Difference Predictive Coding for Training Spiking Neural Networks
Authors: Ville Karlsson
Nicklas Fianda
Joni-Kristian Kämäräinen
Venue: ICLR 2026 Poster (2026)
Keywords: Spiking neural networks
predictive coding
biologically plausible learning
neuromorphic computing
difference predictive coding
local learning rules
energy efficiency
communication efficiency
spike-based learning
surrogate gradient alternatives
Primary area: unsupervised, self-supervised, semi-supervised, and supervised representation learning
TL;DR: We create a spiking neural network compatible learning algorithm based on predictive coding theory

Abstract: Predictive coding networks (PCNs) offer a local-learning alternative to backpropagation in which layers communicate residual errors, aligning well with biological computation and neuromorphic hardware. In this work we introduce Difference Predictive Coding (DiffPC), a spike-native PC formulation for spiking neural networks. DiffPC replaces dense floating-point messages with sparse ternary spikes, provides spike-compatible target and error updates, and employs adaptive threshold schedules for event-driven operation. We validate DiffPC on fully connected and convolutional architectures, demonstrating competitive performance on MNIST (99.3\%) and Fashion-MNIST (89.6\%), and outperforming a backpropagation baseline on CIFAR-10. Crucially, this performance is achieved with high communication sparsity, reducing data movement by over two orders of magnitude compared to standard predictive coding. DiffPC thus establishes a faithful, hardware-aligned framework for communication-efficient training on neuromorphic platforms.

Published as a conference paper at ICLR 2026
DIFFERENCE PREDICTIVE CODING FOR TRAINING SPIKING NEURAL NETWORKS
Ville Karlsson, Nicklas Fianda & Joni-Kristian Kämäräinen Tampere University
Tampere, Finland [ville. [ville.karlsson karlsson, nicklas. facklas..fannn fianda) tunni,
joni- meni-kristian.o kamarainentturi.
ABSTRACT
Predictive coding networks (PCNs) offer a local-learning alternative to backprop- agation in which layers communicate residual errors, aligning well with biolog- ical computation and neuromorphic hardware. In this work we introduce Differ- ence Predictive Coding (DiftPC), spike-native PC formulation for spiking neu- ral networks. DiffPC replaces dense floating-poin point messages with sparse ternary spikes, provides spike-compatible target and error updates, and employs adaptive threshold schedules for event-driven operation. We validate DifiPC on fully con- nected and convolutional architectures, demonstrating competitive performance on MNIST (99.3%) and Fashion-MNIST (89.6%). and outerfforming backprop- agation baseline on CIFAR-10. Crucially, this performance achieved with high communication sparsity, reducing data movement by over two orders of magni- tude compared standard predictive coding. DifiPC thus establishes a faithful, hardware-aligned framework for communication-effiieenn training neuromor- phic platforms.
INTRODUCTION
The error backpropagation algorithm has been fundamental to the success of deep learning, yet its core mechanisms are widely considered biologically implausible [Salvatori et al. (2023). Key limitations include the requirement for global error whhlee synaptic updates depend on information transmitted across multiple layers, far beyond locally available signals- and the reliance on continuous-valued communication and gradients, in contrast to the brain's 's use of discrete, event- driven signals N et al. 2024). These discrepancies create significant gap between conventional artificial neural networks (ANNs) and biological neural systems.
Artificial intelligence does not need to replicate biology-airplanee do not flap their wings wutgs-but certain biological properties are worth emulating. Examples include the brain's 's energy efficiency and its ability to perform robust and adaptive computation with sparse, noisy, and low-precision sig- nals N'dri et al. (2024). These observations align with the development of neuromorphic systems, which address the limitations of conventional von Neumann architectures by co-locating memory and computation to reduce data movement AI Abdul Wahid et al. (2024), thereby enabling substan- tially lower energy consumption-a key biological property worth emulating. Within this hardware setting, Spiking Neural Networks (SNNs) provide natural computational model: information is represented not by continuous activations but by discrete spikes, as observed in the brain Olshausen & Field (1996 (004)). In SNNs, neurons communicate through sparse, asynchronous events rather than dense, synchronous updates Mainen & Sejnowski (1955; Cox al. (2000), making them in- trinsically well suited for low-pon wow-pwwer implementation.
One promising framework for training such systems Predictive Coding (PC), Originating from neuroscience, PC theorizes that the brain functions as prediction machine, continuously generating top-down predictions of sensory input while bottom-up signals convey only the residual prediction errors Rao & Ballard (1999); Friston (2005): Spratling 2017); Huang & Rao (2011) Keller & (lsicc-lloee](((11)) Importantly. PC relies on local learning rules, where synaptic updates depend only on the activity of adjacent pre- and post-synaptic neurons. This locality makes PC highly

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compatible with the parallel and distributed organization of neuromorphic hardware N' et al. 2024 Salvatori et al. (2023).
However, despite its theoretical alignment with neuromorphic principles. the standard formulation PC faces practical computational challenges. To infer neural activities, PC networks typically per- form an iterative settling process, requiring multiple forward and backward passes of information to converge for a single input. In standard implementations, this process relies on dense, floating-pon point message passing, resulting in a computational overhead that notably exceeds that of backpropaga- tion Rosenbaum (2022). ). This reliance on dense, continuous communication during the iterative phase can offset the efficiency gains sought by deploying SNNs on event-driven hardware. There- fore, while PC offers solution to the global transport problem of backpropagation, its standard formulation does not fully exploit the sparsity and efficiency of neuromorphic substrates.
Combining SNNs with PC-based training is natural research direction to address these issues Lan jtt al. (2022); Wacongne al. (2012); Boerlin al. (2013); Ororbia (2023). In this work, we intro- duce Difference Predictive Coding (DifiPC), framework for native implementation in SNNs. DiffPC seeks to address the communication over- head of standard PC by replacing dense, floating- phati message passing with sparse, ternary spike- based communication. By employing schedules, DifiPC ensures that computation and message passing event-driven, occurring when necessary to correct prediction errors. Our results indicate that DifiPC achieves accuracy matching exceeding that of standard predictive coding networks (PCNs) and backpropagation trained mod- els on benchmark datasets, while reducing the number of transmitted bits by more than two orders of magnitude compared to standard PC baselines.
algorithm that reformulates the predictive coding
state update rules and adaptive threshold
We propose Difference Predictive Coding (DiffPC), a spike-native framework based on novel update rules that transmit incremental state updates via sparse ternary spikes rather than broadcasting the full state, resulting in reduced communication costs.
We introduce an adaptive threshold scheduling mechanism that enables the discrete spiking network to closely approximate the dynamics of standard continuous predictive coding with fewer timesteps.
We empirically validate DifiPC fully connected and convolutional architectures, demon- strating that it matches the accuracy of standard predictive coding and matches or exceeds that of Backpropagation on MNIST, Fashion-MNISS and CIFAR-10. while reducing the number of transmitted bits by more than two orders of magnitude compared to standard predictive coding baselines.
