Operator-Splitting Methods for Neuromorphic Circuit Simulation

Amir Shahhosseini; Rodolphe Sepulchre; Thomas Chaffey

REVIEW 3 major objections 4 minor 1 cited by

Operator-Splitting Methods for Neuromorphic Circuit Simulation

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper claims that simulating a spiking neural network can be recast as a difference-of-monotone zero-finding problem and solved by a consensus-based Douglas–Rachford splitting, whose fixed-point iterations converge weakly to the…

desk verdict Novel consensus-based splitting algorithm and a clean proof, but Theorem 2's assumptions force the network operator to be strongly monotone, contradicting the paper's own nonmonotone spiking examples. read the letter →

arxiv 2505.22363 v1 pith:2SSX4F3F submitted 2025-05-28 eess.SY cs.SY

Amir Shahhosseini , Thomas Chaffey , Rodolphe Sepulchre This is my paper

classification eess.SYcs.SY MSC 47H0547J2565K10

keywords operatorsplittingdifference-of-monotoneoperatorsDouglas-Rachfordneuromorphiccircuitsspikingneuralnetworksmonotonetheoryfixed-pointiterationcircuitsimulation

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

Spiking neurons and networks are usually simulated by stepping stiff, nonlinear differential equations forward in time. This paper argues that the same problem can be posed as a zero-finding problem for an operator built from the circuit elements, then split into a difference of monotone operators, one per capacitor, ion-channel branch, and synaptic connection. The split follows the circuit's single-layer architecture, so each piece has a physical meaning and a cheap resolvent. The paper proposes a consensus-based difference-of-monotone Douglas–Rachford algorithm, proves weak convergence to a solution under monotonicity and set-invariance assumptions, and demonstrates spiking, bursting, and half-center-oscillator simulations that agree with conventional integration while allowing continuation, coarse-to-fine refinement, and event detection.

What carries the argument

The central object is the consensus-based difference-of-monotone Douglas–Rachford iteration, a fixed-point map $T$ built from resolvents $J_{\alpha E}$ and $J_{p\alpha F_i}$ and direct evaluations $G_i$, with a projection onto the consensus set so that an arbitrary number of operator pairs can be handled in parallel. The companion mechanism is the lifting of each neuron branch into shifted monotone partners, so that the resolvent of each $F_i$ can be evaluated by a cheap inner fixed-point iteration whose contraction condition is $|\gamma\alpha_+/(1+\lambda\gamma)|<1$, and LTI parts such as $CD$ and $(\tau_x D+\mathrm{Id})^{-1}$ are applied in the frequency domain via FFT. The splitting is determined by circuit topology, so each operator corresponds to a physical element.

What would settle it

Take the bursting-neuron model (39) and run Algorithm 1 with the shift $\lambda$ decreased below the smallest value that makes every $F_i$ strongly monotone; if the iteration still converges to the numerical-integration solution, the assumed monotonicity is not actually needed, while if it diverges, the theorem's conditions are doing the work. A direct check is whether the inner resolvent contraction condition $|\gamma\alpha_+/(1+\lambda\gamma)|<1$ fails for some timescale at the claimed shift lower bound and the inner solver stops converging.

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Extended reading notes

Core claim

The central claim is that the simulation problem $W(v)=i_{\rm ext}$ of a neuromorphic circuit can be rewritten, using shifts $\lambda\,\mathrm{Id}$ that leave the dynamics unchanged, as a difference-of-monotone zero-finding problem $E(x)+\sum_i(F_i(x)-G_i(x))=0$ in a lifted Hilbert space. The single-layer architecture of the neuron and network dictates the splitting: the capacitor gives $E=CD$, each positive-conductance channel gives a shifted monotone $A_x^s$, each negative-conductance channel gives an anti-monotone piece $B_x^s$ whose shift makes it monotone, and each row of synaptic dynamics gives a monotone $C_i$ minus a shift $\lambda_i^{\rm syn}\mathrm{Id}$. The authors prove that the fixed-point iteration of the resulting consensus-based Douglas–Rachford map (Algorithm 1) converges weakly to a solution when $E$ is $\rho$-strongly monotone, each $F_i$ is $\gamma$-strongly monotone, each $G_i$ obeys a two-sided slope bound, the map keeps a closed convex set $D$ invariant, and $\gamma>\beta+1+2/\epsilon$. They then show numerically that the solver reproduces excitability, bursting, and network rebound bursts, and that its global, signal-to-signal nature avoids forward error propagation.

