dynamical systems analysis for systems of spiking neurons
DESCRIPTION
Dynamical Systems Analysis for Systems of Spiking Neurons. Models: Leaky Integrate and Fire Model. CdV/dt= -V/R+I syn Resting Potential V Rest assumed to be 0. CR = Membrane time constant (20 msec for excitatory neurons, 10 msec for inhibitory neurons.) - PowerPoint PPT PresentationTRANSCRIPT
Dynamical Systems Analysis for Systems of Spiking Neurons
Models: Leaky Integrate and Fire Model
CdV/dt= -V/R+Isyn
•Resting Potential VRest assumed to be 0.
•CR = Membrane time constant (20 msec for excitatory neurons, 10 msec for inhibitory neurons.)
•Spike generated when V reaches VThreshold
•Voltage reset to VReset after spike (not the same as VRest)
•Synaptic Current Isyn assumed to be either delta function or alpha function.
Models: Spike-Response Model
Observation: The L-IF-model is linear
CdV1/dt= -V1/R+I1syn
CdV2/dt= -V2/R+I2syn
Cd(V1+V2)/dt= -(V1+V2)/R+I1syn+I2syn
Why not simply take the individual effect of each spike and add them all up?
Result: The Spike response model.
V(t)=effect of previously generated spikes by neuron+
sum over all effects generated by spikes that have arrived at synapses
Background: The Cortical Neuron
Input Output
Threshold
Time
•Absolute Refractory PeriodAbsolute Refractory Period
•Exponential DecayExponential Decay of effect of a spike on membrane potential
•SynapseSynapse
•Dendrites (Input)Dendrites (Input)
•Cell BodyCell Body
•Axon (Output)Axon (Output)
Background: Target System
Neocortical Column:Neocortical Column: ~ 1 mm~ 1 mm22 of the cortex of the cortex
Recurrent networkRecurrent network
~100,000 neurons~100,000 neurons
~10,000 synapses per neuron~10,000 synapses per neuron
~80% excitatory~80% excitatory
~20% inhibitory~20% inhibitory
Input
Recurrent System
Output
Background: The Neocortex
(Healthy adult human male subject)
Source: Dr. Krishna Nayak, SCRI, FSU
Background: The Neocortex
(Area V1 of Macaque Monkey)
Source: Dr. Wyeth Bair, CNS, NYU
Background: Dynamical Systems Analysis
Phase SpacePhase Space•Set of all legal states
DynamicsDynamics•Velocity Field
•Flows
•Mapping
Local & Global propertiesLocal & Global properties•Sensitivity to initial conditions
•Fixed points and periodic orbits
Content:
•ModelModel•A neuron
•System of Neurons: Phase SpacePhase Space & Velocity FieldVelocity Field
•Simulation ExperimentsSimulation Experiments•Neocortical Column
•Qualitative Characteristics: EEG power spectrumEEG power spectrum & ISI frequency distributionISI frequency distribution
•Formal AnalysisFormal Analysis•Local Analysis: Sensitivity to Initial ConditionsSensitivity to Initial Conditions
•ConclusionsConclusions
Time
Model: Single Neuron
1 21 1 11 1 2 2P( ,..., , ,..., ,...., ,..., )mnn n
m mx x x x x x
Each spike represented as: How long since it departed from soma.
t=0
t=0
t=0
11x
12x
11x
12x
21x 3
1x 41x
22x
13x 2
3x
Potential Function
Model: Single Neuron: Potential function
1 21 2 P(
Implicit everywhere bounded fun
, ,..., ) , ,..
cti, , .
.
o
n
,
Effectiveness of a Spike:
Membrane Potential:
inm i i i ix x
C
x x x x x
1... , & 1...
1... , & 1...
P 0
P 0
0
ii
ji
jij
jii
i m j n for
i m j fo
x
xn r
x
x
01... , & 1...
