Lecture 14 Population dynamics and associative memory; stable learning

Transcription

1 Lecture 14 Population dynamics and associative memory; stable learning -Introduction -Associative Memory -Dense networks (mean-ield) -Population dynamics and Associative Memory -Discussion

2 Systems or computing and inormation processing Brain Computer Distributed architecture 10 (10 proc. Elements/neurons) No separation o processing and memory memory CPU input Von Neumann architecture 1 CPU 10 (10 transistors)

3 motor cortex visual cortex association cortex to motor output

4 1mm neurons 3 km wires Signal: action potential (spike) action potential

5 Systems or computing and inormation processing Brain 1mm neurons 3 km wires Distributed architecture neurons 4 10 connections/neurons No separation o processing and memory

6 - recognize/understand images: pattern recognition Noisy image Associative memory/ collective computation Full image Brain-style computation

7 Lecture 14 Population dynamics and associative memory; stable learning -Introduction -Associative Memory -Dense networks (mean-ield) -Population dynamics and Associative Memory -Final discussion

8 Associative memory Elementary pixel S i = +1 S i = -1 w i = +1 w i = +1 w i = -1 dynamics S i ( t 1) sgn w S i t Hopield model Sum over all interactions with i

9 Associative memory interactions w i p i p Prototype p 1 Prototype p 2 S i ( t 1) sgn Sum over all prototypes dynamics w S i t Hopield model Sum over all interactions with i

10 Associative memory interactions w i p i p This rule is very good or random patterns Sum over all prototypes It does not work well or correlated patters Prototype p 1 S i ( t 1) sgn dynamics w S i t Hopield model Sum over all interactions with i

11 Associative memory Interacting neurons Prototype p 1 Hopield model Finds the closest prototype i.e. maximal overlap (similarity) m Computation - without CPU, - without explicit memory unit

12 Hebbian Learning k pr e w i i post When an axon o cell repeatedly or persistently takes part in iring cell i, then s eiciency as one o the cells iring i is increased Hebb, local rule - simultaneously active (correlations)

13 Hebbian Learning Elementary pixel S i = +1 S i = -1 w i = +1 w i = +1 w i = -1

14 Hebbian Learning item memorized

15 Hebbian Learning Recall: Partial ino item recalled

16 Lecture 14 Population dynamics and associative memory; stable learning -Introduction -Associative Memory So ar: neuron=spin=pixel=binary=on/o BUT: what about more realistic spiking neurons?

17 Lecture 14 Population dynamics and associative memory; stable learning -Introduction -Associative Memory -Dense networks (mean-ield) -Population dynamics and Associative Memory -Stable learning

18 Populations o spiking neurons t? I(t) population dynamics? t population activity A( t) n( t; t N t t)

19 Neuron # Activity in a populations o neurons A [Hz] input -low rate -high rate time [ms] 200 Neuron # Population neurons - 20 percent inhibitory - randomly connected u [mv] time [ms] 200

20 Homogeneous network (I&F) Consider 1 neuron

21 Homogeneous network (I&F) Assumption o Stochastic spike arrival: network o exc. neurons, total spike arrival rate A(t) d C u dt All synapses have identical weight I I 0 k I( t) I ion k k I(t) Membrane equation Synaptic current pulses o shape J 0 ( t tk ) N EPSC R I(t)

22 Homogeneous network (I&F) Assumption o Stochastic spike arrival: network o exc. neurons, total spike arrival rate A(t) C d dt u k I ion k I(t) membrane equation I ( t) I o I noise I I 0 R I(t) I 0 A 0 Population activity

23 Analysis o Homogeneous Population Step 1: Single neuron property: Inect noise current I I 0 I(t) Hodgkin-Huxley d C u dt k I ion k I(t) Leaky I&F d dt u ( u u ) R I( t) rest I ( t) I o I noise Measure requency with noise I ( t) I o I noise g ( I 0 ) I 0

24 Analysis o Homogeneous Population Step 2 : consider 1 neuron in the network I ext All neurons receive the same input ( mean ield ): - input rom network - external input I I0 I0 A0 I ext Population activity Mean input rom network prop. to population activity

