Emergent Synaptic Plasticity from Tunable Dynamics of Probabilistic Bits

Pramey Upadhyaya; Sagnik Banerjee; Shiva T. Konakanchi; Supriyo Datta

REVIEW 3 major objections 4 minor 59 references

Emergent Synaptic Plasticity from Tunable Dynamics of Probabilistic Bits

T0 review · 3 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read A hidden p-bit with a tunable flip rate programs the effective coupling between two p-bits, and does so directionally.

desk verdict A coherent theory proposal: a hidden p-bit's fluctuation rate as a new coupling knob, with a clean formula and honest limitations in the fast-regime model. read the letter →

arxiv 2505.00252 v1 pith:H2GFFKMF submitted 2025-05-01 cond-mat.dis-nn physics.app-phphysics.comp-ph

Sagnik Banerjee , Shiva T. Konakanchi , Supriyo Datta , Pramey Upadhyaya This is my paper · ORCID

classification cond-mat.dis-nnphysics.app-phphysics.comp-ph

keywords probabilisticcomputingp-bitstunablecouplingsynapticplasticityrandomtelegraphnoisetimescaletuningdirectional

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 claims that the interaction strength between two probabilistic bits (p-bits) can be programmed by inserting a third 'hidden' p-bit between them and changing how fast that hidden p-bit randomly flips. In the regime where the coupling from the first p-bit to the hidden p-bit is weak while the hidden-to-second coupling is strong, the effective coupling $J_{21}$ rises smoothly from a small value toward saturation as the hidden p-bit's correlation time $\tau_h^r$ is made shorter than the synapse time $\tau_s$, while the reverse coupling $J_{12}$ stays fixed. The authors derive a closed-form expression for $J_{21}$ as a function of $\tau_h^r/\tau_s$ and confirm it numerically, and they map the motif onto an effective two-p-bit model with coupling $J_{\mathrm{eff}}$. Because this 'synaptic plasticity' uses only the intrinsic dynamics of p-bits, it could allow on-chip reprogramming of p-computer networks without additional tuning hardware.

What carries the argument

The central object is the three-p-bit 'hidden p-bit model' of Eqs. (2)-(5): each p-bit outputs $\operatorname{sign}(\tanh(\beta I)+\eta)$ with random telegraph noise $\eta$ of correlation time $\tau_r$, and each synapse is a first-order low-pass filter with time constant $\tau_s$. The tunability mechanism is the filtering of a fast hidden p-bit through a slow synapse: the power spectral density of the hidden p-bit's telegraph noise, $S_h(f) = \frac{2\tau_h^r}{1+(2\pi f\tau_h^r)^2}$, combined with the synapse transfer function $H(f) = \frac{J_2}{1+i2\pi f\tau_s}$, gives the second p-bit an input with mean $\mu = J_1 J_2 \beta$ and standard deviation $\sigma = \frac{J_2}{\sqrt{1+\tau_s/\tau_h^r}}$; the ratio $\mu/\sigma$ is what Eq. (1) exposes as the control knob for the effective coupling.

What would settle it

Measure the correlator $\langle m_1 m_2 \rangle$ in a three-p-bit circuit with $J_1\beta = 0.1$, $J_2\beta = 50$, and computational p-bit times much larger than $\tau_s$, sweeping the hidden p-bit's correlation time from $\tau_h^r/\tau_s = 10$ down to $0.001$. The paper predicts $\langle m_1 m_2 \rangle$ climbs from about 0.1 to near 1 along the erf curve in Eq. (1) when p-bit 1 drives p-bit 2, while the reverse direction stays near 0.1; if the measured curve does not follow that prediction, or if both directions tune, the central claim is refuted.

Watch

Extended reading notes

Core claim

The central discovery is that a hidden p-bit in the fast-fluctuation regime ($\tau_h^r < \tau_s$) does not merely add noise: the synapse's RC-like low-pass filter converts the hidden p-bit's random telegraph output into a Gaussian-distributed input for the second computational p-bit whose mean-to-noise ratio grows as $\sqrt{1+\tau_s/\tau_h^r}$. With $J_1\beta \ll 1$ and $J_1 J_2 \beta^2 \gtrsim 1$, the effective coupling from computational p-bit 1 to p-bit 2 is $J_{21} = \frac{1}{\beta}\tanh^{-1}\left[\operatorname{erf}\left(\frac{J_1\beta}{\sqrt{2}}\sqrt{1+\tau_s/\tau_h^r}\right)\right]$ for $\tau_h^r/\tau_s \lesssim 1$, and $J_{21} = J_1$ for $\tau_h^r/\tau_s \gtrsim 1$, while $J_{12} = J_1$ for all ratios. This yields continuously tunable and directional effective coupling, verified by matching the correlator $\langle m_1 m_2 \rangle$ between the hidden-p-bit model and an effective model with direct coupling $J_{\mathrm{eff}}$.