2 RELATED WORKS
Spiking neural networks. SNNs compute with discrete events and update their state only upon spike arrivals, yielding sparse, asynchronous processing that maps well neuromorphic substrates and modern accelerators Pfeiffer & Pfeil (2018); Tavanaei et al. (2019). This event-driven operation provides a natural match the parallel, low-power architecture of neuromorphic hardware such as TrueNorth Akopyan et al. (2015). Loihi Davies et al. (2018), Loihi 2 Intel Corporation (2021b), and SpiNNaker[Furber et al. (2014). Beyond neuromorphic-viis vissoi benchmarks, deep SNNs now achieve competitive accuracy on static datasets when equipped with convolutional backbones and carefully engineered neuron and normalization layers Hu al.) (2024).
Several software frameworks have been developed to simulate and train SNNs, including Brian2 Stimberg et al. (2019), NEST Gewaltig & Diesmann (2007), SpikingJelly Fang et al. (2023), and LAVA Intel Corporation (222aa) In this work we use LAVA to verify that our methods are com- patible with the Intel hardware chip Loihi Intel Corporation (0221b), but also provide a Py PyTrch implementation for easy verification and faster runtimes.
SNN training. A key challenge in SNN learning is that spike generation is non-differentiabl,, preventing direct application of backpropagation. Contemporary approaches can be divided into three main families, each with distinct accuracy, latency, and efficiency trade-offs Hu et al. (2024).
(i) ANN conversion. In this approach, a ReLU ANN is trained with backpropagation and mapped to an SNN under a rate or latency coding assumption Cao et al. (2015). Practical pipelines reduce activation-rate mismatch via weight and threshold normalization Diehl et al. (2015) reset-
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by-subtraction Rueckauer et al. (017)) and further tighten equivalence using quantization mapping Hu et al. (2023), y clip-floor-shin activation Bu et al. (2023), and post-training parameter calibration Li et al. 2024). Conversion scales well and can match ANN accuracy with few timesteps, but training remains off-chip with dense floating-pont communication, and residual conversion error or latency may erode energy benefits.
(ii) Direct aurrogate-gradien training. By treating SNNs as recurrent systems unrolled over time, the zero derivative of the spike function is replaced with a smooth surrogate, enabling backprop- agation through time (BPTT) Wu et al. (2018); Neftci et al. (2019). Representative methods in- clude temporal-loss formulations SpikePropl Bohte al. (2002) and time-based error reassignment (SLAYER) Shrestha (LLAYER))/rrsthaa & Orchard (2018). Later advances improved optimization and representation by learning neuron dynamics such as time constants Fang et al. (02T). Significant progress was also made in normalizing membrane dynamics across time via threshold-dependent scaling (tdBN) Zheng et al. (20T)) time-varying parameter decoupling (BNTT) Kim & Panda (202T), input rescal- ing for uniform temporal distributions (TEBN) Duan et al. (0022), direct membrane potential regulation (MPBN) Guo et al. (2023). Finally, other methods mitigate surrogate mismatch via gra- dient re-weighting (TET) Deng al. (2022), information maximization objectives (IM-Loss) Guo jet al. (2022). or learnable surrogate shapes Lian et al.] (2023). These methods reach state-of-the-an accuracy with O(1 8) timesteps, but they still rely on global backward signals and dense commu- nication, which can limit viability of on-chip training.
(iii Local plasticity. Purely local rules, such as timing-depenneee plasticity (STDP) and reward-modulated variants, are aligned with both biology and hardware constraints. However, they typically require auxiliary classifiers and tend underperform on complex tasks Diehl & Cook
2013); Kheradpisheh et al. (2018); Ororbia (2023).
Overall, these approaches highlight a central trade-off in SNN training: methods that achieve the highest accuracy typically rely on dense, non-local signals or off-chip training, while methods that are fully local and spiking have yet consistently match this performance on complex tasks.
Predictive coding. Predictive coding (PC) has recently emerged as an alternative to backpropaga- tion that is both biologically motivated and compatible with event-driven computation. In PC, each layer generates predictions of activity in the layer below. while only the residual prediction errors are communicated forward Raao & Ballard (1999): Friston (2005); [Spratling| (2017); Huang & Rao (011). PC has been developed into computational framework with formal links to backpropaga- tion and variational inference Millidge et al. (202I): Rao & Ballard (1999). A central advantage is its use of local learning rules: synaptic updates depend only on the activity of adjacent pre- and post- synaptic neurons, making the framework well suited for distributed neuromorphic implementation
N'dri et al. (2024): Salvatori et al. (2023).
Several works have sought to integrate PC with SNNs Lan et al. (2022) Ororbia (2023 Lee et al. (2024). The PC-SNN algorithm formulates predictive coding in time-to Lime-to-fisste to-hist--siike (TTFS) encod- ing, where each neuron spikes at most once Lan et al. (2022). This achieves unmatched energy efficiency in terms of spikes, but runtime scales exponentially with input precision (2 timesteps for B-bit input) and must be predefined due to the single-spike restriction. Additionally, their training schema remains reliant on transmission of dense floating point numbers and is done on GPU. Recent work toward creating purely spiking predictive coding frameworks Ororbia (2023); Lee et al. (2024) has made significant progress. However, to evaluate performance on discriminative benchmarks like MNIST, these frameworks adopt hybrid approach. The spiking network is first trained in purely unsupervised manner to learn representations. Subsequently, its weights are frozen, and a separate, non-spiking linear classifier is trained post-hoc rate-coded activities extracted from the network's 's final layer. This reliance on an external, non- non-spiking component for the final classification step means the reported accuracies do not reflect the performance of an end-to-end spiking system, complicating a direct assessment of their utility for fully neuromorphic deployment.
Relation to DiffPC. Event-driven gradient-by-spikes approaches approximate backpropagation by discretizing gradients into spikes Bohht et al.] (2000); Cai et al.] (2024), which enables training and inference using only spikes. We apply similar approach in our proposed Difference Predictive Coding. Distinct from prior work, DifiPC integrates these concepts through three key mechanisms: (1) a spike-based message passing protocol that adapts sparse ternary communication specifically for predictive coding error propagation; (2) a difference- update rule that triggers communication
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only upon state changes to minimize redundancy; and (3) cyclic threshold scheduler designed to accelerate the convergence of discrete spiking states toward continuous PCN targets.
3 BACKGROUND
3.1 SPIKING NEURAL NETWORKS
SNNs are a class of artificial neural networks that mimic the behavior biological neurons more closely than conventional neural networks. Neurons in spiking neural networks communicate through discrete spikes, or action potentials, when their membrane potential V reaches a certain activation threshold To. The output of the neuron is function of the potential s,,V). The neu- ron model utilized in this work is based on the difference equation of the Integrate- and---ndd-Fr neuron model:
Vi(1 + = v.(1) Tos: (V.(1)) + Σ mi)8s((), Vi(0) = b
where (i()) is the integration variable (membrane potential), Tg is the threshold, by is the bias and wy is the weight of the synapse connecting the input neuron j to the neuron The spike activation function st) E {-1,0, 0,
V 2 Tg. if Vi(t) S -Te
f
(V((t))
otherwise
Activation of the Integrate-and andttte--nd--aaee and-Fire--nn Fire-neuron output causes voltage drop (damping) in the difference equation (
3.2 PREDICTIVE CODING
In the context of neural networks, PC proposes that each layer generates predictions of the activity in the next layer. The next layer then computes the error between the forward propagated prediction and the target activity. This signal is then propagated backward to update both the targets and synaptic weights of the previous layer. The learning objective is to reduce the overall prediction error in the network. Unlike the separate forward and backward passes of conventional deep learning.
