Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 1 Short-term and Long-term Short-term and Long-term Memory Memory Motivation: very simple circuits can store patterns of activity Short-term: stability of activity patterns due to non- linear activity dynamics Long-term: storage of patterns through modifications of synapses The simplest STM system: 0 : 0 0 : 120 100 3 1 3 1 2 2 2 1 2 2 2 1 1 x x x x x S e S e e e S e e Short-term memory (STM) network Stimulus specific activity in delay period in units in temporal and pre- frontal cortex (after Fuster, 1996):
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Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 1
Short-term and Long-term MemoryShort-term and Long-term Memory
Motivation: very simple circuits can store patterns of activityShort-term: stability of activity patterns due to non-linear activity dynamicsLong-term: storage of patterns through modifications of synapses
The simplest STM system:
0: 0
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eSee
eSee
Short-term memory(STM) network
Stimulus specific activity in delayperiod in units in temporal and pre- frontal cortex (after Fuster, 1996):
Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 2
Stationary points:all 3 satisfy:
This is a cubic equation for e0 with 3 solutions:
Linearizing around these three stationary pointsresults in the 3 linear systems: (τ=20ms)
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1e
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)0,0(
)20,20(
)80,80(
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)3(100
e
eeee
80 , 20 , 0 30
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node" stable"
050050:sEigenvalue
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005.01
., -.-
A
saddle" unstable"
130030:sEigenvalue
05.008.0
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., -.
A
node" stable"
030070:sEigenvalue
05.002.0
02.005.03
., -.-
A
Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 3
Simulation:
started at(14,25):
started at(50,20):
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Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 4
consider additional input K to both units:
K K
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Hysteresis in STM ModelHysteresis in STM Model
Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 5
Problem: there should be some forgettingIdea: incorporate adaptation (fatigue) into units
Forgetting in STM ModelForgetting in STM Model
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ai variables slowly adjust slope of Naka Rushton non-linearity. If unit active for long time, it will experience “fatigue”.(models very slow hyperpolarizing potassium current.) a
Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 6
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Time (ms)
E(t
) (r
ed)
& A
(t)
(blu
e)
Simulation: present brief input K=50 to a1(t) between 200ms < t < 400msObservation: between 5 and 6 seconds after stimulus, network forgets
)(1 te
)(1 ta
Explanation: treat ai(t) as constant (slowly changing variable). Plot isoclines of e-dynamics with ai as parameter: stable and unstable nodes join and vanish
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Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 7
Discussion of STM ModelDiscussion of STM Model
Positive:• simple account of behavior of prefrontal neurons in delayed match to sample tasks
Limitations:• provides only qualitative account• no notion of interference• …
Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 8
Long Term MemoryLong Term Memory(associative memory)(associative memory)
Our simplest model neuron so far: McCulloch Pitts neuron• binary, i.e. two states: -1 (inactive) and +1 (active)
N neurons connected via weighted connections wij that represent different synaptic strengths (positive and negative)
Next activity determined by applying non-linear function to difference of a unit’s weighted sum of inputs andthreshold μi.
else:1
0:1 where,
1
oldnew yyxwx i
N
jjiji
' oldnew μWxx
)(
)(
'1
Ny
y
y
Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 9
Network of McCulloch-Pitts neurons with symmetric all-to-all connections and zero threshold.
connection from unit j to unit i:
symmetry:
zero threshold:
The Hopfield NetworkThe Hopfield Network
ijw
jiij ww
ii 0
1
oldnew
N
jjiji xwx
asynchronous updating: pick unit at random, apply update rule for this unit only, then pick next unit at random and update it, etc.
oldnew Wxx or
Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 10
Figure from Hertz,Krogh, Palmer (1991)
Example:object recognition, each pixelhas a corresponding unit,exhibits pattern completion
Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 11
Idea: activity pattern (state) ξ “stored” if it is fixed point of the update equation, i.e. if network is in this state and the update rule is applied, then state does not change:
Storing a single patternStoring a single pattern
ii
N
jjiji hwx
!
1
new
jiijw Claim: pattern stabilized if weights set according to:
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jjji
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jjiji NN
wh 111
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Proof: let’s be specific and set : jiij Nw 1
and hence: iix new
Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 12
Multiple patterns, storage capacity Multiple patterns, storage capacity
Second term so-called crosstalk term. Can make pattern unstable if big. Happens for large P and small N, i.e. many patterns in small net. Capacity: Pmax ~ 0.138 N
Consider stability of pattern ξp : pi
pi
N
j
pjiji hwx
!
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new
N
j
P
pkk
pj
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pi
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1 Weights:
Split sum over k intotwo parts: k=p and rest
First term alone would meanpattern is stable.
Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 13
Energy Function for Hopfield NetEnergy Function for Hopfield Net
The dynamics of the Hopfield network is governed by a bounded function of the statethat decreases over time (energy function, Lyapunov function).
Metaphor: energy landscape. Every unit update can only bring us downhill or let’s usstay at same level. Slide down to closest local energy minimum.
Note: inaccurate picture because states are not points ona plane but corners of N-dimensional hypercube.
energy
states
ji
N
i
N
jij xxwV
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1stored pattern withits basin of attraction
Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 14
Example: phone book Example: phone book
Idea: code text strings by having anumber of units for each letter/digit
stor
ed p
atte
rns
recall from only partial pattern: spurious states:
Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 15
Positive:- content addressable memory- weights can be computed directly, learning is instantaneous- distributed architecture results in fault tolerance (graceful degradation) if, e.g. some units or connections pruned- many extensions, e.g. for temporal patterns
Negative:- poor biological realism- poor generalization (no invariance, can’t shape basins of attractions)- only qualitative account
Note: more biologically plausible models of specific memory systems(e.g. the CA3 region of the Hippocampus) have been proposed.
Critique of Hopfield memoryCritique of Hopfield memory
Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 16
Positive:
• conceptually simple models, only stereotypic connectivity
• range of “interesting” phenomena, providing qualitative account of cognitive phenomena
• at least: metaphors for how a range of things may work
Negative:
• not necessarily easy to scale up
• need many parameters
• no learning
Conclusions: NeurodynamicsConclusions: Neurodynamics
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