Load-bearing premise

The convergence proof requires each split operator to be strongly monotone on a closed convex signal set, the overall map to keep that set invariant, and a fixed point to exist; the paper relies on physical plausibility that local monotonicity holds for the circuit operators rather than verifying these conditions operator by operator.

Editorial extensions

If this is right

Spiking-network simulation becomes a fixed-point search on the whole voltage signal, so errors do not accumulate forward in time; solving one signal directly is what makes coarse grids locate events correctly.
Because each split operator corresponds to a capacitor, ion channel, or synapse, the solver is modular: resolvents of neurons are computed independently and synaptic effects are added as consensus-paired operators.
Continuation and variability analysis are natural: a converged solution for one parameter value is a valid warm start for a nearby value, and the paper demonstrates this on the maximal-conductance parameter.
Coarse-to-fine and template-based refinement are supported: coarse-resolution runs locate events, then fine-resolution runs or single-neuron spike templates refine them, which is difficult for incremental integrators.
The number of operators grows linearly with network size ($2n+2m+1$ for $n$ neurons and $m$ timescales), enabling parallel implementation, although the consensus set grows with the network and slows convergence.

Reading between the lines

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

If local monotonicity is the operative condition, the method's difficulty is governed by how many and how strong the negative-conductance branches are, not by the number of positive elements; this suggests a complexity scaling argument for neuromorphic circuits with sparse negative conductances.
The event-capturing coarse-to-fine property could be developed into an adaptive multirate scheme that refines only around detected event windows, since the paper demonstrates template substitution but not fully adaptive local refinement.
A testable extension is to apply the solver to circuits with device-level nonidealities, such as memristor variability, where the required shift $\lambda$ would need to grow with the spread of device conductances; the contraction condition gives a concrete bound to check.
The unverified strong-monotonicity and set-invariance assumptions point to a concrete research program: derive explicit bounds on $\lambda$ for tanh- and sigmoid-with-first-order-lag compositions so the assumptions become checkable a priori rather than physically plausible.

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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

The real contribution here is the consensus-based difference-of-monotone Douglas-Rachford algorithm for an arbitrary number of operator pairs, plus a self-contained proof. The lifting construction and the translation of the single-layer circuit architecture into the splitting are genuinely nice, and the examples do show spiking, bursting, and half-center-oscillator behavior that matches numerical integration. That part is worth reading.

The soft spot is load-bearing. Theorem 2's assumptions force the full operator W in (23) to be strongly monotone. Under conditions 1-3 and gamma > beta + 1 + 2/epsilon, each F_i - G_i is strongly monotone with constant gamma - beta - 1/epsilon, so W = E + sum(F_i - G_i) is strongly monotone. But Section II-B2 argues that the spiking network operator W can never be monotone. Both cannot hold. The explicit split (38) is a concrete check: F1 = Id - i_ext has gamma = 1, while G1 = 2 tanh(.) has slope bound 2, so beta + 1/epsilon >= 2 and condition 4 requires gamma > beta + 1 + 2/epsilon >= 3. Adding lambda Id to both F and G does not help because gamma and beta increase by the same amount. So the convergence theorem, as stated, does not apply to any of the paper's examples. Remark 6's appeal to local monotonicity does not save it, because the spiking trajectories are not confined to a set D where the operator could be strongly monotone and still spike.

Secondary issues: the validation is qualitative with no error metrics or runtime comparison to NI, the lambda = 2 and lambda = 4 choices are not checked against the theorem's hypotheses, and the code statement lacks a URL. The authors do honestly acknowledge the scalability limitation. None of these are fatal by themselves; the theorem-application mismatch is.

