Stochastic Model of a Micro Agents population
26
description
Stochastic Model of a Micro Agents population. Dejan Milutinovic [email protected]. Outline. Motivating problem. Introduction to Math. Analysis. Mathematical Analysis. Applications. Biology. Robotics. Motivating problem. T-Cell. CD3. peptide. APC. MHC. - PowerPoint PPT Presentation
Transcript of Stochastic Model of a Micro Agents population
Outline
Motivating problem
Introduction to Math. Analysis
Mathematical Analysis
Applications
Biology Robotics
T-Cell Receptor (TCR) triggering
APCMHC
peptide
T-Cell
CD3
T-Cell, CD3 receptor, Antigen Presenting Cell (APC),
peptide-MHC complex
Motivating problem
T-Cell population
- T-Cell
- APC
Introduction to Math. Analysis
q(123)=never_connectedq(246)=connected
q(235)=disconnected
T-Cell population
- T-Cell
- APC
Introduction to Math. Analysis
Complex System !!!
How the Micro Dynamics of the Individuals propagates to the Dynamics of Macro observations ?
Introduction to Math. Analysis
1 –never connected, 2 - connected, 3- disconnected, a-connection, b-disconnection
q=3
u(t)=aq=1 q=2
u(t)=au(t)=b
)()(
0)(
tCxty
tx
)()(
))(()( 1
tCxty
txftx
)()(
))(()( 2
tCxty
txftx
The Micro Agent model of the T-Cell
Mathematical Analysis
Mathematical Analysis
Micro Agent (A)Initial condition (x0,q0)
Input event
sequence
Au(t)
Continuous output
Y(t)
Deterministic system
u(t) Y(t)
a b c
Mathematical Analysis
Stochastic Micro Agent (SA)
A
A
AStoch.
processDeterminist.
systemStoch.
process
SAStochastic system
Mathematical Analysis
Micro and Macro Dynamics relation
• PDF function describes the state probability of one A
• Looking to the large population of A, PDF is a normalized distribution of the state occupancy by all A
• Micro dynamics of A and macro dynamics of A population are related through the state PDFs
Dual Meaning of the State Probability Density Function
• Statistical Physics reasoning (Boltzman distribution)
A stochastic process (x(t),q(t))X Q is called a Micro Agent Stochastic Execution iff a Micro Agent stochastic input event sequence e(n),nN, 0 = 0 1 2 … generates
transitions such that in each interval [n,n+1), nN, q(t)
q(n).
Micro Agent Stochastic Execution
Remark 1. The x(t) of a Stochastic Execution is a continuous time function since the transition changes only the discrete state of a Micro Agent.
Mathematical Analysis
f(x,N)1 i N
V V V
x1
xn
xn-1. ..
f(x,1)
e(n )
f(x,i) q
e(n+1 )
e(n+2 )
X x Q
A Stochastic Micro Agent is a pair SA=(A,e(t)) where A is a Micro Agent and e(t) is a Micro Agent stochastic input event sequence such that the stochastic process (x(t),q(t))X Q is a Micro Agent Stochastic Execution.
Stochastic Micro Agent (SA)Mathematical Analysis
AA
AStoch.
processDeterminist.
systemStoch.
process
SAStochastic system
Stochastic system
A Stochastic Micro Agent is called a Continuous Time Markov Process Micro Agent iff (x(t),q(t))X Q is a Micro Agent Continuous Time Markov Process Execution.
