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A Gradient-based yet Spike-Timing-Dependent Solution to the Feedback Learning Problem in Neural Microcircuits

Xiangnan Zhang, Jingxin Liu, Ranqi Lu, Jingyu Liu, Qunxi Dong, Fuze Tian, Lixian Zhu, Bin Hu, Björn W. Schuller

arXiv:2609.08070v2cs.NEcs.LGq-bio.NC

TL;DR

Practical feedback learning in neural microcircuits and temporal credit assignment from spike timing remain open challenges. The paper uses state separation and Gradient Tunneling to derive sparse-feedback learning signals from local spike timing, and reports a practical framework compatible with ANN-SNN hybrids. It presents the framework as a biologically plausible solution and a computationally plausible model of cortical learning.

  • Problem

    Practical automatic learning of NMC feedback connections remains open, as does whether spike timing alone can support online temporal credit assignment.

    Source: §1 · p. 2

  • Method

    GT with lead-lag expansion treats credit assignment as extracting task-critical state components induced by historical perturbations from the current neural state.

    Source: §1 · p. 2, §2.1.2 · p. 4

  • Results

    The framework provides a practical gradient-based method for sparse-feedback NMC training and a spike-timing-dependent solution compatible with ANN-SNN hybrid architectures.

    Source: §1 · p. 2

  • Takeaways & Limitations

    The framework suggests a computationally plausible explanation for cortical learning and aligns sparse feedback and stochastic computing with neuromorphic hardware requirements.

    Source: §3 · p. 10

    Fading memory makes state separation difficult for long sequences with complex dynamics, while the lack of spatial feature extraction limits suitability for high-dimensional visual streams.

    Source: §3.3 · p. 11

Abstract

from arXiv · show

The brain uses discrete spikes for dynamic computation, yet, how neural microcircuits (NMCs) solve temporal credit assignment using local spike timing remains a fundamental open question. Dominant spiking neural network (SNN) approaches circumvent this by approximating backpropagation through surrogate gradients, decoupling learning from biological spike timing. Here, we reformulate temporal credit assignment as a state separation problem: extracting task-required components induced by historical perturbations directly from the current neural state. This enables an online feedback learning framework for NMCs through a gradient tunneling (GT) algorithm and the lead-lag expansion technique that derives credit assignment from local synaptic spike timing, while remaining compatible with ANN-SNN hybrid architectures. Experimentally, GT-trained NMCs excel at long-timescale evidence integration and noise-robust memory retention, and perform comparably to leading SNN online learning methods on real-world benchmarks with far fewer parameters. The proposed framework addresses the two-decade-old NMC feedback learning problem and suggests a computationally plausible explanation for the brain's learning mechanisms.

1 Introduction

The paper frames practical feedback learning in neural microcircuits and temporal credit assignment from spike timing as linked open challenges. It proposes state separation and Gradient Tunneling (GT) to address them with sparse feedback and local spike timing.

1 Introduction

  • NMC research spans fixed-recurrent liquid state machines with adaptive readouts and train-all recurrent networks with greater dynamic flexibility but lower biological realism and higher computational demand.NMCs are compelling both for understanding biological intelligence and for bio-inspired computation.Source: §1 · pp. 1–2
  • Automatically learning signals for sparse feedback connections in NMCs remains an open problem despite theoretical results on their universal computational capability.The challenge is to assign output error online to preceding causal synaptic events using spike timing.Source: §1 · p. 2
  • Surrogate-gradient methods approximate backpropagation through time but assign credit through differentiable computation graphs rather than spike timing itself.The paper identifies whether spike timing alone carries the information needed for temporal credit assignment as a second challenge.Source: §1 · p. 2
  • GT with lead-lag expansion treats temporal credit assignment as separating task-critical state components induced by historical perturbations from the current neural state.Trainable feedback amplifies those components, providing learning signals from local spike timing while requiring extremely sparse feedback.Source: §1 · p. 2
  • The framework presents a practical gradient-based method for sparse-feedback NMC training, compatible with ANN-SNN hybrid architectures, and proposes a biologically plausible spike-timing-dependent solution.The authors suggest local spike timing as a computationally plausible basis for supervised dynamic reshaping in cortex.Source: §1 · p. 2

2 Results

The results develop a stochastic, state-separation account of feedback learning and test its gradient estimates under stationary and non-stationary inputs. GT-trained NMCs show evidence integration, robust long-memory performance, and real-world task results, with a nested sub-network identified as their functional core.

