Dynamic Threshold Curve (DTC) Analysis

• Dynamic Threshold Curve (DTC) is a family of threshold constructions where the decision boundary evolves dynamically with state, time, and contextual parameters.
• It is applied across domains such as excitable systems, MOSFET logic, cognitive radio, deep metric learning, and epidemic modeling to delineate shifting regimes.
• Analytical approaches including bifurcation theory, spectral analysis, and recursive dynamics are used to optimize performance and predict transitions in systems with DTC.

Searching arXiv for recent and foundational papers on “Dynamic Threshold Curve” and related threshold concepts. Dynamic Threshold Curve (DTC) denotes a threshold relation that changes with state, time, bias, or environment rather than remaining fixed. In current arXiv usage, the term is explicit in periodically forced excitable systems, where the DTC is a time-dependent effective spike boundary (Rubin et al., 4 Oct 2025). Closely related constructions appear, often without the exact phrase, in Dynamic Threshold MOSFET and Variable Threshold MOSFET logic, where the effective device threshold varies with gate–body bias (Ragini et al., 2010); in cognitive-radio sensing, where the detection threshold is re-estimated from quiet-time noise observations (Salahdine et al., 2016); in piecewise recursive systems whose switching threshold co-evolves with the state (Valenti, 25 Jul 2025); in deep metric learning, where mining and loss thresholds are adapted during training (Jiang et al., 2024); in interconnected-network epidemics, where a threshold curve separates epidemic and non-epidemic regimes in parameter space (Das et al., 2023); and in threshold-based robotic decision dynamics, where switching is governed by bifurcation curves adaptive to physical and environmental constraints (Amorim et al., 2023). Across these settings, a DTC is not a single standardized object but a family of mathematically related threshold constructions.

1. Conceptual scope and terminology

The most stable common feature of a DTC is that the threshold is not treated as a constant scalar. Instead, it is updated or reinterpreted as a function of other variables: terminal voltages in MOS devices, noise realizations and SNR in signal detection, state-threshold coupling in recursive dynamics, batch statistics and meta-gradients in deep metric learning, infection strengths and topology in epidemic spreading, or bifurcation parameters in decision-making dynamics (Ragini et al., 2010, Salahdine et al., 2016, Jiang et al., 2024, Das et al., 2023, Amorim et al., 2023).

The phrase itself is not uniformly standardized. In the VTMOS paper, the authors do not explicitly use “Dynamic Threshold Curve,” but the underlying concept is reconstructed from the dynamic adjustment of threshold voltage under gate–body tracking (Ragini et al., 2010). The cognitive-radio paper likewise does not name a DTC, yet its dynamic threshold estimation naturally induces a family of threshold-performance curves (Salahdine et al., 2016). By contrast, "Dynamic threshold curves and response precision in forced excitable systems" gives a direct formal definition of the DTC as the locus of minimally spike-inducing points along a subthreshold deterministic trajectory (Rubin et al., 4 Oct 2025).

Taken together, these works suggest three recurring geometries of a DTC. In one class, the curve is a state-space locus, as in excitable systems or moving switching manifolds. In a second class, it is a parameter-space phase boundary, as in epidemic threshold curves or bifurcation curves. In a third class, it is a time-varying threshold function or family of operating points, as in cognitive radio, VTMOS biasing, or adaptive deep metric learning. A common misconception is therefore to treat DTC as synonymous with a fixed threshold schedule; the literature instead ties it to feedback, co-evolution, or phase-dependent thresholding (Valenti, 25 Jul 2025, Rubin et al., 4 Oct 2025).

