchical information processing structure has emerged or was embedded or whether a combination of both was studiod.", "questions": "How where the connection matrices initialised? \n\nCan you provide more information about the pruning steps? How are connections selected for pruning? Are the main results based on prunung or is pruning in place only to analyse the network(s). Why are some connecitons later explicitly reinitialized (from line 234 and after line 253)?\n\nCan you provide more information on the statistics of the reconstruction error?\n\nWhat and how many paths have been used for trainnig? Is the path length (appendix) in meters? How smooth were the paths? Did they have perferrential directions (as suggested by Fig. S2).\n\nHow does your model compare quantitatively to esxisting biologically inspired path integration algorithms? How to technical path integration algorithms?", "limitations": "The authors show clearly that they are aware of some of the limitations of the approach.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "“Path integration is essential for spatial navigation.” As spatial navigation is possible also purely visually or based on odour gradients, this statement may not as immediately be acceptable as it would be expected at the beginning of the abstract.\n\n“clarifying”: Do you mean “identifying” or “classifying” here? \n\n“challenges the conventional view of considering only grid cells for path integration” which is not only misleading in the logical sense that grid cells alone will not be sufficent without other cells, but it is also conceptually questionable as simple models should be given preference unless they leave part of the data unexplained, but here no attempt is made to demonstrate this. \n\nThe approach is interesting, but its explainatory power is remain unrealised. Many theoretical studies have been conducted on the hexogonal patters (or stripes or localised activations) since the papers of Amari in the 1970s, so many ideas can be theoretically checked. However to achieve this in a single optimization process is difficult and requires more evidence, statistics and explanation.\n\nHow is the “module” of a particular cell type defined based on the analysis of the network sturcture?\n\nBPTT is known to be a learning algorithms that is not always easy to handle. In order to substantiate your research you need to provide quanitative results on the learning progress and share your experiences related to the parameter optimization process, but you don’t resolve the acromyn and don’t give a reference.\n\nIn particular, it does not seem plausible given the provided informatiohn that the patterns shown in Fig. 1c can be obtained, unless you initialize the matrix J in some localized manner, but initialization of the weight is not properly discussed. So, it seems the content of the “black box” is not entirely unknown although the prior knowledge that was used in the initialisation was not shared with the reader. \n\nFurthermore, it is  confusing that after the presentation of the result, an explicite (i.e. not be training of any kind) setting of some of the connections is described from line 234.\n\nFig. 5d is the only evidence for the function of the proposed model, but no quantitiative results are included. Can you explain why the error remains (more or less, this cannot be guess with certainty from the picture) constant over time although not corrective information is added from the environment apart from the current velocity. Under such conditions the precision would be expected to (more or less quickly) decrease over time.\n\n228: What are “cognitive cells”?\n\nGood discussion, but also this part shows that this contribution may be more relevant for a workshop than for a conference.", "was_revised": true, "final_justification": "While the other questions were reasonably addressed, the paper needs In order to be acceptable additional numerical simulations, which seems possible towards the final version. However, this may not be intended by the authors (the rebuttal should not exclude additional figures), therefore a only borderline acceptance seems justifiable.", "rebuttal": "Thank you for your time and comments. We believe there are some misunderstandings of our manuscript and we will address your comments point by point.\n\n1.  “Path integration is essential for spatial navigation.” As spatial navigation is possible also purely visually or based on odour gradients, this statement may not as immediately be acceptable as it would be expected at the beginning of the abstract.\n\nYes, odour gradients is important for spatial navigation for rodents (those good at using odour information). However, path integration is still essential for spatial navigation, in particular for constructing a cognitive map, where motion information is necessary for connecting far away spatial locations (visual/odour cues can only provide local information). Also, path integration is essential for navigating in cue-poor environments - sea, desert etc, and also in darkness. We believe this is a commonly accepted knowledge in the field.\n \n2.  “clarifying”: Do you mean “identifying” or “classifying” here?