Predictive Coding Networks (PCNs) involve a bidirectional local flow of information; predictions of current targets in direction and prediction errors in another (Figure [ The target activity updated using the received prediction error and the synaptic weights using the target activity. The updates are local, which substantial difference from the backpropagation algorithm, where updates depend on a single error signal calculated at the final output and propagated backward through the entire network.
X
x
Xasa
X
Xpa
X
X
E
E
E
The structure of multi-layer Predictive Coding Network (PCN). Each neuronal unit, bounded by the dashed line, consists the target activity Xy, prediction Xy, and prediction error e. The arrows indicate the flow of information between layers = 0, 1, 1,..., L. where the feedforward path carries the predictions from Xy, to X and the feedhack path conveys the prediction errors which are used to update X7,1-1- Computations and updates can be asynchronous.
Figure 1:
In the PCN literature the terms top/down more common, but opt forward/backward for consistency with the deep learning terminology.
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We review the PC principles for conventional Multi-Layer Perceptron network (MLP) of L dense layers to the imput x to output y. The input and output layers are indexed as / = 0 and L. and between them are the hidden layers 1 to --1. 1. Predictive Coding (PC) makes use of each layer's current activation vectors Xr.1 ("T' referring to "Target'). In addition, the layers include vectors x (F: 'forward') that are the predictions generated by the previous layer I 1. Ideally, the predictions and the targets should be the same.
Energy function. Layer predictions are calculated from the target activity of the layer
xp = W Wio(XT,-1)
(3)
where Wr E R N are the weights, function. The difference between the targets and predictions is fed back as the prediction error
the number of neurons, and o(-) is an activation
6 = XT,I XF.
The errors from all layers are summed to compute the 'free-energy' of a network and PC operates by minimizing this free-energy function
F
Σ Iall
Σ (XT. xF,)²
(5)
Update steps. The main difference between PC and the gradient descent enror backpropagation that PC alternates between the updates, the target and weight updates, which are computed locally and can be asynchronous. The synaptic weight update is
OF OW,
w
w(O))
w (CT((XT)-1)
(6)
where is the learning rate, and the prediction update is
OF OXT
7 ( - Wfi o ((T,))) for
for < L,
(1+1) x T.1 X
(1)
(7)
7Eb
where ? is the prediction learning rate and is the Hadamard product. Because updates only occur between adjacent layers, they must be performed iteratively to allow information to propagate throughout the entire network. See Appendix 7.1 for an intuitive explanation of predictive coding.
THE DIFFERENCE PREDICTIVE CODING ALGORITHM (DIFFPC)
In this section, propose the Difference Predictive Coding (DifPPC) algorithm that implements the standard PC on SNNs. To ensure that our algorithm can be deployed on a neuromorphic chip, used the instruction set of the Intel Loihi 2 neuromorphic chip. The algorithm verified in the official simulator because access to the actual Loihi 2 hardware limited to Intel partners. We provide the full simulator code and the pseudocode is given in Algorithm
4.1 ALGORITIIM OVERVIEW
To adapt the predictive coding framework from Section 3.2 for spiking neural networks, its floating- point computations and information transfer must be converted into discrete spikes (Algorithm [). In this formulation, all information transmitted during the learning steps takes the form of a sequence of ternary values (-1, 0, 1).
In DifiPC, each unit maintains target state 2T and actual state ZA. The target state æy repre- sents the desired (target) activity, while the actual state x attempts to follow TT, aiming to minimize the difference between them. Their difference is reduced incrementally by steps proportional to an adaptive threshold Tg. The difference- based adjustments are communicated as spikes to subsequent layers, which integrate the incoming spiking information.
Two error variables, ér and e function in the same way and represent the errors of the two states. er is the target error and e the actual error that attempts align with er. The error ea is adjusted
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by steps proportional Tg, which are then transmitted subsequent layers spikes. Since the threshold Te determines the step size of these updates, its schedule is critical for convergence. In the next section, will introduce specific scheduling strategies.
Feed Forward Initialization. Before the iterative DiftPC process begins (as detailed in Algo- rithm D. the network undergoes feed forward initialization phase. In this phase, input spikes propagated through the layers in a single pass without feedback error calculation. This rapidly es- tablishes an initial estimate for the target activities XT and predictions XF, reducing the number of subsequent iterative steps required for convergence. This phase effectively mimics standard feed- forward SNN inference step to prime the network state and can be implemented by utilizing graded spikes on the Loihi 2 chip. See section 7.2 of the appendix for more detailed explanation of the algorithm.
Algorithm DifPP Algorithm for Spiking Neural Network Training
Input: Spike signals Sin- Se
Process parameters: threshold (t),,, learning rate r(t), weight Ir a Initialize: Zy, Ir, AA er, Ca. CB, $,, &e
1: Feed Forward Initialization: Propagate input to prime
2: for each time step do
3: s/n + W
> Receive spike imput
4:
Jp xp - alp + + TT 1)
Update forward prediction
5:
if("(() > 0 then
6:
+ 2 2
b Compute target error b Update Target Activity
7:
+ + (- + (4) 0))
8:
if >00000m0 then
9:
G + o/. s/p
Update target error
10:
A + sign(z) - 0 (1 = (t)
Generate spikes
+ s S @ (1' + To(t)= 0)
'Spiking ReLU'
12:
AA x x + (()) s
Update Actual Activity
(ut.send(s (x)
b Send state spikes
14:
Propagate Error: coout.send(s')
15:
> Send error spikes
16:
eni'' + (Wi+I)Ta 1+1
> Receive error input D Accumulate incoming errors D Generate error spikes
17:
éB - epp + (o - 1) ee in'
18:
s + sign(er - e1)) (1cr - c/al > To()))
19:
é + d'A + [o(t)= st
b Update actual error
20:
W
+ W + a Ma 1)
b Update Weights
4.2
To AND SCHEDULES
We present a cyclic scheduler, which allows for accurate approximation of the standard PC algo- rithm,
2m
(o(t)
(() = g(t mod n)
(8)
2 mad n
where
7 if 0, otherwise
Z sets the initial magnitude of the threshold via (0) = 2m, and mod n denotes the modulo operation. The set of steps starting with (t) y is referred to 7-cycle and set of timesteps during which we train the network with single input and output pair is called an iteration. A single iteration therefore consists of multiple 7-cycles.
where t E N+ denote the cycle length and timestep index, respectively,

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To theoretically motivate the precision of this approach, establish the following bound on the quantization eτror demonstrating that, in the absence of error, the spiking states exponentially con- verge their seaating-poin targets within cycle.
Theorem 4.1. Suppose that the target activity for layer I, denoted as sly, satisfies |c/1 s'Al < 2 and zdp > O. Then, after one r-cycle of timesteps of cyclic scheduler, the difference between the target activity and the actual activity 1z/r z'Aal is less than 2m+
Proof. See Appendix 7.3
Though this method is able to attain great precision in the approximation of standard PC when set n large, comes at the cost of extra timesteps and spikes. In practice, we observe that as the PC network converges, the changes in er and 2T become smaller. Thus should also scale the eyclic scheduler to be smaller as the network converges allowing us to use smaller n and still attain high accuracies. This is the motivation behind the eyclic decay scheduler defined
n
(t) d(t mod T) mod
7(1) g(t mod n).
