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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

Performance of the intact model.A. Changes in the output activity of the four centers with increasing neuronal excitation (α). LE and RE indicate left and right extensor centers, respectively; LF and RF indicate left and right flexor centers. For low values of α (α < 0.55) RF activates upon LF shutdown (see vertical dashed lines). Due to asymmetry of flexor and extensor phase duration at low values of α there is a part of each step cycle when both flexor centers are inactive and hence both extensor centers are active. α was changed linearly in time from α = 0 at t = 0 to α = 1.2 at t = 120 s. B. Bifurcation diagrams for different dynamical characteristics. From top to bottom: Frequency—frequency of bursts in Hz; Amplitude (F)—flexor center amplitude (maximum output activity of flexor center); Phase dif. LF-RF—phase difference between left and right flexor centers; Phase dif. F-E—phase difference between ipsilateral flexor and the extensor centers. When α increases there is an increase in frequency (mostly concerned with shortening of extensor phase) as well as a decrease in the amplitude of each flexor center burst (see also LF and RF panels on the left). For relatively high values of excitation (α > 0.55) the phase difference between flexor centers is exactly 0.5. This means that the RF activates exactly one half-period after the LF. At lower values of α two distinct regimes exist: the first has a phase difference between flexor centers less than 0.5 (lower branch) which corresponds to the activation pattern shown in A (LF leads, RF activates on LF’s shutdown, then a pause until LF activates again) and the second regime has a phase difference greater than 0.5 (upper branch) when RF activates first and LF activates on RF’s shutdown (not shown on A). Note that in all diagrams in B, α was changed in both directions, first forward from 0 to 1.2 and then backward from 1.2 to 0. However, in all diagrams in Fig 3B, the graphs for forward (red) and backward (blue on the top) changes of α overlapped. In summary, the intact model demonstrates left-right alternating activity at all levels of excitation (all values of α).
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pcbi.1004270.g003: Performance of the intact model.A. Changes in the output activity of the four centers with increasing neuronal excitation (α). LE and RE indicate left and right extensor centers, respectively; LF and RF indicate left and right flexor centers. For low values of α (α < 0.55) RF activates upon LF shutdown (see vertical dashed lines). Due to asymmetry of flexor and extensor phase duration at low values of α there is a part of each step cycle when both flexor centers are inactive and hence both extensor centers are active. α was changed linearly in time from α = 0 at t = 0 to α = 1.2 at t = 120 s. B. Bifurcation diagrams for different dynamical characteristics. From top to bottom: Frequency—frequency of bursts in Hz; Amplitude (F)—flexor center amplitude (maximum output activity of flexor center); Phase dif. LF-RF—phase difference between left and right flexor centers; Phase dif. F-E—phase difference between ipsilateral flexor and the extensor centers. When α increases there is an increase in frequency (mostly concerned with shortening of extensor phase) as well as a decrease in the amplitude of each flexor center burst (see also LF and RF panels on the left). For relatively high values of excitation (α > 0.55) the phase difference between flexor centers is exactly 0.5. This means that the RF activates exactly one half-period after the LF. At lower values of α two distinct regimes exist: the first has a phase difference between flexor centers less than 0.5 (lower branch) which corresponds to the activation pattern shown in A (LF leads, RF activates on LF’s shutdown, then a pause until LF activates again) and the second regime has a phase difference greater than 0.5 (upper branch) when RF activates first and LF activates on RF’s shutdown (not shown on A). Note that in all diagrams in B, α was changed in both directions, first forward from 0 to 1.2 and then backward from 1.2 to 0. However, in all diagrams in Fig 3B, the graphs for forward (red) and backward (blue on the top) changes of α overlapped. In summary, the intact model demonstrates left-right alternating activity at all levels of excitation (all values of α).

Mentions: The performance of the intact network is shown in Fig 3. Panel A of the figure shows the changes in the output activity, f(V), of all four centers with α changing from 0 to 1.2. The vertical dashed lines in this panel indicate that activities of left (LF) and right (RF) flexor centers alternate at all values of α. Panel B shows how the frequency of oscillations (top diagram), the amplitude of flexor activity (second diagram), and the phase differences between the activities of left and right flexor centers (LF-RF) and left flexor and left extensor centers (F-E) (two bottom diagrams) changed with α. Note that in contrast to panel A, panel B shows α being changed in both directions, first forward from 0 to 1.2 (red) and then backward from 1.2 to 0 (blue). However, in all diagrams in Fig 3B, the graphs for forward and backward changes of α fully overlapped.


