Can Biologically Plausible Temporal Credit Assignment Rules Match BPTT for Neural Similarity? E-prop as an Example
Authors: Yuhan Helena Liu
Guangyu Robert Yang
Christopher J Cueva
Venue: ICML 2025 poster (2025)
Keywords: Computational neuroscience
biologically-plausible learning rules
recurrent neural networks
neural data similarity
neural representations
Primary area: applications->neuroscience_cognitive_science
TL;DR: We investigate whether the pursuit of biologically plausible learning rules prioritizes plausibility at the level of synaptic implementation (e.g., locality) at the expense of reproducing brain-like neural activity.

Abstract: Understanding how the brain learns may be informed by studying biologically plausible learning rules. These rules, often approximating gradient descent learning to respect biological constraints such as locality, must meet two critical criteria to be considered an appropriate brain model: (1) good neuroscience task performance and (2) alignment with neural recordings. While extensive research has assessed the first criterion, the second remains underexamined. Employing methods such as Procrustes analysis on well-known neuroscience datasets, this study demonstrates the existence of a biologically plausible learning rule — namely e-prop, which is based on gradient truncation and has demonstrated versatility across a wide range of tasks — that can achieve neural data similarity comparable to Backpropagation Through Time (BPTT) when matched for task accuracy. Our findings also reveal that model architecture and initial conditions can play a more significant role in determining neural similarity than the specific learning rule. Furthermore, we observe that BPTT-trained models and their biologically plausible counterparts exhibit similar dynamical properties at comparable accuracies. These results underscore the substantial progress made in developing biologically plausible learning rules, highlighting their potential to achieve both competitive task performance and neural data similarity.

Can Biologically Plausible Temporal Credit Assignment Rules Match BPTT for Neural Similarity? E-prop as an Example
Yuhan Helena Liu 12
Guangyu Robert Yang" 34
Christopher J. Cueva 35
Abstract
1. Introduction
Understanding how the brain learns may be in- formed by studying biologically plausible Iearn- ing rules. These rules, often approximating gra- dient descent learning to respect biological con- straints such as locality, must meet two critical cri- teria to be considered an appropriate brain model: (I) good neuroscience task performance and (2) alignment with neural recordings. While exten- sive research has assessed the first criterion, the second remains underexamined. Employing meth- ods such as Procrustes analysis on well-known neuroscience datasets, this study demonstrates the existence of a biologically plausible learning rule namely e-prop, which based on gradient trun- cation and has demonstrated versatility across a wide range of tasks that can achieve neural data similarity comparable to Backpropagation Through Time (BPTT) when matched for task accuracy. Our findings also reveal that model ar- chitecture and initial conditions can play a more significant role in determining neural similarity than the specific learning rule. Furthermore, we observe that BPTT-trained models and their bi- ologically plausible counterparts exhibit similar dynamical properties at comparable accuracies. These results underscore the substantial progress made in developing biologically plausible learn- ing rules, highlighting their potential achieve both competitive task performance and neural data similarity.
Understanding how animals learn complex behaviors that span multiple temporal scales is a fundamental question in neuroscience. Effectively updating synaptic weights to achieve such learning requires solving the temporal credit assignment problem: determining how to assign the contri- bution of past neural states to future outcomes. In pursuit of answers, neuroscientists have increasingly adopted the mathematical framework of training recurrent neural net- works (RNNs) as model for brain learning mechanisms, in- spired by seminal works that laid the foundation for this ap- proach (Zipser, 1991; Fetz, 1992; Moody et al., 1998; Mante et al., 2013). This strategy has driven significant advance- ments in developing biologically plausible (bio- bbo-plausibbe) learning rules, which aim to model learning processes that respect biological constraints (e.g.. locality (Marschall et al., 2019; Bredenberg et al., 2023)) while achieving good task performance (Lillicrap et al., 2020; Richards et al., 2019). Specifically, biological plausibility here refers to learning rules where all signals required for weight updates are phys- ically available at the synapse (Marschall et al., 2019). Be- yond synaptic-level constraints, a learning rule appropriate for the brain must also meet two key criteria: (1) it must support good task performance, and (2) it must produce brain-like activity. While extensive work has addressed the first question, the second - whether bio-plausible learning rules yield bikin-like representations remains underex- amined.
Navigating the vast space of computational models - which vary not only in learning rules but also in architecture and tasks (Richards et al., 2019; Zador, 2019; Yang & Molano- Mazón, 2021) necessitates systematic comparison of model representations with empirical brain data. To address this challenge, variety of methods have been developed, aiming to quantify the similarity between computational models and neural data. Among these, popular method- ologies include linear regression (Yamins et al., 2014), Representational Similarity Analysis (RSA) (Kriegeskorte et al., 2008), Centered Kernel Alignment (CKA) (Kornblith et al., 2019), Singular Vector Canonical Correlation Analy- sis (Raghu et al., 2017), Procrustes distance (Williams et al., 2021; Ding et al., 2021; Duong et al., 2022), and Dynamical
'Center for Statistics and Machine Learning, Princeton Uni- versity, Prineeton, NJ, USA 2Princeton Neuroscience Institute, Princcton University, Princeton, NJ, USA McGGovem Institute for Brain Research, Massachusetts Institute of Technology, Cam- bridge, MA, USA* AAltea,, San Francisco Bay Area, CA, USA 'Department of Cognitive and Psychological Sciences, Brown University, Providence, RI, USA. Correspondence to: Yuhan Helena Liu <h17582@ prinettooonn. Christopher Cueva <ccuevva satevva@gmii..cm.. gmail.com>.
Proceedings of the 42nd International Conference Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Neural Data Similarity and Biologically Plausible Temporal Credit Assignment Rules
Similarity Analysis (DSA) (Ostrow et al., 2024). By com- paring the geometry of state representations the dynamics of neural activity, these methods provide a critical frame- work for evaluating the extent to which models approximate neural systems.
highlights that variations in these factors can surpass differences observed across learning rules (Figure 2).
Third, to explain the comparable similarities between BPTT and e-prop, we investigate their commonalities. We show their increased similarity at a lower learn- ing rate for BPTT (Figure 3). We also analyze their resemblance in terms of their top demixed principal components (Figure 4), post- gost-training weight eigenspec- trum, and dynamical properties, explored via Dynami- cal Similarity Analysis (DSA) in Appendix Figure 8.
Leveraging existing comparison methodologies, we com- pute the similarity scores of RNN models trained with bio- plausible learning rules to experimental data. Specifically, evaluate these similarity scores by comparing them to those achieved by Backpropagation Through Time (BPTT)- trained models, as BPTT-trained RNNs are the predominant method for brain modeling and the benchmark that many bio-plausible learning rules aim to approximate (Yang & Wang. 2020; Richards et al., 2019; Lillicrap et al., 2020). This comparison allows us to assess the efficacy of bio- plausible approximations of gradient-descen learning in capturing neural data similarity. Has the pursuit of bio- logically plausible learning rules obeyed plausibility at the level of synaptic implementation (e.g,, locality) at the expense of reproducing brain-like neural activity - or can some such rules, under the right conditions, yield representations as brain-like as those learned by BPTT?
2. Related Works
Understanding the mechanisms through which the brain learns, utilizing its myriad elements, remains a perennial quest in neuroscience. Recent years have seen a resur- gence of interest proposing biologically plausible learn- ing rules (Lillicrap et al., 2020; Scellier & Bengio, 2017: Hinton, 2022; Laborieux & Zenke, 2022; Greedy al., 2022; Sacramento et al., 2018; Payeur et al., 2021; Roelf- sema & Holtmaat, 2018; Aljadeff et al.. 2019; Meulemans et al., 2022; Murray, 2019; Bellec et al., 2020; Liu et al., 2021; 2022b; Marschall et al., 2020; Ghosh et al., 2023; Wang et al., 2024; Confavreux et al., 2024; Richards et al., 2019; Kaleb et al., 2024), suggesting potential neural algo- rithms that leverage known biological components. Many of these rules are grounded in experimental observations of synaptic plasticity, including STDP (Bi & Poo, 1998), differential anti-Hebbian plasticity (Xie & Seung, 1999), burst-induced plasticity (Remy & Spruston, 2007), and neu- romodulated Hebbian rules shown to approximate backprop- agation (Payeur et al., 2021; Aceituno et al., 2023).
Main contributions: Our findings reveal that the distance between data and models trained with truncation- based bio-plausible learning rules is comparable to the distance achieved by models trained using BPTT. We specifically focus on learning rules that approximate the gradient by truncating bio-implausible terms, as these truncation-bue based bio- bio-plausible rules, such as e-prop. have demonstrated ef- ficacy and versatility in learning non-trivial tasks (Bellec al., 2020; Eyono et al., 2022). Other bio-plausible training strategies for RNNs either have limited versatility or success on non-trivial tasks (see Related Works). Specifically, our contributions include:
Despite these advances, relatively little research has ex- amined whether such rules motivated by neuronal or synaptic-level constraints give rise to brain-like activity at the circuit or network level. A separate line of work fo- cuses on inferring learning rules directly from neural record- ings (Nayebi et al., 2020; Ashwood et al., 2020; Lim et al., 2015; Kepple et al., 2021; Portes et al., 2022), aiming to reconstruct the learning mechanisms underlying synaptic plasticity. In contrast, our work does not seek to identify the brain's actual learning rule an especially difficult task when only post-learning data is available - but in- stead asks: can an existing biologically plausible rule pro- duce neural activity aligned with empirical recordings? Our framework evaluates the outcomes of learning. using flex- ible post hoc comparisons that do not rely on observing the learning process itself. While this approach does not provide direct mechanistic insight, it fills critical gap by assessing whether existing bio-plausible rules can replicate the dynamics observed in the brain. Recent work has simi- larly begun comparing representations learned by different types of algorithms- for example, Codol et al. (2024) ex-
We demonstrate there exists a biologically plausible learning rule- - e-prop, which is a local lcarning rule based on gradient truncation and versatile across tasks thht can achieve neural data similarity comparable to BPTT at equal accuracies. We also examine ModProp in Appendix Figure 7. This is achieved by bench- marking well-known neuroscience datasets - Mante 2013 (Mante et al., 2013) and Sussillo 2015 (Sussillo et al., 2015) - which are primate datasets chosen for their higher task difficulty compared datasets from other species, using state-of-the-ar similarity meth- ods, partieularly Procrustes distance (Figure and Ap- pendix Figure 7).
Second, we show that factors such as architecture and initial conditions - particularly initial weight settings can have a more pronounced impact on neural data similarity than the specific choice of learning rule. This
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Neural Data Similarity and Biologically Plausible Temporal Credit Assignment Rules
plore how brain-like neural dynamics for behavioral control can emerge through reinforcement -aooiinng--ovvingggggggg complementary perspective to our approach.
aside the broader field of reinforcement learning for future work, thus not covering certain learning rules like the one in (Miconi, 2017).
Our research focuses on learning rules for recurrent neu- ral networks, which are extensively used in brain model- ing (Vyas et al., 2020; Perich et al., 2021; Schuessler et al., 2020; Yang et al., 2019; Mante et al.. 2013; Turner al.,
Comparing high- sigh-dimensional neural responses across differ- ent systems and contexts is crucial in neuroscience (Chung & Abbott, 2021) for assessing model quality, determining invariant neural states, and aligning brain-machin inter- face recordings, among other tasks (Schrimpf al., 2018; Chaudhuri et al., 2019; Degenhart et al., 2020; Pagan et al., 2025), Among the myriad of methods developed to quantify representational dissimilarity (Yamins et al., 2014: Schrimpf et al., 2018; Kriegeskorte et al., 2008; Raghu et al., 2017; 7: Shahbazi et al., 2021; Williams et al., 2021; Ding et al., 2021; Duong et al., 2022; Khosla Williams, 2023; Lin & Kriegeskorte, 2023b; Pospisil et al., 2023; Nejatbakhsh et al., 2024) such as linear regression, Canonical Corre- lation Analysis (CCA), Centered Kernel Alignment (CKA), Representational Similarity Analysis (RSA), shape metrics, and Riemannian distance - we focus on Procrustes distance for its ability to provide a proper metric for comparing the geometry of state representations, and because several weak- nesses have been identified in other similarity measures that are, for example, biased due to high dimensionality, or may rely low variance noise components of the data (Korn- blith et al., 2019; Davari et al., 2022; Dujmović et al., 2023; Elmoznino & Bonner, 2024; Cloos et al., 2024). Addition- ally, we extend our investigation to include Dynamical Simi- larity Analysis (DSA (Ostrow et al., 2024)) in the Appendix, assessing system dynamics to complement our geometric analyses. Overall, the value of these existing measures stems from their ability to compare complex systems without fully understanding them by capturing key structures. However, this strength also poses a limitation: they focus on specific structures, and it remains uncertain whether these structures accurately capture the computational properties of interest. Therefore, developing new measures remains a crucial and intriguing endeavor (Sexton & Love, 2022; Sucholutsky et al., 2023; Lin & Kriegeskorte, 2023a; Klabunde et al., 2023; Bowers et al., 2023; Lampinen et al., 2024).