\n\nWe meant \"identifying\". Thanks.\n\n3. “challenges the conventional view of considering only grid cells for path integration” which is not only misleading in the logical sense that grid cells alone will not be sufficent without other cells, but it is also conceptually questionable as simple models should be given preference unless they leave part of the data unexplained, but here no attempt is made to demonstrate this.\n\nThank you for pointing this out. First, we rephrased this sentence to:\n“While prior frameworks posited grid cells as the principal velocity integrator, our results suggest a critical role of band cells in driving path integration.” \n\nIn many theoretical models of path integration (e.g., Burak et al., 2009, Giocomo et al., 2011), grid cells were modelled as the main substrate for integrating velocity inputs to perform path-integration. However, this does not mean grid-cell based model is the simpler one. We have attempted to show this point from two aspects: 1). From the veiw of efficient computation, we compared our hierarchical band–grid network to a pure grid-cell model in performing 2D navigation and showed that the hierarchical network requires only 6N + N² neurons—versus 5N² for the pure grid-cell scheme—to achieve equivalent performance, highlighting a significant efficiency advantage. 2). From the view of our prunning analysis, we found that band cells, rather than grid cells, are primarily responsible for processing velocity information. Additionally, there are biological evidence for band cells (Krupic et al. 2012), located in the parasubiculum, one synapse upstream of MEC. Furthermore, a theoretical work has shown that they could perform 1-D path integration (Burgess 2007; Bush et al, 2014). Our main contribution lies in depicting the band-grid cell relationship in performing path-integration in a normative model (part one in the paper) and inspired from which, we proposed a mechanistic circuit model for performing path-integration in the neural system (part two in the paper). \n\n4. The approach is interesting, but its explainatory power is remain unrealised. Many theoretical studies have been conducted on the hexogonal patters (or stripes or localised activations) since the papers of Amari in the 1970s, so many ideas can be theoretically checked. However to achieve this in a single optimization process is difficult and requires more evidence, statistics and explanation.\n\nWe are not quite sure we understand this comment, but we try to answer it. First of all, we did not try to achieve a mechanistic model from a single optimization of a RNN. This paper can be essentially devided into two parts. Frist is the RNN dissection (Sections 2), where we conducted analyses (cell-type identification, pruning connections, attractor dynamics analysis et al.) to show how a standard path-integration RNN develops band cells, grid cells, attractor dynamics, and the hierarchical information processing. Second is the construction of a neural mechanistic model inspired by the first part (Section 3), where we built continuous-attractor network based hierarchical path integration network, preserving the key computational motifs (band cells and grid cells) in a fully analyzable framework. We will clarify the above points in the revised paper. \n\n5. How is the “module” of a particular cell type defined based on the analysis of the network structure?\n\nThe definition of \"module\" of band cells is described in Section 3.1 and illustrated in Figure 4, specifically: we applied Fourier analysis to each band cell’s receptive field to extract its dominant spatial frequency, which allows us to group band cells into four distinct modules based on spacing and orientation.\n\n6. BPTT is known to be a learning algorithms that is not always easy to handle. In order to substantiate your research you need to provide quanitative results on the learning progress and share your experiences related to the parameter optimization process, but you don’t resolve the acromyn and don’t give a reference.\nIn particular, it does not seem plausible given the provided informatiohn that the patterns shown in Fig. 1c can be obtained, unless you initialize the matrix J in some localized manner, but initialization of the weight is not properly discussed. So, it seems the content of the “black box” is not entirely unknown although the prior knowledge that was used in the initialisation was not shared with the reader.\n\nThanks for your comments. We simply adopted the training paradiagm from Banino et al. (2018), which is a well-established framework to robustly show grid-cell-like pattern in trained RNN, and has been used widely in several other studies (e.g., Sorscher et al., 2023, and Schaeffer et al., 2022). We have not introduced any new training tricks to obtain the grid\u001f and band\u001fcell patterns in Fig. 1c—these patterns emerge naturally under the published RNN framework reported in Banino et al.