P(
.) P(.)
dPP(.) = T( ) 0dt
. Threshold:
jjii
xxii m j n
and
Model: System of Neurons
•Point in the Phase-SpacePoint in the Phase-Space•Configuration of spikes
1 1 2P ( , ,..., )mx x x
2 1 2P ( , ,..., )mx x x
3 1 2P ( , ,..., )mx x x
0t t
11x 2
1x
12x
23x1
3x
14x
•DynamicsDynamics•Birth of a spike
•Death of a spike
Model: Single Neuron: Phase-Space
1 2,Pre ,.limi ..,nary: 0 , iinnx x x
1 2
2
, ,...,Trans
formation 1: i
i
i
n
ix
n
i
z z z
z
T
e
1
2 1
1 1 0
0 1 1
..
( ) *.. * ( ) * ( )
, ,Transforma ...,tion 2:
i
i
i
i
i
i
n
nn
nn
nz z
z
z
z z z
a
z
a
z
a
a a a C
0
0,
Model: Single Neuron: Phase-Space
TheoremTheorem: Phase-Space can be defined formally
Phase-Space for Total Number of Spikes Assigned = 1, 2, & 3.
Model: Single Neuron: Structure of Phase-Space
i
i
i i
i
σin
i j i kn n
i 0n
Phase-Space for fixed number of Dead spikes:
Dead vs. Live Spikes: : is an imbedding
finer t
opologAss y ign to
,
L
• L L
L
j k
Theorem
•Phase-Space for n=3
• 1, 2 dead spikes.
Model: System of Neurons: Velocity Field
i
i
Si 0
ni=1
i
1
Cartesian product of Phase-Spaces;
Surfaces at and
can be defined mathematic
dP 0dally
(when no
t
e
L
P
System:
Birth of Spike:
Veloci
(
ty F
.)
ield :
=
:
T(.)
Ii
i
i
P
V
V
Theorem
2
12
vent) at
(for birth of spike) at
disregarding position on submanifold
p
p
Ii
Iii
ii
P
PV
V V
Simulations: Neocortical Column: Setup
•1000 neurons each connected randomly to 100 neurons.
•80% randomly chosen to be excitatory, rest inhibitory.
•Basic Spike-response model.
•Total number of active spikes in the system ►EEG / LFP recordings
•Spike Activity of randomly chosen neurons ►Real spike train recordings
•5 models: Successively enhanced physiological accuracy•Simplest model
•Identical EPSPs and IPSPs, IPSP 6 times stronger
•Most complex model
•Synapses: Excitatory (50% AMPA, NMDA), Inhibitory (50% GABAA, GABAB)
•Realistic distribution of synapses on soma and dendrites
•Synaptic response as reported in (Bernander Douglas & Koch 1992)
Simulations: Neocortical Column: Classes of Activity
Number of active spikesNumber of active spikes: Seizure-like & Normal Operational Conditions
Normal Operational Conditions (20 Hz)Normal Operational Conditions (20 Hz): Subset (200 neurons) of 1000 neurons for 1 second.
T=0 T=1000 msec
Simulations: Neocortical Column: Chaotic Activity
Simulations: Neocortical Column: Total Activity
Normalized time seriesNormalized time series: Total number of active spikes & Power Spectrum
Simulations: Neocortical Column: Spike Trains
Representative spike trainsRepresentative spike trains: Inter-spike Intervals & Frequency Distributions
Simulations: Neocortical Column: Propensity for Chaos
ISI’s of representative neuronsISI’s of representative neurons: 3 systems; 70%,80%,90% synapses driven by pacemaker
Simulations: Neocortical Column: Sensitive Dependence on Initial Conditions
Spike activity of 2 SystemsSpike activity of 2 Systems: Identical Systems, subset (200) of 1000 neurons, Identical Initial State except for 1 spike perturbed by 1 msec1 spike perturbed by 1 msec..
T=0 T=400 msec
Analysis: Local Analysis
•Are trajectories sensitive to initial conditionssensitive to initial conditions?
•If there are fixed pointsfixed points or periodic orbitsperiodic orbits, are they stablestable?
