The document summarizes key concepts about the Hopfield model, an attractor neural network model inspired by physics. It discusses how memory is stored in the symmetric connectivity matrix through Hebbian learning of stored patterns. During recall, the network dynamics relax toward one of the stored memory patterns as an attractor state. This can be modeled deterministically or stochastically. The number of memories an N-neuron network can reliably store is approximately 0.15N.

1. SS2016 Modern Neural
Computation
Lecture 3: Network
Dynamics
Hirokazu Tanaka
School of Information Science
Japan Institute of Science and Technology

2. Neural network as a dynamical system.
In this lecture we will learn:
• Attractor dynamics
- Hopfield model
- Winner-take-all and Winner-less competition
• Randomly connectivity
- Girko’s circular law
- Phase transition by synaptic variability
• Collective dynamics
- Hebb’s cell assemblies
- Synfire chain, Neuronal avalanche, small-world network
• Recurrent network dynamics
- Echo-state network, liquid-state network
- Self-organizing recurrent network (SORN)
• Synchronization
- Kuramoto model

3. Hopfield model inspired by physics of ferromagnetism.
1
1
iS

 

Hopfield (1982) PNAS; Gerstner (2014) Neuronal Dynamics
“Spin” variable for neuron i
Excited
Rest
General connectivity Symmetric connectivity
S1
S2
S3
S4
S5
S6
S7
S1
S2
S3
S4
S5
S6
S7
ij jiw w ij jiw w
Here we will see that a neural network with symmetric connectivity exhibits an
attractor dynamics.

4. Hopfield model inspired by physics of ferromagnetism.
   i ij j
j
h t w S t 
   
   
 
 
 
 
1
1
Pr 1|
Pr 1|
i
i
i
i
S
i i
S
i
h
t
i
t
h
S t t h t e e
S t t h t ee
 

  
 
       
     
H
H
   
 
   
  1 tanh
Pr 1|
2
i
i i
h t
h t t
i
i hi
h te
S t t h t
e e

 

 

       

 
,
ij i j
i j
S w S S    H1
1
iS

 

      lim Pr 1| sgni i iS t t h t h t
 
      
Hopfield (1982) PNAS; Gerstner (2014) Neuronal Dynamics

5. Associative memory is stored in connection strengths.
1
1 M
ij i jw p p
N
 

 
 1; 1, , , 1, ,ip i N M
   L L
M-stored patterns
Overlap with patterns
 
1 1 1
1 M N M
i ij j i j j i
j j
h t w S p p S p m
N
   
   
    
1
j j
j
m p S
N
 
 
Hebbian learning
Hopfield (1982) PNAS; Gerstner (2014) Neuronal Dynamics

6. Memory recall is a relaxation process to fixed pointes.
Tank & Hopfield (1987) Scientific American
1
1 M
ij i jw p p
N
 

 

7. Memory recall is a relaxation process to fixed points.
M=3 case
Deterministic case
Hopfield (1982) PNAS; Gerstner (2014) Neuronal Dynamics
Suppose that an initial pattern of population activity has a significant similarity with
pattern μ=3:
while there are no overlap with the other patterns:
 3
0 0.4m t 
   1 2
0 0 0.m t m t 
  1 1 2 2 3 3 3 3 3
0 sgn sgn sgni i i i i i iS t t p m p m p m p m p m p 

 
               
 

     
23 3 3
0 0
1 1
1.i i i
i i
m t t p S t t p
N N
       
Therefore, the population activity converges to the pattern μ=3.

8. Memory recall is a relaxation process to fixed points.
M=3 case
Deterministic case
Hopfield (1982) PNAS; Gerstner (2014) Neuronal Dynamics
Suppose that an initial pattern of population activity has a significant similarity with
pattern μ=3:
while there are no overlap with the other patterns:
 3
0 0.4m t 
   1 2
0 0 0.m t m t 
  1 1 2 2 3 3 3 3 3
0 sgn sgn sgni i i i i i iS t t p m p m p m p m p m p 

 
               
 

     
23 3 3
0 0
1 1
1.i i i
i i
m t t p S t t p
N N
       
Therefore, the population activity converges to the pattern μ=3.