This paper is for readers interested in operator splitting for nonmonotone or DC-type problems, and for neuromorphic simulators open to non-integrator methods. I would send it to peer review because the algorithm is novel and the problem is important, but the revision needs major rework. Either the convergence theory must extend to genuinely nonmonotone differences, or the paper should be reframed as a heuristic method with strong empirical validation. As it stands, the central claim is unsupported.

Referee Report

3 major / 4 minor

Summary. The manuscript proposes recasting the simulation of neuromorphic circuits as a zero-finding problem for a difference of monotone operators, deriving a topology-based splitting for the spiking neuron and network models of Section II, and introducing a consensus-based difference-of-monotone Douglas-Rachford algorithm (Algorithm 1). Theorem 2 claims weak convergence of the resulting fixed-point iteration to a solution of (23) under strong-monotonicity and two-sided-bound assumptions on the split operators. Three numerical examples (a single spiking neuron, a bursting neuron, and a half-center oscillator) are compared against numerical-integration solvers, and the paper highlights event-based resolution, continuation for parameter variation, and coarse-to-fine warm-starting. The key claimed contribution is a principled convergent splitting method for nonmonotone spiking circuits.

Significance. If the convergence theorem were applicable, the framework would be of real interest: it gives an architectural splitting with physical meaning, a parallel consensus structure, an event-capturing coarse-to-fine workflow, and reproducible MATLAB code validated against an independent numerical-integration solver. The proof in Appendix I is self-contained and does not rely on fitted quantities, and the validation is external. However, the central theorem's hypotheses are inconsistent with the target problem class, and the tutorial example explicitly fails condition 4 of Theorem 2. The numerical illustrations therefore cannot be interpreted as instances of the proven convergence result, and the claimed theoretical foundation for the algorithm is not established. The significance is accordingly conditional on a substantial revision of the convergence analysis.

major comments (3)

[II-B2 and III-D (Theorem 2)] The assumptions of Theorem 2 imply that the full operator in (23) is strongly monotone on D, contradicting the paper's own statement in Section II-B2 that the spiking network operator can never be monotone. Indeed, combining the gamma-strong monotonicity of F_i (assumption 2) with the upper bound in (25) gives <(F_i-G_i)x-(F_i-G_i)y, x-y> >= (gamma-beta-1/epsilon)||x-y||^2, and condition 4 yields gamma-beta-1/epsilon > 1+1/epsilon > 0; with E rho-strongly monotone, W=E+sum_i(F_i-G_i) is strongly monotone. A strongly monotone operator is monotone and has at most one zero, so the convergence guarantee applies only to monotone problems, not to the excitable/nonmonotone class that Section II-B2 identifies as the subject of the paper. This is not a matter of unverified constants: the theorem's hypotheses exclude the application.
[IV-B1, Eq. (38)] For the tutorial splitting in (38), condition 4 of Theorem 2 fails already for i=1. Here F1=Id-i_ext has gamma=1, while G1=2tanh(.) has derivative 2sech^2 bounded between 0 and 2. If D contains a neighbourhood of the resting potential, the two-sided bound (25) forces beta=0 and 1/epsilon >= 2, so condition 4 would require 1=gamma > beta+1+2/epsilon >= 5, a contradiction. Adding the shift lambda Id to both F1 and G1, as suggested in Section III-B, increases gamma and beta by the same amount and therefore leaves gamma-beta unchanged; it cannot repair the inequality. The same obstruction applies to the bursting-neuron and half-center-oscillator models in Sections IV-B2 and IV-B3, whose negative-conductance terms are handled identically. Consequently no numerical example in the paper operates in the regime required by Theorem 2.
[Remark 6 and Appendix I] The proof of Theorem 2 relies essentially on strong monotonicity of all operators on D, in particular Eq. (77) in Appendix I uses rho>0 and gamma>beta+1+2/epsilon. Remark 6's assertion that local monotonicity conforms to the physics is not a verification of assumptions 1-5 for the concrete operators CD, A_x^s, B_x^s, C_i, and lambda_i Id; no domain D, constants rho, gamma, beta, epsilon, or proof of T(D) subset D and FixT nonempty are provided. Since the contradictions above show that no such constants exist for the explicit splits, the convergence guarantee cannot be transferred to the neuromorphic setting as it stands.

