Pith. sign in

REVIEW 2 major objections 5 minor 1 cited by

Shunting Inhibition and Dendritic Branching Shape Local Credit Assignment

T0 review · 2 major / 5 minor · reviewed 2026-08-02 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read For a conductance-based dendritic tree, every synaptic gradient splits exactly into a synapse-local eligibility term times a path-specific compartment error, so local learning reduces to how faithfully restricted feedback can approximate th

desk verdict Clean derivation, honest empirics, but the figure-ground MNIST headline number is mismatched because LocalCA gets an HSIC auxiliary objective that backprop doesn't. read the letter →

arxiv 2607.03556 v2 pith:2YHBZEMJ submitted 2026-07-03 q-bio.NC

Houman Safaai , Maceo Richards , Bernardo L. Sabatini This is my paper · ORCID
classification q-bio.NC
keywords localcreditassignmentdendriticcomputationshuntinginhibitionpathgainthree-factorrulefeedbackbottleneckconductance-basedmodelbackpropagationapproximation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper derives, for a steady-state conductance-based dendritic tree, the exact gradient of the loss with respect to every synaptic and dendritic conductance. The gradient separates into a synapse-local eligibility—presynaptic activity, input resistance, driving force—and a single non-local compartment error, which is the somatic error transported through a product of branch-to-parent gains. That turns biologically plausible local learning into a concrete approximation problem: how well a restricted broadcast can reproduce the path-specific error field. The paper then shows empirically that shunting inhibition changes these path gains, but under a five-factor local rule with the main restricted feedback scheme, shunting networks still sit 5–6 percentage points below matched backpropagation on MNIST-style tasks; the gap closes when the feedback field is replaced by neuron-wise or exact path-transported errors. The conclusion is that feedback-field fidelity, not the local eligibility structure, is the principal limit in these experiments.

What carries the argument

The load-bearing object is the path gain α̃_n (conductance-only version αcond_n): the product, along the unique compartment-to-soma path, of the child-to-parent transfer factors f'_i(V_i) R_tot_k gden_{i→k}. Theorem 1 shows that the exact compartment error is α̃_n times the somatic error; Corollary 1 then factorizes the synaptic gradient into the local eligibility x_i R_tot_n (E_i − V_n) and that path-transported error. Propagation of the factorized identity makes local credit assignment a problem of approximating the path-specific field, and it is the object the paper manipulates when it tests shunting, neuron-wise feedback, low-rank broadcasts, and exact transport.

What would settle it

Retrain the same shunting and additive architectures on a task where the exact compartment-error field is inherently per-branch (e.g., the cue-routing task) while supplying only a global scalar broadcast: the paper predicts the gap to backpropagation stays large even with unlimited training; if a global scalar matches backpropagation in such a regime, feedback fidelity is not the bottleneck. Alternatively, in a biophysical simulation of a [3,3] tree, measure the distal compartment error under a somatic voltage clamp and compare it with the product α̃_n δ0^V; a mismatch larger than numerical to

Watch

Extended reading notes

Core claim

For a rooted dendritic tree with conductance-based synapses, Theorem 1 and Corollary 1 establish that ∂L/∂g_syn_i = x_i R_tot_n (E_i − V_n) · ∂L/∂V_n, where R_tot_n is the local input resistance and ∂L/∂V_n = α̃_n δ0^V is the product of effective path gains from compartment n to the soma times the somatic error. This is an exact backpropagation equation for a passive, non-spiking tree: the first factor is available synaptically, and the only non-local quantity is the compartment error. Thus every local learning rule in the paper is an attempt to approximate this error with a restricted broadcast; exact reconstruction verifies the identity, and the experiments use path-gain diagnostics, inhib

Load-bearing premise

The whole credit factorization assumes the neuron is a steady-state, non-spiking conductance tree in which voltage is a weighted average set by reversal potentials, child activities, and a fixed leak conductance; if temporal cable dynamics, spiking, or active conductances change the path-gain product, the derived credit geometry and the feedback-bottleneck conclusion need not transfer to real neurons.

