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Fractional-order hardware for neuromorphic computing: Is the order really the problem?

Christof Teuscher

arXiv:2609.10882v1cs.ETcs.NE

TL;DR

Neuromorphic systems need to account for power-law adaptation across many timescales, but fractional operators make direct evaluation costly and the relevant evidence is spread across communities. This review organizes hardware by that cost and examines its numerical, circuit, and device limits. It concludes that the order gap has largely closed, while bandwidth, low-frequency measurement, and energy data remain open issues.

  • Problem

    Power-law neural adaptation spans many timescales, while integer-order models require an additional state variable for each one, and the hardware cost question has not been brought together across communities.

    Source: §1 · pp. 1–2

  • Method

    The review organizes fractional hardware by how it handles non-local history costs and clarifies that the derived storage and fixed-point limits apply to direct-form implementations.

    Source: §1 · p. 2, §2.5 · p. 8

  • Results

    The order gap has largely closed, but a bandwidth gap of about three decades, an unmeasured low-frequency corner, and absent energy data prevent judging whether the approach succeeds.

    Source: §8 · p. 27

  • Takeaways & Limitations

    Whether a physical element supplying multiple relaxation timescales is cheaper in area or energy than explicit heterogeneity remains an unanswered empirical question.

    Source: §7.5 · p. 26

    The history-storage and fixed-point coefficient limits are specific to direct-form implementations, not bounds on what the fractional operator can achieve.

    Source: §2.5 · p. 8

Abstract

from arXiv · show

Does a neuromorphic system need a true power-law memory kernel, and if so, can anyone build one? Neuromorphic systems process signals spanning many timescales at once, from milliseconds to tens of seconds. Integer-order circuits buy each additional timescale with an additional state variable. Fractional-order dynamics offer a different bargain: one operator whose power-law kernel carries a continuum of timescales, tuned by one parameter, the order alpha. A fractional derivative is non-local, so evaluating it costs storage and arithmetic that grow with the retained history, where an integer-order derivative costs a constant. This review organizes the hardware literature around that cost. We derive the retained history needed to hold the truncation error below a tolerance epsilon, show that it scales as epsilon^(-1/alpha), and set beside it a second and independent limit on the direct form: in fixed point the weights themselves underflow, so word length caps the usable history however long the buffer is. The two limits move at very different rates with the order, and where they cross decides whether a word length can serve an order at all. We use both to sort published hardware into three strategies, note a fourth the numerical literature has developed and this hardware has not, and survey digital, analog and device work. Along the way we ask whether the field is worried about the right obstacle. It is not. Fabricated constant-phase devices already span the orders two groups identify as task-optimal, so the order gap has largely closed, leaving a residual gap near 0.1 and at the lower order describing cortical adaptation. What remains is a frequency-band gap of about three decades at the low end. That corner is not empty, since double-layer electrodes work there, but every device in it is discrete, and no integrable thin-film element has been characterized there.

1 Introduction

Neural adaptation spans milliseconds to tens of seconds and decays as a power law, while integer-order circuits require a state variable for each timescale. This review frames fractional-order hardware around the cost of representing that multi-timescale memory.

1 Introduction

  • Neural adaptation spans milliseconds, seconds, and tens of seconds at once; its power-law decay is not described by one time constant.In integer-order models, each added timescale requires another state variable and parameter set.Source: §1 · p. 1
  • A fractional derivative uses a power-law memory kernel, giving one operator a continuum of relaxation timescales controlled by order α.The order interpolates between behaviors rather than selecting from a discrete menu; fractional models recur in neuronal modeling and circuit theory.Source: §1 · pp. 1–2
  • Directly evaluating a fractional derivative requires retained past samples, so storage and arithmetic grow with history length, unlike an integer-order derivative’s constant cost.The review organizes hardware by how it pays this cost: truncating history, using a finite integer-order approximation, or relying on a native power-law physical process.Source: §1 · p. 2
  • Numerical analysis, circuit design, device physics, and computational neuroscience have studied related pieces, but their different vocabularies have kept the hardware cost question fragmented.The review brings these literatures together, grounding its analysis in neuroscience, quantifying non-locality costs, and assessing what device data support.Source: §1 · p. 2
  • The review excludes broad fractional-order control, purely dynamical chaos and image-encryption work, and mathematical theory beyond hardware needs; it includes orders α in (1, 2).That interval reverses the review’s central trade rather than merely extending the α ∈ (0, 1) range common in literature and fabricated devices.Source: §1 · p. 2
  • Fractional viscoelasticity shares the same power-law-kernel mathematics as electrical impedance, but its mechanical literature is excluded because it does not target hardware.Cell mechanics has a reported exponent near 0.17 across five cell types and 0.1–100 Hz; the review cautions that matching exponents do not establish a shared mechanism and identifies diffusive representations as a fourth strategy [27] [28].Source: §1 · p. 2

