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2001/02 Lecture 6. Examples of mathematical and
computational models:
 Predator-prey models;
 Spike generation models;
 Associative memory models (discrete time);
 Neural oscillator models;
Conservative Systems
Newton’s law
mx  F (x ) . Here we have no damping or friction, and we can say that the energy is
conserved (conservative system).
dV
Let V(x) denote the potential energy, F ( x )  
. Then
dx
dV
mx 
 0.
dx
dV
d 1

1

x ( mx 
)  0   mx 2  V ( x )   0  E   mx 2  V ( x )   const
dx
dt  2

2

Energy E is constant.
Pendulum
Lx   g sin x
1
L
x
g
m
t
where x is the angle from downward vertical, g is the acceleration due to gravity, L is
the length of the pendulum.
Dimensionless form:
x   sin x or
 x  y

 y   sin x
Equilibrium points: x  0, y  0, 1, 2  i - Centre.
x   , y  0, 1  1, u1  (1,1); 2  1, u2  (1,1) - Saddle.
Damping to the pendulum:
 x  y

 y  b * y  sin x
Hopfield Neural Network, Attractor Neural Network
Let x(t) is binary vector (01 or +1);
 1, if  wij x j (t )  0

x i (t  1)   1, if  wij x j (t )  0

 x (t ),  wij x j (t )  0
wij  w ji , wii  0
“Energy”
E(t )  1 / 2 wij xi x j
Assynchronose dynamics: update only one unit per time.
E(t  1)  E(t )  1 / 2( x1 (t  1)  x1 (t )) wij x j  0
Associative memory: memorise vectors α, β, ...
wij   i  j   i  j  ...
Continous time:
xi   xi   wij g ( x j )  Ii , g () is the sigmoid function
E  1 / 2 wij gi g j   Ii gi 
Predator-Prey Models
Bazykin’s model of a predator-prey ecosystem:
xy
 
2
 x  x  1  x  x

 y   y  xy  y
1  x

2
x and y are (scaled) numbers of a prey and predator, respectively.  ,  ,  ,  are
nonnegative parameters describing the behaviour of isolated populations and their
interaction.
Three population model:
E   E  EC  EW

C   C  EC
W   W  EW

where E(t) is the elk population, C(t) is the coyote population, W(t) is the wolf
population. All populations are measured in thousands of animals. t is measured in
years (from 1995).
Oscillating Chemical Reactions
4 xy
 
 x  a  x  1  x 2

 y  bx1  y 
2

 1 x 
where x and y are the dimensioless concentrations, a and b are positive parameters.
(See Appendix 1 bellow).
Spike generation models
Slow-fast equations:
x  y  F ( x)

 y  a  kx
Van der Pol equation
x  ax( x 2  1)  x  0
also is an example of slow-fast equations. Notice that
d
x  ax( x 2  1)  [ x  a( 13 x 3  x)] .
dt
So, if we let
F ( x)  13 x 3  x, z  x  aF ( x) ,
the Van der Pol equation implies that
z  x  ax( x 2  1)   x
Hence the Van der Pol equation may be rewritten as
 x  z  aF ( x)

 z   x
d
d

,   1 / a 2 then
Let y  z / a,
dt d
a
3
 x  a[ y  F ( x)]

1
 y   a x
x  y  F ( x)
.

 y   x
Nullclines: y  F ( x), x  0 .
Equilibrium point is a point of the nullclines intersection.
Vectors, vector field. Fast and slow time scales (movements).
Relaxation oscillations.
Spike generation effect:
x   x * (1  x )(1  x )  y

 y  x  a
x(0)  I , y (0)  y0
where x is potential, I is external input, a is constant.
Threshold.
Hodgkin-Huxley model
(See appendix 2 bellow).
Neural oscillator model
 1 E   E  f ( wee E  wei I  P)

 2 I    I  f ( wie E  wii I  Q)
where E(t) is the average activity of excitatory neural population, I(t) is the average
activity of inhibitory neural population, w are the connection strengths, P and Q are
the external inputs, f ( . ) is the sigmoid function.
Lorenz equation (chaotic behaviour)
 x   ( y  x )

 y  rx  y  xz
 z   xy  bz

where x, y, z are variable to describe convection in the fluid layer,  is the Prandtl
number, r is the Rayleigh number, b is related to the height of the fluid layer.
(See Appendix 3 bellow).
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