Load-bearing premise

The load-bearing premise is that the behavioral model of Eqs. (2)-(5), previously validated for slow p-bits with fast synapses, remains quantitatively accurate when the hidden p-bit fluctuates faster than the synapse; if that extension fails, Eq. (1) and the extracted $J_{\mathrm{eff}}$ lose their support.

Editorial extensions

If this is right

P-bit networks could achieve on-chip reprogrammability by adjusting fluctuation rates of hidden p-bits, without adding dedicated coupling-tuning hardware.
The scheme naturally produces asymmetric (non-reciprocal) couplings $J_{21} \neq J_{12}$, which could be exploited in directed networks for machine learning and Bayesian inference.
Pairing two hidden p-bits, one for each direction, would allow independently programming $J_{ij}$ and $J_{ji}$, enabling symmetric programmable couplings for Boltzmann sampling and annealing.
Existing FPGA-based p-bit implementations could emulate the scheme using tunable clock rates, while antiferromagnetic p-bits with picosecond flipping rates are identified as promising physical candidates.

Reading between the lines

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

The same mean-to-noise filtering mechanism may extend beyond p-bits to any stochastic binary unit with tunable noise correlation, suggesting a generic design principle for programmable interactions in neuromorphic hardware.
Because the tuning curve in Eq. (1) is monotone in $\tau_h^r/\tau_s$, it could serve as a calibration function: measuring the correlator at one ratio fixes the effective coupling, allowing closed-loop programming of a p-bit network.
Since the directionality arises from asymmetry in $J_1$ vs $J_2$ rather than in the network topology, one could create networks with arbitrary directed graphs by choosing the weak and strong synapses appropriately; this is implicit but not developed in the paper.
A straightforward experimental check would be an FPGA-based p-bit network where the hidden p-bit's flip rate is set by a clock; deviations from the erf prediction would reveal whether the behavioral model holds in the fast-p-bit regime.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Short version: the paper is a coherent theory proposal with a genuinely new knob – using the correlation time of an intermediate hidden p-bit to tune effective couplings in p-bit networks – and a clean analytic formula. It's worth engaging with, but not yet proof that the mechanism survives in hardware.

The main result is Eq. (1), which gives J21 as a function of τ_h^r/τ_s, with J12 fixed. The derivation is transparent: in the slow-hidden limit you get tanh(βJ1), in the fast limit you filter the hidden p-bit's RTN through the synapse RC, get a Gaussian, and the correlator becomes an erf. The mapping to a direct effective coupling is by definition, but that's not circular in the fitting sense; no parameters are fitted to data. The numerics cover the phase diagram and the analytic curves match the simulations. That part is solid.

The soft spot is exactly where the stress-test lands: the fast-p-bit regime. The paper acknowledges that prior analytical update rules are invalid for τ_h^r < τ_s and solves the ODEs numerically. But it's the same behavioral model equations (2)–(5) being integrated – there's no independent device-level check that the model remains faithful when the hidden p-bit flips faster than the synapse. The assumption of instantaneous p-bit response (Eq. (2)) is stated, not validated. So Eq. (1) is conditional on that model being right in this regime. That is a real limitation, but it's not a fatal one for a proposal paper. The authors are transparent about it, and the direction of the effect – noise creating a rectified coupling – is plausible and worth testing.

Minor issues: no error bars on the numerical correlators, no code released, and the piecewise formula has a discontinuity at τ_h^r/τ_s = 1 that is glossed over (though it's small in practice). The citation pattern looks fine; the prior p-bit work is cited and the paper clearly distinguishes its regime from the standard fast-synapse assumption.

Who is this for: anyone building p-circuit architectures, especially FPGA or magnetic p-bit implementations, and people thinking about using noise structure as a computational resource. It deserves a serious referee. A referee should push for either an experiment, a device-level simulation, or at least a crisper statement that Eq. (1) is a model-level prediction rather than a hardware guarantee.

Referee Report

3 major / 4 minor

Summary. The paper proposes a mechanism for programmable synaptic coupling in probabilistic computers: insert a "hidden" p-bit between two computational p-bits and tune the hidden p-bit's fluctuation time tau_h^r relative to the synapse time tau_s. Using a behavioral model of p-bit dynamics (Eqs. (2)-(5)) with random telegraph noise and RC-type synapses, the authors simulate the three-p-bit motif and derive analytical expressions for the time-averaged correlator <m1 m2>. In the limit J1*beta << 1 and J1*J2*beta^2 >= 1, they obtain Eq. (1), which gives a tunable effective coupling J21 that increases as tau_h^r/tau_s decreases, while J12 remains fixed at J1. They map the motif onto an effective two-p-bit model with coupling Jeff and verify the mapping numerically in Fig. 3(c). The central claim is that coupling strength and direction can be programmed by modulating fluctuation rates alone.

