Codimension-2 saddle-node loop bifurcations as critical switch in neuronal excitability, filtering and susceptibility to stochastic ion channels

Abstract

Information processing in neurons is governed by the voltage dynamics of the cells’ membrane. Filtering of synaptic stimuli and synchronisation to other neurons depends on the interplay of the bifurcation structure and the stochasticity of ion channels. Changes in filtering are organised by higher order bifurcations. 
We prove and derive conditions for the generic existence of a global codimension-2 saddle-node loop bifurcation in conductance-based neuronal models, like theВ Traub-Miles model. This bifurcation is reached by system parameters that affect the timescale separation between voltage and gating dynamics, such asВ temperature, capacitance or leak conductance [1]. The immediate implicationsВ for neuronal filtering are given by linear response theory of the phase reductionВ of the accompanying stochastic limit cycle [2]. The hallmark of the saddle-nodeВ loop transition is that, compared to the Bogdanov-Takens point, already inВ finitesimal parameter changes modify filtering. Around this critical point, weВ can furthermore prove that the tangent space to the isochrons [3] is spanned byВ the strongly stable manifold. This changes the systems susceptibility to different ion channel kinetics, suggesting that the impact of different noise sources inВ neurons can be derived from the bifurcation mediating the onset of spiking.