Network-Optimised Spiking (NOS)

• NOS is a compact two-state spiking unit where the fast state v(t) encodes normalized congestion and the slow state u(t) represents recovery dynamics.
• It integrates mechanisms such as finite-buffer saturation, service-rate leak, graph-local delayed inputs, and differentiable resets to manage network traffic efficiently.
• NOS is applied across diverse domains—from 6G scheduling and intrusion detection to neural sampling and hardware co-design—ensuring stability and efficient event-driven optimization.

Network-Optimised Spiking (NOS) most specifically denotes the compact two-state spiking unit introduced for event-driven networking, in which a fast state $v(t)$ encodes a normalised queue or congestion proxy and a slow state $u(t)$ encodes a recovery or slowdown resource. In that formulation, NOS combines finite-buffer saturation, service-rate leak, graph-local delayed inputs, stochastic burst arrivals aligned with telemetry smoothing, and differentiable reset schemes, while admitting equilibrium, stability, and topology-aware analysis through the Perron eigenvalue of the coupling matrix (Bilal, 27 Sep 2025). In the broader literature represented here, the same label is also used, or retrospectively applied, to network-level optimisation of spiking dynamics in scheduling, intrusion detection, combinatorial optimisation, neural sampling, learning, and accelerator co-design. The term is therefore context-dependent rather than fully uniform across fields (Bilal et al., 13 Oct 2025, Bilal et al., 1 Jan 2026, Jonke et al., 2014).

1. Definitions and terminological scope

The principal and most concrete definition is the 2025 networking model, which introduces NOS as “a compact, two-state spiking unit whose variables, parameters, and inputs map directly to event-driven networking semantics: finite buffers, service, recovery, per-link delays and gates, and stochastic burst arrivals consistent with telemetry smoothing” (Bilal, 27 Sep 2025). In that sense, NOS is a specific dynamical system, not merely a design slogan.

The supplied literature also documents broader uses. In the neural-sampling framework of Jonke, Habenschuss, and Maass, the term “Network-Optimised Spiking (NOS)” is explicitly not used in the original paper; it is presented in the synthesis as a conceptual label for network-level design of energy landscapes over spike-defined states (Jonke et al., 2014). In the optical coherent Ising-machine work, NOS denotes co-design of network coupling and spiking dynamics for combinatorial search (Lu et al., 2022). In SpikeX, NOS refers to network–hardware co-optimisation that shapes spike statistics and accelerator dataflow jointly (Xu et al., 18 May 2025).

Source	Scope	NOS characterisation
(Bilal, 27 Sep 2025)	Event-driven networking	Compact two-state unit with queue, service, recovery, delays, and gates
(Bilal et al., 13 Oct 2025)	6G O-RAN scheduling	Bounded two-state kernel coupled to a proportional-fair grant head
(Bilal et al., 1 Jan 2026)	Streaming IDS at consumer gateways	Discrete-time per-flow unit with sigmoid event surrogate and WFQ gating
(Jonke et al., 2014)	Neural sampling and CSPs	Conceptual label for network-level energy-landscape design
(Xu et al., 18 May 2025)	SNN accelerator co-design	Joint optimisation of spike patterns and accelerator architecture

A persistent source of confusion is therefore terminological. In the networking papers, NOS is a concrete two-state model with explicit queue semantics; in several other works it is a broader optimisation paradigm for spiking systems. The supplied data support both usages, but they should not be conflated.

2. Core dynamical formulation for event-driven networking

In the networking formulation, NOS has two states per node. The fast state $v_i(t)$ is a dimensionless normalised queue or congestion proxy with $v\in[0,1]$, where $v=1$ corresponds to a full buffer. The slow state $u_i(t)$ is a dimensionless recovery or slowdown resource on the same time base as $v$. Time can be continuous, $t$ in seconds, or discretised into bins $\Delta t$, such as $5\,\mathrm{ms}$ (Bilal, 27 Sep 2025).