\n\nFurthermore, our recurrent weight matrix J is initialized randomly using PyTorch's default scheme: a uniform distribution with range scaled by the inverse square root of the input dimension (i.e., U(−1/in_features, 1/in_features)). There are no hand\u001fcrafted or localized initialization tricks. Details of this procedure and all hyperparameters are provided in Supplementary Section S1.2. Other works (e.g., Sorscher et al., 2023, Fig. S3B) has already reported Fig. 1c–style activity using the same protocol, hence no special initialization is required. But please note that previous works only found band cells, but they did not go further to analyze the properties and computational roles of band cells, and the hierarchical pathway for path-integration as we did in this work.\n\n\n7. Furthermore, it is confusing that after the presentation of the result, an explicite (i.e. not be training of any kind) setting of some of the connections is described from line 234.\n\nAs we replied in the comment 4, our study includes two stages:\n●Section 2 ( neuroscience-driven interpretation of the black box of RNN):\u000bWe analyzed the RNN trained for path integration, using pruning, Fourier/Isomap, and connectivity probes et al. to interpret the emergent cell types, attractor dynamics, and the information processing pathway. \n●Section 3 (RNN-inspired neural modelling):\u000bInspired by the RNN study and incooperating previous neuroscience modellings, we constructed a neural circuit model based on continuous attractor networks for path integration. In this part, we only took the key computational motifs (band cells, grid cells, and the hierarchical pathway) found in the RNN study to propose mechanistic circuit model for neural path-integration.\n\n8. Fig. 5d is the only evidence for the function of the proposed model, but no quantitiative results are included. Can you explain why the error remains (more or less, this cannot be guess with certainty from the picture) constant over time although not corrective information is added from the environment apart from the current velocity. Under such conditions the precision would be expected to (more or less quickly) decrease over time.\n\nWe apologize for not making this point explicit: our hierarchical model’s zero-drift performance in Fig 5d is not a lucky simulation result but a direct consequence of the continuous-attractor-network (CANN) theory. By splitting the 2D integration into two independent 1D attractors, each subnetwork supports a perfectly stable activity “bump” that integrates velocity inputs exactly in the absence of noise—a fact rigorously proved previously (Amari 1977; Samsonovich & McNaughton 1997; Burak & Fiete 2009). We will add a brief theoretical note in the revised manuscript to clarify why no error accumulation is expected under these ideal conditions. To validate our model further, we will also include quantitative RMSE-overtime curves for distinct moving trajectories. \n\n\n9. 228: What are “cognitive cells”?\n\nSorry for the typos. We meant \"conjunctive cells\", which refers to the cells that are both tuned to moving direction and spatial location.\n\nWe hope our replies clarify our contributions and resolve misunderstandings. We hope the reviewer will find the revised manuscript improved and we are looking forward to your feedback."}, {"review_id": "I54sVt2Zag", "reviewer": "Reviewer_bx3s", "summary": "This paper analyzes the circuits and computations behind RNNs performing path integration. Through a pruning analysis, the authors reveal the central role of band cells, challenging the predominant role of grid cells that are thought to be central for path integration. Through that same pruning analysis, the authors extract the circuit of path integration, where band cells receive the velocity inputs, then grid cells and band cells interact to process the inputs, and then grid cell output the result. Lastly, the authors focus on band cells, study the organization of their modules, and finally propose a dynamic neural mechanism for the band cells, that is tested with simulations. The authors propose testable experiments for neuroscience.", "questions": "Questions/Suggestions for actionable improvements that would convince me to update my score:\n\n1. Discuss the different RNN architectures that have been used to study path integration and explain why you chose [17] over others.\n    \n2. Clarify the role of border cells in path integration, ie what is known from the neuroscience and from RNN studies. Explain why they do not appear in your model, and whether that is a limitation (or not).\n    \n3. Clarify how many RNNs have been trained, and whether the findings from the study are consistent across trained RNNs. \na) Update Figure 2 to show results across trained RNNs, including error bars for the error wrt connections pruned. \nb) Update Figure 3 to show error bars in the band count and grid count across experiments.\n    \n4. Add quantitative results in Figure 5, showing how well the simulation is able to perform path integration, including error bars across several runs.", "limitations": "Yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths\n\nS1. The paper is very well written and easy to follow. The figures are really well done: they strike the perfect balance between being simple enough yet very explanatory.