Stochastic and deterministic dynamics in networks with excitable nodes.

Area of Science:

• Complex Systems
• Nonlinear Dynamics
• Network Science

Background:

• Networks of excitable systems model diverse phenomena across biology, social sciences, and physics.
• Continuous phase transitions are common in these models, often assuming linear node transfer functions for small inputs.
• Previous models implicitly relied on linearity, potentially oversimplifying real-world dynamics.

Purpose of the Study:

• To investigate the impact of cooperative excitations and nonlinear transfer functions on the collective dynamics of excitable networks.
• To understand how nonlinearity qualitatively alters phase transitions and introduces phenomena like hysteresis.
• To analyze finite-size effects in networks with nonlinear dynamics.

Main Methods:

• Development of a mean-field theory to analyze collective dynamics using a one-dimensional map.
• Theoretical analysis of finite-size effects by examining the behavior of single-node initial excitations in large networks.
• Numerical simulations to complement theoretical findings and validate the model's predictions.

Main Results:

• Introduction of any nonlinearity into the transfer function qualitatively changes system dynamics, leading to discontinuous phase transitions and hysteresis.
• The mean-field theory successfully captures the essential features of the dynamics.
• Nonlinear transfer functions create a rich effective phase diagram for finite networks, deviating from predictions of linear models.

Conclusions:

• Nonlinearities in excitable networks fundamentally alter collective dynamics, inducing discontinuous transitions and hysteresis.
• Mean-field theory and finite-size analysis provide valuable insights into these complex behaviors.
• Researchers must carefully consider nonlinear effects and avoid assumptions of noncooperative excitations when modeling excitable systems.