1.1 Scope, and how this corpus was assembled

  • The review corpus was assembled by citation chasing from an existing classification and related neuroscience and device-physics work, then closed when searches yielded no new implementations.It is narrative, not systematic: it used no protocol-registered database search and does not claim to enumerate the field.Source: §1.1 · p. 3
  • The review is designed for neuromorphic engineers and fractional-circuit researchers, beginning with operators and costs before turning to biology, models, implementations, devices, and research targets.The electrochemical double-layer literature is cited but not tabulated because it is extensive and uses different impedance-spectroscopy conventions.Source: §1.1 · p. 3

2 Fractional operators and what they cost

Fractional operators provide power-law memory across a continuum of timescales, but direct evaluation makes retained history costly. This section separates truncation and fixed-point precision limits and uses them to organize implementation strategies.

2.1 Definitions, and which one you actually build

  • Caputo dynamics use ordinary physical initial conditions, while the discrete Grünwald–Letnikov form is directly implementable and avoids spike-discontinuity artifacts reported for truncated Caputo models.Fractional derivatives generally do not compose as integer-order derivatives do, and the operator depends on its lower terminal.Source: §2.1 · pp. 3–4

2.2 The memory kernel

  • The power-law kernel decays more slowly than an exponential, so remote history remains relevant; smaller α gives a heavier tail and longer memory.As α approaches 1 the operator approaches an ordinary derivative, while as α approaches 0 it approaches the identity.Source: §2.2 · p. 4

2.3 The cost of non-locality

  • At ε = 10^-2, the history bound requires about 3200 samples at α = 0.5, but about 10^6 at α = 0.3, making direct truncation impractical at low orders.The worst-case bound is a ceiling: spoken-digit accuracy at α = 0.5 saturates near 150 samples, while truncating a neuron model’s history can still substantially change firing rate.Source: §2.3 · pp. 5–6, Table 1 · p. 6
  • Task-optimal orders can fall within the digitally affordable and fabricated-device ranges, but orders used to describe cortical adaptation do not; the bound is worst-case over bounded signals.For spoken-digit classification, accuracy saturates around 150 retained samples at α = 0.5, whereas neuron firing rate remains sensitive to history removed even after 100 s [32, 33].Source: §2.3 · pp. 5–6

2.4 Coefficient precision, a second and independent limit

  • Fixed-point weights eventually round to zero, creating an independent direct-form limit: at ε = 10^-2, 32-bit coefficients reach α = 0.27, but lower orders exceed representable history.The history and representable-weight limits move at very different rates with α; the word-length threshold does not apply to schemes that aggregate far-tail weights before storage.Source: §2.4 · pp. 6–7

2.5 Three ways to pay

  • Hardware pays for fractional dynamics by truncating history, substituting a finite-order approximation valid over a design band, or embodying power-law response in a physical element.Embodiment removes the sample buffer and per-timestep arithmetic, but device order and usable frequency band are fabrication outcomes, and physical relaxation cannot be read or reset.Source: §2.5 · pp. 7–8
  • Fast and oblivious convolution offers a fourth, unimplemented hardware route: it evaluates the power-law convolution in O(N log N) operations with O(log N) active memory.Unlike direct-form fixed point, logarithmic binning aggregates far-tail weights before storage, so the coefficient-underflow rule does not carry over.Source: §2.5 · pp. 7–8

2.6 Properties that break engineering intuition

  • Fractional systems have distinct stability conditions; short-memory truncation can destabilize a stable system, and embodied relaxation cannot be cleared between trials.A physical relaxation process carries history, making initialization a physical constraint rather than only a modeling choice.Source: §2.6 · p. 8

3 Does biology already do this?

Neural recordings support power-law, multi-timescale adaptation, but do not establish that its underlying mechanism is a true fractional kernel. The review connects this biology to hardware by examining adaptation’s function, competing mechanisms, and the evidence’s limits.

