Download The Spike Activity of Neocortical Columns: A Dynamical Systems

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
•Synapse
•Dendrites (Input)
•Cell Body
•Axon (Output)
Output
Input
Threshold
•Absolute Refractory Period
•Exponential Decay of effect of a
spike on membrane potential
Time
Background: Target System
Neocortical Column:
~ 1 mm2 of the cortex
Output
Recurrent network
~100,000 neurons
~10,000 synapses per neuron
~80% excitatory
~20% inhibitory
Recurrent System
Input
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 Space
•Set of all legal states
Dynamics
•Velocity Field
•Flows
•Mapping
Local & Global properties
•Sensitivity to initial conditions
•Fixed points and periodic orbits
Content:
•Model
•A neuron
•System of Neurons: Phase Space & Velocity Field
•Simulation Experiments
•Neocortical Column
•Qualitative Characteristics: EEG power spectrum & ISI frequency distribution
•Formal Analysis
•Local Analysis: Sensitivity to Initial Conditions
•Conclusions
Model: Single Neuron
x11
t=0
x12 x13 x14
t=0
x21
t=0
x22
x31 x32
Potential
Function
Each spike represented as: How
long since it departed from soma.
1
1
n1
1
1
2
n2
2
1
m
x21
nm
m
P( x ,..., x , x ,..., x ,...., x ,..., x )
x11
Time
Model: Single Neuron: Potential function
Membrane Potential: Implicit, C  , everywhere bounded function.
P( x1 , x2 ,..., xm )
xi   xi1 , xi2 ,..., xin 
i
Effectiveness of a Spike:
i  1...m, &j  1...ni P
i  1...m, &j  1...ni P
xij
xi
i  1...m, &j  1...ni P(.)
Threshold:
P(.) = T(.) and
dP
dt
 0 for xij  0
 0 for xij  
j
j
xi  0
0
 P(.)
j
xi  
Model: System of Neurons
x11
x12
P1 ( x1 , x2 ,..., xm )
x21
P2 ( x1 , x2 ,..., xm )
x31
P3 ( x1, x2 ,..., xm )
x32
x41
t 0
•Dynamics
•Birth of a spike
•Death of a spike
t
•Point in the Phase-Space
•Configuration of spikes
Model: Single Neuron: Phase-Space
Preliminary: x1 , x2 ,..., xni 0,  i
n
0
Transformation 1:
 zi  e
2 ixi
z1 , z2 ,..., zni  T ni

Transformation 2:
 z i  an 1 z
n
i
ni 1
a0 , a1 , ..., ani 1  C ni
 ..  a1 z  a0 
( z  zn ) *.. * ( z  z2 ) * ( z  z1 )
i

0, 
Model: Single Neuron: Phase-Space
Theorem: 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
σ
 Phase-Space for fixed number of Dead spikes: L nii
•Dead vs. Live Spikes: Theorem: j  k , i Ljni  i Lkni is an imbedding
i
0
 Assign finer topology to L ni
•Phase-Space for n=3
• 1, 2 dead spikes.
Model: System of Neurons: Velocity Field
System:
S
 Cartesian product of Phase-Spaces;
 Birth of Spike:
 Surfaces
Pi I at Pi (.)=T(.) and dPi

i
L0n
i
i=1
dt
0
Velocity Field: Theorem: Vi can be defined mathematically
 Vi1 (when no event) at p  Pi I
 Vi 2 (for birth of spike) at p  Pi I
 Vi 2  Vi1 disregarding position on submanifold
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 spikes: Seizure-like & Normal Operational Conditions
Simulations: Neocortical Column: Chaotic Activity
T=0
T=1000 msec
Normal Operational Conditions (20 Hz): Subset (200 neurons) of 1000 neurons for 1 second.
Simulations: Neocortical Column: Total Activity
Normalized time series: Total number of active spikes & Power Spectrum
Simulations: Neocortical Column: Spike Trains
Representative spike trains: Inter-spike Intervals & Frequency Distributions
Simulations: Neocortical Column: Propensity for Chaos
ISI’s of representative neurons: 3 systems; 70%,80%,90% synapses driven by pacemaker
Simulations: Neocortical Column: Sensitive Dependence
on Initial Conditions
T=0
T=400 msec
Spike activity of 2 Systems: Identical Systems, subset (200) of 1000 neurons, Identical
Initial State except for 1 spike perturbed by 1 msec.
Analysis: Local Analysis
•Are trajectories sensitive to initial conditions?
•If there are fixed points or periodic orbits, are they stable?
Analysis: Setup: Riemannian Metric
 Riemannian Metric  Symmetric Bilinear Form  Orthonormal Basis
S
S
: T( L ni )  T( i Lnii1 )  R
 i 1
i
i=1
i=1
Volume and Shape Preserving between events (birth/death of spikes)
Orthonormal Basis:

