Time-Dependent ISI Distribution

• Time-dependent ISI distribution is the probability density of intervals between events in systems with time-varying statistics, central to both neuronal and communication models.
• Analytical methods, including Fokker–Planck equations, moment hierarchies, and eigenfunction expansions, capture the dynamic behaviors and feedback effects inherent in these systems.
• Practical applications span adaptive filtering in optical/RF communications to enhanced spike detection in neural networks, improving system performance through precise modeling.

A time-dependent inter-symbol interval (ISI) distribution quantifies the probability density of intervals between consecutive discrete events—such as spikes in neuronal models or transmitted symbols in communication channels—when the statistical properties of these intervals vary with respect to time or state. This concept underpins both the characterization of temporal correlations in stochastic systems and the development of optimal detection and decoding algorithms in non-stationary environments. Analytical frameworks for time-dependent ISI arise in neurodynamics, feedback-modulated spiking processes, nonlinear communication channels, and colored-noise decoding schemes.

1. Mathematical Definitions and Frameworks

Multiple domains implement time-dependent ISI distributions via domain-specific stochastic processes:

• Neuronal Population Density: For integrate-and-fire neurons under time-dependent input, the joint state distribution $p(V,s,t)$—with $V$ the membrane potential and $s$ the age since last spike—evolves according to a two-dimensional Fokker–Planck equation (Falorsi et al., 30 Dec 2025):

$$\partial_t p(V,s,t) = -\partial_V[a(V,t) p(V,s,t)] - \partial_s p(V,s,t) + D(t)\, \partial^2_{VV} p(V,s,t) + \nu(t)\, \delta(V-V_{\text{reset}})\, \delta(s)$$

The time-dependent ISI density is extracted as the threshold-crossing flux

$$f(t,s) = \frac{\phi(s,t)}{\nu(t)},\quad \text{where}\ \phi(s,t) := J(V_{\text{th}}, s, t)$$

• Thresholded Feedback Neuronal Models: For a threshold-two neuron under Poisson input and delayed excitatory feedback, the ISI-PDF $p(t)$ incorporates both the no-feedback ISI distribution $p^0(t)$ and instantaneous/delayed feedback mechanisms (Shchur et al., 2018):

$p(t) = \int_0^\Delta p(t\mid s) f(s) ds$

with explicit cases for $t < \Delta$, $t = \Delta$, and $t > \Delta$, and a nontrivial Dirac delta peak at the delay $t = \Delta$.

• Communication Channel ISI Noise: In WDM and dispersive channels, received symbols are modeled as linearly filtered sums with slowly time-varying ISI taps $h_k^{(n)}$ (Dar et al., 2013):

$y_n = a_n + \sum_k h_k^{(n)}\, a_{n-k}$

Taps $h_k^{(n)}$ themselves are stochastic processes with explicit autocorrelation and variance structure.

• ISI Channels With Colored Noise: In digital communications, post-equalization noise sequences exhibit time-dependent, autoregressive structure (e.g., AR(2) for RFView, AR(1) for dicode) with a joint Gaussian distribution parameterized by covariance matrices evolving in time (Duffy et al., 16 Oct 2025).

2. Statistical Properties and Distributions

Key statistical characterizations exhibit universality across domains:

• Gaussianity and Central Limit Approximation: In both optical channel ISI taps and decoded noise vectors post-equalization, time-dependent ISI processes often admit complex or real Gaussian approximations, with explicit zero means and temporal covariance structure. For WDM cross-phase modulation,

$$h_k^{(n)} \sim \mathcal{CN}(0, \sigma_k^2),\quad f_{h_k}(h) = \frac{1}{\pi \sigma_k^2} e^{-|h|^2/\sigma_k^2}$$

Variances $\sigma_k^2$ derive from link parameters, modulation, and pulse shapes (Dar et al., 2013).

• Autocorrelation and Temporal Evolution: Time-dependent ISI components typically exhibit autocorrelation decay over tens or hundreds of symbol intervals (WDM), or lag-1/lag-2 persistence under AR(1)/AR(2) noise models in digital communication. In neuronal models, persistence manifests as slow relaxation and modulated spectral response via age-moment hierarchies and Green's functions (Falorsi et al., 30 Dec 2025).
• Singular Features: In feedback-modulated neurons, the ISI-PDF displays Dirac delta peaks at feedback delays, indicating spike-time clustering and resonance effects (Shchur et al., 2018).