The continuous-time node dynamics are

$u(t)$0

$u(t)$1

The saturating nonlinearity

$u(t)$2

encodes convex backlog rise under increasing load while enforcing finite buffers. Its small-signal approximation is $u(t)$3 as $u(t)$4, and its global ceiling is $u(t)$5. The derivative

$u(t)$6

is bounded, and its maximum is

$u(t)$7

The remaining terms have direct queueing semantics. The parameter $u(t)$8 is a service leak on $u(t)$9, with $v_i(t)$0 in discrete implementations. The parameter $v_i(t)$1 is subthreshold damping to $v_i(t)$2, modelling EWMA-style queue relaxation and telemetry smoothing. The parameters $v_i(t)$3 and $v_i(t)$4 capture residual linear drift and offset. Recovery drag enters through $v_i(t)$5, while $v_i(t)$6 integrates recent congestion at rate $v_i(t)$7, sensitivity $v_i(t)$8, and passive decay $v_i(t)$9.

Graph-local input takes the form

$v\in[0,1]$0

with optional per-link occupancy gates

$v\in[0,1]$1

where typical gate choices are $v\in[0,1]$2 for $v\in[0,1]$3 or a logistic gate. This preserves causality through delays $v\in[0,1]$4, keeps attribution graph-local, and allows only congested links to attenuate (Bilal, 27 Sep 2025).

Exogenous arrivals follow a compound Poisson shot-noise model aligned with telemetry smoothing: $v\in[0,1]$5 with $v\in[0,1]$6. The moments and spectrum are

$v\in[0,1]$7

$v\in[0,1]$8

NOS supports two reset mechanisms. The first is an event-based exponential soft reset,

$v\in[0,1]$9

and the second is a continuous differentiable pullback,

$v=1$0

A homotopy on reset sharpness,

$v=1$1

with $v=1$2 and $v=1$3, is used to make training smooth early and event timing crisp later.

3. Equilibria, stability, topology, and spectral thresholds

The primary analysis of NOS is unusually explicit for a spiking networking model. Eliminating $v=1$4 at equilibrium yields

$v=1$5

and the subthreshold equilibrium condition

$v=1$6

A sufficient existence–uniqueness condition on $v=1$7 is

$v=1$8

with the conservative bound

$v=1$9

Operationally, this means that net drain must dominate the steepest admissible excitability slope (Bilal, 27 Sep 2025).

For a single node, the Jacobian is

$u_i(t)$0

with trace

$u_i(t)$1

and determinant

$u_i(t)$2

The Routh–Hurwitz conditions $u_i(t)$3 and $u_i(t)$4 give local asymptotic stability.

A distinctive result is that saturation enlarges the stable region. Since

$u_i(t)$5

increasing $u_i(t)$6 decreases the trace and increases the determinant for $u_i(t)$7, which expands the region $u_i(t)$8.

At network scale, linearisation about equilibrium gives the block Jacobian

$u_i(t)$9

Projecting homogeneous scaling $v$0 onto the Perron eigenpair yields the effective $v$1 Jacobian

$v$2

The critical coupling is

$v$3

Equivalently, with

$v$4

the stability proxy is

$v$5

This separates topology, through $v$6, from node physics, through $v$7 and $v$8. The paper also reports finite-size smoothing of synchrony onsets: coherence rises sharply near $v$9, the transition sharpens with $t$0, hub-dominated topologies track the Perron prediction closely, and chains show stronger resilience (Bilal, 27 Sep 2025).

A later O-RAN scheduler extends this line of analysis to delayed control by defining a delay-dependent threshold $t$1 and the spectral margin

$t$2

which compresses topology, controller gain, and delay into a single design parameter. Under light assumptions on arrivals, $t$3 yields geometric ergodicity and sub-Gaussian backlog and delay tail bounds with exponents proportional to $t$4 (Bilal et al., 13 Oct 2025).

4. Calibration, training, implementation, and empirical behaviour

NOS is designed to be calibrated directly from traffic statistics rather than only from labels. For light-load mean matching to $t$5, the service leak is set by $t$6, the input is mapped as $t$7, and a multiplicative output scale is fitted so that NOS predictions align with the analytic light-load mean $t$8 in the small-$t$9 regime. The reported run gives sample per-node output scales $\Delta t$0 and $\Delta t$1 (Bilal, 27 Sep 2025).

Under bursty input generated by MMPP, the bounded excitability and differentiable pullback truncate deep tails relative to $\Delta t$2: the CCDF decays more sharply, with fewer deep queues, lower tail latency, and less marking or dropping during ON phases. In zero-shot, label-free forecasting of congestion onsets using arrival-only calibration and known $\Delta t$3, $\Delta t$4, and $\Delta t$5, NOS reports

• AUROC $\Delta t$6,
• AUPRC $\Delta t$7 for top-10% bursts.