Significance. If the behavioral model remains valid in the fast-p-bit regime, this is a conceptually simple and potentially hardware-friendly way to implement programmable, directional synapses in p-computers without adding external tuning circuitry. The analytic results are parameter-free within the stated model, and the numerical simulations are internally consistent with that model, which is a genuine strength of the paper. The main limitation is that the central quantitative prediction, Eq. (1), relies on extending the behavioral model into the tau_h^r < tau_s regime, a regime the paper itself notes lies outside previously validated update rules, and no device-level or experimental validation is provided. The effective-model comparison in Fig. 3(c) is also partly a mapping by construction. The work is best read as a model-based proposal that could stimulate hardware experiments, but the load-bearing fast-regime assumption needs to be addressed before the quantitative claims can be fully accepted.

major comments (3)

[Section III and Appendix A, Eq. (8)] The fast-hidden-p-bit branch of the central result, Eq. (1), is derived from the behavioral model Eqs. (2)-(5) in the regime tau_h^r < tau_s, yet the paper itself states in Section III that the previously validated fast-synapse update rules are not valid in this regime and that Eq. (2) assumes instantaneous p-bit response. No device-level or experimental validation is provided for the extended model in this regime, so the quantitative predictions of Eq. (1) for tau_h^r/tau_s <= 1 rest on an unvalidated assumption. Please either validate the extended model (e.g., against device-level simulations of stochastic magnetic tunnel junctions in the fast-p-bit regime) or explicitly frame Eq. (1) as a model-based prediction whose fast-regime branch depends on the instantaneous-response assumption.
[Section IV B, Fig. 3(c)] The effective-coupling extraction is a mapping by definition: Jeff is defined as tanh^{-1}(<m1 m2>) of the hidden-p-bit model, and the effective model then reproduces that same correlator by construction. The agreement between the hidden and effective models in Fig. 3(c) is therefore an internal consistency check, not an independent validation of the hidden-p-bit dynamics. The authors should state this limitation and, if possible, validate the mapping with a quantity not used in its definition, such as the conditional distribution P(m2 | m1) or the cross-correlation function C(t) = <m1(t) m2(t+tau)>.
[Eq. (1)] The two branches of Eq. (1) do not match at tau_h^r/tau_s = 1: for small J1*beta the fast branch approaches (1/beta) atanh(erf(J1*beta)) ~ 1.128 J1, whereas the slow branch gives J1, a relative discontinuity of about 13%. Since the abstract and Section II claim continuous tunability, the authors should address this mismatch, e.g., by quantifying the width of the crossover region in Fig. 3(c) or by providing a single expression that interpolates smoothly between the two branches.

minor comments (4)

[Section III, Eq. (4)] The discrete-time update in Eq. (4) uses the flip probability 1 - exp(-dt/tau_i^r); the authors should state the simulation time step dt and confirm that it is small enough that the Poisson approximation does not affect the reported correlation times.
[Fig. 2 caption] The caption states tau_r = 10 and tau_s = 1, so tau_r/tau_s = 10, whereas Section IV A asserts tau_r/tau_s >> 1 and the analytics in Appendix A assume this limit; Fig. A1 indicates noticeably better agreement for tau_r/tau_s = 100. Please reconcile the parameter choice or explain why tau_r/tau_s = 10 is adequate.
[Appendix A, Eq. (A4)] Rh(tau) is called the autocorrelation function but is used as the autocovariance, since the mean J1*beta is separated into mu; rename it to autocovariance or explicitly state that it is the centered autocorrelation.
[Eq. (8) and Appendix A] The Gaussian approximation for the filtered hidden-p-bit output is invoked for tau_h^r/tau_s <= 1; near tau_h^r ~ tau_s the number of flips per RC time is order unity, so the central-limit justification is weak. A short quantification of the approximation error would strengthen the derivation.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Eq. (1) is derived from the adopted behavioral model and the effective-coupling extraction is an explicit mapping from computed correlators, not a fitted or self-referential prediction.

full rationale

The paper's derivation is self-contained after adopting the behavioral model of Eqs. (2)-(5) from Refs. [13,54]. No parameters are fitted to the target result: the analytic correlator in Eq. (8) is obtained by approximating the low-pass filtered hidden-p-bit output as Gaussian in Appendix A, and the numerical simulations solve the same ODEs without using Eq. (8) or Eq. (1). The effective coupling Jeff is introduced operationally in Sec. IV.B as the coupling in a two-p-bit model that reproduces the computed correlator; equating tanh(beta Jeff) to the hidden-p-bit correlator is a mapping by definition, not a hidden fit. The 'validation' in Fig. 3 compares the full model with its own analytic approximation and with the equivalent effective model; this is an internal consistency check, not independent experimental evidence, but internal consistency is not circularity. The acknowledged limitation that Eqs. (2)-(5) are extended to the fast-p-bit regime without device-level validation is a correctness/validity risk, not a circularity. The self-citations for the behavioral model point to prior device-level simulations and are not the sole justification of the present derivation. No claim reduces by construction to its inputs.