9. Memory recall is a relaxation process to fixed pointes.
Stochastic case
Hopfield (1982) PNAS; Gerstner (2014) Neuronal Dynamics
       3 3
0Pr 1|i i i iS t t h t g p m t g p m t 

 
            
 

     3
0Pr 1|i iS t t h t g m t         
     3
0Pr 1|i iS t t h t g m t          
3
1ip  For
3
1ip  For
   
   
3 3
3 3
0 0
3 3
0 03 3
1 1
1
1 1
i i
i i
i
i i
i i
p p
m t t p S t t
N
N N
S t t S t t
N N N N
 
 
 
    
     

 

10. Memory recall is a relaxation process to fixed pointes.
Stochastic case
Hopfield (1982) PNAS; Gerstner (2014) Neuronal Dynamics
   
   
3 3
3 3
0 0
3 3
0 03 3
1 1
1
1 1
i i
i i
i
i i
i i
p p
m t t p S t t
N
N N
S t t S t t
N N N N
 
 
 
    
     

 
        
3
3 3 3
03 3
1
1 1
(*) Pr 1 Pr 1 2 1
i
i i i
i
p
S t t N S t t N S t t g m
N N
 
 

                    
        
3
3 3 3
03 3
1
1 1
(**) Pr 1 Pr 1 2 1
i
i i i
i
p
S t t N S t t N S t t g m
N N
 
 

                     
     3 3 3
0 0 0m t t g m t g m t          
Update rule
(*) (**)

11. Memory recall is a relaxation process to fixed pointes.
Figure 17,8 in Gerstner (2014) Neuronal Dynamics
     3 3 3
0 0 0m t t g m t g m t          
    1
1 tanh
2
g m m 
   3 3
0 0tanhm t t m t     
If we assume a sigmoid function for activation:
When β>1, the network is attracted to the pattern.

12. Memory recall is a relaxation process to fixed pointes.
Tank & Hopfield (1987) Scientific American

13. Demo: Matlab example.
N=25, β=3

14. Demo: Matlab example.
N=25, β=0.8

15. Exercise: fill the Matlab code.
%% parameters
N = 5^2; % # neurons
beta = 3; % inverse temperature
T = 9; % # simulation steps
%% M=3 patterns
P = [ [1,1,1,1,-1, -1,1,-1,-1,1, -1,1,-1,-1,1, -1,1,-1,-1,1, -1,1,1,1,-1]; ...
[1,1,1,1,1, -1,-1,-1,1,-1, -1,-1,-1,1,-1, 1,-1,-1,1,-1, 1,1,1,-1,-1]; ...
[-1,1,1,1,1, 1,-1,-1,-1,-1, 1,-1,-1,-1,-1, 1,-1,-1,-1,-1, -1,1,1,1,1] ];
figure(1);
subplot(131); imagesc(reshape(P(1,:),5,5)); title('pattern 1');
subplot(132); imagesc(reshape(P(2,:),5,5)); title('pattern 2');
subplot(133); imagesc(reshape(P(3,:),5,5)); title('pattern 3');
% connectivity matrix
W = 1/N*(P'*P);
%% simulation
S = 2*(rand(N,1)>0.5)-1; % initial pattern
figure(2); subplot(1,9,1); imagesc(reshape(S,5,5)); title(['t=1']);
for t=2:T
h = W*S; % inputs
p = 1/2*(1+tanh(beta*h)); % prob(S=+1)
S = 2*(rand(N,1)<p)-1; % stochastic Glaubner dynamics
figure(2); subplot(1,9,t);
imagesc(reshape(S,5,5)); title(['t=' num2str(t)]);
end
Write your own code here.

