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Memory Network Maintenance Using Spike-Timing Dependent Plasticity David Jangraw, ELE ’07 Advisor: John Hopfield, Department of Molecular Biology category i(mA) 10 • Each neuron is modeled as a parallel RC circuit with a firing threshold 5 6 0 0 0.02 15 Br Bl 2 Gr 4 6 property 0.08 0.1 time(s) output voltage 0.12 8 10 • The firing frequency ‘f’ is our measure of activity. 5 Each black square in this grid represents an active neuron encoding something about a person. For example, if the bottom row of neurons encodes eye color, and the ‘green’ neuron is active, this person has green eyes. -5 0 0.02 0.04 0.06 0.08 time(s) 0.1 0.12 70 frequency (Hz) Synaptic Drift (δT): Small, random changes in synaptic strength due to “noisy” cellular processes • This will make the neuron fire more quickly than the neurons connected to it (-δt) 50 40 • STDP would transform this spike timing delay into negative changes in synaptic strength (-δT) 30 • This would produce a proportional negative change in input current (+δI) that could stabilize the memory! 20 10 Normal memories have equal activity in each participating neuron, but synaptic drift causes unequal activity (a corrupted memory) or even memory loss. 0 0 2 4 6 8 10 12 Io = DC input current (mA) 14 16 18 20 Non-Memory +δT +δI –δt –δT –δI -2 • STDP applied based on average delay after 50 spikes of each neuron -3 -4 • STDP successfully used to synchronize firing of many neurons -5 -6 -7 0 0.5 1 1.5 2 2.5 3 applications of STDP rule (x50 spikes) 3.5 • Slope of STDP rule affects speed and stability of synchronization 4 Synchronization of two neurons 0.4 0.2 1 -0.6 -0.8 m=5 m=30 m=50 m=80 m=100 -1 -1.2 1 2 3 4 5 6 7 8 applications of STDP rule (x50 spikes) 0.6 18 0.2 delta T - Neuron Activity 0.4 12 -0.6 -0.8 11.5 12 12.5 Io (mA) 13 13.5 14 Changes in input current (while still on the f-I plateau) lead to changes in spike timing. 13 12.8 12.4 12.2 12 11.8 11.6 11.4 11.2 1 1.5 2 2.5 3 3.5 applications of STDP 4 Future Directions -0.4 10 current through active neurons 0 -0.2 14 10 Drift Correction • This indicates that spiking neurons could correct for synaptic drift using STDP 20 16 9 Two neurons with slightly different input currents are synchronized using STDP rules with different slopes (m). Low m produced slow convergence; high m produced damped oscillation; extremely high m (not shown) produced instability. • Each drifted memory did converge to an ideal memory 0.8 22 time of spike relative to input max (ms) -0.4 • Applied random synaptic drift (≤5%) to each active connection in a memory. A Simple STDP Rule 24 8 11 -0.2 12.6 final timing of spikes: A=5, w=30 Figure (except titles) from J. J. Hopfield (2006). “Searching for memories, Sudoku, implicit check-bits, and the iterative use of not-always-correct rapid neural computation.” arXiv.org, 19 Sep 2006. -1 • Simplified memory network using continuous-variable neurons with spiking neuron’s f-I curve and spike timing patterns AC input produces a plateau on the neuron’s f-I curve. …Can the system correct itself? Drifted Memory IAC 60 The Problem: Ideal Memory 0.16 Stability Analysis: • A small positive change in T (synaptic drift, +δT) will produce a proportional positive change in the input current (+δI) to the receiving neuron frequency response of neuron: A=5mA w=30Hz I = Io + 0.05 sin(2π wt) Synaptic Strength (matrix T): T21 is proportional to the size of the electrical response in neuron 2 evoked by an electrical spike in neuron 1 T12 0.14 Spike-Timing Dependent Plasticity Synapse: A connection between two neurons 2 • If we feed an AC input of the form: I=Io + A sin(2π wt) into a neuron, a large range of DC offsets (Io) will drive the cell to fire at a frequency w, creating a plateau on the f-I curve (below left). • Modeled disconnected neurons given slightly different input currents, causing varying delays in spike timing 0 80 1 • When the membrane voltage exceeds threshold, the cell fires. 0.16 0 Synaptic Drift T21 0.14 10 … v (mV) 2 0.06 t 4 Eyes: 0.04 Synchronization of 51 neurons, m=0.1 0 input current (mA) • We will model the activity of these neurons using MATLAB 15 8 • Each property of the person or thing being remembered is represented by the activity of one neuron • All neurons active in a memory are connected, so thinking of one thing about a memory (i.e. a name) will recall other things (i.e. height, eye color) 20 10 • Associative memory is not fully understood; here we refine a simplified model of it Spike-Timing Dependent Plasticity (STDP): Experimentally observed phenomenon by which relative spike timing changes synaptic strength input current Sample Memory Synchronization average delay (t2-t1), in ms Associative Memory: Learned connections between ideas one remembers as being associated Stabilization of Firing average delay (t2-t1), in ms Associative Memory -1 -4 • Load multiple, overlapping memories into network and test performance -3 -2 -1 0 delay (ms) 1 2 3 Changes in spike timing are converted to proportional changes in synaptic strength. Spikes too far separated are ignored. 4 • Create converging memory network of spiking neurons • Use more realistic STDP rule 4.5 5