Type I membranes, phase resetting curves, and synchrony

Area of Science:

• Computational Neuroscience
• Mathematical Biology
• Systems Neuroscience

Background:

• Type I membrane oscillator models, like the Connor and Morris-Lecar models, exhibit very low-frequency oscillations near critical applied currents.
• Previous numerical studies indicated difficulties in achieving synchrony and strictly positive phase resetting curves for these models.

Purpose of the Study:

• To analytically demonstrate that difficult synchrony and strictly positive phase resetting are general properties of Type I membrane models.
• To investigate the occurrence of Type II resetting in models utilizing Hopf bifurcations.
• To differentiate the effects of fast and slow synapses and derive a canonical form for phase interactions.

Main Methods:

• Application of singular perturbation methods.
• Utilizing averaging techniques.
• Analysis of models exhibiting rhythmicity via Hopf bifurcations.

Main Results:

• Singular perturbation and averaging confirm that difficult synchrony and strictly positive phase resetting are general properties of Type I membrane models.
• Type II resetting is shown to occur in models that achieve rhythmicity through a Hopf bifurcation.
• Differences between rapid and slow synapses are elucidated, and a canonical form for phase interactions is derived.

Conclusions:

• The study provides analytical evidence for the inherent properties of Type I membrane models regarding synchrony and phase resetting.
• A distinction is made between Type I and Type II resetting behaviors based on the underlying bifurcation mechanisms.
• The findings offer insights into synaptic dynamics and provide a generalized framework for phase interactions in neural models.