16. How many patterns can N-neuron network remember?
Hopfield (1982) PNAS; Gerstner (2014) Neuronal Dynamics
Stability condition
 0i iS t p

Assume that the network represents the pattern ν at time t0:
Then, in deterministic case, the network at time t0+Δt is determined by
 
 
 
0
1
1 1
1
1
1
sgn
1 1
sgn
1
sgn 1
1
sgn 1
N M
i j
j
N N
j j
j j
N
j
i j
i j i j
i i i j
i
j
S t t p p p
N
p p p p p p
N N
p p p p p
N
p
N
  

     
 
    
 


  



 
    
 
  
   
   
  
    
   
 

  
 
 1
i i
j
j
N
jp p p p   
 
 
 
  
 

17. How many patterns can N-neuron network remember?
     error
2 1
Pr 1 Pr 1 Pr 1| 1 erf
2
0
2
,i i i i
N
P a a a a
M
    
  
              
   
: N
Hopfield (1982) PNAS; Gerstner (2014) Neuronal Dynamics
Stability condition
   0 sgn 1ii iS t t p a
   
 1
1 N
i ji i j
j
a p p p p
N
   

 
  
   
1
E 0, Vari i
M
a a
N
 

 
Therefore, the number of patterns M must be small enough
compared to the number of neurons N.

18. Physics of spin systems.
Ising model
Spin glass
 
,
i j
i j
S J S S    H
 
,
ij i j
i j
S J S S    H
, : nearest neighbori j
2
0
,;ij ij
J J
N N
J J
 
 
 
: N
 
,
ij i j
i j
S J S S    H
Short-range interaction
(Edwards-Anderson model)
Long-range interaction
(Sherrington-Kirkpatrick
model)
Ising (1925); Edwards & Anderson (1975); Sherrington & Kirkpartick (1975)

19. Simple dynamics with random connections.
 tanh
d
dt
   
x
x Wx
2
0,ijW
N
 
 
 
: N
 N
d
dt
   
x
I W x
Dynamics of N interconnected neural network with random connections
Linearized dynamics
The origin x=0 is a fixed point. Whether it is stable or unstable is
determined by the eigenvalues of the connectivity matrix W.
S1
S2
S3
S4
S5
S6
S7

20. Semi-circle law of eigenvalue density function.
1
0,ij
N
W
 
 
 
: NAll components are
normally distributed:
And symmetric: ij ijW W
Semi-circle law: Wigner (1951)
In the limit of infinite n, the eigenvalues of an nxn random symmetric matrix W
follow a semi-circle distribution:
  21
4
2
p  

 
%% Parameters
n =10000; t =1; v = [ ]; dx = 0.1;
%% Experiment
for i=1:t,
a=randn(n); % random nxn matrix
s =(a+a')/2 ; % symmetrized matrix
v =[v ; eig(s)] ; % eigenvalues
end
v=v/sqrt (n/2) ;
%% Plot
[count , x]= hist(v , -2:dx:2) ;
cla reset; hold on ;
%% Theory
plot(x , sqrt (4-x.^2)/(2*pi) , 'k-', 'LineWidth' , 2);
bar (x , count/(t*n*dx) , 'facecolor', [0.7 0.7 0.7]) ;
Wigner (1951)

21. Circular law of eigenvalue density function.
1
0,ij
N
W
 
 
 
: NAll components are
normally distributed:
Circular law: Girko (1985)
In the limit of infinite n, the eigenvalues of an nxn random (not necessarily
symmetric) matrix W follow a uniform distribution in a unit circle in the complex
plane.
N = 20000;
sigma = 1.01;
W = randn(N,N)/sqrt(N)*sigma;
figure(k); clf; hold on;
plot(eig(W)-1, 'k.');
theta = linspace(0, 2*pi, 100);
plot(cos(theta)-1, sin(theta), 'k');
plot([0 0], [-1 1], 'r')
set(gca, 'color', [0.9400 0.9400 0.9400]); axis equal;
Girko (1984) Teor. Veroyatnost. i Primenen.

22. Circle law of eigenvalue density function.
1.00 0.99  1.01 

23. Dale’s law: neurons are either excitatory or inhibitory.
Rajan & Abbott (2006) Phys Rev Lett
Excitatory neuron
inhibitory neuron
0 forijW i 
0 forijW i 
11 1
1
1 1j
n n
i n
ni nj n
W W
W W
W W
W W
 
 
  
 
 
W
L L L
L L L
M MM M
Exercise: Examine the dynamics of the neural network when Dale’s law is
imposed on the random connection matrix, i.e., all components in a column are
either positive (excitatory) or negative (inhibitory). This problem is already
analyzed by Rajan and Abbott (2006).

