Growing Critical: Self-Organized Criticality in a Developing Neural System.

Experiments in various neural systems found avalanches: bursts of activity with characteristics typical for critical dynamics. A possible explanation for their occurrence is an underlying network that self-organizes into a critical state. We propose a simple spiking model for developing neural networks, showing how these may "grow into" criticality.

Hybrid discrete-time neural networks.

Hybrid dynamical systems combine evolution equations with state transitions. When the evolution equations are discrete-time (also called map-based), the result is a hybrid discrete-time system. A class of biological neural network models that has recently received some attention falls within this category: map-based neuron models connected by means of fast threshold modulation (FTM).

Bursting regimes in map-based neuron models coupled through fast threshold modulation.

A system consisting of two map-based neurons coupled through reciprocal excitatory or inhibitory chemical synapses is discussed. After a brief explanation of the basic mechanism behind generation and synchronization of bursts, parameter space is explored to determine less obvious but biologically meaningful regimes and effects. Among them, we show how excitatory synapses without any delays may induce antiphase synchronization; that a synapse may change its character from excitatory to inhibitory and vice versa by changing its conductance, without any change in reversal potential; and that small variations in the synaptic threshold may result in drastic changes in the synchronization of spikes within bursts.

Patterns in inhibitory networks of simple map neurons.

We study the dynamics of networks of inhibitory map-based bursting neurons. Linear analysis allows us to understand how the patterns of bursting are determined by network topology and how they depend on the strength of synaptic connections, when inhibition is balanced. Two kinds of patterns are found depending on the symmetry of the network: slow cyclic patterns riding on subthreshold oscillations where almost all neurons contribute bursts in a sparse manner and fast patterns of bursts in which only one of two mutually exclusive groups of neurons take part.

Sensitivity versus resonance in two-dimensional spiking-bursting neuron models.

Through phase plane analysis of a class of two-dimensional spiking and bursting neuron models, covering some of the most popular map-based neuron models, we show that there exists a trade-off between the sensitivity of the neuron to steady external stimulation and its resonance properties, and how this trade-off may be tuned by the neutral or asymptotic character of the slow variable. Implications of the results for the suprathreshold behavior of the neurons, both by themselves and as part of networks, are presented in different regimes of interest, such as the excitable, regular spiking, and bursting regimes. These results establish a consistent link between single-neuron parameters and resulting network dynamics, and will hopefully be useful as a guide for modeling.

Synchronization and propagation of bursts in networks of coupled map neurons.

The present paper studies regular and complex spatiotemporal behaviors in networks of coupled map-based bursting oscillators. In-phase and antiphase synchronization of bursts are studied, explaining their underlying mechanisms in order to determine how network parameters separate them. Conditions for emergent bursting in the coupled system are derived from our analysis.