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Entrainment Ranges for Chains of Forced Neural and Phase Oscillators.

Massarelli N, Clapp G, Hoffman K, Kiemel T - J Math Neurosci (2016)

Bottom Line: This results in a simpler model yet maintains some properties of the neural model.Entrainment ranges for chains with nonuniform intersegmental coupling asymmetry are larger when forcing is applied to the middle of the chain than when it is applied to either end, a result that is qualitatively similar to the experimental results.In the limit of weak coupling in the chain, the entrainment results of the neural model approach the entrainment results for the derived phase model.

View Article: PubMed Central - PubMed

Affiliation: Department of Mathematics and Statistics, University of Maryland, Baltimore County, 1000 Hilltop Circle, Baltimore, MD, 21250, USA.

ABSTRACT
Sensory input to the lamprey central pattern generator (CPG) for locomotion is known to have a significant role in modulating lamprey swimming. Lamprey CPGs are known to have the ability to entrain to a bending stimulus, that is, in the presence of a rhythmic signal, the CPG will change its frequency to match the stimulus frequency. Bending experiments in which the lamprey spinal cord has been removed and mechanically bent back and forth at a single point have been used to determine the range of frequencies that can entrain the CPG rhythm. First, we model the lamprey locomotor CPG as a chain of neural oscillators with three classes of neurons and sinusoidal forcing representing edge cell input. We derive a phase model using the connections described in the neural model. This results in a simpler model yet maintains some properties of the neural model. For both the neural model and the derived phase model, entrainment ranges are computed for forcing at different points along the chain while varying both intersegmental coupling strength and the coupling strength between the forcer and chain. Entrainment ranges for chains with nonuniform intersegmental coupling asymmetry are larger when forcing is applied to the middle of the chain than when it is applied to either end, a result that is qualitatively similar to the experimental results. In the limit of weak coupling in the chain, the entrainment results of the neural model approach the entrainment results for the derived phase model. Both biological experiments and the robustness of non-monotonic entrainment ranges as a function of the forcing position across different classes of CPG models with nonuniform asymmetric coupling suggest that a specific property of the intersegmental coupling of the CPG is key to entrainment.

No MeSH data available.


Related in: MedlinePlus

Illustration of two-parameter continuation used to find entrainment ranges as a function of the forcing strength. The dotted circles denote how the values of  and  are updated at each continuation step. The plus signs denote points on the entrainment range that are detected by the continuation circles. This allows us to detect sharp corners that may be missed with standard continuation where we only look at vertical slices of parameter space
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Fig7: Illustration of two-parameter continuation used to find entrainment ranges as a function of the forcing strength. The dotted circles denote how the values of and are updated at each continuation step. The plus signs denote points on the entrainment range that are detected by the continuation circles. This allows us to detect sharp corners that may be missed with standard continuation where we only look at vertical slices of parameter space

Mentions: We used a series of one-parameter continuations instead of two-parameter continuations, because a two-parameter continuation can become inaccurate near degenerate bifurcations [35]. In order to follow these curves in any direction in parameter space, the one-parameter continuations were performed along ellipses in parameter space rather than straight lines, as illustrated in Fig. 7. The larger dotted ellipses represent the path of the continuation steps in the parameter space. These ellipses indicate how the parameters and are updated at each continuation step. We choose the size of the ellipse so that it is large enough to cover a relatively large area in parameter space in order to decrease computation time and also small enough to capture sharp corners of the entrainment range. The small red circles indicate the center of continuation ellipses. To choose the next center, we take a step in the same direction as the previous entrainment point. The points on the entrainment range are indicated by blue plus signs. To better explain this process, consider entrainment points 2 and 3 in Fig. 7. We start with entrainment point 2, which is a known point on the entrainment rage. To get the next center, indicated by the small red circle between points 2 and 3, we step in the same direction as the vector from point 1 to point 2. We then move around the large ellipse, plotted in magenta, and find new fixed points with slightly different values of and δ. To determine points on the boundary of the entrainment range, we compute the stability of the fixed points in each model. Fig. 7


Entrainment Ranges for Chains of Forced Neural and Phase Oscillators.