(10)
where d(t) is a decreasing function. We
d(t) = (1-(1-0))
(11)
where a E (0, 1] such that d(t mod T) (0, 1] and T the length of the iteration. In addition to the cyclic decay scheduler, we introduce the constant decay scheduler, defined as
To(t) = d(t mod c) 7(1) - g( mod nt),
(12)
where e R This schedule maintains fixed threshold that scaled by the decay function d(t), therehy reducing the update size over the course of an iteration.
EXPERIMENTS
5.1 DATA AND SETTINGS
We evaluate our method using Multi-Layer Perceptron (MLP) architectures as MLPs provide a clean and well-studied baseline for predictive coding, and we extend our evaluation to Convolutional Neu- ral Networks (CNNs) on the CIFAR-I0 dataset to demonstrate the method's applicability to convo- -NIIT,, and CIFAR-10
lutional networks. For empirical validation, we use the MNIST, benchmarks. MNIST and Bushion-MNIST comprise 60,000 training and 10,000 test grayscale im- ages size 28 x 28 across classes. CIFAR-10 consists 50,000 training and 10,000 test color images of size 32 32 across 10 classes. We train fully connected networks with one or two hidden layers for the simpler tasks, utilizing dropout. For CIFAR-10, we utilize a convolutional architec- ture. consisting of two convolutional layers with 5 x 5 kernels and stride 2 (with 10 and 5 filters respectively). followed by three fully connected layers. All models are optimized using Adam W. Data augmentation includes random translation jitter for MNIST and random horizontal flips for CIFAR-10 10. We assess performance by test-set classification accuracy and by spike efficiency, quan- tified the average number of activity and error spikes per neuron per sample during training. Our CIFAR-10 implementation is based on code from Rosenbaum (022): Millidge et al. (202 Whittington & Bogacz (2017).
Selected baselines For comparison, we focus on Convolutional and MLP networks and include both conventional floating soating-point and spiking implementations. In addition, report results from spiking networks trained with alternative learning rules beyond predictive coding, providing context on how our approach compares to state-of-the-art non-PC methods

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5.2 RESULTS
Classification accuracy On MNIST, DiffPC achieves high accuracy that matches previously re- ported results for non-convolutional spike-based methods. For example, DiffPC-L. attains 99.3% accuracy and DifIPC-S reaches 98.2%, placing them par with above several recent SNN mod- els, as shown in Table T On Fashion-MNIST, which presents a greater challenge, DifIPC also achieves competitive accuracy (Table 2 Finally, CIFAR- 10, DiffPC demonstrates effective scal- ing to convolutional architectures: DifiPC-Long achieves 65.6% 6% accuracy, surpassing the standard backpropagation baseline of 63.5%, while DiffCC-Efficien reaches 63.3% (Table)4).
Communication efficiency Communication efficiency provides further insight into the advan- tages of DiffPC. On modern hardware, the energy cost of moving data is often comparable to, and in many workloads higher than, the cost of arithmetic operations Horowitz (2014); [Lian et al.] (2023)- Because memory access and interconnect traffic can be orders of magnitude more energy-intenas than a multiply-accumulate, the number of floating-point values transmitted during training and inference is key proxy for communication energy. In addition, spiking implementations require computation to unfold in discrete timesteps, and the number of timesteps needed for convergence strongly predicts runtime performance Li al. (023)).
Table reports the average number bits transmitted per neuron during error propagation and the corresponding timestep counts the MNIST task. Standard backpropagation transmits 32 bits per neuron a single timestep, while predictive coding (PC-SE) requires 960 bits across 15 timesteps. SNN-based predictive coding (PC-SNN) is similar they use floating point numbers during the training stage of their network. In contrast, DifIPC achieves urders-of-magniuus DiffPC-L transmits only 0.18 bits (0.09 spikes) per neuron on average across 120 timesteps, and DifiPC-S reduces this further to 0.08 bits (0.04 spikes) per neuron across 75 timesteps. These results demonstrate that DifiPC combines competitive accuracy with substantially improved efficiency in terms of communication.
improvements:
We observe similar trends on the CIFAR-10 dataset, as detailed Table 4 Here, we compare two configurations: DiffPC-Long, which utilizes scheduler cycle length of = 16, and DiffPC- Efficient, which employs a shorter cycle length of = 12. Both configurations run for 15 cycles per sample. While the convolutional architecture utilizes higher-fidelity error messaging compared to the MLP used for MNIST, DiffPC retains a substantial efficiency advantage. DifiPC-Long requires only 1.9 bits per neuron, and DifiPC-Efficien further reduces this to 0.7 bits. Although these values are higher than those for MNIST, they remain significantly lower than the 32 bits required by back- propagation the 960 bits used by standard PC, demonstrating that the communication sparsity of DifiPC scales effectively to convolutional networks.
Table 1: Comparison the Test Accuracy of Different SNN and PC Models on the MNIST dataset. (FC denotes fully connected layers)
Method
Network Architecture
Acc, ( (%)
Backpropagation
99.3
PC-SE (Standard PC) Pinchetti et al.
2024
98.3
STDD- (Mirsadeghi et al. (0211)
40C5-P2-1000FC--I
99.2
SSTDP Liu et al. (202T)
784FC-300FC-10FF
98.1
PC-SNN Lan et al. (2022
284FC-200FC-10FC
98.1
SRC-RNN DDe Geeter et al.
2024
98.4
FastSNN Taylor et al. 2022
784FC-1000FC.
97.9
FastSNN Taylor et al. 2022
02-64C5-P2-1000FE
99.3
DiffPC-L (Ours)
784FC-1024FCC
99.3
DifIPC-S (Ours)
184FC-400FC-1OFC
98.3
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Table 2: Comparison of the Test Accuracy models on the Fashion-MNIST dataset.
Method
Network Architecture
Acc. (%)
FastSNN
Taylor
2022
784FC-1000FC. -OFF
89.1
FastSNN
Taylor et al. [022
10A-----6-C5-2--10000 784FCC512FC-5112C--3
90.6
SRC-RNN
[De Geeter et al.
2024)
88.5
DiffPC-M (Ours)
784FC-1000FC-00FG
89.6
DifP (Ours)
C84FC-400CC
89.2
Table 3: Average bits transferred during the error propagation stage of different models, along with the timesteps used the MNIST task. (fp: floating-point operations; sp: spikes)
Method
Ops
Network Architecture
Bits/N
Timesteps
Backpropagation
fp
784FC-1024FC--122CC-000
32
PC-SE (Pinchetti al. (0244)
Ip
784FC-1024FC-512FCC
960
15
PC-SNN (Lan et al. (2022)
ip
784FC-200FC-00FC
960
15
DiffPC-L (Ours)
sp
7B4FC-1024FC-552F---066
0.18
120
DifPPC-S (Ours)
sp
784FC-400FC-10FC
0.08
75
Table 4: Comparison of Test Accuracy and Efficiency (average bits per neuron during error propa- gation) on the CIFAR-10 dataset. (fp: floating-pon point: sp: spikes)
Method
Ops
Network Architecture
Acc. (%)
Bits/N
Timesteps
Backpropagation
fp
10C352-5C582-50FF -FFEE5S22--OCC------1FFF
63.5
32
PC-SE (Pinchetti et al. 2024))
fp (CC552-5CK22-50FC--0R-----
65.3
960
15
DifTPC-Lngg (Ours)
sp
D0CSS2-SCS22--00C3300----
65.6
1.9
240
DifPCC-ffiiient (Ours)
sp
10CCSS--CCS2--5FCC:00C.000
63.3
0.7
180
Numerical precision We evaluated the numerical precision of DiffPC by comparing the final states 2y obtained with standard predictive coding (PCN) and with our method. To quantify the approximation, measured the absolute difference between the hidden-layer activations produced by the two algorithms.