\n\nS2. The authors reveal a circuit of computations between band cells and grid cells, challenging the known role of both cells.\n\nS3. The authors propose a neural mechanism for the role of band cells.\n\nS4. The authors propose testable experiments for experimental neuroscience.\n\nWeaknesses\n\nW1. The results are only valid for one type of RNN trained to do path integration [17]. It is unclear whether they would extend to other types of RNNs that have also been proposed to do path integration.\n\nW2. Some RNNs trained to do path integration are known to exhibit border cells: how do these cells enter the theory proposed by the authors? Why haven’t they been observed in the experiments? Are they included in the \"undefined category\"?\n\nW3. It is unclear how many RNNs have been trained for this analysis, and whether results are stable from one RNN to the next. Some figures, eg Fig 2 and 4), refer to “the” trained RNN: how can we know that this RNN hasn’t been cherry picked?\n\nW4. Likewise, it is unclear how many experiments have been run for the simulation of the proposed neural mechanism for path integration, as the simulation results section is very short in the main text.\n\nMinor:\n\nThe two sections “neuroscience for AI” and “AI for neuroscience” do not really help the reader follow the paper, and the wording “AI” seem unnecessarily buzzword-y.\n\nThe acronym CANN has not been defined.", "was_revised": true, "final_justification": "The authors have provided a very comprehensive rebuttal. Their addition of new RNN architectures and several seeds for each run convinced me of the significance of the work. I have increased my score.", "rebuttal": "We thank Reviewer bx3s for insightful comments. Below we address them point by point:\n\n1. **The results are only valid for one type of RNN trained to do path integration [17]. It is unclear whether they would extend to other types of RNNs that have also been proposed to do path integration.**\n\n   Results presented in our paper extend to many other types of RNNs.  \n   The RNN we used is based on the one reported in Banino et al. (2018), which has been used and extensively analyzed in several other studies such as Sorscher et al., 2023, to explain the emergence of grid-cell-like firing patterns. To check the flexibility of our results, we trained three additional path-integration RNN variants:\n   1. **Vanilla RNN → LSTM transition** (different in net structure).  \n   2. **Xu et al. (2022) training paradigm** (different in net structure (RNN and LSTM), place field (Gaussian) and loss function (conformal isometry and path integration loss)).  \n   3. **Petterson et al. (2024) training paradigm** (different in initial state (MLP), loss function (distance preservation and capacity loss) and decoder (none)).  \n\n   Importantly, all the results are preserved while changing the RNNs. Specifically:  \n   1. We observed the emergence of band cells.  \n   2. Pruning analysis revealed a hierarchical information flow from band cells to grid cells.  \n\n   These results indicate that our findings are not simply an idiosyncrasy of a single RNN implementation, but rather a robust and general computational strategy for path integration across diverse architectures and training regimes.\n\n   We thank the referee for this insightful comment and will add these results in the revised paper.\n\n2. **Some RNNs trained to do path integration are known to exhibit border cells: how do these cells enter the theory proposed by the authors? Why haven’t they been observed in the experiments? Are they included in the \"undefined category\"?**\n\n   Thanks for pointing this out. We did observe border cells in the trained RNN (and also in several RNNs we included for response to comment 1). They belonged to the \"undefined category\". We think border cells (boundary vector cells) may not directly link to the path-integration theory we focus on here, but they do have functions in spatial navigation such as error correction of path integration (Hardcastle et al., 2015). Due to space limits, we did not report them in the main text.\n\n   We will add the analysis of border cells in the Supplementary Information (SI). In more detail, we identified 201 border cells and performed pruning analyses similar to those for other cell types, and found:\n\n   a. **Identification**: Used the standard boundary‑vector‑cell metric (Solstad et al., 2008)—we computed each unit’s firing‑rate correlation with the agent’s distance to the nearest wall and applied a significance threshold to classify border‑cell–like units.  \n   b. **Pruning protocols** (Section 2.2.2, Fig. 2):\n   - **Velocity‑input pruning**: Removing velocity inputs to border cells has negligible impact on path‑integration error—unlike pruning band‑cell inputs, which causes a large performance drop.  \n   - **Place‑cell readout pruning**: Eliminating border‑cell→place‑cell connections significantly alters decoded‑position accuracy, indicating border cells contribute boundary information to the place‑cell map.  \n   - **Recurrent‑connection pruning**: Among all recurrent projections, pruning band→border connections yields the greatest increase in integration error, showing border cells primarily read out band‑cell dynamics to encode proximity to walls.