Cui J, Canavier CC, Butera R. Functional phase response curves: a method for understanding synchronization of adapting neurons. J Neurophysiol 102: 387-398, 2009. First published May 6, 2009 doi:10.1152/jn.00037.2009. Phase response curves (PRCs) for a single neuron are often used to predict the synchrony of mutually coupled neurons. Previous theoretical work on pulse-coupled oscillators used single-pulse perturbations. We propose an alternate method in which functional PRCs (fPRCs) are generated using a train of pulses applied at a fixed delay after each spike, with the PRC measured when the phasic relationship between the stimulus and the subsequent spike in the neuron has converged. The essential information is the dependence of the recovery time from pulse onset until the next spike as a function of the delay between the previous spike and the onset of the applied pulse. Experimental fPRCs in Aplysia pacemaker neurons were different from single-pulse PRCs, principally due to adaptation. In the biological neuron, convergence to the fully adapted recovery interval was slower at some phases than that at others because the change in the effective intrinsic period due to adaptation changes the effective phase resetting in a way that opposes and slows the effects of adaptation. The fPRCs for two isolated adapting model neurons were used to predict the existence and stability of 1:1 phase-locked network activity when the two neurons were coupled. A stability criterion was derived by linearizing a coupled map based on the fPRC and the existence and stability criteria were successfully tested in two-simulated-neuron networks with reciprocal inhibition or excitation. The fPRC is the first PRC-based tool that can account for adaptation in analyzing networks of neural oscillators.
The synchronization of two distributed Belousov-Zhabotinsky systems is experimentally and theoretically investigated. Symmetric local coupling of the systems is made possible with the use of a video cameraprojector scheme. The spatial disorder of the coupled systems, with random initial configurations of spirals, gradually decreases until a final state is attained, which corresponds to a synchronized state with a single spiral in each system. The experimental observations are confirmed with numerical simulations of two identical Oregonator models with symmetric local coupling, and a systematic study reveals generalized synchronization of spiral waves. Several different types of synchronization attractors are distinguished. DOI: 10.1103/PhysRevE.68.026205 PACS number͑s͒: 82.40.Qt, 05.45.Xt, 05.65.ϩb, 47.54.ϩr Synchronization phenomena are of fundamental importance in physical, chemical, biological, and technical systems. The synchronization of coupled chaotic oscillators has attracted much attention in recent years, and complete, phase, lag, and generalized synchronization have been distinguished in such systems ͓1͔. These concepts have been applied in the analysis and interpretation of experimental data from the cardiorespiratory system ͓2͔, paddlefish cells ͓3͔, the human brain ͓4͔, and in the contexts of population dynamics ͓5͔ and communication with chaotic lasers ͓6͔. Recently, synchronization phenomena in spatially extended systems have attracted increasing attention. Identical synchronization and phase synchronization have been observed in systems exhibiting spatiotemporal chaos ͓7-9͔.In this paper, we describe experimental and theoretical studies of the synchronization of two locally coupled domains of excitable media exhibiting spiral wave behavior. We have used the photosensitive Belousov-Zhabotinsky ͑BZ͒ reaction ͓10͔, which is particularly convenient for studies on influencing existing patterns or generating new ones by the application of various types of external forcing ͓11-13͔, or by local ͓14͔, nonlocal ͓15͔, or global ͓16͔ feedback mechanisms. The identical synchronization of chemical wave patterns has been previously observed in a BZ system with diffusive cross-membrane coupling ͓17͔. Here we study two domains of excitable media that are locally coupled to each other by means of a video camera-video projector setup through a coupling algorithm, which leads to weaker than identical synchronization ͓18͔.Prior to each experiment, the projected image was adjusted at each pixel by an iterative algorithm to ensure a spatially uniform illumination field, upon which all subsequent projected images were based. The local concentration of oxidized catalyst was recorded with a video camera, and the recorded image was divided into an array of square cells. In all experiments, the lateral size of each cell was much smaller than the spiral wavelength. The medium was partitioned into two square regions separated and surrounded by an unexcitable boundary generated with high-intensity light. The corresponding...
Spatiotemporal networks are studied in a photosensitive Belousov-Zhabotinsky medium that allows both local and nonlocal transmission of excitation. Local transmission occurs via propagating excitation waves, while nonlocal transmission takes place by nondiffusive jumps to destination sites linked to excited sites in the medium. Static, dynamic, and domain link networks are experimentally and computationally characterized. Transitions to synchronized behavior are exhibited with increasing link density, and power-law relations are observed for first-coverage time as a function of link probability.
In order to study the ability of coupled neural oscillators to synchronize in the presence of intrinsic as opposed to synaptic noise, we constructed hybrid circuits consisting of one biological and one computational model neuron with reciprocal synaptic inhibition using the dynamic clamp. Uncoupled, both neurons fired periodic trains of action potentials. Most coupled circuits exhibited qualitative changes between one-to-one phase-locking with fairly constant phasic relationships and phase slipping with a constant progression in the phasic relationships across cycles. The phase resetting curve (PRC) and intrinsic periods were measured for both neurons, and used to construct a map of the firing intervals for both the coupled and externally forced (PRC measurement) conditions. For the coupled network, a stable fixed point of the map predicted phase locking, and its absence produced phase slipping. Repetitive application of the map was used to calibrate different noise models to simultaneously fit the noise level in the measurement of the PRC and the dynamics of the hybrid circuit experiments. Only a noise model that added history-dependent variability to the intrinsic period could fit both data sets with the same parameter values, as well as capture bifurcations in the fixed points of the map that cause switching between slipping and locking. We conclude that the biological neurons in our study have slowly-fluctuating stochastic dynamics that confer history dependence on the period. Theoretical results to date on the behavior of ensembles of noisy biological oscillators may require re-evaluation to account for transitions induced by slow noise dynamics.
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