3.1 Neurons perform fractional differentiation

  • Rat cortical neurons tracked a fractional derivative of order near 0.15 over roughly two decades, while earlier vestibulo-ocular work had identified fractional-order behavior.The same group later reproduced the behavior in vivo.Source: §3.1 · pp. 8–9

3.2 Power-law adaptation and what it computes

  • Power-law adaptation across sensory systems is linked to whitening approximately 1/f inputs, which can maximize rate-coded information per spike.Engineering-grade auditory-periphery models also require adaptation of this kind.Source: §3.2 · p. 9

3.3 The mechanism question

  • The biophysical mechanism remains unsettled: fractal gating and diffusion along state chains compete, while available fitting may not distinguish them.Scale-free recovery is better supported, and anomalous diffusion provides a formal route from microscopic kinetics to fractional macroscopic descriptions.Source: §3.3 · p. 9

3.4 Constant-phase behavior of biological interfaces

  • Biological tissue exhibits Cole–Cole constant-phase behavior, but its fitted exponent describes impedance behavior rather than identifying a unique physical mechanism.The fractional-capacitor interpretation treats an ideal capacitor as a special case.Source: §3.4 · p. 9

3.5 Structure produces the exponent

  • Dendritic diffusion’s anomaly exponent varies with spine density, showing that geometry can tune the order α in a physical medium.This is a biological proof of principle for a designable exponent.Source: §3.5 · p. 9

3.6 But is a power law really needed?

  • Over bounded timescale ranges, a small number of exponentials can approximate power-law adaptation, weakening the case that devices must embody a true power-law kernel.The fractional operator is a compact limiting description; the consequential engineering question is how many exponentials meet a specified tolerance.Source: §3.6 · p. 10

3.7 What the biology does not establish

  • Evidence limits include the small number of cell types and laboratories, no atlas of order variation, and no measured consequence of changing the order for coding or behavior.The reported order near 0.15 spans about two decades of timescale; its functional importance remains assumed rather than shown.Source: §3.7 · p. 10

4 Fractional neuron and network models

Fractional neuron and network models report behaviors unavailable to classical models, while implementation depends on where memory is placed and which order is chosen. The review also identifies limits in computational cost, physical realizability, and the evidence behind some model claims.

4.1 Integrate-and-fire family

  • The fractional integrate-and-fire model adds no state or adaptation current, while orders below about 0.2 yield power-law spike adaptation; orders above one produce downward adaptation.Resetting the memory trace at each spike removes the adaptation effect.Source: §4.1 · p. 10
  • Higher orders lighten the digital memory tail, but passive single elements cannot realize orders above one; nonuniform spike grids also preclude FFT acceleration.The weight-decay direction above unity is established without the truncation bound derived for orders below one.Source: §4.1 · pp. 10–11

4.2 Conductance-based models: where does the memory live?

  • Fractional Hodgkin–Huxley models place memory either in gating variables or membrane capacitance, distinct physical claims not tested on the same data.The reported comparison between placements uses published figures rather than a controlled study.Source: §4.2 · p. 11

4.3 FitzHugh–Nagumo, Izhikevich and Hindmarsh–Rose

  • Incommensurate-order stability results support robustness to order mismatch, while one measured realization used DSP firmware rather than analog hardware.Fractional stability can decay algebraically rather than exponentially.Source: §4.3 · p. 12
  • Fractional Izhikevich models add bursting and history-sustained spiking, but the direction of order-dependent firing changes with parameter set [86, 87].For one parameter set, firing stops below γ = 0.86.Source: §4.3 · p. 12

4.4 Recurrent and associative networks

  • Fractional order shifts network stability boundaries and can make low-dimensional networks chaotic when their integer-order counterparts cannot be.This changes which behaviors are reachable at a given network size.Source: §4.4 · p. 12

4.5 Two problems worth naming

  • The conformable derivative is local and carries no memory kernel, so results using it do not establish the history dependence central to fractional dynamics.The review also distinguishes neuromorphic relevance from papers centered on encryption or unevaluated attractors.Source: §4.5 · p. 14

4.6 Position against prior surveys

  • Unlike earlier neural-network surveys, this review quantifies operator cost, foregrounds device physics, and grounds the hardware case in experimental neuroscience.It also considers whether biology requires a true power law.Source: §4.6 · p. 14

5 Digital and analog implementation

Fractional-order hardware either approximates the operator over a chosen band, truncates its memory, or uses other numerical strategies; each choice trades accuracy, resources, and preserved dynamics. The surveyed implementation reports leave major costs and settings unreported or incomparable.