Editorial extensions

If this is right

  • Theorem 1 and Corollary 1 give an exact, closed-form expression for dendritic credit in a steady-state conductance tree; any local rule can be assessed by how well its broadcast field approximates the path-transported compartment error.
  • With exact path-transported errors (the oracle), the theorem-derived 3F rule reaches the matched backpropagation ceiling, so the local eligibility structure is not the limiting factor.
  • Switching the main matched-width/scalar-fallback broadcast to one neuron-indexed teaching coordinate per neuron's compartments raises MNIST 3F accuracy by 6.40 points in shunting and 5.56 points in additive networks and closes most of the gap.
  • Under the 5F rule with the main restricted feedback, shunting LocalCA stays 5.1–6.0 percentage points below matched backpropagation on the three classification tasks, and the gap is present with matched-width/scalar-fallback but not with higher-fidelity feedback.
  • Inhibition is not universally beneficial: the paper's five-seed replication does not reproduce an initial shunting advantage in branch-gradient direction, and the cross-core sign depends on initialization and activation policy.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A plausible biological reading of the bottleneck is that a single global teaching broadcast cannot carry branch identity; the results suggest the brain would need something like per-neuron instructive signals (or path-routed transport) to approach backpropagation-like credit, consistent with recent observations of neuron-specific instructive signals during learning.
  • Because input resistance appears both in the eligibility and in the path gain, any neuromodulator or plasticity rule that adjusts local conductance effectively tunes the entire credit cascade for a branch, not just a single synapse; one could test this by pairing local conductance changes with measurements of descendant-branch plasticity.
  • The same factorization suggests a design principle for engineered dendritic learning hardware: if the path gains are approximately known or slowly varying, a cheap recursive transport (depth-modulated broadcast) should capture most of the benefit of full backpropagation; the paper's path-propagation and rank-ladder results are a start but leave the optimal structured transport open.
  • The shunting versus additive comparison shows that whether inhibition helps is an empirical, operating-point-sensitive question; in regimes where inhibition is needed to balance or shape the forward computation, its path-gain effect matters, so one should expect results to depend on task and initialization rather than on a universal rule.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper derives exact gradients for conductance-based dendritic trees, showing that the synaptic gradient factorizes into a synapse-local eligibility term (presynaptic activity, driving force, input resistance) and a path-specific compartment error equal to the somatic error multiplied by a dendritic path gain (Theorem 1, Corollary 1). It then studies local credit-assignment (LocalCA) rules with restricted somatic feedback, comparing 3F/4F/5F variants under shunting versus additive integration. Simulations on MNIST, Fashion-MNIST, figure-ground MNIST, and noise-resilience tasks, with extensive controls (exact transported error, neuron-wise feedback, low-rank feedback, inhibition interventions), lead to the conclusion that feedback-field fidelity, rather than the local eligibility factorization, is the principal limit, and that shunting provides no general advantage under matched initialization. The authors are explicit that the theorem assumes steady-state, non-spiking trees with one path per compartment and that transported errors are diagnostic upper bounds, not proposed biological signals.

Significance. The theoretical factorization is clean and appears correct; the verification against autograd is a valuable check. The empirical study is unusually careful: five-seed replications, paired feedback substitutions, an explicit 2x2 initialization factorial, and an independent replication that overturns the authors' own initial three-checkpoint ordering. The conclusion that restricted feedback is the bottleneck is supported by within-model controls (neuron-wise feedback, exact transport). The paper also offers falsifiable predictions for dendritic physiology. However, the abstract's headline empirical comparison is not fully matched for figure-ground MNIST because the LocalCA rows use an HSIC auxiliary objective that the backpropagation reference lacks, understating the reported gap. This issue is local to the empirical summary, not the theoretical core, but it needs correction.