In this experiment, used fixed multilayer perceptron (MLP) with architecture 128-200-11 10. For each trial, we initialized the synaptic weights randomly but shared them between PCN and DiffPC, ensuring that differences arise solely from the approximation scheme. As inputs, we used di
i..d. random vectors sampled uniformly from [-1, 1j and as output we similarly had i.i.d. ran- dom vectors sampled uniformly from [-1, 1|20 initializations and inputs.
We repeated the evaluation over 300 random weight
For each random trial, we computed the absolute difference between the final states ør of PCN and DiffPC. We observed that the error depends systematically on the scheduler parameters. Specifically, the error larger when the number of approximation steps small and the limit-decay value is large, and it decreases consistently as n increases and a decreases. This trend was robust across weight initializations and random inputs, showing that approximation precision can tuned directly through scheduler parameters.
This constitutes general test of numerical fidelity within the specified architecture and activation function for three reasons. First, it eliminates datasct-specific structure and labels, the comparison probes only the update rules rather than task semantics. Sccond, by combining random bounded in- puts with many random weights, it explores wide region of the state space. Third, the use of shared weights across both algorithms isolates the approximation error from any modeling differences.
The results seen in Table 5 demonstrate that DiffPC provides close approximation of standard PCN dynamics under random input conditions, confirming that the method faithfully reproduces PCN across broad range of states for the given architecture.

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Table Mean absolute difference between DifIPC and standard PC, averaged over three seeds. The value after the + symbol represents the sample standard deviation. Lower is better.
a
1.0
0.5
0.25
0.1
7
3
0.1506 + 0.0051
0.0627
0.0014
0.0292 + 0.0014
0.0168 + 0.0004
4
0.0750 + 0.0018
0.0312 +: 0.0005
0.0158 + 0.0005
0.0106 - 0.0003
5
0.0373 = 0.0006
0.0169 + 0.0006
0.0102 + 0.0004
0.0083 + 0.0006
6
0.0197 = 0.0005
0.0107 = 0.0005
0.0083 = 0.0005
0.0075 = 0.0006
7
0.0117 = 0.0002
0.0085 + 0.0005
0.0075 = 0.0006
0.0072 = 0.0006
6
CONCLUSION
In this work, we presented Difference Predictive Coding, a learning framework that reformulates standard predictive coding for native implementation in spiking neural networks. By replacing dense floating-polt communication with sparse, event-driven ternary spikes, DifPPC addresses the data movement bottleneck that typically constrains on-chip training.
Our results on MNIST, Mushion-MNISSY and CIFAR-10 indicate that DiffPC approximates continu- ous predictive coding dynamies with high precision. Crucially, achieves competitive classification accuracy while greatly reducing the number of transmitted bits compared to backpropagation and standard predictive coding baselines. These findings suggest that DiffPC offers a viable pathway for spiking based learning neuromorphic systems.
Looking forward, a primary direction for future research this domain is the evaluation of DiffPC on significantly deeper architectures. Recent advancements. such as μ-PCC] Innocenti et al. (2025a) have shown that predictive coding can scale to deep ResNets when inference dynamics are stabilized. Since DiffPC is designed as faithful discretization of PC, it is reasonable to hypothesize that these stabilization techniques would transfer to this spike-based framework. A valuable extension this work would be quantify the layer-wise deviation between DiffPC and continuous PC states in deep networks, establishing how the spike-communication window must scale to maintain approximation fidelity.
A complementary future direction concerns temporally correlated data. Recent work shows that when inputs evolve smoothly over time, PC inference can be warm-started from previous states, re- ducing inference iterations by half and substantially lowering the number of weight updates Zadeh- Jousdani et al. (2025). Prototype-basee continual-lcarinng methods similarly demonstrate that re- ducing update frequency yields large energy benefits on neuromorphic hardware such as Loihi 2 Hajizada et al.] (2024) 2025). Since DifiPC is intrinsically event-driven- Serriiinnrrmmininng silent during steady states and emitting spikes only on changes-leveraginnn temporal priors may further reduce both inference steps and plasticity operations. Quantifying these lemporal-sparsity benefits repre- sents another natural extension of the present work.
Finally, beyond algorithmic scalability and temporality, transitioning DifiPC from simulation to physical neuromorphic hardware remains critical milestone. Deployment platforms such as Intel Loihi 2 would allow for the assessment the protocol under real-world hardware constraints and provide rigorous verification of its potential energy efficiency advantages.
ACKNOWLEDGMENTS
This work was supported by the Academy of Research Platform).
Finland (project no. 336357, PROFI 6 TAU Imaging
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APPENDIX
7.1 INTUITION OF PREDICTIVE CODING TRAINING
To clarify the mechanics of the learning process, we provide step-by-step intuition of how Pre- dictive Coding Network (PCN) learns to classify inputs. Unlike Backpropagation, which calculates gradients of a loss function with respect to weights, PCN frames learning as energy minimization problem involving local neuronal activities.
The training process for single input-label pair (x,y) proceeds as follows:
1. Prediction (Forward Pass): The input is clamped to the input layer. The network propagates activity forward layer-by-layer to generate a prediction at the output layer.
2. Constraint (Clamping): During training, the output layer is clamped to the correct label
y. This immediately creates prediction error at the output layer (since the network's 's initial guess likely did not match y).
3. Relaxation (Bidirectional state/error flow): This the core PC, The error at the output layer implies that the penultimate layer's activity was "wrong." This error flows backward, pulling the hidden layer neurons away from their original values toward states that would have produced the correct output. This happens iteratively across all layers. The network "relaxes" " into a low-energy state where the activities are consistent with both the input and the correct label.
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4. Weight Update (Learning): Once the neuron activities have shifted to this better configu- ration, the synaptic weights are updated locally. The update rule effectively says: "Change the weight so that next time, this input naturally produces this 'better' hidden activity."
7.2 ALGORITHM BREAKDOWN
Update forward prediction Forward prediction Xx is updated using the incoming spike signal S, from the previous layer,
(H+)) t xp + W To(l- -
(13)
Update Target Activity and Generate Spikes
The core of the DiffPC algorithm lies in iteratively updating the target activity vector Xy for each layer and communicating changes via spikes. The process begins by adjusting xT minimize prediction error, followed by generating spikes based on the discrepancy between this new target and the layer's current state.
First, the target activity xr is updated based on both the local prediction error eg and the error propagated from the subsequent layer, which accumulated in eg. This update, performed only when the learning rate y(t) is active, is defined as:
Xy X + (t)) (- (Xy 0) O cp)
This rule closely mirrors the standard PC update in Equation The term -er corrects for local prediction error, while the second term incorporates feedback from the next layer. The element-wise condition (xT >> 0) serves the derivative of the ReLU activation function, ensuring that updates are only applied to active neurons. During training, the target activities the input and output layers clamped to the provided data and labels, respectively. During inference, only the input layer is clamped.