2021: Valente et al., 2021: Langdon & Engel, 2022; Barak,
2017; Song et al., 2016; Maheswaranathan et al., 2019; Yang & Molano-Mazón, 2021). This study specifically investi- gates local learning rules that truncate gradients, as these have shown promising results in task learning and offer ver- satility across various network architectures. A systematic review (Marschall et al., 2020) recognized random feed- back local online (RFLO) as the only fully local (hence bio- plausible) rule. Post-review developments include e-prop, an adaptation of RFLO for non-vanilla (particularly spike- based) RNNs (Bellec et al., 2020), and MDGL (Liu et al., 2021) with its extension ModProp (Liu et al., 2022b), which further refine the gradient approximation by considering local modulatory signals (Smith et al., 2020). These rules are notable for their effectiveness in bio-plausible temporal credit assignment, matching the performance of the more traditional BPTT in many settings (Eyono et al., 2022). Our study will, therefore, concentrate on these specific learning rules due to their demonstrated efficacy and bio-plausibility. Further details of these rules are explained in Appendix A.2.
Alternative training strategies for RNNs exist, but they ei- ther face cio-plausibility issues, lack versatility across set- tings, or struggle to scale to complex tasks. For instance, equilibrium propagation and related rules depend on the equilibrium assumption (Scellier& & Bengio, 2017; Meule- mans et al., 2022); although there are architectures similar to equilibrium propagation or deep feedback control that do not make the equilibrium assumption (Gilra & Gerstner, 2017; Kaleb et al.. 2024). Within truncation-baat based methods, the SnAP-n algorithm introduced in (Menick et al., 2020) allows customization by selecting the truncation level n. While SnAp-1 aligns closely with e-prop/RFL.O, SnAp-2 and higher n require storing a triple tensor, which poses ((N3) storage demands not yet proven feasible for neu- ral circuits. Therefore, SnAp-n (n 2 2) remains biologi- cally implausible, while SnAp-1 effectively reduces to e- prop/RFLO under certain conditions. Beyond truncation, the KeRNL algorithm approximates long-term dependen- cies using first-order low-pass filters and updates parameters via node perturbation, yet this also challenges biological plausibility by requiring frequent meta mera--aa updates. Other strategies like FORCE learning (Sussillo & Abbott, 2009) offer alternatives, but our scope assumes recurrent weight adjustment and the non-reservoir version faces is- sues with locality (Marschall et al., 2019; Bredenberg et al., 2023). This study focuses on supervised learning, setting
3. Preliminaries
3.1. RNN Training Setup
Our RNN architecture consists of No. input units, N hidden units, and Nout readout units. The update mechanism for the hidden state at time h2 E RN, follows the equation:
h = 3hg + - (W) + Waz,),
(I)
where G 1 de
T E R is the leak factor determined by the simulation time step dt and membrane time con- stant Tm f(-) : : + R represents the activation func- tion (We used retanh to mimic type-1 neuronal firing and avoid negative firing rates but also explored Re LU activa-
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Neural Data Similarity and Biologically Plausible Temporal Credit Assignment Rules
tion, as shown in Appendix Figure 7): Wh E RRN and W₂ E NxNN are the recurrent and input weight matrices, respectively; and 24 E IR- is the input at time t. The read- out, de E R. Nout, is calculated as a linear combination of the hidden state's activation, [(ba), with the readout weights W E IR X
and is quantified as
(
(<Ho, (HO2) (1llllaall)
O(H, H) = min arccos
(4)
QEO
where O is the group of orthogonal linear transforma- tions (Ding et al., 2021; Harvey et al., 2023).
To train this RNN for the specific tasks in the datasets, we used synthetic input and target output detailed in Ap- pendix A.2. Our objective is to minimize the scalar loss L E R. For loss minimization, we examine various learn- ing rules, including BPTT (our benchmark) that computes the exact gradient, Vw (W,, E NN +- as well as bio-plausible learning rules that apply approximate gradients, Vw I.(W) E R. x +N +Noul));
4. Results
In our study, we analyze the similarity between task- tusk-trained RNN models and two neural datasets: Sussillo 2015 (Sus- sillo et al., 2015) and Mante 2013 (Mante al., 2013). An overview of our pipeline is provided in Figure 1A, with detailed information about our RNN model setup, similar- (2) ity measure, and datasets in Appendix A. We examine the similarity of RNN models, across different learning rules, (3) to neural data, leveraging Procrustes analysis. Figure 1B shows that multiple learning rules, specifically BPTT and its truncation- based biologically plausible alternative (e- (e-prop), achieve similar Procrustes distances from neural data across two distinct tasks: Sussillo 2015 (Sussillo et al., 2015) and Mante 2013 (Mante et al., 2013). Although the error bars for BPTT and e-prop do not appear to overlap near perfect accuracy in the Sussillo 2015 task, we demonstrate that such differences are minimal compared to other potential confounding factors in the brain, shown in Figure 2 and Appendix Figure 7. We further demonstrate the key trend using additional similarity measures in Table 2.
AW- = VWV
AW LWW,,
where W = [Wr Wh w²) E R. +N Nouc) and ?7 E R is the learning rate.
The learning rules investigated in this study are elaborated upon in Appendix A.2. Our analysis centers on how training RNNs with different algorithms influences their similar- ity to neural data. Predominantly, we concentrate on the iiacaatio-bbsede bio-plausible rule known as e-prop (Bel- lec et al., 2020), which simplifies the gradient by retaining only those terms that align with three-factor learning rule. This includes a Hebbian eligibility trace modulated by top-down instructive factor, potentially attributable to neu- romodulators (Magee & Grienberger, 2020; Gerstner et al., 2018). It is noteworthy that e-prop is equivalent to the RFLO learning rule introduced in (Murray, 2019) under most conditions. Additionally, we explore ModProp (Liu et al.. 2022b), which incorporates cell-type-specifi local modulatory signals (Smith et al., 2020) recover terms omitted by e-prop. However, due to ModProp's 's limitations (constrained to the adherence of Dale's law and employing the ReLU activation function), our examination of this rule is restricted to such specific contexts in Appendix Figure? 7.
Also, to see the progress over time in developing more bi- ologically plausible learning rules has led to improvements in aligning models with neural activity, we also evaluated older learning methods such node perturbation and evolu- tionary strategies. Results show that these methods resulted in greater Procrustes distances compared to the aforemen- tioned rules at equivalent accuracy levels, demonstrating that not all learning rules are equally effective. This also indicates the effectiveness of newer bio-plausible gradient- approximating learning rules over some of the older methods (Appendix Figure 9).
Additionally, Figure 2 delves into the impact of initial weight settings on model-data distances, revealing that such initial condition nuances exert more pronounced influence than the choice of learning rule itself. Initial weight gain is a crucial attribute, as it significantly affects the dynamical properties of RNNs, particularly the Lyapunov exponents that govern the rates of expansion and contraction. It can also interpolate between rich and lazy learning regimes, im- parting distinct inductive biases (Braun et al., 2022; Flesch et al., 2023; Chizat et al., 2019; Woodworth et al., 2020; Schuessler et al., 2023; Bordelon & Pehlevan, 2022; Liu et al., 2023; Paccolat et al., 2021). This finding further underscores the significant role of model initialization in
3.2. Similarity Measures
As mentioned in the Introduction, we utilize the metric Procrustes distance (Williams et al., 2021) to quantify the similarity between the hidden states of RNN models, de- noted by И E F B. and the experimentally recorded neural responses, represented as Il B<TN"' Here, B represents the number of trials or experimental conditions, T denotes the number of time steps in each trial, and N and N' correspond to the number of RNN hidden units and recorded neurons, respectively. The metric Procrustes dis- tance can be viewed as the residual distance after the two neural representations are aligned with an optimal rotation,

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Neural Data Similarity and Biologically Plausible Temporal Credit Assignment Rules
A
Primate datasets
B
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Sussillo 2015
Bettel Distance to neural data
0.475
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Accuinn bouch
Torget crced
Go cuM
Plaach
0.400
Blue: truncated BPTT
Mante 2013
Fixation (0.3s)
targets (0.35s)
dots (0.755)
delay (0.3-1 55s
saccade
0.375
Black: BPTT
Magenta: e-prop
motion context
0.350
fixation point
0.0
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Worse
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Distance to neural data
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Input x
Output 9
Targety
AW LearningRule)()
0.4
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Worse Normalized accuracy
Better
C
Painwise distance
Weight W!!
dimension
Inputx
Output 9
Targety
RNN activity
Orange: neural data
LearningRule()
similarity(RNN activity, neural data)?
UMAP dimension 1
Figure 1. (A) Sctup overview; analysis of two neural datascts. We computed similarity scores between RNN activity and electrode recordings from (I) Mante (2013) and (2) Sussillo et al. (2015). Schematics have been modified from those original papers. RNNs are trained on these respective tasks using various learning rules, including BPTT and bio-plausible alternatives. Subsequently, we evaluate similarity between RNN activity post-training and the neural recordings to compare model-data similarity across different learning rules. (B) The Procrustes distance accuracy plots for the Sussillo 2015 (top) and Mante 2013 (bottom) tasks illustrate that multiple learning rules achieve comparable data similarity. At each training iteration, both task accuracy and neural similarity are computed, and the resulting pairs plotted-showing showing the progression of models during training from low accuracy/high distance (upper left) to high accuracy/low distance (bottom right). This procedure applied uniformly across all learning rules. Here, magenta denotes e-prop, blue denotes truncated BPTT (truncation length 10), and black denotes BPTT. The standard deviations across four random seeds. The x-axis, normalized accuracy, is defined Appendix A.3. (C) UMAP embedding of trained model representations with Sussillo 2015 as an illustrative example; additional training snapshots shown in Appendix Figure 6. Additional similarity measures are given in Table 2.
plotted with bars denoting
shaping learning outcomes, with particular initial conditions facilitating closer approximation to neural data than others.
Given that e-prop be decomposed into lower learning rate BPTT and an approximation error (Liu et al., 2022a), which is further illustrated here by the similarity between lower learning rate BPTT and e-prop (Figure 3B), this shared component of lower learning rate BPTT could partly explain their similar distances. Additionally, post-
Figure 3 explores the impact of learning rates on model-data distances across learning rules. Figure 3A, Procrustes distances remain consistent across learning rates for BPTT.
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Neural Data Similarity and Biologically Plausible Temporal Credit Assignment Rules
A
BPTT
B
E-prop
0.65
Distance to neural data
0.60
Distance to neural data
0.60
0.55
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0.45
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Normalized accuracy
Blue: initial weight gain = 0.0; Black: initial weight gain 0.5; Magenta: initial weight gain 1.0;
Green: initial weight gain = 1.5; Cyan: initial weight gain = 2.0
Figure Impact Initial Weight Magnitude on Model-Data Distances Exceeds Variation from Learning Rules. Model-data
distances normalized accuracy for various initial gain values (depicted by different colors) for (A) BPTT and e-poop Initial weight gain refers to the multiplier applied to the default initializations for recurrent and readout weights. The results shown are for the Sussillo 2015 task, with similar trends observed for the Mante 2013 task. The mean is plotted with bars representing the standard deviation. Table further illustrates the significant influence of gain on distances.
