Spike-timing-dependent plasticity

Hebb’s postulate has a direction in it: cell A gains efficiency at cell B by repeatedly taking part in firing it, which only an input that arrives before B’s spike can do. The rule as usually written, a product of two activities, has no time in it at all. In 1997 a pair of patch pipettes on two connected neurons put the time back, and the order of two spikes, within a few tens of milliseconds, turned out to set the sign of the change.

The window, measured

Markram, Lübke, Frotscher and Sakmann recorded from pairs of connected layer 5 pyramidal neurons in slices of neocortex, evoking the postsynaptic spike by injecting current. Input 10 ms ahead of that spike potentiated the connection, the reverse order depressed it, and spikes 100 ms apart did nothing; blocking NMDA receptors, or keeping the postsynaptic cell from firing, abolished the potentiation. In the same issue of Science, Magee and Johnston showed in hippocampal CA1 cells that pairing input with action potentials that travelled back into the dendrites induced potentiation: the back-propagating spike is how the synapses hear that the cell has fired.

Bi and Poo mapped the window in 1998 in cultured rat hippocampal neurons, pairing input and postsynaptic spike 60 times at 1 Hz. A spike up to 20 ms after the input potentiated the synapse and one up to 20 ms before depressed it; outside that 40 ms span almost nothing happened, and the switch between maximal depression and maximal potentiation was about 5 ms wide. Only relatively weak synapses potentiated. Zhang and colleagues found the same 20 ms windows that year in living tadpoles, between retina and tectum.

Models condense this into a rule for pairs of spikes. With $\Delta t = t_{\text{post}} - t_{\text{pre}}$ the time from a presynaptic spike to a postsynaptic one,

$$\Delta w = \begin{cases} A_+ \, e^{-\Delta t / \tau_+} & \Delta t > 0 \\ -A_- \, e^{\Delta t / \tau_-} & \Delta t < 0 \end{cases}$$

where $\Delta w$ is the change in the synapse’s weight, $A_+$ and $A_-$ are the largest potentiation and depression (which may depend on the weight itself), and $\tau_+$ and $\tau_-$ set how fast each half falls away. Potentiation typically reaches out to about 20 ms; depression, depending on the synapse, to between 20 and 100 ms.

Trains, a depolarisation and an owl

Order effects had been seen with coarser tools. In 1983 Levy and Steward, in the living hippocampus, found that a weak input’s train of stimuli was potentiated if it came before a strong input’s train and depressed if it came after. In 1994 Debanne, Gähwiler and Thompson depressed synapses onto CA1 cells in slice cultures by depolarising the cell shortly before each input, by an amount that depended on the interval. Neither timed single spikes.

Theory came by way of the barn owl, which locates sounds from interaural time differences of a few microseconds using neurons at least ten times slower. Gerstner, Kempter, van Hemmen and Wagner asked how development could wire such neurons precisely enough for that temporal code, and in 1996 answered with a learning window: strengthen a synapse whose spike arrives shortly before the neuron fires, weaken one whose spike arrives shortly after. A model integrate-and-fire neuron given 600 inputs with delays scattered around 2.5 ms, trained on a 5 kHz tone, kept 154 of them, their delays differing roughly by whole periods of the tone, and fired phase-locked to within 25 µs: a bank of delay lines trimmed until the survivors add in step.

The paper preceded Markram’s by four months, and Feldman’s 2012 review credits it with predicting the window. Its depressing half, though, cited Debanne’s result and the abstract in which Markram and Sakmann had first reported theirs, at the Society for Neuroscience meeting in 1995: theory and experiment were converging rather than one waiting on the other. Song, Miller and Abbott named the rule in 2000 and found in simulations that it balances synaptic strengths by itself: synapses compete to control when the cell fires, and inputs that fire it early, or in correlated groups, win.