2.1 NMC Feedback Learning

  • The stochastic NMC account treats population state as expected spiking activity, enabling amplification of task-required state components to separate relevant history.The framework uses feedback connections to reinforce trajectories from historical perturbations toward task-required terminal states.Source: §2.1 · p. 2, Figure 1 · p. 3

2.1.1 Intrinsic Stochastic Rate-coding

  • Recurrent dynamics act as intrinsic white-noise injection, while firing rates estimated by an iterative moving average approximate expected spikes within a short stationary window.The recurrent component’s autocorrelation rapidly vanishes with increasing time lag; this rate-coding property is a prerequisite for the framework.Source: §2.1.1 · p. 4

2.3.1 Evidence Integration on the Cognitive Task

  • On the T-maze task, feedback NMCs learned evidence integration from terminal feedback, whereas the Lyapunov-tuned LSM failed; denser feedback improved final loss and accuracy but increased instability.The authors attribute the instability to stronger shared recurrent input and cross-neural correlations, motivating sparse uniform feedback.Source: §2.3.1 · p. 6, Figure 3 · p. 7

2.3.2 Exceeding the Memory Limit under Extreme Input Noise

  • At L = 75 (SNR ≈−44.76 dB), feedback NMCs returned to low error while LSM error failed to recover after sequence-length increases.The authors interpret this comparison as exceeding fixed-reservoir LSM memory limits under severe noise.Source: §2.3.2 · p. 7, Figure 4 · p. 8

2.3.3 Generalization Capability on Real-world Temporal Pattern Recognition Tasks

  • On SHD, GT-trained feedback NMCs achieved 73.61% accuracy, exceeding e-prop (67.54%) and FPTT (67.24%), but not pp-prop.Setting the GT learning rate to zero significantly reduced accuracy, and training reshaped firing-rate trajectories.Source: §2.3.3 · p. 8, Figure 5 · p. 9
  • On LibEER, GT-trained NMCs led all baselines on SEED accuracy (60.22%) and F1 (58.04%), DEAP-V accuracy (73.93%), and DEAP-A precision (64.37%), recall (65.21%), and F1 (63.36%).The authors report these results across three EEG emotion-recognition tasks under low SNR, non-stationarity, and inter-subject variability.Source: Figure 5 · p. 9, §2.3.3 · p. 10
  • Trainable recurrent connections fell to 0.43% of the fully connected SRNN count; training was about 2.0× faster than LTC-SNN with FPTT and 6.9× faster than SRNN with D-RTRL.The reported speedup was modest relative to the parameter reduction.Source: Figure 5 · p. 9, §2.3.3 · p. 10

2.4 Nested Sub-network of a Feedback NMC

  • The nested sub-network’s output consistently exceeded peripheral-neuron output across three sessions, with negligible accuracy loss versus the full model; peripheral output degraded substantially.The authors identify this smaller network of interconnected feedback neurons as the functional core for temporal pattern recognition.Source: Figure 5 · p. 9, §2.4 · p. 10

3 Discussion

The discussion presents state separation as a solution to practical NMC feedback learning and spike-timing credit assignment. It frames the framework as computationally plausible, while noting that biological use remains to be investigated.

