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Mechanisms of left-right coordination in mammalian locomotor pattern generation circuits: a mathematical modeling view.

Molkov YI, Bacak BJ, Talpalar AE, Rybak IA - PLoS Comput. Biol. (2015)

Bottom Line: The locomotor frequency was controlled by a parameter defining the excitation of neurons and commissural pathways mimicking the effects of N-methyl-D-aspartate on locomotor frequency in isolated rodent spinal cord preparations.We used bifurcation theory and fast-slow decomposition methods to analyze network behavior in the above regimes and transitions between them.The model reproduced, and suggested explanation for, a series of experimental phenomena and generated predictions available for experimental testing.

View Article: PubMed Central - PubMed

Affiliation: Department of Mathematical Sciences, Indiana University-Purdue University, Indianapolis, Indiana, United States of America.

ABSTRACT
The locomotor gait in limbed animals is defined by the left-right leg coordination and locomotor speed. Coordination between left and right neural activities in the spinal cord controlling left and right legs is provided by commissural interneurons (CINs). Several CIN types have been genetically identified, including the excitatory V3 and excitatory and inhibitory V0 types. Recent studies demonstrated that genetic elimination of all V0 CINs caused switching from a normal left-right alternating activity to a left-right synchronized "hopping" pattern. Furthermore, ablation of only the inhibitory V0 CINs (V0D subtype) resulted in a lack of left-right alternation at low locomotor frequencies and retaining this alternation at high frequencies, whereas selective ablation of the excitatory V0 neurons (V0V subtype) maintained the left-right alternation at low frequencies and switched to a hopping pattern at high frequencies. To analyze these findings, we developed a simplified mathematical model of neural circuits consisting of four pacemaker neurons representing left and right, flexor and extensor rhythm-generating centers interacting via commissural pathways representing V3, V0D, and V0V CINs. The locomotor frequency was controlled by a parameter defining the excitation of neurons and commissural pathways mimicking the effects of N-methyl-D-aspartate on locomotor frequency in isolated rodent spinal cord preparations. The model demonstrated a typical left-right alternating pattern under control conditions, switching to a hopping activity at any frequency after removing both V0 connections, a synchronized pattern at low frequencies with alternation at high frequencies after removing only V0D connections, and an alternating pattern at low frequencies with hopping at high frequencies after removing only V0V connections. We used bifurcation theory and fast-slow decomposition methods to analyze network behavior in the above regimes and transitions between them. The model reproduced, and suggested explanation for, a series of experimental phenomena and generated predictions available for experimental testing.

No MeSH data available.


Related in: MedlinePlus

Fast-slow decomposition of the model dynamics when the inhibitory V0D pathways are deleted.Projection of the phase portrait of the model onto the (V, hNaP) plane at α < X6. Sample positions, or image points, of flexor centers are depicted by the black (RF) and blue (LF) circles. A. Input from the right extensor center (RE) to the left flexor centers (LF) is represented as a sleeve of clines ranging from the maximal RE activity (lowest green cline) to minimal RE activity (highest green cline). The maximal and minimal RE activities occur at the beginning and end of the RF inactive phase, respectively. The red cline represents the maximal excitation to the LF from the RF when the RF is active (labeled as “RF on/RE off”). This maximum occurs immediately after the RF activates and begins to excite the LF via the V3 connection. Hence, for any of the LF's initial conditions above the left knee of the red cline (the blue area) the activation of the RF will immediately result in activation of the LF as well (release-on-escape). For low enough values of α the left knee of the green band is higher than the left knee of the red cline. This makes it possible for the LF to climb high enough during the inactive phase of the RF to find itself in the blue area by the time of escape of the latter. Once the RF activates, the LF activates as well thus stabilizing the regime of synchronous oscillations. B.α > X8 the red and green clines interchange their positions so that the left knee of the red band is now higher than the left knee of the green cline. Accordingly, it is no longer possible for the LF to get above the left knee of the low red cline, while moving along the left branch of the green cline, thus making the release-on-escape mechanism impossible and excluding synchronized behavior (see Fig 7B). Instead, such a configuration enables the release-on-shutdown mechanism, because starting at initial conditions above the left knee of the green cline (the yellow area) the LF immediately activates upon the deactivation of the RF. Symmetric alternation (Δφ = 0.5) is stabilized by the release-on-shutdown mechanism given that each flexor center’s image point climbs high enough along the left branch of the red band before the contralateral flexor center deactivates. C. Overlap of the flexor and extensor center cline bands is denoted with a checkered pattern and underlies the bistability observed in the diagram “Phase dif. LF-RF” in Fig 7B. This specific scenario corresponds to the area between X6 and X7 where flexor center synchronization co-exists with alternation in a manner dependent on the system’s initial conditions. Because of the large overlap of the cline bands both the escape-on-release and escape-on-shutdown mechanisms are possible (see text for more detailed description).
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pcbi.1004270.g008: Fast-slow decomposition of the model dynamics when the inhibitory V0D pathways are deleted.Projection of the phase portrait of the model onto the (V, hNaP) plane at α < X6. Sample positions, or image points, of flexor centers are depicted by the black (RF) and blue (LF) circles. A. Input from the right extensor center (RE) to the left flexor centers (LF) is represented as a sleeve of clines ranging from the maximal RE activity (lowest green cline) to minimal RE activity (highest green cline). The maximal and minimal RE activities occur at the beginning and end of the RF inactive phase, respectively. The red cline represents the maximal excitation to the LF from the RF when the RF is active (labeled as “RF on/RE off”). This maximum occurs immediately after the RF activates and begins to excite the LF via the V3 connection. Hence, for any of the LF's initial conditions above the left knee of the red cline (the blue area) the activation of the RF will immediately result in activation of the LF as well (release-on-escape). For low enough values of α the left knee of the green band is higher than the left knee of the red cline. This makes it possible for the LF to climb high enough during the inactive phase of the RF to find itself in the blue area by the time of escape of the latter. Once the RF activates, the LF activates as well thus stabilizing the regime of synchronous oscillations. B.α > X8 the red and green clines interchange their positions so that the left knee of the red band is now higher than the left knee of the green cline. Accordingly, it is no longer possible for the LF to get above the left knee of the low red cline, while moving along the left branch of the green cline, thus making the release-on-escape mechanism impossible and excluding synchronized behavior (see Fig 7B). Instead, such a configuration enables the release-on-shutdown mechanism, because starting at initial conditions above the left knee of the green cline (the yellow area) the LF immediately activates upon the deactivation of the RF. Symmetric alternation (Δφ = 0.5) is stabilized by the release-on-shutdown mechanism given that each flexor center’s image point climbs high enough along the left branch of the red band before the contralateral flexor center deactivates. C. Overlap of the flexor and extensor center cline bands is denoted with a checkered pattern and underlies the bistability observed in the diagram “Phase dif. LF-RF” in Fig 7B. This specific scenario corresponds to the area between X6 and X7 where flexor center synchronization co-exists with alternation in a manner dependent on the system’s initial conditions. Because of the large overlap of the cline bands both the escape-on-release and escape-on-shutdown mechanisms are possible (see text for more detailed description).

