Computational understanding of

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Computational understanding of the neural
circuit for the central pattern generator for
locomotion and its control in lamprey
Li Zhaoping, Alex, Lewis, and Silvia
Scarpetta University College London
University of Salerno
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Lamprey swimming
one wavelength, approx. 100 segments.
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Experimental data
Spontaneous oscillations in decapitated sections
with a minimum of 2-4 segments, from anywhere
along the body.
Three types of neurons E (excitatory), C
(inhibitory), and L (inhibitory). Connections as
in diagram
E ,C neurons shorter range connections (a few
segments), L longer range
Head-to-tail (rostral-to-caudal) descending
connections stronger
E and L oscillate in phase, C phase leads.
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Previous works
Grillner et al Simulation of CPG with detailed
cellular properties.
Kopell, Ermentrout, et al Mathematical model of
CPG simplified as a chain of
phase oscillators.
Many others e.g., Ijspeert et al genetic
algorithms to design part of the networks for
desired behavior.

• Current Work analytical study of the neural
  circuit.
• How do oscillations emerge when single segment
  does not oscillate? --- no previous studies (?)

• How are inter-segment phase lags determined by
  connections

• How do network connections control swimming
  direction?

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Neurons modeled as leaky integrators
-
external inputs
d/dt


Membrane potentials
Decay (leaky) term
E
E
C
C
L
L
Left-right symmetry in connections
E
C
E
C
L
L
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Neurons modeled as leaky integrators
-
external inputs
d/dt


Membrane potentials
E
E
C
C
L
L
E
C
E
C
L
L
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Left and right sides are coupled
Left
Right
E
E
C
C
L
L
Swimming mode always dominant
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The swimming mode
Experimental data show E L synchronize, C phase
leads
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Oscillator equation
d2/dt2 E (2-J-B) d/dt E (1-J)(1-B) K(H-Q)
E 0
Single segment Jii Bii lt 2
Self excitation does not overcome damping
An isolated segment does not oscillate (unlike
previous models)
Inter-segment interaction
When driving forces feed energy from one
oscillator to another, global spontaneous
oscillation emerges.
d2/dt2 Ei a d/dt Ei wo2 Ei Sj Fij
Driving force from other segments.
ith damped oscillator segment of frequency wo
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Controlling swimming directions
Given Fji gt Fij, (descending connections
dominate)
Prediction 2 swimming direction could be
controlled by scaling connections H, (or Q ,K,
B, J), e.g., through external inputs
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More rigorously
Its dominant eigenvector E(x) eltikx
e-i(wt-kx) determines the global phase gradient
(wave number) k
For small k, Re(?) const k function of
(4K(H-Q) (B-J)2)
ve k forward
swimming
-ve k backward
swimming
Eg. Rostral-to-caudal B tends to increase the
head-to-tail phase lag (kgt0) while
Rostral-to-caudal H tends to reduce or reverse it
(klt0).
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Simulation results
Forward swimming
Backward swimming
Increase H,Q
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Turning
Amplitude of oscillations is increased on one
side of the body.
Achieved by increasing the tonic input to one
side only (see also Kozlov et al., Biol. Cybern.
2002)
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Summary
Analytical study of a CPG model of suitable
complexity gives new insights into
How coupling can enable global oscillation from
damped oscillators
How each connection type affects phase
relationships
How and which connections enable swimming
direction control.