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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

Simulation of a single segment within a chain of oscillators defined by (1a)–(1d) for two cycles without forcing. Plot shows the cell voltages within the first segment (). Weak intersegmental coupling, defined by , , , was used to connect segments. Thus, the solution for the oscillator in the chain closely approximates the solution for a single, uncoupled oscillator. Note the spatiotemporal symmetry between left and right cells. The voltage of the left cells is the same as the voltage of the right cells except for a phase shift of half a period
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Fig2: Simulation of a single segment within a chain of oscillators defined by (1a)–(1d) for two cycles without forcing. Plot shows the cell voltages within the first segment (). Weak intersegmental coupling, defined by , , , was used to connect segments. Thus, the solution for the oscillator in the chain closely approximates the solution for a single, uncoupled oscillator. Note the spatiotemporal symmetry between left and right cells. The voltage of the left cells is the same as the voltage of the right cells except for a phase shift of half a period

Mentions: Recall that intersegmental connections have the same connectivity as the intrasegmental connections shown in Fig. 1. For example, given coupling length r, there are 12 nonzero corresponding to 2 connections for each of 6 connection types: E to C, E to L, L to C, C to E, C to L, and C to C. Due to the right–left symmetry of the neural model and the left–right spatiotemporal symmetry of the segmental oscillator’s limit cycle, two connections of the same type have the same connection strength and same coupling function. Note these symmetries can be seen in Fig. 2, which depicts the steady state of the neural model for one segment, simulated without forcing. The left and right cells have the same voltage with a phase shift of half a period. Therefore, we can write 3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \sum_{j=1}^{6} \sum _{l=1}^{6} \alpha_{r}^{lj} H^{lj} = \sum_{c \in\mathcal{C}} \alpha_{r,c} H_{c}, \quad \mbox{where $\mathcal{C} = \{\mathrm{EL},\mathrm{EC},\mathrm{LC},\mathrm{CE},\mathrm{CL},\mathrm{CC}\}$} $$\end{document}∑j=16∑l=16αrljHlj=∑c∈Cαr,cHc,where C={EL,EC,LC,CE,CL,CC} where, for example, and . Let be the mean of for . We define , the coupling function of the length r, as 4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ H_{r} = \frac{1}{\alpha_{r}} \sum_{j=1}^{6} \sum_{l=1}^{6} \alpha_{r}^{lj} H^{lj} = \sum_{c \in\mathcal{C}} \frac{\alpha_{r,c}}{\alpha_{r}} H_{c}. $$\end{document}Hr=1αr∑j=16∑l=16αrljHlj=∑c∈Cαr,cαrHc.Fig. 2


Entrainment Ranges for Chains of Forced Neural and Phase Oscillators.

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

Simulation of a single segment within a chain of oscillators defined by (1a)–(1d) for two cycles without forcing. Plot shows the cell voltages within the first segment (). Weak intersegmental coupling, defined by , , , was used to connect segments. Thus, the solution for the oscillator in the chain closely approximates the solution for a single, uncoupled oscillator. Note the spatiotemporal symmetry between left and right cells. The voltage of the left cells is the same as the voltage of the right cells except for a phase shift of half a period
© Copyright Policy - OpenAccess
Related In: Results  -  Collection

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

Fig2: Simulation of a single segment within a chain of oscillators defined by (1a)–(1d) for two cycles without forcing. Plot shows the cell voltages within the first segment (). Weak intersegmental coupling, defined by , , , was used to connect segments. Thus, the solution for the oscillator in the chain closely approximates the solution for a single, uncoupled oscillator. Note the spatiotemporal symmetry between left and right cells. The voltage of the left cells is the same as the voltage of the right cells except for a phase shift of half a period
Mentions: Recall that intersegmental connections have the same connectivity as the intrasegmental connections shown in Fig. 1. For example, given coupling length r, there are 12 nonzero corresponding to 2 connections for each of 6 connection types: E to C, E to L, L to C, C to E, C to L, and C to C. Due to the right–left symmetry of the neural model and the left–right spatiotemporal symmetry of the segmental oscillator’s limit cycle, two connections of the same type have the same connection strength and same coupling function. Note these symmetries can be seen in Fig. 2, which depicts the steady state of the neural model for one segment, simulated without forcing. The left and right cells have the same voltage with a phase shift of half a period. Therefore, we can write 3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \sum_{j=1}^{6} \sum _{l=1}^{6} \alpha_{r}^{lj} H^{lj} = \sum_{c \in\mathcal{C}} \alpha_{r,c} H_{c}, \quad \mbox{where $\mathcal{C} = \{\mathrm{EL},\mathrm{EC},\mathrm{LC},\mathrm{CE},\mathrm{CL},\mathrm{CC}\}$} $$\end{document}∑j=16∑l=16αrljHlj=∑c∈Cαr,cHc,where C={EL,EC,LC,CE,CL,CC} where, for example, and . Let be the mean of for . We define , the coupling function of the length r, as 4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ H_{r} = \frac{1}{\alpha_{r}} \sum_{j=1}^{6} \sum_{l=1}^{6} \alpha_{r}^{lj} H^{lj} = \sum_{c \in\mathcal{C}} \frac{\alpha_{r,c}}{\alpha_{r}} H_{c}. $$\end{document}Hr=1αr∑j=16∑l=16αrljHlj=∑c∈Cαr,cαrHc.Fig. 2

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