Mentions: At low levels of excitation (α < X6), activities of flexor centers are synchronized signifying that the reciprocal excitation provided by the excitatory V3 pathways is stronger than inputs from the contralateral extensor centers. We found that there are prerequisites for the release-on-escape mechanism, which are further explained using fast-slow decomposition (Fig 8A). The existence of this regime depends on the maximum voltage (Vmax) achieved by the leading flexor center upon activation.


Mechanisms of left-right coordination in mammalian locomotor pattern generation circuits: a mathematical modeling view.

Molkov YI, Bacak BJ, Talpalar AE, Rybak IA - PLoS Comput. Biol. (2015)

Fast-slow decomposition of the model dynamics when the inhibitory V0D pathways are deleted.Projection of the phase portrait of the model onto the (V, hNaP) plane at α < X6. Sample positions, or image points, of flexor centers are depicted by the black (RF) and blue (LF) circles. A. Input from the right extensor center (RE) to the left flexor centers (LF) is represented as a sleeve of clines ranging from the maximal RE activity (lowest green cline) to minimal RE activity (highest green cline). The maximal and minimal RE activities occur at the beginning and end of the RF inactive phase, respectively. The red cline represents the maximal excitation to the LF from the RF when the RF is active (labeled as “RF on/RE off”). This maximum occurs immediately after the RF activates and begins to excite the LF via the V3 connection. Hence, for any of the LF's initial conditions above the left knee of the red cline (the blue area) the activation of the RF will immediately result in activation of the LF as well (release-on-escape). For low enough values of α the left knee of the green band is higher than the left knee of the red cline. This makes it possible for the LF to climb high enough during the inactive phase of the RF to find itself in the blue area by the time of escape of the latter. Once the RF activates, the LF activates as well thus stabilizing the regime of synchronous oscillations. B.α > X8 the red and green clines interchange their positions so that the left knee of the red band is now higher than the left knee of the green cline. Accordingly, it is no longer possible for the LF to get above the left knee of the low red cline, while moving along the left branch of the green cline, thus making the release-on-escape mechanism impossible and excluding synchronized behavior (see Fig 7B). Instead, such a configuration enables the release-on-shutdown mechanism, because starting at initial conditions above the left knee of the green cline (the yellow area) the LF immediately activates upon the deactivation of the RF. Symmetric alternation (Δφ = 0.5) is stabilized by the release-on-shutdown mechanism given that each flexor center’s image point climbs high enough along the left branch of the red band before the contralateral flexor center deactivates. C. Overlap of the flexor and extensor center cline bands is denoted with a checkered pattern and underlies the bistability observed in the diagram “Phase dif. LF-RF” in Fig 7B. This specific scenario corresponds to the area between X6 and X7 where flexor center synchronization co-exists with alternation in a manner dependent on the system’s initial conditions. Because of the large overlap of the cline bands both the escape-on-release and escape-on-shutdown mechanisms are possible (see text for more detailed description).
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Related In: Results  -  Collection