3 Discussion

  • State separation answers practical NMC feedback-learning and spike-timing credit-assignment questions with a sparse, spike-timing-dependent learning rule.The framework is compatible with ANN-SNN hybrid architectures and generalizable to arbitrary NMC dynamical systems.Source: §3 · p. 10

3.1 Searching and Emerging Mechanism for Long Timescale

  • GT updates jointly shape the global attractor and optimize terminal output, selecting feedback changes that benefit both objectives despite short stationary windows.This implicit selection can produce robust feedback connectivity without direct output feedback.Source: §3.1 · p. 10

3.2 Bridging the Contradiction Between Gradient and Spike-timing

  • GT bridges gradient learning and spike timing by operating on spiking-activity distributions; whether cortex uses this mechanism remains untested neurophysiologically.The proposed eligibility trace formalizes timing-dependent plasticity for supervised temporal credit assignment.Source: §3.2 · pp. 10–11

3.3 Limitations and Future Works on Spiking Neural Circuits

  • Fading memory complicates state separation in long, complex sequences, while missing spatial feature extraction limits suitability for high-dimensional visual streams such as DVS inputs.The theoretical breakdown boundary for the stationary-window approximation is also uncharacterized.Source: §3.3 · p. 11
  • The current framework trains feedback within one NMC; supervised feedback across multiple NMCs is proposed as a future hierarchical Spiking Neural Circuits architecture.The feedback NMC is described as the minimal realization of an SNC.Source: §3.3 · p. 11

4 Methods

The methods develop Gradient Tunneling for temporal credit assignment in NMCs using local spike timing, lead-lag feedback, and a deterministic LIF implementation. They also evaluate feedback learning on SHD speech recognition and EEG emotion-recognition tasks through controlled online-learning comparisons.

4.1 Stochastic NMC Model

  • The stochastic NMC model treats spikes as Bernoulli-process realizations and internal state as a stochastic process, while simulations use a deterministic LIF system adhering to Dale’s principle.A stationary window supports the theoretical analysis; in practice its duration is a firing-rate-estimation hyperparameter and should be narrow for bursty, non-stationary inputs.Source: §4.1 · p. 12

4.2.1 Causality-Gradient Theorem

  • The causality-gradient theorem expresses the postsynaptic-rate Jacobian as a difference of conditional firing probabilities, assuming the conditional firing mechanism is invariant to presynaptic-rate changes.The assumption is more plausible when presynaptic neurons converging on a target are not strongly recurrently connected; sparse uniform feedback suppresses cross-neural correlations.Source: §4.2.1 · p. 12, Theorem 1 (Causality-Gradient Theorem) · p. 13

4.2.2 Implementable Form Based on Local Spike-Timing

  • Under stationarity, conditional-mechanism invariance, and rapid decorrelation across time lags, the implementable theorem estimates the desired gradient from a local stochastic process within a stationary window.Its stability condition constrains the network, not the input stream.Source: §4.2.2 · p. 13, Theorem 2 (Implementable Form Based on… · p. 13

4.3 Gradient Tunneling Algorithm

  • GT combines local tracing, which forms eligibility and Jacobian traces, with global tunneling of learning signals; batch size one makes it strictly online.The eligibility trace links local tracing to global tunneling and supports temporal credit assignment, while the Jacobian trace approximates the Jacobian for back-propagating learning signals.Source: §4.3.1 · p. 14, §4.3 · p. 14

4.3.3 Firing Rate Regularization

  • GT adds a locally implemented firing-rate regularizer that steers each neuronal population toward a predefined target rate.The regularization is treated as approximately independent of upstream microcircuits and does not enter learning-signal back-propagation.Source: §4.3.3 · p. 15

4.4 Scheme for Feedback Connection

  • Lead-lag expansion models feedback from the previous spiking state using sparse, widely separated neurons to reduce firing-mechanism correlations and make feedback weights trainable by GT.The lag layer can receive a subset of prior spikes, while the lead layer represents current activity; arbitrary dynamic-pattern inputs can be used with randomly generated, frozen input weights.Source: §4.4 · p. 16

4.5.1 Evidence Integration Task

  • The evidence-integration setup uses 100 input channels over 360 time steps and a 1,000-neuron NMC, with a 20-step stationary window and residual readout.Cue, recall, and noise channels have distinct firing schedules; neurons are spatially embedded in a cube for distance-based recurrent weights.Source: §4.5.1 · p. 16