Analysis: Setup: Riemannian Metric
i i
i i
S S1 1i i
n ni=1 i=1
birth/dea
Riemannian Metric Symmetric Bilinear Form Orthonormal Basis
Volume and Shape Preserving between even th ts o( p f s i
: T( L ) L T( )
R
1 1 2 2 11 11 1 1
1 1 2 2
1
, ..., , , ..., , .., .., , ...,
)
Orthonormal Basis:
is a constant velo
kes
volume and shape prcity field eservi )n( g
S Snn n
S Sx x x x x x
V
Analysis: Setup: Riemannian Metric
0t t t 0t
•Discrete Dynamical SystemDiscrete Dynamical System
•Event ► Event ►Event….
•Event: birth/deathbirth/death of spike
Analysis: Measure Analysis
Birth of a SpikeBirth of a Spike
Death of a SpikeDeath of a Spike
PI
1 1
1 1
112 11 1
12 11 1
11
1
,..., ,...., ,...,
, ,..., ,...., ,...,
Death:
Bir
th: S S
S S
nnS S
nnS S
x x
xx
x x
x x x
1 1
1 1
2 11 1
2 11 1
1..2
11
1
..
1
, ..., ,...., ,...,
,...Perturbation
, ,...., ,...,
Analysis:
S S
S S
i i
nnS S
nnS S
ji
j
i Sj n
i
x x x x
x x x x
x
x x
1..2.. 1
1..1 2.. 1
1
i i
j
i
j
i S ij n
i i
i Sj
i
n
j
ji
Px
Px
Analysis: Perturbation Analysis
Analysis: Perturbation Analysis
Positive ji Negative j
i
What is ?ji
0t t
Birth
Analysis: Local Cross-Section Analysis
B CAT
Births: { }'ji s Deaths
If then sensitivesensitive to initial conditions.
If then insensitiveinsensitive to initial conditions.
limT TB A C
lim 0T TB A C
Death
2
i,
l
j
ow hi
Critical Quantity: ( )
Let t be a trajectory not drawn into the trivial fixed point.
withou For a system t i 2 2>1+ <1+M M
nput, if
j
x
i
Theorem :
gh
low
then
t is ( ) to initial condition.
For a system inp 2+O(1sensitive inse
ut, if then
nsiti/M )
v1> -1
e
wi
t
< -1
s
t
i
h
x
x
almost surely
almo
Stationary conditions, input and internal spikes have identical effect statistically. =number of spikes M
( ) to initial condition.
sensitive insensitivest surely
Assumptions :in the system at any time.
=ratio of number of internal spikes to number of total spikes in the system.
Analysis: Local Cross-Section Analysis
0
1
2
3
4
5
6
7
Synchronized Random Synfire Chains
2Hz Background; 200Spikes/volley2Hz Background; 100Spikes/volley20Hz Background; 200Spikes/volley20Hz Background; 100Spikes/volley
0
2
4
6
8
10
Uncorrelated Poisson Input
2 Hz20 Hz40 Hz
0
1
2
3
4
5
6
7
Synchronized Regular Synfire Chains
2Hz Background; 200Spikes/volley2Hz Background; 100Spikes/volley20Hz Background; 200Spikes/volley20Hz Background; 100Spikes/volley
0123456789
Dispersed Regular Synfire Chains
2Hz Background; 200Spikes/volley2Hz Background; 100Spikes/volley20Hz Background; 200Spikes/volley20Hz Background; 100Spikes/volley
Analysis: Local Cross-Section Analysis: Prediction
>1 =1 <1 =1
Normal SeizureSpike rate
Neocortical ColumnNeocortical Column
Analysis: Local Cross-Section Analysis: Prediction
Analysis: Discussion
2
i,j
( )ji•Existence of time average
•Systems without Input and with Stationary Input
Transformation invariant (StationaryStationary) Probability measure exists.
System has ErgodicErgodic properties.
•Systems with Transient Inputs
?•Information Coding (Computational State vs. Physical State)
•Attractor-equivalentAttractor-equivalent of class of trajectories.