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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

For each type of neural connection between E, L, and C cells, an  function is computed to represent the effects of neurons on the voltage of the neuron within the oscillator. The six  functions are computed for connections from L to E cells, C to E cells, C to L cells, E to C cells, L to C cells, and C to C cells. Here we show only half of the neuron-to-neuron connections in Fig. 1 because of the left–right symmetry within the oscillator
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Fig4: For each type of neural connection between E, L, and C cells, an function is computed to represent the effects of neurons on the voltage of the neuron within the oscillator. The six functions are computed for connections from L to E cells, C to E cells, C to L cells, E to C cells, L to C cells, and C to C cells. Here we show only half of the neuron-to-neuron connections in Fig. 1 because of the left–right symmetry within the oscillator

Mentions: Now, using (4) and (5), we can write the phase model (2a), (2b) as 6a\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{aligned} \dot{\theta}_{i} &= \omega_{0} + \sum _{\substack{k=1\\k \neq i}}^{n} \alpha_{i-k} H_{i-k}( \theta_{k} - \theta_{i}) + \delta_{im} \alpha_{\mathrm{f}} H_{\mathrm{f}} (\theta_{\mathrm {f}} - \theta_{i}), \quad\mbox{for }i=1,\ldots,n, \end{aligned}$$ \end{document}θ˙i=ω0+∑k=1k≠inαi−kHi−k(θk−θi)+δimαfHf(θf−θi),for i=1,…,n,6b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{aligned} \dot{\theta}_{\mathrm{f}} &= \omega_{\mathrm{f}}. \end{aligned}$$ \end{document}θ˙f=ωf. Model (6a), (6b) has the standard form of a chain of coupled phase oscillators forced at one location. To specify this model, two choices remain. First, for each connection length r we must specify the connection strength ratios in (4) that determine the coupling function . We defer this specification until we have computed the coupling function for each connection type c (see Fig. 4 below). Second, we must specify how coupling strength depends on r. Experimental evidence does not provide the exact form of this dependence but does indicate an asymmetry in ascending and descending coupling strengths [7, 26, 27]. Among the possible modeling choices in the literature (e.g. [11, 28]), we will follow Varkonyi et al. [29] and assume that the coupling strength decays exponentially with coupling length: 7\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\alpha}_{r} = \textstyle\begin{cases} A_{d} e^{-/r//\lambda_{d}} & \mbox{for $r>0$ (descending connections),} \\ A_{a} e^{-/r//\lambda_{a}} & \mbox{for $r< 0$ (ascending connections),} \\ 1 & \mbox{for $r=0$ (intrasegmental connections),} \end{cases} $$\end{document}αr={Ade−/r//λdfor r>0 (descending connections),Aae−/r//λafor r<0 (ascending connections),1for r=0 (intrasegmental connections), where , and , are the amplitudes and length constants for descending and ascending coupling, respectively. Representative parameter values can be found in the caption of Fig. 8.


Entrainment Ranges for Chains of Forced Neural and Phase Oscillators.

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

For each type of neural connection between E, L, and C cells, an  function is computed to represent the effects of neurons on the voltage of the neuron within the oscillator. The six  functions are computed for connections from L to E cells, C to E cells, C to L cells, E to C cells, L to C cells, and C to C cells. Here we show only half of the neuron-to-neuron connections in Fig. 1 because of the left–right symmetry within the oscillator
© Copyright Policy - OpenAccess
Related In: Results  -  Collection

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

Fig4: For each type of neural connection between E, L, and C cells, an function is computed to represent the effects of neurons on the voltage of the neuron within the oscillator. The six functions are computed for connections from L to E cells, C to E cells, C to L cells, E to C cells, L to C cells, and C to C cells. Here we show only half of the neuron-to-neuron connections in Fig. 1 because of the left–right symmetry within the oscillator
Mentions: Now, using (4) and (5), we can write the phase model (2a), (2b) as 6a\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{aligned} \dot{\theta}_{i} &= \omega_{0} + \sum _{\substack{k=1\\k \neq i}}^{n} \alpha_{i-k} H_{i-k}( \theta_{k} - \theta_{i}) + \delta_{im} \alpha_{\mathrm{f}} H_{\mathrm{f}} (\theta_{\mathrm {f}} - \theta_{i}), \quad\mbox{for }i=1,\ldots,n, \end{aligned}$$ \end{document}θ˙i=ω0+∑k=1k≠inαi−kHi−k(θk−θi)+δimαfHf(θf−θi),for i=1,…,n,6b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{aligned} \dot{\theta}_{\mathrm{f}} &= \omega_{\mathrm{f}}. \end{aligned}$$ \end{document}θ˙f=ωf. Model (6a), (6b) has the standard form of a chain of coupled phase oscillators forced at one location. To specify this model, two choices remain. First, for each connection length r we must specify the connection strength ratios in (4) that determine the coupling function . We defer this specification until we have computed the coupling function for each connection type c (see Fig. 4 below). Second, we must specify how coupling strength depends on r. Experimental evidence does not provide the exact form of this dependence but does indicate an asymmetry in ascending and descending coupling strengths [7, 26, 27]. Among the possible modeling choices in the literature (e.g. [11, 28]), we will follow Varkonyi et al. [29] and assume that the coupling strength decays exponentially with coupling length: 7\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\alpha}_{r} = \textstyle\begin{cases} A_{d} e^{-/r//\lambda_{d}} & \mbox{for $r>0$ (descending connections),} \\ A_{a} e^{-/r//\lambda_{a}} & \mbox{for $r< 0$ (ascending connections),} \\ 1 & \mbox{for $r=0$ (intrasegmental connections),} \end{cases} $$\end{document}αr={Ade−/r//λdfor r>0 (descending connections),Aae−/r//λafor r<0 (ascending connections),1for r=0 (intrasegmental connections), where , and , are the amplitudes and length constants for descending and ascending coupling, respectively. Representative parameter values can be found in the caption of Fig. 8.

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