minor comments (4)

[III-D2, Eq. (36)] The contraction argument for the inner fixed-point iteration omits the induced norm of the lag operator L^{-1}; the displayed condition should include ||L^{-1}|| or an explicit statement that L^{-1} is nonexpansive on the chosen signal space.
[IV-A.4] The normalized residual formula after Remark 4 is missing norm bars: as printed, 1/L (E(x_sol)+sum(...)) is a vector, not the scalar quality metric described in the text.
[IV-B1 and IV-B2] The hyperparameter choices lambda=4 and lambda=2 are said to ensure monotonicity, but no derivation or exact reference is given for these values for the specific operators; a brief verification or citation to the relevant result in [12] would improve reproducibility.
[Throughout] There are several notation and typographical issues, including the ambiguous 'gamma alpha+' in Eq. (36), 'descrived' in the abstract, 'nP' in Section IV-A.4, and 'ZFPS' in Remark 7; these should be corrected during revision.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the convergence proof is self-contained and the simulations are checked against an independent NI solver; heavy self-citation and unverified monotonicity hypotheses are correctness concerns, not circular reductions.

full rationale

The claimed derivation chain, from the network equations (13) through the algebraic splitting (16)-(22) to the fixed-point formulation (23)-(27) and the convergence theorem (Theorem 2, with proof in Appendix I), does not reduce to its inputs. Theorem 2 is a conditional statement whose proof is self-contained: it derives averagedness of the fixed-point operator from the stated strong-monotonicity and two-sided slope bounds using standard Hilbert-space inequalities, and none of its intermediate estimates (e.g., (65), (77)) depend on a fitted quantity or on the numerical examples. The illustrative simulations are validated against an independent numerical-integration solver (Figs. 5, 7, 8), so there is no case of a fitted parameter being renamed as a prediction. Self-citations are frequent ([7], [12], [13], [22], [24], [35]), and the lower bound on the shift lambda that is said to ensure monotonicity is deferred to [12]; however, [12] is a prior monotone-circuit theorem rather than a restatement of this paper's convergence claim, and the main proof does not import it. The paper's genuine weakness is that the monotonicity/strong-monotonicity assumptions of Theorem 2 are not verified for the spiking examples (for the explicit split (38), F1 has strength gamma=1 while the tanh term G1 makes the condition gamma > beta+1+2/epsilon impossible), and Remark 6 only appeals to physical plausibility. This is a soundness or correctness gap in the application of the theorem, not a circular derivation.

Assumptions & free parameters 5 free parameters · 4 assumptions · 0 invented entities

The algorithm's convergence rests on strong monotonicity of E, F_i and the two-sided bound (25) on G_i over a closed convex set D, plus invariance T(D)⊆D and nonemptiness of FixT. These are stated as assumptions and argued physically, but not verified for the specific operators of the neuromorphic model. The hyperparameters λ, α, F_s, iteration limits and simulation times are chosen by hand for each example; they are not fitted to data, but they do affect the reported behavior.

free parameters (5)

λ (shift parameter) = 4 (spiking), 2 (bursting)
Chosen by hand to ensure monotonicity of shifted operators; large enough values slow convergence.
α (stepsize) = 0.5 (spiking), 0.15 (bursting)
Controls the fixed-point iteration step; bounded above for stability.
F_s (sampling frequency) = 10, 4, 0.1, 2 Hz across examples
Time-discretization resolution; affects signal length and accuracy.
max_iterations = 1000, 7000, etc.
Termination cap for the outer iteration.
simulation_time = 1200, 1500, 12000, 8000 ms
Length of the simulated event; must be long enough for rest-to-rest signals.

assumptions (4)

domain assumption E is ρ-strongly monotone and F_i is γ-strongly monotone on D
Theorem 2 assumption; for E = CD the operator is only lossless (not strongly monotone) under rest-to-rest signals, so Remark 8's relaxation is needed but is not part of the stated theorem.
domain assumption G_i satisfies (β+1/ε)∥y−ŷ∥² ≥ ⟨G_i y − G_i ŷ, y−ŷ⟩ ≥ β∥y−ŷ∥² on D
Theorem 2 assumption; the bound is plausible for bounded tanh/sigmoid branches but not derived for the circuit operators.
domain assumption T(D) ⊆ D and FixT nonempty
Theorem 2 assumption; the paper does not exhibit D or prove invariance.
domain assumption Signals are rest-to-rest (zero boundary conditions)
Section IV-A-2 states simulations start and end at rest; this makes D and the first-order lags monotone/nonexpansive and enables the FFT-based periodicity assumption.

