WINNERLESS COMPETITION PRINCIPLE IN NEUROSCIENCEWINNERLESS COMPETITION PRINCIPLE IN NEUROSCIENCE

Mikhail RabinovichMikhail Rabinovich

INLS University of California, San DiegoINLS University of California, San Diego

’

competition stimulus Winnerless competition stimulus Winnerless without + dependent = Competitionwithout + dependent = CompetitionWINNER WINNER clique Principleclique Principle

Hierarchy of the ModelsHierarchy of the Models

Network with realistic H-H model neurons & Network with realistic H-H model neurons & random inhibitory & excitatory connectionsrandom inhibitory & excitatory connections

Network with FitzHugh-Nagumo spiking Network with FitzHugh-Nagumo spiking neuronsneurons

Lotka-Volterra type model to describe the Lotka-Volterra type model to describe the spiking rate of the Principal Neurons (PNs)spiking rate of the Principal Neurons (PNs)

From standard rate equations From standard rate equations to Lotka-Volterra type modelto Lotka-Volterra type model

Stimulus dependent Rate ModelStimulus dependent Rate Model

(...)ij

ik 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

)(tH

)(tS

dt

dtStHaF

aGaa

i

N

ikikik

j

N

jijii

)();(])()(

)()()([1

HH

ia

ia Is the firing rate of neuron i

Canonical L-V model (N>3)Canonical L-V model (N>3)

)](1[ j

N

jiijiii aaaa

1)1( 1,1

ikiiik 1

1

1

)1(

)1(

N

i ii

ii

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

forstability:

i+1i

Canonical Lotka-Volterra model Canonical Lotka-Volterra model Rigorous results (N=3)Rigorous results (N=3)

Then the heteroclinic contour is a global attractor if

A noise transfer theheteroclinic contour to a stable limit cycle with the same order of a sequential switching

1321

)1/()1(

10

1

1

1

)(

33

22

11

iii

ii

ij

Consider the matrix

WLC Principle & SHS WLC Principle & SHS (rate model)(rate model)

Geometrical image of the switching activity in Geometrical image of the switching activity in the phase space is the orbit in the vicinity of the phase space is the orbit in the vicinity of the the heteroclinicheteroclinic sequence sequence

P

Q

R

P

Q

R

WLC Principle & SHS WLC Principle & SHS (H-H neurons)(H-H neurons)

Geometrical image of the switching activity in Geometrical image of the switching activity in the phase space is the orbit in the vicinity of the phase space is the orbit in the vicinity of the the heteroclinicheteroclinic contour contour

02

4

02

4

0

2

4

0

2

4

WLC in a network of three WLC in a network of three spiking-bursting neuronsspiking-bursting neurons

The main questions:The main questions:

How does sensory information How does sensory information transform into behavior in a robust and transform into behavior in a robust and reproducible way? reproducible way?

Do neural systems generate new Do neural systems generate new information based on their sensory information based on their sensory inputs?inputs?

Can transient dynamics be Can transient dynamics be reproducible? reproducible?

WLC dynamics of the piloric CPG: WLC dynamics of the piloric CPG: experiment & theoryexperiment & theory

Real timeReal timeClione’s Clione’s hunting behaviorhunting behavior

Clione’s Clione’s hunting behaviorhunting behavior

Clione’s Clione’s neural neural circuitcircuit

WLC can generate an irregular WLC can generate an irregular but reproducible sequencebut 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

)())(),((1

tStHaSHaa i

N

jiiijii

Projection of the strange attractorProjection of the strange attractorfrom the 6D phase space of the from the 6D phase space of the

statocyst networkstatocyst network

Weak reciprocal excitation stabilizes Weak reciprocal excitation stabilizes WLCWLC dynamics:dynamics: Birth of the stable limit cycle in Birth of the stable limit cycle in

the vicinity of the former heteroclinic sequencethe vicinity of the former heteroclinic sequence

3

6

1

)1(

iij

N

jijii aaaaa

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

WinnerlessWinnerless

Winner Winner take alltake all

Sequential dynamics of Sequential dynamics of statocyst neuronsstatocyst neurons

Motor outputMotor output dynamicsdynamics

Firing rates of 4 different tail motorneurons at different burst episodes

In spite of the irregularity the sequence is preserved

IMAGES OF THE DYNAMICAL SEQUENCES

Spatio-temporal coding in Spatio-temporal coding in the Antennal Lobe of Locustthe Antennal Lobe of Locust

(space = odor space)(space = odor space)

Lessons from the experiments:

