Transient spatiotemporal chaos collapses into periodic and steady states in an electrically-coupled neural ring network

Chaotic behavior in a spatially extended system is often referred to as spatiotemporal chaos. The trajectories of a system as it evolves through state space are described by irregular spatial and temporal patterns. In mathematical biology, spatiotemporal chaos has been demonstrated in chemotaxis models (Painter & Hillen, 2011) predator-prey models (Sherratt, J. & Fowler, A., 1995) and the Hogdkin-Huxley neural model (Wang, Lu, & Chen, 2006). Transient chaos is a special case of chaotic dynamics in which the system dynamics collapses without external perturbation. Rather, collapse is an intrinsic property of the system. Here, we diff usively couple many spiking neurons into a ring network and fi nd that the network dynamics can collapse on to two diff erent species of attractor: the limit cycle and the steady-state solution

Persistent oscillations in the Aplysia bag cell network

Persistence is a phenomena by which a resting neuron enters a state of persistent behavior following a brief stimulus. Persistent neural systems can exhibit long-term responses that remain after the stimulus is removed, switching from excitable, steady-state dynamics to a period of tonic spiking or bursting. In Aplysia, such behavior, known as the afterdischarge, is exhibited by the bag cell neuron and regulated by second messenger calcium dynamics. In this thesis, we construct a model for the electrical activity of the Aplysia bag cell neuron is constructed based on experimental data. The model includes many features of the bag cell, including use-dependence, non-selective cation channels, a persistent calcium current, and the afterdischarge. Each of these features contributes to the onset of afterdischarge. Several methods are used to fit experimental data and construct the model, including hand tuning, parameter forcing, genetic algorithms for optimization, and continuation analysis. These methods help to address common modeling issues such as degeneracy and sensitivity. Use-dependence in the calcium channels of Aplysia is thought to depend on calcium. The model developed in this thesis verifies calcium as a viable driver for use-dependence. The literature often emphasizes two potassium channels in the context of bag cell afterdischarge, but we show that afterdischarge behavior is produced with only a single potassium channel in simulations. Experimental evidence suggests that nonselective cation channels are a primary driver of afterdischarge behavior. In the model developed here, the nonselective current is required for in silica afterdischarge to take place. Continuation analysis is used to determine and tune the location and stability of fixed points in the model. For exploratory analysis, an electrically-coupled network model is constructed to simulate the observed dynamics in bag cell clusters in vivo. A simple two-neuron network reproduces some experimental results. Larger networks are considered. Little is known about the topology of Aplysia bag cell cluster. The final chapter of this thesis explores different topologies in a 100-neuron network, including a ring topology, a cluster ring topology, and a randomly-connected scatter network, exploring how the coupling constant, topology, and size of the network affect the ability of the network to synchronize

Transient spatiotemporal chaos in a Morris-Lecar neuronal ring network collapses to either the rest state or a traveling pulse

Thesis (M.S.) University of Alaska Fairbanks, 2012Transient spatiotemporal dynamics exists in an electrically coupled Morris-Lecar neuronal ring network, a theoretical model of an axo-axonic gap junction network. The lifetime of spatiotemporal chaos was found to grow exponentially with network size. Transient dynamics regularly collapses from a chaotic state to either the resting potential or a traveling pulse, indicating the existence of a chaotic saddle. For special conditions, a chaotic attractor can arise in the Morris-Lecar network to which transient chaos can collapse. The short-term outcome of a Morris-Lecar ring network is determined as a function of perturbation configuration. Perturbing small clusters of nearby neurons in the network consistently induced chaos on a resting network. Perturbation on a chaotic network can induce collapse in the network, but transient chaos becomes more resistant to collapse by perturbation when greater external current is applied.1. Introduction -- 1.1. The physics of neurons -- 1.2. Transient spatiotemporal chaos -- 1.3. Synopsis -- 2. Model -- 3. Transient spatiotemporal chaos -- 4. Perturbations on a network at rest -- 5. Perturbations on a nework in the neighborhood of the chaotic saddle -- 6. Conclusions -- 6.1. Outview -- Bibliography