\n\n   **Table 1.** Path integration error increase of pruning velocity inputs to each cell group and pruning read‑out connections from each cell group. Data are shown as mean ± SD across 30 trials when 201 connections are pruned (units: m).\n\n   |                                     | band cell      | grid cell      | border cell     | undefined cell  |\n   | ----------------------------------- | -------------- | -------------- | --------------- | --------------- |\n   | pruning velocity inputs             | 0.0087 ± 0.0024 | 0.0009 ± 0.0023 | 0.0006 ± 0.0019 | 0.0006 ± 0.0017 |\n   | pruning read‑out connections        | 0.0034 ± 0.0029 | 0.0022 ± 0.0024 | 0.0176 ± 0.0048 | 0.0015 ± 0.0022 |\n\n   **Table 2.** Path integration error increase of pruning the connections *from* border to band and *to* border to grid cell, respectively. Data are shown as mean ± SD across 30 trials when 201 connections are pruned (units: m).\n\n   |                    | band cell       | grid cell       |\n   | ------------------ | --------------- | ---------------- |\n   | from border cell   | 0.0027 ± 0.0016 | 0.0015 ± 0.0018  |\n   | to border cell     | 0.1741 ± 0.0152 | 0.0065 ± 0.0022  |\n\n   These supplementary analyses confirm that border cells—with their environment‑tuned boundary coding—play a complementary role (distilling band cell outputs into boundary signals). We will include these results in the revised SI and a brief discussion of border-cell analysis in the revised manuscript.\n\n3. **It is unclear how many RNNs have been trained for this analysis, and whether results are stable from one RNN to the next. Some figures (e.g., Fig 2 and 4) refer to “the” trained RNN: how can we know that this RNN hasn’t been cherry‑picked?**\n\n   Thank you for pointing this out. This RNN was not cherry‑picked. We tested our network under multiple random initializations using the same analysis pipeline and found highly consistent results, demonstrating that the hierarchical path integration we discovered is a general mechanism for RNN‑based path integration. This consistency holds even across different training paradigms (see response to comment 1). We will include in the SI analyses of three additional RNNs, and we will release all training and analysis code to guarantee full reproducibility.\n\n4. **Likewise, it is unclear how many experiments have been run for the simulation of the proposed neural mechanism for path integration, as the simulation results section is very short in the main text.**\n\n   The proposed neural mechanism is grounded in continuous attractor network theory (see Amari (1977), Samsonovich & McNaughton (1997), and Burak & Fiete (2009)), which has been rigorously studied. This is why we did not focus extensively on simulation details in the main text—these models are well established. However, we will add more discussion in the revised paper to clarify this.\n\n   We will also include in the SI results of multiple simulation experiments across a variety of velocity input sequences (different movement trajectories) and a broad parameter regime to help readers better understand our hierarchical CANN model.\n\nWe hope our responses have addressed your concerns. Please let us know if anything remains unclear."}, {"review_id": "TrFCUAWdWT", "reviewer": "Reviewer_pJcG", "summary": "Previous studies have implicated various neural substrates, especially the medial entorhinal cortex (MEC), in path integration. While previous RNN-based models showed that networks trained for path integration could exhibit grid-cell-like activity, later work suggested that path integration can occur even in the absence of grid patterns, leaving the precise neural mechanism for path integration largely unknown\n\nIn this work, the authors trained a recurrent neural network (RNN) on a path integration task and conducted ablation analyses to probe the functional roles of different emergent cell types. By selectively pruning connections, they found that band cells process velocity input, while grid cells aggregate band responses to decode position, suggesting a hierarchical structure.\n\nTo better understand this hierarchy beyond the RNN, which is usually a black box, the authors proposed a mechanistic neural circuit model. The model consists of pure and conjunctive band cells forming 1D continuous attractors, which project into grid cells arranged on a 2D toroidal manifold. A subsequent simulation showed that the model reproduced band and grid cell patterns of varying orientation and spacing, and could accurately infer movement trajectory, supporting its validity for path integration.