The same label-free setup reports the following comparators:

• Physics Fluid: AUROC $\Delta t$8, AUPRC $\Delta t$9, MAE $5\,\mathrm{ms}$0,
• MovingAvg: AUROC $5\,\mathrm{ms}$1, AUPRC $5\,\mathrm{ms}$2,
• TGNN-smooth: AUROC $5\,\mathrm{ms}$3, AUPRC $5\,\mathrm{ms}$4,
• LIF-leaky: AUROC $5\,\mathrm{ms}$5, AUPRC $5\,\mathrm{ms}$6.

The interpretation given is that bounded, event-driven dynamics align spikes to burst onsets while avoiding prolonged false positives between bursts (Bilal, 27 Sep 2025).

For supervised or train-calibrated forecasting, the reported setup uses chain, star, and scale-free topologies with $5\,\mathrm{ms}$7, per-link delays $5\,\mathrm{ms}$8 ms, and $5\,\mathrm{ms}$9 normalised to $u(t)$00 before scaling by $u(t)$01. The training protocol is residual-based and label-free at event-definition time: per-node standardisation on the train split, residual $u(t)$02-scores, event starts inferred by thresholding residuals, per-node thresholds chosen on validation but calibrated with train-only statistics, and a minimum episode duration to suppress one-bin blips. The optimiser is Adam, with gradient clipping by global norm in $u(t)$03, fast-sigmoid surrogate derivative

$u(t)$04

and reset homotopy from $u(t)$05 to $u(t)$06. Truncated BPTT is required to cover the dominant recovery time scale,

$u(t)$07

Across chain, star, and scale-free graphs, NOS is reported to yield the highest F1 with strong precision and improved recall, the lowest MAE and RMSE, and the earliest median start-latency relative to MLP, simple RNN, GRU, and adjacency-constrained temporal GNN. Exact F1 and latency tables are not reported, but the qualitative gains are stated to be consistent across topologies under the same label-free protocol (Bilal, 27 Sep 2025).

For deployment, the paper gives explicit stability checks and safe parameter ranges. The event-driven update cost is $u(t)$08 per node per tick for input accumulation, plus fixed-cost updates for $u(t)$09, $u(t)$10, and thresholding; memory stores delay buffers per incoming edge with integer ticks $u(t)$11. It also gives admissible per-bin ranges such as $u(t)$12, $u(t)$13, $u(t)$14, $u(t)$15, $u(t)$16, $u(t)$17, $u(t)$18, $u(t)$19, $u(t)$20, and $u(t)$21 (Bilal, 27 Sep 2025).

5. Derived networking systems and domain-specific instantiations

The most direct descendants of NOS in the supplied corpus are a 6G O-RAN scheduler and a consumer-gateway streaming IDS. Both preserve the bounded two-state kernel while changing the control surface and readout (Bilal et al., 13 Oct 2025, Bilal et al., 1 Jan 2026).

In the O-RAN scheduler, each bearer $u(t)$22 has backlog $u(t)$23, PRB fraction $u(t)$24, and per-PRB rate $u(t)$25, with service

$u(t)$26

Interference is encoded by a non-negative matrix $u(t)$27, clique feasibility is enforced by $u(t)$28, and the NOS kernel injects delayed neighbour spikes through $u(t)$29. The proportional-fair grant head smooths requests,

$u(t)$30

attenuates them by delayed neighbour activity,

$u(t)$31

and computes PF weights

$u(t)$32

Continuous per-clique allocations are then normalised and mapped to integer PRBs through floor, drop, renormalise, and water-fill steps.

The scheduler’s analysis defines the delay-dependent threshold $u(t)$33, the spectral margin $u(t)$34, and a sufficient test of $u(t)$35 for geometric ergodicity. Under that margin, backlog and delay tails satisfy sub-Gaussian bounds. The numerical study uses $u(t)$36, $u(t)$37, $u(t)$38, hence $u(t)$39, and examines pair2, line4, and ring8 topologies over $u(t)$40 ms. With a single gain fixed at worst $u(t)$41 and $u(t)$42 ms, NOS sustains the highest utilisation AUC, the smallest $u(t)$43th-percentile delay, and clique-feasible integer PRBs under a $u(t)$44 per-slot DU budget (Bilal et al., 13 Oct 2025).