Rule
gain=0.0
gain=0.5
gain=1.0
gain=1.5
1.0 vs 0.0
1.0 vs 0.5
BPTT
0.461 +: 0.006
0.423 + 0.005
0.398 + 0.007
0.428 +: 0.010
p=2.07e-5
p= 1=9.94e-3 5=5.7le-3
e-prop
0.461 + 0.007
0.437 + 0.009
0.407 + 0.008
0.432 + 0.008
p=1.30e-4
e-prop vs BPTT
p=0.985
p=0.078
p=0.173
p=0.587
Noise ceiling
0.565 + 0.002
0.529 + 0.005
0.467 + 0.005 0.508 + 0.017
Table 1. Procrustes distance across gains and learning rules for Figure 2. Distances measured at normalized accuracy R 0.8. Values are bars across gains suggest gain influences neural distances; overlapping bars within columns suggest similarity across rules (for fixed gain). P-values (uncorrected t-tests due to only small number of comparisons) highlight significant gain-dependent differences but insignificant differences across rules.
std across seeds. Noise ceiling corresponds untrained models. Non-overlapping
A
Distance to neural data
0.475
B
Distance E-prop
0.285
0.280
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0.275
0.425
0270
0.400
0.655
Black: Ir=1e-3
0.200
0.375
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Cyan: 1r=1e-4
0.255
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0.0
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10t4
30
BPTT Learning
Figure 3. (A) Procrustes distances remain consistent across various learning rates when employing the same rule (BPTT). Different color shades represent different lcarning rates: le - 3, 3e 4, and le These rates result in nearly indistinguishable Procrustes distances. The analysis in this figure is done using the Sussillo 2015 task. (B) E-prop - has been viewed as BPTT with a reduced learning rate plus some degree of gradient approximation error (Liu et al., 2022a) aligns more closely with BPTT lower learning rate (le compared to the default setting (1e 3). Here, the mean distance from BPTT to e-prop is plotted, with error hars denoting the standard deviation.
training weight cigenspectrums and distances, analyzed via Dynamical Similarity Analysis (DSA), further reinforce the similarity between BPTT and e-prop (Appendix Figure 8).
This similarity is further explored in Figure 4, where top demixed principal components show a qualitative match between the neural data and the models. We also display the
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Neural Data Similarity and Biologically Plausible Temporal Credit Assignment Rules
Data
BPTT
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Data projected ento time componentt
Datia pro projected ontos time component.
Data projected onto Dere component
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Cata pryccted onto condi on corganent
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Time movement onset (ms)
Time relative movement onset (ms)
Time relative to movement onset (me))
Figure 4. Demixed principal component analysis (dPCA) shows qualitative match between model and data when projected onto the first time component and first condition component. The time component captures variance aligned with task time, while the condition component captures variance attributable to input condition differences. Here, the Sussillo 2015 dataset illustrated; similar trends observed for the Mante 2013 dataset. Each color represents a different reach condition.
CCA
CKA
the simplest cases can depend strongly on initialization.
BPTT
0.272 0.009
0.152 + 0.015
Proposition 4.1. (Informal) For a ID linear RNN trained using e-prop, the weight updates can converge to (or diverge from) the solution W found via an arbitrary algorithm (e.g., BPTT), depending on the initialization of the recurrent weight W W()). (The formal statement and proof are provided in Appendix B.)
e-prop
0.273 = 0.009
0.154 + 0.006
p-val
p=0.953
p=0.817
Table Additional similarity measures yield trends consistent with those in Figure 1. We note that Sussillo 2015 is used here as an example. All p-vauues (uncorrected, due to the small number of comparisons) show significant differences between trained BPTT and e-prop models across metrics (at matched accuracies).
It is noteworthy that if all models equally far from the data, it might also suggest random noise. However, that is not the case, as our models are significantly closer to the neural data after training (Figure 5 and Table 3). These results are also consistent with the existing studies that reported the correlation between task performance and neural data similarity (Yamins et al., 2014). Additionally, what does it mean for model to be close to the data and how do we interpret the range of alignment values? To interpret those, we need a baseline based on data-to to-aata similarity, which reflects how close the models are to the data relative to other data points (subsamples within the dataset). Due to limited subjects, we generated this baseline by splitting the data by neurons within the same subject, matched across timepoints and experimental conditions, though this may create an overly stringent baseline due to potential neuron dependence (details in Appendix A.3). We also explain why we didn't 't use trial-based subsampling in Appendix A.3. For the Hatsopoulos 2007 dataset (Hatsopoulos et al., 2007), the final trained models match the neural data as closely as other neurons (Figure 5). For the Sussillo 2015 and Mante 2013
similarity among models terms of their pairwise distances and their embeddings across different sampled training snap- shots in Appendix Figure 6.
Building the observed similarities between BPTT and e-prop, we derive Proposition 4.1, which demonstrates that e-prop can converge to W* (the solution obtained by BPTT) in highly simplified setting, highlighting the critical role of initialization. Specifically, we present a ID linear RNN example where e-prop matches BPTT under certain initial- izations but diverges under others. This toy case illustrates the sensitivity of e-prop to initialization, a key factor also observed in empirical results. Extending such theoreti- cal analysis to higher-dimensional RNNs would be valuable, though likely on the order of (Schuessler et al., 2020), and is left for future work. Our aim is not to offer a comprehensive theoretical account of e-prop, but rather to motivate future investigation by showing that convergence behavior in even

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Neural Data Similarity and Biologically Plausible Temporal Credit Assignment Rules
Hatsepoulos2007
o o2015
Mante2013
Data-data and model-data distance
Data-data and model-data distance
Data-data and model-data distance
Subsampiee neural data
subsampted neural data
subsampled neural data
0.82
9,923
Distance to
9.55-
Distance to
0:78
Distance
0.50
4:45
9.225
6.74
EXTT
6.72
OPTLING persmaterr DOYAA parureden
9-373
EFTT, Eares ta parreem
Iridal paremeter Rainns vnluu pncasetion
Figure 5. Data-to-date distance (noise floor) vs model-to-deee -ata distance (BPTT and e-prop before and after training). Left: Hatsopolous 2007; middle: Sussillo 2015; right: Mante 2013. The data-splitting procedure for obtaining the baseline (i.e. noise floor) detailed in Appendix A.3. We also tabularize these values in Table 3. We note that these distances are computed with fewer neurons (about half) and units than the previous plots, so the exact distance values here may differ.
Dataset
Noise floor
BPTT (trained)
e-prop (trained)
Untrained
BPTT vs. floor
Hatsopoulos 2007
0.401 + 0.024
0.395 + 0.020
0.442 + 0.030
0.538 + 0.041
p=0.311 p=5.02c-6 p=1.29e-5
Sussillo 2015
0.373 = 0.009
0.406 + 0.012
0.411 = 0.012
0.498 = 0.015
Mante 2013
0.704 + 0.004
0.721 + 0.008
0.727 + 0.010
0.800 + 0.008
Table 3. Neural similarity scores from Figure 5. Trained models outperform their untrained initializations and the noise floor. P-values (uncorrected t-tests) compare BPTT-trained models to the noise floor; similar trend was observed for e-prop trained model as well, Significant differences from the noise floor are found for Sussillo 2015 and Mante 2013, but not for Hatsopoulos 2007.
datasets, trained models approach the noise floor compared to untrained models; the remaining differences from the baseline offer insights for improving learning algorithms and architectures in future work.
While BPTT exhibits questionable biological plausibility, it has been widely used for brain modeling, especially in seminal works (Mante et al., 2013; Sussillo et al., 2015; Yamins et al., 2014; Yang & Wang, 2020). Comparing BPTT to e-prop allows us examine how biologically motivated gradient truncation affects neural similarity, and whether BPTT-based models, despite their widespread use, might under- or overestimate representational alignment relative to plausible alternatives. Importantly, it was not obvious a priori whether e-prop would yield representations more or less brain-like than BPTT. Our finding that e-prop can match BPTT under certain conditions is therefore nontrivial and highlights the promise of such rules. More broadly, framework is designed to eventually evaluate models that surpass BPTT as a benchmark for neural similarity.
5. Discussion
Investigating biologically plausible learning rules is impor- tant to understanding how the brain learns. While many such rules adhere to synaptic-level constraints (Richards et al., 2019; Lillicrap et al., 2020), their ability to reproduce brain-like neural activity remains unclear. Here, we assess whether RNNs trained with approximate, gradient-basee bio-plausible rules- focusing e-prop (and ModProp in the Appendix) - can match the neural similarity of stan- dard BPTT- models. Using Procrustes analysis, we show that at matched task accuracy, --rrp-rrrined RNNs exhibit neural similarity comparable to BPTT. Notably, ar- chitectural choices and initialization have a larger influence on motel-data alignment than the choice of learning rule (Figure 2, Appendix Figure 7). Further, BPTT exhibits in- creased similarity to bio-plausible models at lower learning rates (Figure 3) and shares post- post-training properties in weight eigenspectrum and dynamics, as shown via DSA (Appendix Figure 8). These findings demonstrate that bio-plausibl rules can, under certain conditions, achieve both strong task performance and brain-like dynamics motivating broader reevaluation of how architecture and initialization shape neural similarity.
Limitations and future works. Extending our approach to include a broader range of learning rules, architectures, datasets, and comparison methods represents an important direction for future research. comprehensive evaluation across these dimensions is beyond the scope of single paper, especially in a rapidly evolving field. Our pipeline is flexible, allowing for expansion across these various facets in future investigations. Moreover, while our study is lim- ited in the learning rules and architectures examined, it demonstrates the existence of scenarios where biologically plausible learning rules can achieve neural data similari- ties comparable to their deep learning counterparts. Our focus on gradient truncation- learning rules is due to their effectiveness in task learning and versatility (see Re-
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Neural Data Similarity and Biologically Plausible Temporal Credit Assignment Rules
lated Works). Despite focusing primarily on e-prop (and ModProp in the Appendix), our findings suggest that it's pos- sible for biologically plausible learning approximations to preserve synaptic-level constraints without substantially sac- rificing neural data alignment. That said, existing findings do not suggest that maintaining similarity at the synaptic level necessarily translates to similarity at the neural activity level, possibly due to differences in scale and the emergent nature of neural dynamics. Additionally, while this work emphasizes learning via synaptic credit assignment, alter- native frameworks such as biologically plausible in-context learning (Von Oswald et al., 2023) offer complementary perspectives and merit deeper investigation.
demonstrate that handcrafted solutions can produce more brain-like dynamics than BPTT. These observations under- score the limitations of BPTT as benchmark and moti- vate future research on alternative learning frameworks and auleaarchteecture co-design. While in our study we inten- tionally fixed the architecture, initialization, and task - fol- lowing prior work (e.g., (Liu et al., 2022a)) - to isolate the effect of the learning rule on neural similarity, many biologi- cally plausible rules are designed with specific architectural constraints in mind, A more systematic investigation of learning rule and architecture interactions is an important direction for future work.
Furthermore, we chose to focus on Procrustes distance for its ability to provide a proper metric for comparing the ge- ometry of state representations, and its stringency in only allowing for rotations and a global stretching align neural trajectories (although we also explored additional measures in Table 2 and Appendix Figure 8B). Beyond its geometric grounding, recent work (Cloos et al., 2024) found that opti- mizing Procrustes may better preserve task-relevant features than other metries, suggesting it can capture more meaning- ful neural structure. They also showed that Procrustes is stricter than CKA: high Procrustes was observed to imply high CKA, but not vice versa (see also Harvey et al. (2024)). We were also motivated to emphasize Procrustes distance because several weaknesses have been identified in other similarity measures that are, for example, biased due to high dimensionality, or may rely on low variance noise compo- nents of the data (Kornblith et al., 2019; Davari et al., 2022: Dujmović et al., 2023; Elmoznino & Bonner, 2024; Cloos et al., 2024). That said, like all scalar measures, it focuses specific structures, and it remains uncertain whether these structures accurately capture the computational prop- erties of interest. Relatedly, certain deep learning architec- tures, under sufficient conditions, may lead many models to exhibit similar internal representations (Huh et al., 2024). Therefore, developing new measures remains a crucial and intriguing endeavor (Sexton & Love, 2022; Sucholutsky et al., 2023; Lin & Kriegeskorte, 2023a; Klabunde et al., 2023; Bowers et al., 2023; Lampinen et al., 2024). Alto- gether, this vibrant area invites further work on learning aele-trchitectre interactions, task design, and evaluation methods.
