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Lecture 13: Associative Memory References: D Amit, N Brunel, Cerebral Cortex 7, 237-252 (1997) N Brunel, Network 11, 261-280 (2000) N Brunel, Cerebral Cortex 13, 1151-1161 (2003) J Hertz, in Models of Neural Networks IV (L van Hemmen, J Cowan and E Domany, eds) Springer Verlag, 2002; sect 1.4

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What is associative memory?

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“Patterns”: firing activity of specific sets of neurons (Hebb: “assemblies”)

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What is associative memory? “Patterns”: firing activity of specific sets of neurons (Hebb: “assemblies”) “Store” patterns in synaptic strengths

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What is associative memory? “Patterns”: firing activity of specific sets of neurons (Hebb: “assemblies”) “Store” patterns in synaptic strengths Recall: Given input (initial activity pattern) not equal to any stored pattern, network dynamics should take it to “nearest” (most similar) stored pattern

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What is associative memory? “Patterns”: firing activity of specific sets of neurons (Hebb: “assemblies”) “Store” patterns in synaptic strengths Recall: Given input (initial activity pattern) not equal to any stored pattern, network dynamics should take it to “nearest” (most similar) stored pattern (categorization, error correction, …)

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Implementation in balanced excitatory-inhibitory network Model (Amit & Brunel): p non-overlapping excitatory subpopulations

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Implementation in balanced excitatory-inhibitory network Model (Amit & Brunel): p non-overlapping excitatory subpopulations each of size n = fN (fp < 1)

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Implementation in balanced excitatory-inhibitory network Model (Amit & Brunel): p non-overlapping excitatory subpopulations each of size n = fN (fp < 1) stronger connections within subpopulations (“assemblies”)

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Implementation in balanced excitatory-inhibitory network Model (Amit & Brunel): p non-overlapping excitatory subpopulations each of size n = fN (fp < 1) stronger connections within subpopulations (“assemblies”) weakened connections between subpopulations

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Implementation in balanced excitatory-inhibitory network Model (Amit & Brunel): p non-overlapping excitatory subpopulations each of size n = fN (fp < 1) stronger connections within subpopulations (“assemblies”) weakened connections between subpopulations Looking for selective states: higher rates in a single assembly

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Model Like Amit-Brunel model (Lecture 9) except for exc-exc synapses:

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Model Like Amit-Brunel model (Lecture 9) except for exc-exc synapses: From within the same assembly:

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Model Like Amit-Brunel model (Lecture 9) except for exc-exc synapses: From within the same assembly:

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Model Like Amit-Brunel model (Lecture 9) except for exc-exc synapses: From within the same assembly: (strengthened, “Hebb” rule)

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Model Like Amit-Brunel model (Lecture 9) except for exc-exc synapses: From within the same assembly: From outside the assembly: (strengthened, “Hebb” rule) (weakened, “anti-Hebb”)

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Model Like Amit-Brunel model (Lecture 9) except for exc-exc synapses: From within the same assembly: From outside the assembly: Otherwise:no change (strengthened, “Hebb” rule) (weakened, “anti-Hebb”)

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Model Like Amit-Brunel model (Lecture 9) except for exc-exc synapses: From within the same assembly: From outside the assembly: Otherwise:no change (strengthened, “Hebb” rule) (weakened, “anti-Hebb”) To conserve average strength:

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Model Like Amit-Brunel model (Lecture 9) except for exc-exc synapses: From within the same assembly: From outside the assembly: Otherwise:no change (strengthened, “Hebb” rule) (weakened, “anti-Hebb”) To conserve average strength: =>

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Mean field theory Rates: active assembly inactive assemblies rest of excitatory neurons inhibitory neurons ext input neurons

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Mean field theory Input current to neurons in the active assembly: Rates: active assembly inactive assemblies rest of excitatory neurons inhibitory neurons ext input neurons

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Mean field theory Input current to neurons in the active assembly: Rates: active assembly inactive assemblies rest of excitatory neurons inhibitory neurons ext input neurons to rest of assemblies:

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Mean field theory Input current to neurons in the active assembly: Rates: active assembly inactive assemblies rest of excitatory neurons inhibitory neurons ext input neurons to rest of assemblies: to other excitatory neurons:

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Mean field theory Input current to neurons in the active assembly: Rates: active assembly inactive assemblies rest of excitatory neurons inhibitory neurons ext input neurons to rest of assemblies: to other excitatory neurons: to inhibitory neurons:

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Mean field theory (2) Noise variances (white noise approximation):

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Mean field theory (2) Noise variances (white noise approximation):

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Mean field theory (2) Noise variances (white noise approximation):

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Mean field theory (2) Noise variances (white noise approximation):

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Mean field theory (2) Noise variances (white noise approximation):

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Mean field theory (2) Noise variances (white noise approximation): Rate of an I&F neuron driven by white noise:

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Mean field theory (2) Noise variances (white noise approximation): Rate of an I&F neuron driven by white noise:

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Spontaneous activity: All assemblies inactive:

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Spontaneous activity: All assemblies inactive:

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Spontaneous activity: All assemblies inactive: becomes

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Spontaneous activity: All assemblies inactive: becomes Similarly,

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Spontaneous activity: All assemblies inactive: becomes Similarly,