2. Mathematical constructions

The most explicit formalization appears in periodically forced excitable systems. For a non-autonomous system

$\dot{x}(t)=F(x(t),I(t)), \qquad x\in\mathbb{R}^d,$

with spike section $\{x_1=0\}$, deterministic subthreshold trajectory $X(t;0,x_0)$, and perturbation direction $p$, the dynamic threshold function is

$$\kappa(t)=\inf\left\{k>0:\sup_{s\ge t}x_1\big(s;t,X(t;0,x_0)+kp\big)\ge 0\right\},$$

and the DTC is

$\Phi(t)=X(t;0,x_0)+\kappa(t)p.$

Here $\kappa(t)$ is the minimal instantaneous kick that will eventually cause a spike, and $\Phi(t)$ is the corresponding threshold locus in state space (Rubin et al., 4 Oct 2025). This definition makes the threshold phase-dependent and future-dependent; it is not the static threshold associated with the instantaneous input value.

In discrete-time threshold dynamics, the threshold itself becomes a state variable. The paper on piecewise recursive sequences studies

$$a_{n+1}= \begin{cases} f(a_n), & a_n\le c_n,\ g(a_n), & a_n>c_n, \end{cases} \qquad c_{n+1}=h(a_n,c_n),$$

so the switching threshold $c_n$ co-evolves with the primary state $\{x_1=0\}$0 (Valenti, 25 Jul 2025). In this setting, the line $\{x_1=0\}$1 is the switching manifold, while the threshold sequence $\{x_1=0\}$2 acts as a moving threshold. The Common Limit Theorem shows that if both regimes are visited infinitely often and both sequences converge, then state and threshold converge to the same limit (Valenti, 25 Jul 2025).

In detection theory, the threshold is adapted directly from data. For matched-filter detection in cognitive radio, the paper defines the matched-filter statistic

$\{x_1=0\}$3

uses quiet-time noise-only observations to estimate

$\{x_1=0\}$4

and applies the dynamic threshold

$\{x_1=0\}$5

with threshold factors $\{x_1=0\}$6 (Salahdine et al., 2016). The threshold thus changes with the current noise projection onto the pilot sequence.

In epidemic spreading on interconnected networks, the threshold curve is defined spectrally. With

$\{x_1=0\}$7

the epidemic threshold condition is $\{x_1=0\}$8, and after reduction one obtains

$\{x_1=0\}$9

The normalized epidemic threshold $X(t;0,x_0)$0 can then be plotted against $X(t;0,x_0)$1, producing the epidemic threshold curve (Das et al., 2023).

In adaptive decision dynamics, the threshold is realized as a bifurcation boundary rather than a direct comparator. For the supercritical pitchfork normal form

$X(t;0,x_0)$2

the saddle-node thresholds are

$X(t;0,x_0)$3

which define the critical bias values at which commitment to one task disappears and the system must switch to the other (Amorim et al., 2023).

3. Domain-specific realizations

The following summary organizes the principal DTC realizations documented in the cited literature.

Domain	Threshold object	Dynamic dependence
DTMOS/VTMOS logic	Effective $X(t;0,x_0)$4 or switching threshold	Gate, body, and bias voltages
Cognitive radio	Sensing threshold $X(t;0,x_0)$5	Quiet-time noise estimate and factor $X(t;0,x_0)$6
Piecewise recursive systems	Switching threshold $X(t;0,x_0)$7	Coupled update $X(t;0,x_0)$8
Deep metric learning	Mining and loss thresholds	Pair ratios, batch statistics, meta-learning
Epidemics on networks	Threshold curve in parameter space	Infection strengths, topology, interconnection
Coupled decision/control	Switching bias thresholds $X(t;0,x_0)$9	Attention $p$0, physical gain $p$1, environment
Forced excitable systems	Dynamic threshold function $p$2 and curve $p$3	Input phase and future trajectory

In sub-threshold CMOS logic, the DTC is tied to body-bias modulation. In DTMOS, the substrate is directly tied to the gate, so the threshold voltage adjusts dynamically with gate voltage. In VTMOS, the substrate differs from the gate by a constant bias: positive $p$4 for NMOS and negative $p$5 for PMOS (Ragini et al., 2010). The paper describes this as an extension of DTMOS in which the body tracks the gate with an offset. Device-level $p$6–$p$7 families and inverter voltage transfer characteristics imply a tunable family of dynamic threshold curves parameterized by $p$8 and $p$9 (Ragini et al., 2010).

