Computational understanding of the neural circuit for the central pattern generator for locomotion and its control in l

1. 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

2. 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.

3. 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.

4. 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?

5. ( ) ( ) ( ( ) ) 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

6. 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

7. ( ( ( ( ( ( ( ( ( ) ) ) ) ) ) ) ) ) ( ( ( ) ) ) ) ( 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

8. 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

9. 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

10. 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

11. ( ) 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).

12. Simulation results: Forward swimming Backward swimming Increase H,Q

13. 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)

14. 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.