Another limitation worth noting is that our pipeline is not de- signed identify the brain's 's actual learning rule. This stems from identifiability issues inherent in under-consttained sys- tems: multiple learning rules may converge to similar ter- minal activity patterns when only post-learning data are available. Constraining the architecture using experimen- tal data- - or incorporating neural data during learning could help narrow the solution space, and we view this as a promising direction for future work. That said, our goal is not recover the precise rule used by the brain, but rather to test whether any existing biologically plausible rule (c.g.. prop) can yield neural dynamics comparable to those trained with BPTT, the de facto benchmark for brain-like models. Our findings suggest that such rules can indeed achieve competitive neural similarity and task performance without sacrificing biological plausibility.
Beyond learning rules, factors such architecture and ini- tialization - highlighted in Figure 2 and Appendix Figure 7 are critical areas for future exploration. While our results show that e-prop can match BPTT in neural similarity at equal task accuracy, this does not imply the two are function- ally indistinguishable. In fact, e-prop struggles on certain tasks (Liu et al., 2021), motivating future efforts to gather experimental data where bio-plausible rules diverge, abling more targeted comparisons. This consideration partly motivated our focus on primate datasets, which typically involve more complex tasks than those used during rodent experiments. Expanding the analysis to more open- source primate datasets and other modalities is an important next step. Relatedly, our study centers on temporal tasks with rich dynamics, aligning with our goal of investigating bio- logically plausible credit assignment over time. That said, extending these analyses to non-temporal domains such as vision, where many recent bio-plausible rules are formu- lated - is a valuable direction for future work.
Acknowledgement
We are grateful to Mante, Hatsopoulos, and Churchland for sharing their neural datasets. We also thank Nathan Cloos and Katheryn Zhou for their preprocessing code for the Mante 2013 dataset, which builds on Valerio Mante's original scripts. This research was supported by FRQNT B3X (Y.H.L.); Pearson Fellowship (Y.H.L...
As recent studies have noted, BPTT-trained models often converge to limited subset of solutions (Pagan et al., 2025), and it remains unclear whether the brain uses BPTT at all (Song et al., 2024); Zahorodnii et al. (2025) further
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Neural Data Similarity and Biologically Plausible Temporal Credit Assignment Rules
Impact Statement
Chaudhuri, R., Gerçek, B., Pandey, B., Peyrache, A., and Fiete, I. The intrinsic attractor manifold and population dynamics of canonical cognitive circuit across waking and sleep. Nature neuroscience, 22(9):1112-122) 2019,
This research advances our understanding of biologically plausible learning models in recurrent neural networks, with no immediate ethical or societal impacts expected. Over time, the findings could influence related fields like neu- roscience and deep learning, potentially affecting society based on how these disciplines evolve.
Chizat, L., Oyallon, E., and Bach, F. On lazy training in differentiable programming. Advances in Neural Infor- mation Processing Systems, 32, 2019.
Chung, S. and Abbott, L. F. Neural population geometry: An approach for understanding biological and artificial neural networks. Current opinion in neurobiology, 70: 137-144, 2021.
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Neural Data Similarity and Biologically Plausible Temporal Credit Assignment Rules
A. Methods
A.1. Further Details on the Neural Datasets and Synthetic Data for RNN Training
The Mante 2013 dataset was downloaded from
ini. uzh. wttp//rr/reeee.rccggorooooo /
data. html. We trained RNNs using synthetic task setup from Neurogym (Molano- al., 2022), which included 350 ms fixation period, 750 ms stimulus presentation period, 300 ms delay period, and 300 ms decision period, The activity of the trained RNNs during the stimulus period then compared the downloaded neural dataset using the aforementioned similarity measures. A grid search on the fixation and decision interval durations revealed only minor differences in distances and consistent trend across learning rules.
The Sussillo 2015 dataset consisted of electrode recordings from primary motor (M1) and dorsal premotor cortex (PMd) taken while monkey performed maze-reaching task consisting of 27 differerent reaching conditions (Sussillo et al., 2015). To assess the similarity between the neural activity and RNNs we compared activity from -1450 ms to 400 ms relative to movement onset. The inputs and outputs train the models were described in Sussillo et al. 2015, but in brief, for each reach condition there were 16 inputs and target outputs. The 7 outputs
the electromyographic (EMG) signals recorded from 7 muscles as the monkey performed reaching movement. 15 inputs specified the upcoming reach condition, and were derived from preparatory period neural activity. The remaining input was hold-cue that took a value of +1 before movemement onset and then a value of 0 to initiate the movement, whereupon the model generated the 7 EMG signals.
A.2. Further Details on the Learning Rule
This subsection aims to clarify the approximation mechanisms employed by each bio-plausible learning rule. For com- prehensive descriptions, we recommend consulting the detailed references provided. We begin by expressing the gradient via real-time recurrent learning (RTRL) factorization (an equivalent but causal alternative to the BPTT factorization the gradient):
dL dWn.s
ÖL dhi,s
Σ 1,
(5)
abu, dWa.3s"
We follow the total versus partial derivative notation (d vs. θ) as in (Bellec et al., 2020). The primary challenge with RTRL. dhi.c
in terms of biological plausibility and computational efficiency lies in the term aW..s; from the gradient decomposition (Eq. 5). This term tracks all recursive dependencies h4, on the weight Wh,s due to recurrent connections, calculated recursively as:
dhu, (Wn.s OWn,s
Thg,
Ohu dhm.t-1
ahm,t-1 dWn.3j
ahy +
ahu, dhy,-1
Emzr Wh.m (hmmtt 1) dh
+
(6)
OWn.i
Th1a-11 dWnis
dW..is
involving all weights Wh.tm
Consequently, aWa.1 presents a significant challenge for biological plausibility as it includes nonlocal terms, necessitating knowledge of all other network weights for updating each Wn.ig- For learning rule to be biologically plausible, all information required to update synaptic weight must be physically accessible to that synapse. However, it remains unclear how neural circuits could make such extensive information readily available to every synapse.
dhr.
Approaches like e-prop (Bellec et al., 2020) and equivalently, RFLO (Murray, 2019), address this by truncating the problematic nonlocal terms in Eq. 6, ensuring that updates to Wn,s follow a tnree-factor framework - the updates rely solely on local pre- and post-synaptic activity and a third top-down instructive signal (e.g. from neuromodulators).
Thg
ah.t aW.. 0,
ab., ahat-1 WWa,ss
Ther-
aW h.
which yields a much simpler factor than the comprehensive tensor depicted in Eq. 6. This truncation can be achieved in PyTorch using h.detach(), preventing gradient propagation through the recurrent weights.
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Neural Data Similarity and Biologically Plausible Temporal Credit Assignment Rules
Putting this together, e-prop can be written in terms of known biological processes including eligibility trace e and top-down instructive signals -- (Bellec et al., 2020):
A AW lepooppppppp prrpp
Σ Jasei
(8)
where 4,, aL_ is the top-down instructive signal (e.g. from neuromodulator dopamine, neuronal firing, etc. (Gerstner Shg
ah.,, 0het-1 is the eligibility Dhut ah.,, + 8h,,e-1 aW.,i5
et al., 2018; Bellec et al., 2020)) sent to neuron i at time t, and Cij, = aW,,, DW trace for synapse (ij) at time f. This is three-factor rule, with the pre-and postsynaptic neuron factors in the eligibility trace as well as third factor from the instructive signal.
Besides eligibility traces and top-down instructive signals, recent transcriptomics data (Smith et al., 2020) suggest the presence of widespread cell-type-specific local modulatory signals that could convey additional information for guiding synaptic weight updates. ModProp is developed to incorporate these processes and restore the gradient terms truncated by e-prop, thereby improving the approximation of the gradient. Specifically, the ModProp update rule is described as follows (Liu et al., 2022b):
AWh,i ModProp & 1: x Cij +
Σ aE
Σ Ulaa
Tity x Fas
Cij.
Fag,a p p
where I and again denote the top-down learning signal and the eligibility trace, respectively. Here, neuron belongs type o, neuron p type 3, and C denotes the set of cell types. Fa, is hypothesized represent type-specific filter taps of GPCRs expressed by cells of type | in response to precursors secreted by cells of type a. The operator * denotes convolution, and indexes the filter taps. The hyperparameter μı, set to 0.25 in this study, and the genetically predetermined (W*)as values for different filter taps Fas,s could be optimized over evolutionary timescales (Liu et al., 2022b).
We also explored an older learning rule, node perturbation (Werfel et al., 2003; Lillicrap et al., 2016), which is known to have trouble scaling beyond small-scale networks and simple tasks. Specifically, implemented by
AWhasl x Σ Ehuaa
(10)
where I, = (Li(he - Li(he))/)) provides an estimate to Th. zero-mean Gaussian distribution with variance o².
elements of & are chosen independently from a aL.
In addition, we explored evolutionary strategies (Salimans et al., 2017) for parameter updates in our model. This method, for a Gaussian distribution, is implemented as follows:
s
AWhai ES
(+)
(11)
o
where c(+) is sampled from a standard normal distribution (0, for 1, ..., S. Here, (() represents the loss function evaluated after perturbing the parameter by (e(*), the standard deviation of the perturbations, and is the number of samples. Due to computational constraints, we set S to 50 for our experiments.
A.3. Additional Details on Training and Analysis
Our model-data comparison method utilizes Procrustes distances, as implemented in https://github.com Ahwillia/netrep with the configuration set to metric Linear-eetic(alphh True). Additionally, in Appendix Figure 8, we employed Dynamical Systems Analysis (DSA), available at ht nttps:///ithtii
1.0, www.eee.coooumatti
For this analysis, we tested with hyperparame- ters n. n_delays E (5, 10, 15, 20} and rank E {10, 20, 30, 40). observing consistent trends across settings. We did not test values beyond these ranges due to computational resource limitations. For the loss used in training RNNs, we used cross-entropy loss for the Mante 2013 task and mean-squared error for the Sussillo 2015 task (with EMG
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Neural Data Similarity and Biologically Plausible Temporal Credit Assignment Rules
outputs as the targets (Cloos et al., 2022)). As mentioned, the Mante 2013 dataset was downloaded from https: httpaa/arrrcc/rrropprraaaaeee//e
/ / /www. ini. w.h.
html. The Hatsopoulos 2007 dataset was down-
loaded from https: htt:://aaadyaaade org http:/aaasee////eeeeeeee / doi 10:5556611/rrr xs jm However, we obtained the Sussillo 2015 dataset from the original authors and do not have permission to redistribute it.
Our code is available at https://githii main. Both YHL and CJC contributed to the code development. We utilized PyTorch Version 1.10.2 (Paszke et al., 2019). Simulations were executed on server equipped with two 20-core Intel(R) Xeon(R) CPU ES-2698 v4 2.20GHz. The average training duration for tasks was about 10 minutes, and the analysis pipeline required approximately minutes per model. Training employed the Adam optimizer. Unless otherwise noted, the learning rate was set at le - 3,, optimized through a grid search of {3e - 3, le 3, 3e le 4). We used a batch size of around 100; changes in this parameter led to negligible differences in the results. The number of time steps, T, for the Sussillo task was set to 186, matching the original data. The number of time steps T, for the Mante task was 34, based on dt = 50 ms from the original Mante paper and the total task duration in the Neurogym setting. Similar trends were observed when we varied dt and the durations of the fixation and delay periods. We employed 200 hidden units for the Sussillo 2015 task and 400 hidden units for the Mante 2013 task; doubling these numbers resulted in similar trends. Each simulation was repeated with four different seeds (except for 10 seeds for Figure 3B), and results for each seed were plotted as separate lines in our figures. Training involved 1000 SGD iterations for Sussillo 2015 and 3000 for Mante 2013, with input, recurrent, and readout weights all trainable. Local learning rule approximations were specifically applied to input and recurrent weights, due to the locality issues discussed in Section A.2. Unlike these weights, readout weights do not encounter such issues; hence, by default, the same readout weights were used for both forward and backward computations. However, as verified in Appendix Figure 10, employing random feedback readout weights for training (i.e., feedback alignment (Lillicrap et al., 2016)) resulted in comparable distances.