In deep metric learning, the DTC appears as adaptive mining and loss boundaries. The Asymmetric Sample Mining Strategy uses distinct thresholds for positives and negatives,

$$\kappa(t)=\inf\left\{k>0:\sup_{s\ge t}x_1\big(s;t,X(t;0,x_0)+kp\big)\ge 0\right\},$$0

and Adaptive Tolerance ASMS updates these thresholds according to mined pair imbalance through

$$\kappa(t)=\inf\left\{k>0:\sup_{s\ge t}x_1\big(s;t,X(t;0,x_0)+kp\big)\ge 0\right\},$$1

while the loss threshold $$\kappa(t)=\inf\left\{k>0:\sup_{s\ge t}x_1\big(s;t,X(t;0,x_0)+kp\big)\ge 0\right\},$$2 in Soft Contrastive Loss is updated by a meta-learning-based threshold generator (Jiang et al., 2024). The full method, Dual Dynamic Threshold Adjustment Strategy, reports competitive performance on CUB200, Cars196, and SOP, with full DDTAS reaching CUB200 $$\kappa(t)=\inf\left\{k>0:\sup_{s\ge t}x_1\big(s;t,X(t;0,x_0)+kp\big)\ge 0\right\},$$3, $$\kappa(t)=\inf\left\{k>0:\sup_{s\ge t}x_1\big(s;t,X(t;0,x_0)+kp\big)\ge 0\right\},$$4, Cars196 $$\kappa(t)=\inf\left\{k>0:\sup_{s\ge t}x_1\big(s;t,X(t;0,x_0)+kp\big)\ge 0\right\},$$5, $$\kappa(t)=\inf\left\{k>0:\sup_{s\ge t}x_1\big(s;t,X(t;0,x_0)+kp\big)\ge 0\right\},$$6, and SOP $$\kappa(t)=\inf\left\{k>0:\sup_{s\ge t}x_1\big(s;t,X(t;0,x_0)+kp\big)\ge 0\right\},$$7 (Jiang et al., 2024).

4. Dynamics, synchronization, and phase transitions

One major role of a DTC is to separate qualitatively different dynamical regimes. In epidemics on interconnected networks, the threshold curve partitions parameter space into epidemic and non-epidemic regions. The paper states that when normalized infection strengths are below the threshold curve, the infection is not spreading, whereas when the infection strengths are above the threshold curve, the infection is spreading, and that this is true for any level of interconnection (Das et al., 2023). In the spillover setting, a second threshold curve separates major and minor spillover regions in the space of inter-population link density and inter-network infection rate (Das et al., 2023).

In piecewise recursive dynamics, the DTC acts as a moving switching boundary whose interaction with the state can yield convergence, bistability, periodic orbits, spirals, and chaos in the affine case (Valenti, 25 Jul 2025). The threshold persistence parameter $$\kappa(t)=\inf\left\{k>0:\sup_{s\ge t}x_1\big(s;t,X(t;0,x_0)+kp\big)\ge 0\right\},$$8 is decisive for stability. The paper emphasizes that component-wise contraction may be insufficient: contracting regime maps do not guarantee convergence if the threshold map destabilizes the coupled two-dimensional system (Valenti, 25 Jul 2025). This shifts the DTC from a passive decision boundary to an active generator of dynamics.

