Transcript rsws3 7032

Biomimetic searching strategies
Massimo Vergassola
CNRS, URA 2171
Institut Pasteur, Unit “In Silico Genetics”
Source Odors
Direction and
velocity of
the wind are
determined
by air currents
and visual
clues.
wind
Zigzag
Casting:
Extended
crosswind
Zigzagging and casting
(J.S. Kennedy, e.g. in Physiological
Entomology,1983)
2m
Male
Moth released
Sniffers
Olfactory
robots with
applications
to the
detection of
chemical
leaks, drugs,
bombs, land
and/or sea
mines.
D. Martinez “On the right scent” Nature, 445, 371-372,
2007 (N&V).
Micro vs macro-organisms:
the role of size and transport
Chemotaxis of living organisms
Temporal or
spatial
gradients are
sensed and
either climbed
or descended.
Crucial that the chemoattractant field be smooth and the
concentration high enough to be measurable.
Gradients ought to provide a reliable local cue.
Physical constraints on
concentration measurements
(Berg & Purcell, Biophys. J., 1977)
Smoluchowski’s diffusion-limited rate of encounters
J(r)  4Dac(r)
Reliable measurement of concentration requires:
Measured hits in the time Tint >> fluctuations:

Dac(r)Tint 1
Bottomline: Chemotaxis requires exponential
integration times for exponentially small
concentrations
Searches by macroorganisms
Responses times are O(ms)
Away from the source, gradients
are not effectively traceable and
do not always point to the source.
Odor encounters are sparse and
sporadic.
Yet, birds respond Km’s away and
moths locate females hundreds of
meters away.
Existing sniffers rely on micro-organism
mimetic strategies
• Chemotactic methods, e.g. Ishida et al. (1996); Kuwana et
al. (1999); the robolobster by Grasso & Atema et al. (2000);
Russell et al. (2003).
• Plume-tracking, e.g. Belanger & Arbas (1998); Li, Farrell,
Cardé (2001); Farrell, Pang, Li (2003)&(2005); Ishida et al. (2005);
Pang, Farrell (2006).
Effective in dense conditions (relatively
close to the source)
Sniffer front view
Sniffer in action
Strategies for
searches starting
far away from the
source, in dilute
conditions?
M.V., E. Villermaux, B.I. Shraiman
Infotaxis as a strategy for searching
without gradients. Nature, 445: 4069, 2007.
In a nutshell
Concentration is not a good local clue in dilute
conditions.
What else could we track in the “desert”,
when nothing is detected?
1. Build a map of probability for the source
position on the basis of the history of
receptions.
2. Move locally to make the map sharp as fast as
possible, i.e. maximize the rate of entropy
reduction.
The message of odor encounters
r3, t3
r2, t2
r1, t1
The source emits particles that are
transported in the (random)
environment.
Consider them as a message sent
to the searcher.
Message in a random medium.
Use the trace of odor encounters
experienced by the searcher to
infer the position of the source.
Decoding the message
As in message decoding, construct the posterior
distribution Pt(r0) for the position of the source r0
from the trace ((r1,t1),(r2,t2),…,(rH,tH)) of the hits.
Rr | r0 
Hit rate at position r if source located at r0 .
t

e
 R(r(s)|r0 )ds
0
Pt r0  
H
 R(r(t ) | r )
h
0
h1
t
 dze

 R(r(s)|z )ds
0
H
 R(r(t ) | z)
h
h1
A simple model of random medium
“Particles” are patches of odors where mixing has not
dissipated them below the detectibility threshold.
Particles emitted at rate R, advected by a mean wind V,
having a finite lifetime  and diffused with diffusivity D.
1
0  V  c(r | r0 )  Dc(r | r0 )  c(r | r0 )  Rr  r0 

After some algebra
R
c(r | r0 ) 
eV (yy0 ) 2Der 
4D(r  r0 )
D

1 V 2 4D
R(r | r0 )  4Dac(r | r0 )
General problem: How should we exploit the
posterior and deal with its uncertainties?
t

e
 R(r(s)|r0 )ds
0
Pt r0  
H
 R(r(t ) | r )
h
0
h1
t
 dze

 R(r(s)|z )ds
0
H
 R(r(t ) | z)
h
h1
The “unusual” feature is that the field cannot
be quite trusted and is continuously updated.
ML is not suitable.
Search time-entropy relationship
N points to visit. Probability at the j-th visited point
is pj and neighborhood constraints dismissed.




1


   jp j   
p
1


 j 
 p j ln p j  S

j
 j

 j

Gibbs distribution

S  T ln T  (T 1)ln( T 1)


p j e j
reducing to (T>>1)
T e
S1
 N /e
Search times vs entropy
T e
S1
N e
Note the exponential dependence on S, contrary
to the “standard” optimal code length inequality:

l p
j
j
S
j
The reason is that the “search alphabet” is
degenerate, i.e. made of a single letter. Words are
discriminated by their length only

(no coalescence as in Huffman coding)
Infotaxis
Choose the local direction of motion maximizing the
rate of information acquired: Maximum expected
reduction <S> of the entropy of the field Pt(r0) .


S r  rj  Pt rj 0  S  1 Pt rj  0S0  1S1  ...
h k eh
k 
k!
With the
expected hit rate
hrj  t  Pt rRrj | rdr
Exploitation vs exploration
Gradients of concentration in chemotaxis
Rate of acquisition of information, i.e. reduction of
entropy of the posterior field Pt(r0).


S r  rj  Pt rj 0  S  1 Pt rj  0S0  1S1  ...
Exploitation:
maximum likelihood.
Exploration: passive
gathering of information.
RS Sutton, AG Barto Reinforcement Learning MIT Press, 1998.
Infotactic trajectories
pM sperm responding (sea urchin)
Kaupp et
al., Nature
Cell
Biology,
2003
Search time statistics
Infotaxis is the most robust and rapid
among a set of alternative strategies
Robustness to inaccuracies in the
model of the environment
Independent
detection
model in a
real jet flow
Spatial maps in animal brains
Microstructure of
a spatial map in
the entorhinal
cortex
Nature, 2005
and following
papers by E.I.
Moser and colls.
Spatial cues are
transmitted to
the hippocampus
J. O’Keefe, J. Dostrovsky Brain
Research 1971 discovery of place
cells in hippocampus (see also The
Hippocampus as a Cognitive Map,
1978)
In collaboration with
Boris Shraiman (Kavli Inst. Theor. Phys., UCSB)
Emmanuel Villermaux (IRPHE, Marseille)
A simple possible way to account for
time correlations
A model where consecutive detections have
a space-independent rate give:

e
  R(r(s)|r0 )ds
i
Vi
Pt r0  
 dze


i
Vi
H
 R(r(t
h
) | r0 )
h1
R(r(s)|z )ds H
 R(r(t
h
) | z)
h1
Consecutive detections are counted just once
Learning about the source and the
medium
Start the searcher with rough estimates of the
parameters which make the rate function
R(r|r0) flatter than in realitynot stuck.
The searcher will get to the source slowly but
steadily. Once there, infer from its odor
encounter trace the parameters of the medium
and the source.
Learning about the source and the
medium
Loglikelihood of
the
experienced
series of
odor
encounters.