Cluster synchronization in networks of identical oscillators with α-function pulse coupling.

Area of Science:

• Computational Neuroscience
• Complex Systems Dynamics
• Mathematical Biology

Background:

• Neural networks exhibit complex dynamics, with focus on synchronized and asynchronous states.
• Coupling parameters (K) dictate network behavior, distinguishing excitation from inhibition.
• Previous studies concentrated on fully synchronized or asynchronous states.

Purpose of the Study:

• To uncover and characterize a broader range of attracting states in networks of identical leaky integrate-and-fire neurons.
• To investigate the stability and bifurcations of these newly identified partially synchronized states.
• To develop a framework for distinguishing neuron models based on their attractor sets.

Main Methods:

• Utilized a dimensional reduction strategy exploiting K=0 dynamics.
• Analyzed a simplified continuous flow on a codimension 3 subspace.
• Employed high-precision numerical simulations for N=2-4 neurons.
• Investigated bifurcations and stability of fixed points and limit cycles.

Main Results:

• Identified a rich set of attractors including (N-1,1) fixed states and equal-sized splay states.
• Discovered limit cycles clarifying previously observed quasiperiodic behavior.
• Demonstrated that the sign of K determines the direction of the simplified flow.
• Characterized the complete bifurcation sequence and stability for small N.

Conclusions:

• The identified attracting states offer a new perspective on neural network dynamics.
• Partially synchronized states, beyond (N-1,1), are possible in integrate-and-fire networks but not Kuramoto models.
• The framework provides a method for classifying different neuron models.
• Generalization to non-identical neurons shows attracting fixed points where neurons need not fire simultaneously.