24. Hierarchical modular connectivity structure.
Exercise: Examine the dynamics of the neural network with a hierarchical
modular connectivity matrix. This problem has NOT been analyzed so far, to my
knowledge.
Anatomical studies suggest that cortical neurons are not randomly connected and that
they are connected with a modular and hierarchical manner.
regular network random network small world
HM stochastic HM Cat visual cortex
n×n Hierarchical network:
(1) m on-diagonal blocks of size s are
connected with prob pm and n=ms.
(2) 1st level of off-diagonal blocks of
size s are connected with prob pc.
(3) Subsequent levels of off-diagonal
blocks are of size 2s, 4s, 8s, … and
connected with prob pcq, pcq2, pcq3, …
Robinson et al. (2009) Phys Rev Lett; Aljadeff, Stern, Sharpee (2015) Phys Rev Lett

25. Firing-rate equation.
Chapter 7 in Dayan & Abbott (2000) Theoretical Neuroscience
 r s
dv
v F I
dt
   
1
uN
s
s s b b s
b
dI
I w u I
dt


       w uSynaptic input dynamics
Firing rate dynamics
s r ? s
s s
dI
I
dt
    w u  sv F I
s r =  r
dv
v F
dt
    w u

26. Feedforward and recurrent networks.
Feedforward network
Chapter 7 in Dayan & Abbott (2000) Theoretical Neuroscience
 
d
dt
   
v
v F Wu
Recurrent network
 
d
dt
    
v
v F Wu Mv

27. Excitatory-inhibitory network.
 
 
E
E E E EE E EI I
I I IE E III I
I
d
dt
d
dt


  
  


F h
v
v vM M
F vh M M
v
v
v v
Chapter 7 in Dayan & Abbott (2000) Theoretical Neuroscience
Ev IvEE 0M II 0M
IE 0M
EI 0M
: Excitatory population activity : Inhibitory population activityEv Iv

28. Continuously labeled network.
Chapter 7 in Dayan & Abbott (2000) Theoretical Neuroscience
   1 v
T
a Nv v v v LDiscretely labeled network
Continuously labeled network
 v v 
 
11 1
1
v
v v v
N
ab
N N N
M M
M
M M
 
 
   
 
 
M
L
M O M
L
 ,M M  
 
          , ,r
dv
v F d W u M v
dt



         

        
  

29. Linear network: Selective amplification.
Chapter 7 in Dayan & Abbott (2000) Theoretical Neuroscience
 
       ,r
dv
v h d M v
dt




       

      
   1
, cosM

 


 

  

30. Nonlinear network: Gain modulation.
Chapter 7 in Dayan & Abbott (2000) Theoretical Neuroscience
 
       ,r
dv
v h d M v
dt

 

       

     

 

   1
, cosM

 


 

  

31. Nonlinear network: Winner-takes-all selection.
Chapter 7 in Dayan & Abbott (2000) Theoretical Neuroscience
 
       ,r
dv
v h d M v
dt

 

       

     

 

   1
, cosM

 


 

  

32. Nonlinear network: Working memory.
Chapter 7 in Dayan & Abbott (2000) Theoretical Neuroscience
 
       ,r
dv
v h d M v
dt

 

       

     

 

   1
, cosM

 


 

  

33. Non-symmetric connectivity: Winner-less competition.
- Dynamics in phase space connecting saddle points (heteroclinic connections).
- Memories are represented in terms of heteroclinic trajectories.
         i
i i ij j i
j
da
a t a t S t
dt
 
 
   
 
S S
Rabinovich et al. (2001) Phys Rev Lett

34. Winnerless competition: Coupled FN neurons.
   
 
i
x i i i i
i
i i
i
z i ij j
j
dx
f x y z x
dt
dy
x by a
dt
dz
z g G x
dt
 

   
  