By default, zero-mean Gaussian noise with standard deviation of 0.1 was added to the hidden activity, except in cases where the noise was removed to assess its impact. Typically, no connectivity constraints were applied, except for settings in Figure 7B where 80% of the neurons were enforced as strictly excitatory and 20% as inhibitory. To enforce Dale's 's law, we used the same masking procedure in (Yang & Wang, 2020). Additionally, the Sussillo 2015 plot in Figure I applies a sparsity constraint, limiting only 25% of the recurrent weights to be nonzero and trainable. To initialize the weights, we initialized with random Gaussian distributions where each weight element Wh. ~ *N (0. N2),), with an initial weight variance of g: unless otherwise mentioned, we set g 1.0. Input and readout weights were initialized similarly as in (Yang & Wang, 2020) (see their F.T RN N.R.NNN.pppll notebook).
Normalized accuracy, which appears as the x-axis in several plots, is defined such that a value of 1 corresponds to perfect performance. For Sussillo 2015, normalized accuracy is calculated as 1 - normalized mean squared error, as used in (DePasquale et al., 2018). In the case of Mante 2013, which involves a classification task where mean squared error is not applicable, normalized accuracy is computed as - cross entropy loss to maintain consistency with the definition where 1 indicates the best performance. We also applied x-axis limits to constrain the range between 0 and 1 for uniformity. While we match task accuracies for neural similarity comparisons, we note that as observed in prior work (Bellec et al., 2020) e-prop typically requires more training iterations than BPTT to reach the same accuracy.
We detail the data-splitting procedure used for generating the noise floor, i.e. the baseline, in Figure 5. We split the neural data into nonoverlapping groups each containing Neempie neurons (ineuronsl, ineurons2). We sample Neample units from the RNN model (iunits). We compute the distance between two samples of neural data dl = D(ineurons2, inewronsl). ). d1 the lowest we can hope to get given the variability in the neurons that were recorded. We compute the distance between samples of the model and neural data d2 = Diiunits, ineuronsl). For each iteration of this procedure we get new estimate for the distance between the model and data, and the data-to-data distance.
Number of trials averaged
1
5
10
15
20
Distance
0,513 # 0.008 0.358 # 0.005
0.280 + 0.004
0.238 + 0.003
0.212 +: 0.003
Table 4. Illustrating greater data-data distance when computed from single trials; note that these numbers are not comparable those in
Figure 5 due to different ways of subsampling.
Although the neural datasets used in this study contain multiple trials per condition, we did not compute data-data similarity via bried- subsampling. This decision reflects the substantial trial-to-trial variability in single-trial firing
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Neural Data Similarity and Biologically Plausible Temporal Credit Assignment Rules
rate estimates, which dominate the distance metric and obseure meaningful similarities between similar neural responses (Cunningham al., 2009; Kay et al., 2024). As illustrated in Table A.3, data-data distances are substantially higher when computed from single trials, but decrease with trial averaging. This trend reflects how averaging across trials helps recover stable, condition-specifee firing patterns. While existing methods (e.g., LFADS (Pandarinath et al., 2018)) have improved single-trial inference, accurate estimation of latent dynamics from single trials remains an open challenge. Therefore, following standard practice systems neuroscience, we computed similarity based on trial-averaged responses. Our chosen data-data baseline (used in Figure 5) compares held-out subsets neurons recorded under matched task conditions and timepoints. and serves assess practical question: if more neurons had been recorded from the same brain region, would they be distinguishable from model units?
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Neural Data Similarity and Biologically Plausible Temporal Credit Assignment Rules
B. Convergence and Divergence of E-prop Updates in a Toy Setting
We present the formal statement and proof for Proposition 4.1 in Proposition B.I.
Consider I-dimensional linear recurrent neural network (RNN) with a single scalar recurrent weight parameter W. imput sequence { C R, and output ý defined the readout of the last hidden state hy!
h₂ Why-1 + Fz, ① hr,
(12)
leading to
5(W) Σ WT- 0
(13)
The loss function is the least-squares loss I, = (0- Assume target 0, making T. 102. We note that this simple setup effectively reduces to polynomial root finding problem, with { C IR as the coefficients. The corresponding gradient-based updates are modeled as a continuous-time dynamical system (CTDS), with specific forms for BPTT and e-prop updates:
(W)) dt BPTT dw (W)) dW(1)
dl.
=- W--------
(14)
dt
le-prop
Proposition B.1. Consider the seuup in Eqs. 12-14 and assume the polynomial ((W) has at least one real root W*, where (WW)) O, and BPTT converges to W*; the Jacobian Je-prop (W) evaluated W satisfies Re(X) < where is the eigenvalue of Je-prop (W*); the imput sequence { satisfies IT-1 + O, ensuring non non-degenerate e-prop updates.
If (0) satisfies I (0) WII < 8, where 5 > 0 is the radius of neighborhood around W within which Re(X) of Je-pap(W (W) < O for all W, requiring sign(j((W)) sign(Er-1) upon initialization, then the e-prop update will asymptotically converge to W*. If (0) is initialized such that (ign(j(WW) = -sign(xr (T))) and sign( (W (t)) remains constant for t 2 0, then e-prop will diverge.
Proof. We analyze the system governed by the e-prop updates as a continuous-ting dynamical system (CTDS):
dt
(WW)
where the Jacobian is:
ow dt - -W
a
(dW(t)
Je-prop (W)
By definition, since Je-prop (W*) is a scalar for I-dimensional RNN, its eigenvalue is itself, and W is an asymptotically stable fixed point and only (W*) < 0, ensuring that trajectories initialized sufficiently close to W' converge to W*
We start by proving the existence of case for convergence. Evaluate the Jacobian W*:
Je-poop(W (W*) = T
(W*) )xr-1
Having (W²) < 0 implies:
d'(W") OIT >0.
As provided in the Proposition statement, suppose (O)) is initialized such that [| WI 5 for some 8 0 (within the asymptotic stability region of W*), within which the Jacobian remains negative Je-prop (W (t)) < 0 Vt > 0. In this region, the CTDS is contracting flow:
d (tt) 2-1 2(W - w*) dW(t)
dt
dt
Substitute the update rule:
d
[[()) - (w))
dt
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Neural Data Similarity and Biologically Plausible Temporal Credit Assignment Rules
Using Taylor expansion near W' (()) (WW()(((1) W* Thus:
d dt [I 2=1 -22- - -W)).
Since s/ (WT) O, this ensures [[([1) W* 12 decreases monotonically, proving convergence W*. As side note, since W (t) and W are scalars, [tW w= (W W
We next prove the existence of for failure to converge. By the Proposition statement, if ((0) is initialized such that sign(i((W) = -ign(xr-1)) sign(ū' (eg(ji(W (t) remains constant for t 2 0 and TT-1 is a constant, then
J((W))=T-1 0
Vt M 0.
This would imply that
Je peop (W(1)) 0
41 2 0.
Since the eigenvalue of Jacobian Je-prop ( (t)) (i.e. itself, since J is scalar here) has strictly positive real part for all t 2 0 along the trajectory W()), ((1) does not converge to any stable fixed point, including W*.
To help visualize this theoretical result (facilitated by the simplicity of the polynomial root-finding setup), consider the following scenarios. For stable initialization, requiring the initialization to satisfy sign( (/ (W)) sign (2y-1) would mean that the trajectory begins on rising edge when sign(zr-1) > 0 falling edge when sign(xr-1) < 0. For the unstable initialization case, requiring the initialization to satisfy sign( (W)) A sign( would mean that the trajectory starts on an edge with mismatched signs, leading to initial divergence. However, if (r' (W (t)) changes during the trajectory (e.g.. by entering a different region), the system may settle into a different basin of attraction; this scenario would violate the assumption of a constant sign (j'((())) for all 1 2 0. particular case of instability occurs when sign(ær-1) at initialization and the trajectory initialized on far-right (or far-left) rising edge. Here, the weight updates diverge without entering any basin.
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Neural Data Similarity and Biologically Plausible Temporal Credit Assignment Rules
C. Additional Simulations
In Appendix Figure 6 displays the similarity among models in terms of their pairwise distances and their embeddings across different sampled training snapshots. In Appendix Figure 7, we demonstrate consistent patterns when recurrent noise is removed and Dale's law is enforced. We also explore ModProp (Liu al., 2022b), which incorporates cell-type-speceee local modulatory signals reintroduce terms omitted by e-prop; however, ModPProp is effective only under specific conditions (Dale's law and Re LU activation), confining Appendix Figure 7B to these settings. Further analysis of post-training weight eigenspectrums and distances, conducted using Dynamical Similarity Analysis (DSA), reinforces the similarity between BPTT and e-prop, as shown in Appendix Figure 8.
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Neural Data Similarity and Biologically Plausible Temporal Credit Assignment Rules
A
Painwise distance
Painkise disiacc
0.41
UMAP dimension
Distance to noural data
UMAP dimension 1
B
Painwise distance
0.42
UMAP dimension 2
Distance to neura dat
UMAP dimension 1
C
Pairwise distance
0.43
Distancc neunal cats
UMAP dimension
0,095
dimension
D
Pairwise distance
Patneise dasance
UMAP dimension 2
Detancee neura dats
0.893
UMAP dimension
Blue: truncated BPTT; Black: BPTT; Magenta: e-prop; Orange: neural
Figure 6. UMAP embedding and pairwise distance matrix heatmap for different models when (A) best e-prop accuracy, (B) 80%, (C) 60%. and (D) 10% accuracies reached. Here, the Sussillo 2015 dataset is illustrated. Black: BPTT, blue: truncated BPTT, magenta: e-prop, orange: neural data. The pairwise distances show similarities across learning rules relative to data, indicated by lower distances between models as compared model-date distance.
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Neural Data Similarity and Biologically Plausible Temporal Credit Assignment Rules
A
Without recurrent noise
B
Dale's law
0.50
0.575
Distance to neural data
Distance to neural data
0.550
0.45
0.525
0.500
0.40
0.475
0.450
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
Normalized accuracy
Normalized accuracy
Blue: truncated BPTT; Black: BPTT; Magenta: e-prop; Green: ModProp
Figure 7. This figure replicates the core trend from Figure 1 - namely, comparable similarities across rules - under different settings. The settings include: (A) removal RNN hidden activity noise, and (B) enforcement of Dale's law. This plot uses the Sussillo2015 dataset as an illustrative example and compares Procrustes distances versus accuracy for three learning rules: BPTT (black), e-prop (magenta), and ModProp (green) - the latter functioning exclusively under Dale's law constraint and ReLU activation. Plotting conventions mirroring those in Figure 1.
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Neural Data Similarity and Biologically Plausible Temporal Credit Assignment Rules
A
B
BPTT
e-prop
The base setting
Distance neural data
0.0200
0.0175
0.0150
0.0125
0.0100
Black: BPTT; Magenta: e-prop
0.0
0.2
0
0.6
0.8
1.0
Normalized accuracy
Initial ga0
initial gain=2.0
maac pie
Figure 8. (A) presents the eigenvalues of the recurrent weight matrix post-training, with columns representing BPTT and e-prop respectively. Each row displays a different training setting: the base setting (referenced Figure 1), initial weight standard deviation set and initial weight standard deviation to 2/ /.. Notably, cigenvalue distributions appear learning rules (BPTT vs. e-prop) than across different settings for the same learning rule, further highlighting the similarity between BPTT and e-prop. B) The Dynamical Similarity Analysis (DSA), which evaluates systems based on their dynamical characteristics, also unable distinguish between learning rules when considering their proximity to neural data, Similar to previous figures, e-prop plotted magenta and BPTT plotted in black.
similar within each setting across
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Neural Data Similarity and Biologically Plausible Temporal Credit Assignment Rules
A
Sussillo 2015
B
Mante 2013
Distance to neural data
0.50
Distance to neural data
0.80
0.78
0.45
0.76
0.40
0.74
0.72
0.2
0.4
0.6
1.0
0.0
0.2
0.6 Normalized accuracy
1.0
Normalized accuracy
Figure Node perturhation (cyan) and evolutionary strategies (yellow) lead higher Procrustes distances from the neural data compared to BPTT (black) and e-prop (magenta) when accuracies are equivalent. This figure presents the Procrustes distance versus accuracy plots, adhering to plotting conventions established Figure for (A) Sussillo 2015 (B) Mante 2013 task. Here, simulations are done without recurrent noise, in Appendix Figure 7A, for more stable performance of some learning rules.
