The Influence of the A-Current on the Dynamics of an Oscillator-Follower Inhibitory Network

Abstract

The transient potassium A-current is present in almost all neurons and plays an essential role in determining the timing and frequency of action potential generation. We use a three-variable mathematical model to examine the role of the A-current in a rhythmic inhibitory network, as is common in central pattern generation. We focus on a feed-forward architecture consisting of an oscillator neuron inhibiting a follower neuron. We use separation of time scales to demonstrate that the trajectory of the follower neuron within each cycle can be tracked by analyzing the dynamics on a 2-dimensional slow manifold that as determined by the two slow model variables: the recovery variable and the inactivation of the A-current. The steady-state trajectory, however, requires tracking the slow variables across multiple cycles. We show that tracking the slow variables, under simplifying assumptions, leads to a one-dimensional map of the unit interval with at most a single discontinuity depending on gA, the maximal conductance of the A-current, or other model parameters. We demonstrate that, as the value of gA is varied, the trajectory of the follower neuron goes through a set of bifurcations to produce n:m periodic solutions where the follower neuron becomes active m times for each n cycles of the oscillator. Using a generalized Pascal triangle, each n:m trajectory can be constructed as a combination of solutions from a higher level of the triangle.

Cross-frequency coupling explains the preference for simple ratios in rhythmic behaviour and the relative stability across non-synchronous patterns

“…Speculatively, the plausibility of the phase-attractive circle map needs to be discussed in light of related phenomena in neural dynamics. There is growing evidence that cross-frequency hierarchical coupling is intrinsic to, and may help organize, neural dynamics in individual neurons [32] as well as populations of neurons [21], and is linked to diverse scenarios involving many different cognitive and motor functions [33,34], including rhythm and musical meter [8,35]. In this context, it is worth raising the question whether intrinsic neural CFC implements a low-level prior for rhythmic behaviour in general.…”

Short-term synaptic dynamics control the activity phase of neurons in an oscillatory network

“…For example, the follower ventral dilator (VD) and PY neurons have a much higher levels of I A , which in turn has a larger effect on the timing of post-inhibitory spiking. In a set of computational studies, we addressed the role of I A in determining the burst phase in response to periodic inputs (Zhang et al, 2008; Zhang et al, 2009) and in conjunction with short-term depression in the synaptic input (Bose et al, 2004). An experimental clarification of the relative contribution of intrinsic properties vs. synaptic input could be done with controlled dynamic clamp synaptic input, such as those used in the current study, injected in PY or VD neurons.…”

Framing of grid cells within and beyond navigation boundaries

“…Because they essentially work by turning velocity-modulated temporal oscillations into grid-like spatial oscillations, synchronizing the former might in principle ‘synchronize’ the latter, thus spatially interlocking the grids. Interlocked grids at different scales would require ‘n:m’ reciprocal entrainment, by which n cycles of one rhythm correspond to m cycles of the other, possibly at different phase-lags ( Zhang et al, 2009 ; Deshmukh et al, 2010 ; Belluscio et al, 2012 ; Brandon et al, 2013 ). Such a framework, to our knowledge, has not been explored (but Zilli and Hasselmo, 2010 ; Blair et al, 2014 investigated the synchronization of velocity-modulated oscillators for path-integration error-correction).…”