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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

Relative strengths  of different connection types as a function of the connection length r
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Fig5: Relative strengths of different connection types as a function of the connection length r

Mentions: The model is connectionist with one variable per cell: is the “voltage” of cell j in segment i, scaled to be unitless and lie between −1 and 1. (For convenience we use the term “cell” to refer to a class of cells.) When the cell does not fire action potentials and represents the membrane voltage of the cell body. When the cell fires action potentials and represents the normalized firing rate. Although the model is connectionist, its form is similar to conductance-based models such as the Hodgkin–Huxley model [25] with the time derivative of voltage proportional to the sum of “currents”, each with its own reversal potential. The reversal potentials are in the range from −1 to 1, so that voltage remains in this same range. The model is 1a\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{aligned} \dot{v}_{ij} = & -G_{R}v_{ij} + G_{T}^{j}(1 - v_{ij}) + \sum _{k=1}^{n} \sum_{l=1}^{6} \alpha_{i-k}^{lj} G_{0}^{lj}h(v_{kl}) \bigl(V_{\mathrm{syn}}^{l} - v_{ij}\bigr) \\ &{}+ \delta_{im} \alpha_{\mathrm{f}} \sum _{s=1}^{2} G_{\mathrm{f}}^{sj} h \bigl(v_{\mathrm{ec}}^{s}(\theta_{\mathrm{f}})\bigr) \bigl(V_{\mathrm{syn},\mathrm {ec}}^{sj} - v_{ij}\bigr), \\ &\hspace{-12pt}\mbox{for }i=1,\ldots,n; j=1,\ldots,6, \end{aligned}$$ \end{document}v˙ij=−GRvij+GTj(1−vij)+∑k=1n∑l=16αi−kljG0ljh(vkl)(Vsynl−vij)+δimαf∑s=12Gfsjh(vecs(θf))(Vsyn,ecsj−vij),for i=1,…,n;j=1,…,6,1b\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, where 1c\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ h(x) = \sigma\log \bigl(1 + e^{{x}/{\sigma}} \bigr) $$\end{document}h(x)=σlog(1+ex/σ) is a smooth threshold function and 1d\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ v_{\mathrm{ec}}^{s}(\theta_{\mathrm{f}}) = (-1)^{s} \sin(2\pi\theta _{\mathrm{f}}) $$\end{document}vecs(θf)=(−1)ssin(2πθf) is the edge cell voltage with s denoting the left or the right side as illustrated in Fig. 1. (See Table 1 for a list of the model parameters and their values.) In Eqs. (1a)–(1d), n represents the number of spinal cord segments in the experimental preparation being modeled. We choose as a compromise between required computation time and approximating the large number of segments in experimental preparations, where n can approach 50. On the right side of (1a), the first term represents the resting conductance that drives the voltage toward 0. The second term represents the tonic excitatory conductance that drives the voltage toward 1. The third term, the double summation, represents the influence of other neurons on , which occurs via the intrasegmental () and intersegmental () connections. The term is the maximal synaptic conduction of the connection from cell l of oscillator k to cell j of oscillator i. Cell indices are indicated in Fig. 1. Note that the maximal synaptic conductance does not depend on the absolute positions of the two oscillators in the chain, but only on the signed distance between them, . Note for convenience, we refer to r as the connection length, where negative values correspond to ascending connections and positive values correspond to descending connections. For intrasegmental connections, , and is the maximal synaptic conductance. For intersegmental connections, expresses the maximal synaptic conductance as a fraction of the maximal synaptic conductance of the intrasegmental connection of the same type. Figure 5 illustrates the synaptic conductances for connections between E and C cells, L and C cells, and all other cellular connections. We refer to as connection strength and describe how connection strengths are specified when we consider the phase-model approximation in Sect. 3. The threshold function h given by (1c) describes how coupling depends on the voltage of the presynaptic cell. This function represents an activation threshold, where once the voltage of the neuron reaches a certain threshold it becomes “active.” In contrast to the models of Buchanan [23] and Williams [24], which use a piecewise-linear h, we chose a smooth h to facilitate our computational analysis. As σ decreases to 0, the smooth function approaches the non-smooth version of [23] and [24]. We used in our simulations. A connection between cells drives the postsynaptic cell’s voltage toward the synaptic reversal potential , which depends on the type of the presynaptic cell l. If cell l is an E cell, which is excitatory, then ; if cell l is an L or C cell, which are inhibitory, then . Table 1


Entrainment Ranges for Chains of Forced Neural and Phase Oscillators.