\n\nThis study provides important computational insights and a solid theoretical framework that bridges RNN models with a structured and testable neural circuit for path integration. I believe this is particularly valuable for understanding the black-box nature of both RNNs and biological neural systems. However, I do have some concerns regarding the pruning methodology and the scope of the work, especially given that it builds directly on previous models and involves several carefully selected engineering choices.", "questions": "1. It seems to me that the network configuration and training setup for path integration in this paper is similar to previous papers, particularly Sorscher et al. (2019). However, the current network exhibits band cells and velocity cells, which were not reported in Sorscher et al. 2019. I wonder if there are any differences in settings that might have led to this? Or is it simply the previous study did not report cells other than grid cells.\n2. I'm curious if the authors have compared the number of band cell groups with the number of grid cell modules. If the periodicity (i.e., grid and band spacing) is aligned and the number of scales matches, that would be a very strong piece of evidence supporting the proposed band-cell-to-grid-cell hierarchy.", "limitations": "The paper has addressed many of its limitations well. However, one additional point I would like to raise is that both the RNN model and the subsequent theoretical model are built upon an RNN trained with Difference-of-Softmax place fields and cross-entropy loss. These two design choices are important but not clearly justified. Moreover, this setup emphasizes a unidirectional flow of information from MEC to place cells, while anatomically, the MEC and hippocampus are known to be bidirectionally connected. I believe the paper would benefit from a discussion of how the proposed band-cell-to-grid-cell model might fit within alternative models of the MEC-hippocampus loop, or at least from acknowledging its reliance on the assumptions of the underlying RNN training setup.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths**\n1. Path integration is a core component of navigation. While several previous studies have revealed training methods that lead to the emergence of grid cells, it remains arguably unclear whether grid cells are the sole cell type supporting path integration. In this context, an ablation study on the trained RNN is, in my view, particularly important for elucidating the precise mechanisms underlying path integration in such models.\n2. It is very interesting that the band cells also cluster into modules (groups), with each group exhibiting 1D continuous attractor dynamics.\n3. For most of the theoretical sections, both the visual and textual presentations are exceptionally clear, which I greatly appreciate. That said, some implementation details are omitted from the main text, which I believe are equally important—particularly given the known sensitivity of RNN training to specific choices in implementation.\n4. Section 3 is clear and well-structured. I appreciate the effort to construct a simplified numerical model to interpret the behavior of the trained AI system.\n\n**Weaknesses**\n1. The presentation of the pruning section is a bit unclear. Specifically, is pruning performed on a single trained network? If so, how consistent are the results across different independently trained models?\n2. I have a major concern regarding the section that prunes different connections to dissect the information flows between neuron groups: RNN model dynamics are typically quite fragile. In particular, the input, output, and recurrent connection matrices require careful initialization in terms of their magnitude; otherwise, training tends to become unstable. This sensitivity often depends heavily on the exact scale and dimensionality of the input signals each projection matrix is mapping from. Pruning connections reduces the effective input dimensionality received by different parts of the RNN, which will in turn alter the network’s internal dynamics and may potentially destabilize them. Have the authors accounted for this issue in their analysis?\n3. The model uses the cross-entropy loss, and the difference-of-softmax to model the place fields. To my knowledge, these two setups are particularly important for grid cell emergence but not necessary for the path integration, as argued by a previous paper by Schaeffer, Khona, and Fiete (***NeurIPS*** 2022). Given the ANN model is directly built upon previous models that include these settings, I'm concerned if the scope of this paper will also limited by these specific training choices?", "was_revised": true, "final_justification": "The paper attempted to unfold the black box of RNNs for path integration. Specifically, it aimed to reveal this black box by (1) pruning a trained RNN containing grid cells used for path integration, and (2) proposing a theoretical framework in which velocity inputs first project to band cells before projecting to grid cells. The paper thus puts forward a potential hierarchical formation pathway for grid cells.\n\nFor the points addressed in the rebuttal:\n- The authors have clarified the precise implementation steps of the pruning process.\n- They also added additional experiments pruning random cell types as a baseline, allowing a better comparison with pruning specific cell types. \n- In their response to other reviewers, the authors tested their findings on other RNN variants, including LSTMs, and also on alternative path-integration models that do not rely heavily on cross-entropy loss and difference-of-Gaussian place fields, which helps strengthen the generalizability of their findings.