IEEE Circuits and Systems Magazine - Q3 2019 - 17

preventing cohesion loss and making sure fast and robust transfer of information are critical to collective decision making. So how a birds flock flies as one and how
does a bird in the flock moves as if it knew exactly where it
would go? Recently, a remarkable paper, i.e., Ref. [98], has
provided excellent answer for this problem. "This is one
of the first studies that gets to the details of how groups
move in union," Professor David Sumpter from Uppsala
University in Sweden commented. We will review their
work in the following section.
Attanasi et al. used high-resolution cameras to film
starlings and utilized tracking software to reconstruct
the three dimensional trajectory of individual bird in a
flock [98]. Based on the spatiotemporal data, researchers have conducted a very detailed study of information
transfer and decision-making turning of birds. Using
the analysis methods of image processing, the tracking
data showed that the information about changing directions started from a few leaders and swept through
the whole starling flock with an approximately linear
dispersion law, that is, the speed of propagation attenuation is negligible, hence minimizes the decoherence of flock [98]. The speed of information transfer is
surprising, it is an approximately constant value which
is in the range of twenty to forty meters per second,
which means that the decision to turn takes just a little
more than half a second to sweep through a flock of
400 birds [98].
So what could be the fundamentally theoretical mechanism for these phenomena? And what would be the functions of decision making in the turning process of birds?
Almost all the theoretical descriptions of collective
behavior are based on align dynamics [48]: individuals
in a flock tend to keep consistent with the moving directions of their neighbors,
v i (t + Tt ) = v i (t ) + J

/

v j (t ) Tt,

(7)

j ! N i (t)

in which they have ignored the noise and assumed that
the alignment strength J is large enough to achieve a
deeply ordered phase as natural flocks are. Let Tt " 0,
Eq. (7) equals to
dv i
=- 2H , H =-J / v i · v j,
dt
2v i
i, j

(8)

where H is the corresponding Hamiltonian of the flock
system, which represents the total energy. The obtained data indicated another interesting fact that the
trajectories of starlings lie approximately on a plane
during a turning, hence maximally promotes the synchronization of these birds. This allows one to use two
THIRD QUARTER 2019

parameters to describe v i as v 0 e i{. Where { is the angle
between the moving direction of bird i and the flock.
The alignment strength J is very large and the velocity of bird i is very close to the velocity of the flock, so
that { is extremely small. Then the Hamiltonian can be
expressed as
H =-J / v @i v j =-J / v 20 e -i{i e i{ j
i, j

i, j

. -v 20 J / cos ({ i - { j)
i, j

= J / ({ i - { j) 2 .
2 i, j

(9)

Denote the average minimal distance between two agents
as a, Eq. (9) is equivalent to
2

H=a J
2
2

=a J
2
2

=a J
2

/ ( d{ ( x, t )) 2
i, j

xj

/#
i, j x i

#

( d{ (x, t ))2 3
dx
a3

dx 3 ( d{ (x, t ))2 .
a3

(10)

The corresponding motion equation is
{o =-

2 H = a 2 J d 2 {.
2{

(11)

We recast Eq. (11) to Langevin equation formation
0·

d2 {
d{
+1 ·
- a 2 J d 2 { = 0,
dt
dt 2

from above equation one can get the dispersion law
0 · ~ 2 - i · ~ - a 2 Jk 2 = 0,
that is, ~ = ik 2 . The frequency is pure imaginary, which
leads to an overdamped occasion of information transfer in birds' turning. Concurrently, the general solution
of heat Eq. (11) is
{ (x, t ) =

( p - x) 2

1
2a rJt

# }(p ) e - 4a Jt
2

dp,

in which x 2 /(4a 2 Jt ) is the damping rate. The information
will travel sublinearly, i.e., x + t . These two consequences are shapely contrast with the obtained data [98].
The theory itself has two significant defects to describe the turning of bird flocks. Firstly, Hamiltonian
(9) is apparently invariant under a uniform rotation of
the velocities,
H = J / ({ i - { j)2 = J / (({ i + d{) - ({ j + d{))2 .
2 i, j
2 i, j
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IEEE Circuits and Systems Magazine - Q3 2019

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