Spiking Neuron Excitability Dynamics

• Spiking neuron excitability is the property by which neurons remain quiescent under subthreshold stimuli yet produce all-or-nothing action potentials when a critical threshold is crossed.
• It is characterized mathematically through bifurcation mechanisms such as saddle-node on invariant circle and Hopf, differentiating Type I and Type II excitable behaviors.
• This dynamic property underlies efficient event-based processing in biological neural circuits as well as in synthetic systems including neuromorphic and photonic devices.

Spiking neuron excitability is the dynamical property by which a neuron responds quiescently to sub-threshold stimuli but generates an all-or-nothing action potential (“spike”) whenever its input crosses a well-defined threshold in stimulus space or state space. This concept, pioneered in neurobiology and now fundamental to neuromorphic, photonic, and electronic circuits, underlies all event-based information processing in spiking systems. Excitability is formalized both at the circuit (input–output) level in terms of well-defined dynamic thresholds, and at the dynamical systems level in terms of bifurcation structures, manifolds, and separatrix geometry that distinguish the resting and spiking responses.

1. Mathematical and Dynamical Characterization of Excitability

Excitability is canonically formalized in two complementary frameworks: the behavioral (input–output) view and the dynamical-systems (phase-space) view. In the behavioral perspective, an excitable system is defined by the input-output set of current-voltage trajectories $\{I(t), V(t)\}$ such that subthreshold stimuli elicit only small perturbations, but stimuli above a threshold $(A^*, \sigma^*)$ in amplitude and duration induce a large, stereotyped voltage excursion (spike). The resulting operation is characterized by the locus of threshold curves in the $(A, \sigma)$ plane, with a sharp transition separating the no-spike and single-spike regimes (Sepulchre et al., 2017).

Dynamically, excitability arises near bifurcations that separate a unique stable equilibrium (resting state) from oscillatory (spiking) attractors. For single neurons, two principal mechanisms are:

• Type I (Class I) excitability: Onset of spiking via a saddle-node on invariant circle (SNIC) or homoclinic bifurcation. Here, as the input (e.g., applied current $I$) reaches threshold, the limit-cycle period diverges ($T \to \infty$), and arbitrarily low firing rates emerge—classic zero-frequency onset.
• Type II (Class II) excitability: Onset via a (subcritical or supercritical) Hopf bifurcation. Spiking begins at a finite, nonzero minimum frequency; the threshold manifold is less sharply defined and subthreshold oscillations are prominent.

The mathematical signatures are seen in canonical reduced models such as the FitzHugh–Nagumo (FHN) and Morris–Lecar equations. The FHN, in its mirrored/generalized form, can be unfolded to reveal five distinct types of excitability, including hybrid cases with bistability or spike latency, critically shaped by the presence or absence of cooperative gating variables (Franci et al., 2012, Broek et al., 2020).

2. Spike Initiation, Threshold Manifolds, and Refractory Dynamics

The onset of an action potential is governed by the system’s threshold manifold (separatrix), which divides initial conditions returning to rest from those that undertake a large “excitable excursion.” In type I systems, the manifold is set by the unstable manifold of a saddle or saddle-focus. The spike-initiation process is determined by:

• Voltage or state-space separatrix: For each system, a well-localized manifold (often 1D in 2D models) separates subthreshold return from supra-threshold spiking trajectories (Yelo-Sarrión et al., 2022, Coomans et al., 2011).
• Intrinsic threshold criteria: In engineered CMOS or SNN systems, both state thresholds (e.g., critical membrane voltage $V_{th}$) and energy/charge thresholds (e.g., $Q_c$) can be extracted; only the former is input-independent and intrinsic to the neuron (Brandt et al., 16 Nov 2025).
• Refractory period characterization: After the spike, system variables require a recovery period before a new spike can be evoked. Mechanistically, this is often realized by the slow regulator dynamics (e.g., potassium activation, threshold adaptation, or thermal slow-timescales in photonic systems) (Li et al., 22 Sep 2025, Xiang et al., 2019).

Dynamical systems positioned just beyond the destruction of a limit cycle via homoclinic bifurcation exhibit classic excitable dynamics: logarithmic divergence of the limit-cycle period near threshold, sharp threshold manifolds, and singular relaxation trajectories back to rest (Yelo-Sarrión et al., 2022).

3. Canonical Models Across Modalities: Biological, Electronic, and Photonic Realizations

Excitability is a modality-independent property, appearing in diverse physical instantiations:

• Biological neurons: Hodgkin–Huxley, Morris–Lecar, and FitzHugh–Nagumo models produce both type I (SNIC) and type II (Hopf) excitability, governed by the interplay of fast-positive (e.g., $Na^+$) and slow-negative (e.g., $K^+$) feedback. The critical role of the slow conductance $g_s$ is explicit; experimental modulation of $g_s$ moves neurons between types (Broek et al., 2020).
• SNN architectures: In neuromorphic computing, variants of integrate-and-fire and LIF neurons abstract the excitable spike mechanism. Extension to RPLIF (refractory period LIF) incorporates spike-triggered, multiplicative threshold adaptation, strictly enforcing forbidden intervals (absolute/relative refractory) and improving coding efficiency and noise robustness (Li et al., 22 Sep 2025).
• CMOS analog neurons: Excitability in subthreshold-analog silicon neurons is defined by two equivalent criteria: a critical supplied charge (stimulus-dependent) and a membrane-potential threshold (state-space intrinsic). Only the latter unequivocally reflects the neuron’s nonlinear dynamics, associated with a saddle-node bifurcation and a robust separatrix (Brandt et al., 16 Nov 2025).
• Photonic neurons: Diverse photonic structures—including Bose–Hubbard dimers, RTD neurons, asymmetric ring lasers, and passive microresonators—display all strict signatures of excitability: threshold manifolds, all-or-nothing spike excursions, refractory-limited recovery, and coherence resonance under noise. Their phase-space topology (folded separatrix, cubic nullclines) closely imitates 2D biological neuron models (Yelo-Sarrión et al., 2022, Owen-Newns et al., 28 Jul 2025, Coomans et al., 2011, Xiang et al., 2019).

