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.