\n\nHowever, a few points remain not fully addressed:\n- First, the paper’s main goal is to understand the black box of RNNs for path integration. This inherently constrains its scope to a specific class of models that assume unidirectional information flow. This is both a limitation and a strength. The limitation is that most existing grid cell emergence models belong to this same general category of models with unidirectional information flow, which makes the scope of the paper narrowly focused on a specific, though widely used, family of models. However, this focus also comes with strengths. As these models increasingly shape current theoretical frameworks of grid cells, there is a clear need to better understand the mechanisms underlying them.\n- Second, while pruning a trained network remains a pragmatic approach, it is also somewhat constrained in interpretability. Nevertheless, it is a reasonable methodological choice given the goals of the paper.\n\nI think the two unresolved points might be inherent in the question this paper attempted to resolve, and would be difficult for the authors to fully address within a single work. Still, the claims in the paper point to a possible direction on the formation of grid cells. Thus, I recommend borderline accept. That is, the scope of the paper may be limited due to the nature of the questions it aims to address, but the work could still have some impact in the relevant field and be useful for future studies.", "rebuttal": "Thank you for the valuable and encouraging comments and the recognition of our contributions in this work. Below we address your concerns point by point.\n\n1. The presentation of the pruning section is a bit unclear. Specifically, is pruning performed on a single trained network? If so, how consistent are the results across different independently trained models?\n\nYes, all pruning experiments shown in the main text were performed on a single, trained RNN. The results from different independently trained RNNS are highly consistent, and they are not idiosyncratic. For example, the property that removing velocity inputs to band cells producing a substantially larger degradation in path-integration accuracy than pruning velocity inputs to grid cells or any other cell type holds across all trained RNNs. We are sorry for missing reporting this important technical detail. We will include those additional RNN model pruning analyses in the Supplementary Information and publicly release all training and analysis codes to guarantee full reproducibility and demonstrate that our conclusions do not depend on a cherry-picked network (see also our response to Reviewer bx3s). \n\n2.  I have a major concern regarding the section that prunes different connections to dissect the information flows between neuron groups: RNN model dynamics are typically quite fragile. In particular, the input, output, and recurrent connection matrices require careful initialization in terms of their magnitude; otherwise, training tends to become unstable. This sensitivity often depends heavily on the exact scale and dimensionality of the input signals each projection matrix is mapping from. Pruning connections reduces the effective input dimensionality received by different parts of the RNN, which will in turn alter the network’s internal dynamics and may potentially destabilize them. Have the authors accounted for this issue in their analysis?\n\nThanks for raising this insightful question. Indeed, training RNN can be sensitive to perturbations—removing a few connections may destabilize the network dynamics unpredictably. But on the other hand, if we remove some components and it has little affect on the network performance, this readily indicates that these components are not important. This is what we observed when we pruning velocity inputs to grid cells, or pruning readout weights from band cells; together they suggest that there exist a hierarchical pathway in the trained RNN for path integration. Meanwhile, to ensure a fair comparison, we always removed the same number of connections to/within each cell group (e.g., velocity inputs, recurrent weights, or readout).  \n\n3. The model uses the cross\u001fentropy loss, and the difference \u001fof \u001fsoftmax to model the place fields. To my knowledge, these two setups are particularly important for grid cell emergence but not necessary for the path integration, as argued by Schaeffer, Khona, and Fiete (NeurIPS 2022). Given the ANN model is directly built upon previous models that include these settings, I'm concerned if the scope of this paper will also be limited by these specific training choices?