Mathematical Analysis
Continuous Time Markov Process Micro Agent (CTMPA)
SAq=3
u(t)=aq=1 q=2
u(t)=au(t)=b
)()(
0)(
tCxty
tx
)()(
))(()( 1
tCxty
txtx
)()(
))(()( 2
tCxty
txtx
)),,((),(
)),2,(()2,(
)),1,(()1,(
),(),(
tNxNxf
txxf
txxf
txLt
tx T
Mathematical Analysis
)()( tPLtP T
TN tPtPtPtP )()()()( 21
NNT
ijL ij
TtNxtxtxtx )),,(()),2,(()),1,((),( )),,(( tix
),( ixf
The Continuous Time Markov Chain Micro Agent with N discrete state and state probability given by
where
state i and
is the probability of discrete
is transition rate matrix andis rate of transition from discrete state i to discrete state j.The vector of probability density functions
where is probability density function of state (x,i) attime t, satisfies
is the vector of vector fields value at state (x,i).where
Biological application
The Micro Agent model of the T-Cell
u(t)=a
q=1
q=2
q=3
u(t)=au(t)=b
12
23
32
b-disconnection, ij – event rate which leads to transition from state i to state j
0 –never connected, 1 - connected, 2- disconnected, a-connection,
)()(
)()( 3
txty
txktx
)()(
)()( 2
txty
txktx
)()(
0)(
txty
tx
Biological application
0
1
2 02
4
0
2
4
6
8
Time [s]TCR quantity (x)[nor.]
((x
,1),
t)
0
1
2 02
4
0
2
4
6
8
Time [s]TCR quantity (x)[nor.]
((x
,2),
t)
01
2 02
4
0
2
4
6
8
Time [s]TCR quantity (x)[nor.]
((x
,3),
t)
01
2 02
4
0
2
4
6
8
Time [s]TCR quantity (x)[nor.]
(y,
t)
Case I solution
12 =0.9, 23= 0 , 32 =0.5, k2 =0.5, k3=0.25
0
1
2 02
4
0
2
4
6
8
Time [s]TCR quantity (x)[nor.]
((x
,1),
t)
0
1
2 02
4
0
2
4
6
8
Time [s]TCR quantity (x)[nor.]
((x
,2),
t)
01
2 02
4
0
2
4
6
8
Time [s]TCR quantity (x)[nor.]
((x
,3),
t)
01
2 02
4
0
2
4
6
8
Time [s]TCR quantity (x)[nor.] (
y,t)
Biological application
Case II solution
12 =0.9, 23= 0.8 , 32 =0.9, k2 =0.5, k3=0.05
0
1
2 02
4
0
2
4
6
8
Time [s]TCR quantity (x)[nor.]
((x
,1),
t)
0
1
2 02
4
0
2
4
6
8
Time [s]TCR quantity (x)[nor.]
((x
,2),
t)
01
2 02
4
0
2
4
6
8
Time [s]TCR quantity (x)[nor.]
((x
,3),
t)
01
2 02
4
0
2
4
6
8
Time [s]TCR quantity (x)[nor.] (
y,t)
Biological application
Case III solution
12 =0.9, 23= 0.8 , 32 =0.9, k2 =0.5, k3=0.25
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
2
4
6
(y,
t)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
2
4
6
TCR quantity (x)[nor.]
(y,
t)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
2
4
6
(y,
t)
Case I
Case II
Case III
0s
0s
0s
3.4s
3.4s
3.4s
Biological application
Biological application
Biological application
Robotics application
Source 1 Source 3Source 2
x2
x1
x1
x2
x1 x1
=/4 =-/4 =0
Population
a)
b)
q=3
2321
3212q=1
q=2
Txxy
vx
vx
][
)0sin(
)0cos(
21
2
1
Txxy
vx
vx
][
)4sin(
)4cos(
21
2
1
Txxy
vx
vx
][
)4sin(
)4cos(
21
2
1
Robotics application
Milutinovic, D., Athans, M., Lima, P., Carneiro, J. “Application of Nonlinear Estimation Theory in T-Cell Receptor Triggering Model Identification”, Technical Report RT-401-02, RT-701-02, 2002, ISR/IST Lisbon, Portugal
Milutinovic D., Lima, P., Athans, M. “Biologically Inspired Stochastic Hybrid Control of Multi-Robot Systems”, submitted to the 11th International Conference on Advanced Robotics ICAR 2003,June 30 - July 3, 2003 University of Coimbra, Portugal
Milutinovic D., Carneiro J., Athans, M., Lima, P. “A Hybrid Automata Modell of TCR Triggering Dynamics” , submitted to the 11th Mediterranian Conference on Control and Automation MED 2003,June 18 - 20 , 2003, Rhodes, Greece
Milutinovic, D., “Stochastic Model of a Micro Agents Population”, Technical Report ISR/IST Lisbon, Portugal (working version)
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