5.1 Approximation methods

  • Approximation methods differ, but all are accurate only over a chosen band, so stage count must be allocated to the application’s timescales.Oustaloup distributes poles and zeros geometrically; Charef prescribes error in decibels, while Matsuda fits selected frequency points.Source: §5.1 · p. 14

5.2 Truncation in practice

  • Truncating memory creates an accuracy trade-off, and fixed-point coefficient representability imposes an independent limit on usable history.The review identifies coefficient representation as an additional constraint beyond choosing the window length.Source: §5.2 · pp. 14–15

5.3 Does the discretization preserve the dynamics?

  • The Adomian decomposition reproduced one trajectory in 0.033 s, versus 3.33 s for Grünwald–Letnikov and 559 s for Adams–Bashforth–Moulton, but its four-term accuracy was not tested.It uses four terms per state variable and only the current state; the paper reports no convergence study or error bound.Source: §5.3 · p. 15

5.4 FPGA realizations

  • FPGA implementations include Grünwald–Letnikov operators, approximated coefficients, and neuron models; the most fully reported example uses 32-bit fixed point and runs at 9.33 MHz.Monir and colleagues’ Artix-7 design uses window lengths of 32, 512, and 1024 and occupies 64 DSP slices and 36% of the device.Source: §5.4 · p. 15

5.5 Analog and mixed-signal

  • Analog implementations realize approximations as physical networks, but the two surveyed CMOS-node designs are simulations or layouts, not fabricated circuits.The surveyed works include ladder and tree topologies, current-mode implementations, and an RC ladder derived from Charef’s method.Source: §5.5 · pp. 15–16

5.6 What does the implementation literature not report?

  • Across the surveyed reports, none gives energy per fractional operation, only two report power, and resource counts are not comparable across platforms.Window length is stated in four of eight entries; three reports omit clock rate, and the authors leave missing values unfilled rather than infer them.Source: §5.6 · p. 16, Table 4 · p. 16

6 Devices with intrinsic fractional dynamics

The device literature shows that constant-phase behavior can arise through several physical routes, and fabricated devices span many task-optimal orders. The remaining mismatch is chiefly at low frequencies, where characterized double-layer devices are discrete rather than integrable.

6.1 Where constant phase comes from

  • A constant-phase element has impedance proportional to s^-α, giving a phase angle of -απ/2 across a band rather than an ideal capacitor’s -π/2.Proposed physical accounts differ, and no single origin is considered universal.Source: §6.1 · p. 17

6.2 Why power-law relaxation is generic in solids

  • Broad distributions of relaxation times produce constant phase and power-law memory, making such behavior close to generic in disordered solids rather than exclusive to specially prepared materials.Dispersive transport in amorphous semiconductors supplies a proposed physical mechanism for power-law relaxation in amorphous oxides.Source: §6.2 · p. 17, Figure 3 · p. 23

6.3 Fabricated fractional-order capacitors

  • Fabricated constant-phase devices span diverse materials, but reported order ranges usually combine separate variants rather than one tunable device, and four of six bands end at instrument sweep limits.The reported phase ripple can consume or exceed the ten-percent order tolerance used for comparison, before unit-to-unit spread or drift is measured.Source: §6.3 · pp. 17–18, Table 5 · p. 18

6.4 Supercapacitors and double-layer devices

  • Porous double-layer electrodes produce long time constants and low-frequency constant-phase behavior, and supercapacitor stacks were used in the field’s end-to-end neuron demonstration.These are discrete millifarad components, not integrable elements.Source: §6.4 · pp. 18–19

6.5 Memristors, memcapacitors and ionic devices

  • Neuromorphic ionic and memristive devices are generally characterized for memory figures of merit, not whether their relaxation is power-law, stretched-exponential, or exponential.The relaxation shape determines whether a device embodies a fractional kernel or only a slow response.Source: §6.5 · p. 19, Figure 3 · p. 23

6.6 The order and bandwidth mismatch

  • Task-optimal orders vary by workload, with sweeps reporting optima near 0.3–0.5 for spoken digits, 0.5–0.7 for cart-pole, and 0.1 for physiological prediction.An independent cart-pole sweep also found an optimum at α = 0.5 [31]; the order gap has largely closed, though a residual gap remains near 0.1 and for biological orders around 0.15–0.3.Source: §6.6 · pp. 19–21
  • Thin-film devices hold constant phase from about 100 Hz upward, while adaptation extends toward hundredths of a hertz; no tabulated device has been characterized below about 10 Hz.Double-layer electrodes reach the millihertz range, but the characterized low-frequency devices are discrete rather than integrable.Source: §6.6 · p. 20

6.7 What the embodiment argument actually buys

  • Embodiment removes runtime history updates, but fixes the device’s order and band at fabrication and leaves initialization, drift, and unit-to-unit variability as concerns.A fractional controller required an integer-order network during its first six decisions to recover up to thirty-five seconds of balance time at α = 0.5.Source: §6.7 · pp. 21, 24

7 What is the field not measuring?

The review finds no hardware implementation of fractional learning, while identifying physical fractional reservoirs as a promising experimental direction. It catalogs unresolved measurement and comparison gaps, including whether fractional dynamics outperform integer-order alternatives at matched cost.

