STDP - spike-timing-dependent plasticity

1. STDP - spike-timing-dependent plasticity What Can a Neuron Learn with Spike-Timing-Dependent Plasticity? STDP Finds the Start of Repeating Patterns in Continuous Spike Trains

2. Abstract • Spiking neurons are flexible computational modules. • Enable implement with their adjustable synaptic parameters an enormous variety of different transformations from input spike trains to output spike trains. • The perceptronconvergence theorem asserts the convergence of a supervised learning algorithm. • In contrast, no guarantee for the convergence of STDP with teacher forcing that holds for arbitrary input spike patterns.

3. Abstract • On the other hand, hold for STDP in the case of Poisson input spike trains. • The resulting necessary and sufficient condition can be formulated in terms of linear separability. • In case of perceptrons (McCulloch-Pitts neurons): threshold gates with static synapses, static batch inputs and outputs. • In case of STDP: time-varying input and output streams.

4. Abstract • The theoretically predicted convergence of STDP with teacher forcing also holds for more realistic neurons models, dynamic synapses, and more general input distributions. • The positive learning results hold for different interpretations of STDP where: • changes the weights of synapses • modulates the initial release probability of dynamic synapses

5. Introduction • STDP is related to various important learning rules and learning mechanisms. • Question: To what extent STDP might support a more universal type of learning where a neuron learns to implement an “arbitrary given” map? • There exist many maps from input spike trains to output spike trains that can’t be realized by a neuron for any setting of its adjustable parameters. • For example, no values of weight could enable a generic neuron to produce a high-rate output spike train in the absence of any input spikes.

6. Introduction • A neuron learn to implement transformations in a stable manner with a parameter setting that represents a equilibrium point for the learning rule under consideration (STDP). • STDP always produces bimodal distribution of weights, the minimal or maximal possible. • Need to consider such conditions.

7. Introduction • Which of the many parameters that influence the input-output behavior should be viewed as being adjustable for a specific protocol for inducing synaptic plasticity (i.e., “learning”)? • STDP adjust the following parameters: • scaling factors w of the amplitudes • initial release probabilities U • Whereas: • An increase of parameter U will increase the amplitude of the EPSP for the first spike. • An increase of the scaling factor w tends to decrease the amplitudes of shortly following EPSPs.

8. Introduction • Assumption: during learning, the neuron is taught to fire at particular points in time via extra input currents, • which could represent synaptic inputs from other cortical or subcortical areas. • SNCC – spiking neuron convergence conjecture: STDP enables neurons to learn under this protocol: • starting with arbitrary initial values • any input-output transformation that the neuron could implement • in a stable manner for some values of its adjustable parameters.

9. Models for Neurons, Synapses, and STDP • A standard leaky integrate-and-fire neuron model: • = membrane potential • = membrane time constant • = membrane resistance • = the current supplied by the synapse • = a constant background current • = currents induced by a “teacher”

10. Models for Neurons, Synapses, and STDP • If Vmexceeds the threshold voltage, it is reset and held there for the length of the absolute refractory period.

11. Models for Neurons, Synapses, and STDP • The model proposed in Markram, Wang and Tsodyks (1998), predicts the amplitude of the excitatory postsynaptic current (EPSC) for the kth spike in a spike train with interspike intervals through the equations:

12. Models for Neurons, Synapses, and STDP • The variables are dynamic, whose initial values for the first spike are • The parameters U, D, and F were randomly chosen from gaussian distributions that were based on empirically found data for such connections: • If the input was excitatory (E) the mean values of these three parameters (with D, F expressed in seconds) were chosen to be 0.5, 1.1, 0.05. • If the input was inhibitory (I) then 0.25, 0.7, 0.02. • The SD of each parameter was chosen to be 10% of its mean.

13. Models for Neurons, Synapses, and STDP • The effect of STDP is commonly tested by measuring in the postsynaptic neuron the amplitude A1 of the EPSP for a single spike from the presynaptic neuron. • The interpretations for any change in the amplitude of can caused by : • A proportional change of the parameter w • A proportional change of the initial release probability u1 = U • A change of both w and U

14. Models for Neurons, Synapses, and STDP • According to Abbott & Nelson (2000), the change in the amplitude A1 of EPSPs (for the first spike in a test spike train) that results from pairing of: • a firing of the presynaptic neuron at some time • a firing of the postsynaptic neuron at time can be approximated for many cortical synapses by terms of the form:

15. Models for Neurons, Synapses, and STDP • with constants W+,W−, τ+, τ− > 0 • with an extra clause that prevents the amplitude A1 from growing beyond some maximal value Amax or below 0.