Massarelli N, Clapp G, Hoffman K, Kiemel T - J Math Neurosci (2016)

Illustration of two-parameter continuation used to find entrainment ranges as a function of the forcing strength. The dotted circles denote how the values of  and  are updated at each continuation step. The plus signs denote points on the entrainment range that are detected by the continuation circles. This allows us to detect sharp corners that may be missed with standard continuation where we only look at vertical slices of parameter space
© Copyright Policy - OpenAccess
Related In: Results  -  Collection

Show All Figures
getmorefigures.php?uid=PMC4835419&req=5

Fig7: Illustration of two-parameter continuation used to find entrainment ranges as a function of the forcing strength. The dotted circles denote how the values of and are updated at each continuation step. The plus signs denote points on the entrainment range that are detected by the continuation circles. This allows us to detect sharp corners that may be missed with standard continuation where we only look at vertical slices of parameter space
Mentions: We used a series of one-parameter continuations instead of two-parameter continuations, because a two-parameter continuation can become inaccurate near degenerate bifurcations [35]. In order to follow these curves in any direction in parameter space, the one-parameter continuations were performed along ellipses in parameter space rather than straight lines, as illustrated in Fig. 7. The larger dotted ellipses represent the path of the continuation steps in the parameter space. These ellipses indicate how the parameters and are updated at each continuation step. We choose the size of the ellipse so that it is large enough to cover a relatively large area in parameter space in order to decrease computation time and also small enough to capture sharp corners of the entrainment range. The small red circles indicate the center of continuation ellipses. To choose the next center, we take a step in the same direction as the previous entrainment point. The points on the entrainment range are indicated by blue plus signs. To better explain this process, consider entrainment points 2 and 3 in Fig. 7. We start with entrainment point 2, which is a known point on the entrainment rage. To get the next center, indicated by the small red circle between points 2 and 3, we step in the same direction as the vector from point 1 to point 2. We then move around the large ellipse, plotted in magenta, and find new fixed points with slightly different values of and δ. To determine points on the boundary of the entrainment range, we compute the stability of the fixed points in each model. Fig. 7

Bottom Line: This results in a simpler model yet maintains some properties of the neural model.Entrainment ranges for chains with nonuniform intersegmental coupling asymmetry are larger when forcing is applied to the middle of the chain than when it is applied to either end, a result that is qualitatively similar to the experimental results.In the limit of weak coupling in the chain, the entrainment results of the neural model approach the entrainment results for the derived phase model.

View Article: PubMed Central - PubMed

Affiliation: Department of Mathematics and Statistics, University of Maryland, Baltimore County, 1000 Hilltop Circle, Baltimore, MD, 21250, USA.

ABSTRACT
Sensory input to the lamprey central pattern generator (CPG) for locomotion is known to have a significant role in modulating lamprey swimming. Lamprey CPGs are known to have the ability to entrain to a bending stimulus, that is, in the presence of a rhythmic signal, the CPG will change its frequency to match the stimulus frequency. Bending experiments in which the lamprey spinal cord has been removed and mechanically bent back and forth at a single point have been used to determine the range of frequencies that can entrain the CPG rhythm. First, we model the lamprey locomotor CPG as a chain of neural oscillators with three classes of neurons and sinusoidal forcing representing edge cell input. We derive a phase model using the connections described in the neural model. This results in a simpler model yet maintains some properties of the neural model. For both the neural model and the derived phase model, entrainment ranges are computed for forcing at different points along the chain while varying both intersegmental coupling strength and the coupling strength between the forcer and chain. Entrainment ranges for chains with nonuniform intersegmental coupling asymmetry are larger when forcing is applied to the middle of the chain than when it is applied to either end, a result that is qualitatively similar to the experimental results. In the limit of weak coupling in the chain, the entrainment results of the neural model approach the entrainment results for the derived phase model. Both biological experiments and the robustness of non-monotonic entrainment ranges as a function of the forcing position across different classes of CPG models with nonuniform asymmetric coupling suggest that a specific property of the intersegmental coupling of the CPG is key to entrainment.

No MeSH data available.


Related in: MedlinePlus