major comments (2)
  1. [Abstract; Table S8; Fig. S5B] The abstract states that shunting LocalCA remains 5–6 percentage points below matched backpropagation on MNIST, Fashion-MNIST, and figure-ground MNIST. For figure-ground MNIST, Table S8 reveals that the LocalCA rows use an HSIC auxiliary objective (weight 0.01), while the shunting BP reference is the standard cross-entropy reference. The appendix's own ablation (Fig. S5B) shows this HSIC term improves figure-ground MNIST by about 3.2 pp (from 77.3±0.9 to 80.5±1.5 on the main seeds). Thus the reported 5.8 pp gap is not an objective-matched comparison; without HSIC the gap is approximately 9 pp. This does not invalidate the feedback-bottleneck conclusion, which is supported by the within-model neuron-wise and exact-transport controls, but the abstract's empirical summary is inaccurate for one of the three tasks and should be corrected or qualified.
  2. [Section 4, 'Matched-capacity performance'; Table S8] The phrase 'matched backpropagation' is used repeatedly, but the figure-ground MNIST comparison is not matched in objective. In addition, the additive backpropagation reference for figure-ground MNIST is absent from Table S8, so the cross-architecture performance comparison for that task is incomplete. The main text and figure captions should distinguish 'architecture-matched/capacity-matched' from 'objective-matched' and explicitly state when an auxiliary objective is used for one method only.
minor comments (5)
  1. [Discussion, 'Scope' paragraph] The authors appropriately state: 'The theorem assumes steady-state, non-spiking trees with one path per compartment.' This limitation should be echoed in the abstract or introduction, since the title's general claim about 'Shunting Inhibition and Dendritic Branching' could be misread as covering temporal/spiking dynamics. The current hedge is good but easy to miss.
  2. [Section 4, paragraph beginning 'Matched-capacity performance'] 'remains close to matched backprop' is too strong given the reported 5–6 pp gaps and the HSIC mismatch; suggest 'within 5–6 percentage points' or similar with the appropriate caveat.
  3. [Appendix B.3, additive control] The text says 'Rtot_n ≡ 1 by definition since there is no denominator.' Since Rtot is not actually defined in the additive core, consider saying 'effectively fixed at 1' or 'treated as 1' to avoid a definitional contradiction.
  4. [References] Reference [11] appears as 'V ogels' (with a space), which should be 'Vogels'.
  5. [Fig. 3B caption] The caption says the comparison is 'descriptive rather than evidence for a cross-core ordering.' This is appropriate, but the error bars are said to be s.d. over checkpoints; clarify whether these are across checkpoints or seeds, as the two have different inferential meaning.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem 1 is the chain rule applied to the stated conductance model; the transported-error oracle is explicitly an upper-bound diagnostic; the compression hypothesis is tested and fails, which is the opposite of assuming it.

full rationale

The derivation chain is self-contained. Theorem 1 and Corollary 1 start from the steady-state voltage equation V_n = (sum E_j x_j g^syn_j + sum a_j g^den_j)/(sum x_j g^syn_j + sum g^den_j + 1) (Eq. 1) and apply the ordinary chain rule over the rooted tree, producing dL/dV_n = dL/dV_{p(n)} f'_n(V_n) R^tot_{p(n)} g^den_{n->p(n)} and the eligibility/compartment-error factorization (Eq. 6). No term in these equations is defined in terms of the conclusion; the theorem is a consequence of the forward pass. The local 3F/5F updates are introduced as approximations of that factorization with broadcast errors e_n, not as fitted predictions of the exact gradient. The exact-transport condition e^transport_n = alpha_n delta^V_0 = dL/dV_n is explicitly labeled an 'analysis upper bound' (Section 3), so its closeness to backpropagation is a sanity check, not a claimed discovery. The shunting path-gain-compression hypothesis is not smuggled in: Appendix A reports the direct log-gain covariance margin as negative at NI=5,20,40 and states 'we therefore do not use this covariance condition as a positive main result.' Feedback-bottleneck conclusions rest on within-model neuron-wise and exact-transport controls. References are to external textbooks and unrelated prior work; there are no load-bearing self-citations. The one notable flaw is empirical rather than circular: for figure-ground MNIST, Table S8's note says 'the figure-ground LocalCA rows use the HSIC auxiliary objective (weight 0.01)' while the BP reference uses standard cross-entropy, and Fig. S5B shows HSIC adds about 3.2 pp, so the abstract's '5-6 pp' range understates that gap. This affects comparability of one headline number but does not make any derivation reduce to its inputs. The Discussion also states scope limits ('steady-state, non-spiking trees with one path per compartment'), further confirming that the claims are conditional and not tautological.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The central derivation rests on the steady-state conductance-tree model (Eq. 1), the tree-topology assumption, and the specific reactivation/nonnegativity choices; the empirical claims also lean on hand-set stabilizers (5F clamps, EMA rates) and an HSIC auxiliary objective in one benchmark. No new physical entities are postulated. The paper's own replication and initialization controls reduce the risk that these choices force the headline result.