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Spontaneous activity: All assemblies inactive: becomes Similarly, and

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Spontaneous activity: All assemblies inactive: becomes Similarly, and

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Spontaneous activity: All assemblies inactive: becomes Similarly, and Solve for

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Simplified model (Brunel 2000)

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pf << 1

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Simplified model (Brunel 2000) pf << 1 g + ~1/f >> 1

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Simplified model (Brunel 2000) pf << 1 g + ~1/f >> 1 variances + = act, 1 as in spontaneous-activity state

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Simplified model (Brunel 2000) pf << 1 g + ~1/f >> 1 variances + = act, 1 as in spontaneous-activity state Define L = fJ 11 g +

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Simplified model (Brunel 2000) pf << 1 g + ~1/f >> 1 variances + = act, 1 as in spontaneous-activity state Define L = fJ 11 g + Then (1) spontaneous activity state has r + = r 1,

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Simplified model (Brunel 2000) pf << 1 g + ~1/f >> 1 variances + = act, 1 as in spontaneous-activity state Define L = fJ 11 g + Then (1) spontaneous activity state has r + = r 1, (2) In recall state with r act > r +, r 1 and r 2 are same as in spontaneous activity state

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Simplified model (Brunel 2000) pf << 1 g + ~1/f >> 1 variances + = act, 1 as in spontaneous-activity state Define L = fJ 11 g + Then (1) spontaneous activity state has r + = r 1, (2) In recall state with r act > r +, r 1 and r 2 are same as in spontaneous activity state (3) r act is determined by

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Graphical solution (This L = (our L ) x m ) ( r -> )

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Graphical solution (This L = (our L ) x m ) ( r -> ) 1-assembly memory/recall state stable for big enough L (or g + ) ~ describes “working memory” in prefrontal cortex

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Capacity problem In this model, memory assemblies were non-overlapping. This is unrealistic.

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Capacity problem In this model, memory assemblies were non-overlapping. This is unrealistic. Alternative model: neurons in each assembly independently chosen A single neuron can be in many assemblies

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Capacity problem In this model, memory assemblies were non-overlapping. This is unrealistic. Alternative model: neurons in each assembly independently chosen A single neuron can be in many assemblies How many patterns can be stored using N neurons before interference between patterns destroys the recall ability?

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Capacity problem In this model, memory assemblies were non-overlapping. This is unrealistic. Alternative model: neurons in each assembly independently chosen A single neuron can be in many assemblies How many patterns can be stored using N neurons before interference between patterns destroys the recall ability? Here: solve this for a simplified model (binary neurons, can be either excitatory on inhibitory, “Hebbian” synapse formula)

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Model N Binary neurons:

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Model N Binary neurons: Assemblies/patterns : p sets of n = fN neurons with S i = 1

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Model N Binary neurons: Assemblies/patterns : p sets of n = fN neurons with S i = 1

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Model N Binary neurons: Assemblies/patterns : p sets of n = fN neurons with S i = 1

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Model N Binary neurons: Assemblies/patterns : p sets of n = fN neurons with S i = 1 (Synchronous) dynamics:

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Model N Binary neurons: Assemblies/patterns : p sets of n = fN neurons with S i = 1 (Synchronous) dynamics: Synapses:

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Model N Binary neurons: Assemblies/patterns : p sets of n = fN neurons with S i = 1 (Synchronous) dynamics: Synapses: (global inhibitory term makes average J ij = 0 )

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Order parameters

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(normalized) overlap with pattern 1:

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Order parameters (normalized) overlap with pattern 1: Total average activity:

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Net input to neuron i:

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Fluctuations

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with = p/N

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Fluctuations with = p/N (recall nf = O(1) )

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Mean field equations For neurons in pattern 1, h = m + Gaussian noise

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Mean field equations For neurons in pattern 1, h = m + Gaussian noise =>

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Mean field equations For neurons in pattern 1, h = m + Gaussian noise => with

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Mean field equations For neurons in pattern 1, h = m + Gaussian noise => with

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Mean field equations For neurons in pattern 1, h = m + Gaussian noise => with For other neurons, h = Gaussian noise

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Mean field equations For neurons in pattern 1, h = m + Gaussian noise => with For other neurons, h = Gaussian noise =>

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Mean field equations For neurons in pattern 1, h = m + Gaussian noise => with For other neurons, h = Gaussian noise => Solve for m and Q

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Graphical interpretation

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Capacity estimate Weight in tail ( h > m ) of big gaussian centered at 0 must be < weight in small one centered at m

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Capacity estimate Weight in tail ( h > m ) of big gaussian centered at 0 must be < weight in small one centered at m Can take

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Capacity estimate Weight in tail ( h > m ) of big gaussian centered at 0 must be < weight in small one centered at m Can take =>

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Capacity estimate Weight in tail ( h > m ) of big gaussian centered at 0 must be < weight in small one centered at m Can take => i.e.,

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Capacity estimate Weight in tail ( h > m ) of big gaussian centered at 0 must be < weight in small one centered at m Can take => i.e., Use asymptotic form of H :

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Capacity estimate Weight in tail ( h > m ) of big gaussian centered at 0 must be < weight in small one centered at m Can take => i.e., Use asymptotic form of H : => capacity estimate

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