In adaptive decision-making and robotic control, the DTC is the bifurcation structure governing switching between two tasks. The opinion–position dynamics couple a nonlinear opinion variable $$\kappa(t)=\inf\left\{k>0:\sup_{s\ge t}x_1\big(s;t,X(t;0,x_0)+kp\big)\ge 0\right\},$$9 to the physical state $\Phi(t)=X(t;0,x_0)+\kappa(t)p.$0, and task switching occurs when the bias $\Phi(t)=X(t;0,x_0)+\kappa(t)p.$1 crosses saddle-node thresholds $\Phi(t)=X(t;0,x_0)+\kappa(t)p.$2 or $\Phi(t)=X(t;0,x_0)+\kappa(t)p.$3 (Amorim et al., 2023). Because these thresholds depend on attention $\Phi(t)=X(t;0,x_0)+\kappa(t)p.$4, physical gain $\Phi(t)=X(t;0,x_0)+\kappa(t)p.$5, and environmental conditions, the effective decision threshold is adaptive rather than fixed. The paper further shows that increasing $\Phi(t)=X(t;0,x_0)+\kappa(t)p.$6 widens the bistable region in $\Phi(t)=X(t;0,x_0)+\kappa(t)p.$7, while numerical results indicate that increasing $\Phi(t)=X(t;0,x_0)+\kappa(t)p.$8 moves the saddle-node values toward the origin, shrinking that bistable region (Amorim et al., 2023).

The synchronization result in recursive systems gives a particularly strong structural statement. If both regimes are visited infinitely often and both the state and threshold sequences converge, then $\Phi(t)=X(t;0,x_0)+\kappa(t)p.$9 and $\kappa(t)$0 converge to the same value (Valenti, 25 Jul 2025). This implies that a DTC can govern persistent regime alternation, but if convergence still occurs, the threshold and the state cannot maintain a permanent asymptotic gap.

5. Performance, trade-offs, and interpretive use

Because a DTC modulates accessibility of events or regimes, it typically induces an explicit trade-off between sensitivity and selectivity. In matched-filter detection, increasing the threshold factor $\kappa(t)$1 reduces $\kappa(t)$2 but also reduces $\kappa(t)$3; for a fixed $\kappa(t)$4, increasing SNR increases $\kappa(t)$5 and decreases $\kappa(t)$6 (Salahdine et al., 2016). The paper therefore interprets its families of $\kappa(t)$7–SNR and $\kappa(t)$8–SNR curves as the operational manifestation of dynamic threshold selection. The quiet-time estimate makes the threshold track actual noise conditions more closely than a static design based on fixed $\kappa(t)$9 (Salahdine et al., 2016).

In VTMOS logic, shifting the dynamic threshold curve upward in effective $\Phi(t)$0 lowers both $\Phi(t)$1 and $\Phi(t)$2, reducing power at the expense of some delay (Ragini et al., 2010). At a 0.2 V supply and 100 kHz, average power reduces by about 54% when going from CMOS to VTMOS with $\Phi(t)$3, while the highest VTMOS delay is still almost equal to CMOS delay (Ragini et al., 2010). The paper also reports that above approximately 8 MHz, dynamic power dominates static power dissipation and there appears to be no advantage of VTMOS compared to CMOS circuits (Ragini et al., 2010). Here the DTC is a design knob for the power–delay trade-off.

In forced excitable systems, the DTC determines response precision. The spike-time distribution is well captured by the first passage time of a simple Gaussian stochastic process to the distance-to-threshold function $\Phi(t)$4, and the paper shows that peaks, troughs, and slopes of the DTC all convey fine information about spike timing (Rubin et al., 4 Oct 2025). Type 3 excitable cells produce deep, narrow, strongly skewed troughs in $\Phi(t)$5, leading to tight phase locking, whereas Type 2 cells exhibit shallower and broader troughs and therefore lower precision (Rubin et al., 4 Oct 2025). The histogram-based response precision index $\Phi(t)$6 is then linked to the geometry of the DTC rather than merely to local linearization around rest (Rubin et al., 4 Oct 2025).