free parameters (5)
  • 5F preconditioner clamp bounds = r4F∈[0.1,2.0], φn∈[0.25,4.0]
    Hand-set stabilizer bounds for the 4F covariance proxy and 5F predictability factor; sensitivity to tightening/widening is checked (Fig S1) but the bounds are chosen by hand.
  • 4F/5F EMA smoothing rate = 0.1 (default; checked over 0.05–0.2)
    EMA rate for branch-variance statistics; robustness to variation is shown but the default is a choice.
  • HSIC auxiliary weight = 0.01 (self/target weights 0.3, RBF σ=1.0, warmup 5 epochs)
    Tuned auxiliary objective used in figure-ground MNIST LocalCA runs; adds ≈3 pp in ablation and is not given to the matched backprop reference.
  • Inhibitory synapse count NI = 20 in main runs; swept over {0,5,10,20,40}
    Per-branch inhibitory synapse count is an architecture choice; the regime sweeps show performance is not monotonic in NI.
  • Numerical stabilizers ε = ε = 10^-8 (denominators), ridge λ = 10^-3
    Floating-point stabilizers in conductance denominators and residual regression; minor but hand-set.
assumptions (5)
  • domain assumption Steady-state passive-cable voltage equation (Eq. 1): Vn = (Σ Ej xj gsyn_j + Σ aj gden_j) / gtot_n, with leak conductance fixed to 1.
    The entire derivation and all path-gain diagnostics rest on this steady-state conductance-weighted model; real dendrites have temporal dynamics, spiking and active conductances. Acknowledged in Discussion: 'The theorem assumes steady-state, non-spiking trees with one path per compartment.'
  • domain assumption Dendritic morphology is a rooted tree with a unique path from each compartment to the soma.
    Theorem 1 uses the unique parent p(n) per compartment; recurrent paths or gap-junction loops would invalidate the path-gain product form (Eq. 3–5).
  • ad hoc to paper Nonnegative conductances g=softplus(θ) and reactivation f_n(V)=(tanh(m(V−b))+1)/2 are the branch transfer.
    Nonnegativity and the specific learnable bounded-tanh are implementation choices; the identity-transfer control (Table S11) shows results change with f′, confirming sensitivity to this choice.
  • ad hoc to paper The 3F/5F batch-averaged updates (Eqs. 8–9) are the plasticity rule synapses implement.
    The paper assumes learning follows the derived conductance-space gradient with batch averaging; biological plasticity rules are far richer (STDP, calcium dynamics, etc.).
  • standard math Chain rule on the computation graph (standard calculus).
    Theorem 1 is a direct application of the chain rule to the tree-structured forward model; no nonstandard mathematical background is invoked.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Shunting Inhibition and Dendritic Branching Shape Local Credit Assignment." pith.science (2026). https://pith.science/paper/2YHBZEMJ

@misc{pith2026260703556,
  author       = {Pith},
  title        = {Pith review of: Shunting Inhibition and Dendritic Branching Shape Local Credit Assignment},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2YHBZEMJ}},
  note         = {Machine review of arXiv:2607.03556}
}
read the original abstract