Built into the synapse

What crossed to engineering was a learning rule that a synapse can carry out by itself. Each synapse needs only the recent spike history on either side, and a leaky integrator can hold that. Let every presynaptic spike kick a trace that decays exponentially, and every postsynaptic spike another; when the cell fires, add the presynaptic trace’s present value to the weight, and when an input arrives, subtract the postsynaptic trace’s (Morrison, Diesmann and Gerstner, 2008). That is two first-order low-pass filters and two sampling events, with nothing computed elsewhere and routed in — what neuromorphic designers want from learning rules. Giacomo Indiveri made the case in 2002: rules built on mean firing rates are hard to implement in analogue circuits and usually need a separate weight-normalisation step, while spike-based ones map directly onto silicon.

In his circuit each presynaptic spike starts a voltage that relaxes linearly; a postsynaptic spike, while it lasts, gates a current set by that voltage onto the weight capacitor, and a mirror-image path removes charge when the order is reversed. Below threshold a transistor’s current is exponential in its gate voltage, so the linear relaxation becomes an exponential decay of current: the exponential in the window comes from device physics rather than arithmetic. Four bias voltages set the height and width of each half. In 2006 Indiveri, Chicca and Douglas put leaky integrate-and-fire neurons and bistable synapses with spike-timing-dependent plasticity on one chip, sending and receiving spikes as address events.

A memristor goes further. Its conductance shifts with the voltage across it and stays shifted, and placed between a presynaptic and a postsynaptic line it sees the difference of the two spike waveforms. Give it a threshold below which nothing changes — a dead zone, which also lets it hold its value — set so that a lone spike stays under it, and shape each spike as a brief, tall pulse with a long, shallow negative tail. Then only overlapping spikes cross the threshold, with a sign set by which came first and a size that grows as they come closer. Integrated, the change against the interval resembles Bi and Poo’s curve, and reshaping the spike reshapes the rule (Zamarreño-Ramos and colleagues, 2011). Snider had proposed synchronous versions by 2008, and Jo and colleagues at the University of Michigan demonstrated one in 2010, with CMOS neurons driving silicon-based memristors. In the idealised device even the weight dependence is physics: each update changes the conductance in proportion to its square.

No single window

That curve is one synapse type under one protocol; Feldman’s review counts more than 20 types of synapse with some timing dependence, and the rules differ. In a mormyrid electric fish, Bell and colleagues found in 1997 that input followed within 60 ms by a postsynaptic spike is depressed — the sign inverted, suiting a structure that learns to cancel predictable sensory input. At several synapses the depressing half is the broader one.

Position matters. A back-propagating spike loses about half its amplitude within several hundred micrometres of the soma as it travels out along the dendritic cable. In layer 2/3 pyramidal cells Froemke, Poo and Dan found that distal synapses potentiate less and depress over a broader window; in layer 5 the textbook window holds within about 100 µm of the soma, and beyond about 500 µm pairing gives only anti-Hebbian depression. Rate matters. A single connection in slices shows the window only around 10–20 Hz, giving only depression below 10 Hz and potentiation in either order above about 30 Hz (Sjöström, Turrigiano and Nelson, 2001, among others). A pair rule predicts the opposite trend; Pfister and Gerstner’s triplet rule fixes it with a second, slower postsynaptic trace. Chemistry matters. In slices of adult visual cortex, Seol and colleagues found that the potentiating half needed receptors coupled to adenylyl cyclase, such as β-adrenergic ones, and the depressing half receptors coupled to phospholipase C, such as muscarinic ones. Dopamine, whose neurons signal reward prediction error, gates the rule at several synapses and can reverse it.

Feldman’s reading is that timing is one factor in a rule that also depends on rate, depolarisation and neuromodulation. Worth being clear, then, that the circuits above build the pair rule, because two traces can carry it — and Indiveri’s bistable synapses keep one bit each in the long run. The memristor version has a power bill as well. By Zamarreño-Ramos and colleagues’ estimate, a square centimetre of a million neurons, each driving 10,000 synapses of 1 MΩ at 1 V for 20 ms spikes ten times a second, would dissipate about 2 kW in its synapses alone; the resistances would have to rise at least a hundredfold.