3. Analytical and Numerical Computation

Various analytical approaches are used to derive and compute time-dependent ISI distributions:

• Moment Hierarchies: In neuronal dynamics, moment hierarchies condense the evolution of ISI statistics to PDEs for age-moments $M_n(t)$, reducing computational complexity and enabling direct calculation of mean and variance (Falorsi et al., 30 Dec 2025).
• Eigenfunction and Green’s Function Expansion: Linear-response and perturbative methods yield frequency-domain kernels relating non-stationary input modulations to ISI spectral shifts, facilitated by spectral decomposition of governing linear operators (Falorsi et al., 30 Dec 2025).
• Convolution and Conditioning Integrals: The full ISI-PDF under delayed feedback is given in piecewise integral form, encoding explicit conditional distributions for pre-delay, delay, and post-delay regimes (Shchur et al., 2018).
• Autoregressive Noise Fitting: In ISI-decoding, covariance estimation via the Yule–Walker equations calibrates AR coefficients to empirical autocorrelations, yielding time-dependent conditional and joint Gaussian densities for colored noise (Duffy et al., 16 Oct 2025).

Domain	ISI Process Model	Distribution Example
Neuronal (Pop.)	FP equation for $p(V,s,t)$	Age-dependent flux $f(t,s)$
Feedback neuron	Piecewise with delay, Poisson input	Dirac at delay plus continuous PDF
WDM channel	Time-varying linear ISI filter	Zero-mean complex Gaussian tap
Communication	AR(1)/AR(2) colored noise, block model	Blockwise joint Gaussian noise

4. Time Dependence: Origins and Mechanisms

The time dependence in ISI distributions arises from multiple mechanistic sources:

• Input Modulation and Network State: Non-stationarity driven by time-varying drift, diffusion, or network oscillations directly modulates the population density and ISI output in neuronal systems (Falorsi et al., 30 Dec 2025).
• Feedback and Delay: Feedback loops with non-zero delay impose periodic clustering in ISI values, producing singularities and resetting effects linked to network architecture parameters (Shchur et al., 2018).
• Cross-Channel Nonlinear Interference: In WDM optical systems, slow evolution of cross-channel data symbols maps into slowly time-varying linear ISI coefficients, with time dependence essentially reflecting the underlying stochastic data streams (Dar et al., 2013).
• Impulse Response Variation: In radio frequency channels, empirical sampling of dispersive impulse responses (RFView) generates tap sequences whose noise coloring must be dynamically tracked and fit by autoregressive models (Duffy et al., 16 Oct 2025).

A plausible implication is that layered modeling—from underlying input statistics through nonlinear system dynamics to observable ISI output—can be systematically exploited for both statistical prediction and algorithmic adaptation.

5. Applications and Implications

Time-dependent ISI distribution formalism supports critical advances in modeling and signal processing:

• Adaptive Filtering and Cancellation: For WDM systems, expressing nonlinear interference as a time-varying linear ISI channel enables adaptive linear filtering for interference cancellation; tap statistics provide closed-form analytic performance bounds (Dar et al., 2013).
• Blockwise Decoding With Colored Noise: Retaining time-dependent colored noise structure via AR-model fitting, as implemented in ORBGRAND-AI, grants performance gains over interleaved schemes—information theory calculations and empirical simulations confirm 2–6 dB reductions in block error rate (Duffy et al., 16 Oct 2025).
• Non-stationary Network Phenomena: Population density Fokker–Planck approaches for neural ISI modeling deliver exact, time-resolved distributions under arbitrary network feedback, input modulation, and oscillatory dynamics, with agreement observed in large-scale spiking simulations (Falorsi et al., 30 Dec 2025).
• Resonant Spiking and Feedback Effects: Exact solutions for feedback-modulated ISI reveal spike-time resonance phenomena, non-Poissonian clustering, and explicit analytical relationships for ISI moments, parameterized by input rate and feedback delay (Shchur et al., 2018).

6. Parameter Dependence and Control

ISI distribution statistics are strongly modulated by system parameters:

• Variance and Correlation: In communication channels, tap variance scales with channel number, spacing, nonlinearity coefficients, and system length (Dar et al., 2013); AR parameter estimation is crucial where colored noise entropy reduction yields direct decoding gains (Duffy et al., 16 Oct 2025).
• Delay and Input Rate: In feedback neuronal models, the location and weight of delta peaks in ISI-PDF depend non-linearly on delay $\Delta$ and input rate $\lambda$, with critical regimes for resonance and smoothing (Shchur et al., 2018).
• Modulation and Drift: For neuronal populations, time-varying mean input and diffusion coefficients imprint directly onto time-dependent ISI dynamics, with analytic linear response available for weak modulations (Falorsi et al., 30 Dec 2025).

This suggests that explicit modeling of time dependence in ISI distributions enables precise control and optimization of both system behavior and algorithmic performance in stochastic, nonlinear, and feedback-rich environments.