The universality of the excitable bifurcation and the geometric structure of the separatrix (e.g., folded manifold in SRLs and high-Q microresonators) enable deterministic multi-spike, refractory, and bursting behaviors in both biological and synthetic hardware (Coomans et al., 2011, Razvan et al., 7 Jan 2024).

4. Stochastic Excitability and Coherence Resonance

Realistic neuron and circuit implementations contend with various sources of intrinsic and extrinsic noise. Additive white Gaussian noise, whether from synaptic input or device-level fluctuations, can induce random spike emission below deterministic threshold, broadening interspike-interval distributions.

• Stochastic threshold crossing: Even subthreshold stimuli, when combined with noise, may drive the system across the separatrix.
• Coherence resonance: There exists an optimal noise intensity at which spike emission becomes most regular, minimizing the coefficient of variation $R$ of the interspike interval. Both electronic (FHN circuits), photonic (Bose–Hubbard dimers), and biological neurons display this stochastic resonance phenomenon (Yelo-Sarrión et al., 2022, Medeiros et al., 2011).
• Poisson statistics and dynamic range: In the low-noise regime, spike trains are Poisson-like; mean firing rate curves vs. stimulus intensity are sigmoidal with dynamic range ($\sim$6–10 dB) comparable to biological sensory neurons (Medeiros et al., 2011).

The combination of deterministic and stochastic excitability defines the operational robustness and coding capacity of both biological and artificial spiking systems.

5. Excitability at the Network Level: Population and Cascadability Effects

At the SNN or population level, excitability governs the network’s ability to amplify, encode, and temporally process incoming information:

• Dynamic range maximization: Balanced excitatory/inhibitory (E/I) regimes optimize the network’s dynamic range $\Delta$, quantifiable as the monotonic range of stimuli eliciting nonsaturating but robust responses. Maximal $\Delta$ arises for E/I ratios $\sim$4:1 and moderate connection sparseness ($p \sim 0.02$), enabling robust coding over several log-units of input (Seyed-allaei, 2013).
• Parameter-phase diagrams: Systematic mapping of network excitability (as dynamic range, maximal rate, or oscillatory regime) in the parameter space of E/I ratios, sparseness, and synaptic weights reveals wide robustness plateaus and sharp transitions into oscillatory or quiescent regimes.
• Cascadability and signal propagation: In photonic architectures, excitability entails that the output “spike” from one neuron reliably triggers spike emission in downstream units, contingent on the relative placement of thresholds and refractoriness, ensuring robust multilayer propagation (Xiang et al., 2019).

Population-level excitability is thus an emergent, but formally quantifiable, property depending critically on the interplay of single-neuron dynamics and network connectivity.

6. Excitability Types, Bifurcation Structures, and Extensions

Generalized planar neuron models identify a taxonomy of excitability types, including both classical and hybrid classes, organized by codimension-3 unfoldings such as pitchfork bifurcations. Cooperative gating variables (e.g., $Ca^{2+}$ activation) introduce mirrored nullcline branches, yielding novel dynamical features:

• Type I: SNIC/homoclinic, arbitrarily low-frequency onset, robust threshold.
• Type II: Hopf, finite-frequency onset, less sharply defined threshold.
• Type III–V: Slope detectors, saddle–homoclinic cycles, bistable plateau potentials and pronounced latency, especially in systems with cooperative or mixed gating.

The critical parameter for switching between types is the slow conductance $g_s$, measurable by dynamic input conductance protocols; tuning $g_s$ enables experimental or synthetic modulation of excitability class (Broek et al., 2020, Franci et al., 2012).

7. Functional Implications and Implementation-Specific Considerations

The excitability property is fundamental for fault-tolerance, event-based computation efficiency, and temporal coding in both biological and artificial spiking systems:

• Noise and over-excitation protection: Spike-triggered threshold adaptation (e.g., RPLIF) enforces interspike intervals, suppresses noise-driven chattering, and contributes to energy efficiency (Li et al., 22 Sep 2025).
• Temporal precision and coding: Dynamic modification of threshold or refractoriness tunes temporal sensitivity and selectivity, facilitating the detection of salient temporal features and minimizing redundancy.
• Hardware constraints: In solid-state neurons, noise introduces threshold jitter, and intrinsic device dynamics (e.g., negative-differential-resistance in RTDs, slow thermal relaxation in photonic microresonators) set operational speed and power consumption limits while preserving classical excitability features (Brandt et al., 16 Nov 2025, Owen-Newns et al., 28 Jul 2025).

In summary, excitability emerges from a confluence of fast-positive and slow-negative feedbacks in neuronal and synthetic devices, is transduced via clear threshold geometry and bifurcation structure, and is exploited at both the single-neuron and network level for robust, efficient, temporally precise event coding (Sepulchre et al., 2017, Yelo-Sarrión et al., 2022, Owen-Newns et al., 28 Jul 2025, Li et al., 22 Sep 2025, Broek et al., 2020, Franci et al., 2012, Xiang et al., 2019, Brandt et al., 16 Nov 2025, Medeiros et al., 2011, Razvan et al., 7 Jan 2024, Seyed-allaei, 2013, Coomans et al., 2011).