25 Step 3: assume Stationary State/Asynchronous State A 0 I g( 0 ) All neurons are the same A(t)=const Step 4: close equation calculate A0 A 0 g( I0) g( A0 I ext ) I0 A0 I ext 1 A0 I0 I ext typical mean ield (Curie Weiss) A 0 g( I 0 ) requency (single neuron) Blackboard

26 Exercise (some time ago): ind stationary state A(t)=const I0 A0 I ext 1 A0 I0 I ext ully connected coupling J0/N typical mean ield A 0 requency (single neuron) g( I 0 ) close equations calculate A0

27 Lecture 14 Population dynamics and associative memory; stable learning -Introduction -Associative Memory review -Dense networks (mean-ield) -Population dynamics and Associative Memory

28 Back to Associative memory -Possible with spiking neurons -Calculation: mean-ield -Prototypes = random patterns Interacting neurons Computation - without CPU, - without explicit memory unit

29 Associative memory - simple model -Rate model/population activity -Calculation: mean-ield -Prototypes = random patterns Interacting neurons rate model Single-neuron iring rate Ai g( wi A ) population rate g( I 0 ) w i N 2 max p i p

30 Associative memory For comparison with Spin S=+/-1 1 m N p S Prototype p 1 Task: Find the prototype with maximal overlap m 1 N p (2 +/-1 max 1) Blackboard

31 Associative memory g ( I 0 ) Input current g( h ) 0 Input potential Prototype p 1 m 1 N p (2 max 1) overlap 2 1 m max p N N +/-1 p with prob /-1

32 Associative memory main idea requency (single neuron) m g( I 0 ) Pattern recognized Prototype p 1 m Pattern not recalled Task: Find the prototype with maximal overlap 1 N p (2 max 1) +/-1

33 Conclusion - Associative memory -Possible with spiking neurons -Calculation: mean-ield -Prototypes = random patterns Interacting neurons Computation - without CPU, - without explicit memory unit

34 Associative memory interactions w i p i p This rule is very good or random Patterns (a=0.5) Prototype p 1 Sum over all prototypes It does not work well or correlated patters or low activity random Patterns (e.g., a=0.1) wi ( pi c)( p a) e.g., in each pattern exactly 10 percent o neurons are active

35 Associative memory 1 wi K( pi, p ) N Locally stored inormation Prototype p 1 Blackboard or low activity random Patterns (e.g., a=0.1) wi ( pi c)( p a) e.g., in each pattern exactly 10 percent o neurons are active

36 Associative memory 1 wi K( pi, p ) N Blackboard Locally stored inormation wi ( pi c)( p a) e.g., a=0.1 means in each pattern exactly 10 percent o neurons are active Lz ( ) h( z) K( z, z ') ( s) A( z ', t s) ds z ' N A( z ') g( h( z '))

37 Associative memory For a given pattern only 2 populations! A requency (single neuron) g( h ) 0 Blackboard h in h act Lz ( ) h( z) K( z, z ') ( s) A( z ', t s) ds z ' N 1 1 wi K( pi, p ) ( pi c)( p a) N N A( z ') g( h( z '))

38 Lecture 14 Population dynamics and associative memory; stable learning -Introduction -Associative Memory -Dense networks (mean-ield) -Population dynamics and Associative Memory -Final discussion

39 Nearly the end: what can I improve or the students next year? Integrated exercises? Miniproect? Overall workload?(4 credit course = 6hrs per week) Background/Prerequisites? -Physics students -SV students -Math students

40 Exam: -written exam, 23.6 rom 8:15-11:15 - miniproects counts 1/3 towards inal grade For written exam: -bring 1 page A5 o own handwritten notes The end

41 The end

42 Exercise now: Associative memory g( I 0 ) i g ( wi w i ) N 2 max p i p i g( p m i ) Prototype p 1 Assume 4 patterns. At time t=0, overlap with Pattern 3, no overlap with other patterns. discuss temporal evolution (assume that patterns are orthogonal) m 1 N p (2 max 1)

43 The end