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Cite this review

Pith. "Pith review of Operator-Splitting Methods for Neuromorphic Circuit Simulation." pith.science (2026). https://pith.science/paper/2SSX4F3F

@misc{pith2026250522363,
  author       = {Pith},
  title        = {Pith review of: Operator-Splitting Methods for Neuromorphic Circuit Simulation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2SSX4F3F}},
  note         = {Machine review of arXiv:2505.22363}
}

read the original abstract

A novel splitting algorithm is proposed for the numerical simulation of neuromorphic circuits. The algorithm is grounded in the operator-theoretic concept of monotonicity, which bears both physical and algorithmic significance. The splitting exploits this correspondence to translate the circuit architecture into the algorithmic architecture. The paper illustrates the many advantages of the proposed operator-theoretic framework over conventional numerical integration for the simulation of multiscale hierarchical events that characterize neuromorphic behaviors.

Figures

Figures reproduced from arXiv: 2505.22363 by the authors.

**Figure 1.** Modeling the neuron’s membrane with a capacitive element is a classical approach in neurodynamics. The first-order lags filter the voltage and generate voltages at different timescales (vf , vs, and vµs). The interaction of positive and negative conductances at different timescales captures the essence of excitability. In the model of [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

**Figure 2.** The spiking network also has the same single-layer architecture. It can be seen that the state (voltage) of all neurons can contribute to the dynamics of the k th neuron through synaptic currents. The effect of these synaptic connections is combined with the internal dynamics (denoted by int superscripts) of the k th neuron (stemming from its own ion channels) and equates the externally injected current to the k th … view at source ↗

**Figure 3.** Addition of a positive shift operator to the composition of monotone operators makes the upper block monotone. The same shift is subtracted from the composition of the anti-monotone operator and the monotone operator to make it anti-monotone. This virtual addition does not alter the dynamics of the system. same linear resistor but with negative conductance to the I − x branch of the model of [PITH_FULL_IMAGE:figure… view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: The effect of the all the synaptic connections of the network on the k th neuron is demonstrated here. The green link represents the aggregate current that the k th neuron receives from its synaptic connections. Given the active nature of the synaptic connections, they…

**Figure 5.** Figure 5: Simulation of a spiking neuron subjected to external current. The external injected current cannot excite the system if it is small and only creates small bumps in the potential. In contrast, by having a slightly larger external current, the neuron spikes. As it can be…

**Figure 6.** Figure 6: illustrates continuation on the modulation of the maximal conductance parameter alpha from G1 = 2 tanh(·) to G1 = 2.4 tanh(·) and the question is how this changes the behavior. The algorithm is first implemented to simulate the behavior of the spiking neuron with G1 = …

**Figure 7.** Figure 7: The excitability of a bursting neuron is captured using this simulation framework. The simulation begins at rest and from early on, begins to detect the occurrence of events. By continuing the iterations, the solver refines the signal further until its convergence. NI …

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Forward citations

Cited by 1 Pith paper

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

Circuit realization and hardware linearization of monotone operator equilibrium networks
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Resistor-diode circuits realize ReLU monotone operator equilibrium networks, and their exact gradient can be computed in the same hardware by linearizing the diodes.