Assumptions & free parameters 3 free parameters · 3 assumptions · 0 invented entities

The central claim is derived within a borrowed behavioral model, so the main ledger items are model-validity and control assumptions rather than fitted constants. The numbers 0.1, 50, and 100 are chosen operating points, not fits to data. No new physical entities are introduced: the hidden p-bit is an existing p-bit assigned a coupling role.

free parameters (3)

J1*beta operating point = 0.1
Chosen to satisfy J1*beta << 1 in the numerical demonstrations (Figs. 2d-e, 3); not fitted to data, but the central plots use this value.
J2*beta operating point = 50
Chosen to satisfy J1*J2*beta^2 >= 1 with J1*beta = 0.1 in the numerical demonstrations; not fitted.
computational p-bit correlation time tau_r/tau_s = 10 (Fig. 2), 100 (Fig. 3 and Fig. A1)
Chosen to approximate the slow limit tau_r >> tau_s; agreement with analytics improves as this ratio grows, as shown in Fig. A1.

assumptions (3)

domain assumption The behavioral model of Refs. [13,54] (Eqs. 2-5) is quantitatively valid for a p-bit whose noise correlation time is shorter than the synapse response time.
Used throughout Section IV; the paper explicitly notes prior analytical update rules fail in this regime and instead solves the ODEs numerically, but no independent device-level validation is given for the extension.
domain assumption The p-bit output responds instantaneously to input and noise, as in Eq. (2); additional timescales such as analog-to-binary conversion are negligible.
Explicitly stated as an ignored effect in Section III; at ultra-fast fluctuation rates relevant to the scheme, such delays could alter the effective coupling.
domain assumption The hidden p-bit's correlation time tau_h^r can be tuned externally over orders of magnitude without changing the computational p-bits or the synaptic strengths.
The proposal depends on this tunability knob; the paper cites spin-orbit-torque tuning of magnetic fluctuation rates (Ref. [51]) and exponential barrier-height dependence, but no experiment is presented in this paper.

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Cite this review

Pith. "Pith review of Emergent Synaptic Plasticity from Tunable Dynamics of Probabilistic Bits." pith.science (2026). https://pith.science/paper/H2GFFKMF

@misc{pith2026250500252,
  author       = {Pith},
  title        = {Pith review of: Emergent Synaptic Plasticity from Tunable Dynamics of Probabilistic Bits},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/H2GFFKMF}},
  note         = {Machine review of arXiv:2505.00252}
}

read the original abstract

Probabilistic (p-) computing, which leverages the stochasticity of its building blocks (p-bits) to solve a variety of computationally hard problems, has recently emerged as a promising physics-inspired hardware accelerator platform. A functionality of importance for p-computers is the ability to program-and reprogram-the interaction strength between arbitrary p-bits on-chip. In natural systems subject to random fluctuations, it is known that spatiotemporal noise can interact with the system's nonlinearities to render useful functionalities. Leveraging that principle, here we introduce a novel scheme for tunable coupling that inserts a ''hidden'' p-bit between each pair of computational p-bits. By modulating the fluctuation rate of the hidden p-bit relative to the synapse speed, we demonstrate both numerically and analytically that the effective interaction between the computational p-bits can be continuously tuned. Moreover, this tunability is directional, where the effective coupling from one computational p-bit to another can be made different from the reverse. This synaptic-plasticity mechanism could open new avenues for designing (re-)configurable p-computers and may inspire novel algorithms that leverage dynamic, hardware-level tuning of stochastic interactions.

Figures

Figures reproduced from arXiv: 2505.00252 by the authors.

**Figure 1.** FIG. 1. (a) Schematic representation of an interconnected p-bit network illustrating hidden p-bits (colored red) introduced [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

**Figure 2.** FIG. 2. (a) Schematic representation of the hidden p-bit model under investigation. Here, the input of p-bit 1 has been [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** FIG. 3. Directional asymmetry in the hidden p-bit model illustrated for two cases: (a) input to p-bit 1 fixed to [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

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