Biological neurons assign credit across branching dendrites, where synaptic drive, conductance, local voltage, and somatic teaching signals interact to shape plasticity. We study conductance-based dendritic networks with excitatory and inhibitory synapses, shunting inhibition, and tree-structured branch-to-soma coupling, asking when restricted somatic feedback can approximate compartment-specific backpropagated errors. Exact gradients factor into a synapse-local eligibility term, set by presynaptic activity, driving force, and input resistance, and a path-specific compartment error obtained by transporting a somatic error through dendritic gains. This turns local learning into a credit-signal approximation problem. We test whether shunting improves learning when its effect on dendritic gain makes compartment errors more compatible with restricted feedback. Exact-gradient reconstruction verifies the factorization, while path-gain, feedback-fidelity, inhibition-intervention, and transported-error controls probe the mechanism and its limits. With nonnegative conductances and a five-factor rule using matched-width feedback with scalar fallback, shunting LocalCA remains 5 to 6 percentage points below matched backpropagation on MNIST, Fashion-MNIST, and figure-ground MNIST, showing that feedback fidelity remains a major bottleneck. A three-factor rule approaches matched backpropagation with exact transported feedback in the shunting model and with neuron-wise feedback in both architectures, but shunting has no general advantage under matched initialization. These results show how conductance and dendritic branching enter the exact credit equation and identify restricted feedback as a principal limit.

Figures

Figures reproduced from arXiv: 2607.03556 by the authors.

Figure 1
Figure 1. Model and credit assignment. (A) Dendritic E/I unit with branch synapse banks. (B) Layer of such units projecting to a task readout; δ 0 = ∂L/∂V0 is the soma/core error used for feedback. (C) Exact updates multiply local eligibility by path-specific errors δu,n; LocalCA replaces them with broadcast estimates eu,n of increasing fidelity. The compartment error is path-specific: under identity upward transfer it is the… view at source ↗
Figure 2
Figure 2. Exact reconstruction and gradient diagnostics. (A) Exact-factorization reconstruction matches autograd. (B,C) Final LocalCA–backpropagation cosine and scale mismatch; dots show seeds. (D) Layer-soma factorial diagnos￾tic separating exact vs. reused soma teaching signals and exact path transport vs. per-soma sharing. Metrics/protocol: Appendix A. Error bars in (B,C) are ±1 s.d. eligibility and path-error factorizatio… view at source ↗
Figure 3
Figure 3. Mechanistic chain from path gains to learning. (A) Conductance-stage path-gain field α cond. (B) MNIST exact-error geometry and implemented-feedback compatibility: unrestricted rank-1 SVD residual ρ1 (lower is better) and actual per-soma broadcast cosine (higher is better). (C) Post-training inhibitory-conductance interventions. (D,E) Broadcast fidelity and transported-error learning on noise resilience. CV, ρ1, per… view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Matched-capacity performance and regime dependence. (A) Matched shunting backpropagation reference vs. per-soma 5F LocalCA; additive backpropagation references are reported in Table S4 where matched five￾seed runs are available. (B) Inhibitory dose-response. (C) Morpho…
Figure 5
Figure 5. Figure 5: Rule and feedback controls. (A) Rule ablation. (B) Error-source ablation. (C) Noise-resilience feedback ladder: PS, path proxy, low-rank K, and transported oracle. (D) Cue-routing rank diagnostic with pathway-vector feedback (PV). Details: Appendix Fig. S1; Table S8. 5…

Review history (2 revisions) →

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. When Branch-Local Shunting Helps: A Gain-Load-Alignment Principle for Dendritic E/I Networks

    q-bio.NC 2026-07 conditional novelty 7.0 of 10

    Shunting helps population readout only under gain–load–alignment: reliable, support-matched divisors that suppress aligned multiplicative gain without excessive signal loss or denominator variability.

