Threshold-Based Firing Mechanism

• Threshold-based firing mechanisms are systems where a state variable, such as membrane potential, triggers a spike emission when it crosses a preset threshold, fundamental to neural computation.
• The models include deterministic and stochastic formulations, with rigorous mathematical treatment ensuring well-posedness by approximating hard thresholds with steep sigmoid functions.
• Adaptive and dynamic thresholds, as used in signal processing and neuromorphic hardware, facilitate robust encoding, synchronization, and energy-efficient computation in both biological and artificial networks.

A threshold-based firing mechanism refers to the class of systems—biological neurons, artificial neurons in spiking neural networks, and event-based signal processing elements—whose output transitions (typically spike emission) are triggered when a state variable (e.g., membrane potential, integrated input, population activity, etc.) crosses a defined threshold. The threshold may be fixed or dynamically adaptive, continuous or discrete, deterministic or stochastic. This mechanism is foundational for spike-based computation in biological systems and is widely implemented and analyzed in theoretical neuroscience, neuromorphic hardware, signal processing, and mathematical models of excitable dynamics.

1. Canonical Threshold Mechanisms in Point-Neuron Models

The archetypal threshold-based firing model is the (leaky) integrate-and-fire (LIF) neuron, where a continuous-time membrane potential variable $V(t)$ accumulates input until it reaches a firing threshold $\Theta$, at which point a spike is emitted, $V$ is reset, and possibly clamped during a refractory period. In the deterministic LIF,

$$\tau_m \frac{dV}{dt} = - [V - V_\mathrm{rest}] + I_\mathrm{in}(t), \quad \text{spike if}~ V(t) \geq \Theta,$$

with reset $V \to V_\mathrm{reset}$ after spiking (Xu, 2018, Kreutz-Delgado, 2015). Variants include (a) stochastic thresholds (e.g., $\Theta(t)$ as an Ornstein-Uhlenbeck process), (b) shot-noise driven inputs, and (c) network generalizations (Braun et al., 2015, Xu, 2018).

In firing-rate models, the Heaviside step function $H(x)$ or its steep sigmoid approximations $S_\beta(x) = \frac12 (1 + \tanh(\beta x))$ are used to model population or single-neuron firing as an instantaneous stochastic process (Nielsen, 2017, Radulescu, 2010). The sharpness parameter $\beta$ allows for continuous regularization, yielding a family of well-posed ODEs whose limiting case is the discontinuous, "all-or-none" idealization.

2. Mathematical Formulation, Regularization, and Well-posedness

A central challenge with hard threshold functions such as the Heaviside $H(x)$ is ill-posedness: ODEs with $H(x)$ in the right-hand side can lose uniqueness and continuity with respect to initial conditions, as the system's vector field is discontinuous at threshold (Nielsen, 2017). For a network with $N$ neurons,

$$\tau_i \frac{du_i}{dt} = -u_i + \sum_{j=1}^N \omega_{ij} S_\beta(u_j - u_\theta) + q_i(t),$$

the limit $\beta \to \infty$ recovers the hard threshold $H(x)$, but it may yield multiple solutions, with discontinuity in the initial-value map. To ensure existence and uniqueness (well-posedness), one uses large but finite $\beta$, guaranteeing globally Lipschitz right-hand sides (Picard–Lindelöf theorem) and robust numerical integration. Convergence to the discontinuous limit is only guaranteed when solutions spend zero Lebesgue measure time exactly at the threshold—the "threshold-simple" condition (Nielsen, 2017).

3. Adaptive, Stochastic, and Dynamic Thresholds

Thresholds in biological and artificial settings are in general not static. Mechanisms of adaptation and modulation include:

• Homeostatic and activity-driven adjustment: In SNNs, thresholds can be updated using activity proxies (e.g., low-pass filtered spike counts) to keep firing rates within target bands (Nomura et al., 2024). For neuron $i$, with calcium trace $C_i(t)$,

  $$\tau_{IP} \frac{dC_i}{dt} = -C_i + \sum_{\text{spikes}} \delta(t - t_f^{(i)}),$$

  $$\theta_i^f \to \theta_i^f \pm \mathrm{LR}_{thr}, \quad \text{if}~ C_i \notin [(1-o/2) C_{IP}, (1+o/2) C_{IP}].$$

• Time- and rate-dependent modulation: The BDETT model combines energy-based and temporal terms, with thresholds made functions of the local average membrane potential and the recent depolarization rate

  $$\theta_i(t+1) = \frac{1}{2} \left( E_i(t) + T_i(t+1) \right),$$

  where $E_i(t)$ depends positively on the deviation from the mean potential, and $T_i(t+1)$ depends negatively on the rate of depolarization (Ding et al., 2022).

• Stochastic threshold models: The firing threshold itself is a stochastic process, frequently modeled as an OU process

  $$d\theta = -\gamma(\theta - \theta_0) dt + \varepsilon \sqrt{D} dW(t),$$

  with reset at each spike (Braun et al., 2015). This introduces nontrivial correlations, variability, and phenomena such as inverse stochastic resonance.