Next, the algorithm generates spikes communicate the necessary adjustments for bringing the layer's actual state, X,. in line with the newly updated larget state, Xy. Instead of transmitting dense floating-point values, DiffPC sends sparse ternary spikes. spike is generated only if the magnitude of the difference between the target and actual activity for given neuron exceeds the adaptive threshold (t).
The activity spike vector S is computed as follows:
$ = sign(xr- xxA) @ ([x-- - > (t)))
(14)
where @ denotes the Hadamard product. The sign(-) function determines the spike's polarity (+1 or -1), while the comparison operator produces a binary mask, ensuring that spikes are only generated when the required update is significant. This event-driven mechanism ensures that communication is sparse, as spikes are only transmitted to correct meaningful deviations from the target state.
Spiking ReLU To implement non-linear transfer function similar to the Rectified Linear Unit (ReLU) in conventional neural networks, we propose a masking operation that effectively prevents the actual neural activity X from becoming negative.
x + SA O (xx +SA To(t) > 0)
(15)
The spiking ReLU ensures that only correction spikes in SA maintaining X 0 are allowed, effectively implementing ReLU-like activation function. We can also implement a clipped ReLU activation function similar manner by setting an additional constraint:
t + SA 3 @ X + SA [t()) > 0)
(16)
Update activation The spiking ReLU produces simple update step to update the actual activity x,:
xA f- AA + To(() st
(17)
The activation update operation adjusts X towards Xy.
Error encoding in spikes The target update is the same as in the standard PC in Sec. 3.2
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The target error vector e is computed using 4 Similarly to the original PC, this error term received from the following layer serves as a measure of how well the current forward prediction matches the target activity. However, unlike the standard PC cannot directly error encoded in the form of spikes. Instead, after computing the target error er <t, the DiffPC algorithm generates error spikes S, based on the difference between er and the actual error e사.
the update rule 7 since the
Se sign(er CA) @ (lor cal > To(t))
(18)
The spikes Se are then sent to the following layers as error signals.
Accumulate incoming errors- Errors from the next layers are integrated into the network using the accumulated error vector ég. This vector represents the sum of the incoming error signals weighted by the threshold (t).
eB - EB + (t) e in
where e_in denotes the incoming error signals from the next layer. The accumulated error vector helps refine the target activity ør by incorporating feedback from different layers of the network.
The actual error ε is then updated using previously generated error spikes Se.
ea - Ca + (t(t) 8e
Weight Update Mechanism The update rule for a single sample, derived from minimizing free energy, is computed as AW;; sj X er.; Φ(FT,), where ey, is the post-synaptic error state and ZT,j the pre-synaptic activity state. This computation is compatible with neuromorphic hardware like Loihi 2, which supports fixed exed-precision multiplication and accumulation of local variables.
7.3 PROOF OF CONVERGENCE
Here we provide the proof for Theorem 4.1 regarding the convergence of the eyclic scheduler.
Proof. Consider the cyclic scheduler where (0) 2 At the first timestep t 0, if 2 |cdp x'Al > 2 then r is updated by 2m. Consequently, after the update, we have:
|ot - - < 2
The same trivially holds if 1 - r'Al < 2 already held the first timestep. At the next timestep t = 1, with (1) 2mm-1 the difference |at 2'x1 can again be reduced by 2 if it exceeds 2m-1 Repeating this process over n timesteps, each reduction step halves the threshold compared to the previous timestep from which the result follows by induction.
7.4 HYPERPARAMETER ANALYSIS
To understand the impact of our key scheduler hyperparameters, the cycle length and the decay factor a, we performed extensive grid search. The results, visualized in Figure [ reveal clear trade-off between classification accuracy, communication cost (spikes), and runtime (timesteps).
Figure 2a shows that, with fixed decay (a=1.0), increasing the cycle length n generally improves performance, On both datasets, the accuracy gains diminish as performance saturates for sufficiently large n. However, this accuracy gain comes at a direct cost. By definition, larger value increases the number of timesteps per iteration, which in turn increases both the total runtime and the number of spikes transmitted.
The role of the adaptive decay parameter a is to tune the precision of spike-based communication within a fixed number of timesteps. Figure 2b illustrates this trade-off effectively. For smaller cycle lengths, reducing from 1.0 to smaller values provides notable accuracy boost. This performance gain is achieved by allowing the adaptive threshold to decrease more over the iteration, which in turn generates more spikes represent the error signals with higher fidelity. This increase communication cost, but does not increase the runtime as the number of timesteps per iteration is fixed by n.
Crucially, the benefit of a smaller a diminishes as n increases. For large n, the performance is already high and stable, and varying has little effect on the final accuracy. This suggests that long
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Dataset: FMNIST
Dataset: MN:ST
100
Averoges Best Test Accuraiy 150
80
70-
(Cyycle Longth)
(Cycle Longth)
(a) Effect of cycle length n (see Eq. g both MNIST and Fashion-MNIST though returns diminish for sufficiently large
on test accuracy with fixed decay (a=1.0). Performance increases with
Datanet: FRINIST
Dataset: MNIST
ei leigh)
takes Roumn
7000 Spites
(b) Trade-off between test accuracy and communication cost (spikes). Varying the decay parameter (marker size) improve accuracy, particularly for smaller cycle lengths (marker shape/color), at the cost of in- creased spike activity.
Analysis of scheduler hyperparameters and on MNIST and figure isolates the effect of n, while the bottom figure shows the interplay between n, a, accuracy, and spike cost.
Figure 2:
The top
cycle length n already provides sufficient timesteps for the network's states to converge with high precision. In this regime, the fine-tuning offered by the adaptive decay (a; 1.0) becomes redundant, as the inherent precision of the long spike train is already maximal for the task.
7.5 HYPERPARAMETER SELECTION HEURISTICS
Choosing optimal hyperparameters for DiffPC, as with many complex models, a non-trivial task without closed-fome solution. However, we have identified several heuristics that provide strong starting point for tuning the network for a new task.
From standard predictive coding theory, functional network requires minimum number of laxation steps, typically at least twice the depth of the network, to allow information to propagate fully between the input and output layers (see e.g. A.3.2 of Innocenti al. (2025b)). This principle provides a useful guideline for the minimum number of timesteps required. For DifiPC, we offer the following more specific guidance.
Choosing m For the cyclic scheduler, a robust choice is m=2 when using clipped activation function like ReLU6. The parameter m sets the initial and largest threshold value cycle, which is 2mh If this value is significantly larger than the maximum possible activation (e.g., 6 for ReLU6), the initial timesteps will generate no spikes, as the difference AT AAl will never exceed the threshold. Setting m=2 yields initial threshold of 4, which is the same order of magnitude as the activation range, ensuring that the spike generation process is active from the beginning of the cycle.
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Choosing and The parameters and a jointly control the trade-off between runtime, commu- nication cost, and precision. A practical approach to tuning them is a two-step process.
Find an effective eycle length n. First, set a=1.0 (disabling adaptive decay) and incre- mentally increase while monitoring test accuracy. Continue until performance saturates, establishing a baseline for the required precision.
2. Optimize for efficiency. Once a saturation point Rsat is found, one can attempt to reduce the eycle length T'ew Tsat k for some small integer k, thereby reducing runtime. To compensate for the potential loss of precision, the decay factor can be set to a Ra 1 1/2*.