Reference graph

Works this paper leans on

46 extracted references · 1 canonical work pages · cited by 1 Pith paper

  1. [1]

    (1999).Biophysics of Computation: Information Processing in Single Neurons

    Koch, C. (1999).Biophysics of Computation: Information Processing in Single Neurons. Oxford University Press

    1999

  2. [2]

    Dayan, P., & Abbott, L. F. (2001).Theoretical Neuroscience: Computational and Mathematical Modeling of Neural Systems. MIT Press

    2001

  3. [3]

    Poirazi, P., Brannon, T., & Mel, B. W. (2003). Pyramidal neuron as two-layer neural network.Neuron, 37(6), 989–999. https://doi.org/10.1016/S0896-6273(03)00149-1

    doi:10.1016/s0896-6273(03)00149-1 2003

  4. [4]

    London, M., & Häusser, M. (2005). Dendritic computation.Annual Review of Neuroscience, 28, 503–532

    2005

  5. [5]

    Spruston, N. (2008). Pyramidal neurons: dendritic structure and synaptic integration.Nature Reviews Neuro- science, 9, 206–221. https://doi.org/10.1038/nrn2286

    doi:10.1038/nrn2286 2008

  6. [6]

    Branco, T., & Häusser, M. (2010). The single dendritic branch as a fundamental functional unit in the nervous system.Current Opinion in Neurobiology, 20(4), 494–502. https://doi.org/10.1016/j.conb.2010.07.009

    doi:10.1016/j.conb.2010.07.009 2010

  7. [7]

    Larkum, M. (2013). A cellular mechanism for cortical associations: an organizing principle for the cerebral cortex.Trends in Neurosciences, 36(3), 141–151. https://doi.org/10.1016/j.tins.2012.11.006

    doi:10.1016/j.tins.2012.11.006 2013

  8. [8]

    Urbanczik, R., & Senn, W. (2014). Learning by the dendritic prediction of somatic spiking.Neuron, 81(3), 521–528

    2014

Show all 46 references
  1. [9]

    Carandini, M., & Heeger, D. J. (2012). Normalization as a canonical neural computation.Nature Reviews Neuroscience, 13(1), 51–62

    2012

  2. [10]

    R., & Koch, C

    Holt, G. R., & Koch, C. (1997). Shunting inhibition does not have a divisive effect on firing rates.Neural Computation, 9(5), 1001–1013. 11

    1997

  3. [11]

    P., Sprekeler, H., Zenke, F., Clopath, C., & Gerstner, W

    V ogels, T. P., Sprekeler, H., Zenke, F., Clopath, C., & Gerstner, W. (2011). Inhibitory plasticity balances excitation and inhibition in sensory pathways and memory networks.Science, 334(6062), 1569–1573

    2011

  4. [12]

    Frémaux, N., & Gerstner, W. (2016). Neuromodulated spike-timing-dependent plasticity, and theory of three- factor learning rules.Frontiers in Neural Circuits, 9, 85. https://doi.org/10.3389/fncir.2015.00085

    2016

  5. [13]

    J., & V ogels, T

    Hennequin, G., Agnes, E. J., & V ogels, T. P. (2017). Inhibitory plasticity: balance, control, and codependence. Annual Review of Neuroscience, 40, 557–579. https://doi.org/10.1146/annurev-neuro-072116-031005

    2017 doi

  6. [14]

    P., Cownden, D., Tweed, D

    Lillicrap, T. P., Cownden, D., Tweed, D. B., & Akerman, C. J. (2016). Random synaptic feedback weights support error backpropagation for deep learning.Nature Communications, 7, 13276

    2016

  7. [15]

    P., Santoro, A., Marris, L., Akerman, C

    Lillicrap, T. P., Santoro, A., Marris, L., Akerman, C. J., & Hinton, G. (2020). Backpropagation and the brain. Nature Reviews Neuroscience, 21(6), 335–346. https://doi.org/10.1038/s41583-020-0277-3

    2020 doi

  8. [16]