   
0
1
1
0
0
0
1
1
0
0
0
0
0
1
0
0
0
0
1
0
0
0
0
1
0
0
0
1
0
0
1
0
1
0
0
0
0
1
0
0
0
1
1
0
1
0
1
1
0
0
0
1
1
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
0
0
0
1
0
0
0
0
1
1
0
0
0
0
0
0
1
0
0
1
1
0
0
0
1
1
0
0
0
0
0
1
0
0
0
0
0
0
0
1
0
0
0
0
1
Rabinovich et al. (2001) Phys Rev Lett

35. Winnerless competition: Coupled FN neurons.
   
 
i
x i i i i
i
i i
i
z i ji j
j
dx
f x y z x
dt
dy
x by a
dt
dz
z g G x
dt
 

   
  
   
Rabinovich et al. (2001) Phys Rev Lett
15 52 21 24 45 65 26 36
53 74 57 84 58 86 89 95 2
g g g
g
g g g g g
g g g g g gg
    
     
 
  
4
2
9
1
3
5
8
7
6

36. Winnerless competition: Matlab simulation.
Rabinovich et al. (2001) Phys Rev Lett
function Y = odeWLC(t, X)
% parameters
tau1 = 0.08; tau2 = 3.1;
a = 0.7; b = 0.8;
nu = -1.5;
% stimulus
S = [0.1; 0.15; 0.0; 0.0; 0.15; 0.1; 0.0; 0.0; 0.0];
% connectivity
g = zeros(9, 9); g0=2;
g(1,5)=g0; g(5,2)=g0; g(2,1)=g0; g(2,4)=g0;
g(4,5)=g0;
g(6,5)=g0; g(2,6)=g0; g(3,6)=g0; g(5,3)=g0;
g(7,4)=g0;
g(5,7)=g0; g(8,4)=g0; g(5,8)=g0; g(8,6)=g0;
g(8,9)=g0;
g(9,5)=g0;
% differential equations
x = X(1:9); y = X(10:18); z = X(19:27);
dxdt = ((x-x.^3/3)-y-z.*(x-nu)+0.35+S)/tau1;
dydt = x-b*y+a;
dzdt = (g'*(x>=0)-z)/tau2;
Y = [dxdt; dydt; dzdt];
x0 = [-1.2*ones(9,1); -0.62*ones(9,1);
0*ones(9,1)];
[T,X] = ode45(@odeWLC,[0 500], x0);
% all FHN neurons
figure(1);
for n=1:9
subplot(9,1,n); plot(T,X(:,n),'k');
end
% PCA
[y, s, l] = pca(X(:,1:9));
figure(3); plot3(s(:,2), s(:,3), s(:,4)); grid on;

37. Winnerless competition: Matlab simulation.
Rabinovich et al. (2001) Phys Rev Lett

38. Example: Olfactory processing in insects.
Mazor & Laurent (2005) Neuron

39. Cell assemblies: functional units of brain computation.
Definition
Cell assembly: a group of neurons that perform a given action or
represent a given percept.
Hebb (1949) The Organization of Behavior; Harris (2005) Nature Rev Neurosci

40. Cell assemblies: functional units of brain computation.
Definition
Cell assembly: a group of neurons that perform a given action or
represent a given percept.
Hebb (1949) The Organization of Behavior; Harris (2005) Nature Rev Neurosci
1
1
1
1
2
2
2
2
3
3
3
3
4
4
4
4
1
1
1
1
2
2
2
2
3
3
3
3
4 4
4
4

41. Synfire chain in a feedforward network.
Diesmann et al. (1999) Nature

42. Synfire chain in a feedforward network.
Brian Spiking Neural Network Simulator,http://briansimulator.org/

43. Feedforward synfire chain requires activity tuning.
Diesmann et al. (1999) Nature
   , 30 spikes, 2 msn      , 40 spikes, 2 msn      , 50 spikes, 2 msn  
   , 60 spikes, 2 msn      , 70 spikes, 2 msn      , 80 spikes, 2 msn  

44. Feedforward synfire chain requires activity tuning.
Diesmann et al. (1999) Nature
   , 80 spikes,1msn      , 80 spikes, 2 msn      , 80 spikes, 3 msn  
   , 80 spikes, 4 msn      , 80 spikes, 5 msn      , 80 spikes,10 msn  

