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Conditions for Multi-functionality in a Rhythm Generating Network Inspired by Turtle Scratching.

Snyder AC, Rubin JE - J Math Neurosci (2015)

Bottom Line: We show through simulation that the proposed network can achieve the desired multi-functionality, even though it relies on hip unit generators to recruit appropriately timed knee extensor motoneuron activity, including a delay relative to hip activation in rostral scratch.Furthermore, we develop a phase space representation, focusing on the inputs to and the intrinsic slow variable of the knee extensor motoneuron, which we use to derive sufficient conditions for the network to realize each rhythm and which illustrates the role of a saddle-node bifurcation in achieving the knee extensor delay.This framework is harnessed to consider bistability and to make predictions about the responses of the scratch rhythms to input changes for future experimental testing.

View Article: PubMed Central - PubMed

Affiliation: Department of Mathematics, University of Pittsburgh, 301 Thackeray Hall, Pittsburgh, PA, 15260, USA, acs73@pitt.edu.

ABSTRACT
Rhythmic behaviors such as breathing, walking, and scratching are vital to many species. Such behaviors can emerge from groups of neurons, called central pattern generators, in the absence of rhythmic inputs. In vertebrates, the identification of the cells that constitute the central pattern generator for particular rhythmic behaviors is difficult, and often, its existence has only been inferred. For example, under experimental conditions, intact turtles generate several rhythmic scratch motor patterns corresponding to non-rhythmic stimulation of different body regions. These patterns feature alternating phases of motoneuron activation that occur repeatedly, with different patterns distinguished by the relative timing and duration of activity of hip extensor, hip flexor, and knee extensor motoneurons. While the central pattern generator network responsible for these outputs has not been located, there is hope to use motoneuron recordings to deduce its properties. To this end, this work presents a model of a previously proposed central pattern generator network and analyzes its capability to produce two distinct scratch rhythms from a single neuron pool, selected by different combinations of tonic drive parameters but with fixed strengths of connections within the network. We show through simulation that the proposed network can achieve the desired multi-functionality, even though it relies on hip unit generators to recruit appropriately timed knee extensor motoneuron activity, including a delay relative to hip activation in rostral scratch. Furthermore, we develop a phase space representation, focusing on the inputs to and the intrinsic slow variable of the knee extensor motoneuron, which we use to derive sufficient conditions for the network to realize each rhythm and which illustrates the role of a saddle-node bifurcation in achieving the knee extensor delay. This framework is harnessed to consider bistability and to make predictions about the responses of the scratch rhythms to input changes for future experimental testing.

No MeSH data available.


Related in: MedlinePlus

Phase space views for the KE dynamics in the reduced module shown in Fig. 7 during the pocket rhythm. Top left: full three-dimensional slow phase space. Top right: projections onto the two two-dimensional planes where the trajectory lies. Bottom: single, combined two-dimensional representation. In all plots, black and red curves are projections of parts or all of the trajectory of a periodic pocket scratch solution, with bold black and thin red denoting times when EP is active and bold red and thin black times when ER is active. Green curves denote the fixed point curves for KE (stable, solid),  (unstable, dashed), and  (stable, solid) (in order of increasing ) while EP is active. Magenta curves denote the analogous curves of fixed points for KE while ER is active. The dark blue curve is the curve of jump down knees for KE while EP is active; cyan curves are jump down knees and jump up knees (larger  values) for KE while ER is active. Finally, dashed black curves in the top right indicate points on the two projections that correspond to the same times, when the switches between the EP active phase and the ER active phase occur. Additional labeling on the top right indicates relevant structures defined above. Additional labeling on the bottom indicates key changes in activity of various populations throughout the rhythms. Gray tick marks indicate transitions from activity to silence. This labeling holds for all panels and future figures
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Fig8: Phase space views for the KE dynamics in the reduced module shown in Fig. 7 during the pocket rhythm. Top left: full three-dimensional slow phase space. Top right: projections onto the two two-dimensional planes where the trajectory lies. Bottom: single, combined two-dimensional representation. In all plots, black and red curves are projections of parts or all of the trajectory of a periodic pocket scratch solution, with bold black and thin red denoting times when EP is active and bold red and thin black times when ER is active. Green curves denote the fixed point curves for KE (stable, solid), (unstable, dashed), and (stable, solid) (in order of increasing ) while EP is active. Magenta curves denote the analogous curves of fixed points for KE while ER is active. The dark blue curve is the curve of jump down knees for KE while EP is active; cyan curves are jump down knees and jump up knees (larger values) for KE while ER is active. Finally, dashed black curves in the top right indicate points on the two projections that correspond to the same times, when the switches between the EP active phase and the ER active phase occur. Additional labeling on the top right indicates relevant structures defined above. Additional labeling on the bottom indicates key changes in activity of various populations throughout the rhythms. Gray tick marks indicate transitions from activity to silence. This labeling holds for all panels and future figures