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Show All Figures
getmorefigures.php?uid=PMC4430237&req=5

pcbi.1004270.g008: Fast-slow decomposition of the model dynamics when the inhibitory V0D pathways are deleted.Projection of the phase portrait of the model onto the (V, hNaP) plane at α < X6. Sample positions, or image points, of flexor centers are depicted by the black (RF) and blue (LF) circles. A. Input from the right extensor center (RE) to the left flexor centers (LF) is represented as a sleeve of clines ranging from the maximal RE activity (lowest green cline) to minimal RE activity (highest green cline). The maximal and minimal RE activities occur at the beginning and end of the RF inactive phase, respectively. The red cline represents the maximal excitation to the LF from the RF when the RF is active (labeled as “RF on/RE off”). This maximum occurs immediately after the RF activates and begins to excite the LF via the V3 connection. Hence, for any of the LF's initial conditions above the left knee of the red cline (the blue area) the activation of the RF will immediately result in activation of the LF as well (release-on-escape). For low enough values of α the left knee of the green band is higher than the left knee of the red cline. This makes it possible for the LF to climb high enough during the inactive phase of the RF to find itself in the blue area by the time of escape of the latter. Once the RF activates, the LF activates as well thus stabilizing the regime of synchronous oscillations. B.α > X8 the red and green clines interchange their positions so that the left knee of the red band is now higher than the left knee of the green cline. Accordingly, it is no longer possible for the LF to get above the left knee of the low red cline, while moving along the left branch of the green cline, thus making the release-on-escape mechanism impossible and excluding synchronized behavior (see Fig 7B). Instead, such a configuration enables the release-on-shutdown mechanism, because starting at initial conditions above the left knee of the green cline (the yellow area) the LF immediately activates upon the deactivation of the RF. Symmetric alternation (Δφ = 0.5) is stabilized by the release-on-shutdown mechanism given that each flexor center’s image point climbs high enough along the left branch of the red band before the contralateral flexor center deactivates. C. Overlap of the flexor and extensor center cline bands is denoted with a checkered pattern and underlies the bistability observed in the diagram “Phase dif. LF-RF” in Fig 7B. This specific scenario corresponds to the area between X6 and X7 where flexor center synchronization co-exists with alternation in a manner dependent on the system’s initial conditions. Because of the large overlap of the cline bands both the escape-on-release and escape-on-shutdown mechanisms are possible (see text for more detailed description).
Mentions: At low levels of excitation (α < X6), activities of flexor centers are synchronized signifying that the reciprocal excitation provided by the excitatory V3 pathways is stronger than inputs from the contralateral extensor centers. We found that there are prerequisites for the release-on-escape mechanism, which are further explained using fast-slow decomposition (Fig 8A). The existence of this regime depends on the maximum voltage (Vmax) achieved by the leading flexor center upon activation.

Bottom Line: The locomotor frequency was controlled by a parameter defining the excitation of neurons and commissural pathways mimicking the effects of N-methyl-D-aspartate on locomotor frequency in isolated rodent spinal cord preparations.We used bifurcation theory and fast-slow decomposition methods to analyze network behavior in the above regimes and transitions between them.The model reproduced, and suggested explanation for, a series of experimental phenomena and generated predictions available for experimental testing.

View Article: PubMed Central - PubMed

Affiliation: Department of Mathematical Sciences, Indiana University-Purdue University, Indianapolis, Indiana, United States of America.

ABSTRACT
The locomotor gait in limbed animals is defined by the left-right leg coordination and locomotor speed. Coordination between left and right neural activities in the spinal cord controlling left and right legs is provided by commissural interneurons (CINs). Several CIN types have been genetically identified, including the excitatory V3 and excitatory and inhibitory V0 types. Recent studies demonstrated that genetic elimination of all V0 CINs caused switching from a normal left-right alternating activity to a left-right synchronized "hopping" pattern. Furthermore, ablation of only the inhibitory V0 CINs (V0D subtype) resulted in a lack of left-right alternation at low locomotor frequencies and retaining this alternation at high frequencies, whereas selective ablation of the excitatory V0 neurons (V0V subtype) maintained the left-right alternation at low frequencies and switched to a hopping pattern at high frequencies. To analyze these findings, we developed a simplified mathematical model of neural circuits consisting of four pacemaker neurons representing left and right, flexor and extensor rhythm-generating centers interacting via commissural pathways representing V3, V0D, and V0V CINs. The locomotor frequency was controlled by a parameter defining the excitation of neurons and commissural pathways mimicking the effects of N-methyl-D-aspartate on locomotor frequency in isolated rodent spinal cord preparations. The model demonstrated a typical left-right alternating pattern under control conditions, switching to a hopping activity at any frequency after removing both V0 connections, a synchronized pattern at low frequencies with alternation at high frequencies after removing only V0D connections, and an alternating pattern at low frequencies with hopping at high frequencies after removing only V0V connections. We used bifurcation theory and fast-slow decomposition methods to analyze network behavior in the above regimes and transitions between them. The model reproduced, and suggested explanation for, a series of experimental phenomena and generated predictions available for experimental testing.

No MeSH data available.


Related in: MedlinePlus