Clustering through post inhibitory rebound in synaptically coupled neurons

Post inhibitory rebound is a nonlinear phenomenon present in a variety of nerve cells. Following a period of hyper-polarization this effect allows a neuron to fire a spike or packet of spikes before returning to rest. It is an important mechanism underlying central pattern generation for heartbeat, swimming and other motor patterns in many neuronal systems. In this paper we consider how networks of neurons, which do not intrinsically oscillate, may make use of inhibitory synaptic connections to generate large scale coherent rhythms in the form of cluster states. We distinguish between two cases i) where the rebound mechanism is due to anode break excitation and ii) where rebound is due to a slow T-type calcium current. In the former case we use a geometric analysis of a McKean type model to obtain expressions for the number of clusters in terms of the speed and strength of synaptic coupling. Results are found to be in good qualitative agreement with numerical simulations of the more detailed Hodgkin-Huxley model. In the second case we consider a particular firing rate model of a neuron with a slow calcium current that admits to an exact analysis. Once again existence regions for cluster states are explicitly calculated. Both mechanisms are shown to prefer globally synchronous states for slow synapses as long as the strength of coupling is sufficiently large. With a decrease in the duration of synaptic inhibition both systems are found to break into clusters. A major difference between the two mechanisms for cluster generation is that anode break excitation can support clusters with several groups, whilst slow T-type calcium currents predominantly give rise to clusters of just two (anti-synchronous) populations.

Receptors, sparks and waves in a fire-diffuse-fire framework for calcium release

Calcium ions are an important second messenger in living cells. Indeed calcium signals in the form of waves have been the subject of much recent experimental interest. It is now well established that these waves are composed of elementary stochastic release events (calcium puffs or sparks) from spatially localised calcium stores. The aim of this paper is to analyse how the stochastic nature of individual receptors within these stores combines to create stochastic behaviour on long timescales that may ultimately lead to waves of activity in a spatially extended cell model. Techniques from asymptotic analysis and stochastic phase-plane analysis are used to show that a large cluster of receptor channels leads to a release probability with a sigmoidal dependence on calcium density. This release probability is incorporated into a computationally inexpensive model of calcium release based upon a stochastic generalization of the Fire-Diffuse-Fire (FDF) threshold model. Numerical simulations of the model in one and two dimensions (with stores arranged on both regular and disordered lattices) illustrate that stochastic calcium release leads to the spontaneous production of calcium sparks that may merge to form saltatory waves. Illustrations of spreading circular waves, spirals and more irregular waves are presented. Furthermore, receptor noise is shown to generate a form of array enhanced coherence resonance whereby all calcium stores release periodically and simultaneously.

Calcium oscillations

Changes in cellular calcium concentration control a wide range of physiological processes, from the subsecond release of synaptic neurotransmitters, to the regulation of gene expression over months or years. Calcium can also trigger cell death through both apoptosis and necrosis, and so the regulation of cellular calcium concentration must be tightly controlled through the concerted action of pumps, channels and buffers that transport calcium into and out of the cell cytoplasm. A hallmark of cellular calcium signalling is its spatiotemporal complexity: stimulation of cells by a hormone or neurotransmitter leads to oscillations in cytoplasmic calcium concentration that can vary markedly in time course, amplitude, frequency, and spatial range. In this chapter we review some of the biological roles of calcium, the experimental characterisation of complex dynamic changes in calcium concentration, and attempts to explain this complexity using computational models. We consider the "toolkit" of cellular proteins which influence calcium concentration, describe mechanistic models of key elements of the toolkit, and fit these into the framework of whole cell models of calcium oscillations and waves. Finally, we will touch on recent efforts to use stochastic modelling to elucidate elementary calcium signal events, and how these may evolve into global signals.