Hebbian learning and spiking neurons

Wulfram Gerstner
1999
Article

A correlation-based (Hebbian'') learning rule at the spike level is formulated, mathematically analyzed, and compared with learning in a firing-rate description. As for spike coding, we take advantage of a learning window'' that describes the effect of timing of pre- and postsynaptic spikes on synaptic weights. A differential equation for the learning dynamics is derived under the assumption that the time scales of learning and spiking dynamics can be separated. Formation of structured synapses is analyzed for a Poissonian neuron model which receives time-dependent stochastic input. It is shown that correlations between input and output spikes tend to stabilize structure formation. With an appropriate choice of parameters, learning leads to an intrinsic normalization of the average weight and the output firing rates. Noise generates diffusion-like spreading of synaptic weights.

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Wulfram Gerstner
1999
Article

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https://infoscience.epfl.ch/record/97790?ln=fr

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Spike-Based Reinforcement Learning in Continuous State and Action Space: When Policy Gradient Methods Fail

Nicolas Frémaux, Wulfram Gerstner, Walter Senn, Eleni Vasilaki

Changes of synaptic connections between neurons are thought to be the physiological basis of learning. These changes can be gated by neuromodulators that encode the presence of reward. We study a family of reward-modulated synaptic learning rules for spiking neurons on a learning task in continuous space inspired by the Morris Water maze. The synaptic update rule modifies the release probability of synaptic transmission and depends on the timing of presynaptic spike arrival, postsynaptic action potentials, as well as the membrane potential of the postsynaptic neuron. The family of learning rules includes an optimal rule derived from policy gradient methods as well as reward modulated Hebbian learning. The synaptic update rule is implemented in a population of spiking neurons using a network architecture that combines feedforward input with lateral connections. Actions are represented by a population of hypothetical action cells with strong mexican-hat connectivity and are read out at theta frequency. We show that in this architecture, a standard policy gradient rule fails to solve the Morris watermaze task, whereas a variant with a Hebbian bias can learn the task within 20 trials, consistent with experiments. This result does not depend on implementation details such as the size of the neuronal populations. Our theoretical approach shows how learning new behaviors can be linked to reward-modulated plasticity at the level of single synapses and makes predictions about the voltage and spike-timing dependence of synaptic plasticity and the influence of neuromodulators such as dopamine. It is an important step towards connecting formal theories of reinforcement learning with neuronal and synaptic properties

Public Library of Science2009

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Intrinsic Stabilization of Output Rates by Spike-Based Hebbian Learning

Wulfram Gerstner

We study analytically a model of long-term synaptic plasticity where synaptic changes are triggered by presynaptic spikes, postsynaptic spikes, and the time differences between pre- and postsynaptic spikes. We show that plasticity can lead to an intrinsic stabilization of the mean firing rate of the postsynaptic neuron. Subtractive normalization of the synaptic weights (summed over all presynaptic inputs converging on a postsynaptic neuron) follows if, in addition, the mean input rates and the mean input correlations are identical at all synapses. If the integral over the learning window is positive, firing-rate stabilization requires a non-Hebbian component, whereas such a component is not needed, if the integral of the learning window is negative. A negative integral corresponds to

anti-Hebbian' learning in a model with slowly varying firing rates. For spike-based learning, a strict distinction between Hebbian and

anti-Hebbian' rules is questionable since learning is driven by correlations on the time scale of the learning window. The correlations between presynaptic and postsynaptic firing are evaluated for a piecewise-linear Poisson model and for a noisy spiking neuron model with refractoriness. Whereas a negative integral over the learning window leads to intrinsic rate stabilization, the positive part of the learning window picks up spatial and temporal correlations in the input.

Massachusetts Institute of Technology Press2001

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Generalized Bienenstock-Cooper-Munro rule for spiking neurons that maximizes information transmission

Wulfram Gerstner, Jean-Pascal Théodor Pfister, Taro Toyoizumi

Maximization of information transmission by a spiking-neuron model predicts changes of synaptic connections that depend on timing of pre- and postsynaptic spikes and on the postsynaptic membrane potential. Under the assumption of Poisson firing statistics, the synaptic update rule exhibits all of the features of the Bienenstock-Cooper-Munro rule, in particular, regimes of synaptic potentiation and depression separated by a sliding threshold. Moreover, the learning rule is also applicable to the more realistic case of neuron models with refractoriness, and is sensitive to correlations between input spikes, even in the absence of presynaptic rate modulation. The learning rule is found by maximizing the mutual information between presynaptic and postsynaptic spike trains under the constraint that the postsynaptic firing rate stays close to some target firing rate. An interpretation of the synaptic update rule in terms of homeostatic synaptic processes and spike-timing-dependent plasticity is discussed.

2005

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