Download Spatiotemporal neural coding: WinnerLess Competition Principle

WINNERLESS COMPETITION
PRINCIPLE IN NEUROSCIENCE
’
Mikhail Rabinovich
INLS University of California, San Diego
competition
stimulus
Winnerless
without
+ dependent = Competition
WINNER
clique
Principle
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)
From standard rate equations
to Lotka-Volterra type model
Stimulus dependent Rate Model

N
a i  ai [ (H )    ij (H ) G (a j )
j 1

N
d
   ik F (ak )  H i (t )]  S i (t ); ( ) 
dt
k i
ai
ai
Is the firing rate of neuron i
ij (...) is the strength of inhibition in i by j
 ik
is the strength of excitation in i by k
H (t )
is the excitation from the other neural ensembles
S(t )
is an external action
Canonical L-V model (N>3)

N
a i  ai [1  (ai   ij a j )]
i j
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:
 k i  1,  (i 1)i  1
k  i 1
Necessary
condition
for
stability:
i
i+1
 i (i 1)  1
 
1
i 1   ( i 1) i
N
Canonical Lotka-Volterra model
Rigorous results (N=3)
Consider the matrix
 1 1 1 


(  ij )    2 1  2 
 
1 
3
 3
0  i  1  i
 i  (  i  1) /(1   i )
Then the heteroclinic
contour is a global
attractor if
 1 2 3  1
A noise transfer the
heteroclinic contour to a
stable limit cycle with the
same order of a
sequential switching
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
Q
R
P
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

4
4
2
2
0
0


0
0
2
2
4
4
WLC in a network of three
spiking-bursting neurons
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?
WLC dynamics of the piloric CPG:
experiment & theory
Real time
Clione’s hunting behavior
Clione’s hunting behavior
Clione’s
neural
circuit
WLC can generate an irregular
but reproducible sequence
Model assumptions




All connections
are inhibitory
The SRCs are
asymmetrically
connected
There is 30%
connectivity
among the
neurons
The hunting
neuron excites
allSCHs at
variable strength

N
a i  ai ( ( H , S )   ij ai  H i (t ))  Si (t )
j 1
Projection of the strange attractor
from the 6D phase space of the
statocyst network
Weak reciprocal excitation stabilizes
WLC dynamics: Birth of the stable limit cycle in
the vicinity of the former heteroclinic sequence

N 6
a i  ai (1   ij a j )  ai ai 3
j 1
Conductance-based model for “Winner
take all” and “Winnerless” competition
Winner
take all
Winnerless
Sequential dynamics of
statocyst neurons
Motor output
dynamics
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
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
Winnerless Competition Principle &
New Dynamical Object:
Stable Heteroclinic Sequence
input
output
1
0
1
0
1
2
1
2
8
9
8
9
WLC
&
SHS
1
0
0
1
0
0
0
1
0
1
1
0
Time
Transformation of the identity
input Into spatio-temporal
output based on the intrinsic
sequential dynamics of the
neural ensemble
Transient dynamics of the bee antennal lobe
activity during post-stimulus relaxation
Low dimensional projection of Trajectories
Representing PN Population Response over
Time
Stable Heteroclinic Sequence
Reproducible sequences in
complex networks
N
dai (t )
 ai (t )[ i  ai (t )   ij a j (t )]   (t )
dt
i j
Inequalities for reproducibility:
 k 1
 k 1
 1   ( k 1) k 
k
k
 k 1
 k 1
  ( k 1) k 
1
k
k
Reproducibility of the
heteroclinic sequence
Neuron
Stable manifolds of the saddle points keep the
divergent directions in check in the vicinity of a
heteroclinic sequence
WLC in complex neural
ensembles
Complex network = many elements +
+ disordered connections
Most important phenomena in complex
systems on the edge of reproducibility are:
(i) clustering, and
(ii) competition
Rate model of the Random
network
Q
Is the step function
TWO REGIMES:
A)
B)
What controls the dynamics?
Phase portrait of the
sequential activity
Chaos in random network
Reproducible transient sequence
generated in random network
Reproducibility of the transient
dynamics
Example of sequence
The network of songbird brain
HVC Songbird patterns
Self-organized WLC in a network
with Hebbian learning
WLC in the network with local
learning
WLC networks cooperation:
* synchronization
(i) electrical connections,
(ii) synaptic connections;
(iii) ultra-subharmonic synchronization
** competition
Synchronization of the CPGs
of two different animals
Heteroclinic synchronization:
Ultra-subharmonic locking
Heteroclinic Arnold tongues
Chaos between stairs of
synchronizaton
Heteroclinic synchronization:
Map’s description
Competition between learned
sequences: on line decision making
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.
Thanks to the collaborators
Valentin Afraimovich, Rafael Levi, Allan Selverston,
Valentin Zhigulin,
Henry Abarbanel, Yuri Arshavskii & Gilles
Laurent
Spatio-temporal patterns in
Clione’s nerves
Neuron
WLC: Dynamics of the H-H network
time (ms)
Reproducibility of the dynamics
1
} – 10 trials
14
2
15
3
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6
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9
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25
time
time
Stimulation of statocyst nerve triggers a
dynamical response in the motor neurons
Motor output
electrophysiological
recording
Motor output
firing rates
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