In deep metric learning, dual threshold adaptation is used to regulate the balance of positive and negative pairs and to move the loss boundary during training (Jiang et al., 2024). This suggests that a DTC can also serve as a curriculum-like mechanism: early thresholds admit many informative pairs, whereas later thresholds concentrate training on the overlap region between positive and negative similarity distributions. The paper explicitly frames this as avoiding time-consuming grid search over static thresholds (Jiang et al., 2024).

6. Computation, limitations, and broader significance

Computation of a DTC depends on domain. In excitable systems, $\Phi(t)$7 is obtained by dichotomic search over kick amplitude $\Phi(t)$8, testing whether the perturbed trajectory eventually crosses the spike section; the DTC is then the image $\Phi(t)$9 over one forcing period (Rubin et al., 4 Oct 2025). In cognitive radio, the threshold is recomputed each sensing iteration from quiet-time matched-filter outputs and then scaled by $$a_{n+1}= \begin{cases} f(a_n), & a_n\le c_n,\ g(a_n), & a_n>c_n, \end{cases} \qquad c_{n+1}=h(a_n,c_n),$$0 (Salahdine et al., 2016). In epidemic models, the curve is traced by solving the spectral-radius condition $$a_{n+1}= \begin{cases} f(a_n), & a_n\le c_n,\ g(a_n), & a_n>c_n, \end{cases} \qquad c_{n+1}=h(a_n,c_n),$$1 while varying one infection parameter against another (Das et al., 2023). In adaptive decision-making, the threshold curve is computed from saddle-node conditions in the reduced scalar bifurcation problem $$a_{n+1}= \begin{cases} f(a_n), & a_n\le c_n,\ g(a_n), & a_n>c_n, \end{cases} \qquad c_{n+1}=h(a_n,c_n),$$2 (Amorim et al., 2023). In deep metric learning, part of the threshold update is closed-form from mined-pair statistics and part comes from a single-step meta-gradient update of $$a_{n+1}= \begin{cases} f(a_n), & a_n\le c_n,\ g(a_n), & a_n>c_n, \end{cases} \qquad c_{n+1}=h(a_n,c_n),$$3 (Jiang et al., 2024).

Several limitations recur. Matched-filter dynamic thresholds require prior knowledge of the pilot sequence and accurate quiet times; if the primary user is present during supposed quiet time, the threshold estimate is biased (Salahdine et al., 2016). VTMOS benefits are confined to sub-threshold operation and to bias voltages below the supply voltage; above approximately 8 MHz the leakage advantage is overwhelmed by dynamic power (Ragini et al., 2010). In recursive threshold dynamics, convergence of the threshold requires coordinated contraction of the full coupled system, not just the state component (Valenti, 25 Jul 2025). In excitable systems, the DTC framework is developed primarily for model-based settings, mostly in planar systems, and the first-passage approximation implicitly assumes a weak-noise regime (Rubin et al., 4 Oct 2025). In deep metric learning, the adaptive thresholds reduce but do not eliminate hyperparameter choices, since initial $$a_{n+1}= \begin{cases} f(a_n), & a_n\le c_n,\ g(a_n), & a_n>c_n, \end{cases} \qquad c_{n+1}=h(a_n,c_n),$$4 and the meta-step size remain design variables (Jiang et al., 2024).

The broader significance of the DTC concept lies in its unifying abstraction. It turns thresholding from a fixed decision rule into a dynamical object that can be analyzed geometrically, spectrally, statistically, or through bifurcation theory. Depending on context, the DTC may represent an effective device threshold, a moving sensing boundary, an adaptive mining rule, a switching manifold, a spectral phase boundary, or a phase-dependent spike threshold. The cited literature therefore does not yet yield a single universal theory of Dynamic Threshold Curves, but it does establish a coherent technical pattern: whenever threshold location must respond to evolving internal or external conditions, a DTC becomes the natural analytical representation (Ragini et al., 2010, Salahdine et al., 2016, Das et al., 2023, Amorim et al., 2023, Jiang et al., 2024, Valenti, 25 Jul 2025, Rubin et al., 4 Oct 2025).