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)

Performance of the intact model.A. Changes in the output activity of the four centers with increasing neuronal excitation (α). LE and RE indicate left and right extensor centers, respectively; LF and RF indicate left and right flexor centers. For low values of α (α < 0.55) RF activates upon LF shutdown (see vertical dashed lines). Due to asymmetry of flexor and extensor phase duration at low values of α there is a part of each step cycle when both flexor centers are inactive and hence both extensor centers are active. α was changed linearly in time from α = 0 at t = 0 to α = 1.2 at t = 120 s. B. Bifurcation diagrams for different dynamical characteristics. From top to bottom: Frequency—frequency of bursts in Hz; Amplitude (F)—flexor center amplitude (maximum output activity of flexor center); Phase dif. LF-RF—phase difference between left and right flexor centers; Phase dif. F-E—phase difference between ipsilateral flexor and the extensor centers. When α increases there is an increase in frequency (mostly concerned with shortening of extensor phase) as well as a decrease in the amplitude of each flexor center burst (see also LF and RF panels on the left). For relatively high values of excitation (α > 0.55) the phase difference between flexor centers is exactly 0.5. This means that the RF activates exactly one half-period after the LF. At lower values of α two distinct regimes exist: the first has a phase difference between flexor centers less than 0.5 (lower branch) which corresponds to the activation pattern shown in A (LF leads, RF activates on LF’s shutdown, then a pause until LF activates again) and the second regime has a phase difference greater than 0.5 (upper branch) when RF activates first and LF activates on RF’s shutdown (not shown on A). Note that in all diagrams in B, α was changed in both directions, first forward from 0 to 1.2 and then backward from 1.2 to 0. However, in all diagrams in Fig 3B, the graphs for forward (red) and backward (blue on the top) changes of α overlapped. In summary, the intact model demonstrates left-right alternating activity at all levels of excitation (all values of α).
© Copyright Policy
Related In: Results  -  Collection

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

pcbi.1004270.g003: Performance of the intact model.A. Changes in the output activity of the four centers with increasing neuronal excitation (α). LE and RE indicate left and right extensor centers, respectively; LF and RF indicate left and right flexor centers. For low values of α (α < 0.55) RF activates upon LF shutdown (see vertical dashed lines). Due to asymmetry of flexor and extensor phase duration at low values of α there is a part of each step cycle when both flexor centers are inactive and hence both extensor centers are active. α was changed linearly in time from α = 0 at t = 0 to α = 1.2 at t = 120 s. B. Bifurcation diagrams for different dynamical characteristics. From top to bottom: Frequency—frequency of bursts in Hz; Amplitude (F)—flexor center amplitude (maximum output activity of flexor center); Phase dif. LF-RF—phase difference between left and right flexor centers; Phase dif. F-E—phase difference between ipsilateral flexor and the extensor centers. When α increases there is an increase in frequency (mostly concerned with shortening of extensor phase) as well as a decrease in the amplitude of each flexor center burst (see also LF and RF panels on the left). For relatively high values of excitation (α > 0.55) the phase difference between flexor centers is exactly 0.5. This means that the RF activates exactly one half-period after the LF. At lower values of α two distinct regimes exist: the first has a phase difference between flexor centers less than 0.5 (lower branch) which corresponds to the activation pattern shown in A (LF leads, RF activates on LF’s shutdown, then a pause until LF activates again) and the second regime has a phase difference greater than 0.5 (upper branch) when RF activates first and LF activates on RF’s shutdown (not shown on A). Note that in all diagrams in B, α was changed in both directions, first forward from 0 to 1.2 and then backward from 1.2 to 0. However, in all diagrams in Fig 3B, the graphs for forward (red) and backward (blue on the top) changes of α overlapped. In summary, the intact model demonstrates left-right alternating activity at all levels of excitation (all values of α).
Mentions: The performance of the intact network is shown in Fig 3. Panel A of the figure shows the changes in the output activity, f(V), of all four centers with α changing from 0 to 1.2. The vertical dashed lines in this panel indicate that activities of left (LF) and right (RF) flexor centers alternate at all values of α. Panel B shows how the frequency of oscillations (top diagram), the amplitude of flexor activity (second diagram), and the phase differences between the activities of left and right flexor centers (LF-RF) and left flexor and left extensor centers (F-E) (two bottom diagrams) changed with α. Note that in contrast to panel A, panel B shows α being changed in both directions, first forward from 0 to 1.2 (red) and then backward from 1.2 to 0 (blue). However, in all diagrams in Fig 3B, the graphs for forward and backward changes of α fully overlapped.

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