    E., Hinton, G

    Rumelhart, D. E., Hinton, G. E., & Williams, R. J. (1986). Learning representations by back-propagating errors. Nature, 323, 533–536

    1986

  9. [17]

    Nøkland, A. (2016). Direct feedback alignment provides learning in deep neural networks.NeurIPS, 29, 1037–1045

    2016

  10. [18]

    P., & Richards, B

    Guerguiev, J., Lillicrap, T. P., & Richards, B. A. (2017). Towards deep learning with segregated dendrites.eLife, 6, e22901

    2017

  11. [19]

    P., Bengio, Y ., & Senn, W

    Sacramento, J., Costa, R. P., Bengio, Y ., & Senn, W. (2018). Dendritic cortical microcircuits approximate the backpropagation algorithm.NeurIPS, 31, 8721–8732

    2018

  12. [20]

    C., & Bogacz, R

    Whittington, J. C., & Bogacz, R. (2019). Theories of error back-propagation in the brain.Trends in Cognitive Sciences, 23(3), 235–250

    2019

  13. [21]

    A., & Lillicrap, T

    Richards, B. A., & Lillicrap, T. P. (2019). Dendritic solutions to the credit assignment problem.Current Opinion in Neurobiology, 54, 28–36

    2019

  14. [22]

    Scellier, B., & Bengio, Y . (2017). Equilibrium Propagation: Bridging the Gap between Energy-Based Models and Backpropagation.Frontiers in Computational Neuroscience, 11, 24. https://doi.org/10.3389/fncom.2017.00024

    2017

  15. [23]

    Song, Y ., Millidge, B., Salvatori, T., Lukasiewicz, T., Xu, Z., & Bogacz, R. (2024). Inferring neural activity before plasticity as a foundation for learning beyond backpropagation.Nature Neuroscience, 27, 348–358. https://doi.org/10.1038/s41593-023-01514-1

    2024 doi

  16. [24]

    J., & Schölkopf, B

    Gretton, A., Bousquet, O., Smola, A. J., & Schölkopf, B. (2005). Measuring statistical dependence with Hilbert- Schmidt norms.Lecture Notes in Computer Science, 3734, 63–77. https://doi.org/10.1007/11564089_7

    2005 doi

  17. [25]

    Welford, B. P. (1962). Note on a method for calculating corrected sums of squares and products.Technometrics, 4(3), 419–420. https://doi.org/10.1080/00401706.1962.10490022

    1962

  18. [26]

    Turrigiano, G. G. (2008). The self-tuning neuron: synaptic scaling of excitatory synapses.Cell, 135(3), 422–435

    2008

  19. [27]

    A., & Naud, R

    Payeur, A., Guerguiev, J., Zenke, F., Richards, B. A., & Naud, R. (2021). Burst-dependent synaptic plasticity can coordinate learning in hierarchical circuits.Nature Neuroscience, 24(7), 1010–1019

    2021

  20. [28]

    W., Pemberton, J., Mellor, J., & Ponte Costa, R

    Greedy, W., Zhu, H. W., Pemberton, J., Mellor, J., & Ponte Costa, R. (2022). Single-phase deep learning in cortico-cortical networks.NeurIPS, 35, 24213–24225

    2022

  21. [29]

    Haider, P., Ellenberger, B., Kriener, L., Jordan, J., Senn, W., & Petrovici, M. A. (2021). Latent Equilibrium: A unified learning theory for arbitrarily fast computation with arbitrarily slow neurons.NeurIPS, 34, 17839–17851. 12

    2021

  22. [30]

    Hinton, G. (2022). The Forward-Forward Algorithm: Some Preliminary Investigations.arXiv:2212.13345

    2022 arXiv

  23. [31]

    Dellaferrera, G., & Kreiman, G. (2022). Error-driven input modulation: Solving the credit assignment problem without a backward pass.Proceedings of the 39th International Conference on Machine Learning,Proceedings of Machine Learning Research, 162, 4937–4955

    2022

  24. [32]