Mentions: To reduce dimension further, we identify the interneuron pairs that activate together, and , to form a single half-center oscillator and we consider a reduced model to describe KE activity, illustrated in Fig. 7. With this reduction, using , , and , the synaptic input for knee extensor becomes \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_{\mathrm {syn}}^{\mathit {KE}} = s_{\mathit {ER}}\bigl[ (g_{\mathit {IP}}+g_{\mathit {ER}})V_{\mathit {KE}}-g_{\mathit {IP}}e_{\mathrm {syn}}^{\mathrm {inh}} \bigr]+s_{\mathit {EP}}\bigl[ (g_{\mathit {IR}}+g_{\mathit {EP}})V_{\mathit {KE}}-g_{\mathit {IR}}e_{\mathrm {syn}}^{\mathrm {inh}} \bigr]. $$\end{document}IsynKE=sER[(gIP+gER)VKE−gIPesyninh]+sEP[(gIR+gEP)VKE−gIResyninh]. This step reduces our phase space from five dimensions to three, with variables . The projection of the periodic pocket trajectory of the reduced model to space is shown in the top left of Fig. 8, along with several curves that are important for understanding KE dynamics. These plots are critical to our analysis. When ER is active, , so the corresponding part of the trajectory, color coded red, lies approximately on the plane within phase space, which is the back right face of the cube shown. Similarly, the epoch with EP active has and yields a trajectory, color coded black, near the back left face of the cube. As an alternative to considering a three-dimensional phase space, however, it is convenient to switch between a pair of two-dimensional slow phase planes, corresponding to the back two faces in the top left of Fig. 8, as EP and ER alternate between periods of silence and activity. These are shown in the top right of Fig. 8. For example, while EP is active, evolves and the projection of the trajectory to the plane is shown as the thick black curve. Of course, even after EP switches from active to silent, the projection of the trajectory to the plane still exists; the projected trajectory segment after the switch is shown as the thin black curve. Using similar considerations for the projection to , we in fact plot two copies of the full trajectory, each in its own two-dimensional phase plane, one with the trajectory shown thick while EP is active and thin while ER is active, and the other the opposite. The switch from EP active to ER active occurs abruptly when begins its slow decay from and increases very rapidly (instantly in the singular limit) to , and we switch each curve from thick to thin when occurs. Fig. 7


Conditions for Multi-functionality in a Rhythm Generating Network Inspired by Turtle Scratching.

Snyder AC, Rubin JE - J Math Neurosci (2015)

Phase space views for the KE dynamics in the reduced module shown in Fig. 7 during the pocket rhythm. Top left: full three-dimensional slow phase space. Top right: projections onto the two two-dimensional planes where the trajectory lies. Bottom: single, combined two-dimensional representation. In all plots, black and red curves are projections of parts or all of the trajectory of a periodic pocket scratch solution, with bold black and thin red denoting times when EP is active and bold red and thin black times when ER is active. Green curves denote the fixed point curves for KE (stable, solid),  (unstable, dashed), and  (stable, solid) (in order of increasing ) while EP is active. Magenta curves denote the analogous curves of fixed points for KE while ER is active. The dark blue curve is the curve of jump down knees for KE while EP is active; cyan curves are jump down knees and jump up knees (larger  values) for KE while ER is active. Finally, dashed black curves in the top right indicate points on the two projections that correspond to the same times, when the switches between the EP active phase and the ER active phase occur. Additional labeling on the top right indicates relevant structures defined above. Additional labeling on the bottom indicates key changes in activity of various populations throughout the rhythms. Gray tick marks indicate transitions from activity to silence. This labeling holds for all panels and future figures
© Copyright Policy - OpenAccess
Related In: Results  -  Collection

