isi-adaptive-threshold-neuronal-networks

name	isi-adaptive-threshold-neuronal-networks
description	First-passage-time analysis of inter-spike interval (ISI) statistics for excitatory-inhibitory (EI) integrate-and-fire neurons with depolarizing and hyperpolarizing adaptive thresholds. Use when studying stochastic neuronal firing, ISI variability, adaptive threshold mechanisms, or EI balance effects on spike-time statistics.
metadata	{"arxiv_id":"2607.18428","published":"2026-07-22","authors":["Oliver Gambrell","Abhyudai Singh"],"tags":["neuroscience","spiking-neurons","inter-spike-interval","adaptive-threshold","EI-balance","first-passage-time","stochastic-neuronal-dynamics"]}
license	Complete terms in LICENSE.txt

ISI Statistics in Adaptive-Threshold Neuronal Networks

A first-passage-time methodology for analyzing how depolarizing and hyperpolarizing adaptive thresholds shape the mean and variability of inter-spike intervals in excitatory-inhibitory (EI) integrate-and-fire neurons.

Core Idea

Extend the classical integrate-and-fire model by letting the threshold potential depend on recent presynaptic activity. This captures two biophysical effects:

• Depolarizing adaptive threshold: threshold rises after excitation (spike-frequency adaptation via sodium/calcium-activated potassium channels).
• Hyperpolarizing adaptive threshold: threshold falls after inhibition (post-inhibitory rebound via sodium channel recovery from inactivation).

The framework derives the ISI distribution via first-passage-time (FPT) analysis and quantifies ISI noise with the coefficient of variation (CV).

When to Use

• Modeling spike-time variability in EI circuits.
• Comparing fixed vs. adaptive threshold effects on firing regularity.
• Predicting post-inhibitory rebound spiking.
• Studying how excitatory/inhibitory input balance affects ISI statistics.

Key Findings

1. Depolarizing adaptive thresholds increase ISI noise relative to fixed thresholds at the same mean ISI.
2. ISI noise can be hypo- or hyper-exponential (CV < 1 or CV > 1) depending on excitatory and inhibitory input rates.
3. Hyperpolarizing adaptive thresholds can generate spikes driven purely by inhibition, producing post-inhibitory rebound firing.
4. Quantal content (QC) is modeled as binomial random variables, linking synaptic release stochasticity to postsynaptic firing statistics.

Methodology

1. Model the postsynaptic neuron as a leaky integrate-and-fire neuron receiving Poisson excitatory and inhibitory inputs.
2. Model quantal content of each presynaptic AP as independent binomial random variables with known mean and variance.
3. Fix a threshold potential for the baseline case; then extend to adaptive thresholds:
   • Depolarizing: threshold increases with membrane depolarization.
   • Hyperpolarizing: threshold decreases with membrane hyperpolarization.
4. Apply first-passage-time analysis to derive the mean ISI and the coefficient of variation.
5. Validate analytical predictions with stochastic simulations across input-rate parameter space.

Implementation Sketch

import numpy as np

# Simulate membrane potential with EI Poisson inputs and adaptive threshold
# V(t): membrane potential, tau: leak time constant
# dV/dt = -V/tau + sum_i qe_i * delta(t-te_i) - sum_j qi_j * delta(t-ti_j)
# threshold theta(t) adapts based on recent excitation or inhibition

# First-passage-time estimate: sample trajectories, record crossing times
# ISI = time between successive threshold crossings
# mean_ISI = np.mean(ISIs); CV = np.std(ISIs)/mean_ISI

Parameter Sensitivities

• High, balanced EI rates maximize ISI noise (CV²) in the fixed-threshold model.
• Depolarizing threshold systematically raises CV for matched mean ISI.
• Hyperpolarizing threshold introduces rebound spikes even when net synaptic drive is inhibitory.

Pitfalls

• Adaptive threshold dynamics must be slow enough relative to the membrane time constant; otherwise the model effectively reverts to a fixed threshold.
• Binomial QC approximation assumes independent vesicle releases; correlated release or short-term plasticity require extension.
• The analytical FPT solution assumes diffusion-like statistics; Poisson input with large quanta may need exact simulation.

Related Concepts

• Integrate-and-fire neurons
• Coefficient of variation
• Spike-frequency adaptation
• Post-inhibitory rebound
• First-passage-time analysis
• Excitatory-inhibitory balance
• Quantal content / synaptic vesicle release

Activation

inter-spike interval, ISI statistics, adaptive threshold, EI circuit, first-passage time, spike variability, integrate-and-fire, post-inhibitory rebound, quantal content, neuronal noise