The reasoning for this two-step process is as follows. Reducing the cycle length by k steps removes the k timesteps that have the smallest, and therefore most precise, threshold values. To compensate, a smaller value of is used to scale down the entire threshold schedule within the new, shorter cycle. The heuristic z 1 1/2 l is specifically chosen because it ensures that the final, smallest threshold in the new ~See cycle is approximately equal to what the final threshold was in the original A step cycle with a=1.0. This approach aims recover the necessary representational fidelity while benefiting from a shorter runtime.
TABLE OF NOTATION
Table summarizes the mathematical symbols used in the Difference Predictive Coding (DifiPC) algorithm.
Table 6: Nomenclature and Symbols
Symbol Description
Shared Variables (Standard PC & DiffPC)
I
Layer index, LJ.
W
Synaptic weight matrix connecting layer | to
X
Forward Prediction. The prediction generated by the previous layer.
XT
Target Activity. The ideal state calculated to minimize prediction energy. Prediction Error. The difference between target and prediction (Xr- -x)), Inference learning rate time t.
c
(t)
DiffPC-Specific States (Spiking Implementation)
XA
Actual Activity. The discrete state that tracks the shared target xy. updated
via spikes.
SAA
Activity Spikes. Ternary spikes (-1,0,1) 13 communicating changes XA- Target Error. The local error variable (functionally equivalent to in this context).
eT
CA
Actual Error. The discrete state that tracks ey, updated via spikes. Error Spikes. Ternary spikes communicating changes in the error state, Backward Error. The error signal accumulated from layer + 1.
eB
Scheduler & Thresholds
To
Adaptive firing threshold at time t.
m
Scheduler magnitude parameter (sets max threshold 2"").
n
Scheduler cycle length (periodicity of the steps).
Decay factor for the cyclic decay scheduler.
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Published as a conference paper at ICLR 2026
9 EVENT-DRIVEN RESPONSE TO INPUT CHANGES
To demonstrate the event-driven nature of our method, we conducted an experiment to measure the network's ' spiking activity in response to changing input. We presented static input image from the test set to a trained DifiPC network and monitored the total number of activity spikes (s.) across all layers over time. After initial period of 25 timesteps, introduced abrupt change by shifting the imput image by single pixel.
The results, averaged over 1000 different input images, are shown in Figure Initially, there is burst of spiking activity as the network processes the new image. This activity quickly subsides, and the network becomes nearly silent its internal state converges to a stable representation of the static input. At timestep 25 when the input is shifted, the network immediately responds with another burst of spikes, which then decays as it settles into new stable state.
This behavior highlights key feature of DiffPC: computation is performed only when necessary to process new or changed information. For applications where inputs may remain static for periods of time, this event-driven property suggests the potential for energy savings by eliminating redundant processing, making the approach highly suitable for energy-constrained neuromorphic hardware.
Solkas Time (n-1000
Sp CAFT
Figure 3: Event-driven spiking in a trained DifiPC network. The network shows an initial burst of spikes when image is presented, then falls silent. A second burst of activity is triggered precisely at timestep 25, when the input image is shifted by one pixel, demonstrating that the network only computes in response change.
REPRODUCIBILITY STATEMENT
We are committed to ensuring the reproducibility our research. The complete PyTorch implemen- tation of the Difference Predictive Coding algorithm, including model architectures, schedulers, and training procedures, will be made available as supplementary material. Our core method detailed in Section 7.2 with step-by-step breakdown provided in the Appendix. The source code includes the exact configurations and hyperparameters used to generate all reported results, including clas- sification accuracy MNIST (Table [ and MNNIS (Table 2 and the communication efficiency analysis (Table 3 Further, the codes used to generate the CIFAR-10 results will also be made public. All experiments were conducted using standard public datasets, and the specific data processing pipelines are explicitly defined within our implementation.
THE USE OF LARGE LANGUAGE MODELS (LLMS)
In the preparation of this manuscript, Large Language Models (LLMs) were utilized as general- purpose assistive tool. The authors take full responsibility for all content, ensuring its scientific accuracy and originality. The specific roles of the LLMS outlined below:
19

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Published as a conference paper at ICLR 2026
Writing Assistance: LLMs employed to improve the language and clarity of the manuscript. This included refining sentence structures, correcting grammatical errors, and ensuring overall readability. The core scientific ideas, arguments, and conclusions pre- sented are entirely the work of the authors.
Literature Discovery: L.MMs were used as tool to aid in the literature review process by suggesting potentially related academic papers and summarizing established concepts. All works cited in this paper were subsequently retrieved, read, and critically evaluated by the authors verify their relevance and accuracy.
Coding Support: LLMs assisted in the software development process by generating boil- erplate code, helping to debug specific code segments, and suggesting algorithmic opti- mizations. The overall design of the experiments, the core logic of the implementation, and the final analysis were conceived and performed by the authors.
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Reviews:
[{"review_id": "hfQTqdHwUd", "reviewer": "Reviewer_8biP", "summary": "The manuscript proposes Difference Predictive Coding (DiffPC)—a spike-native reformulation of predictive coding (PC) for spiking neural networks (SNNs). DiffPC replaces dense floating-point message passing with sparse ternary spikes for both state and error communication, introduces adaptive threshold schedules to drive event-based updates, and targets on-chip feasibility by aligning operations with Intel Loihi 2’s instruction model (validated in simulator). On MNIST and Fashion-MNIST with MLP backbones, DiffPC reports up to 99.3% and 89.6% test accuracy, respectively, while dramatically cutting communicated bits per neuron during training (e.g., 0.18 bits across 120 timesteps for DiffPC-L versus 32–960 bits for backprop/PC baselines). The method is positioned as a hardware-aligned, energy-efficient alternative to backpropagation and prior PC-SNN hybrids.", "questions": "1. Exact synaptic update mechanics. How are weight updates computed and applied in the purely spiking formulation at run time? Is Eq. (6) implemented via accumulated spike counts/low-precision accumulators, and how are precision/overflow handled on Loihi 2? Please make this explicit in Algorithm 1 (or add an Algorithm 2 for weight updates).\n\n2. Scheduler sensitivity & autotuning. Can you provide ablations on 𝑛, 𝑎, 𝑇, 𝑐 showing accuracy, spikes/neuron, and timesteps across datasets, and discuss any rule-of-thumb for choosing them at larger scales? Table 4 addresses approximation error only.\n\n3. Hardware-level metrics. Do you have on-chip measurements (energy per inference/training iteration, latency, utilization) or realistic cycle-accurate estimates on Loihi 2 for DiffPC vs. PC-SE and surrogate-gradient SNNs?", "limitations": "", "rating": 4, "confidence": 3, "soundness": 2, "presentation": 1, "contribution": 2, "strengths": "1. Spike-native PC formulation. Clear conceptual shift from continuous residuals to ternary spike streams for both targets and errors, enabling fully spiking training and inference under local rules.\n\n2. Hardware awareness. The algorithm and simulator validation target Loihi 2 semantics, improving credibility for on-chip training pathways.", "weaknesses": "1. Evaluation scope (datasets & models). Experiments are restricted to MNIST/Fashion-MNIST and MLP backbones. The claims of hardware-aligned efficiency would be more compelling on spatiotemporal or event-camera benchmarks, and with convolutional or recurrent SNNs typical of neuromorphic use.\n\n2. Learning rule completeness (weights). The PC weight update (Eq. 6) is reviewed, but Algorithm 1 does not explicitly show the weight-update pathway in the spike-only regime (e.g., how ternary error/state spikes translate into synaptic updates compatible with Loihi). This leaves ambiguity about exact stochasticity/quantization and memory consistency.