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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

Basic simulation results. Example relative population activity for MN populations resulting from simulation of system (1) with the S weights. MN population identified in the legend. The y-axis represents population activity as rescaled voltage, 0 indicates silent, 1 indicates active. Note that the relative timing and durations of activity in the simulation match the recordings (see Fig. 1). The SCE weights produce the desired relative timing and durations as well (not shown)
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Fig6: Basic simulation results. Example relative population activity for MN populations resulting from simulation of system (1) with the S weights. MN population identified in the legend. The y-axis represents population activity as rescaled voltage, 0 indicates silent, 1 indicates active. Note that the relative timing and durations of activity in the simulation match the recordings (see Fig. 1). The SCE weights produce the desired relative timing and durations as well (not shown)

Mentions: Given these considerations, our model for each interneuron population takes the form 1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \begin{aligned} C_{m} \dot{V_{i}} &= -I_{\mathit {Na}P}(V_{i},h_{i})-I_{L}(V_{i})- \sum_{j \ne i}I_{\mathrm {syn}}(V_{i},s_{j})-I_{\mathrm {ext}}(V_{i}) \equiv F_{i}(V_{i},h_{i},\mathbf{s}), \\ \dot{h_{i}} &= \bigl(h_{\infty}(V_{i})-h_{i} \bigr)\tau_{h}(V_{i}) \equiv g_{i}(V_{i}, h_{i}), \\ \dot{s_{i}} &= \alpha(1-s_{i})s_{\infty}(V_{i})- \beta s_{i}, \end{aligned} $$\end{document}CmVi˙=−INaP(Vi,hi)−IL(Vi)−∑j≠iIsyn(Vi,sj)−Iext(Vi)≡Fi(Vi,hi,s),hi˙=(h∞(Vi)−hi)τh(Vi)≡gi(Vi,hi),si˙=α(1−si)s∞(Vi)−βsi, where denotes voltage, the inactivation of the persistent sodium current , the fraction of the maximal synaptic conductance that is induced by the population’s activity, and s the vector of s variables of all populations in the network (although the evolution of does not depend directly on ). In the voltage equation for population i, , is a leak current, for denotes synaptic current induced by population j, denotes excitatory synaptic current with conductance from a source outside the network, , , and are monotone sigmoidal functions given by , with and increasing and decreasing, and for . All synaptic inputs are defined with ; whether a synaptic input is excitatory or inhibitory is determined by its reversal potential . Default parameter values used in simulations are listed in Table 2; values of are varied and are discussed as they arise in our analysis. Simulations of the above system give physiologically realistic voltage ranges with the parameters used in Table 2. However, because we are interested in relative durations of activity, it is more useful to consider rescaled voltage as a representation of population activity. That is, the population activity, PA, is related to voltage, V, as follows: . This can be seen in Figs. 6, 15, and 16. Table 2


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

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

Basic simulation results. Example relative population activity for MN populations resulting from simulation of system (1) with the S weights. MN population identified in the legend. The y-axis represents population activity as rescaled voltage, 0 indicates silent, 1 indicates active. Note that the relative timing and durations of activity in the simulation match the recordings (see Fig. 1). The SCE weights produce the desired relative timing and durations as well (not shown)
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Related In: Results  -  Collection

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Fig6: Basic simulation results. Example relative population activity for MN populations resulting from simulation of system (1) with the S weights. MN population identified in the legend. The y-axis represents population activity as rescaled voltage, 0 indicates silent, 1 indicates active. Note that the relative timing and durations of activity in the simulation match the recordings (see Fig. 1). The SCE weights produce the desired relative timing and durations as well (not shown)
Mentions: Given these considerations, our model for each interneuron population takes the form 1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \begin{aligned} C_{m} \dot{V_{i}} &= -I_{\mathit {Na}P}(V_{i},h_{i})-I_{L}(V_{i})- \sum_{j \ne i}I_{\mathrm {syn}}(V_{i},s_{j})-I_{\mathrm {ext}}(V_{i}) \equiv F_{i}(V_{i},h_{i},\mathbf{s}), \\ \dot{h_{i}} &= \bigl(h_{\infty}(V_{i})-h_{i} \bigr)\tau_{h}(V_{i}) \equiv g_{i}(V_{i}, h_{i}), \\ \dot{s_{i}} &= \alpha(1-s_{i})s_{\infty}(V_{i})- \beta s_{i}, \end{aligned} $$\end{document}CmVi˙=−INaP(Vi,hi)−IL(Vi)−∑j≠iIsyn(Vi,sj)−Iext(Vi)≡Fi(Vi,hi,s),hi˙=(h∞(Vi)−hi)τh(Vi)≡gi(Vi,hi),si˙=α(1−si)s∞(Vi)−βsi, where denotes voltage, the inactivation of the persistent sodium current , the fraction of the maximal synaptic conductance that is induced by the population’s activity, and s the vector of s variables of all populations in the network (although the evolution of does not depend directly on ). In the voltage equation for population i, , is a leak current, for denotes synaptic current induced by population j, denotes excitatory synaptic current with conductance from a source outside the network, , , and are monotone sigmoidal functions given by , with and increasing and decreasing, and for . All synaptic inputs are defined with ; whether a synaptic input is excitatory or inhibitory is determined by its reversal potential . Default parameter values used in simulations are listed in Table 2; values of are varied and are discussed as they arise in our analysis. Simulations of the above system give physiologically realistic voltage ranges with the parameters used in Table 2. However, because we are interested in relative durations of activity, it is more useful to consider rescaled voltage as a representation of population activity. That is, the population activity, PA, is related to voltage, V, as follows: . This can be seen in Figs. 6, 15, and 16. Table 2

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