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

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

3. 3 Outline Materi Hopfield Model. Lyapnov Function.

4. 4 Hopfield Model

5. 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. 6 Network Format Define: Vector Form:

7. 7 Hopfield Network

8. 8 Lyapunov Function V a  1 2 --- a T Wa – f 1– u  ud 0 a i       i1= S  b T a –+=

9. 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. 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. 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. 12 Example

13. 13 Example Lyapunov Function V a  1 2 --- a T Wa –f 1– u  ud 0 a i       i1= S  b T a –+=

14. 14 Example Network Equations

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

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

17. 17 Convergence to a Saddle Point a 1 a 2

18. 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  1 2 --- a T Wa – f 1– u  ud 0 a i       i1= S  b T a –+=

19. 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. 20 Effect of Gain n a

21. 21 Lyapunov Function V a  1 2 --- 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  ------ 2  ---  a i 2 --------   cos   log 4  2 ---------  a i 2 --------   coslog–== 4  2 ---------  a 2 --------   coslog–

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

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