\n\n3. Inference protocol and latency. Inference appears iterative/event-driven; it would help to quantify latency–accuracy trade-offs (how many timesteps to reach stable predictions) and to compare against rate-coded SNNs with fixed horizons.\n\n4. Energy model granularity. Bits-per-neuron is informative, but ignores NoC hops, memory hierarchies, spike routing costs, and neuron state updates on specific hardware; without this, absolute energy claims remain provisional.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "VbXSsipdYV", "reviewer": "Reviewer_VdSN", "summary": "The paper presents a spike-based version of the Predictive Coding (PC) for training, equivalent to standard PC and called DiffPC. The method consists of tracking the changes of the continuous signal by sending \"corrective\" ternary spikes, by iteratively adjusting the \"weight\" of the spikes using an adaptative threshold. Equivalence with PC is empirically demonstrated on MNIST and Fashion-MNIST benchmarks.", "questions": "The paper lacks a conclusion/perspective section, is it on purpose?\nRegarding implementation efficiency, could the author provide some insight or benchmark that compares standard PC and DiffPC?\n\nOverall, the quality of the paper could still be improved: more explanation/insight on the method scheduling could be provided. The paper also lacks a conclusion. Table 4 and related content could be improved, perhaps with a graph better illustrating the convergence between PC and DiffPC as the threshold becomes smaller and an absolute scale or a relative one. But the mean absolute difference is not very meaningful...", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- The paper is well written and easy to read;\n- The DiffPC implements a relatively classic spike coding method (based on tracking a continuous signal with spike corrections) that has already been successfully applied to backpropagation as well;\n- The method is mathematically equivalent to standard PC, modulus the final threshold value approximation.", "weaknesses": "- The cyclic scheduler equations are not very well explained. Parameter m is not defined (equation 8).\n- The DiffPC relies on scheduling hyperparameters that may be hard to tune to get the best efficiency vs accuracy compromise. A discussion about this is missing from the main paper (although mentionned in the supplementary material);\n- Although some data on energy efficiency are provided, it is not clear of they translates to real world implementation efficiency.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "7Otxipxe2C", "reviewer": "Reviewer_8Q9N", "summary": "The primary contribution of DiffPC lies in being the first predictive coding framework trained fully through spike-based event-driven mechanisms, effectively bridging the gap from theoretical feasibility to hardware-ready neuromorphic deployment. Nevertheless, the current experimental scope remains limited, and the theoretical analysis could be further strengthened. More extensive evaluations on diverse tasks and real neuromorphic hardware measurements are needed to substantiate the method’s generality and energy-efficiency advantages.", "questions": "1. Lines 156–164 state that four innovations are introduced. However, the first contribution—ternary spiking—has already been widely adopted in SNN research. The third contribution—dynamic thresholding—has also been extensively explored in prior SNN literature. Regarding the fourth contribution, the near-exact PCN approximation resembles a conversion or distillation process; therefore, clarification is needed on whether it should be viewed as a form of teacher–student distillation rather than an independent methodological innovation.\n\n2. The manuscript argues that surrogate-gradient-based backpropagation is biologically implausible due to reliance on global gradients, whereas DiffPC uses local plasticity and is thus more brain-like. However, validating DiffPC only on MLP architectures does not substantiate this claim. The experiments do not reveal whether a fully local learning mechanism can scale to larger networks. More importantly, when every layer is trained exclusively through DiffPC, the paper does not analyze how network depth or size influences final model performance.\n\n3. Although the paper emphasizes its relevance to neuromorphic tasks, no temporal benchmarks are included. Evaluation on event-driven datasets would be necessary to support this claim.\n\n4. The authors are encouraged to provide real power measurements on Loihi 2 hardware, and further discuss the quantitative advantages over PC-SNN and surrogate-gradient-based SNN training, along with a clearer articulation of the fundamental differences.", "limitations": "", "rating": 2, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "1. This work is the first to fully discretize both target updates and error transmission in predictive coding into sparse ternary spike events, eliminating the need for floating-point communication and external ANN-based training pipelines.\n\n2. On the MNIST benchmark, DiffPC achieves a test accuracy of 99.3%, comparable to standard predictive coding and backpropagation, while delivering more than a 5,000-fold improvement in communication efficiency.", "weaknesses": "1. The Introduction provides an insufficient and somewhat disorganized treatment of the importance of local learning. I recommend a thorough rewrite of this section to clarify the narrative and strengthen the conceptual motivation.\n\n2. The Introduction lacks a high-level articulation of the motivation for proposing DiffPC. Based solely on this section, readers cannot clearly infer what the authors aim to do, what has been accomplished, or the scholarly significance of the contributions.\n\n3. The experimental evaluation is severely limited, and results on MNIST and Fashion-MNIST are inadequate to substantiate the method’s effectiveness. At minimum, an evaluation on CIFAR-10 is recommended.\n\n4. In the Related Works section, citations to similar training frameworks should be mapped one-to-one with the corresponding methods and claims.\n\n5. There are multiple presentation issues: a symbol error at the end of line 195; missing punctuation after Equation (11); inconsistent decimal precision in the “Acc.” column of Table 2; and numerous other minor formatting and symbol inconsistencies, likely due to a rushed preparation. Please correct these errors throughout.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "pKBOZDWqFV", "reviewer": "Reviewer_htVv", "summary": "This paper proposes DiffPC, a spike-native local learning algorithm for SNNs. DiffPC fits the Predictive Coding algorithm into SNNs, replacing floating-point messages with sparse ternary spikes, providing spike-compatible target and error updates, and employing adaptive threshold schedules for event-driven operation. Experimental results have shown the effectiveness and energy efficiency of DiffPC.", "questions": "1. Could you please introduce Predictive Coding more clearly? How to initialize the parameters such as $x_T$? From Algorithm 1, it seems there are $x_T, x_F, x_A$ for the neuron state. However, only $x_T$ and $x_F$ is introduced in Section 3.2. Does $x_A$ appear in PC? Also in Algorithm 1, the step of updating weights seems not included.\n2. I recommend the authors to place a table illustrating the meaning of each symbol. If the room is not enough in main body, you can put it in appendix.\n3. What is the intuitive behind the PC method? How the network is trained to predict classification result through PC? Please give more detailed intuition.\n4. What is the number of 'the number of approximation steps *n*' in line 426? Is it the number of optimization steps?\n5. There are two 'Energy efficiency' in 5.2 RESULTS. Please change one of them. In line 355, a full stop is missing.\n6. In Figure 2, why there are two peaks of spike activity after input change? Could the authors give some intuition?", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "1. The proposed algorithm only needs local information propagation, which is much more compatible with SNNs than BP-based algorithms.\n2. The proposed algorithm propagates information through spikes during training and inference, which suits the characteristic of SNNs.", "weaknesses": "1. There are many symbols in this paper, and some are not clearly explained, making it difficult to follow.\n2. The writing of this paper is not clear enough. See Questions for details.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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:

Decision: Accept (Poster)