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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


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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.


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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 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 andL 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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(

)

(

)

(

(

)

)

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

Contra-lateral connections from C neurons

-

+ external inputs

d/dt

=

+

Membrane potentials

Decay (leaky) term

E

E

C

C

L

L

Left-right symmetry in connections

E

E

C

C

L

L


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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


Slide7 l.jpg

(

(

(

(

(

(

(

(

(

)

)

)

)

)

)

)

)

)

(

(

(

)

)

)

)

(

g(EL)

E+

L+

C+

E-

L-

C-

E+

L+

C+

EL

LL

CL

E ±

L ±

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


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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 + (2-J-B) d/dt E + [(1-J)(1-B) +K(H-Q)] E =0

inhibit

C-

(E, L)-

excite

Damping

Restoration force

Experimental data show E &L synchronize, C phase leads


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Segmt. i

Fji

Fij

Segmt. j

Coupling: Fij= (Jij + Bij) d/dt Ej +[B+J] ijEj -[BJ+K(H-Q)]ijEj

Oscillator equation:

d2/dt2 E + (2-J-B) d/dt E + [(1-J)(1-B) +K(H-Q)] E =0

Single segment: Jii + Bii < 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 + wo2Ei = Σj Fij

Driving force from other segments.

ith damped oscillator segment of frequency wo


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Controlling swimming directions

Segmt. i

Fji

Fij

Forward swimming (head phase leads tail)

B+J > BJ+K(H-Q)

Segmt. j

Coupling: Fij= (Jij + Bij) d/dt Ej +[B+J] ijEj -[BJ+K(H-Q)]ijEj

Backward swimming (head phase lags tail)

B+J < BJ+K(H-Q)

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


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(

)

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 ~ e-i(wt-kx)determines the global phase gradient (wave number) k

For small k, Re(λ) ≈ 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 (k>0); while Rostral-to-caudal H tends to reduce or reverse it (k<0).


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Simulation results:

Forward swimming

Backward swimming

Increase H,Q


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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)


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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.


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