A
Sussillo 2015
B
Mante 2013
0.55
Distance to neural data
Distance to neural data
0.80
0.50
0.78
0.45
0.76
0.74
0.40
Black: BPTT
Red: FA
0.72
0.35
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
Normalized accuracy
Normalized accuracy
Figure The use random feedback readout weights gradient computation (red) (Lillicrap et al.. 2016) resulted distances comparable to those achieved using exact readout weights (black). Plotting conventions consistent with previous figures: task performance varies with training iterations, error hars denote variation across seeds.
25


Reviews:
[{"review_id": "V9OiV7Any7", "reviewer": "Reviewer_23UQ", "summary": "This paper compares the representations learned by BackProp Through Time (BPTT), truncated BPTT (tBPTT), and e-prop–a “biologically plausible” learning rule designed as a model of neural plasticity–to those learned in the brains of monkeys performing several tasks. Specifically, the distance between neural representations and learned model representations are calculated via Procrustes analysis. The authors find that the representational distance between e-prop and neural data is not so much larger than that of BPTT, or tBPTT, and neural data. To further bolster this favourable comparison of e-prop and BPTT, a theorem is proven showing that, in linear models, the solution found by e-prop can match that of BPTT if certain initial conditions are met. Lastly, the authors demonstrate that initial conditions can actually have a larger effect on representational similarity than learning rule itself.\n\n## Update after rebuttal\nThank you to the authors for the many clarifications, the additional experiments and the statistical significance analysis! I still feel that the theoretical results are exceedingly limited but I believe that, with the rebuttal updates, the experiments alone make the paper worthy of consideration. For this reason I have updated by score to a 3.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**Tabular results and statistical test for Fig 2**. We appreciate this reviewer's concrete feedback and tabulate the results:\n\n**Table 1**: Results from Fig2. *Noise ceiling corresponds to untrained models.\n\n| rule         | gain=0.0      | gain=0.5      | gain=1.0      | gain=1.5\n| -------------|---------------|---------------|---------------|---------------\n| *BPTT*       | 0.461 ± 0.006 | 0.423 ± 0.005 | 0.398 ± 0.007 | 0.428 ± 0.010\n| *e-prop*     | 0.461 ± 0.007 | 0.437 ± 0.009 | 0.407 ± 0.008 | 0.432 ± 0.008\n| Noise ceil.* | 0.565 ± 0.002 | 0.529 ± 0.005 | 0.467 ± 0.005 | 0.508 ± 0.017\n\nNon-overlapping error bars across rows suggest differences across gains (fixed rule), while overlapping bars within columns suggest similarity across rules (fixed gain). Values are mean ± standard deviation across seeds. Noise ceiling refers to untrained models. Gain=2.0 is excluded due to poor e-prop performance. Distances were measured at normalized accuracy ≈ 0.8 to ensure matched performance. To further support these trends, we find BPTT at gain=1.0 vs 0.0 differs significantly (p=2.07e-5), as does e-prop (p=1.3e-4). In contrast, BPTT vs e-prop yields no significant difference at fixed gain=1.0 (p=0.173) or gain=0.0 (p=0.985). We report these two gains for brevity (due to rebuttal char limits), but similar trends hold at other values and are available upon request. \n\n**Fig 5 updates**. As suggested, we add tabulated bar values and statistical tests below; we apologize for initially omitting Mante13 (due to an earlier code we wrote for Hats07), but now include it and observe similar trends as Suss15 — trained BPTT/e-prop outperform the noise ceiling but still show a gap from the noise floor: \n\n**Table 2:** Neural similarity scores (Fig5). \n\n| Dataset  | Noise floor   | BPTT (trained) | e-prop (trained) | BPTT (init)   | e-prop (init) |\n|----------|---------------|----------------|------------------|---------------|----------------|\n| Hats07   | 0.401 ± 0.024 | 0.395 ± 0.020  | 0.442 ± 0.030    | 0.538 ± 0.041 | 0.540 ± 0.040  |\n| Suss15   | 0.373 ± 0.009 | 0.406 ± 0.012  | 0.411 ± 0.012    | 0.498 ± 0.015 | 0.494 ± 0.014  |\n| Mante13  | 0.704 ± 0.004 | 0.721 ± 0.008  | 0.727 ± 0.010    | 0.800 ± 0.008 | 0.801 ± 0.008  |\n\nAs noted, distances here use fewer neurons (to compute the noise floor), so values may differ slightly from earlier plots. Consistent with the bar plots, the gap from the noise floor is insignificant for Hats07 (p=0.311, BPTT-trained), but significant for Mante13 (p=1.29e-5) and Suss15 (p=5.02e-6). We will append these results to the updated manuscript. We will also add a noise ceiling to the main plots and visualization using normalized values to the manuscript.    \n  \n**Additional content in the main text**. As suggested, we will add DSA and UMAP results to the main text. We’ve clarified the choice of Procrustes and added new results using CCA and CKA (**see Table 3 for Reviewer AQPQ**). \n\n**Other suggestions and questions**. We will incorporate all these helpful suggestions to improve clarity. Fig2&5 comments are addressed above. For Fig5, the reviewer is correct: the noise floor was computed by comparing different neuron subsets matched across time and condition. Our baseline compares time-varying firing rates across conditions (not trial-to-trial variability), aligning with common practice in systems neuroscience. The variance in Fig. 1B (Mante) is largely due to the absence of learning rate decay; adding decay stabilizes training without altering the main trend — we will include this plot. We will also revise the main text to clarify the toy theorem’s assumptions, limited scope, and illustrative purpose, motivating comprehensive future work (see also Reviewer xVGX). We will cite and discuss **all** references the reviewer mentioned. Finally, we will update Fig1&7 to be on the same footing. \n\n**The significance of e-prop BPTT comparison**. This is an important question. Beyond the motivation in lines 70–73, BPTT remains a widely used benchmark in brain modeling, especially in seminal works (Yamins, DiCarlo, Mante, Sussillo, …). Comparing it to a biologically plausible alternative like e-prop offers broadly relevant insight into neural similarity. While BPTT has known limitations (**see response to Reviewer buDs**), our pipeline aims to evaluate — and eventually surpass — it as a benchmark in future studies. \n\nMore broadly, our study addresses a fundamental and underexplored question: how do bio-plausible gradient approximations affect neural similarity? Using e-prop as a case study (demonstrating existence, not universality), we show that despite gradient truncation, it can match BPTT (before the truncation). This was not obvious a priori; both “e-prop < BPTT” and “e-prop > BPTT” were plausible outcomes (**see further discussions on novelty&contribution with AQPQ**). Our framework enables detecting future models that outperform BPTT. We will add these discussion points to the manuscript.", "claims_and_evidence": "The claims seem reasonably well-supported, but it is at times difficult to parse noise from signal. For this reason the reviewer suggests statistical tests and tabular display of data (see Weaknesses).", "methods_and_evaluation_criteria": "The datasets used seem reasonable.", "theoretical_claims": "The proof was skimmed but not checked in detail.", "experimental_designs_or_analyses": "Experimental design and analysis seems sound.", "relation_to_broader_scientific_literature": "As discussed in the paper, past work in computational neuroscience has focused on proposing “biologically plausible” alternatives to BPTT, as candidate models for learning in the brain–many of which were discussed in the paper (e.g. e-prop, RFLO). Most of this literature has evaluated these models based on (1) whether they are local in space and time (not requiring information about spatially distant synapses, or temporally distant timesteps, to update the given synapse at the given time), and (2) whether they can learn reasonably difficult tasks. As of yet, there has not been a significant amount of work comparing the representations learned by these bio-plausible algorithms. This is the gap that the paper intends to address.", "essential_references_not_discussed": "Recent work has similarly begun comparing representations learned by different types of algorithms–for example Brain-like neural dynamics for behavioral control develop through reinforcement learning by Codol et al. (2024 – biorXiv)–which could be a useful reference. Also, the term “biologically plausible” is somewhat underdetermined, which could be worth discussing. One potential reference for this is Formalizing locality for normative synaptic plasticity models by Bredenberg et al. (2023 – NeurIPs).\n\nA potential issue related to this paper is the recent finding in certain deep learning architectures that, under sufficient conditions, many models exhibit similar internal representations. An example reference here would be The Platonic Representation Hypothesis, Huh et al., 2024.", "other_strengths_and_weaknesses": "## Strengths\nThe paper seems to be addressing a meaningful problem that has been understudied in the computational neuroscience community, and has selected datasets that are well-known and well-studied with which to address the problem. The paper seems to be well written, and the flow of ideas is logical and well-presented. By and large, the authors have done a solid job situating their work, with a respectable background section and citations.\n\n## Weaknesses\nThe primary weaknesses that the reviewer sees with the paper are issues of (1) clarity when it comes to displaying and analyzing the data, (2) significance of differences in data, (3) and depth of content provided in the paper. To elaborate:\n1. The data that is reported in the main body of the paper is relatively noisy and the effects are not huge. It is at times also difficult to compare across plots–for example, if one wishes to compare the effect of initialization using BPTT to e-prop (Fig.2A to Fig2.B). The reviewer would thus suggest (i) statistical testing, and (ii) reports of numerical results, e.g. in a table, to better support the claims. \n2. Critically, it is also difficult to get an idea of what kinds of differences between representational similarity are large and which are small (in part due to the issues mentioned in the above bullet point). Figure 5 takes a good first step towards this by comparing to a noise floor and a kind of '(relevant) noise ceiling'. However, it exhibits the issues of displaying data and statistical testing mentioned above, and doesn’t investigate the 2nd main dataset used in the study (Mante). Including a kind of ‘noise ceiling’ used in Figure 5–a comparison with a matched randomly initialized network (ideally with the best performing initialization scaling) in the main plots (and proposed table) would be helpful. One might also be able to normalize somehow by the difference between this ‘ceiling’ and the ‘noise floor’ provided by a neural data-to-neural data comparison or the best performing model for the given plot.\n3. This point is somewhat more subjective, but the reviewer wonders whether the paper provides enough content to warrant publication at a venue like ICML. The reviewer thinks this perception could be a function of the fact that only the Procrustes results are included in the main body of the paper. If the figures were made more concise so that some alternative tests (like the DSA results and UMAP examples) could be moved into the main text this should help. Also, comparing with another metric (e.g. CCA, or trying to come up with a paradigm where representation similarity analysis could be used), could also be interesting.\n\nIf the authors satisfactorily address these weaknesses and the below points the reviewer would likely raise their score.", "other_comments_or_suggestions": "- Line 13 LHS: first sentence seems a little unnecessarily wordy\n- Line 17 LHS: “approximate” => “approximating”\n- Line 23 RHS: suggestion: “approach” used twice very close together; perhaps use a different word\n- Fig 1B: do the authors know why there is such higher noise on the Mante dataset?\n- Fig 1: could be useful to state truncation length here.\n- Fig 2: it would be useful to have some numerical quantification of how the variance due to initialization exceeds that of the learning rule\n- Fig 3: could be useful to include color in this figure; the greyscale makes it very difficult to tell differences between lines and points\n- Fig 4: could be good to explain what the “condition component 1” represents in this context\n- Fig 5: why use Hatsopoulos dataset and not Mante?\n- Fig 5: if the reviewer understands correctly, the noise floor was calculated by comparing the similarity of different sets of recorded neurons. Is this the case? If so, why choose to compute similarity between different sets of neurons rather than between different sample points (matched for subject and test condition)?\n- Line 420 RHS: final sentence seems rather verbose and doesn’t contribute much insight.\n- The reviewer noted that the theorem is for 1D linear RNNs, and it’s not mentioned that only the 1D case is studied in the main body of the paper. Perhaps this should be mentioned?", "questions_for_authors": "1. The reviewer wonders if the authors could motivate a little more the significance of seeing similar representations between e-prop and BPTT. Given that e-prop tries to approximate BPTT this might not seem that surprising. Moreover, from a neuroscience perspective–given the biological implausibility of BPTT, is it such a good benchmark to compare with?\n2. Why use the Hatsopolous dataset in place of Mante in Fig.5?\n3. The reviewer noted that the objective in the theorem’s proof is to optimize the 1D linear RNN so that its hidden state is zero after $T$ time steps. Of course, given any parameterization with $W < 1$ the hidden state will converge to zero for large $T$. Do the authors not think that, for this reason, the studied case might be too simple?\n4. Would it be possible to plot appendix Fig.7 and Fig.1 on the same footing (i.e. noise or no noise in both cases)?