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Fig8: Phase space views for the KE dynamics in the reduced module shown in Fig. 7 during the pocket rhythm. Top left: full three-dimensional slow phase space. Top right: projections onto the two two-dimensional planes where the trajectory lies. Bottom: single, combined two-dimensional representation. In all plots, black and red curves are projections of parts or all of the trajectory of a periodic pocket scratch solution, with bold black and thin red denoting times when EP is active and bold red and thin black times when ER is active. Green curves denote the fixed point curves for KE (stable, solid), (unstable, dashed), and (stable, solid) (in order of increasing ) while EP is active. Magenta curves denote the analogous curves of fixed points for KE while ER is active. The dark blue curve is the curve of jump down knees for KE while EP is active; cyan curves are jump down knees and jump up knees (larger values) for KE while ER is active. Finally, dashed black curves in the top right indicate points on the two projections that correspond to the same times, when the switches between the EP active phase and the ER active phase occur. Additional labeling on the top right indicates relevant structures defined above. Additional labeling on the bottom indicates key changes in activity of various populations throughout the rhythms. Gray tick marks indicate transitions from activity to silence. This labeling holds for all panels and future figures
Mentions: To reduce dimension further, we identify the interneuron pairs that activate together, and , to form a single half-center oscillator and we consider a reduced model to describe KE activity, illustrated in Fig. 7. With this reduction, using , , and , the synaptic input for knee extensor becomes \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_{\mathrm {syn}}^{\mathit {KE}} = s_{\mathit {ER}}\bigl[ (g_{\mathit {IP}}+g_{\mathit {ER}})V_{\mathit {KE}}-g_{\mathit {IP}}e_{\mathrm {syn}}^{\mathrm {inh}} \bigr]+s_{\mathit {EP}}\bigl[ (g_{\mathit {IR}}+g_{\mathit {EP}})V_{\mathit {KE}}-g_{\mathit {IR}}e_{\mathrm {syn}}^{\mathrm {inh}} \bigr]. $$\end{document}IsynKE=sER[(gIP+gER)VKE−gIPesyninh]+sEP[(gIR+gEP)VKE−gIResyninh]. This step reduces our phase space from five dimensions to three, with variables . The projection of the periodic pocket trajectory of the reduced model to space is shown in the top left of Fig. 8, along with several curves that are important for understanding KE dynamics. These plots are critical to our analysis. When ER is active, , so the corresponding part of the trajectory, color coded red, lies approximately on the plane within phase space, which is the back right face of the cube shown. Similarly, the epoch with EP active has and yields a trajectory, color coded black, near the back left face of the cube. As an alternative to considering a three-dimensional phase space, however, it is convenient to switch between a pair of two-dimensional slow phase planes, corresponding to the back two faces in the top left of Fig. 8, as EP and ER alternate between periods of silence and activity. These are shown in the top right of Fig. 8. For example, while EP is active, evolves and the projection of the trajectory to the plane is shown as the thick black curve. Of course, even after EP switches from active to silent, the projection of the trajectory to the plane still exists; the projected trajectory segment after the switch is shown as the thin black curve. Using similar considerations for the projection to , we in fact plot two copies of the full trajectory, each in its own two-dimensional phase plane, one with the trajectory shown thick while EP is active and thin while ER is active, and the other the opposite. The switch from EP active to ER active occurs abruptly when begins its slow decay from and increases very rapidly (instantly in the singular limit) to , and we switch each curve from thick to thin when occurs. Fig. 7

Bottom Line: We show through simulation that the proposed network can achieve the desired multi-functionality, even though it relies on hip unit generators to recruit appropriately timed knee extensor motoneuron activity, including a delay relative to hip activation in rostral scratch.Furthermore, we develop a phase space representation, focusing on the inputs to and the intrinsic slow variable of the knee extensor motoneuron, which we use to derive sufficient conditions for the network to realize each rhythm and which illustrates the role of a saddle-node bifurcation in achieving the knee extensor delay.This framework is harnessed to consider bistability and to make predictions about the responses of the scratch rhythms to input changes for future experimental testing.

View Article: PubMed Central - PubMed

Affiliation: Department of Mathematics, University of Pittsburgh, 301 Thackeray Hall, Pittsburgh, PA, 15260, USA, acs73@pitt.edu.

ABSTRACT
Rhythmic behaviors such as breathing, walking, and scratching are vital to many species. Such behaviors can emerge from groups of neurons, called central pattern generators, in the absence of rhythmic inputs. In vertebrates, the identification of the cells that constitute the central pattern generator for particular rhythmic behaviors is difficult, and often, its existence has only been inferred. For example, under experimental conditions, intact turtles generate several rhythmic scratch motor patterns corresponding to non-rhythmic stimulation of different body regions. These patterns feature alternating phases of motoneuron activation that occur repeatedly, with different patterns distinguished by the relative timing and duration of activity of hip extensor, hip flexor, and knee extensor motoneurons. While the central pattern generator network responsible for these outputs has not been located, there is hope to use motoneuron recordings to deduce its properties. To this end, this work presents a model of a previously proposed central pattern generator network and analyzes its capability to produce two distinct scratch rhythms from a single neuron pool, selected by different combinations of tonic drive parameters but with fixed strengths of connections within the network. We show through simulation that the proposed network can achieve the desired multi-functionality, even though it relies on hip unit generators to recruit appropriately timed knee extensor motoneuron activity, including a delay relative to hip activation in rostral scratch. Furthermore, we develop a phase space representation, focusing on the inputs to and the intrinsic slow variable of the knee extensor motoneuron, which we use to derive sufficient conditions for the network to realize each rhythm and which illustrates the role of a saddle-node bifurcation in achieving the knee extensor delay. This framework is harnessed to consider bistability and to make predictions about the responses of the scratch rhythms to input changes for future experimental testing.

No MeSH data available.


Related in: MedlinePlus