Phase response theory extended to nonoscillatory network components

Abstract

New tools for analysis of oscillatory networks using phase response theory (PRT) under the assumption of pulsatile coupling have been developed steadily since the 1980s, but none have yet allowed for analysis of mixed systems containing nonoscillatory elements. This caveat has excluded the application of PRT to most real systems, which are often mixed. We show that a recently developed tool, the functional phase resetting curve (fPRC), provides a serendipitous benefit: it allows incorporation of nonoscillatory elements into systems of oscillators where PRT can be applied. We validate this method in a model system of neural oscillators and a biological system, the pyloric network of crustacean decapods.

FIG. 1

(Color online) Schematized protocols for PRC measurement in neurons. (a) Protocol for fPRC measurement: isolated nonoscillatory neuron i is silent at rest, but emits a burst in response to an initial input. An online protocol detects this burst and thereafter delivers a stimulus at fixed delay Δ after the initiation of each subsequent burst for N cycles. Response times t_rk are measured after each input. (b) The fPRC consists of averages t̄_r tabulated across Δ. (c) Definition of terms: stimulus delay Δ and response time t_r during 1:1 phase-locking are shown for bursting neurons i and j . The shaded and hatched regions correspond to the burst duration. (d) Graphical method for prediction: fPRCs for each component neuron are plotted on the appropriate axes such that any point of intersection gives a solution mode of 1:1 phase-locking. Stability is calculated as described in the text. (e) Protocol for spPRC measurement provided for comparison: phase shift to an ongoing oscillatory rhythm with period P₀ is measured in isolated neuron x by delivering a single-pulse stimulus at delay Δ and measuring the resulting interval P₁. (f) The spPRC consists of the relative change in period P₁−P₀ tabulated across delay Δ. Both terms are scaled to phase ϕ on [0,1) by normalizing by P₀.

FIG. 2

Schematic of the pyloric network.

FIG. 3

(Color online) Comparison of (a) model and (b) biological results showing that fPRCs measured in (i) isolated component neurons make (ii) predictions that match (iii) simulated or experimental observations of network activity. Summarized observations from column (iii) are indicated by asterisks (*) plotted in column (ii). Parameter g_s was varied to produce families of three fPRCs for each neuron in (b)(ii). All intersections in (b)(ii) were calculated to be stable. To avoid clutter in (b)(ii), confidence bounds of +/−1 standard error are shown as the shaded area for only one fPRC from each component neuron and for the observed activity. Observations at multiple points in experimental time were necessary in (b) to determine the degree of stationarity of the biological system.

FIG. 4

(Color online) Summary of results. Predicted (Pred) network (a) phase and (b) period are compared to observed (Obs) behavior for four experimental preparations.