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

Relative strengths  of different connection types as a function of the connection length r
© Copyright Policy - OpenAccess
Related In: Results  -  Collection

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

Fig5: Relative strengths of different connection types as a function of the connection length r
Mentions: The model is connectionist with one variable per cell: is the “voltage” of cell j in segment i, scaled to be unitless and lie between −1 and 1. (For convenience we use the term “cell” to refer to a class of cells.) When the cell does not fire action potentials and represents the membrane voltage of the cell body. When the cell fires action potentials and represents the normalized firing rate. Although the model is connectionist, its form is similar to conductance-based models such as the Hodgkin–Huxley model [25] with the time derivative of voltage proportional to the sum of “currents”, each with its own reversal potential. The reversal potentials are in the range from −1 to 1, so that voltage remains in this same range. The model is 1a\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{aligned} \dot{v}_{ij} = & -G_{R}v_{ij} + G_{T}^{j}(1 - v_{ij}) + \sum _{k=1}^{n} \sum_{l=1}^{6} \alpha_{i-k}^{lj} G_{0}^{lj}h(v_{kl}) \bigl(V_{\mathrm{syn}}^{l} - v_{ij}\bigr) \\ &{}+ \delta_{im} \alpha_{\mathrm{f}} \sum _{s=1}^{2} G_{\mathrm{f}}^{sj} h \bigl(v_{\mathrm{ec}}^{s}(\theta_{\mathrm{f}})\bigr) \bigl(V_{\mathrm{syn},\mathrm {ec}}^{sj} - v_{ij}\bigr), \\ &\hspace{-12pt}\mbox{for }i=1,\ldots,n; j=1,\ldots,6, \end{aligned}$$ \end{document}v˙ij=−GRvij+GTj(1−vij)+∑k=1n∑l=16αi−kljG0ljh(vkl)(Vsynl−vij)+δimαf∑s=12Gfsjh(vecs(θf))(Vsyn,ecsj−vij),for i=1,…,n;j=1,…,6,1b\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, where 1c\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ h(x) = \sigma\log \bigl(1 + e^{{x}/{\sigma}} \bigr) $$\end{document}h(x)=σlog(1+ex/σ) is a smooth threshold function and 1d\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ v_{\mathrm{ec}}^{s}(\theta_{\mathrm{f}}) = (-1)^{s} \sin(2\pi\theta _{\mathrm{f}}) $$\end{document}vecs(θf)=(−1)ssin(2πθf) is the edge cell voltage with s denoting the left or the right side as illustrated in Fig. 1. (See Table 1 for a list of the model parameters and their values.) In Eqs. (1a)–(1d), n represents the number of spinal cord segments in the experimental preparation being modeled. We choose as a compromise between required computation time and approximating the large number of segments in experimental preparations, where n can approach 50. On the right side of (1a), the first term represents the resting conductance that drives the voltage toward 0. The second term represents the tonic excitatory conductance that drives the voltage toward 1. The third term, the double summation, represents the influence of other neurons on , which occurs via the intrasegmental () and intersegmental () connections. The term is the maximal synaptic conduction of the connection from cell l of oscillator k to cell j of oscillator i. Cell indices are indicated in Fig. 1. Note that the maximal synaptic conductance does not depend on the absolute positions of the two oscillators in the chain, but only on the signed distance between them, . Note for convenience, we refer to r as the connection length, where negative values correspond to ascending connections and positive values correspond to descending connections. For intrasegmental connections, , and is the maximal synaptic conductance. For intersegmental connections, expresses the maximal synaptic conductance as a fraction of the maximal synaptic conductance of the intrasegmental connection of the same type. Figure 5 illustrates the synaptic conductances for connections between E and C cells, L and C cells, and all other cellular connections. We refer to as connection strength and describe how connection strengths are specified when we consider the phase-model approximation in Sect. 3. The threshold function h given by (1c) describes how coupling depends on the voltage of the presynaptic cell. This function represents an activation threshold, where once the voltage of the neuron reaches a certain threshold it becomes “active.” In contrast to the models of Buchanan [23] and Williams [24], which use a piecewise-linear h, we chose a smooth h to facilitate our computational analysis. As σ decreases to 0, the smooth function approaches the non-smooth version of [23] and [24]. We used in our simulations. A connection between cells drives the postsynaptic cell’s voltage toward the synaptic reversal potential , which depends on the type of the presynaptic cell l. If cell l is an E cell, which is excitatory, then ; if cell l is an L or C cell, which are inhibitory, then . Table 1

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