7.1 Learning, and a null result

  • No surveyed FPGA, ASIC, memristive, or crossbar system implements fractional learning; recent “fractional” training examples use ordinary gradient descent and have not run on hardware.Yin et al. train a per-neuron order parameter with ordinary backpropagation, while Ge et al. train a power-law kernel with integer-order surrogate gradients.Source: §7.1 · p. 24

7.2 Reservoir computing is the better place to plant a flag

  • A physical fractional reservoir could test simulation-based predictions of task-dependent optimal order, but no such reservoir has yet been built and measured.The proposed experiment would test whether order-dependence survives device variability.Source: §7.2 · p. 24

7.3 Gaps, stated as questions

  • Open gaps include neuron-order variation across biological contexts, a controlled comparison of Hodgkin–Huxley formulations, device-order variability, energy data, common-platform approximation sweeps, real workloads, and plasticity rules.Existing order estimates near 0.15 come from a small number of cell types in one preparation; under two percentage points of accuracy loss was reported at ten percent order drift.Source: §7.3 · pp. 24–25

7.4 A metric set and two benchmark tasks

  • The review proposes reporting energy, retained history and error tolerance, effective order and ripple over a stated band, and accuracy against a matched-parameter integer-order baseline.It recommends spoken-digit classification and cart-pole control as hardware-sized benchmark tasks with published order sweeps.Source: §7.4 · pp. 25–26

7.5 The competition this has to beat

  • Integer-order heterogeneity already supports multi-timescale memory, so fractional hardware must demonstrate lower area or energy for the same effect.The reported fractional-network comparison in matched neither parameter count nor training budget, leaving the source of its advantage unsettled.Source: §7.4 · p. 25, §7.5 · p. 26

8 A path forward and a call for action

The review calls for measurements that resolve low-frequency coverage, device variability, relaxation mechanisms, energy, and digital implementation trade-offs. It concludes that the order gap has largely closed, while bandwidth and energy evidence remain insufficient to judge success.

Device measurements

  • Characterize thin-film constant-phase devices below 10 Hz into the millihertz range and test whether impedance-derived order predicts the time-domain kernel under step-like transients.The proposed campaign uses existing devices and equipment common in electrochemistry groups.Source: §8 · p. 26
  • Publish order drift and unit-to-unit variability: under two percentage points of accuracy loss was reported at ten percent order drift, while ±4° ripple consumes about half, nine tenths, or more than the tolerance budget at α = 0.88, 0.5, and 0.33, respectively.Only one fabricated capacitor has published long-term stability data.Source: §8 · p. 26
  • Map whether memristive, memcapacitive, and ionic devices exhibit power-law, stretched-exponential, or exponential relaxation, rather than relying on memory-oriented figures of merit.The review identifies this mechanism-to-relaxation mapping as missing.Source: §8 · p. 26

Digital and systems targets

  • Report energy per fractional operation and jointly sweep order, retained history, coefficient word length, resources, accuracy, and energy on one reproducible platform; build a physical fractional reservoir.The proposed reservoir experiment would test a task-dependent optimal-order prediction already examined in simulation.Source: §8 · p. 26

Conclusion

  • The order gap has largely closed, but a roughly three-decade bandwidth gap, an unmeasured low-frequency corner, and absent energy data prevent judging whether fractional hardware succeeds.These gaps are not evidence that the approach fails.Source: §8 · p. 27

Funding

The work acknowledges National Science Foundation support under grant 2318139.

Funding

  • This material is based on work supported by the National Science Foundation under grant no. 2318139.Source: Funding · p. 27

Conflicts of interest

The author discloses co-authorship of a cited order-sweep study and no financial interest in the devices, materials, or platforms discussed.

Conflicts of interest

  • The author co-authored, on which section 6.6 depends, and reports no financial interest in any discussed device, material, or platform.The independent replication shared grant support but no personnel or analysis with this review.Source: Conflicts of interest · p. 27

Ethics

The review involved no human or animal studies, so no ethics approval was sought or required.

Ethics

  • No human participants, human data or tissue, or animal studies were involved, so ethics approval was not sought or required.No new data were collected.Source: Ethics · p. 27

Data availability

The review generated no experimental data; its numerical results were produced by supplementary scripts, with source-page provenance supplied for tables 3–5.

Data availability

  • No experimental data were generated; numerical results were produced by supplementary scripts, and tables 3–5 have accompanying source-page provenance records.The accompanying materials also include a bibliography-verification record.Source: Data availability · p. 27
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