4.5.2 Incremental Add Task

  • The incremental-add setup increases effective sequence length from 15 to 75 within 90 steps and uses a 512-neuron NMC with 10% feedback and a 15-step stationary window.The GT learning rate starts at 1 and is halved every 4,000 iterations; the readout learning rate remains 10^-4.Source: §4.5.2 · p. 17

4.5.3 Speech Recognition on the SHD Dataset

  • SHD comparisons used a shared 50-step input window, single terminal loss, and 1,000-neuron recurrent branch to isolate online learning rules.Raw SHD streams were binned into 100 steps of 14 ms across 700 channels, with the nearly silent second half discarded; feedback dimensionality was 700. The feedback NMC used a GLU-Residual readout [24] [45], while baseline readouts followed their original implementations; models were selected by test-set criterion and compared using independent-samples t-tests.Source: §4.5.3 · p. 17

4.5.4 EEG-based Emotion Recognition

  • EEG emotion recognition used subject-dependent SEED and DEAP protocols with four frequency-band populations whose final states were classified by Softmax regression.Following LibEER [46], the tasks were three-class SEED and binary valence/arousal DEAP, with 6:2:2 splits and 3-second clips. Four-band frequency-domain entropy features from 14 electrodes were converted to spike trains using improved BSA; each population processed 14 channels, with stationary windows equal to clip length. Reproduced comparisons included offline LSTM/GRU, EEGNet, and SNN baselines: Lyapunov-tuned LSM, LTC-SNN with FPTT, LSNN with e-prop, and SRNN with D-RTRL and pp-prop, using 512 hidden neurons for recurrent models.Source: §4.5.4 · p. 17

A.1 Proof of Theorem 1

The proof derives the gradient expression under an invariance assumption on the conditional firing mechanism, rather than requiring presynaptic spike events to be independent.

A.1 Proof of Theorem 1

  • The derivation expands expectations using total probability and invokes the Bernoulli identity E[X] = P(X = 1) to obtain the desired result.Source: §A.1 · p. 22
  • Under conditional-mechanism invariance, the relevant conditional-probability terms are treated as constants when differentiating with respect to presynaptic firing rates.Source: §A.1 · p. 22
  • The proof requires the postsynaptic conditional firing distribution to remain stable under presynaptic firing-rate perturbations, not presynaptic independence.The authors call this assumption rectification indispensable for the validity of the gradient expression.Source: §A.1 · p. 22

A.2 Proof of Theorem 2

The proof derives the input-weight partial derivative through expectation simplifications under stated conditions, then applies the Causality-Gradient Theorem; the resulting update uses nonpolar firing events.

A.2 Proof of Theorem 2

  • Conditions C1 and C3 support simplifying and factorizing the expectations used in the derivative derivation.The proof also uses C1 to drop temporal indices because the conditional probability is independent of time.Source: §A.2 · p. 23
  • Applying the Causality-Gradient Theorem under Condition C2 gives the final step in deriving the input-weight partial derivative.Source: §A.2 · p. 23
  • The implementable update uses nonpolar firing events, so the synaptic plasticity rule follows a unified update and naturally respects Dale’s principle.Source: §A.2 · p. 24

Generation of Recurrent Weights and Neuron Polarity

The methods specify recurrent connectivity and neuron polarity, initialize input and feedback weights, and configure IMA filters using the expected stationary-window width.