Reference graph

Works this paper leans on

64 extracted references · 56 canonical work pages · cited by 1 Pith paper

[1]

This is the result of theBanach fixed-point Theorem

FixTis non-empty andTis a contractive mapping. This is the result of theBanach fixed-point Theorem

work page
[2]

neuron” block is exactly the dynamics presented in the left- hand side of (9) that describes the internal dynamics of the neuron. For thek th neuron, the internal “neuron

FixTis non-empty andTis an averaged operator. Then, the iterations of (8) are known as theKrasnosel’ski ˘ı- Manniteration. B. Model Definition The model chosen to represent neurons determines the level of conformity with neurophysiology, hardware implementabil- ity and computational complexity. This work adopts the spik- ing neuron model from study [22]. ...

work page
[3]

Among these properties,mono- tonicityholds a special place

Monotonicity—The Property of Interest:Given the struc- ture and properties of the operatorW(·), different methods can be used to solve (14). Among these properties,mono- tonicityholds a special place. This property bridges physical interpretability with computational tractability. From a phys- ical perspective, it indicates the incremental passivity of th...

work page
[4]

The justification for the breakdown of monotonicity is threefold

Beyond Simple Monotone Problems:The dynamics of the spiking neuron (10) and network (12) can never be monotone. The justification for the breakdown of monotonicity is threefold. 1)Nature of Excitability: Excitability and spiking behav- ior is grounded in the interaction of active elements at fast timescales with dissipative elements at slower timescales [...

work page
[5]

To further exploit the structure of our problem and simplify the proof of convergence, further assumptions are put on the operators

Consensus-based Difference of Monotone Douglas- Rachford Algorithm:The problem of interest has the formu- lation E(x) + pX i=1 (Fi(x)−G i(x)) = 0(23) whereE,F i and,G i are monotone operators. To further exploit the structure of our problem and simplify the proof of convergence, further assumptions are put on the operators. The signal space of (23) is con...

work page
[7]

OperatorF i isγ-strongly monotone onDfor alli∈I

work page
[8]

OperatorG i satisfies (β+ 1 ϵ )∥y−ˆy∥ 2 ≥ ⟨Gi y−G i ˆy, y−ˆy⟩ ≥β∥y−ˆy∥2 (25) AUTHORet al.: TITLE 7 for ally,ˆy∈ Dandi∈I

work page
[9]

Then, the fixed-point iteration zk+1 =T(z k)(27) converges to a solution of (23), if such a solution exists

The conditionγ > β+ 1 + 2 ϵ holds on the subsetD 5)T:L 2 →L 2 defined by T(·) = Id(·)−J αE(PC(·)) + JpαF(2 JαE(PC(·))−Id(·) +pαG(J αE(PC(·))) (26) satisfiesT(D)⊆D, whereD= L i∈I Dis the lifted subset. Then, the fixed-point iteration zk+1 =T(z k)(27) converges to a solution of (23), if such a solution exists. Proof:: The proof of convergence for this FPI a...

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[10]

The main bottleneck lies in computing the resolvent operatorJ Fi

Technical Notes on Efficient Computational Methods: Although Algorithm 1 uses first-order methods, this does not guarantee computational efficiency. The main bottleneck lies in computing the resolvent operatorJ Fi. Due to the nonlinear and dynamic nature of theF i operators, a...
[11]

The dynamic operators, including bothDthat represents the capacitor andL that represents the first-order lags, are LTI systems

Time-Frequency Hopping and Exploiting the Structure: Given the structured architecture of the model, it is possible to further accelerate the computational steps. The dynamic operators, including bothDthat represents the capacitor andL that represents the first-order lags, are...
[12]

This matrix is used at every iteration of Algorithm 1 and does not change throughout the simulation

Pre-processing frequent components:Due to the LTI structure of the operatorE, it is possible to construct its matrix representation and consequently, its resolventJ αE in the frequency domain. This matrix is used at every iteration of Algorithm 1 and does not change throughout...
[13]

The first one isαwhich is the stepsize of Algorithm 1 and dictates the change at every iteration

Choice of the hyperparameters and their meaning:This simulation framework has five hyperparameters. The first one isαwhich is the stepsize of Algorithm 1 and dictates the change at every iteration. A small choice ofαtypically corresponds to slower convergence while a largerαal...
[14]

Here, the ∥xk+1−xk∥ ∥xk∥ determines the change of the solution after one iteration

Termination Criterion:The termination criterion for Al- gorithm 1 is similar to most iterative algorithms and relies on the relative change of the solution at every iteration. Here, the ∥xk+1−xk∥ ∥xk∥ determines the change of the solution after one iteration. If this change is...
[15]