NOS-Gate adapts the same design to metadata-only streaming intrusion detection on encrypted traffic. It instantiates a discrete-time NOS unit per directed five-tuple flow with window length $u(t)$45 ms. The evidence drive is

$u(t)$46

where $u(t)$47 is obtained by causal online $u(t)$48-scoring, and optional neighbour coupling is

$u(t)$49

Main results use $u(t)$50. The event surrogate is a bounded sigmoid

$u(t)$51

and the score is

$u(t)$52

with $u(t)$53 in the main experiments.

Actionability is determined by a $u(t)$54-of-$u(t)$55 persistence rule,

$u(t)$56

with defaults $u(t)$57, $u(t)$58. Thresholds are set label-free by burn-in quantiles,

$u(t)$59

using $u(t)$60, after which they are frozen for test. When $u(t)$61, NOS-Gate applies reversible weighted-fair-queueing mitigation by reducing the flow weight from $u(t)$62 to $u(t)$63, $u(t)$64. At an achieved $u(t)$65 false-positive operating point, NOS-Gate attains incident recall $u(t)$66 versus $u(t)$67 for the best baseline in those runs, reduces mean $u(t)$68 queueing delay by approximately $u(t)$69 ms and mean $u(t)$70 collateral delay by approximately $u(t)$71 ms across worlds, and has mean scoring cost of approximately $u(t)$72 per $u(t)$73 row on CPU (Bilal et al., 1 Jan 2026).

6. Broader interpretations across spiking research

Outside networking proper, the supplied literature uses NOS for several distinct but related ideas. In the neural-sampling framework of noisy spiking neurons, NOS is a later conceptual label for designing computations at the level of the stationary distribution $u(t)$74 over spike-defined states, using modular motifs such as WTA and OR circuits to add linearly to the energy. There, noise is a computational resource, precise spike timing rather than rate alone is central, and deterministic off-transitions create asymmetric dynamics that can cross high-energy barriers more readily than symmetric Gibbs sampling (Jonke et al., 2014).

In the optical coherent Ising-machine work, NOS denotes joint design of network coupling and spiking regimes for combinatorial optimisation. Each neuron is a coupled pair of a DOPO pulse and a dissipative pulse with antisymmetric intra-neuron coupling, and the digital measurement-and-feedback path applies

$u(t)$75

The stated rationale is that the saturating nonlinearity mitigates amplitude heterogeneity, while the gain, dissipation, and coupling schedules move the system across Hopf bifurcation boundaries to destabilise local minima. Empirically, the SNN-CIM is reported to approach best-known values on 800-node GSET Max-Cut instances and to outperform simulated annealing, conventional CIM, and CIM with sigmoid filtering on more irregular 2000-node instances (Lu et al., 2022).

In hardware and training work, NOS is again broadened. SpikeX defines it as shaping an SNN’s spatiotemporal spike activity and layer topology so the accelerator can reuse multi-bit synaptic weights across time and space, skip work on zeros, and sustain high array utilisation under unstructured sparsity; its joint network–hardware co-optimisation reports $u(t)$76 to $u(t)$77 reduction in energy–delay product without compromising model accuracy (Xu et al., 18 May 2025). Time-to-first-spike deep SNNs provide another related viewpoint: constant-slope TTFS mapping preserves the training trajectory equivalence with ReLU ANNs, supports global spike minimisation, and yields less than $u(t)$78 spikes per neuron while matching ANN performance on multiple datasets (Stanojevic et al., 2023). Information-theoretic training of stochastic spiking neurons frames NOS as joint optimisation of task likelihood, entropy or stability, and efficiency through explicit point-process objectives, while multi-objective genetic optimisation of cortical SNNs treats network connectivity as the search space for target excitatory and inhibitory firing regimes, often favouring sparse connectivity (Sinyavskiy, 2016, Fitzgerald et al., 2021).

Taken together, these uses show that NOS is not a single universal formalism across all spiking research. The networking model of (Bilal, 27 Sep 2025) is the most explicit and self-contained instantiation: a bounded two-state dynamical unit with queue semantics, differentiable resets, and topology-aware stability rules. The broader literature applies the same phrase, or an explicitly retrospective label, to network-level optimisation of spiking dynamics under other objectives: energy-landscape shaping, optimisation search, accelerator efficiency, statistical learning, or population-level activity control (Jonke et al., 2014, Xu et al., 18 May 2025). A plausible implication is that the common denominator is not one canonical set of equations, but the deliberate co-design of spiking dynamics with network structure, constraints, and operational objectives.