Neuronal population dynamics with post inhibitory rebound:A reduction to piecewise linear discontinuous circle maps

Abstract

Post inhibitory rebound (PZR) is a nonlinear phenomenon present in a variety of nerve cells. It is an important mechanism underlying central pattern generationfor heartbeat, swimming and other motor patterns in many neuronal systems. In this paper, we propose an extension of the binary threshold neuron model to incorporate the efects of PZR. For a single neuron, the dynamics can be described by a piecewise linear circle map with two discontinuities. Both frequency-locking and chaos can occur. The Liapunov exponent of the map is evaluated and used to dejine transitions between these two distinct types of asymptotic behaviour. Hysteresis between periodic orbits is also observed. A small network of these model neurons, with reciprocal inhibition, is shown to exhibit 'self-sustained' anti-phase oscillations, making PZR a plausible mechanism for central pattern generation in neuronal systems. Unlike coupled oscillator theories, network oscillations emerge naturally as a consequence of the biological description from which the neuronal dynamics is derived. The simplicity of the dynamical model allows for the possibility of large population studies in contrast to other classical models of single neuron dynamics that incorporate active processes.

Bifurcations of piecewise smooth flows: Perspectives, methodologies and open problems

“…They are frequently treated by ad hoc modifications of tools borrowed from smooth systems; reviews can be found in [12,70,74,99]. A few of the wide range of applications that exhibit piecewise smooth dynamics include mechanical problems of friction [43,82,35,96,68,56], switched feedback in control theory [97,61,39,28] and electronics [22,5,9,79,32], nonsmooth models in economics [62,50], ecology [69,38,40], neuron signaling [33,34,76], genetic potentials [53,36,21,54], and novel nonlinear effects of superconductors [8,63]. Interest in such diverse applications in the vacuum of an insufficiently developed theory has left behind a nomenclature that is dogged by semantic difficulties.…”

Shared Inputs, Entrainment, and Desynchrony in Elliptic Bursters: From Slow Passage to Discontinuous Circle Maps

“…Using (14), (15) it is straightforward to compute the time (T S ) it takes for h S (t) to reach y J , and then derive values for C P such that h −1 P (y J ) = T S as required by our definition of h p . To compute the jump up point y J , we derive an expression for ỹj (y i ) which is also needed in the definition of our kick map.…”

“…Rigorously assessing the presence of chaos is, however, not a simple task. This was accomplished for related piecewise smooth maps [9,14,45,12] where various definitions of chaos were used, depending on context. For example, Keener [45] showed that piecewise, surjective and non-decreasing maps with overlap have rotation numbers spanning a non-empty interval, an indication of chaos for circle maps.…”

Bifurcations and Chaos in a Periodically Probed Computer Network

“…The resulting map is an example of a circle map because it can be viewed as mapping the circumference of a circle to itself. Discontinuous circle maps like this one have been used to model neurons [Bresloff & Stark, 1990;Coombes & Doole, 1996], cardiac arrhythmia [Bub & Glass, 1995] and power electronic switching circuits [Jain & Banerjee, 2003] among other applications. Several interesting dynamical features of these maps, including frequency locking, devil's staircases in rotation number and Lyapunov exponent, and banded chaos have been explored in [Bresloff & Stark, 1990;Coombes & Doole, 1996;Bub & Glass, 1995;Keener, 1990;Wu et al, 1995;Qu et al, 1997].…”