,...,
1
x1

,
n   1
x1
1
1

x2
1
,...,

n 2  2
x2
,..,..,
1

x S
1
,...,
V 1 is a constant velocity field (volume and shape preserving)

n S  S 1
x S
Analysis: Setup: Riemannian Metric
t 0
t
t 0
t
•Discrete Dynamical System
•Event ► Event ►Event….
•Event: birth/death of spike
Analysis: Measure Analysis
Death of a Spike
PI
Birth of a Spike
Analysis: Perturbation Analysis
x12 ,..., x1n1  1 1 ,...., x1S ,..., xSnS  S 1
Birth:

x11 , x12 ,..., x1n1 1 1 ,...., x1S ,..., xSnS  S 1
x11 , x12 ..., x1n1 1 ,...., x1S ,..., xSnS  S
Death:

x12 ,..., x1n1 1 ,...., x1S ,..., xSnS  S
Perturbation Analysis:
 x11 

i 1.. S
j  2.. ni  i 1
i j xij
P
i 
j


i 1.. S
j  2.. ni  i 1
xi
j

i 1.. S
j  2.. ni  i 1
P
xi
j
i j  1
Analysis: Perturbation Analysis
t 0
t
What is i j ?
Positive i
j
Negative i j
Analysis: Local Cross-Section Analysis
Births: {i j }' s
Deaths
AT
B
Birth
C
Death
If limT  B  AT  C   then sensitive to initial conditions.
If limT  B  AT  C  0
then insensitive to initial conditions.
Analysis: Local Cross-Section Analysis
Critical Quantity:  
j 2
(

 i)
i,j
Theorem : Let  x  t  be a trajectory not drawn into the trivial fixed point.

2 
  <1+
 then
M high 

 x  t  is almost surely sensitive (insensitive) to initial condition.
2
 For a system without input, if  >1+
M low

1 
-1   < -1 then

 

 x  t  is almost surely sensitive (insensitive) to initial condition.
 For a system with input, if  >
 Assumptions :
2+O(1/M low )
Stationary conditions, input and internal spikes have identical effect statistically.
M=number of spikes 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: Prediction
10
8
2 Hz
20 Hz
40 Hz
6
4
2
0
Uncorrelated Poisson Input
7
2Hz Background; 200
Spikes/volley
2Hz Background; 100
Spikes/volley
20Hz Background; 200
Spikes/volley
20Hz Background; 100
Spikes/volley
6
5
4
3
2
1
0
9
8
2Hz Background; 200
Spikes/volley
2Hz Background; 100
Spikes/volley
20Hz Background; 200
Spikes/volley
20Hz Background; 100
Spikes/volley
7
6
5
4
3
2
1
0
Synchronized Regular Synfire Chains
Dispersed Regular Synfire Chains
7
2Hz Background; 200
Spikes/volley
2Hz Background; 100
Spikes/volley
20Hz Background; 200
Spikes/volley
20Hz Background; 100
Spikes/volley
6
5
4
3
2
1
0
Synchronized Random Synfire Chains
Analysis: Local Cross-Section Analysis: Prediction
Seizure
Normal
Spike rate
>1
=1
<1
Neocortical Column
=1
Analysis: Discussion
j 2
(

•Existence of time average  i )
i,j
•Systems without Input and with Stationary Input
Transformation invariant (Stationary) Probability measure exists.
System has Ergodic properties.
•Systems with Transient Inputs
?
•Information Coding (Computational State vs. Physical State)
•Attractor-equivalent of class of trajectories.