\n\nThanks for raising this important question. Our findings do not depend on the cross-entropy loss, the use of softmaxed place-cell readout, or other different settings of the model. Actually, we found that our results hold for alomost all network models, once they are trained for path integration. The only exception is the model of Schaeffer, Khona & Fiete (2022), where they considered velocity modulates synaptic weights directly, rather than serve as inputs to neurons; in their model, band cells do not emerge (since it is hard for synaptic strengths tracking animal velovity instantaneously, we tend to believe their model is not biologically practical). \n\nIn the revised manuscript,we will add the resuts of applying the same analysis to extra two RNN models, one from Xu et al, (2022), another from Petterson et al, (2024). In both cases, we observed: robust emergence of band-cell–like units, pruning velocity inputs to band cells causing a significantly larger degradation in path‑integration accuracy than pruning velocity inputs to grid cells or any other cell type, and other key findings in this work. \n\n4. It seems to me that the network configuration and training setup for path integration in this paper is similar to previous papers, particularly Sorscher et al. (2019). However, the current network exhibits band cells and velocity cells, which were not reported in Sorscher et al. 2019. I wonder if there are any differences in settings that might have led to this? Or is it simply the previous study did not report cells other than grid cells.\n\nThanks for raising this point. In fact, Sorscher et al. (2023) included in their Figure S3b some examples of band-cell–like units. However, they focused mainly on grid-cell-like units and did not go further to analyze these band-cell-like units and their roles in path integration. Our main contributions in this work are that we analyzed the detailed properties of band cells and unveiled a hierarchical pathway formed by band and grid cells for 2D path inegration. \n\n5. I'm curious if the authors have compared the number of band cell groups with the number of grid cell modules. If the periodicity (i.e., grid and band spacing) is aligned and the number of scales matches, that would be a very strong piece of evidence supporting the proposed band-cell-to-grid-cell hierarchy.\n\nThank you for the great question. In our current RNN paradigm, grid cells form periodic fields but don’t organize into distinct modules, so we haven’t yet compared band‑cell group counts to grid‑cell modules. However, recent work (e.g. Schaeffer et al. 2023) shows that with more biologically inspired training, networks can develop multiple grid modules. In future work we will adopt those regimes, measure band and grid spacings across scales, and test whether band‑cell group numbers align with grid‑cell module counts.\n\n6. Moreover, this setup emphasizes a unidirectional flow of information from MEC to place cells, while anatomically, the MEC and hippocampus are known to be bidirectionally connected. I believe the paper would benefit from a discussion of how the proposed band-cell-to-grid-cell model might fit within alternative models of the MEC-hippocampus loop, or at least from acknowledging its reliance on the assumptions of the underlying RNN training setup.\n\nThis a great point and we totally agree with you! The experimental studies have suggested that MEC receives self-motion cue and is primarily responsible for path-integration, while HPC primarilly receives environment cues (e.g., Chen et al., 2019); and they have intensive reciprocal connections. In this study, for simplicity, we focused on the path-intgeration function by MEC only and largely ignored the feedfack from HPC. As suggested by the reviewer, we should discuss about this issue. \n\nMeanwhile, we would like to point out that in the current model, we actually implicitly incorporated some feedback interaction from HPC, that is, at the first time step of each path‐integration trial, the RNN receives an input from the place‐cell population activity, which serves to initialize the RNN state before the velocity‐driven integration (a common operation used in training RNN for path integration, see, e.g., Banino et al., 2018) . This initialization may be regarded as mimicking the hippocampal → entorhinal feedback required to anchor the path‐integration estimate to a visual object. We will discuss more about this point in the revised manuscript. \n\nWe hope we have addressed your concerns and please let us know if there is anything unclear."}]

Author rebuttal:

Meta-review: This paper presents an analysis of how recurrent neural networks learn to perform path integration. Their analysis connects to the literature on grid cells and band cells. The authors then use pruning to analyze the circuitry of the RNN. Based on the intuition they grained here, they then propose a neural circuit model for path integration that is effective but has key differences compared to the existing neuroscience models.

After the rebuttal, the overall consensus that this paper still has some weaknesses (like the efficacy of pruning as a method of interpretability) it presents an interesting and valuable contribution.

The AC kindly asks the authors to update the final manuscript based on the discussions with the reviewers. As an examples, please do make the change to the statement "challenges the conventional view of considering only grid cells for path integration" that was proposed in the rebuttal.
Decision: Accept (poster)