The key role of the inhibition

Nonsymmetric connections

No direct connection between PNs

1

2

8

9

10

1

0

Time

1

2

8

9

10

01

input output

Transformation of the Transformation of the identityidentity input Into spatio-temporalinput Into spatio-temporaloutput based on the intrinsicoutput based on the intrinsicsequential dynamics of the sequential dynamics of the neural ensembleneural ensemble

0

1

0

1

00

10

Winnerless Competition Principle &Winnerless Competition Principle & New Dynamical Object: New Dynamical Object:

Stable Heteroclinic Sequence Stable Heteroclinic Sequence

WLC

&

SHS

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

Low dimensional projection of Trajectories Low dimensional projection of Trajectories Representing PN Population Response over Representing PN Population Response over

TimeTime

Stable Heteroclinic SequenceStable Heteroclinic Sequence

1

1

1)1(

1

1)1(

1

k

kkk

k

k

k

kkk

k

k

Reproducible sequences in complex Reproducible sequences in complex

networksnetworks

)()]()()[()(

ttatatadt

tdaj

N

jiijiii

i

Inequalities for reproducibility:

Reproducibility of the Reproducibility of the heteroclinic sequenceheteroclinic sequence

Neuron

Stable manifolds of the saddle points keep the Stable manifolds of the saddle points keep the divergent directions in check in the vicinity of a divergent directions in check in the vicinity of a

heteroclinic sequenceheteroclinic sequence

WLC in complex neural WLC in complex neural ensemblesensembles

Complex network = many elements +Complex network = many elements + + disordered connections+ disordered connections

Most important phenomena in complex Most important phenomena in complex systems on the edge of reproducibility are:systems on the edge of reproducibility are: (i) (i) clusteringclustering, and, and (ii) (ii) competitioncompetition

Rate model of the Random Rate model of the Random networknetwork

Is the step function

TWO REGIMES:

A)

B)

What controls the dynamics?What controls the dynamics?

Phase portrait of the Phase portrait of the sequential activitysequential activity

Chaos in random networkChaos in random network

Reproducible transient sequence Reproducible transient sequence generated in random networkgenerated in random network

Reproducibility of the transient Reproducibility of the transient dynamicsdynamics

Example of sequenceExample of sequence

The network of songbird brainThe network of songbird brain

HVC Songbird patternsHVC Songbird patterns

Self-organized WLC in a network Self-organized WLC in a network with Hebbian learningwith Hebbian learning

WLC in the network with local WLC in the network with local learninglearning

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

** ** competitioncompetition

Synchronization of the CPGs Synchronization of the CPGs of two different animalsof two different animals

Heteroclinic synchronization: Heteroclinic synchronization: Ultra-subharmonic lockingUltra-subharmonic locking

Heteroclinic Arnold tonguesHeteroclinic Arnold tongues

Chaos between stairs of Chaos between stairs of synchronizatonsynchronizaton

Heteroclinic synchronization: Heteroclinic synchronization: Map’s descriptionMap’s description

Competition between learned Competition between learned sequences: on line decision makingsequences: on line decision making

The main messages:The main messages: The WLC principle & SHS do not depend on the The WLC principle & SHS do not depend on the

level of the neuron & synapse description and level of the neuron & synapse description and can be realized by many different kinds of can be realized by many different kinds of network architectures.network architectures.

The WLC principle is able to solve a The WLC principle is able to solve a fundamental contradiction between robustness & fundamental contradiction between robustness & sensitivity.sensitivity.

The transient sequence can be reproducible.The transient sequence can be reproducible. SHS can interact with each others: compete,SHS can interact with each others: compete,

synchronized & generate chaos.synchronized & generate chaos.

Thanks to theThanks to the collaboratorscollaborators

Valentin Afraimovich, Rafael Levi, Allan Selverston, Valentin Zhigulin,

Henry Abarbanel, Yuri Arshavskii & Gilles

Laurent

Spatio-temporal patterns in Spatio-temporal patterns in Clione’Clione’s nervess nerves

WLC: Dynamics of the H-H networkWLC: Dynamics of the H-H network

time (ms)

Neu

ron

Reproducibility of the dynamicsReproducibility of the dynamics14

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} – 10 trials

time time

Stimulation of statocyst nerve triggers a Stimulation of statocyst nerve triggers a dynamical response in the motor neuronsdynamical response in the motor neurons

Motor output electro-physiological recording

Motor output firing rates

Statocyst receptor activity during Statocyst receptor activity during hunting episodeshunting 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