45. Synfire chain can be made robust by feedback connections.
Moldakarimov et al. (2015) PNAS
1
1
1
1
2
2
2
2
3
3
3
3
4
4
4
4

46. Synfire chain can be made robust by feedback connections.
Moldakarimov et al. (2015) PNAS
   , 80 spikes, 5 msn      , 80 spikes, 5 msn      , 80 spikes, 5 msn  
excitatory feedback: 0.1 excitatory feedback: 0.2 excitatory feedback: 0.3

47. Neural avalanche with scale-free dynamics.
Beggs & Plenz (2003) J Neurosci
 
3
size where
2
P



 :

48. Neural avalanche as a branching process.
Zapperi et al. (1995) Phys Rev Lett
   , , s
n n
s
f x p P s p x     , ,n ng x p Q p x

 
     2
1 , 1 ,n nf x p x p pf x p
          2
1 , 1 ,n ng x p p pg x p   
 ,nP s p
 ,nQ p
probability of avalanche of size s
probability of avalanche boundary
of size s
Generating function of avalanche size Generating function of avalanche
boundary size
Recursive relation of fn(x,p) Recursive relation of gn(x,p)

49. Neural avalanche as a branching process.
Zapperi et al. (1995) Phys Rev Lett
     2
, 1 ,f x p x p pf x p      
2
1 1 4
,
2
pqx
f x p
px
  

 
 2 2 3 4 5
2
2 3 !!1 1 1 5 7 1
1 1 1
2 8 16 128 256 2 2 !
s
s
s
s
x x x x x x x x
s



           L
 
 
 
2
2 1
2
2 3 !!1 1 4 1
, 2
2 2 !
s s
s
spqx
f x p qx pq x
px p s



  
   
 
 
 
 
  
 
 
3
2
2
2 3 !! 2 2 !1 1
2 1 4 2 exp
2 ! 2 ln 41 !
s ss s s
P s pq pq s
p s ps pqs
   
       
:
Recursive relation of fn(x,p) when n is large enough
Using a Taylor expansion
The generating function can be expanded as
Therefore the probability of avalanche size s is given as:

50. Echo-state network: harnessing chaotic units.
Jaeger & Haas (2004) Science

51. Echo-state network.
% load the data
trainLen = 2000; testLen = 2000; initLen = 100;
data = load('MackeyGlass_t17.txt');
% generate the ESN reservoir
inSize = 1; outSize = 1;
resSize = 1000;
a = 0.3; % leaking rate
rand( 'seed', 42 );
Win = (rand(resSize,1+inSize)-0.5) .* 1;
W = rand(resSize,resSize)-0.5;
opt.disp = 0;
rhoW = abs(eigs(W,1,'LM',opt));
disp 'done.'
W = W .* ( 1.25 /rhoW);
% allocated memory for the design (collected states)
matrix
X = zeros(1+inSize+resSize,trainLen-initLen);
% set the corresponding target matrix directly
Yt = data(initLen+2:trainLen+1)';
% run the reservoir with the data and collect X
x = zeros(resSize,1);
for t = 1:trainLen
u = data(t);
x = (1-a)*x + a*tanh( Win*[1;u] + W*x );
if t > initLen
X(:,t-initLen) = [1;u;x];
end
end
% train the output
reg = 1e-8; % regularization coefficient
X_T = X';
Wout = Yt*X_T * inv(X*X_T +
reg*eye(1+inSize+resSize));
Y = zeros(outSize,testLen);
u = data(trainLen+1);
for t = 1:testLen
x = (1-a)*x + a*tanh( Win*[1;u] + W*x );
y = Wout*[1;u;x];
Y(:,t) = y;
u = y;
end
http://minds.jacobs-university.de/mantas/code

52. Echo-state network.

53. Summary
• Population neural dynamics can be formulated by using
the techniques developed in physics and dynamical
systems, including spin models, phase transitions, scale-
free dynamics and so on.
• Population activity exhibits a variety of emergent
phenomena such as attractor memory dynamics, winner-
takes-all process, short-term memory, winner-less
competition, and so on.
• Population neural dynamics is a very active field, and a
number of novel researches keep going on today.