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WINNERLESS COMPETITION PRINCIPLE IN NEUROSCIENCEWINNERLESS COMPETITION PRINCIPLE IN NEUROSCIENCE
Mikhail RabinovichMikhail Rabinovich
INLS University of California, San DiegoINLS University of California, San Diego
’
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competition stimulus Winnerless competition stimulus Winnerless without + dependent = Competitionwithout + dependent = CompetitionWINNER WINNER clique Principleclique Principle
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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)
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From standard rate equations From standard rate equations to Lotka-Volterra type modelto Lotka-Volterra type model
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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
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Canonical L-V model (N>3)Canonical L-V model (N>3)
)](1[ j
N
jiijiii aaaa
1)1( 1,1
ikiiik 1
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)1(
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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
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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
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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
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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
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WLC in a network of three WLC in a network of three spiking-bursting neuronsspiking-bursting neurons
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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?
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WLC dynamics of the piloric CPG: WLC dynamics of the piloric CPG: experiment & theoryexperiment & theory
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Real timeReal timeClione’s Clione’s hunting behaviorhunting behavior
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Clione’s Clione’s hunting behaviorhunting behavior
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Clione’s Clione’s neural neural circuitcircuit
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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
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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
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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
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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
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Sequential dynamics of Sequential dynamics of statocyst neuronsstatocyst neurons
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Motor outputMotor output dynamicsdynamics
Firing rates of 4 different tail motorneurons at different burst episodes
In spite of the irregularity the sequence is preserved
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IMAGES OF THE DYNAMICAL SEQUENCES
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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
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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
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Transient dynamics of the bee antennal lobe Transient dynamics of the bee antennal lobe activity during post-stimulus relaxationactivity during post-stimulus relaxation
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Low dimensional projection of Trajectories Low dimensional projection of Trajectories Representing PN Population Response over Representing PN Population Response over
TimeTime
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Stable Heteroclinic SequenceStable Heteroclinic Sequence
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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:
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Reproducibility of the Reproducibility of the heteroclinic sequenceheteroclinic sequence
Neuron
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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
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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
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Rate model of the Random Rate model of the Random networknetwork
Is the step function
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TWO REGIMES:
A)
B)
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What controls the dynamics?What controls the dynamics?
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Phase portrait of the Phase portrait of the sequential activitysequential activity
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Chaos in random networkChaos in random network
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Reproducible transient sequence Reproducible transient sequence generated in random networkgenerated in random network
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Reproducibility of the transient Reproducibility of the transient dynamicsdynamics
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Example of sequenceExample of sequence
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The network of songbird brainThe network of songbird brain
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HVC Songbird patternsHVC Songbird patterns
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Self-organized WLC in a network Self-organized WLC in a network with Hebbian learningwith Hebbian learning
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WLC in the network with local WLC in the network with local learninglearning
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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
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Synchronization of the CPGs Synchronization of the CPGs of two different animalsof two different animals
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Heteroclinic synchronization: Heteroclinic synchronization: Ultra-subharmonic lockingUltra-subharmonic locking
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Heteroclinic Arnold tonguesHeteroclinic Arnold tongues
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Chaos between stairs of Chaos between stairs of synchronizatonsynchronizaton
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Heteroclinic synchronization: Heteroclinic synchronization: Map’s descriptionMap’s description
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Competition between learned Competition between learned sequences: on line decision makingsequences: on line decision making
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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.
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Thanks to theThanks to the collaboratorscollaborators
Valentin Afraimovich, Rafael Levi, Allan Selverston, Valentin Zhigulin,
Henry Abarbanel, Yuri Arshavskii & Gilles
Laurent
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Spatio-temporal patterns in Spatio-temporal patterns in Clione’Clione’s nervess nerves
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WLC: Dynamics of the H-H networkWLC: Dynamics of the H-H network
time (ms)
Neu
ron
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Reproducibility of the dynamicsReproducibility of the dynamics14
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} – 10 trials
time time
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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
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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