1 Pertemuan 17 HOPFIELD NETWORK Matakuliah: H0434/Jaringan Syaraf Tiruan Tahun: 2005 Versi: 1

2 Learning Outcomes Pada akhir pertemuan ini, diharapkan mahasiswa akan mampu : Menjelaskan konsep dari Jaringan Hopfield

3 Outline Materi Hopfield Model. Lyapnov Function.

4 Hopfield Model

5 Equations of Operation n i - input voltage to the ith amplifier a i - output voltage of the ith amplifier C- amplifier input capacitance I i - fixed input current to the ith amplifier

6 Network Format Define: Vector Form:

7 Hopfield Network

8 Lyapunov Function V a  a T Wa – f 1– u  ud 0 a i       i1= S  b T a –+=

9 Individual Derivatives td d f 1– u  ud 0 a i       a i d d f 1– u  ud 0 a i       td da i f 1– a i  td i n i td i === d dt f 1– u  ud 0 a i       i1= S  n T d a dt = Third Term: Second Term: First Term:

10 Complete Lyapunov Derivative  – a i d d f 1– a i    td da i   2 i1= S  = From the system equations we know: So the derivative can be written: Ifthen a i d d f 1– a i  0 

11 Invariant Sets This will be zero only if the neuron outputs are not changing: Therefore, the system energy is not changing only at the equilibrium points of the circuit. Thus, all points in Z are potential attractors:

12 Example

13 Example Lyapunov Function V a  a T Wa –f 1– u  ud 0 a i       i1= S  b T a –+=

14 Example Network Equations

15 Lyapunov Function and Trajectory a 1 a 2 a 2 a 1 V(a)V(a)

16 Time Response tt a 1 a 2 V(a)V(a)

17 Convergence to a Saddle Point a 1 a 2

18 Hopfield Attractors V  a 1   V a 2   V... a S   V T 0 == The potential attractors of the Hopfield network satisfy: How are these points related to the minima of V(a)? The minima must satisfy: Where the Lyapunov function is given by: V a  a T Wa – f 1– u  ud 0 a i       i1= S  b T a –+=

19 Hopfield Attractors Using previous results, we can show that: The ith element of the gradient is therefore: a i   V a  – td dn i  – td d f 1– a i  ()  – a i d d f 1– a i  td da i === Since the transfer function and its inverse are monotonic increasing: All points for whichwill also satisfy Therefore all attractors will be stationary points of V(a).

20 Effect of Gain n a

21 Lyapunov Function V a  a T Wa – f 1– u  ud 0 a i       i1= S  b T a –+=  1.4=  0.14=  14= a f 1– u  ud 0 a i  2   ---  a i   cos   log 4   a i   coslog–== 4   a   coslog–

22 High Gain Lyapunov Function where As  the Lyapunov function reduces to: The high gain Lyapunov function is quadratic:

23 Example a 1 a 2 a 2 a 1 V(a)V(a)