These mechanisms establish threshold adaptation as a critical element for maintaining robust, homeostatic firing and enabling effective coding in fluctuating or varying environments (Nomura et al., 2024, Girardi-Schappo et al., 4 Sep 2025, Ding et al., 2022).

4. Synaptic Plasticity and the Role of Synchronized Thresholds

Threshold-based firing is deeply linked to both intrinsic and synaptic plasticity. In hardware-oriented spiking networks, stepwise, event-driven rules synchronize firing (intrinsic) and synaptic thresholds:

• Firing threshold $\theta^f$ adapts according to spike-driven activity metrics.
• Learning thresholds $\theta^{up}$ and $\theta^{down}$, governing potentiation/depression in synaptic weight updates, are updated in lockstep with $\theta^f$ (Nomura et al., 2024).

This unified adjustment preserves the necessary inequalities ($\theta^{down} < \theta^{up} < \theta^f$) for stable event-driven operation, facilitates binary quantization (single-bit threshold/weight hardware), and results in minimal hardware complexity for neuromorphic implementation. Such synchronization is essential for robust anomaly detection and accurate temporal credit assignment in recurrent networks.

5. Network and Population Phenomena: Criticality and Coding

In coupled networks, threshold adaptation determines population-level firing patterns and coding regimes. Around dynamical phase transitions (e.g., between quiescent, critical, and active phases), self-suppressive threshold feedback induces optimized dual coding: strong inputs are rate coded (mean firing), while weak inputs are represented in pattern (variance) coding (Girardi-Schappo et al., 4 Sep 2025). The adapted model yields:

• Firing probability and mutual information metrics that are optimized not only exactly at criticality (as in non-adaptive networks) but over a wide range of coupling strengths when threshold recovery times match experimentally observed timescales (100–1000 ms).
• Homeostasis via $\theta$ ensures the network resides near criticality, maximizing susceptibility and pattern entropy for input representations.

Table: Dual Coding Metrics and Threshold Adaptation (based on (Girardi-Schappo et al., 4 Sep 2025))

Coding Metric	Adapted Network (large $\tau$)	Nonadaptive Network
Rate Coding $A_r$	High and $J$-independent	Maximized only at $J=J_c$
Pattern Coding $H$	Peaks at biologically observed $\tau$	Flat (or zero) except at $J=J_c$
Mutual Info $I$	Robust, $\tau$-peaked	Sharp maximum at $J=J_c$

This establishes a mechanistic link between threshold recovery/adaptation and functional encoding capacity in both biological and artificial systems.

6. Threshold Mechanisms in Signal Processing and Neuromorphic Engineering

Threshold-based firing is mathematically formalized in event-based signal sampling (Send-on-Delta, SOD) and in the IF model with reset-to-mod, both of which select minimal sufficient spike representations subject to Alexiewicz-norm (integral supremum) constraints (Moser et al., 20 Jan 2025). Specifically,

$$\dot{V}(t) = f(t), \text{ emit spike when } |V(t^-)| \geq \Theta, \ V(t^+) \leftarrow V(t^-) - q_\Theta(V(t^-)).$$

These models have quantifiable reconstruction errors ($\|f - \mathrm{IF}_\Theta^M(f)\|_A < \Theta$) and provably maximize sparsity (minimum $\ell_1$-norm) within fidelity balls. Practical hardware implementations exploit event-driven, sparse, threshold-based processing for energy-efficient neuromorphic computation (Nomura et al., 2024, Moser et al., 20 Jan 2025).

7. Dynamical Systems Perspective: Bifurcation, Synchronization, and Population Oscillations

Thresholds correspond to bifurcation points in reduced neuron and population dynamics, notably saddle-node on invariant circle and Hopf bifurcations. For example, the onset of spiking in the artificial axon and canonical neuron models occurs at a saddle-node—the critical point of excitability (Pi et al., 2020). The threshold can exhibit a well-defined scaling law:

$$\tau \propto (V_\mathrm{clamp} - V_\mathrm{crit})^{-1/2},$$

signifying critical slowing.

In firing-rate models, the emergence of tonic versus burst firing in populations is governed by folds (saddle-node bifurcations) of the fast subsystem, with threshold surfaces (e.g., in adaptation) predicting transitions between dynamical regimes (Radulescu, 2010). In QIF networks, the spike-synchronization term (bilinear in population mean voltage and firing rate) arises precisely because threshold crossing by many neurons in synchrony leads to coherent oscillations (as in $\gamma$-band rhythms), which Wilson-Cowan type models cannot capture (Devalle et al., 2017).

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Threshold-based firing mechanisms thus provide both a minimal and universal framework for describing and analyzing excitable dynamics, from subcellular to population and network scales. The mathematical (ODE, SDE, and Fokker-Planck), computational (event-driven simulation, hardware realization), and theoretical (bifurcation, coding, criticality) analyses collectively delineate the profound role of dynamic thresholding in neural systems and neuromorphic devices.