[Paper Review] Analysis of biological integrate-and-fire oscillators

Marat Akhmet|arXiv (Cornell University)|Nov 25, 2010

Nonlinear Dynamics and Pattern Formation30 references3 citations

TL;DR

This paper proposes a novel map-based analysis for non-identical, pulse-coupled integrate-and-fire oscillators with continuous inter-oscillator coupling, demonstrating that synchronization occurs under specific conditions even when oscillators are not identical. The key contribution is significant progress toward solving Peskin's second conjecture by identifying synchronization regions and showing that synchronization time increases as the initial condition set expands.

ABSTRACT

We consider discontinuous dynamics of integrate-and-fire models, which consist of pulse-coupled biological oscillators. A thoroughly constructed map is in the basis of the analysis. Synchronization of non-identical oscillators is investigated. Significant advances for the solution of second Peskin's conjecture have been made. Examples with numerical simulations are given to validate the theoretical results. Perspectives are discussed.

Motivation & Objective

• To address the unresolved second Peskin conjecture concerning synchronization in non-identical, pulse-coupled integrate-and-fire oscillators.
• To develop a dynamical systems framework that accounts for continuous coupling between oscillators during non-firing intervals, enhancing biological realism.
• To construct a specialized map (not a Poincaré map) that tracks state transitions across oscillators, enabling analysis of synchronization and periodic motion.
• To identify conditions under which non-identical oscillators synchronize, even when initial conditions are not identical.
• To provide theoretical and numerical foundations for studying periodic, quasi-periodic, and almost periodic motions in discontinuous dynamical systems with variable discontinuity moments.

Proposed method

• A prototype map $ L $ is defined that maps the state of one oscillator after a firing event to the state of another, with roles interchanged in successive mappings.
• The map $ L $ is constructed using solutions of the differential equation $ u' = f(u) $, with $ s(v) $ denoting the time to reach threshold and $ \bar{L}(v) = u(s, 0, 0) $, which tracks the state of the other oscillator.
• The analysis relies on monotonicity of the dynamics and properties of the function $ g $, particularly when $ g $ is of $ \mathcal{K} $ type and satisfies condition (A2).
• The system is modeled with all-to-all coupling, where each firing induces a state jump in all non-firing oscillators, with jump size $ \epsilon + \epsilon_i $.
• Perturbations are introduced via parameters $ \mu_i, \xi_i, \epsilon_i $, allowing non-identical dynamics while preserving synchronization under specific conditions.
• Theoretical results are validated through numerical simulations of two- and multi-oscillator systems, with focus on synchronization time and region size.

Experimental results

Research questions

• RQ1 Under what conditions do non-identical, pulse-coupled integrate-and-fire oscillators synchronize, even when initial conditions are not identical?
• RQ2 How does the synchronization time depend on the size of the initial condition region, particularly as it expands?
• RQ3 Can a map-based approach be constructed that captures the interplay between firing events and state transitions across oscillators in a non-identical system?
• RQ4 What role do perturbations (in coupling strength, thresholds, and dynamics) play in preserving or disrupting synchronization?
• RQ5 Can the proposed map framework be extended to analyze periodic, almost periodic, or quasi-periodic motions in discontinuous dynamical systems?

Key findings

• Synchronization is achieved within a finite time interval $[t_0 + \frac{m}{2}\tilde{T}, t_0 + mT]$ when $ x_2(t_0+) \in S_m $, provided $ g $ is of $ \mathcal{K} $ type and condition (A2) holds.
• If $ g $ is of $ \mathcal{K} $ type but (A2) fails, the system does not synchronize, indicating a necessary condition for synchronization.
• For multi-oscillator systems, synchronization occurs within $[t_0, t_0 + T]$ if all non-firing oscillators are in $ S_0 $, and within a longer interval if some are in higher $ S_{k_i} $ sets.
• The time to synchronization increases without bound as the region of initial conditions requiring synchronization expands, indicating that non-synchronized initial values exist.
• The measure of non-synchronized initial values is not proven to be zero (unlike in the identical case), but the paper identifies and characterizes such regions.
• The method is generalizable to systems with time-varying thresholds $ 1 + \phi(t,x) $, state-dependent jumps, and discontinuous right-hand sides, provided the map $ L $ has required properties.