    Lee, D.-H., Zhang, S., Fischer, A., & Bengio, Y . (2015). Difference target propagation.Machine Learn- ing and Knowledge Discovery in Databases: ECML PKDD 2015, 498–515. https://doi.org/10.1007/ 978-3-319-23528-8_31

    2015

  25. [33]

    Meulemans, A., Tristany Farinha, M., García Ordóñez, J., Vilimelis Aceituno, P., Sacramento, J., & Grewe, B. F. (2021). Credit assignment in neural networks through deep feedback control.NeurIPS, 34, 4674–4687

    2021

  26. [34]

    K., & Buckley, C

    Millidge, B., Seth, A. K., & Buckley, C. L. (2021). Predictive coding: A theoretical and experimental review. arXiv:2107.12979

    2021 arXiv

  27. [35]

    Koch, C., Poggio, T., & Torre, V . (1983). Nonlinear interactions in a dendritic tree: localization, timing, and role in information processing.PNAS, 80(9), 2799–2802. https://doi.org/10.1073/pnas.80.9.2799

    1983 doi

  28. [36]

    Silver, R. A. (2010). Neuronal arithmetic.Nature Reviews Neuroscience, 11(7), 474–489

    2010

  29. [37]

    Max, K., Kriener, L., Pineda García, G., Nowotny, T., Jaras, I., Senn, W., & Petrovici, M. A. (2024). Learning efficient backprojections across cortical hierarchies in real time.Nature Machine Intelligence, 6, 619–630. https://doi.org/10.1038/s42256-024-00845-3

    2024 doi

  30. [38]

    Ma, C., Wu, J., Si, C., & Tan, K. C. (2024). Scaling supervised local learning with augmented auxiliary networks. International Conference on Learning Representations (ICLR)

    2024

  31. [39]

    Lv, C., Xu, J., Lu, Y ., Wang, X., Wang, Z., Xu, Z., Yu, D., Du, X., Zheng, X., & Huang, X. (2025). Dendritic Localized Learning: Toward Biologically Plausible Algorithm.Proceedings of the 42nd International Conference on Machine Learning,Proceedings of Machine Learning Resear...

    2025

  32. [40]

    Erdogan, M., Pehlevan, C., & Erdogan, A. T. (2025). Error Broadcast and Decorrelation as a Potential Artificial and Natural Learning Mechanism.Advances in Neural Information Processing Systems, 38

    2025

  33. [41]

    Kao, C.-H., & Hariharan, B. (2024). Counter-Current Learning: A Biologically Plausible Dual Network Approach for Deep Learning.Advances in Neural Information Processing Systems, 37. https://doi.org/10.52202/079017-2265

    2024 doi

  34. [42]

    Xiao, H., Rasul, K., & V ollgraf, R. (2017). Fashion-MNIST: A novel image dataset for benchmarking machine learning algorithms.arXiv:1708.07747

    2017 arXiv

  35. [43]

    LeCun, Y ., Bottou, L., Bengio, Y ., & Haffner, P. (1998). Gradient-based learning applied to document recognition. Proceedings of the IEEE, 86(11), 2278–2324

    1998

  36. [44]

    Krizhevsky, A. (2009). Learning multiple layers of features from tiny images. Technical report, University of Toronto

    2009

  37. [45]

    D., Toloza, E

    Francioni, V ., Tang, V . D., Toloza, E. H. S., Ding, Z., Brown, N. J., & Harnett, M. T. (2026). Vectorized instructive signals in cortical dendrites.Nature, 652, 1254–1263. https://doi.org/10.1038/s41586-026-10190-7

    2026 doi

  38. [46]

    O., Forest, J., & Ahmad, S

    Iyer, A., Grewal, K., Velu, A., Souza, L. O., Forest, J., & Ahmad, S. (2022). Avoiding catastrophe: Active dendrites enable multi-task learning in dynamic environments.Frontiers in Neurorobotics, 16, 846219. 13 A Supplementary Empirical Results This appendix records the suppor...

    2022

Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.