1. WINNERLESS COMPETITION PRINCIPLE IN NEUROSCIENCE Mikhail Rabinovich INLS University of California, San Diego ’

2. competition stimulus Winnerless without + dependent = Competition WINNER clique Principle

3. Hierarchy of the Models  Network with realistic H-H model neurons & random inhibitory & excitatory connections  Network with FitzHugh-Nagumo spiking neurons  Lotka-Volterra type model to describe the spiking rate of the Principal Neurons (PNs)

4. From standard rate equations to Lotka-Volterra type model

5. Stimulus dependent Rate Model is the strength of excitation in i by k is the excitation from the other neural ensembles is an external action is the strength of inhibition in i by j Is the firing rate of neuron i

6. Canonical L-V model (N>3) A heteroclinic sequence consists of finitely many saddle equilibria and finitely many separatrices connecting these equilibria. The heteroclinic sequence can serve as an attracting set if every saddle point has only one unstable direction. The condition for this is: Necessary condition for stability: i+1 i

7. Canonical Lotka-Volterra model Rigorous results (N=3) Then the heteroclinic contour is a global attractor if A noise transfer the heteroclinic contour to a stable limit cycle with the same order of a sequential switching Consider the matrix

8. WLC Principle & SHS (rate model) Geometrical image of the switching activity in the phase space is the orbit in the vicinity of the heteroclinic sequence Geometrical image of the switching activity in the phase space is the orbit in the vicinity of the heteroclinic sequence

9. WLC Principle & SHS (H-H neurons) Geometrical image of the switching activity in the phase space is the orbit in the vicinity of the heteroclinic contour Geometrical image of the switching activity in the phase space is the orbit in the vicinity of the heteroclinic contour

10. WLC in a network of three spiking-bursting neurons

11. The main questions: The main questions:  How does sensory information transform into behavior in a robust and reproducible way?  Do neural systems generate new information based on their sensory inputs?  Can transient dynamics be reproducible?

12. WLC dynamics of the piloric CPG: experiment & theory

13. Real time Clione’s hunting behavior

14. Clione’s hunting behavior

15. Clione’s neural circuit

16. WLC can generate an irregular but reproducible sequence   All connections are inhibitory   The SRCs are asymmetrically connected   There is 30% connectivity among the neurons   The hunting neuron excites allSCHs at variable strength Model assumptions

17. Projection of the strange attractor from the 6D phase space of the statocyst network

18. Weak reciprocal excitation stabilizes WLC dynamics: Birth of the stable limit cycle in the vicinity of the former heteroclinic sequence

19. Conductance-based model for “Winner take all” and “Winnerless” competition Winnerless Winner take all

20. Sequential dynamics of statocyst neurons

21. Motor output dynamics Firing rates of 4 different tail motorneurons at different burst episodes In spite of the irregularity the sequence is preserved

22. IMAGES OF THE DYNAMICAL SEQUENCES

23. Spatio-temporal coding in the Antennal Lobe of Locust (space = odor space) Spatio-temporal coding in the Antennal Lobe of Locust (space = odor space) Lessons from the experiments: The key role of the inhibition Nonsymmetric connections No direct connection between PNs

24. 1 2 8 9 1 0 1 0 Time 1 2 8 9 1 0 0 1 inputoutput Transformation of the identity input Into spatio-temporal input Into spatio-temporal output based on the intrinsic sequential dynamics of the neural ensemble 0 1 0 1 0 0 1 0 Winnerless Competition Principle & New Dynamical Object: Stable Heteroclinic Sequence WLC & SHS

25. Transient dynamics of the bee antennal lobe activity during post-stimulus relaxation

26. Low dimensional projection of Trajectories Representing PN Population Response over Time

27. Stable Heteroclinic Sequence

28. Reproducible sequences in complex networks Inequalities for reproducibility:

29. Reproducibility of the heteroclinic sequence Neuron

30. Stable manifolds of the saddle points keep the divergent directions in check in the vicinity of a heteroclinic sequence

31. WLC in complex neural ensembles Complex network = many elements + + disordered connections + disordered connections Most important phenomena in complex systems on the edge of reproducibility are: (i) clustering, and (i) clustering, and (ii) competition (ii) competition

32. Rate model of the Random network   Is the step function

33. TWO REGIMES: A) B)

34. What controls the dynamics?

35. Phase portrait of the sequential activity

36. Chaos in random network

37. Reproducible transient sequence generated in random network

38. Reproducibility of the transient dynamics

39. Example of sequence

40. The network of songbird brain

41. HVC Songbird patterns HVC Songbird patterns

42. Self-organized WLC in a network with Hebbian learning

43. WLC in the network with local learning

44. WLC networks cooperation: * synchronization (i) electrical connections, (ii) synaptic connections; (iii) ultra-subharmonic synchronization ** competition

45. Synchronization of the CPGs of two different animals

46. Heteroclinic synchronization: Ultra-subharmonic locking

47. Heteroclinic Arnold tongues

48. Chaos between stairs of synchronizaton

49. Heteroclinic synchronization: Map’s description

50. Competition between learned sequences: on line decision making

51. The main messages:  The WLC principle & SHS do not depend on the level of the neuron & synapse description and can be realized by many different kinds of network architectures.  The WLC principle is able to solve a fundamental contradiction between robustness & sensitivity.  The transient sequence can be reproducible.  SHS can interact with each others: compete, synchronized & generate chaos. synchronized & generate chaos.

52. Thanks to the collaborators Thanks to the collaborators Valentin Afraimovich, Rafael Levi, Allan Selverston, Valentin Zhigulin, Henry Abarbanel, Yuri Arshavskii & Gilles Laurent

53. Spatio-temporal patterns in Clione’s nerves

54. WLC: Dynamics of the H-H network time (ms) Neuron

55. Reproducibility of the dynamics 14 15 16 17 18 19 20 21 22 23 24 25 1 2 3 4 5 6 7 8 9 10 11 12 13 } – 10 trials time

56. Stimulation of statocyst nerve triggers a dynamical response in the motor neurons Motor output electro- physiological recording Motor output firing rates

57. Statocyst receptor activity during hunting episodes   The constant statocyst receptor activity turns into bursting in physostigmine   The activity is variable between episodes   A single receptor is active during different phases of the hunting episodes