Generation of Recurrent Weights and Neuron Polarity

  • Recurrent connectivity probability depends on neuron distance; connected weights are sampled from a truncated Gaussian, with 20% of neurons assigned inhibitory polarity.The parameters are C = 0.2 and σ = 3; connected weights have mean 0.1 and standard deviation 0.05 before negative values are truncated.Source: Generation of Recurrent Weights and… · p. 24

Initialization of Input Weights

  • Input weights are independently sampled from a Gaussian, with their mean chosen so saturated input is expected to make the postsynaptic neuron fire.Feedback connectivity is set so that most postsynaptic neurons receive trainable feedback; the no-feedback probability is q = (1 − p)^d_l−1.Source: Initialization of Input Weights · p. 24

Setup of IMA Filters

  • IMA filter coefficients are configured from the expected stationary-window width, treated as the filter’s settling time, with a 5% error setting.The stated condition le ≥ 14.2 is satisfied in this research.Source: Setup of IMA Filters · pp. 24–25

Appendix C Readout Configuration for SHD Dataset

The SHD feedback NMC uses a GLU-Residual readout, while the baselines retain their source-native readouts so the comparison isolates recurrent-layer online learning rules.

Appendix C Readout Configuration for SHD Dataset

  • The feedback NMC maps terminal firing rates to class logits with a GLU-Residual readout trained by backpropagation, while GT updates recurrent weights.The readout combines a gated linear unit compression and a residual block.Source: Appendix C · p. 25
  • Source-native baseline readouts avoid temporal dynamics modeling so the SHD comparison isolates online learning rules in the recurrent layer.Baseline outputs use terminal logits at step 50 without summing or averaging per-step outputs.Source: Appendix C · p. 25
  • The e-prop LSNN uses 20 non-spiking leaky-integrator output units with a 15-step readout time constant and eligibility traces decaying across the 50-step evidence interval.Its output weights have shape 1000×20; the eligibility-trace decay is exp(−1/33) ≈ 0.97.Source: Appendix C · p. 25
  • The FPTT LTC-SNN uses the official shallow DVS-Gesture leaky-integrator readout with 20 learnable, time-shared leak parameters initialized to produce λ = 0.5.Its leak parameters are initialized to zero.Source: Appendix C · p. 25
  • The pp-prop SRNN uses a fixed low-pass filter followed by a static affine projection, with terminal cross-entropy training and no eligibility traces.Baselines retain their source-native optimizers and learning-rate schedules.Source: Appendix C · p. 25

Appendix D The Improved BSA Spike Coding Scheme

The improved BSA scheme addresses coefficient selection and unbounded inputs by using a two-parameter FIR filter and normalizing and squeezing the analog signal before encoding.

Appendix D The Improved BSA Spike Coding Scheme

  • Traditional BSA uses a finite impulse response filter whose optimal coefficients are difficult to determine and cannot encode inputs without a predetermined bound.These are the two limitations motivating the improved spike-coding scheme.Source: Appendix D · p. 25
  • The FIR coefficients use τ1 = 3, τ2 = 2, and length l = 20, requiring tuning only two parameters while producing a distribution similar to recommended settings.The coefficients can be interpreted as the postsynaptic potential of a LIF neuron with first-order synapses.Source: Appendix D · p. 26
  • Because normalized FIR coefficients restrict BSA inputs to [0, 1], the method normalizes the original signal to mean 0 and standard deviation 1 before squeezing it.Under a Gaussian assumption, most normalized values fall in [0, 1], but bounds are not guaranteed.Source: Appendix D · p. 26
  • The squeezing step sets κ = 0.45 and β = 0.55 to constrain encoded inputs to (0.1, 1), leaving a lower margin to mitigate small-value encoding artifacts.The BSA definition domain is [0, 1].Source: Appendix D · p. 26

Appendix E Additional Experimental Results

Additional results compare gradient tunneling with Lyapunov-based tuning across neural population scales and report benchmark metrics and resource consumption across several datasets.

Appendix E Additional Experimental Results

  • The appendix reports SHD learning-algorithm test metrics averaged over four seeds, EEG pattern-recognition results, SEED classification results, and DEAP training resource consumption.Source: Table E2 · p. 26, Table E3 · p. 27, Table E4 · p. 28
  • Gradient tunneling outperforms Lyapunov-based tuning from edge length 7, with the advantage increasing as neuron amount expands.The comparison is reported across neural population scales.Source: Figure E1 · p. 27

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