In doing so, it is possible to feed this solution to the initial ZFP of (23)

Verification of the Solution:It is important to verify if the solutionx sol of Algorithm 1 is close enough to the true solution. In doing so, it is possible to feed this solution to the initial ZFP of (23). In an ideal scenario, the solution must be the zero vector of appropri...
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Single Spiking Neuron:A single spiking neuron requires two timescales (instantaneous and fast). The numerical values for a spiking neuron are directly extracted from [7] and the model is CDv+v−2 tanh(v) + 2 tanh( 1 50 D + Idv)−i ext = 0(37) where againDis the differentiation o...
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OperatorEisρ-strongly monotone onD
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OperatorTdefined in (26) satisfiesT(D)⊆D
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The conditionγ > β+ 1 + 2 ϵ holds on the subsetDfor operator pairsF i andG i
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In fact, from early on, the existence of an event at around t= 6000mscan be seen and, the later iterations only refine the details. 0 2000 4000 6000 8000 10000 12000 Time [ms] -3 -2 -1 0 1 2 3 Voltage [v] Iteration 50 Iteration 300 Iteration 6400 NI Solver 0 2000 4000 6000 800...

2000

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

This is the result of theBanach fixed-point Theorem

FixTis non-empty andTis a contractive mapping. This is the result of theBanach fixed-point Theorem

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[2] [2]

neuron” block is exactly the dynamics presented in the left- hand side of (9) that describes the internal dynamics of the neuron. For thek th neuron, the internal “neuron

FixTis non-empty andTis an averaged operator. Then, the iterations of (8) are known as theKrasnosel’ski ˘ı- Manniteration. B. Model Definition The model chosen to represent neurons determines the level of conformity with neurophysiology, hardware implementabil- ity and computational complexity. This work adopts the spik- ing neuron model from study [22]. ...

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[3] [3]

Among these properties,mono- tonicityholds a special place

Monotonicity—The Property of Interest:Given the struc- ture and properties of the operatorW(·), different methods can be used to solve (14). Among these properties,mono- tonicityholds a special place. This property bridges physical interpretability with computational tractability. From a phys- ical perspective, it indicates the incremental passivity of th...

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[4] [4]

The justification for the breakdown of monotonicity is threefold

Beyond Simple Monotone Problems:The dynamics of the spiking neuron (10) and network (12) can never be monotone. The justification for the breakdown of monotonicity is threefold. 1)Nature of Excitability: Excitability and spiking behav- ior is grounded in the interaction of active elements at fast timescales with dissipative elements at slower timescales [...

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[5] [5]

To further exploit the structure of our problem and simplify the proof of convergence, further assumptions are put on the operators

Consensus-based Difference of Monotone Douglas- Rachford Algorithm:The problem of interest has the formu- lation E(x) + pX i=1 (Fi(x)−G i(x)) = 0(23) whereE,F i and,G i are monotone operators. To further exploit the structure of our problem and simplify the proof of convergence, further assumptions are put on the operators. The signal space of (23) is con...

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[6] [7]

OperatorF i isγ-strongly monotone onDfor alli∈I

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[7] [8]

OperatorG i satisfies (β+ 1 ϵ )∥y−ˆy∥ 2 ≥ ⟨Gi y−G i ˆy, y−ˆy⟩ ≥β∥y−ˆy∥2 (25) AUTHORet al.: TITLE 7 for ally,ˆy∈ Dandi∈I

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[8] [9]

Then, the fixed-point iteration zk+1 =T(z k)(27) converges to a solution of (23), if such a solution exists

The conditionγ > β+ 1 + 2 ϵ holds on the subsetD 5)T:L 2 →L 2 defined by T(·) = Id(·)−J αE(PC(·)) + JpαF(2 JαE(PC(·))−Id(·) +pαG(J αE(PC(·))) (26) satisfiesT(D)⊆D, whereD= L i∈I Dis the lifted subset. Then, the fixed-point iteration zk+1 =T(z k)(27) converges to a solution of (23), if such a solution exists. Proof:: The proof of convergence for this FPI a...

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