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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 . Lamprey swimming. Head oscillation leads tail
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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
Lamprey swimming
Head oscillation leads tail
forward swimming
Tail leads head
backward swimming
Fictive Swimming: Spontaneous oscillations in isolated section of spinal cord, with phase lag of ~1% of a cycle per segment. The network that generates the oscillations is the CPG (Central Pattern Generator).
one wavelength, approx. 100 segments.
Experimental data
To motor neurons
E
C
C
E
L
L
Two segments in CPG
E
C
C
E
L
L
Spontaneous oscillations in decapitated sections with a minimum of 24 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
Headtotail (rostraltocaudal) descending connections stronger
E andL oscillate in phase, C phase leads.
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.
(
)
(
)
(
(
)
)
JWQ
0
K
g(EL)
EL
LL
CL
EL
LL
CL
0
A
g(LL)
H
B
g(CR)
Connection strengths
Firing rates
Neurons modeled as leaky integrators
Contralateral connections from C neurons

+ external inputs
d/dt
=
+
Membrane potentials
Decay (leaky) term
E
E
C
C
L
L
Leftright symmetry in connections
E
E
C
C
L
L
Neurons modeled as leaky integrators
(
)
(
)
(
(
)
)
JWQ
0
K
g(EL)
EL
LL
CL
EL
LL
CL
0
A
g(LL)
H
B
g(CR)
Connection strengths
E1L
E2L
E,L,C are vectors: EL=
E3L
J,K,etc are matrices:
:
.
J11
J12
J13
.
.
J=
J21
J22
.
.
.
J31

+ external inputs
d/dt
=
+
Membrane potentials
E
E
C
C
L
L
E
E
C
C
L
L
(
(
(
(
(
(
(
(
(
)
)
)
)
)
)
)
)
)
(
(
(
)
)
)
)
(
g(EL)
E+
L+
C+
E
L
C
E+
L+
C+
EL
LL
CL
E ±
L ±
C±
ER
LR
CR
EL
LL
CL
EL
LL
CL
E
L
C
0
0
0
K
+K
K
JWQ
JWQ
JWQ
0
0
0
+A
A
A
+ external inputs
g(LL)
+
+
+
d/dt
= 
H
H
H
B
+B
B
g(CR)
Linear approximation leads to decoupling
±
=
“+” mode
“” mode
)
)
(
(
E
E+
+ external inputs
d/dt
d/dt
= 
= 
+
+
+
+
+
+
L
L+
E
E
C
C
C+
C
L
L
Swimming mode
C becomes excitatory.
The connections scaled by the gain g’(.) in g(.), controlled by external inputs.
Left and right sides are coupled
Left
Right
E
C
C
E
L
L
Swimming mode always dominant
The swimming mode
(
)
K
0
JWQ
(
(
(
)
)
)
E
L
C
E
L
C
E
C
A
0
H
B
E=L, J=W, K=A, simplification
(
)
(
)
J
Q
1
K
E
=
d/dt
(
)
H
B
1
E
C
=

L
d/dt
+
Prediction 1: H>Q needed for oscillations!
C
Oscillator equation:
d2/dt2 E + (2JB) d/dt E + [(1J)(1B) +K(HQ)] E =0
inhibit
C
(E, L)
excite
Damping
Restoration force
Experimental data show E &L synchronize, C phase leads
Segmt. i
Fji
Fij
Segmt. j
Coupling: Fij= (Jij + Bij) d/dt Ej +[B+J] ijEj [BJ+K(HQ)]ijEj
Oscillator equation:
d2/dt2 E + (2JB) d/dt E + [(1J)(1B) +K(HQ)] E =0
Single segment: Jii + Bii < 2
Self excitation does not overcome damping
An isolated segment does not oscillate (unlike previous models)
Intersegment interaction:
When driving forces feed “energy” from one oscillator to another, global spontaneous oscillation emerges.
d2/dt2 Ei + a d/dt Ei + wo2Ei = Σj Fij
Driving force from other segments.
ith damped oscillator segment of frequency wo
Controlling swimming directions
Segmt. i
Fji
Fij
Forward swimming (head phase leads tail)
B+J > BJ+K(HQ)
Segmt. j
Coupling: Fij= (Jij + Bij) d/dt Ej +[B+J] ijEj [BJ+K(HQ)]ijEj
Backward swimming (head phase lags tail)
B+J < BJ+K(HQ)
Feeds energy when Ei & Ej in phase
Feeds energy when Ei lags Ej
Feeds energy when Ei leads Ej
Given Fji > 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
(
)
E
C
(
)
(
)
J
Q
1
K
E
=
d/dt
H
B
1
C
Backward Swimming
So, increasingH (e.g, via input toLneurons)
More rigorously:
Its dominant eigenvector E(x) ~ elt+ikx ~ ei(wtkx)determines the global phase gradient (wave number) k
For small k, Re(λ) ≈ const + k · function of (4K(HQ) –(BJ)2)
+ve k forward swimming
ve k backward swimming
Eg. Rostraltocaudal B tends to increase the headtotail phase lag (k>0); while Rostraltocaudal H tends to reduce or reverse it (k<0).
Simulation results:
Forward swimming
Backward swimming
Increase H,Q
Forward swim
Turn
Simulation
Left
EL,R
Right
Time
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)
Further work:
Control of swimming speed (oscillation frequency) over a larger range
Include synaptic temporal complexities in model
Summary
Analytical study of a CPG model of suitable complexity gives new insightsinto
How coupling can enable global oscillation from damped oscillators
How each connection type affects phase relationships
How and which connections enable swimming direction control.