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "1pc8SA3JBl", "reviewer": "Reviewer_AQPQ", "summary": "This paper investigates whether biologically plausible learning rules (e-prop) in RNNs can achieve neural activity similarity to biological data comparable to models trained with BPTT. Using primate datasets and Procrustes distance as a similarity metric, the authors demonstrate that e-prop-trained models match BPTT in neural data similarity at equivalent task accuracies. Additionally, they show that architecture and initialization significantly influence similarity, often surpassing differences caused by learning rules.", "questions": "", "limitations": "", "rating": 2, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**Alternative metrics**. We thank this reviewer for their constructive comments and now add additional metrics to the manuscript:\n\n**Table 3:** Additional measures. \n\n| Rule    | CCA            | CKA            | Duong et al., 2023 |\n|---------|----------------|----------------|---------------------|\n| BPTT    | 0.283 ± 0.011  | 0.160 ± 0.018  | 0.349 ± 0.038       |\n| e-prop  | 0.281 ± 0.012  | 0.155 ± 0.009  | 0.394 ± 0.072       |\n\nThese additional results again support the insignificant differences across the rules (p=0.834, 0.683, 0.373, for the respective measures).\n\nWhile we provided initial justification for using Procrustes in the Discussion, we will expand on it in the revised manuscript. In addition to its interpretability and geometric grounding, a recent study (Cloos et al., ICLR 2025) found that optimizing Procrustes may better preserve task-relevant features than other metrics. This suggests it can capture more meaningful neural structure. They also observed that Procrustes is stricter than CKA: high Procrustes implies high CKA, but not vice versa (see also Harvey et al., UniReps 2024). We will add this discussion.  \n\n**Novelty & contribution**. We thank the reviewer and respectfully clarify that while we use existing tools, our contribution lies in addressing a fundamental, underexplored question: How do biologically plausible gradient approximations affect neural activity similarity? Using e-prop as an example, we show that despite gradient truncation, it is possible to match the similarity of BPTT (before the truncation). This is not obvious a priori—e-prop < BPTT or > BPTT were both plausible. The finding of matched similarity is thus nontrivial, and we further analyze why this occurs (Figs. 3 & Appx Fig. 8), as well as how confounds like architecture/initialization dominate (Fig. 2). These insights also help refine the precise future questions that can be systematically explored across learning rules, architectures, datasets, and metrics — all of which our flexible pipeline is designed to support. We will revise the Discussion to reflect these points and believe our insights meaningfully advance this emerging area, warranting consideration for ICML. \n\n**Why focus mainly on one bio-plausible rule**. We thank the reviewer for raising this point. Our goal is not to claim universality — that all biologically plausible rules match BPTT in neural similarity — but to demonstrate existence, by showing that e-prop can do so under certain conditions. To support this existence claim, showing a single such example should suffice, which we believe is both non-trivial and novel, especially given how underexplored representational similarity is in this context. \n\nThat said, we sympathize with the reviewer and acknowledge that our current title may overstate the scope. To avoid this confusion, we plan to revise the title to better reflect our focus on a single illustrative case — e.g., “*Can biologically plausible temporal credit assignment rules match BPTT for neural similarity? E-prop as an example*”. Additional planned updates are outlined in our response to Reviewer buDs.\n\n**Questions**. We thank the reviewer for those meaningful questions and will add them to our manuscript:  \n1. This is a nuanced question that touches on how task timescale and complexity modulate the roles of both initialization and learning rules. Due to rebuttal word limits, we focus here on timescale. Prior work shows that performance gaps between BPTT and e-prop widen with longer task timescales (Liu et al., 2021). If neural similarity correlates with task performance—as supported by Yamins et al. (2014) and our results—we would expect a growing similarity gap as well (although we'd be violating the assumption of matched accuracy for comparison). Testing this directly would require datasets with varying temporal dependencies, which we view as a promising direction for future work.\n2. Intuitively, if both metrics show similar results for e-prop and BPTT, a composite score (based on weighted or normalized average) would reflect the same conclusion. However, we prefer to report each metric separately to avoid arbitrary weighting and allow for clearer interpretation. We’ll add composite scores to the manuscript for completeness.\n3. This is a profound question that touches on translational relevance: to what extent do findings from one species (e.g., primates) generalize to others (e.g., rodents or humans)? The widespread use of model organisms in neuroscience is based on shared circuit motifs and learning mechanisms. While we expect our conclusions may extend to other species or even SNNs with similar dynamics, systematic cross-species and model-type studies will be essential. We view this as an exciting direction for future work.\n\n**Other comments**. We will cite and discuss BrainScale and others in Related Works. We didn’t use Tanh for type-1 firing for nonnegative firing rate; we will clarify this point in the manuscript.", "claims_and_evidence": "The evidence is robust for the claims overall, except reliance on Procrustes distance alone may miss nuances captured by other metrics. The study lacks experimental comparisons to demonstrate the superiority of Procrustes distance over alternative similarity metrics.", "methods_and_evaluation_criteria": "Yes.", "theoretical_claims": "Theorem 4.1 is proven in Appendix B,", "experimental_designs_or_analyses": "Baseline comparisons (Fig. 5) and ablation studies (Fig. 7) strengthen validity. But tasks are limited to two datasets; generalization to other species or tasks is unexplored.", "relation_to_broader_scientific_literature": "The work extends prior studies on bio-plausible learning (e.g., Lillicrap et al., 2020; Richards et al., 2019) by systematically evaluating neural similarity—a less-explored aspect. It aligns with Yamins et al., 2014)on performance-similarity correlations but adds insights into initialization’s role.", "essential_references_not_discussed": "The article omits some of the latest biologically plausible algorithms such as BrainScale: Enabling scalable online learning in spiking neural networks.", "other_strengths_and_weaknesses": "**Strengths**: \n\nThe article is fluent and easy to understand overall and provide insights into initialization.\n\n**Weaknesses**: \n\n1. Using existing similarity evaluation metrics to assess the algorithm's performance on neural datasets, the article lacks significant contributions and novelty.\n2. Limited discussion of alternative similarity metrics (e.g., CKA, RSA).\n3. The analysis is restricted to comparing only one biologically plausible algorithm (e-prop) with BPTT, leading to limited validity and generalizability of the evaluation.", "other_comments_or_suggestions": "'We used retanh to mimic type-1 neuronal firing ‘ should it be '*Tanh*'?", "questions_for_authors": "1. Does the dominance of initialization over learning rules hold across tasks with varying temporal dependencies (e.g., memory-intensive tasks)?\n2. How would results change if combining Procrustes with dynamical metrics (e.g., DSA) into a composite score?\n3. Could the conclusions extend to SNN or non-primate datasets?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {"ethical_review_concerns": "none"}}, {"review_id": "GGXzK9DZQq", "reviewer": "Reviewer_xVGX", "summary": "This paper examines the properties of recurrent neural networks trained on experimentally motivated tasks using both Backpropagation Through Time (BPTT) and biologically plausible learning rules, in particular e-prop, a truncated approxiamtion of BPTT which uses only local information for weight updates.  They demonstrate that e-prop (as well as mod-prop), can achieve competitive similalrity with neural recordings when matched for task performance with BPTT.  They also inversigate the effects of learning rate and weight initialization scale on performance-matched similarity scores, finding a profound impact of initialization, and a small effect of learning rate.  Comparing neural distances to a baseline constricted by splitting neural recordings into multiple populations, they show that RNNs can reach this baseline level of similarity for a simpler, but not a more complex task.  To explain the similarity between E-Prop and BPTT, they show that E-Prop will converge to the same solution as BPTT in a linear RNN with a single recurrent unit solving an effective root-finding task.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**Theoretical claims**. We appreciate the reviewer’s constructive feedback and acknowledge that the main text did not clearly convey the limited scope of our theoretical result. We agree that the theorem is restrictive in two key ways: (1) it applies only to a 1D setting, and (2) W* can be the solution of any algorithm Z, not specifically BPTT.\n\nWe recognize that our presentation may have unintentionally suggested a broader theoretical contribution than intended, potentially distracting from our main message. **In response, we will make the following updates to our manuscript to (A) clarify the theorem’s assumptions and scope in the main text, and (B) re-center the reader’s attention on our core empirical focus and existence-based framing**:\n- Remove all phrases referring to the theoretical results in abstract and Contributions, including “*to support this further, we provide theoretical results …*”, to avoid implying that we seek to establish a general theoretical framework for e-prop convergence, which we do not aim to do. \n- Clarify the restrictive assumptions in the main text’s informal theorem statement by replacing “*linear RNN*” with “*1D linear RNN*”, and “*found via BPTT*” with “*found via an arbitrary algorithm (e.g., BPTT)*”, to acknowledge that our result does not specifically hinge on BPTT alone. \n- Update Appendix B heading to “*Convergence and divergence of e-prop in a toy setting*” \n- Explain the theorem’s purpose in the surrounding text, clarifying that it serves as an existence demonstration in a toy setting to motivate future theoretical work: “*We provide a 1D linear RNN example showing that e-prop can match BPTT under certain initializations, but diverge under others. While this illustrates key ideas—such as e-prop’s sensitivity to initialization—extending the analysis to higher-dimensional RNNs is a substantial effort (e.g., Schuessler et al., NeurIPS’20) and beyond the scope of this paper.*”\n\nTo offer more information about the last point, our rationale for focusing on a toy (1D) setup is twofold. First, to our knowledge, there is no existing theoretical framework for e-prop convergence that we could readily build upon, and developing such a framework (e.g., in the spirit of Schuessler et al.) would be a major effort in its own right and beyond the scope of this work. Second, the toy setting lets us illustrate the key **existence story** we wish to highlight: there are indeed cases where e-prop converges to the same solution as BPTT, and there are cases where it fails. This demonstration is **not** meant to be comprehensive, but rather to motivate future investigations (e.g., broader architectures, higher-dimensional RNNs, etc.).  \n\nBy making these changes, we hope our updated manuscript can better convey that our theorem is meant primarily to provide an example—“e-prop can match BPTT under certain initial conditions, but not always”—rather than to establish a general theoretical foundation for e-prop, as well as motivating future work on theoretical work to examine insights for the simple toy model across more realistic settings. We hope these revisions clarify that the theoretical result plays a limited, illustrative role and is not essential to our main aims.\n\n**Amount of data/training required to achieve equal performance**. We agree this is an important point. While we did not include these comparisons originally, they have been addressed in prior work evaluating the “(1) good neuroscience task performance” criterion (see Fig. 3, Bellec et al., 2020), which shows that e-prop typically requires more training iterations than BPTT to reach the same accuracy. Our findings are consistent with this, and we will include this information and citation in the revised manuscript.\n\n**Additional questions**. Indeed, task performance is varied by changing the number of training iterations. At each iteration, we compute both task accuracy and neural similarity, and plot one against the other. As training progresses, models move from low accuracy/high distance (upper left) to high accuracy/low distance (bottom right) in Figures 1-3. The same procedure is applied to all learning rules. We will add this explanation to the caption for clarity. \n\n**Longer captions**. In addition to the above explanation, we will ensure all figure captions include complete axis label explanations and refer to the relevant appendix sections. We acknowledge that short captions are more common in the appendix, and will revise those to match the clarity of the main text, including color legend descriptions (e.g., Fig. 8B).\n\n**Other comments**. We thank the reviewer for catching the indexing error and will correct it in the updated version.", "claims_and_evidence": "Empirical claims are well supported.", "methods_and_evaluation_criteria": "The evaluation that is missing is the amount of data / training time required to achieve equal performance across learning rules.", "theoretical_claims": "I found the main text to be very misleading about the extent of the theoretical claims proven in the SI.  You can't say in the abstract that you \"provide theoretical results demonstrating that e-prop can converge to the same solution as BPTT, depend- ing on the initialization,\" give only an informal statement of the theorem in the main text which glosses over the fact that the dimensionality of the RNN considered in the theorem is only one-dimensional, and finally get to that limitation in the S.I.  There's no argument given for higher-dimensional systems, and this should be made clear in the main text.\n\nFurthermore, the assumptions of theorem 4.1 are so restrictive that they actually bypass any meaningful consideration of the BPTT algorithm.  The same exact theorem could be applied by replacing BPTT with \"algorithm Z\" and assuming that \"algorithm Z\" converges to some fixed point $w^*$ which solves the task, then so will E-Prop if converged within a small neighborhood of $w^*$.  This is more a statement about the task considered than the e-prop algorithm, which only converges in this case when $\\hat{y}'(W(0)) and $x_{T-1}$ have the same sign, which is essentially putting in as a hypothesis that the e-prop updates will move in the right direction.  \n\nI think these limitations in the theoretical claims need to be mentioned more transparently in the text, and I would urge the authors to extend their theorem to the multivariate linear case, where similar tasks have been explored theoretically (see e.g. https://arxiv.org/abs/2006.11036)", "experimental_designs_or_analyses": "I didn't check beyond reading what's in the paper.", "relation_to_broader_scientific_literature": "This paper relates to a large body of literature comparing the representations learned by neural networks to neural activity measured in behaving animals.  Also to a large body of work seeking and analyzing biologically plausible learning rules that can achieve competitive performance compared to non-plausible algorithms, namely Backpropagation.", "essential_references_not_discussed": "none that are obvious to me", "other_strengths_and_weaknesses": "Strengths: \n\nThe introduction is well-written with a good set of references to related work.\n\nThe main point demonstrated here -- the bio-plausible learning rules can achieve competitive neural data similarity to BPTT, is a useful fact that will interest many people in the field.\n\nWeaknesses: \nThe captions of the figures are a bit short, and including more detail could enhance readablity.\n\nTheoretical contribution is not consistent with the message presented in the abstract or main text\n\nSome details are missing or not elaborated clearly in the main text (see questions).", "other_comments_or_suggestions": "I think there's an indexing error in equation 6 in the SI.  As it currently reads, you get $\\frac{\\partial h_{l, t}}{\\partial W_{h, ij}} = \\frac{\\partial h_{l, t}}{\\partial W_{h, ij}}  + \\cdots $ implying the rest of the expression is zero.", "questions_for_authors": "1) What parameter is varied to change task performance along the curves in figures 1B, 2AB. and 3A?  Is sample size or training time changing?  How does this compare for the different training algorithms?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "u1VR1rcw6h", "reviewer": "Reviewer_buDs", "summary": "The authors use a the procustes method to compare how well do recurrent neural networks trained with eligibility propagation compare with networks trained with backpropagation through time on matching data from neural recordings on  monkeys.\nAs secondary results, the authors add a theoretical point suggesting that eligibility propagation can find the same solutions as backpropagation through time, and they show that other features such as initialization can be more relevant than learning rules to macht neural dynamcis.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**A more focused title/claim and why mainly one bio-plausible rule**. We thank the reviewer for this thoughtful and detailed feedback. We agree that our study does not aim to characterize all biologically plausible learning rules; rather, our goal is to show that at least one such rule—e-prop—can match BPTT in neural similarity when trained to similar performance, thereby **demonstrating existence rather than universality**; as the reviewer noted, universality would be beyond the scope of a short paper. To clarify our focus, we will:\n- Update the title to: “*Can biologically plausible temporal credit assignment rules match BPTT for neural similarity? E-prop as an example*”\n- Modify lines 70–73 to end with: “*— or can some such rules, under the right conditions, yield representations as brain-like as those learned by BPTT?*”\n- Revise the first contribution point to state explicitly that this is an existence result\n- Emphasize in the Discussion that e-prop serves as a concrete example, not a general case\n\nWith this reframing, we believe that providing a single positive example is sufficient for the contribution we aim to make, though we do explore additional rules in appendices for curiosity.\n\n**Focusing on tasks with dynamics**. While vision tasks are important, our study focuses on temporal credit assignment—a critical challenge in biological learning. This focus, aligned with prior work on dynamic tasks (e.g., Bellec et al., 2020), allows us to evaluate how well models capture evolving neural activity over time rather than static averages, making it ideal for representational similarity analyses. Our updated title now reflects this focus, and we will note that extension to vision tasks is an interesting direction for future work. \n\n**Coupling of learning rule and architecture**. We agree that learning rules are often developed with specific architectures in mind, and we acknowledge this broader issue. In our study, we intentionally fix the task, architecture, and initialization to isolate the effect of the learning rule on neural similarity, following prior work (e.g., Liu et al., 2022). The architecture used is a standard RNN widely adopted in computational neuroscience (e.g., Yang & Wang, 2020). We will add a discussion point noting the importance of studying rule–architecture interactions more systematically in future work.\n\n**Additional similarity measures**. Please see our response under “Alternative Metrics” to Reviewer AQPQ, including new results using additional metrics **Table 3**. \n\n**Theoretical claim**. We agree the result may appear intuitive, but BPTT and e-prop do not generally share the same fixed points (e.g., \\(\\hat{y}'(W) = 0\\) yields a fixed point for BPTT but not generally for e-prop); even when fixed points overlap, the dynamics of convergence may differ due to distinct basins of attraction. Also, what's a bit more precise than saying they sometimes converge to different ones is the sign condition on initialization for e-prop, which determines whether it converges to W*. That said, we recognize the theoretical result is limited in scope and may distract from our main contributions. As noted in our response to Reviewer xVGX, we will revise the manuscript to de-emphasize the theoretical component and clarify that the theorem serves only as a simple illustrative example, not a general convergence result. Our aim is to highlight, in a concrete toy setting, e-prop’s sensitivity to initialization as a motivation for future theoretical work.\n\n**BPTT limitations**. We agree. While BPTT-trained RNNs are widely used in brain modeling, recent studies (e.g., Pagan et al., 2024) show that BPTT tends to favor a limited subset of solutions. Zahorodnii et al. (bioRxiv, 2025) further demonstrate that handcrafted solutions can yield more brain-like dynamics. We will also cite Song et al. (2024) and others who question the usage of BPTT by the brain (see also significance of the e-prop–BPTT comparison in response to Reviewer 23UQ). Together, these points highlight the limitations of BPTT as a benchmark and motivate future work on alternative rules and rule–architecture co-design; we will update the manuscript accordingly.\n\n**Additional references**. In the updated manuscript, we will discuss and cite **all** references the reviewer mentioned. \n\n**Is the question ill-posed?** This is an excellent point. In under-constrained systems with unknown architectural variables, multiple learning rules may converge to similar solutions. Constraining the architecture using experimental data—or incorporating data during learning—could help narrow the search space, making this a promising direction for future work. We will update the manuscript with this point. That said, our goal is not to pinpoint the brain’s actual learning rule, but to test whether any existing biologically plausible rule (e.g., e-prop) can match BPTT in neural similarity, the de facto benchmark for brain-like models.", "claims_and_evidence": "There are two main problems with the claims:\n\\textbf{Focus on one specific case while the question is general}\nI think that the question they formulate is too broad for a short article. The paper simply shows that e-prop converges to a similar solution to BPTT, and even that is only for certain parameters. \nThis is not the claim of the title, and only narrowly answers their main motivating question (lines 70 to 73). Two issues:\n- The single theoretical argument is about e-prop resembling BPTT, not about any other rule. I would not claim to be answering the more general question from this.\n- It examines only e-prop, but the claim is much more general. If the authors pipeline is very flexible, they should test some other rules to validate their method (even if they perform worse). I don't fully see why did they not include ModProp in the main text.\n\n\\textbf{Learning rules and architectures}\nThe authors argue that they evaluate bio-plausible learning rules. However, e-prop is a combination of architecture and learning rules, and so are other methods that they cite (equilibrium propagation, for example, requires a specific architecture). The learning rule has a role to play, but only in combination with a specific architecture.\n\n\\textbf{Suggestion}\nThe results seem correct and interesting, but the title and some of the key words are misleading. I find it hard to conclude anything about bioplausible learning rules, other than there is one model that includes one specific rule that seems to converge to a similar solution to BPTT. I would change the title to e-prop converges to BPTT or something along those lines. In the discussion the authors can make a point that this is an example that shows that there are bioplausible learning rules that can do as well as BPTT.", "methods_and_evaluation_criteria": "It's not clear, given the question, why they focus only on task with dynamics, rather than the very common set-up including vision, where most recent bioplausible models have been formulated. I mention this because, the original source for using BPTT (Mante and Sussillo, 2014) has limitations that are noted by Valerio Mante himself (Pagan, M., Tang, V.D., Aoi, M.C. et al. Individual variability of neural computations underlying flexible decisions. Nature (2024))\nAlso, nothing wrong with procustes analysis, but it seems to be a choice among many, and it is unclear if the results would be different with other methods. \nNone of those points mean that the methods are wrong, just that the claims need to be clarified.", "theoretical_claims": "Theorem 1 is correct. \nHowever, the way I understand the result seems trivial, and I would like to see what am I missing. If the point is to argue that e-prop converges to BPTT, it should suffice to state that both have the same fixed points (without error there won't be gradients). Also, it is clear that in a problem with two different solutions, it is perfectly possible that BPTT/e-prop converges to one or the other, so it is clear that they could converge to different ones.", "experimental_designs_or_analyses": "Seems correct. More examples of learning rules would be necessary.", "relation_to_broader_scientific_literature": "While the authors argue that learning rules should match recordings, so I would point out that there is a very wide literature on neural recording investigating learning rules. The whole idea of testing whether a learning rule makes sense should at the very least mention that such rules should be compatible with the experimental evidence aimed at investigating precisely which are the learning rules found in biological neurons (not only theoretical properties such as locality). Just a few examples: \nSTDP (Bi and Poo, 1998) or differential Hebbian learning can be used in supervised learning (Xie & Seung, Neurips 1999), and to approximate backprop in some settings (Aceituno et al. Front. Comp. Neur. 2023)\nBurst-induced plasticity (Remy & Spruston, PNAS 2007) is definitely observed in the brain and can also approximate BP (Payeur et al Nat. Neuro. 2021).\n\nAlso, the authors should at least mention that there is a debate about whether the brain uses backpropagation at all, which seems to be the main benchmark used. For example, (Song et al., Nat. Neuro. 2024).", "essential_references_not_discussed": "Pagan, M., Tang, V.D., Aoi, M.C. et al. Individual variability of neural computations underlying flexible decisions. Nature (2024)\n\nThere are architectures similar to the equilibrium-propagation or deep feedback control that do not make the equilibrium assumption (assuming by that they mean static inputs and responses)\nAditya Gilra Wulfram Gerstner (2017) Predicting non-linear dynamics by stable local learning in a recurrent spiking neural network eLife\nKaleb, Klara, et al. \"Feedback control guides credit assignment in recurrent neural networks.\" The Thirty-eighth Annual Conference on Neural Information Processing Systems.", "other_strengths_and_weaknesses": "I think it's a good point to show that the hyperparameters matter more than the learning rules. However, it seems to indicate that the question is ill-posed, as it seems to be the learning capacity that matters, rather than whether the rule is really biological or not. The authors could comment on that point", "other_comments_or_suggestions": "N/A. It is well written and good graphics.\nAlso, no need to be overtly polite in the responses, I would simply like clarifications or corrections (also if I am wrong)", "questions_for_authors": "N/A", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: The authors address an interesting problem, namely the biological plausibility of representations built by various learning algorithms. 

Most reviewers suggested acceptance. While one reviewer leaned towards rejection, this decision seemed to stem largely from a different appreciation of / interest in the submission's general problem. 

Indeed, one possible drawback is that this submission is perhaps less suited to ICML than to, say, ICLR or NeurIPS, due to being oriented towards biological plausibility rather than actual "machine learning" per se. While I believe the overlap in interest is sufficient to recommend acceptance, I would understand a different decision.
Decision: Accept (poster)